course_content.txt

# Unit1_README.md
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## Unit 1: Introduction to Astronomy

This unit sets the stage for understanding the vastness and complexity of the cosmos by introducing students to the historical development of astronomy, the immense scale of the universe, the night sky, and the effects of light travel time.

![Banner Image](../Unit1/figures/unit1_banner.png)

### Teacher Assessment
- Summary Notes
- Assignment/Project

### Self-Assessment
- Each section of this unit includes a set of "Check Your Understanding" questions that are designed to prepare you for the midterm exam.

### [Summary Notes](https://teaghan.github.io/astronomy-12/Unit1/Unit1_Summary_Notes.pdf)

- While going through the lessons, working on the Check Your Understanding problems, and completing any assignment/project, you are expected to take meaningful notes for your future self. These notes can include concepts, diagrams, examples, etc.
- **You will be allowed to access to these notes during the test!**
- Print out the summary notes file and fill it with useful, hand-written notes while working through this unit and submit your notes at the end of the unit to be assessed for completion.
- There is a suggested set of topics as well as empty boxes to give you the opportunity to summarize additional topics.

### [1.1 A Brief History of Astronomy](../md_files/1_1_history.html)
- Explore the evolution of astronomical understanding from ancient civilizations to modern times.
- Significant contributions from astronomers such as Ptolemy, Copernicus, Galileo, and Newton.

### [1.2 The Scale of the Universe](../md_files/1_2_scale.html)
- Understand the vast distances in space using scientific notation.
- Learn about the structure and size of the solar system, the Milky Way, and the observable universe.

### [1.3 The Celestial Sphere](../md_files/1_3_the_sky.html)
- Concept of the celestial sphere and its components: celestial poles, celestial equator, and ecliptic.
- Understand the apparent motion of stars and constellations.

### [1.4 The Brightness of Stars](../md_files/1_4_brightness.html)
- Differentiate between luminosity, flux, and apparent brightness.
- Understand the inverse square law of light.
- Explore the magnitude scale and the distance modulus.

### [1.5 Consequences of Light Travel Time](../md_files/1_5_light_travel.html)
- Effect of the finite speed of light on observations.
- Concept of looking back in time.

### [Assignment: Night Sky Observations](https://teaghan.github.io/astronomy-12/Unit1/Unit1_Assignment.pdf)
- Print off the attached assignment.
- Follow the steps and complete the questions.
- Submit your document with the questions answered.

### Course Resources
- **Free Textbook**: [**Astronomy**](https://openstax.org/books/astronomy/pages/1-introduction) by OpenStax.
- **AI Tutor**: [**Astronomy Tutor**](https://chatgpt.com/g/g-10CjMHMvk-astronomy-tutor) to support you with this class.

### Science Curricular Connections
- **Earth Science 11:** The nebular hypothesis, Stars as the center of a solar system

### Learning Standards
- I can use scientific notation to express large astronomical distances.
- I can describe the scale of the universe and its components.
- I can explain the concept of a light-year and its importance in measuring cosmic distances.
- I can describe how parallax is used to measure the distance to nearby stars.
- I can explain the concept of the celestial sphere and its components.
- I can explain the apparent daily and annual motion of the Sun, Moon, and stars.
- I can define flux and luminosity and explain their relationship.
- I can apply the inverse square law to calculate the apparent brightness of stars.
- I can distinguish between apparent magnitude and absolute magnitude.
- I can use the distance modulus formula to determine the distance to stars.
- I can explain how observing distant objects allows us to look back in time.
- I can describe the methods used to measure cosmic distances, including parallax and standard candles.

# Unit2_README.md
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## Unit 2: Gravity and Our Solar System

This unit delves into the laws governing the motion of celestial bodies, including planets, moons, and satellites. It covers gravitational forces and orbital mechanics, providing a comprehensive understanding of the principles that dictate the movement and interaction of objects in space.

![Banner Image](../Unit2/figures/unit2_banner.png)

### Teacher Assessment
- Summary Notes
- Assignment/Project

### Self-Assessment
- Each section of this unit includes a set of "Check Your Understanding" questions that are designed to prepare you for the midterm exam.

### [Summary Notes](https://teaghan.github.io/astronomy-12/Unit2/Unit2_Summary_Notes.pdf)

- While going through the lessons, working on the Check Your Understanding problems, and completing any assignment/project, you are expected to take meaningful notes for your future self. These notes can include concepts, diagrams, examples, etc.
- **You will be allowed to access to these notes during the test!**
- Print out the summary notes file and fill it with useful, hand-written notes while working through this unit and submit your notes at the end of the unit to be assessed for completion.
- There is a suggested set of topics as well as empty boxes to give you the opportunity to summarize additional topics.

### [2.1 Kepler’s Laws of Planetary Motion](../md_files/2_1_keplers_laws.html)
   - Law of Orbits: All planets move in elliptical orbits, with the Sun at one focus.
   - Law of Areas: A line that connects a planet to the Sun sweeps out equal areas during equal intervals of time.
   - Law of Periods: The square of the period of any planet is proportional to the cube of the semi-major axis of its orbit.

### [2.2 Newton's Law of Universal Gravitation](../md_files/2_2_gravity.html)
   - Definition and formula
   - Gravitational constant
   - Inverse square law
   - Applications of gravitational theory

### [2.3 Circular Motion](../md_files/2_3_circular_motion.html)
   - Definition and formula for centripetal force
   - Examples of centripetal forces in daily life and astronomy
   - Orbital speed and its derivation

### [2.4 The Structure and Components of Our Solar System I](../md_files/2_4_solar_system_1.html)
   - Formation of the Solar System
   - The Sun: structure and composition
   - Terrestrial planets: characteristics and notable features
   - Jovian planets: characteristics and notable features

### [2.5 The Structure and Components of Our Solar System II](../md_files/2_5_solar_system_2.html)
   - Dwarf planets: Pluto, Eris, Ceres, Haumea, Makemake
   - Asteroids and the asteroid belt: origin and significant examples
   - Comets and the Kuiper Belt: structure and notable comets
   - Meteoroids, meteors, and meteorites: definitions and historical impacts
   - Moons of the Solar System: notable examples and formation theories
   - The Oort Cloud: hypothetical cloud of icy bodies and its role in the solar system

### [Assignment: Investigating Galaxy Data](https://teaghan.github.io/astronomy-12/Unit2/Unit2_Assignment.pdf)
- Print off the attached assignment.
- Follow the steps and complete the questions.
- Submit your document with the questions answered.

### Course Resources
- **Free Textbook**: [**Astronomy**](https://openstax.org/books/astronomy/pages/1-introduction) by OpenStax.
- **AI Tutor**: [**Astronomy Tutor**](https://chatgpt.com/g/g-10CjMHMvk-astronomy-tutor) to support you with this class.

### Science Curricular Connections
- **Physics 12:** uniform circular motion, centripetal force and acceleration, changes to apparent weight, gravitational field and Newton’s law of universal gravitation, gravitational dynamics
- **Physics 11:** mass, force of gravity, apparent weight, Newton’s laws of motion, horizontal uniform and accelerated motion
- **Earth Sciences 11:** astronomy seeks to explain the origin and interactions of Earth and its solar system, impacts of the Earth-moon-sun system

### Learning Standards
- I can state Kepler's First Law and explain the concept of elliptical orbits with the Sun at one focus.
- I can describe the characteristics of an ellipse and its relevance to planetary orbits.
- I can explain Kepler's Second Law and the concept of equal areas in equal time periods.
- I can describe the relationship between a planet's distance from the Sun and its orbital speed according to Kepler's Second Law.
- I can use Kepler's Third Law to calculate the orbital period of a planet given its semi-major axis.
- I can verify the consistency of Kepler's Third Law for different planets.
- I can use Newton's Second Law to calculate the force acting on an object given its mass and acceleration.
- I can explain the relationship between mass, force, and acceleration according to Newton's Second Law.
- I can calculate the gravitational force between two masses using Newton's Law of Universal Gravitation.
- I can explain the concept of the inverse square law in the context of gravitational force.
- I can describe the importance of the gravitational constant G in Newton's law.
- I can explain the concept of weight as the gravitational force acting on an object.
- I can describe the role of gravity in maintaining the orbits of planets and satellites.
- I can describe the formation of the solar system from the solar nebula.
- I can explain the role of gravity and circular motion in the formation of the Sun and planets.
- I can calculate the centripetal force acting on an object moving in a circular path.
- I can describe the relationship between gravitational force and centripetal force in orbital motion.
- I can calculate the centripetal acceleration of an object moving in a circular orbit.
- I can derive the formula for orbital speed and explain its significance in astronomy.
- I can calculate the orbital speed of moons and satellites using the given distance and mass of the central body.
- I can calculate the orbital period of a moon or satellite using Kepler's Third Law.
- I can calculate the surface gravity on a moon or planet given its mass and radius.

# Unit3_README.md
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## Unit 3: Stars

This unit explores the fascinating life cycles of stars, from their formation in molecular clouds to their ultimate demise as white dwarfs, neutron stars, or black holes. It covers the processes that power stars, their chemical compositions, and the physical principles that govern their radiation.

![Banner Image](../Unit3/figures/unit3_banner.png)

### Teacher Assessment
- Summary Notes
- Assignment/Project

### Self-Assessment
- Each section of this unit includes a set of "Check Your Understanding" questions that are designed to prepare you for the midterm exam.

### [Summary Notes](https://teaghan.github.io/astronomy-12/Unit3/Unit3_Summary_Notes.pdf)

- While going through the lessons, working on the Check Your Understanding problems, and completing any assignment/project, you are expected to take meaningful notes for your future self. These notes can include concepts, diagrams, examples, etc.
- **You will be allowed to access these notes during the test!**
- Print out the summary notes file and fill it with useful, hand-written notes while working through this unit and submit your notes at the end of the unit to be assessed for completion.
- There is a suggested set of topics as well as empty boxes to give you the opportunity to summarize additional topics.

### [3.1 Star Formation](../md_files/3_1_star_formation.html)
   - Molecular Clouds and Protostars: The initial stages of star formation from molecular clouds to protostars.
   - Angular momentum
   - Observational Evidence: Infrared images and other observational data supporting star formation.

### [3.2 Atoms and Elements](../md_files/3_2_atoms_particles.html)
   - Basic atomic structure
   - Protons, neutrons, electrons
   - Neutrinos, photons
   - Isotopes
   - Chemical Reactions

### [3.3 Nuclear Fusion](../md_files/3_3_nuclear_fusion.html)
   - Fusion Processes: The proton-proton chain and the CNO cycle.
   - Energy Generation: How nuclear fusion powers stars, focusing on temperature and pressure conditions in the stellar core.
   - Creating Heavier Elements

### [3.4 Chemical Composition of Stars](../md_files/3_4_chemical_composition.html)
   - Stellar Spectroscopy: Techniques to determine the chemical composition of stars.
   - Abundance of Elements: How the relative abundance of elements provides clues about star formation and evolution.
   - Nucleosynthesis: The process of nucleosynthesis in stars, creating elements during different fusion stages and supernovae.

### [3.5 Life Cycle of Stars](../md_files/3_5_life_cycle.html)
   - Main Sequence Evolution: The life of a star on the main sequence and the changes as it exhausts its hydrogen fuel.
   - Post-Main Sequence Evolution: Evolutionary paths after the main sequence, including red giants, supernovae, white dwarfs, neutron stars, and black holes.
   - Stellar Remnants: Types and observational properties of stellar remnants.
   - The Hertzsprung-Russell (H-R) Diagram: Understanding the H-R diagram and its significance in studying stellar evolution.

### [Assignment: Stellar Evolution](https://teaghan.github.io/astronomy-12/Unit3/Unit3_Assignment.pdf)
- Print off the attached assignment.
- Follow the steps and complete the questions.
- Submit your document with the questions answered.

### Course Resources
- **Free Textbook**: [**Astronomy**](https://openstax.org/books/astronomy/pages/1-introduction) by OpenStax.
- **AI Tutor**: [**Astronomy Tutor**](https://chatgpt.com/g/g-10CjMHMvk-astronomy-tutor) to support you with this class.

### Science Curricular Connections

**Physics 11:**
- mass, force of gravity
- balanced and unbalanced forces in systems
- conservation of energy; principle of work and energy
- generation and propagation of waves
- properties and behaviors of waves

**Physics 12:**
- gravitational field and Newton’s law of universal gravitation
- gravitational dynamics and energy relationships
- Electric field 
- electrostatic dynamics and energy relationships
- electromagnetic induction
- electrostatic dynamics and energy relationships 
- applications of electromagnetic induction
- imomentum
- conservation of momentum

**Chemistry 11:**
- quantum mechanical model and electron configuration
- valence electrons
- bonds/forces
- reactions

**Chemistry 12:**
- energy change during a chemical reaction

**Earth Sciences 11:**
- solar radiation interactions
- the nebular hypothesis (explanation of the formation and properties of our solar system)

### Learning Standards
- I can describe the process of star formation, from molecular clouds to protostars.
- I can explain the role of angular momentum in the formation of stars.
- I can calculate the angular momentum and moment of inertia when given the shape and mass of an object.
- I can use the conservation of angular momentum to determine how the rotational speed of an object changes when the radius changes.
- I can explain the basic structure of atoms, including protons, neutrons, and electrons.
- I can identify the role of isotopes in stellar chemistry.
- I can describe the importance of neutrinos and photons in stellar processes.
- I can explain the conditions necessary for nuclear fusion to ignite within protostars.
- I can describe the proton-proton chain reaction and the CNO cycle in stars.
- I can explain the process of nucleosynthesis and apply it to the creation of heavier elements in stars.
- I can calculate the energy output of a nuclear fusion reaction.
- I can use the Hertzsprung-Russell (H-R) diagram to classify stars and predict their future evolution.
- I can describe how the mass of a star influences its evolutionary path and final state, such as becoming a white dwarf, neutron star, or black hole.
- I can explain the life cycle of stars from the main sequence to their end stages, comparing the different paths taken by low-mass and high-mass stars.

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## Unit 4: Light and Spectroscopy

Spectroscopy is a crucial tool in astronomy, enabling us to decode the light from stars, galaxies, and distant worlds to uncover their secrets. In this unit, you'll explore the nature of light, from the electromagnetic spectrum to wave-particle duality. Discover how blackbody radiation reveals the temperatures of celestial objects, how atomic energies produce the spectral lines that serve as cosmic fingerprints, and how spectroscopy helps astronomers determine the composition, temperature, and motion of objects across the universe.

![Banner Image](../Unit4/figures/unit4_banner.png)

### Teacher Assessment
- Summary Notes
- Assignment/Project

### Self-Assessment
- Each section of this unit includes a set of "Check Your Understanding" questions that are designed to prepare you for the midterm exam.
- 
### [4.1 The Nature of Light](../md_files/4_1_nature_of_light.html)
- **Electromagnetic Waves**: Introduction to the electromagnetic spectrum, including types of electromagnetic radiation from radio waves to gamma rays and their significance in astronomy.
- **Wave Properties**: Exploration of key properties of light such as amplitude, wavelength, and frequency, with a detailed explanation of the relationship between wavelength and frequency through the wave equation.
- **Wave-Particle Duality**: Discussion of light's dual nature as both a wave and a particle (photons) and how photon energy is calculated using Planck's equation.
- **Interaction of Light with Matter**: Detailed coverage of how light interacts with matter through reflection, refraction, and absorption, and their applications in telescope design and astronomical observations.
- **Applications in Astronomy**: The use of different wavelengths across the electromagnetic spectrum to study celestial phenomena, including radio waves, infrared, ultraviolet, and X-rays.

### [4.2 Blackbody Radiation](../md_files/4_2_blackbody.html)
- **Blackbody Spectrum**: Examination of the radiation emitted by blackbodies and how it varies with temperature.
- **Wien’s Displacement Law**: Explanation of how temperature relates to the peak wavelength of blackbody radiation, revealing the color-temperature relationship of stars.
- **Stefan-Boltzmann Law**: Discussion on how the total energy radiated by a blackbody per unit area depends on its temperature, helping to explain the luminosity of stars.
- **Color Indices and Stellar Classification**: Overview of how color indices are used to classify stars by temperature and color, with references to the Hertzsprung-Russell diagram.

### [4.3 Atomic Energies](../md_files/4_3_atomic_energies.html)
- **Quantum Nature of Light**: Review of light's wave-particle duality and the concept of photon energy using Planck’s equation.
- **Electron Volts**: Introduction to electron volts (eV) as a unit of energy, with practical conversion examples relevant to atomic physics.
- **Bohr Model and Atomic Energy Levels**: Explanation of the quantized energy levels in atoms and the transitions between these levels that give rise to emission and absorption spectra.
- **Hydrogen Spectrum**: Detailed exploration of the Lyman, Balmer, and Paschen series in hydrogen and their importance in astronomical spectroscopy.
- **Ionization Energy**: The role of ionization energy in determining the conditions within stars and other celestial bodies.

### [4.4 Spectroscopy I: Principles of Spectroscopy](../md_files/4_4_spectroscopy_1.html)
- **Types of Spectra**: Introduction to continuous, emission, and absorption spectra, with a focus on how they are produced under different astrophysical conditions.
- **Kirchhoff's Laws of Spectroscopy**: An explanation of Kirchhoff’s three laws that govern the production of spectra and their applications in astronomy.
- **Spectral Line Broadening**: Examination of broadening mechanisms such as Doppler broadening and their astrophysical significance in understanding stellar and galactic motions.

### [4.5 Spectroscopy II: Applications of Spectroscopy](../md_files/4_5_spectroscopy_2.html)
- **Identifying Elements in Stars**: How the analysis of absorption and emission lines in spectra is used to identify the chemical composition of stars and other celestial objects.
- **Measuring Stellar Velocities**: Detailed discussion on the Doppler effect and its role in measuring the radial velocities of stars and galaxies, with applications to detecting exoplanets and studying stellar motion.
- **Practical Spectroscopic Analysis**: Overview of spectral line catalogs, databases, and tools astronomers use to analyze stellar spectra and determine the physical properties of celestial objects.

### [Assignment: Spectroscopy in Action](https://teaghan.github.io/astronomy-12/Unit4/Unit4_Assignment.pdf)
- Print off the attached assignment.
- Follow the steps and complete the questions.
- Submit your document with the questions answered.

### Course Resources
- **Free Textbook**: [**Astronomy**](https://openstax.org/books/astronomy/pages/1-introduction) by OpenStax.
- **AI Tutor**: [**Astronomy Tutor**](https://chatgpt.com/g/g-10CjMHMvk-astronomy-tutor) to support you with this class.

### Science Curricular Connections

**Physics 11:**
- Properties and behaviors of waves
- Generation and propagation of waves

**Physics 12:**
- Electromagnetic induction and its applications
- Doppler effect

**Chemistry 11:**
- Quantum mechanical model and electron configuration
- Analysis techniques (e.g., spectroscopy)

**Chemistry 12:**
- Energy change during a chemical reaction

**Earth Sciences 11:**
- Solar radiation interactions

### Learning Standards
- I can describe the electromagnetic spectrum, including the types of electromagnetic radiation and their importance in astronomy.
- I can explain wave-particle duality and calculate the energy of photons using Planck’s equation.
- I can describe the principles of blackbody radiation and explain how it relates to the temperature and energy output of stars.
- I can apply Wien’s Displacement Law to determine the peak wavelength of radiation and the Stefan-Boltzmann Law to calculate the total energy radiated by stars.
- I can explain the structure of atoms and how the interaction of electromagnetic radiation with electrons produces emission and absorption spectra.
- I can distinguish between continuous, absorption, and emission spectra, and describe the conditions under which each is produced.
- I can apply Kirchhoff’s laws of spectroscopy to explain the formation of continuous, emission, and absorption spectra in various astrophysical environments.
- I can explain the mechanisms of spectral line broadening, including Doppler broadening, and understand their significance in the study of stellar and galactic motions.
- I can use spectroscopy to identify the chemical composition, temperature, and other properties of stars and celestial objects.
- I can explain the Doppler effect and apply it to measure the radial velocities of stars and galaxies.
- I can describe how line lists, spectral catalogs, and practical spectroscopy tools are used to analyze the composition and motion of celestial bodies.

# Unit5_README.md
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## Unit 5: Galaxies and the Universe

This unit takes students on an expansive journey through the cosmos, exploring the large-scale structure of the universe, the diverse types of galaxies, the mysteries of black holes, and the nature of dark matter and dark energy. Students will uncover how galaxies form and evolve, understand the dynamic processes within black holes, and learn about the possible futures of the universe driven by dark matter and dark energy.

![Banner Image](../Unit5/figures/unit5_banner.png)

### Teacher Assessment
- Summary Notes
- Assignment/Project

### Self-Assessment
- Each section of this unit includes a set of "Check Your Understanding" questions that are designed to prepare you for the midterm exam.

### [5.1 The Big Bang and the Early Universe](../md_files/5_1_early_universe.html)
- **The Big Bang Theory**: Learn about the evidence supporting the Big Bang, including cosmic microwave background radiation and nucleosynthesis.
- **The Early Universe**: Understand the first few moments after the Big Bang and the formation of the first atomic nuclei.

### [5.2 Cosmology](../md_files/5_2_cosmology.html)
- **Structure of the Universe**: Explore the organization of matter into galaxies, clusters, and superclusters.
- **Hubble’s Law**: Understand how redshift reveals the expansion of the universe and the relationship between distance and velocity for galaxies.
- **The Cosmological Principle**: Learn about the homogeneous and isotropic nature of the universe on large scales.
- **Hubble’s Constant**: Examine the significance of Hubble’s constant and current efforts to refine its value.

### [5.3 Types of Galaxies](../md_files/5_3_types_of_galaxies.html)
- **Spiral, Elliptical, and Irregular Galaxies**: Investigate the different types of galaxies, their structures, and their characteristics.
- **Theories of Formation and Evolution of Galaxies**: Learn about how galaxies form, evolve, and interact over time.
- **Quasars**: Study the energetic centers of galaxies and their importance in understanding the early universe.

### [5.4 Black Holes](../md_files/5_4_black_holes.html)
- **Escape Velocity**: Introduction to the concept of escape velocity and its connection to black holes.
- **Event Horizon**: Explore the boundary around a black hole from which no light or matter can escape.
- **Introduction to General Relativity**: Learn how Einstein’s theory of general relativity explains the warping of spacetime around massive objects.
- **Evidence for Black Holes**: Examine the observational evidence for black holes, including star movements and X-rays from accretion disks.
- **Gravitational Waves**: Discover how gravitational waves provide new insights into black holes and cosmic collisions.

### [5.5 Dark Matter, Dark Energy, and the Fate of the Universe](../md_files/5_5_dark_matter_and_energy.html)
- **Dark Matter**: Understand the role of dark matter in galaxy formation and its observational evidence.
- **Dark Energy**: Learn about dark energy and how it drives the accelerated expansion of the universe.
- **Possible Futures of the Universe**: Explore how dark matter and dark energy influence the ultimate fate of the universe—whether it will expand forever, slow down, or collapse.

### [Assignment: The Universe in Motion](https://teaghan.github.io/astronomy-12/Unit5/Unit5_Assignment.pdf)
- Print off the attached assignment.
- Follow the steps and complete the questions.
- Submit your document with the questions answered.

### Course Resources
- **Free Textbook**: [**Astronomy**](https://openstax.org/books/astronomy/pages/1-introduction) by OpenStax.
- **AI Tutor**: [**Astronomy Tutor**](https://chatgpt.com/g/g-10CjMHMvk-astronomy-tutor) to support you with this class.

### Science Curricular Connections

**Physics 11:**
- Mass, force of gravity, and Newton’s laws of motion
- Energy conservation and the principle of work and energy

**Physics 12:**
- Gravitational field and Newton’s law of universal gravitation
- Relativistic effects and the conservation of momentum

**Earth Sciences 11:**
- The nebular hypothesis and the formation of galaxies

**Chemistry 12:**
- Energy change during reactions and nuclear processes

### Learning Standards
- I can describe the evidence supporting the Big Bang and the key events in the early universe.
- I can explain Hubble’s Law and how it provides evidence for the expanding universe.
- I can describe the different types of galaxies and explain the theories of galaxy formation and evolution.
- I can identify quasars and their significance in the study of galaxies.
- I can explain the basic properties of black holes, including escape velocity and event horizons.
- I can describe the role of general relativity in understanding black holes.
- I can summarize the evidence for black holes and describe how gravitational waves help detect cosmic events.
- I can explain the concept of dark matter and its role in the structure of the universe.
- I can describe the effect of dark energy on the expansion of the universe.
- I can analyze the possible futures of the universe based on the balance of dark matter and dark energy.

# 1_1_history.md
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# 1.1 A Brief History of Astronomy

### Introduction
Imagine looking up at the night sky thousands of years ago, without the light pollution of modern cities. The stars would be a dazzling spectacle, a cosmic mystery waiting to be unraveled. This lesson will take you on a journey through time, exploring how humanity has sought to understand the universe, from ancient stargazers to modern astronomers. Join us as we discover the fascinating story of astronomy—a tale of curiosity, ingenuity, and the relentless pursuit of knowledge.

<div class="alert alert-block alert-success">
  <h4>Video</h4>
  <p>Watch <a href="https://youtu.be/0rHUDWjR5gg?si=NJL_XCQq9rRs723c" target="_blank">this video</a> for an excellent introduction to the study of Astronomy. Importantly, the video includes a discussion on the various careers related to astronomy.</p>
</div>

---

### The Ancient Roots of Astronomy

> **Did You Know?**
> The ancient Babylonians were among the first to create detailed records of celestial events, which helped them develop a calendar and predict lunar eclipses.

Our story begins in ancient Babylon and Egypt, where early astronomers meticulously observed the skies. These civilizations recorded the movements of celestial bodies, using their observations to predict events like eclipses and the flooding of the Nile. The Babylonians developed star catalogs, while the Egyptians created a 365-day calendar based on the heliacal rising of Sirius.

| Civilization | Key Contributions                                         |
|--------------|-----------------------------------------------------------|
| Babylon      | Star catalogs, lunar eclipse predictions                  |
| Egypt        | 365-day calendar based on Sirius                          |
| Greece       | Mathematical harmony, spherical Earth, Ptolemaic model    |
| Mayans       | Accurate calendars, predictions of solar and lunar cycles |
| Chinese      | Earliest recorded supernova, detailed star maps           |

As we move forward in time, we encounter the Greeks, who made significant contributions to our understanding of the cosmos. Pythagoras proposed a mathematical harmony in nature, influencing later cosmologies. Aristotle argued for a spherical Earth, based on the shape of Earth's shadow during lunar eclipses.

<img src="https://raw.githubusercontent.com/teaghan/astronomy-12/main/Unit1/figures/Aristotle.png" alt="Aristotle Image" width="1000" style="display: block; margin-left: auto; margin-right: auto;">

Eratosthenes, a brilliant mathematician, measured Earth's circumference with remarkable accuracy using the angles of shadows in different locations.

<img src="https://github.com/teaghan/astronomy-12/blob/main/Unit1/figures/Eratosthenes.png?raw=true" alt="Eratosthenes Image" width="1000" style="display: block; margin-left: auto; margin-right: auto;">

In Alexandria, Claudius Ptolemy developed the geocentric model, which placed Earth at the center of the universe, with planets and stars orbiting it in complex patterns of circles within circles. This model dominated astronomical thought for over a thousand years.

<img src="https://github.com/teaghan/astronomy-12/blob/main/Unit1/figures/Ptolemy.png?raw=true" alt="Ptolemy Image" width="1000" style="display: block; margin-left: auto; margin-right: auto;">

---

### The Copernican Revolution

The scene shifts to Renaissance Europe, where a quiet revolution was brewing. Nicolaus Copernicus, a Polish astronomer, proposed a heliocentric model that placed the Sun at the center of the universe. This radical idea challenged the long-held geocentric model and set the stage for a seismic shift in our understanding of the cosmos.

> **Interesting Side Note**:
> Copernicus hesitated to publish his work due to fear of criticism and persecution. His groundbreaking book, "De revolutionibus orbium coelestium," was published just before his death in 1543.

<img src="https://github.com/teaghan/astronomy-12/blob/main/Unit1/figures/Copernicus.png?raw=true" alt="Copernicus Image" width="1000" style="display: block; margin-left: auto; margin-right: auto;">

Enter Galileo Galilei, the Italian polymath who used a telescope to observe the heavens like never before. Galileo discovered moons orbiting Jupiter and phases of Venus, providing concrete evidence for the heliocentric model. His findings sparked a clash with the Catholic Church, illustrating the tension between science and established beliefs.

<img src="https://github.com/teaghan/astronomy-12/blob/main/Unit1/figures/Galileo.png?raw=true" alt="Galileo Image" width="1000" style="display: block; margin-left: auto; margin-right: auto;">

Johannes Kepler, a contemporary of Galileo, formulated three laws of planetary motion, describing the orbits of planets as ellipses rather than circles. Kepler's laws revolutionized our understanding of planetary motion and paved the way for further discoveries.

<img src="https://github.com/teaghan/astronomy-12/blob/main/Unit1/figures/Kepler.png?raw=true" alt="Kepler Image" width="1000" style="display: block; margin-left: auto; margin-right: auto;">

Isaac Newton, the English mathematician and physicist, provided the theoretical foundation for Kepler's laws with his laws of motion and universal gravitation. Newton's work unified the physical description of the universe and laid the groundwork for classical mechanics.

---

### The Birth of Modern Astronomy

Our journey continues into the 18th century with William Herschel, who discovered Uranus and used large telescopes to catalog thousands of stars and nebulae. Herschel's work expanded our understanding of the cosmos and revealed the vastness of the universe.

The development of spectroscopy in the 19th century marked a new era in astronomy. Joseph Fraunhofer discovered dark lines in the solar spectrum, leading to the identification of chemical elements in stars. Gustav Kirchhoff and Robert Bunsen further advanced this field, allowing astronomers to determine the composition of stars and other celestial bodies.

<img src="https://github.com/teaghan/astronomy-12/blob/main/Unit1/figures/Fraunhofer.png?raw=true" alt="Fraunhofer Image" width="1000" style="display: block; margin-left: auto; margin-right: auto;">

Edwin Hubble, in the early 20th century, made a groundbreaking discovery that the universe is expanding. By observing the redshift of galaxies, Hubble provided the first evidence for the Big Bang theory, reshaping our understanding of the universe's origins and evolution.

---

### Recent Advances

The 20th and 21st centuries have seen incredible advancements in astronomy. The Hubble Space Telescope has provided stunning images of deep space, transforming our understanding of the universe's structure and origins.

<img src="https://github.com/teaghan/astronomy-12/blob/main/Unit1/figures/HST.png?raw=true" alt="HST Image" width="1000" style="display: block; margin-left: auto; margin-right: auto;">

Radio and X-ray astronomy have revealed exotic objects like pulsars and quasars, and the cosmic microwave background radiation has given us a glimpse into the universe's early stages.

<img src="https://github.com/teaghan/astronomy-12/blob/main/Unit1/figures/CMB.png?raw=true" alt="CMB Image" width="1000" style="display: block; margin-left: auto; margin-right: auto;">

The discovery of exoplanets has expanded our search for potentially habitable worlds, with missions like the Kepler Space Telescope identifying thousands of planets around other stars.

<img src="https://github.com/teaghan/astronomy-12/blob/main/Unit1/figures/transit.png?raw=true" alt="transit Image" width="1000" style="display: block; margin-left: auto; margin-right: auto;">

---

### Check Your Understanding

1. **Interdisciplinary Nature**: Based on the video and lesson content, explain how astronomy overlaps with other scientific disciplines. Provide examples of how different fields contribute to our knowledge of the cosmos.
<br><br>

2. **Scientific Method in Astronomy**: Reflect on the video’s discussion of science. How does the scientific method apply specifically to the field of astronomy, and why is it important for scientific progress?

3. **Galileo's discoveries**: Do a bit of research on Galileo's key contributions to astronomy (there are several) and summarize each.
<br><br>

4. **Telescopes in space**: The Hubble Space Telescope orbits Earth and takes images of the sky. Why do you think it is beneficial to have a telscope take images of space *from* space? 
<br><br>

---

### Resources

- Astronomy (2016). Andrew Fraknoi, David Morrison, and Sidney C. Wolff.


# 1_2_scale.md
---
layout: embed_default
---

# 1.2 The Scale of the Universe

## Introduction

In the winter of 1995, scientists pointed the Hubble Space Telescope at a seemingly empty patch of sky near the Big Dipper. Over ten consecutive days, the telescope took close to 150 hours of exposure of this dark, unobtrusive spot. What came back was nothing short of spectacular: an image of over 1,500 distinct galaxies glimmering in a tiny sliver of the universe. This was just a fraction of the sky, about the size of a ballpoint pen tip held at arm's length, revealing an incredible truth about the vastness of our cosmos. This lesson will explore the immense scales of the universe, from our solar system to the farthest reaches observed by humanity, illustrating the vast distances and sizes that define our universe.

<div class="alert alert-block alert-success">
  <h4>Video</h4>
  <p>Watch <a href="https://www.youtube.com/watch?v=WYQ3O8U6SMY" target="_blank">this video</a> for a discussion on the scale of the universe. It includes the distances and sizes of celestial objects and provides a visual representation of the concepts covered in this lesson.</p>
</div>

---

## Scientific Notation

To understand the vast distances and sizes in the universe, we need to use scientific notation. This method allows us to express very large and very small numbers in a manageable way. For example, the distance from the Earth to the Sun is about 150,000,000 kilometers, which can be written as $1.5 \times 10^8$ kilometers.

> One way to think of scientific notation is to think of how many digits to you have to move the decimal place (postivie exponents to the right, negative exponents to the left). For example, $3.8 \times 10^{3}=3800$, whereas $3.8 \times 10^{-3}=0.0038$

---

## The Size and Scale of the Solar System

The solar system is our cosmic neighborhood, and its scale is immense. The Sun, the central star, has a diameter of about 1.4 million kilometers. Earth, the third planet from the Sun, has a diameter of about 12,700 kilometers. The average distance from Earth to the Sun is approximately 150 million kilometers, also known as 1 Astronomical Unit (AU).

> 1 Astronomical Unit (AU) is equivalent to $1.496 \times 10^{11}$ meters.

Imagine if the Sun were the size of a basketball. In this model, Earth would be a small apple seed 30 meters away.

<img src="https://raw.githubusercontent.com/teaghan/astronomy-12/main/Unit1/figures/Our_Solar_Family.png" alt="Our Solar Family" width="1000" style="display: block; margin-left: auto; margin-right: auto;">

---

## Interstellar Distances

Beyond our solar system, the distances between objects become even more staggering. A light-year, the distance light travels in one year, is about 9.46 trillion kilometers. The nearest star to the Sun, Proxima Centauri, is approximately 4.24 light-years away.


> **Derivation of a Light Year**:
> 
> Light travels at a speed of approximately 299,792 kilometers per second (km/s) in a vacuum.
> There are 60 seconds in a minute, 60 minutes in an hour, and 24 hours in a day.
>
> $$ \text{Speed of light} = 299,792 \, \text{km/s} $$
>
> $$ \text{Seconds in a minute} = 60 $$
>
> $$ \text{Minutes in an hour} = 60 $$
>
> $$ \text{Hours in a day} = 24 $$
>
> $$ \text{Days in a year} = 365.25 $$
>
> To find the distance light travels in one year:
>
> $$ \text{Distance} = \text{Speed} \times \text{Time} $$
>
> First, calculate the number of seconds in a year:
>
> $$ 60 \times 60 \times 24 \times 365.25 \approx 31,557,600 \, \text{seconds/year} $$
>
> Then, multiply the speed of light by the number of seconds in a year:
>
> $$ 299,792 \, \text{km/s} \times 31,557,600 \, \text{s/year} \approx 9.46 \times 10^{12} \, \text{kilometers/year} $$
>
> Therefore, one light-year is approximately 9.46 trillion kilometers.


### Parallax: Measuring Stellar Distances

Parallax is a method used to measure the distances to nearby stars by observing their apparent movement against the background of more distant stars as Earth orbits the Sun. The parallax angle is half the angle that a star appears to move over six months.

<img src="https://raw.githubusercontent.com/teaghan/astronomy-12/main/Unit1/figures/Parallax.png" alt="Parallax" width="1000" style="display: block; margin-left: auto; margin-right: auto;">

### Development of Parallax

The concept of parallax dates back to ancient Greece, where Hipparchus, a Greek astronomer, used it to estimate the distance to the Moon around 150 BCE. Hipparchus observed the apparent shift in the position of the Moon against the background stars when viewed from different locations on Earth. This method relies on the geometric principle that the farther an object is, the smaller its apparent shift when observed from two different points.

However, the method was not applied to stars until the 19th century due to the extremely small angles involved. Friedrich Bessel was the first to successfully measure the parallax of a star, 61 Cygni, in 1838. Bessel's achievement provided the first direct and reliable measurement of the distance to a star other than the Sun.

### What is Parallax?

Parallax is the apparent displacement of an object because of a change in the observer's point of view. In astronomy, it specifically refers to the apparent shift of a star's position against the background of distant stars due to Earth's orbit around the Sun. This shift can be measured at different points in Earth's orbit, six months apart, to form the maximum parallax angle.

<div class="alert alert-block alert-success">
  <h4>Video</h4>
  <p>Watch <a href="https://www.youtube.com/watch?v=XUQAIldqPww" target="_blank">this video</a> for an more thorough explanation on parallax.</p>
</div>

### Arcseconds and Parsecs

**Arcseconds:** When measuring parallax angles, astronomers use very small units called arcseconds. One arcsecond is \(1/3600\) of a degree. Because the angles involved in stellar parallax are so tiny, using arcseconds allows for more precise measurement. 

**Parsecs:** The distance to a star can be determined using its parallax angle. A parsec (pc) is defined as the distance at which a star would have a parallax angle of one arcsecond. It is derived from the formula:

> $$ \text{Distance (parsecs)} = \frac{1}{\text{parallax (arcseconds)}} $$

> One parsec is approximately equal to 3.26 light-years. This unit of measurement is especially useful in astronomy for calculating vast interstellar distances.


**More angle units**: In addition to arcseconds, astronomers also use arcminutes and hours to measure angles and time. One arcminute is $\frac{1}{60}$ of a degree, and one hour is $\frac{1}{24}$ of a day.

<div class="alert alert-block alert-warning">
<b>Example:</b> Calculate the distance to a star with a parallax angle of 0.1 arcseconds. 
</div>

> Using the formula: 
>
> $$ \text{Distance (parsecs)} = \frac{1}{\text{parallax (arcseconds)}} $$
> $$ \text{Distance} = \frac{1}{0.1} = 10 \text{ parsecs} $$
>
> To convert parsecs to light-years, use the conversion factor: 
> $$ 1 \text{ parsec} = 3.26 \text{ light-years} $$
> $$ 10 \text{ parsecs} = 10 \times 3.26 = 32.6 \text{ light-years} $$

### Applications and Limitations

Parallax is a fundamental method for measuring distances to nearby stars **within a few hundred light-years from Earth**. Beyond this range, the parallax angles become too small to measure accurately with current technology. However, advancements in space telescopes, like the Gaia mission, have significantly increased the precision of parallax measurements, allowing astronomers to map stars up to tens of thousands of light-years away.

<img src="https://raw.githubusercontent.com/teaghan/astronomy-12/main/Unit1/figures/Star_Cluster.png" alt="Star Cluster" width="1000" style="display: block; margin-left: auto; margin-right: auto;">

---

## The Milky Way Galaxy

The Milky Way is our galaxy, a massive collection of stars, gas, and dust bound together by gravity. It contains between 100 to 400 billion stars, including our Sun. Our solar system is located in one of the spiral arms of the Milky Way, about 27,000 light-years from the galactic center. The Milky Way itself is about 100,000 light-years in diameter.

---

## The Local Group and Other Galaxies

The Milky Way is part of a larger collection of galaxies known as the Local Group, which contains about 54 galaxies. The Andromeda Galaxy, approximately 2.5 million light-years away, is the closest spiral galaxy to the Milky Way. Galaxies come in various shapes and sizes, including spiral, elliptical, and irregular.

> **Did You Know?**
> The Andromeda Galaxy and the Milky Way are on a collision course and are expected to merge in about 4.5 billion years to form a new galaxy, sometimes dubbed "Milkomeda."

<img src="https://raw.githubusercontent.com/teaghan/astronomy-12/main/Unit1/figures/Closest_Spiral_Galaxy.png" alt="Closest Spiral Galaxy" width="1000" style="display: block; margin-left: auto; margin-right: auto;">

---

## The Observable Universe

Beyond the Local Group lies the observable universe, a vast expanse containing billions of galaxies. The observable universe is about 93 billion light-years in diameter. Superclusters and voids are large-scale structures within the universe, with superclusters containing thousands of galaxies.

> **Interesting Side Note**:
> The Hubble Deep Field image, taken in the mid-1990s, revealed over 1,500 galaxies in a tiny area of the sky, approximately one two-millionth of the night sky. This image showed the immense scale and richness of the universe.

<img src="https://raw.githubusercontent.com/teaghan/astronomy-12/main/Unit1/figures/Fornax_Cluster_of_Galaxies.png" alt="Fornax Cluster of Galaxies" width="1000" style="display: block; margin-left: auto; margin-right: auto;">

---

## Methods of Measurement

In this lesson, we studied the use of parallax to measure interstellar distances.

Astronomers use various other methods to measure distances in the universe, each suited to different scales and types of objects. We will study these more throughout the course.

1. **Parallax**:
   - **Useful For**: Measuring distances to nearby stars within a few hundred light-years.
   - **How It Works**: Observes the apparent movement of a star against the background of more distant stars as Earth orbits the Sun. The parallax angle is half the angle that a star appears to move over six months. This method is most effective for stars relatively close to us.
   - **Further Insight**: Parallax measurements become increasingly difficult for stars that are farther away due to the very small angles involved. Advanced instruments, such as the Gaia space observatory, have significantly improved the accuracy of parallax measurements, enabling the mapping of stars up to tens of thousands of light-years away.

2. **Standard Candles**:
   - **Useful For**: Measuring distances to stars and galaxies up to tens of millions of light-years away.
   - **How It Works**: Uses objects with known luminosity, such as Cepheid variables and Type Ia supernovae, to determine distance based on their observed brightness. By comparing the known intrinsic brightness with the observed brightness, the distance to the object can be calculated.
   - **Further Insight**: Cepheid variables have a well-defined relationship between their luminosity and pulsation period, making them reliable distance indicators. Type Ia supernovae, known for their consistent peak brightness, are used to measure distances to faraway galaxies.

3. **Redshift**:
   - **Useful For**: Measuring distances to very distant galaxies and the overall scale of the universe.
   - **How It Works**: Measures the shift in the spectral lines of distant galaxies toward the red end of the spectrum. This redshift is due to the expansion of the universe and can be used to determine the galaxy's distance. The greater the redshift, the farther away the galaxy is.
   - **Further Insight**: Redshift measurements are fundamental to cosmology, providing crucial data for understanding the expansion rate of the universe (Hubble's Law) and the distribution of galaxies on a large scale.

---

## Check Your Understanding

1. **Unit Conversion**: The distance from Earth to Proxima Centauri is 4.24 light-years. Convert this distance to km, using appropriate scientific notation. How many Astonomical Units (AUs) would this be?
<br><br>

2. **Speed of light**: The distance from Earth to the Sun  is 149 million kilometers. How long does it take the light from the Sun to reach Earth's surface?
<br><br>

3. **Solar System Scale**: If the Sun were the size of a basketball, how far would Neptune be in your model?
<br><br>

4. **Parallax**: Explain how parallax is used to measure the distance to nearby stars.
<br><br>

5. **Measuring Distances**: If a star has a parallax angle of 0.23 arcseconds, what is its distance from Earth in parsecs? Convert this distance to light-years and kilometers (using scientific notation).
<br><br>

## Resources

- Astronomy (2016). Andrew Fraknoi, David Morrison, and Sidney C. Wolff.


# 1_3_the_sky.md
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---

# 1.3 The Celestial Sphere

### Introduction

Have you ever wondered why the stars seem to move across the sky each night, or why certain constellations are only visible during specific times of the year? To ancient astronomers, the heavens appeared as a vast, rotating dome with the Earth at its center. This imaginary dome, or *celestial sphere*, is a concept that has helped astronomers understand and describe the positions and movements of celestial objects for centuries. Today, we'll explore the celestial sphere and discover its significance in modern astronomy.

<div class="alert alert-block alert-success">
  <h4>Video</h4>
  <p>Watch <a href="https://www.youtube.com/watch?v=1Toya19H12w" target="_blank">this video</a> for a quick run-down of the celestial sphere and the motion of the stars.</p>
</div>

---

### The Celestial Sphere

Imagine lying back on a clear night, gazing up at the stars. It feels as if you are inside a gigantic, hollow dome with the stars stuck onto its inner surface. This is the celestial sphere, an imaginary sphere of arbitrarily large radius, centered on the Earth, on which all celestial objects can be considered to lie. Although the celestial sphere is a simplification, it remains incredibly useful for locating stars, planets, and other celestial bodies in the sky.

<div class="alert alert-block alert-info">
<b>Tip:</b> Think of the celestial sphere as a giant globe surrounding the Earth, with stars projected onto its inner surface.
</div>

Key features of the celestial sphere include:

- **Zenith**: The point directly overhead.
- **Horizon**: The line where the sky appears to meet the Earth.
- **Celestial Poles**: Extensions of Earth's North and South Poles onto the celestial sphere.
- **Celestial Equator**: The projection of Earth's equator onto the celestial sphere.

<img src="https://raw.githubusercontent.com/teaghan/astronomy-12/main/Unit1/figures/Figure2_2.png" alt="The Sky around Us" width="1000" style="display: block; margin-left: auto; margin-right: auto;">

---

### Celestial Poles and Celestial Equator

To understand the celestial sphere, imagine extending Earth's axis points into the sky. The points where this line intersects the celestial sphere are the north celestial pole and the south celestial pole. As Earth rotates about its axis, the sky appears to turn in the opposite direction around these celestial poles.

> The celestial equator divides the sky into the northern and southern celestial hemispheres, similar to how Earth's equator divides our planet.

For example, if you stood at the North Pole, you would see the north celestial pole overhead at your zenith. The celestial equator would lie along your horizon. As you watch the stars during the course of the night, they would all circle around the celestial pole, with none rising or setting. Conversely, at the equator, you would see the celestial equator pass overhead through your zenith, and the celestial poles at the north and south points on your horizon.

<img src="https://raw.githubusercontent.com/teaghan/astronomy-12/main/Unit1/figures/Figure2_3.png" alt="Circles on the Celestial Sphere" width="1000" style="display: block; margin-left: auto; margin-right: auto;">

<img src="https://raw.githubusercontent.com/teaghan/astronomy-12/main/Unit1/figures/Figure2_4.png" alt="Circling the South Celestial Pole" width="1000" style="display: block; margin-left: auto; margin-right: auto;">

---

### Rising and Setting of the Sun

The Sun, Moon, and stars appear to rise in the east and set in the west due to Earth's rotation. This daily movement, known as diurnal motion, is one of the most familiar astronomical phenomena. The path the Sun appears to take across the celestial sphere over the course of a year is called the ecliptic. This path is inclined to the celestial equator due to the tilt of Earth's axis, leading to seasonal changes.

> Imagine standing in San Francisco, where the latitude is 38° N. The north celestial pole would be 38° above the northern horizon. As Earth turns, the whole sky seems to pivot about the north celestial pole. Stars within 38° of the North Pole never set, while stars within 38° of the south celestial pole never rise.

<img src="https://raw.githubusercontent.com/teaghan/astronomy-12/main/Unit1/figures/Figure2_7.png" alt="The Celestial Tilt" width="1000" style="display: block; margin-left: auto; margin-right: auto;">

---

### Interactive Exploration

The simulator below gives an interactive visualization of the celestial sphere and the visible sky. Anything above the green plane in the right hand screen is visible to the observer. Try playing around with the settings and clicking "start animation".

<div style="width: 100%; height: 0; padding-bottom: 100%; position: relative;">
    <iframe src="https://astro.unl.edu/naap/motion2/animations/ce_hc.html" 
            style="position: absolute; width: 100%; height: 100%; top: 0; left: 0;" 
            allowfullscreen>
    </iframe>
</div>

---

### Fixed and Wandering Stars

Ancient astronomers categorized celestial objects into "fixed stars" and "wandering stars." Fixed stars maintain fixed patterns relative to each other and form constellations. Wandering stars, known today as planets, move relative to the fixed stars.

The fixed stars include most of the stars we see in the night sky. They form patterns that have been recognized and named as constellations. These patterns appear to move across the sky in a predictable way each night and throughout the year, making them useful for navigation and timekeeping.

> The ancient Greeks called the planets "asteres planetai," which means "wandering stars." Unlike the fixed stars, the planets change their positions relative to the background stars over days, weeks, and months.

---

### Constellations

Constellations are patterns of stars that have been identified and named by various cultures throughout history. Today, astronomers recognize 88 constellations that divide the sky into sectors, making it easier to locate celestial objects.

The modern constellations are used not only to identify patterns of stars but also to define specific areas of the sky. Each constellation covers a region of the celestial sphere, so every point in the sky belongs to one of the 88 constellations. This system helps astronomers communicate about locations in the sky with precision.

> **Did You Know?** The modern meaning of a constellation includes not just the pattern of stars but also the region of the sky around it. For example, the constellation Orion is not just the familiar hunter figure but also the area of the sky where this pattern is found.

<img src="https://raw.githubusercontent.com/teaghan/astronomy-12/main/Unit1/figures/Figure2_6.png" alt="Constellations on the Ecliptic" width="1000" style="display: block; margin-left: auto; margin-right: auto;">

---

### Check Your Understanding

1. How many degrees does the Sun move per day relative to the fixed stars? How many days does it take for the Sun to return to its original location relative to the fixed stars?
<br><br>

2. The Moon's orbital period around Earth is approximately 27.3 days. How many degrees does the Moon move per day relative to the fixed stars? 
<br><br>

3. Is the ecliptic the same thing as the celestial equator? Explain.
<br><br>

4. Explain why more stars are circumpolar for observers at higher latitudes.
<br><br>

5. What is the altitude of the north celestial pole in the sky from your latitude? (The latitude of Salt Spring Island, BC is approximately 49 degrees N.)
<br><br>

6. Suppose you are on a strange planet and observe, at night, that the stars do not rise and set, but circle parallel to the horizon. Next, you walk in a constant direction for 8000 miles, and at your new location on the planet, you find that all stars rise straight up in the east and set straight down in the west, perpendicular to the horizon. How could you determine the circumference of the planet without any further observations? What is the circumference, in miles, of the planet?
<br><br>

---


### Resources

- Astronomy (2016). Andrew Fraknoi, David Morrison, and Sidney C. Wolff.


# 1_4_brightness.md
---
layout: embed_default
---

# 1.4 The Brightness of Stars

## Introduction

Have you ever gazed up at the night sky and wondered why some stars shine brighter than others? The study of stellar brightness not only allows us to appreciate the beauty of the cosmos but also provides crucial information about the stars themselves. In this lesson, we'll explore how astronomers measure the brightness of stars, the physics behind their luminosity, and the historical context of the magnitude scale.

<div class="alert alert-block alert-success">
  <h4>Video</h4>
  <p>Watch <a href="https://youtu.be/JIXFXGiDa4Y?si=x7IidSPRQEKKfjgL" target="_blank">this video</a> for an excellent overview of the brightness of stars, including practical examples and historical context.</p>
</div>

---

### Flux and Luminosity

To understand stellar brightness, we first need to define a few key concepts: flux and luminosity.

**Flux** is the amount of light energy received per unit area per unit time. Imagine standing under a streetlamp at night. The light that hits you is the flux.

**Luminosity**, on the other hand, is the total amount of energy a star emits per unit time. Think of it as the star's total power output.

### The Inverse Square Law

As light travels from a star, it spreads out over an ever-increasing area. This spreading out follows the **Inverse Square Law**, which states that the flux decreases with the square of the distance from the source. Mathematically, we express this as:

> $$ F = \frac{L}{4\pi r^2} $$

where:
- $F$ is the flux (in watts per square meter, $\text{W/m}^2$),
- $L$ is the luminosity (in watts, $\text{W}$),
- $r$ is the distance from the star (in meters, $\text{m}$).

> This law explains why stars appear dimmer the farther away they are. It's not because they're less powerful, but because their light spreads out over a larger area.

<img src="https://raw.githubusercontent.com/teaghan/astronomy-12/main/Unit1/figures/inv_sq_law.png" alt="Inverse Square Law" width="1000" style="display: block; margin-left: auto; margin-right: auto;">

<div class="alert alert-block alert-warning">
<b>Example:</b> Calculating the Solar Flux at Different Distances
</div>

> The luminosity of the Sun is $ 3.839 \times 10^{26} $ W and is at a distance of 1 AU (astronomical unit) away from Earth, which is $ 1.496 \times 10^{11} $ meters.
> 
> Earth receives a radiant flux above its absorbing atmosphere given by:
> 
> > $$ F = \frac{L}{4\pi r^2} $$
> > 
> > $$ F = \frac{3.839 \times 10^{26}}{4\pi (1.496 \times 10^{11})^2} $$
> > 
> > $$ F \approx 1365 \, \text{W/m}^2 $$
>
>This value of the solar flux is known as the **solar irradiance**, sometimes also called the solar constant.

---

## Apparent Brightness and the Magnitude Scale

### Apparent Magnitude

The **apparent magnitude** of a star is a measure of its brightness as seen from Earth. This concept is crucial because it takes into account both the intrinsic luminosity of the star and its distance from us.

### Historical Background of the Magnitude Scale

The magnitude scale dates back to the ancient Greek astronomer Hipparchus, who classified stars according to their apparent brightness. This system was refined by Ptolemy and later standardized to a logarithmic scale, where a difference of 5 magnitudes corresponds to a factor of 100 in brightness.

The logarithmic nature of the scale is represented by:

> $$ m_1 - m_2 = -2.5 \log_{10} \left( \frac{F_1}{F_2} \right) $$

where $ m_1 $ and $ m_2 $ are the magnitudes of two stars, and $ F_1 $ and $ F_2 $ are their respective fluxes.

Another way to write this equation is:

> $$ \frac{F_2}{F_1} = \left(100^{0.2}\right)^{m_1 - m_2} $$

which allows us to compare the flux of two objects (and the flux is what we would typically call "brightness" in every day conversation).

> The star Vega was historically used as the standard for apparent magnitude, set at 0. Modern measurements have slightly adjusted this value, but Vega remains a key reference point in the magnitude system.

<div class="alert alert-block alert-info">
<b>Recall:</b> A logarithm is a mathematical operator that tells us the power to which a base number must be raised to obtain a given number. When the base is not specified, it is assumed to be 10.
</div>

> For example, $ \log_{10}(100) = 2 $ because $ 10^2 = 100 $.

Logarithmic scales are used in astronomy to manage the huge differences in the brightness and distances of celestial objects.

<img src="https://raw.githubusercontent.com/teaghan/astronomy-12/main/Unit1/figures/Apparent_Magnitudes.png" alt="Apparent Magnitudes of Well-Known Objects" width="1000" style="display: block; margin-left: auto; margin-right: auto;">

<div class="alert alert-block alert-warning">
<b>Example:</b> Suppose an astronomer has discovered something special about a dim star with a magnitude of 8.5 and wants to explain how much dimmer this star is compared to Sirius. We'll use the dim star as Star 1 and Sirius as Star 2 in our equation.
</div>

> Given:
> 
> > Dim star magnitude, $ m_1 = 8.5 $
> > 
> > Sirius magnitude, $ m_2 = -1.5 $
> 
> Solution:
> 
> > $$ \frac{F_2}{F_1} = \left(100^{0.2}\right)^{8.5 - (-1.5)} $$
> > 
> > $$ \frac{F_2}{F_1} = \left(100^{0.2}\right)^{10} $$
> > 
> > $$ \frac{F_2}{F_1} = (100)^2 = 100 \times 100 = 10,000 $$
> > 
> > Therefore, the dim star is 10,000 times dimmer than Sirius.

<div class="alert alert-block alert-warning">
<b>Example:</b> It is a common misconception that Polaris (magnitude 2.0) is the brightest star in the sky. However, Sirius, with a magnitude of -1.5, holds that title. Let's compare the apparent brightness of Sirius to Polaris.
</div>

> Given:
> 
> > Polaris magnitude, $ m_1 = 2.0 $
> > 
> > Sirius magnitude, $ m_2 = -1.5 $
> 
> Solution:
> 
> > $$ \frac{F_2}{F_1} = \left(100^{0.2}\right)^{2.0 - (-1.5)} $$
> > 
> > $$ \frac{F_2}{F_1} = \left(100^{0.2}\right)^{3.5} $$
> > 
> > $$ \frac{F_2}{F_1} = 100^{0.7} = 25 $$
> 
> Our calculation shows that Sirius’ apparent brightness is 25 times greater than Polaris’ apparent brightness.

---

## Absolute Magnitude

Absolute magnitude is the apparent magnitude (brightness) a star would have if it were placed at a standard distance of 10 parsecs from Earth. This standardization allows astronomers to compare the true brightness of stars without the confounding effect of distance.

The relationship between a star's absolute magnitude $ M $, apparent magnitude $ m $, and distance $ d $ in parsecs is given by:

> $$ M = m + 5 - 5 \log_{10}(d) $$

---

## Distance Modulus

### Explanation of Distance Modulus

The **distance modulus** provides a way to calculate the distance to a star based on its apparent and absolute magnitudes. The formula is:

> $$ m - M = 5 \log_{10}(d) - 5 $$

where:
- $ m $ is the apparent magnitude,
- $ M $ is the absolute magnitude,
- $ d $ is the distance in parsecs.

This equation is incredibly useful for determining stellar distances, making it a cornerstone of astronomical measurement.

<div class="alert alert-block alert-warning">
<b>Example:</b> Calculating the Absolute Magnitude of the Sun
</div>

> The apparent magnitude of the Sun, $ m_{\text{Sun}} $, is $ -26.83 $, and its distance from Earth is 1 AU, which is $ 4.848 \times 10^{-6} $ parsecs. Using the formula for absolute magnitude:
> 
> $$ M_{\text{Sun}} = m_{\text{Sun}} - 5 \log_{10}(d) + 5 $$
> 
> we can calculate the absolute magnitude of the Sun.
> 
> $$ M_{\text{Sun}} = m_{\text{Sun}} - 5 \log_{10}(d) + 5 $$
> 
> $$ M_{\text{Sun}} = -26.83 - 5 \log_{10}(4.848 \times 10^{-6}) + 5 $$
> 
> Since $ \log_{10}(4.848 \times 10^{-6}) \approx -5.315 $:
> 
> $$ M_{\text{Sun}} = -26.83 - 5(-5.315) + 5 $$
> 
> $$ M_{\text{Sun}} = -26.83 + 26.575 + 5 $$
> 
> $$ M_{\text{Sun}} = 4.74 $$
>
>> Thus, the absolute magnitude of the Sun is $ +4.74 $.

---

## Check Your Understanding

1. **Brightness**: Other than the distance to a star, what characteristics of a star do you think would make it more or less bright to us?
<br><br>
2. **Flux and Luminosity**: Given a star with a luminosity of $ 4 \times 10^{26} $ watts and a distance of 2 parsecs, calculate the flux received on Earth.
<br><br>
3. **Magnitude Scale** Use the Magnitude Scale equation below to explain why a magnitude 2 compared to a magnitude 5 star is about 16 times brighter.

> $$ m_1 - m_2 = -2.5 \log_{10} \left( \frac{F_1}{F_2} \right) $$

4. **Absolute Magnitude**: If a star has an apparent magnitude of 7 and is located 50 parsecs away, what is its absolute magnitude?
<br><br>
5. **Apparent Brightness**: Two stars have apparent magnitudes of 3 and 5. Calculate the ratio of their fluxes (*i.e.* $\frac{F_2}{F_1}$).
<br><br>
6. **Distance Modulus**: Calculate the distance to a star with an apparent magnitude of 10 and an absolute magnitude of 2.
<br><br>
7. **Flux**: At what distance from a 100-W light bulb is the radiant flux equal to the solar irradiance?
<br><br>
8. **Algebra Challenge**: Derive the relation

> $$m = M_{Sun} − 2.5 \log_{10}(\frac{F}{F_{10,Sun}})$$

where $M_{Sun}$ is the absolute magnitude of the Sun and $F_{10,Sun}$ is the flux received from the Sun if it were 10 parsecs away from the observer.

---

### Resources

- Astronomy (2016). Andrew Fraknoi, David Morrison, and Sidney C. Wolff.
- An Introduction to Modern Astrophysics (2017). Bradley W. Carroll and Dale A. Ostlie.


# 1_5_light_travel.md
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---

# 1.5 Consequences of Light Travel Time

## Introduction

Imagine looking up at the night sky and seeing stars that are no longer there. This isn't a sci-fi movie plot but a fascinating consequence of light travel time. When we look at the stars, we see them not as they are now, but as they were in the past. This phenomenon arises because light takes time to travel across the vast distances of space. Understanding light travel time not only helps us measure cosmic distances but also allows us to peer back into the history of the universe.

<div class="alert alert-block alert-success">
  <h4>Video</h4>
  <p>Watch <a href="https://www.youtube.com/watch?v=MohKd8vSdBA" target="_blank">this video</a> for a deeper understanding of light travel time and its implications in astronomy. This video explains the fundamental concepts and fascinating consequences of observing the universe through the lens of light travel time.</p>
</div>

---

## Historical Context

> **Did You Know?** Early theories about the speed of light date back to ancient India. Scholars in the 14th century speculated that light might travel like wind. Although they couldn't prove it, their calculations were surprisingly close to modern measurements of the speed of light.

### Key Historical Contributions

- **Ole Rømer**: In 1676, Rømer was the first to demonstrate that light travels at a finite speed by observing the moons of Jupiter. He noticed discrepancies in the timings of the eclipses of Jupiter's moons, which he attributed to the varying distance between Earth and Jupiter as both planets orbited the Sun.
- **Albert A. Michelson**: In the late 19th and early 20th centuries, Michelson's experiments refined the measurement of the speed of light to its modern value. Using rotating mirrors and precise timing devices, he determined the speed of light with remarkable accuracy.

### The Speed of Light

The most important point to remember is that light travels at a constant speed in a vacuum, approximately 299,792 kilometers per second. This immense speed still takes time to cover the vast distances in space. 

> Light from the Sun takes about 8 minutes and 20 seconds to reach Earth. When you look at the Sun, you see it as it was over 8 minutes ago.

---

## Measuring Distances in Astronomy

Light travel time is crucial in measuring astronomical distances. By knowing how fast light travels, we can determine how far away celestial objects are.

### Concepts of Light-Years and Parsecs

Recall these concepts from a previous lesson:

- **Light-Year**: The distance light travels in one year, approximately 9.46 trillion kilometers. It's a convenient unit for expressing astronomical distances because it directly relates to the speed of light.
- **Parsec**: A unit of distance used in astronomy, equivalent to about 3.26 light-years. One parsec is the distance at which one astronomical unit subtends an angle of one arcsecond.

### Techniques for Measuring Distances

- **Parallax**: By observing the apparent shift in a star's position as Earth orbits the Sun, astronomers can calculate its distance. The greater the shift, the closer the star is.

- **Standard Candles**: These are astronomical objects with a known luminosity. By comparing the known luminosity of these objects to their observed brightness, astronomers can calculate their distance. Here are two main types of standard candles:

  1. **Cepheid Variables**:
     - Cepheid variables are stars that pulsate in a regular cycle. The period of their pulsation is directly related to their luminosity; this relationship is known as the period-luminosity relation.
     - By observing the period of a Cepheid variable, astronomers can determine its true luminosity. Comparing this with the apparent brightness (how bright it appears from Earth) allows astronomers to calculate the distance to the star.
     - Cepheid variables are crucial for measuring distances to nearby galaxies and are often used to establish the distance scale of the universe.

  2. **Type Ia Supernovae**:
     - Type Ia supernovae occur in binary systems where a white dwarf accretes material from its companion star until it reaches a critical mass and explodes.
     - These supernovae have a consistent peak luminosity, making them excellent standard candles for measuring cosmic distances.
     - Because they are incredibly bright, Type Ia supernovae can be observed in distant galaxies, allowing astronomers to measure vast distances across the universe.

<div class="alert alert-block alert-warning">
<b>Example:</b> Imagine astronomers observe a Cepheid variable star in a nearby galaxy. The observed apparent magnitude of the star is 15.2, and its absolute magnitude is known to be -3.4. Calculate the distance to the Cepheid variable star.
</div>

> **Solution**:
>
> 1. **Formula**: The distance modulus formula relates the apparent magnitude ($m$) and absolute magnitude ($M$) of a star to its distance in parsecs ($d$):
>    $$
>    m - M = 5 \log_{10}(d) - 5
>    $$
> 2. **Rearrange the Formula to Solve for Distance**:
>    $$
>    m - M + 5 = 5 \log_{10}(d)
>    $$
>    $$
>    \frac{m - M + 5}{5} = \log_{10}(d)
>    $$
>    $$
>    d = 10^{\left( \frac{m - M + 5}{5} \right)}
>    $$
> 3. **Substitute the Given Values**: $m = 15.2$, $M = -3.4$
>    $$
>    d = 10^{\left( \frac{15.2 - (-3.4) + 5}{5} \right)}
>    $$
>    $$
>    d = 10^{\left( \frac{15.2 + 3.4 + 5}{5} \right)}
>    $$
>    $$
>    d = 10^{\left( \frac{23.6}{5} \right)}
>    $$
>    $$
>    d = 10^{4.72}
>    $$
> 4. **Calculate the Distance**:
>    $$
>    d \approx 52,480 \text{ parsecs}
>    $$
> 
>> Therefore, the distance to the Cepheid variable star is approximately 52,480 parsecs.

---

This example uses the distance modulus formula to calculate the distance to a Cepheid variable star, incorporating the given apparent and absolute magnitudes. This approach helps students apply their knowledge of standard candles and the distance modulus in a practical scenario.

---

## Observable Universe and Light Travel Time

When we observe distant galaxies, we are looking back in time. This concept is fundamental to understanding the observable universe.

### Observing the Past

- **Light from the Past**: The light from distant stars and galaxies takes millions or even billions of years to reach us. Thus, we see these objects as they were in the distant past. For example, when you see the Andromeda Galaxy, you're seeing it as it was about 2.5 million years ago.
- **Cosmic Horizon**: The farthest we can see is limited by the age of the universe and the speed of light. This boundary is known as the cosmic horizon. Beyond this, the light hasn't had enough time to reach us since the beginning of the universe.

### Implications for Astronomy

Observing the past allows astronomers to study the evolution of the universe, including the formation of galaxies, stars, and other cosmic phenomena. It provides a unique window into the history of the cosmos, allowing us to understand how it has changed over billions of years.

> **Interesting Side Note**: The Hubble Space Telescope has observed galaxies as they were just a few hundred million years after the Big Bang, providing a glimpse into the early universe.

<img src="https://raw.githubusercontent.com/teaghan/astronomy-12/main/Unit1/figures/Telescope_in_Orbit.png" alt="Hubble Space Telescope" width="1000" style="display: block; margin-left: auto; margin-right: auto;">

---

## Astronomical Phenomena and Light Travel Time

### Supernovae

When a star explodes as a supernova, the light from the explosion can take years to reach us. By studying these explosions, astronomers can learn about the life cycles of stars. For instance, the supernova observed in the Large Magellanic Cloud in 1987 (SN 1987A) provided invaluable insights into the processes occurring at the end of a star's life.

### Space Observations

- **Hubble Space Telescope**: In 2016, Hubble observed a galaxy 32 billion light-years away. Due to the expanding universe, we see this galaxy as it was 13.4 billion years ago, offering a glimpse into the early stages of galaxy formation.

- **James Webb Space Telescope**: Set to revolutionize our understanding by observing in the infrared spectrum, allowing us to see even further back in time. This telescope will help us study the formation of the first stars and galaxies, pushing the boundaries of our knowledge about the early universe.

<div class="alert alert-block alert-info">
<b>Tip:</b> Remember, every time you look at the stars, you are looking back in time. The further away an object is, the older the light we are seeing.
</div>

---

## Check Your Understanding

1. When astronomers observe the Andromeda Galaxy, which is about 2.5 million light-years away, what are they seeing?
<br><br>
2. Imagine astronomers observe a supernova in a distant galaxy. The light from the supernova, which occurred 163 years ago, has just reached Earth. Determine the parallax angle of the supernova.
<br><br>
3. If a spacecraft could travel at 1% of the speed of light, how many years would it take to travel to a star 10 light-years away?
<br><br>
4. Describe the concept of the cosmic horizon and explain why we cannot see beyond it.
<br><br>

---

### Resources

- **Astronomy (2016)**. Andrew Fraknoi, David Morrison, and Sidney C. Wolff.
- **An Introduction to Modern Astrophysics (2017)**. Bradley W. Carroll and Dale A. Ostlie.


# 2_1_keplers_laws.md
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---

# 2.1 Kepler's Laws: The Story of Planetary Motion

---

#### Introduction

Today we will uncover the story of Kepler's Laws of Planetary Motion. These laws revolutionized our understanding of the solar system and laid the foundation for modern astronomy.

<div class="alert alert-block alert-success">
  <h4>Video</h4>
  <p>Watch <a href="https://www.youtube.com/watch?v=Dvoe8Ib5D1o" target="_blank">this video</a> for an overview of Kepler's Laws of Planetary Motion.</p>
</div>

---

### Historical Context

#### Johannes Kepler: The Mathematician of the Heavens

Johannes Kepler was a German mathematician and astronomer who lived in the late 16th and early 17th centuries. Born in 1571, Kepler is best known for formulating the three laws of planetary motion, which described the orbits of planets around the Sun with unprecedented accuracy.

#### Tycho Brahe: The Observational Genius

Tycho Brahe, a Danish nobleman, was a meticulous observer of the stars and planets. His detailed astronomical data provided the foundation upon which Kepler built his laws. Brahe's observations were the most accurate of his time, and his collaboration with Kepler was crucial to the development of the laws of planetary motion.

#### The Copernican Revolution

The heliocentric model proposed by Nicolaus Copernicus placed the Sun at the center of the solar system, challenging the long-held geocentric model. Kepler's laws provided the mathematical evidence needed to support Copernicus's revolutionary idea, cementing the heliocentric model as the correct representation of our solar system.

---

### Kepler’s Laws of Planetary Motion

#### First Law (Law of Ellipses)

Imagine a time when the prevailing belief was that planets orbited the Earth in perfect circles. This view dominated until Johannes Kepler, through meticulous analysis of Tycho Brahe's precise observations, discovered something remarkable. Kepler found that the planets, including our Earth, do not travel in perfect circles but rather in ellipses—an elegant, stretched circle.

> **Kepler's First Law** states that planets move in elliptical orbits with the Sun at one focus.

An ellipse has a unique shape characterized by two focal points (foci). For planetary orbits, one of these foci is occupied by the Sun. The degree of this "stretching" is determined by the eccentricity of the ellipse. An eccentricity of 0 indicates a perfect circle, while values approaching 1 indicate increasingly elongated shapes. This breakthrough explained the varying distances between the planets and the Sun, a phenomenon that could not be reconciled with circular orbits.

<img src="https://raw.githubusercontent.com/teaghan/astronomy-12/main/Unit2/figures/ellipse.png" alt="Ellipse" width="1000" style="display: block; margin-left: auto; margin-right: auto;">

<div class="alert alert-block alert-info">
  <b>Tip:</b> Remember, an ellipse has two foci. For planetary orbits, one of these foci is occupied by the Sun.
</div>

Kepler's First Law shattered the long-held notion of circular orbits and paved the way for a deeper understanding of celestial mechanics. It was a testament to the power of careful observation and the willingness to challenge established beliefs.

<img src="https://raw.githubusercontent.com/teaghan/astronomy-12/main/Unit2/figures/7_3.png" alt="Orbits of the Planets" width="1000" style="display: block; margin-left: auto; margin-right: auto;">

#### Second Law (Law of Equal Areas)

Kepler found that, when a planet was moving along its elliptical path, it accelerated as it got closer to the Sun, covering a greater distance in a shorter time. Conversely, when it was further away from the Sun, it slowed down. Kepler’s Second Law captures this beautifully by asserting that the imaginary line connecting a planet to the Sun sweeps out equal areas in equal time periods, regardless of where the planet is in its orbit.

> **Kepler's Second Law** states that a line segment joining a planet and the Sun sweeps out equal areas during equal intervals of time.

<img src="https://raw.githubusercontent.com/teaghan/astronomy-12/main/Unit2/figures/3_5.png" alt="Kepler's Second Law: The Law of Equal Areas" width="1000" style="display: block; margin-left: auto; margin-right: auto;">

For instance, when Earth is at perihelion (the point closest to the Sun), it travels faster, zipping through its orbit. When it reaches aphelion (the point farthest from the Sun), its pace slows. This variation in speed explained why planets appeared to change their velocity as observed from Earth—an observation that had puzzled astronomers for centuries.


Kepler's Second Law not only provided insights into planetary motion but also hinted at the gravitational forces at play, a concept that would later be fully developed by Sir Isaac Newton.

#### Third Law (Law of Harmonies)

Kepler also found that there is a precise mathematical relationship between the time a planet takes to orbit the Sun (its orbital period) and the size of its orbit (the semi-major axis). Specifically, the square of a planet’s orbital period is proportional to the cube of the semi-major axis of its orbit.

> **Kepler's Third Law** states that the square of the orbital period of a planet is proportional to the cube of the semi-major axis of its orbit.

Mathematically, this can be written as:

> $$ P^2 \propto a^3 $$

or 

> $$ P^2 = k \cdot a^3 $$

where $ P $ is the orbital period, $ a $ is the semi-major axis, and $ k $ is a constant.

To put this in perspective, consider Earth and Mars. Earth, with a semi-major axis of 1 astronomical unit (AU), completes an orbit in one year. Mars, with a semi-major axis of about 1.52 AU, takes approximately 1.88 Earth years to complete its orbit. Plugging these values into Kepler’s formula, we find that:

> $$ P_{\text{Earth}}^2 = 1^2 = 1 $$
> $$ a_{\text{Earth}}^3 = 1^3 = 1 $$
> 
> $$ P_{\text{Mars}}^2 \approx 1.88^2 = 3.53 $$
> $$ a_{\text{Mars}}^3 \approx 1.52^3 = 3.52 $$

This near-perfect correlation between $ P^2 $ and $ a^3 $ demonstrated the universal applicability of Kepler’s Third Law across different planets, from Mercury to the distant gas giants.

<div class="alert alert-block alert-info">
  <b>Note:</b> When using Kepler's Third Law, ensure that the units are consistent. Typically, the orbital period is measured in Earth years, and the semi-major axis is measured in astronomical units (AU). This consistency is crucial for maintaining the correct proportionality and making accurate predictions. In these units, the proportionality constant, k, is equal to 1.
</div>

Kepler's Third Law unified the celestial mechanics of our solar system, offering a powerful tool for predicting planetary positions and understanding the vast cosmic clockwork. It set the stage for Newton's laws of motion and universal gravitation, which would further explain the forces driving these harmonious motions.

#### Orbital Data for the Planets

To see Kepler's laws in action, let's examine the orbital data for the planets in our solar system:

| Planet   | Semimajor Axis (AU) | Period (years) | Eccentricity |
|----------|----------------------|----------------|--------------|
| Mercury  | 0.39                 | 0.24           | 0.21         |
| Venus    | 0.72                 | 0.6            | 0.01         |
| Earth    | 1.00                 | 1.00           | 0.02         |
| Mars     | 1.52                 | 1.88           | 0.09         |
| (Ceres)  | 2.77                 | 4.6            | 0.08         |
| Jupiter  | 5.20                 | 11.86          | 0.05         |
| Saturn   | 9.54                 | 29.46          | 0.06         |
| Uranus   | 19.19                | 84.01          | 0.05         |
| Neptune  | 30.06                | 164.82         | 0.01         |

---

### Mathematical Examples

<div class="alert alert-block alert-warning">
<b>Example:</b> Calculate the orbital period of an object with a semimajor axis of 50 AU.
</div>

> **Solution**:
>
> 1. **Formula**: Kepler's Third Law relates the square of the orbital period ($P$) to the cube of the semimajor axis ($a$):
>    $$
>    P^2 = a^3
>    $$
> 2. **Substitute the Given Value**: $a = 50 \text{ AU}$
>    $$
>    P^2 = 50^3
>    $$
>    $$
>    P^2 = 125000
>    $$
> 3. **Solve for $P$**:
>    $$
>    P = \sqrt{125000}
>    $$
>    $$
>    P \approx 353.55 \text{ years}
>    $$
>
>> Therefore, the orbital period of an object with a semimajor axis of 50 AU is approximately 353.55 years.

---

<div class="alert alert-block alert-warning">
<b>Example:</b> Verify $ P^2 $ and $ a^3 $ for Venus and Earth using given orbital periods and semimajor axes.
</div>

> **Solution**:
>
> **For Venus**:
>
> 1. **Given**: 
>    $$
>    P = 0.62 \text{ years}, \quad a = 0.72 \text{ AU}
>    $$
> 2. **Calculate $P^2$**:
>    $$
>    P^2 = 0.62^2 \approx 0.3844
>    $$
> 3. **Calculate $a^3$**:
>    $$
>    a^3 = 0.72^3 \approx 0.3732
>    $$
>
>> For Venus, $ P^2 \approx 0.3844 $ and $ a^3 \approx 0.3732 $.
>
> **For Earth**:
>
> 1. **Given**:
>    $$
>    P = 1.00 \text{ year}, \quad a = 1.00 \text{ AU}
>    $$
> 2. **Calculate $P^2$**:
>    $$
>    P^2 = 1.00^2 = 1.00
>    $$
> 3. **Calculate $a^3$**:
>    $$
>    a^3 = 1.00^3 = 1.00
>    $$
>
>> For Earth, $ P^2 = 1.00 $ and $ a^3 = 1.00 $.

---

### Applications and Implications

#### Predicting Planetary Positions

Kepler’s laws allow astronomers to predict the positions of planets in their orbits with great accuracy. This capability is crucial for space navigation and understanding the dynamics of our solar system.

In the early 17th century, predicting the precise locations of planets was an arduous task filled with inaccuracies. Before Kepler, astronomers relied on complex systems of epicycles and deferents to explain planetary motion, a method rooted in the Ptolemaic geocentric model. However, these systems often fell short in matching observations, leading to significant errors in predictions.

With the advent of Kepler's laws, the process of predicting planetary positions became not only simpler but also far more precise. Kepler's First Law clarified that planets travel in elliptical orbits, which accounted for the varying distances between planets and the Sun. His Second Law described how planets speed up and slow down in their orbits, allowing for accurate calculation of their velocities at any given point. The Third Law provided a direct relationship between a planet's orbital period and the size of its orbit, making it possible to predict how long it takes for a planet to complete one revolution around the Sun.

Let’s consider the example of Mars, which was particularly puzzling to astronomers before Kepler. Using Tycho Brahe's detailed observations, Kepler determined that Mars follows an elliptical path. By applying his laws, he could predict Mars' position in the sky at any given time. This was a groundbreaking achievement that not only matched but also exceeded the accuracy of previous models.

#### Kepler’s Legacy in Modern Astronomy

Kepler's laws are not confined to our solar system alone. They are used to study exoplanets—planets orbiting stars other than the Sun. By observing the dimming of a star as a planet passes in front of it (a method known as the transit method), astronomers can apply Kepler's Third Law to determine the exoplanet’s orbital period and distance from its star. This has led to the discovery of thousands of exoplanets, expanding our understanding of the universe.

#### Historical Anecdote: Halley’s Comet

A fascinating historical application of Kepler’s laws involves Halley's Comet. Edmond Halley, using Kepler's laws, calculated the orbit of this famous comet and predicted its return in 1758. Although Halley did not live to see the comet's return, his prediction proved accurate, solidifying the laws' reliability in predicting celestial events. Halley's Comet continues to be a testament to the enduring power of Kepler's laws.

---

### Conclusion

Kepler’s laws revolutionized our ability to predict planetary positions with unmatched precision. From guiding early astronomers to paving the way for modern space missions and the discovery of distant worlds, these laws are a cornerstone of our understanding of the cosmos. The accuracy and elegance of Kepler's laws continue to inspire astronomers and scientists, driving humanity's quest to explore and comprehend the vast universe.

---

### Check Your Understanding

1. According to Kepler’s second law, where in a planet’s orbit would it be moving fastest? Where would it be
moving slowest?
<br><br>
2. Pluto’s orbit is more eccentric than any of the major planets. What does that mean?
<br><br>
3. What is the average distance from the Sun (in astronomical units) of an asteroid with an orbital period of
8 years?
<br><br>
4. In 1996, astronomers discovered an icy object beyond Pluto that was given the designation 1996 TL 66. It
has a semimajor axis of 84 AU. What is its orbital period according to Kepler’s third law?
<br><br>

---

### Resources

- **Astronomy (2016)**. Andrew Fraknoi, David Morrison, and Sidney C. Wolff.
- **Foundations of Astrophysics (2010)**. Barbara Ryden and Bradley M. Peterson.


# 2_2_gravity.md
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# 2.2 Newton's Law of Universal Gravitation

Imagine a quiet afternoon in the English countryside, where a young scholar named Isaac Newton sits pondering under an apple tree. Suddenly, an apple breaks free from its branch and tumbles to the ground. This seemingly mundane event sparks a revelation in Newton's mind—a revelation that would transform our understanding of the universe. What if the same force that pulled the apple to the earth also governed the motion of the moon and the planets? This epiphany led Newton to formulate the law of universal gravitation, a groundbreaking theory that connects the motions of celestial bodies to the everyday experiences of objects falling to the ground. 

<div class="alert alert-block alert-success">
  <h4>Video</h4>
  <p>Watch <a href="https://www.youtube.com/watch?v=kxkFaBG6a-A" target="_blank">this video</a> for an explanation of Newton's law of universal gravitation.</p>
</div>

---

### Historical Context

At the age of 23, while the Great Plague of London raged and Cambridge University was shut down, Newton retreated to his family home. It was during this period, known as his "Annus Mirabilis" or "Year of Wonders," that Newton made groundbreaking discoveries. He formulated the principles that would later be published in his seminal work, "Philosophiæ Naturalis Principia Mathematica" in 1687. This book laid the foundation for classical mechanics and introduced the world to Newton's three laws of motion.

> **Did You Know?**
> 
> Newton's productive isolation led to the development of his laws of motion and universal gravitation, demonstrating that even in the darkest times, human ingenuity can shine brightly.

#### Newton's Laws of Motion

Newton's laws of motion describe the relationship between a body and the forces acting upon it. Among these, his second law is fundamental to understanding motion:

> **Newton's Second Law:**
>
> $F = ma$

This law states that the force acting on an object is equal to the mass of that object multiplied by its acceleration. It provides a quantitative description of the motion of objects and is essential for understanding how forces affect motion.

But what does acceleration actually mean? Acceleration is a change in the velocity of an object over time. This could be a change in the object's speed, its direction of motion, or both. When a force is applied to an object, it causes the object to accelerate in the direction of the force. The amount of acceleration depends on the mass of the object and the magnitude of the force.

> **Implications of Newton's Second Law:**
> 
> - **Direct Proportionality to Force:** If you apply a greater force to an object, it will accelerate more. For example, if you push a shopping cart, the harder you push, the faster it accelerates.
> 
> - **Inverse Proportionality to Mass:** If the object has more mass, it will accelerate less for the same amount of force. This is why a heavy truck accelerates more slowly than a small car when the same force is applied.

<figure style="display: block; margin-left: auto; margin-right: auto; width: 600px;">
  <img src="https://raw.githubusercontent.com/teaghan/astronomy-12/main/Unit2/figures/newton_2nd.png" alt="Newton's Second Law" width="600">
  <figcaption style="text-align: center;">Figure 1: Newton's Second Law. With the same applied force, a more massive object undergoes less acceleration.</figcaption>
</figure>

**Examples to Illustrate Newton's Second Law:**

1. **Launching a Satellite into Orbit**

   <div class="alert alert-block alert-warning">
   <b>Example:</b> A rocket is being used to launch a 2,000 kg satellite into orbit. If the rocket engines provide a constant thrust of 50,000 N, calculate the acceleration of the satellite.
   </div>

   > Given:
   > 
   > > Mass of satellite, $ m = 2000 \, \text{kg} $
   > > 
   > > Thrust (force) provided by rocket engines, $ F = 50,000 \, \text{N} $
   > 
   > Solution:
   > 
   > Using Newton's second law, $ F = ma $, we can solve for acceleration, $ a $:
   > 
   > > $$ a = \frac{F}{m} $$
   > > 
   > > $$ a = \frac{50,000 \, \text{N}}{2000 \, \text{kg}} $$
   > > 
   > > $$ a = 25 \, \text{m/s}^2 $$
   > 
   > Therefore, the acceleration of the satellite is $ 25 \, \text{m/s}^2 $.
   >
   >> This means that every second, the satellite's speed increases by 25 meters per second, as long as the force is applied.

2. **Calculating the Force on an Astronaut in Space**

   <div class="alert alert-block alert-warning">
   <b>Example:</b> An astronaut with a mass of 90 kg is floating in space. During a spacewalk, the astronaut uses a jetpack to apply a force of 180 N. Calculate the acceleration experienced by the astronaut.
   </div>

   > Given:
   > 
   > > Mass of astronaut, $ m = 90 \, \text{kg} $
   > > 
   > > Force applied by jetpack, $ F = 180 \, \text{N} $
   > 
   > Solution:
   > 
   > Using Newton's second law, $ F = ma $, we can solve for acceleration, $ a $:
   > 
   > > $$ a = \frac{F}{m} $$
   > > 
   > > $$ a = \frac{180 \, \text{N}}{90 \, \text{kg}} $$
   > > 
   > > $$ a = 2 \, \text{m/s}^2 $$
   > 
   > Therefore, the acceleration experienced by the astronaut is $ 2 \, \text{m/s}^2 $.
   >
   >> This means that every second, the astronaut's speed increases by 2 meters per second in the direction of the force applied by the jetpack.

---

Understanding acceleration is crucial for comprehending how objects move in response to forces. It explains why astronauts float in space (lack of a significant external force acting on them) and how spacecraft maneuver in the vacuum of space. Newton's second law provides the foundation for analyzing and predicting these motions, making it a cornerstone of physics and engineering.

### Development of Gravitational Theory

Before Newton's insights, the understanding of motion and planetary orbits was fragmented. Galileo Galilei had shown through his experiments that objects fall at the same rate regardless of their mass. Meanwhile, Johannes Kepler had formulated his laws of planetary motion, describing how planets orbit the sun in elliptical paths. But it was Newton who synthesized these ideas into a unified theory of gravitation.

#### The Universal Force of Gravity

Newton realized that the force causing an apple to fall from a tree is the same force that governs the motion of the moon and planets. This universal force, which he called gravity, acts on all objects with mass. He articulated this in his law of universal gravitation:

> **Newton's Law of Universal Gravitation:**
>
> $$ F = G \frac{m_1 m_2}{r^2} $$

where:
- $ F $ is the gravitational force,
- $ G $ is the gravitational constant ($6.67430 \times 10^{-11} \, \text{m}^3 \text{kg}^{-1} \text{s}^{-2}$),
- $ m_1 $ and $ m_2 $ are the masses of the two objects,
- $ r $ is the distance between the centers of the two masses.

| Quantity | Unit    | Description                                   |
|----------|---------|-----------------------------------------------|
| $ F $  | Newtons (N) | Gravitational force                     |
| $ m_1, m_2 $ | Kilograms (kg) | Masses of the objects            |
| $ r $  | Meters (m) | Distance between the centers of the masses |
| $ G $  | $ \text{m}^3 \text{kg}^{-1} \text{s}^{-2} $ | Gravitational constant       |

<figure style="display: block; margin-left: auto; margin-right: auto; width: 600px;">
  <img src="https://raw.githubusercontent.com/teaghan/astronomy-12/main/Unit2/figures/gravity.png" alt="Newton's Gravity" width="600">
  <figcaption style="text-align: center;">Figure 2: Newton's Universal Law of Gravity. Both objects are "pulled" with the same amount of force, which depends on the two masses and the distance between them.</figcaption>
</figure>

This formula expresses that the gravitational force ($F$) between two masses ($m_1$ and $m_2$) is directly proportional to the product of their masses and inversely proportional to the square of the distance ($r$) between their centers. Here, $G$ is the gravitational constant, a value that makes this equation work universally.

<div class="alert alert-block alert-info">
<b>Note:</b> Gravity is always attractive and acts along the line joining the centers of two masses.
</div>

<div class="alert alert-block alert-info">
  <b>Also Note:</b> The gravitational force is experienced by both masses. However, when considering the motion of planets orbiting the sun or the moon orbiting Earth, the smaller mass undergoes much more acceleration and, consequently, more noticeable motion.
</div>


---

### Inverse Square Law

Newton's law of universal gravitation states that the gravitational force between two masses is inversely proportional to the square of the distance between their centers.

This equation reveals that as the distance between two objects increases, the gravitational force between them decreases rapidly. Specifically, if the distance is doubled, the gravitational force becomes one-fourth as strong; if the distance is tripled, the force is reduced to one-ninth, and so on. This inverse square relationship is a fundamental concept in physics.

> **Recall:**
> 
> You have encountered an inverse square relationship before when learning about the flux from a star. The flux, or the apparent brightness of a star as seen from Earth, decreases with the square of the distance from the star. Just as with gravitational force, if you move twice as far from a star, the flux you observe becomes one-fourth as bright. This similarity can help you intuitively understand how gravity behaves over distance.

---

### Gravitational Constant (G)

The gravitational constant $ G $ is a key quantity in Newton's law of universal gravitation. It represents the strength of gravity and its value is approximately $ 6.67430 \times 10^{-11} \, \text{m}^3 \text{kg}^{-1} \text{s}^{-2} $.
 
> The value of $G= 6.67430 \times 10^{-11} \, \text{m}^3 \text{kg}^{-1} \text{s}^{-2} $ is determined experimentally and remains constant across the universe.


<div class="alert alert-block alert-warning">
<b>Example:</b> Calculate the gravitational force between Earth and the Moon.
</div>

> Given:
> 
> > Mass of Earth, $ m_1 = 5.972 \times 10^{24} \, \text{kg} $
> > 
> > Mass of the Moon, $ m_2 = 7.348 \times 10^{22} \, \text{kg} $
> > 
> > Distance between Earth and Moon, $ r = 384,400 \, \text{km} = 384,400,000 \, \text{m} $
> > 
> > Gravitational constant, $ G = 6.67430 \times 10^{-11} \, \text{m}^3 \text{kg}^{-1} \text{s}^{-2} $
> 
> Using Newton's universal law of gravitation:
> 
> > $$ F = G \frac{m_1 m_2}{r^2} $$
> > 
> > $$ F = 6.67430 \times 10^{-11} \, \frac{(5.972 \times 10^{24} \, \text{kg}) (7.348 \times 10^{22} \, \text{kg})}{(384,400,000 \, \text{m})^2} $$
> > 
> > $$ F \approx 1.982 \times 10^{20} \, \text{N} $$
> 
> Therefore, the gravitational force between Earth and the Moon is approximately $ 1.982 \times 10^{20} \, \text{N} $.
>
>> This immense force keeps the Moon in orbit around Earth, demonstrating the power of gravitational attraction even over vast distances.

---

### Applications and Implications

#### Gravity on Earth

Imagine standing on a scale to measure your weight. What you're actually measuring is the gravitational force that Earth exerts on you. This force, which keeps your feet firmly planted on the ground, is a manifestation of gravity. Newton's second law, $F = ma$, helps us calculate this force. On Earth, the acceleration due to gravity, $g$, is approximately $9.8 \, \text{m/s}^2$. Therefore, your weight can be calculated using the equation:

> $$ F = mg $$

For example, let's calculate the weight of a 70 kg person on Earth:

> $$ F = 70 \, \text{kg} \times 9.8 \, \text{m/s}^2 = 686 \, \text{N} $$

This calculation shows that the gravitational force acting on a 70 kg person is 686 newtons. This force is what we commonly refer to as "weight."

---

#### Orbital Motion

Now, let's take a journey beyond Earth, into the vast expanse of space. Here, gravity plays a crucial role in orchestrating the cosmic dance of planets and moons. The same force that keeps us grounded also keeps planets in orbit around the Sun and moons around their respective planets.

Imagine Earth orbiting the Sun. Despite the immense distance between them, the Sun's gravitational pull provides the centripetal force necessary to keep Earth in its nearly circular orbit. This delicate balance of forces ensures that Earth doesn't spiral into the Sun or drift away into the cold void of space.

This gravitational pull isn't limited to planets and stars. It also governs the motion of artificial satellites orbiting Earth. Engineers and scientists must carefully calculate the gravitational forces to ensure these satellites maintain their intended orbits, allowing us to enjoy modern conveniences like GPS and satellite television.

---

#### Tides

Let's shift our gaze back to Earth and focus on the rhythmic rise and fall of ocean tides. This mesmerizing phenomenon is a direct result of the gravitational interactions between Earth, the Moon, and the Sun.

As the Moon orbits Earth, its gravitational pull tugs at the planet's oceans, creating bulges of water known as high tides. On the side of Earth facing the Moon, the gravitational pull is strongest, causing the water to be pulled towards the Moon. Simultaneously, on the opposite side of Earth, inertia creates another bulge. These opposing bulges result in two high tides each day.

The Sun, though much farther away, also influences the tides. During full and new moons, the gravitational forces of the Moon and Sun align, producing exceptionally high and low tides known as spring tides. Conversely, during quarter moons, when the gravitational forces of the Moon and Sun are perpendicular to each other, we experience neap tides, which are less extreme.

> **Did You Know?**
> 
> The interplay of gravitational forces between Earth, the Moon, and the Sun not only creates tides but also plays a crucial role in the Earth's rotational dynamics. This gravitational dance subtly influences the length of our days over long periods.

---

## Check Your Understanding

1. Calculate the gravitational force between Earth (mass $ 5.972 \times 10^{24} \, \text{kg} $) and a 70 kg person standing on its surface (distance from the center of Earth is approximately 6,371,000 m).
<br><br>
2. How does the force of gravity in the International Space Station (orbiting an average of 400 km above Earth’s surface) compare with that
on the ground?
<br><br>
3. Two asteroids begin to gravitationally attract one another. If one asteroid has twice the mass of the other, which one experiences the greater force? Which one experiences the greater acceleration?
<br><br>
4. If there is gravity where the International Space Station (ISS) is located above Earth, why doesn’t the space station get pulled back down to Earth?
<br><br>
5. If identical spacecraft were orbiting Mars and Earth at identical radii (distances), which spacecraft would be moving faster? Why?
<br><br>
6. Suppose astronomers find an earthlike planet that is twice the size of Earth (that is, its radius is twice that of Earth’s). What must be the mass of this planet such that the gravitational force (Fgravity) at the surface would be identical to Earth’s?
<br><br>

---

### Resources

- **Astronomy (2016)**. Andrew Fraknoi, David Morrison, and Sidney C. Wolff.
- **An Introduction to Modern Astrophysics (2017)**. Bradley W. Carroll and Dale A. Ostlie.
- **RESOURCE GK-12 Program**. College of Engineering, University of California Davis.


# 2_3_circular_motion.md
---
layout: embed_default
---

# 2.3 Circular Motion

## Introduction

Circular motion is a fundamental concept in astronomy, playing a crucial role in understanding the behavior of celestial bodies. When studying the orbits of planets around the Sun, moons around their planets, and artificial satellites around Earth, we observe that these objects follow curved paths. This motion is not only key to the structure of our solar system but also to the mechanics of many astronomical phenomena. In this lesson, we will delve into the principles of circular motion, examining its effects on the orbits of various celestial objects and the forces that maintain these orbits.

<div class="alert alert-block alert-success">
  <h4>Video</h4>
  <p>Watch <a href="https://www.youtube.com/watch?v=5vtH1uBaoBY" target="_blank">this video</a> to get a visual introduction to the concepts of circular motion. The first half is particularly useful for our discussion; you can ignore the discussion on centrifugal force.</p>
</div>

---

## Uniform Circular Motion

Uniform circular motion occurs when an object moves along a circular path at a constant speed. Despite the constant speed, the object's velocity is continuously changing because its direction is always changing. This constant change in direction results in an acceleration towards the center of the circular path, known as centripetal acceleration.

**Characteristics:**
- **Constant Speed:** The magnitude of the velocity remains unchanged.
- **Changing Direction:** The direction of the velocity vector changes continuously.
- **Centripetal Acceleration:** The acceleration is directed towards the center of the circular path.

For simple calculations regarding the orbits of planets, moons, satellites, and other celestial bodies, we can approximate these orbits by assuming they are in circular motion. This simplification allows us to use the principles of uniform circular motion to understand and predict their behavior.

**Examples in Astronomy:**
- **Planets orbiting the Sun:** The gravitational pull of the Sun provides the necessary centripetal force.
- **Moons orbiting planets:** Moons remain in orbit due to the gravitational pull of their parent planets.
- **Satellites orbiting Earth:** Artificial satellites stay in orbit due to the Earth's gravitational force.

---

## Centripetal Force

Centripetal force is the force required to keep an object moving in a circular path, directed towards the center of the circle. This force is essential because it counters the object's inertia, which would otherwise cause it to move in a straight line. The word "centripetal" comes from Latin, meaning "center-seeking," highlighting the force's direction toward the center of the circular path.

> Centripetal force keeps an object moving in a circular path by constantly pulling it towards the center.

In the context of circular motion, any force that keeps an object on its curved path acts as a centripetal force. This could be tension in a string for a ball being swung around, friction between tires and the road for a car turning a corner, or gravity for celestial bodies in orbit.

**Mathematical Expression:**

> $$ F_c = \frac{mv^2}{r} $$
where:
- $ F_c $ is the centripetal force
- $ m $ is the mass of the object
- $ v $ is the velocity of the object
- $ r $ is the radius of the circular path

<figure style="display: block; margin-left: auto; margin-right: auto; width: 600px;">
  <img src="https://raw.githubusercontent.com/teaghan/astronomy-12/main/Unit2/figures/circular_motion.png" alt="Circular Motion" width="600">
  <figcaption style="text-align: center;">Figure 1: Circular motion in orbit. The centripetal force here is caused by gravity which causes the satellite to continually be "pulled" into the center, creating circular motion.</figcaption>
</figure>

### Relation to Gravitational Force

In astronomy, the most common centripetal force is gravity. For planetary motion, gravity provides the necessary centripetal force to keep a planet in orbit around the Sun. Similarly, gravity keeps moons orbiting their planets and artificial satellites orbiting the Earth.

> In orbital motion, the centripetal force is the force of gravity.

When we look at the gravitational force acting as the centripetal force, the expressions can be equated:

> $$ F_g = F_c $$

This leads to the equation:

> $$ \frac{GMm}{r^2} = \frac{mv^2}{r} $$

Here, $ G $ is the gravitational constant and $ M $ is the mass of the central body (such as the Sun or Earth). Simplifying this equation allows us to understand the relationships between the various forces and motions involved in orbital mechanics.

<div class="alert alert-block alert-warning">
<b>Example:</b> Calculate the centripetal force acting on the Earth due to its orbit around the Sun.
</div>

> Given:
> 
> > Mass of Earth, $ m = 5.97 \times 10^{24} \, \text{kg} $
> > 
> > Velocity of Earth, $ v = 29.78 \, \text{km/s} = 2.978 \times 10^4 \, \text{m/s} $
> > 
> > Radius of Earth's orbit, $ r = 1.496 \times 10^{11} \, \text{m} $
> 
> Solution:
> 
> > $$ F_c = \frac{(5.97 \times 10^{24} \, \text{kg})(2.978 \times 10^4 \, \text{m/s})^2}{1.496 \times 10^{11} \, \text{m}} $$
> > 
> > $$ F_c \approx 3.54 \times 10^{22} \, \text{N} $$

In this example, the immense centripetal force required to keep Earth in its orbit is provided by the gravitational pull of the Sun. This force constantly pulls Earth towards the Sun, balancing the inertia of Earth's motion, which attempts to move it in a straight line. This delicate balance is what maintains the stable orbit of our planet.

---

## Centripetal Acceleration

Centripetal acceleration is the acceleration experienced by an object moving in a circle at a constant speed, directed towards the center of the circle. This acceleration is what causes the object to change direction continuously.

**Formula:**
> $$ a_c = \frac{v^2}{r} $$

### Application to Circular Orbits

In circular orbits, the centripetal acceleration is provided by the gravitational force acting on the orbiting body.

<div class="alert alert-block alert-warning">
<b>Example:</b> Calculate the centripetal acceleration of the Earth due to its orbit around the Sun.
</div>

> Given:
> 
> > Velocity of Earth, $ v = 29.78 \, \text{km/s} = 2.978 \times 10^4 \, \text{m/s} $
> > 
> > Radius of Earth's orbit, $ r = 1.496 \times 10^{11} \, \text{m} $
> 
> Solution:
> 
> > $$ a_c = \frac{(2.978 \times 10^4 \, \text{m/s})^2}{1.496 \times 10^{11} \, \text{m}} $$
> > 
> > $$ a_c \approx 5.93 \times 10^{-3} \, \text{m/s}^2 $$

---

## Gravity and Circular Motion Simulator

Feel free to play around with the simulator below to get a feel for how gravity impacts orbits of planets, moons, and satellites!

<div style="width: 100%; height: 0; padding-bottom: 75%; position: relative;">
    <iframe src="https://phet.colorado.edu/sims/html/gravity-and-orbits/latest/gravity-and-orbits_en.html" 
            style="position: absolute; width: 100%; height: 100%; top: 0; left: 0;" 
            allowfullscreen>
    </iframe>
</div>

---

## Orbital Speed

Understanding the speed at which an object orbits a celestial body is crucial in astronomy. The orbital speed determines how long it takes for an object, such as a planet, moon, or satellite, to complete one full orbit around its primary body.

By combining Newton's law of universal gravitation with the concept of centripetal force, we can derive the formula for orbital speed. This derivation provides insight into how gravitational forces and the mass of celestial bodies influence the motion of orbiting objects.

### Derivation of Orbital Speed

To derive the orbital speed, we start with the fact that the gravitational force ($F_g$) acting on an object in orbit provides the necessary centripetal force ($F_c$) to keep it in circular motion.

**Step-by-Step Derivation:**

1. **Equate Gravitational Force to Centripetal Force:**
> $$ F_g = F_c $$

2. **Express Gravitational Force:**
> $$ F_g = \frac{GMm}{r^2} $$
   - Where $G$ is the gravitational constant, $M$ is the mass of the central body, $m$ is the mass of the orbiting object, and $r$ is the radius of the orbit.

3. **Express Centripetal Force:**
> $$ F_c = \frac{mv^2}{r} $$
   - Where $v$ is the orbital speed.

4. **Set the Expressions Equal to Each Other:**
> $$ \frac{GMm}{r^2} = \frac{mv^2}{r} $$

5. **Simplify the Equation:**
   - Cancel $m$ from both sides:
> $$ \frac{GM}{r^2} = \frac{v^2}{r} $$
  
   - Multiply both sides by $r$:
> $$ \frac{GM}{r} = v^2 $$

6. **Solve for Orbital Speed ($v$):**
> $$ v = \sqrt{\frac{GM}{r}} $$

This formula shows that the orbital speed of an object depends on the gravitational constant ($G$), the mass of the central body ($M$), and the radius of the orbit ($r$). 

<figure style="display: block; margin-left: auto; margin-right: auto; width: 600px;">
  <img src="https://raw.githubusercontent.com/teaghan/astronomy-12/main/Unit2/figures/orbital_speed.png" alt="Orbital Speed" width="600">
  <figcaption style="text-align: center;">Figure 2: Orbital speed. The orbital speed of a satellite, planet, moon, etc. depends on the size of its orbit. The greater the radius, the slower the orbital speed.</figcaption>
</figure>

### Importance of Orbital Speed in Astronomy

Understanding orbital speed is essential for:
- **Predicting Orbital Periods:** The time it takes for an object to complete one orbit.
- **Satellite Deployment:** Ensuring satellites remain in stable orbits.
- **Planetary Motion:** Studying the dynamics of planets and their moons.
- **Space Missions:** Planning trajectories for space probes and missions.

Orbital speed helps astronomers and scientists comprehend the delicate balance of forces that keep celestial bodies in motion and ensures the stability of their orbits.

By mastering the concept of orbital speed, we can better appreciate the intricate dance of planets, moons, and satellites in our universe.

<div class="alert alert-block alert-warning">
<b>Example:</b> Calculate the orbital speed of a satellite orbiting the Earth at a height of 300 km above the surface.
</div>

> Given:
> 
> > Mass of Earth, $ M = 5.97 \times 10^{24} \, \text{kg} $
> > 
> > Radius of Earth, $ R = 6.371 \times 10^6 \, \text{m} $
> > 
> > Height above the surface, $ h = 300 \times 10^3 \, \text{m} $
> > 
> > Gravitational constant, $ G = 6.674 \times 10^{-11} \, \text{N m}^2/\text{kg}^2 $
> 
> Solution:
> 
> > $$ r = R + h = 6.371 \times 10^6 \, \text{m} + 300 \times 10^3 \, \text{m} = 6.671 \times 10^6 \, \text{m} $$
> > 
> > $$ v = \sqrt{\frac{6.674 \times 10^{-11} \, \text{N m}^2/\text{kg}^2 \times 5.97 \times 10^{24} \, \text{kg}}{6.671 \times 10^6 \, \text{m}}} $$
> > 
> > $$ v \approx 7.73 \times 10^3 \, \text{m/s} = 7.73 \, \text{km/s} $$

---

## Check Your Understanding

1. Calculate the centripetal force acting on a 1000 kg satellite orbiting the Earth at a height of 500 km above the surface.
> **Hint**: The height above the surface is not the *entire* radius of the orbit).
<br><br>
3. Given a satellite orbiting Earth at a height where the radius of the orbit is 7,000 km, calculate the orbital speed of the satellite. Assume Earth's mass is $5.97 \times 10^{24} \, \text{kg}$ and the gravitational constant $G$ is $6.674 \times 10^{-11} \, \text{N m}^2/\text{kg}^2$.
<br><br>
4. Describe the relationship between the radius of an orbit and the orbital speed of a satellite. What happens to the speed if the radius increases?
<br><br>
5. A moon orbits a planet at a radius of $3.84 \times 10^8 \, \text{m}$ with an orbital speed of $1.02 \times 10^3 \, \text{m/s}$. Calculate the mass of the planet.
<br><br>
6. Calculate the centripetal acceleration of the Earth in its orbit around the Sun. Use the given values: velocity of Earth $v = 29.78 \, \text{km/s}$ and radius of Earth's orbit $r = 1.496 \times 10^{11} \, \text{m}$.
<br><br>
7. Explain why satellites need to be launched at a specific speed to remain in stable orbit around the Earth. What would happen if the speed were too low or too high?
<br><br>
11. A satellite completes one full orbit around Earth in 90 minutes. If the radius of the orbit is $7.0 \times 10^6 \, \text{m}$, calculate the orbital speed of the satellite.
> **Hint:** Use the formula for the circumference of a circle, $C = 2 \pi r$, to find the distance traveled in one orbit.
<br><br>

### Resources

- **Astronomy (2016).** Andrew Fraknoi, David Morrison, and Sidney C. Wolff.
- **An Introduction to Modern Astrophysics (2017).** Bradley W. Carroll and Dale A. Ostlie.


# 2_4_solar_system_1.md
---
layout: embed_default
---

# 2.4 The Structure and Components of Our Solar System: Part 1

In this lesson, we will explore the fascinating structure and components of our solar system. This will include an examination of the formation of the solar system, the central role of the Sun, and the various planets that orbit within our celestial neighborhood. This topic ties in with our previous discussions on gravity, circular motion, Kepler's Laws, and Newton's Law of Universal Gravitation, providing a deeper understanding of how these principles govern the dynamics of our solar system.

<div class="alert alert-block alert-success">
  <h4>Video</h4>
  <p>Watch <a href="https://www.youtube.com/watch?v=x1QTc5YeO6w" target="_blank">this video</a> for an excellent overview of the formation of the solar system.</p>
</div>

---

## Formation of the Solar System

Imagine a vast cloud of gas and dust floating in space, a remnant of ancient stellar explosions. About 4.5 billion years ago, this seemingly unremarkable cloud began to collapse under its own gravity. This was the solar nebula, and its collapse initiated the birth of our solar system.

<img src="https://raw.githubusercontent.com/teaghan/astronomy-12/main/Unit2/figures/nebula.png" alt="Solar Nebula" width="1000" style="display: block; margin-left: auto; margin-right: auto;">

### The Role of Gravity and Circular Motion
- **Gravity**: As the solar nebula collapsed, gravity pulled the gas and dust towards the center, creating a dense core that would become the Sun.
- **Circular Motion**: Conservation of angular momentum caused the collapsing cloud to spin faster and flatten into a rotating disc. This rotating disc of gas and dust is known as the protoplanetary or circumstellar disc.

### Formation of Planetesimals and Protoplanets

Within the circumstellar disc, particles of dust and ice began to stick together through electrostatic forces, forming clumps called planetesimals. These planetesimals, ranging from meters to kilometers in size, acted as the building blocks of planets. Through the process of accretion, these planetesimals collided and stuck together, gradually growing larger.

- **Planetesimals**: Small bodies formed from dust and ice that collided and stuck together. Over time, these planetesimals grew through further collisions and accumulation of material.
- **Protoplanets**: As planetesimals continued to collide and merge, they formed larger bodies known as protoplanets. These protoplanets, several hundred kilometers in diameter, had sufficient gravity to influence their surroundings and attract more material.

### Formation of Planets

The protoplanets eventually grew into the planets we see today through continued accretion and collisions. The solar wind from the young Sun blew away much of the remaining gas in the disc, halting further growth of the gas giants but allowing the terrestrial planets to continue forming from rocky materials.

- **Terrestrial Planets**: Closer to the Sun, where temperatures were higher, protoplanets formed from rocky and metallic materials. These include Mercury, Venus, Earth, and Mars.
- **Gas Giants**: Further from the Sun, where temperatures were cooler, protoplanets could accumulate lighter gases like hydrogen and helium, forming the gas giants Jupiter and Saturn.
- **Ice Giants**: In the outer regions of the disc, protoplanets formed from ices and gases, resulting in Uranus and Neptune, which have significant amounts of water, ammonia, and methane.

<div class="alert alert-block alert-info">
<b>Interesting Fact:</b> Jupiter's strong gravity influenced the formation of other planets, preventing the formation of a planet in the asteroid belt and influencing the orbits of nearby protoplanets.
</div>

---

## The Sun: The Central Star

At the heart of our solar system lies the Sun, a G-type main-sequence star that accounts for more than 99% of the solar system's total mass. It is the source of energy that sustains life on Earth and drives the climate and weather.

### Structure of the Sun
- **Core**: The innermost part where nuclear fusion occurs, converting hydrogen into helium and releasing immense amounts of energy.
- **Radiative Zone**: Energy from the core travels outward by radiation.
- **Convective Zone**: Energy is transported to the surface by convection currents.
- **Photosphere**: The visible surface of the Sun, emitting the light we see.
- **Chromosphere**: A layer above the photosphere, visible during solar eclipses as a reddish glow.
- **Corona**: The Sun's outer atmosphere, extending millions of kilometers into space and visible during total solar eclipses.

<img src="https://raw.githubusercontent.com/teaghan/astronomy-12/main/Unit2/figures/sun_layers.png" alt="Layers of the Sun" width="1000" style="display: block; margin-left: auto; margin-right: auto;">

The Sun is primarily composed of hydrogen (about 74%) and helium (about 24%), with trace amounts of heavier elements. The nuclear fusion process in the core converts hydrogen into helium, releasing energy that radiates outward and eventually reaches Earth, sustaining life and driving our climate. We will study this process more in our unit on stars!

The Sun's immense gravitational pull keeps the planets in their orbits, and its energy output creates the dynamic environment of the solar system.

---

## The Planets: Types and Characteristics

The planets in our solar system are divided into two main categories: Terrestrial and Jovian, each with distinct characteristics and fascinating features.

### Terrestrial Planets

The inner solar system is home to the terrestrial planets, rocky worlds forged from the dense materials left behind in the protoplanetary disc. These planets, closest to the Sun, are primarily composed of rock and metal, featuring solid surfaces and a variety of geological landscapes.

#### Mercury
Mercury, the smallest and innermost planet, orbits the Sun at a breakneck speed. Despite its proximity to the Sun, Mercury experiences extreme temperature variations. Daytime temperatures soar to scorching heights, while nighttime temperatures plummet to icy lows due to its thin atmosphere, which lacks the ability to retain heat. The surface of Mercury is heavily cratered, resembling our Moon, and its lack of significant atmosphere allows us to glimpse its ancient, unaltered landscape.

<div class="alert alert-block alert-info">
<b>Did You Know?</b> A day on Mercury (one full rotation) lasts 59 Earth days, while a year (one orbit around the Sun) takes just 88 Earth days.
</div>

<img src="https://raw.githubusercontent.com/teaghan/astronomy-12/main/Unit2/figures/Mercury_surface.png" alt="Terrestrial Planets" width="1000" style="display: block; margin-left: auto; margin-right: auto;">

#### Venus
Venus, often called Earth's "sister planet" due to its similar size and composition, presents a stark contrast in conditions. Shrouded in thick clouds of sulfuric acid, Venus has the hottest surface temperatures in the solar system, reaching up to 467 degrees Celsius (872 degrees Fahrenheit). This intense heat is the result of a runaway greenhouse effect, where a dense atmosphere rich in carbon dioxide traps solar radiation. The surface pressure on Venus is crushing, equivalent to being 900 meters underwater on Earth.

<div class="alert alert-block alert-info">
<b>Interesting Fact:</b> The rotation of Venus is unique—it spins in the opposite direction to most planets, and its rotation period is longer than its orbit around the Sun. A day on Venus lasts 243 Earth days, while a year takes 225 Earth days.
</div>

#### Earth
Earth, our home planet, stands out as the only known world with liquid water on its surface and life in abundance. Earth’s diverse climate and geological features, including mountains, oceans, and forests, create a rich and dynamic environment. The presence of liquid water is crucial, supporting a biosphere teeming with life. Earth's atmosphere, composed of nitrogen, oxygen, and trace gases, provides the right conditions for life and shields the surface from harmful solar radiation.

#### Mars
Mars, known as the Red Planet due to its iron oxide-rich soil, has fascinated scientists and explorers for centuries. With surface features reminiscent of both Earth and the Moon, Mars hosts the largest volcano and canyon in the solar system—Olympus Mons and Valles Marineris, respectively. Mars has polar ice caps composed of frozen water and carbon dioxide. Evidence of ancient river valleys and lake beds suggests that liquid water once flowed on its surface, raising the possibility that Mars might have supported microbial life in the past.

<div class="alert alert-block alert-info">
<b>Interesting Fact:</b> Mars' Olympus Mons is the tallest volcano in the solar system, standing nearly three times the height of Mount Everest.
</div>

---

### Jovian Planets

As we venture further from the Sun, we encounter the majestic Jovian planets, also known as gas giants. These massive planets, composed primarily of hydrogen, helium, and other volatiles, present a stark contrast to the rocky terrestrial worlds. Their immense sizes and unique characteristics offer a fascinating glimpse into the outer reaches of our solar system.

<img src="https://raw.githubusercontent.com/teaghan/astronomy-12/main/Unit2/figures/Giant_Planets.png" alt="Jovian Planets" width="1000" style="display: block; margin-left: auto; margin-right: auto;">

#### Jupiter
Jupiter, the largest planet in our solar system, is a colossal world with a diameter of about 11 times that of Earth. Its most famous feature is the Great Red Spot, a gigantic storm larger than Earth itself, which has been raging for centuries. Jupiter's atmosphere is a dynamic blend of hydrogen and helium, with swirling clouds of ammonia crystals and complex weather patterns. 

<img src="https://raw.githubusercontent.com/teaghan/astronomy-12/main/Unit2/figures/Jupiter.png" alt="Jupiter" width="1000" style="display: block; margin-left: auto; margin-right: auto;">

Jupiter is also home to an impressive array of moons, with over 79 known satellites. The largest of these, Ganymede, is the largest moon in the solar system, even surpassing Mercury in size. Other notable moons include Europa, which may harbor a subsurface ocean, and Io, the most volcanically active body in the solar system.

<div class="alert alert-block alert-info">
<b>Interesting Fact:</b> Jupiter has a strong magnetic field, about 20,000 times stronger than Earth's, which traps charged particles and creates intense radiation belts.
</div>

#### Saturn
Saturn, the second-largest planet, is renowned for its stunning ring system, a collection of countless ice and rock particles that orbit the planet. These rings, which can be seen through a small telescope, vary in density and composition, creating a breathtaking spectacle. Saturn's atmosphere, like Jupiter's, is predominantly hydrogen and helium, with upper cloud layers of ammonia, methane, and water ice.

Saturn's numerous moons, including Titan and Enceladus, add to its intrigue. Titan, the second-largest moon in the solar system, has a thick atmosphere and lakes of liquid methane, while Enceladus, with its geysers of water-ice, hints at the presence of a subsurface ocean.

<div class="alert alert-block alert-info">
<b>Did You Know?</b> Saturn's rings are incredibly thin, with an average thickness of only about 10 meters (33 feet), despite their vast horizontal expanse.
</div>

#### Uranus
Uranus stands out with its extreme axial tilt of about 98 degrees, causing it to rotate on its side. This unique tilt leads to unusual seasonal variations, with each pole experiencing 42 years of continuous sunlight followed by 42 years of darkness. Uranus is classified as an ice giant due to its composition, which includes water, ammonia, and methane ices, in addition to hydrogen and helium.

Uranus's faint ring system and 27 known moons, including Miranda and Ariel, offer further points of interest. Miranda, in particular, features one of the most varied terrains in the solar system, with massive canyons and patchwork regions suggesting a history of intense geological activity.

<div class="alert alert-block alert-info">
<b>Interesting Fact:</b> Uranus was the first planet discovered with a telescope, by William Herschel in 1781.
</div>

#### Neptune
Neptune, the farthest planet from the Sun, is a striking blue world due to the presence of methane in its atmosphere, which absorbs red light and reflects blue. Neptune is known for its supersonic winds, which can reach speeds of up to 2,100 kilometers per hour (1,300 miles per hour), the fastest in the solar system.

Neptune has a faint ring system and 14 known moons, with Triton being the most notable. Triton is unique for its retrograde orbit, meaning it orbits Neptune in the opposite direction of the planet's rotation. Triton also exhibits geysers of nitrogen gas, suggesting active geological processes.

<div class="alert alert-block alert-info">
<b>Did You Know?</b> Triton is thought to be a captured Kuiper Belt object, which may explain its unusual orbit.
</div>

---

## Solar System Simulator

Interested in learning more about the solar system? Check out the simulation below to get an interactive representation of the Sun and palnets.

<div style="width: 100%; height: 0; padding-bottom: 75%; position: relative;">
    <iframe src="https://www.solarsystemscope.com/iframe" 
            style="position: absolute; width: 100%; height: 100%; top: 0; left: 0;" 
            allowfullscreen>
    </iframe>
</div>

---

## Check Your Understanding

Apply the concepts you learned in the previous lessons to the solar system!

1. **Formation of the Solar System**: Calculate the gravitational force between a planetesimal (mass = $1 \times 10^{21}$ kg) and a protoplanet (mass = $5 \times 10^{23}$ kg) that are 5,000 km apart.
<br><br>
2. **Kepler's Second Law**: If Mars sweeps out an area of $2 \times 10^8$ square kilometers in 15 days, what area will it sweep out in 30 days?
<br><br>
3. **Kepler's Third Law**: Calculate the orbital period of Jupiter, which is 5.20 AU from the Sun.
<br><br>
4. **Gravitational Acceleration on a Planet**: Calculate the gravitational acceleration on the surface of Saturn (mass = $5.68 \times 10^{26}$ kg, radius = 58,232 km).
<br><br>
5. **Orbital Speed of Mercury**: Calculate the orbital speed of Mercury, given that its average distance from the Sun is 57.9 million km. (Mass of the Sun = $1.989 \times 10^{30}$ kg).
<br><br>
6. **Energy Output of the Sun**: Given that the Sun's luminosity is $3.828 \times 10^{26}$ watts, calculate the total energy output of the Sun in one day.
> **Hint:** 1 Watt is the same as 1 Joule/second. Joules are a unit of energy.
<br><br>



---

## Resources

- **Astronomy** by Andrew Fraknoi, David Morrison, and Sidney C. Wolff
- **Foundations of Astrophysics** by Barbara Ryden and Bradley M. Peterson


# 2_5_solar_system_2.md
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---

# 2.5 The Structure and Components of Our Solar System: Part 2

In this lesson, we will explore the smaller celestial bodies, moons, and the outer regions of our solar system. This will include an examination of dwarf planets, asteroids, comets, meteoroids, and the vast regions beyond the major planets. This topic ties in with our previous discussions on the planets, providing a deeper understanding of the diverse and dynamic components that make up our solar system.

<div class="alert alert-block alert-success">
  <h4>Video</h4>
  <p>Watch <a href="https://www.youtube.com/watch?v=MD7Zt2cGXRc" target="_blank">this video</a> for a deeper discussion on Pluto, Asteroids, Comets, and more!</p>
</div>

---

## Dwarf Planets

Beyond the eight major planets, our solar system is populated with a diverse array of dwarf planets and smaller celestial bodies. These objects add richness to the tapestry of our cosmic neighborhood, each with its own story and significance.

Dwarf planets are celestial bodies that orbit the Sun and are spherical in shape, but they differ from the major planets in one key aspect: they have not cleared their orbital paths of other debris. This distinction sets them apart and places them in a unique category within our solar system.

### Pluto
Pluto, perhaps the most famous of the dwarf planets, has a fascinating history. Discovered in 1930, Pluto was originally classified as the ninth planet in our solar system. However, in 2006, the International Astronomical Union (IAU) redefined what it means to be a planet, and Pluto was reclassified as a dwarf planet. This decision was made because Pluto shares its orbit with other objects in the Kuiper Belt, a region beyond Neptune filled with icy bodies.

<div class="alert alert-block alert-info">
<b>Did You Know?</b> Pluto's largest moon, Charon, is so large relative to Pluto that the two bodies are often considered a double dwarf planet system, with their center of mass lying outside of Pluto.
</div>

<img src="https://raw.githubusercontent.com/teaghan/astronomy-12/main/Unit2/figures/Pluto.png" alt="Dwarf Planets" width="1000" style="display: block; margin-left: auto; margin-right: auto;">

### Other Notable Dwarf Planets
- **Eris**: Discovered in 2005, Eris is slightly smaller than Pluto but more massive, and its discovery was a key factor in the reclassification of Pluto.
- **Ceres**: Located in the asteroid belt between Mars and Jupiter, Ceres is the closest dwarf planet to Earth and the only one located in the inner solar system.
- **Haumea**: Known for its elongated shape due to its rapid rotation, Haumea also has two moons and is one of the fastest rotating large objects in the solar system.
- **Makemake**: Similar in size to Haumea, Makemake was discovered in 2005 and is one of the brightest objects in the Kuiper Belt.

---

## Asteroids and the Asteroid Belt

Asteroids are rocky remnants from the early solar system, providing a glimpse into the conditions and materials present during its formation. These ancient objects offer valuable clues about the processes that shaped our planetary neighborhood.

### Origin of the Asteroid Belt
The asteroid belt, located between Mars and Jupiter, is a region teeming with millions of rocky objects. But where did these asteroids come from? The prevailing theory suggests that the asteroid belt is composed of material that never coalesced into a planet. During the formation of the solar system, gravitational perturbations from Jupiter, the giant planet nearby, prevented the material in this region from forming a single, larger body. Instead, the material remained as numerous smaller objects, ranging in size from tiny dust particles to the dwarf planet Ceres, the largest object in the belt.

> **Interesting Fact:**
>
> The total mass of the asteroid belt is less than that of Earth's Moon, despite containing millions of individual asteroids.

### Significant Asteroids

- **Ceres**: The first asteroid discovered, Ceres is now classified as a dwarf planet. It has a diameter of about 940 kilometers and contains a significant amount of water ice.
- **Vesta**: The second-largest object in the asteroid belt, Vesta has a differentiated structure with a core, mantle, and crust, similar to terrestrial planets.
- **Pallas and Hygiea**: Other significant asteroids, each with unique features and compositions.

<img src="https://raw.githubusercontent.com/teaghan/astronomy-12/main/Unit2/figures/asteroids.png" alt="Asteroid Belt" width="1000" style="display: block; margin-left: auto; margin-right: auto;">

---

## Comets and the Kuiper Belt

Comets, often described as "dirty snowballs," are icy bodies that provide valuable clues about the outer regions of the solar system. These ancient travelers originate from the distant reaches of our solar system, specifically the Kuiper Belt and the Oort Cloud, and offer insights into the primordial materials that formed our planetary system.

### Structure of Comets

- **Nucleus**: The solid core of a comet is composed of ice, dust, and organic compounds. This nucleus is typically only a few kilometers across but contains the bulk of the comet's mass.
- **Coma**: When a comet nears the Sun, the heat causes the ices to vaporize, forming a glowing cloud of gas and dust around the nucleus known as the coma.
- **Tails**: Comets usually develop two tails as they approach the Sun. The ion tail, composed of charged particles, and the dust tail, made of small solid particles. These tails always point away from the Sun due to the pressure of the solar wind.

### The Kuiper Belt
Located beyond Neptune, the Kuiper Belt is a vast, disk-shaped region filled with icy bodies and dwarf planets, including Pluto. It is a remnant of the early solar system and is also the source of short-period comets, which have orbits that bring them into the inner solar system at regular intervals. These comets, originating from the Kuiper Belt, help us understand the composition and dynamics of the outer solar system.

> **Origins of Comets:**
> 
> While the Kuiper Belt is the birthplace of short-period comets, the Oort Cloud, a distant spherical shell surrounding the solar system, is thought to be the origin of long-period comets. These comets have elongated orbits that can take them far beyond the planets before they return to the inner solar system.

### Famous Comets
- **Halley's Comet**: Perhaps the most famous comet, Halley's Comet is visible from Earth approximately every 76 years. It was last seen in 1986 and will return in 2061. Halley's periodic appearances have been recorded for millennia, making it a well-documented celestial phenomenon.
- **Comet Hale-Bopp**: One of the brightest comets of the 20th century, Hale-Bopp was visible to the naked eye for 18 months in 1996-1997. Its remarkable brightness and longevity provided a spectacular show for observers worldwide.

> **Did You Know?**
> Comet tails can extend for millions of kilometers and are visible due to the sunlight reflecting off the dust and ionized gases.

<img src="https://raw.githubusercontent.com/teaghan/astronomy-12/main/Unit2/figures/Comet.png" alt="Comet" width="1000" style="display: block; margin-left: auto; margin-right: auto;">

---

## Meteoroids, Meteors, and Meteorites

These small bodies provide a link between space and Earth's surface, offering insights into the materials and processes that shape our solar system.

- **Meteoroids**: Small rocky or metallic bodies traveling through space.
- **Meteors**: The streaks of light we see when meteoroids enter Earth's atmosphere and burn up, commonly known as "shooting stars."
- **Meteorites**: Meteoroids that survive their passage through the atmosphere and reach Earth's surface.

> Throughout Earth's history, meteorite impacts have had significant effects. One of the most famous impact events is the Chicxulub impact, which is believed to have caused the mass extinction of the dinosaurs about 66 million years ago.

<img src="https://raw.githubusercontent.com/teaghan/astronomy-12/main/Unit2/figures/meteor.png" alt="Meteor" width="1000" style="display: block; margin-left: auto; margin-right: auto;">

---

## Moons of the Solar System

The solar system is not only home to planets but also to a diverse array of moons, each with unique characteristics and fascinating features. These natural satellites offer intriguing insights into the processes that govern planetary systems and the potential for extraterrestrial life.

### Notable Moons

#### Earth's Moon
Earth's Moon, our closest celestial companion, plays a crucial role in influencing Earth's tides and stabilizing its axial tilt. It is the only celestial body beyond Earth where humans have set foot.

#### Ganymede
Ganymede, one of Jupiter's moons, is the largest moon in the solar system, even surpassing the planet Mercury in size. It boasts a magnetic field and features a mix of older, heavily cratered regions and younger, tectonically resurfaced areas.

<img src="https://raw.githubusercontent.com/teaghan/astronomy-12/main/Unit2/figures/Jupiter_moon.png" alt="Moons" width="1000" style="display: block; margin-left: auto; margin-right: auto;">

#### Titan
Titan, Saturn's largest moon, is known for its thick atmosphere and surface lakes of liquid methane and ethane. With a dense, nitrogen-rich atmosphere similar to early Earth's, Titan is a prime candidate for studying prebiotic chemistry and the potential for life.

#### Europa
Europa, another of Jupiter's moons, is a key target in the search for extraterrestrial life. Beneath its icy crust lies a global ocean, potentially containing more water than all of Earth's oceans combined.

<div class="alert alert-block alert-info">
<b>Interesting Fact:</b> Europa's surface is marked by a complex network of cracks and ridges, created by the tidal forces exerted by Jupiter's immense gravity.
</div>

#### Enceladus
Enceladus, a smaller moon of Saturn, has captured scientific interest due to its active geysers that emit water-ice and organic compounds from a subsurface ocean beneath its icy crust. These geysers suggest that Enceladus may harbor conditions suitable for microbial life.

<div class="alert alert-block alert-info">
<b>Did You Know?</b> The Cassini spacecraft flew through the plumes of Enceladus, analyzing their composition and confirming the presence of organic molecules.
</div>

### Moon Formation Theories

The origins of moons are diverse, with several theories explaining how these natural satellites came to be:

- **Giant Impact Hypothesis**: This widely accepted theory suggests that Earth's Moon formed from the debris of a colossal collision between the early Earth and a Mars-sized body named Theia. The resulting debris coalesced to form the Moon.
- **Capture Theory**: This theory proposes that some moons, particularly those with irregular orbits, were once wandering bodies that were captured by a planet's gravitational pull.
- **Co-formation Theory**: According to this theory, some moons formed alongside their parent planets from the same primordial accretion disc of the solar nebula, developing simultaneously as double systems.

---

## The Oort Cloud

The Oort Cloud is a vast, hypothesized spherical shell of icy bodies that surrounds our solar system, extending far beyond the Kuiper Belt. It represents the distant frontier of the Sun's gravitational influence.

### Hypothetical Cloud of Icy Bodies

The Oort Cloud is believed to stretch from about 2,000 to 100,000 astronomical units (AU) from the Sun. It is thought to be the source of long-period comets, which have highly eccentric orbits that can take them deep into the inner solar system and far out into the distant reaches beyond the planets. These comets spend most of their time in the cold, dark outer regions, only becoming visible when their orbits bring them close to the Sun, where the heat causes their icy surfaces to vaporize, forming glowing comas and tails.

### Its Role in the Solar System

The Oort Cloud marks the outer boundary of the Sun's gravitational influence and is considered the edge of the solar system. This distant region serves as a reservoir of icy bodies that occasionally get nudged by passing stars or gravitational interactions into trajectories that bring them into the inner solar system, where they become visible as comets.

> **Interesting Fact:**
> The existence of the Oort Cloud was first proposed by Dutch astronomer Jan Oort in 1950. While it has not been directly observed, its presence is inferred from the behavior and origins of long-period comets.

---

## Check Your Understanding

Apply the concepts you learned in the previous lessons to the solar system!

1. **Newton's Law of Universal Gravitation**: Calculate the gravitational force between Jupiter (mass = $1.898 \times 10^{27}$ kg) and its moon, Europa (mass = $4.80 \times 10^{22}$ kg), given the average distance between them is 670,900 km.
<br><br>
2. **Circular Motion and Gravitational Force**: Calculate the orbital speed of Enceladus orbiting Saturn at an average distance of 238,000 km. Assume the orbit is circular. (Mass of Saturn = $5.68 \times 10^{26}$ kg).
<br><br>
3. **Newton's Second Law**: A comet with a mass of $1.0 \times 10^{12}$ kg is subjected to a gravitational force of $3.0 \times 10^{10}$ N when it is near the Sun. Calculate its acceleration.
<br><br>
4. **Orbital Period**: Calculate the orbital period of Triton orbiting Neptune at an average distance of 354,800 km. (Mass of Neptune = $1.02 \times 10^{26}$ kg).<br><br>
5. **Orbital Speed of a Moon**: Calculate the orbital speed of Jupiter's moon, Io, which orbits Jupiter at an average distance of 421,700 km.Assume the orbit is circular.
<br><br>
6. **Orbital Speed of Charon**: Calculate the orbital speed of Charon around Pluto, given that the average distance between them is 19,570 km. (Mass of Pluto = $1.31 \times 10^{22}$ kg).
<br><br>
7. **Cometary Orbits**: A comet in the Kuiper Belt orbits the Sun at an average distance of 50 AU. Using Kepler's Third Law, calculate its orbital period.
<br><br>
8. **Surface Gravity on Titan**: Calculate the acceleration due to gravity on the surface of Titan. (Mass of Titan = $1.345 \times 10^{23}$ kg, radius of Titan = 2,575 km).
<br><br>

---

## Resources

- **Astronomy** by Andrew Fraknoi, David Morrison, and Sidney C. Wolff
- **Foundations of Astrophysics** by Barbara Ryden and Bradley M. Peterson



# 3_1_star_formation.md
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# 3.1 Star Formation

Molecular clouds are vast, dark expanses of space, filled with cold, dense clouds of gas and dust. These molecular clouds serve as nurseries for star formation. Over millions of years, these clouds will collapse under their own gravity, eventually igniting nuclear fusion and giving birth to new stars. The story of star formation is a tale of transformation, from the silent, shadowy depths of a molecular cloud to the blazing brilliance of a newborn star.

<div class="alert alert-block alert-success">
  <h4>Video</h4>
  <p>Watch <a href="https://www.youtube.com/watch?v=lI57XIZ17hE" target="_blank">this video</a> for a detailed description of the star formation process.</p>
</div>

---

## The Birthplace of Stars: Molecular Clouds

The Orion Nebula is a glowing cloud visible even to the naked eye. This stellar nursery, like many others, is a molecular cloud—a region teeming with the raw materials for star formation. These clouds, composed mainly of molecular hydrogen, are frigid, with temperatures barely above absolute zero, and dense, harboring the seeds of countless future stars.

Deep within these clouds, the force of gravity begins to pull the gas and dust inward. This inward pull creates regions of higher density, known as clumps. These clumps are the starting point of a star’s life.

> **Did You Know?**
> The Orion Nebula, visible to the naked eye, is one of the most famous star-forming regions and serves as an excellent example of a molecular cloud.

<img src="https://raw.githubusercontent.com/teaghan/astronomy-12/main/Unit3/figures/orion.png" alt="Orion Nebula" width="1000" style="display: block; margin-left: auto; margin-right: auto;">

**Characteristics of Molecular Clouds:**
- **Temperature**: 10-20 K, colder than any place on Earth.
- **Composition**: Mostly hydrogen, with helium and trace elements.
- **Density**: High compared to the surrounding space, a fertile ground for star birth.

**Examples of Molecular Clouds:**
- **Orion Nebula**: A well-known star-forming region.
- **Eagle Nebula**: Famous for the "Pillars of Creation," towering columns of gas sculpted by intense radiation from young, hot stars.

<img src="https://raw.githubusercontent.com/teaghan/astronomy-12/main/Unit3/figures/M16.png" alt="Eagle Nebula" width="1000" style="display: block; margin-left: auto; margin-right: auto;">

---

## Stages of Star Formation

The birth of a star is a cosmic tale of transformation, where the interplay of gravity, pressure, and temperature orchestrates a stunning sequence of events. Let's delve into the fascinating stages that lead from a cold molecular cloud to a radiant main sequence star.

**1. Gravitational Collapse:**
In the vast expanses of space, molecular clouds, teeming with gas and dust, float in silence. This peace can be shattered by a powerful event, such as a supernova explosion, sending shockwaves through the cloud. These shockwaves compress regions within the cloud, causing them to begin to collapse under their own gravity. This marks the first step in the birth of a star.

**2. Accretion Disk and Protostar Formation:**
As the cloud collapses, it fragments into dense clumps. The core of each clump continues to contract, spinning faster due to the conservation of angular momentum—similar to a figure skater pulling in their arms to spin faster. This rapid rotation prevents material from falling directly onto the core, instead forming a rotating disk around it. At the center of this disk, a hot, dense region forms: the **protostar**.

<div class="alert alert-block alert-info">
<b>Ignition of Nuclear Fusion</b> As the protostar continues to contract and heat up, the core temperature eventually reaches the point where hydrogen nuclei begin to fuse into helium. This nuclear fusion releases immense energy, causing the protostar to shine brightly. At this stage, the protostar becomes a main sequence star, where it will spend the majority of its life.
</div>

<img src="https://raw.githubusercontent.com/teaghan/astronomy-12/main/Unit3/figures/formation.png" alt="Star Formation" width="1000" style="display: block; margin-left: auto; margin-right: auto;">

**3. Stellar Wind and Angular Momentum Transfer:**
The young star generates a stellar wind—a stream of charged particles that escapes along the magnetic field lines. This wind carries away excess angular momentum, allowing the star to continue accreting material from the disk without spinning too rapidly. The wind escapes most effectively along the poles, creating striking bipolar outflows that can be observed as jets of high-speed gas.

<img src="https://raw.githubusercontent.com/teaghan/astronomy-12/main/Unit3/figures/protostar_jets.png" alt="Protostar Jets" width="1000" style="display: block; margin-left: auto; margin-right: auto;">

**4. Planet Formation within the Disk:**
Within the disk, dust grains begin to stick together, forming planetesimals—small, solid objects. These planetesimals collide and merge, gradually building up into planets. Meanwhile, the remaining gas and dust are gradually blown away by the stellar wind, clearing the surroundings and leaving a nascent planetary system orbiting the young star.

---

### The Role of Angular Momentum

Angular momentum is a measure of the amount of rotation an object has, considering its mass, shape, and speed. It's crucial for understanding the behavior of rotating systems, from spinning tops to orbiting planets.

Think of a figure skater performing a spin. When the skater pulls their arms close to their body, they spin faster. This happens because **angular momentum is conserved**. For the skater, pulling in their arms reduces their **moment of inertia**, which a measure of how spread out their mass is relative to the axis of rotation. To keep the angular momentum constant, their rotation speed increases.

<div class="alert alert-block alert-info">
<b>Rotating Objects:</b> As an object's size decreases, its rotation rate must increase to conserve angular momentum.
</div>

Similarly, as the cloud of gas and dust contracts, it spins faster due to the conservation of angular momentum. 

> **Angular Momentum Formula:**
> $$
> L = I \cdot \omega
> $$
> where $L$ is angular momentum
>
> $I$ is the moment of inertia
>
> and $\omega$ is the angular velocity (how fast the object is spinning, measured in radians per second).

#### Understanding Moment of Inertia

The moment of inertia ($I$) is a measure of an object's resistance to changes in its rotation and it depends on both the mass of the object and how that mass is distributed relative to the axis of rotation. For example, it's easier to spin a figure skater with their arms close to their body than when their arms are extended. This is because pulling the arms in reduces the moment of inertia, allowing for faster rotation.

- **High Moment of Inertia**: Objects with mass distributed far from the axis of rotation, which makes it harder to spin.
- **Low Moment of Inertia**: Objects with mass concentrated close to the axis of rotation, which makes it easier to spin.

> **Common Moment of Inertia Formulas:**
> 
> - **Solid Sphere**: $I = \frac{2}{5} m r^2$
> - **Thin Disk**: $I = \frac{1}{2} m r^2$
> - **Thin Rod (rotating about its end)**: $I = \frac{1}{3} m L^2$
> - **Thin Rod (rotating about its center)**: $I = \frac{1}{12} m L^2$

<div class="alert alert-block alert-info">
<b>Remember:</b> The more spread out the mass, the harder it is to change the object's rotational speed.
</div>

#### Conservation of Angular Momentum

The conservation of angular momentum is a fundamental principle in physics. It states that if no external torque (twisting force) acts on a system, the total angular momentum of that system remains constant. This principle plays a crucial role in the formation of stars.

<div class="alert alert-block alert-info">
<b>Conservation of Momentum:</b> If no external torque (twisting force) acts on a system, the total angular momentum of that system remains constant.
</div>

As a molecular cloud collapses, it starts to spin faster, much like our figure skater. This increase in rotation speed is due to the conservation of angular momentum.

<div class="alert alert-block alert-warning">
<b>Example:</b> Calculate the final rotation speed of a protostar's accretion disk when its radius decreased from 1000 AU to 100 AU. Assume the disk has approximately the same mass as our Sun and is initially rotating at 0.001 rad/s.
</div>

> Given:
> 
> > Mass of the accretion disk, $ m = 2 \times 10^{30} \, \text{kg} $ (approximately the mass of the Sun)
> > 
> > Initial radius of the disk, $ r_1 = 1000 \, \text{AU} $ (1 AU = $1.496 \times 10^{11} \, \text{m}$)
> > 
> > Final radius of the disk, $ r_2 = 100 \, \text{AU} $
> > 
> > Initial rotation speed, $ \omega_1 = 1 \times 10^{-3} \, \text{rad/s} $
> 
> Solution:
> 
> First, convert the radii to meters:
> 
> > $$ r_1 = 1000 \, \text{AU} = 1000 \times 1.496 \times 10^{11} \, \text{m} = 1.496 \times 10^{14} \, \text{m} $$
> >
> > $$ r_2 = 100 \, \text{AU} = 100 \times 1.496 \times 10^{11} \, \text{m} = 1.496 \times 10^{13} \, \text{m} $$
> 
> Use the conservation of angular momentum:
> 
> > $$ L_1 = L_2 $$
> > 
> > $$ I_1 \cdot \omega_1 = I_2 \cdot \omega_2 $$
> > 
> > The moment of inertia for a thin disk is given by:
> > 
> > $$ I = \frac{1}{2} m r^2 $$
> > 
> > Substitute the moment of inertia into the angular momentum equation:
> > 
> > $$ \frac{1}{2} m r_1^2 \cdot \omega_1 = \frac{1}{2} m r_2^2 \cdot \omega_2 $$
> > 
> > Since the mass $m$ and the constant $\frac{1}{2}$ cancel out, we get:
> > 
> > $$ r_1^2 \cdot \omega_1 = r_2^2 \cdot \omega_2 $$
> > 
> > Solve for the final rotation speed $\omega_2$:
> > 
> > $$ \omega_2 = \frac{r_1^2 \cdot \omega_1}{r_2^2} $$
> > 
> > $$ \omega_2 = \frac{(1.496 \times 10^{14} \, \text{m})^2 \cdot 1 \times 10^{-3} \, \text{rad/s}}{(1.496 \times 10^{13} \, \text{m})^2} $$
> > 
> > $$ \omega_2 = \frac{2.238 \times 10^{25} \, \text{m}^2 \cdot \text{rad/s}}{2.238 \times 10^{26} \, \text{m}^2} $$
> > 
> > $$ \omega_2 = 0.1 \, \text{rad/s} $$
>
> Therefore, the final rotation speed of the protostar's accretion disk is $0.1 \, \text{rad/s}$.

In this example, we examined a **thin disc that changed its radius, but did not change its mass**. We used our algebra skills to find that the new rotational speed can be calculated with the equation:

> $$ \omega_2 = \frac{r_1^2 \cdot \omega_1}{r_2^2} $$

which always works for this scenario!! Hopefully this exemplifies why using algebra and variables is much more powerful than just plugging numbers into a calculator :)

#### Angular Momentum in Star Formation

**Formation of Accretion Disks**: As the molecular cloud contracts, its rotation speed increases, and the material flattens into an accretion disk around the protostar. This disk is essential for the growth of the protostar, channeling material onto it and setting the stage for planet formation.

**Bipolar Outflows**: The rotating disk and the magnetic fields generate jets of gas that are ejected along the poles of the protostar. These jets carry away excess angular momentum, preventing the protostar from spinning too fast and allowing more material to accrete onto the star.

These outflows can be observed as dazzling jets extending far into space, often lighting up surrounding gas clouds and creating stunning visual spectacles known as Herbig-Haro objects. These objects are markers of young, forming stars, providing astronomers with clues about the processes occurring within these hidden stellar nurseries.

<img src="https://raw.githubusercontent.com/teaghan/astronomy-12/main/Unit3/figures/protostar_disk.png" alt="Protostar" width="1000" style="display: block; margin-left: auto; margin-right: auto;">

By understanding the role of angular momentum in star formation, we can appreciate the delicate balance of forces that shape the birth of stars. From the initial collapse of a molecular cloud to the formation of accretion disks and the ejection of bipolar outflows, each step is a testament to the intricate dance of physics that governs our universe.

---

### Observation and Evidence of Star Formation

Stars are born in regions shrouded by thick clouds of gas and dust, making them invisible to the naked eye. However, by using a variety of sophisticated tools and techniques, astronomers can unveil these stellar nurseries and gather critical evidence of star formation.

#### Methods of Observing Star Formation

**1. Infrared Astronomy:**
Infrared light is a powerful tool in the astronomer’s toolkit. Unlike visible light, which is easily blocked by dust, infrared light can penetrate these thick clouds, allowing us to see what lies within. By observing in the infrared spectrum, astronomers can detect the faint glow of young stars and protostars still nestled in their natal environments. These infrared observations have revolutionized our understanding of star formation, revealing entire regions of space teeming with stars in various stages of development.

> **Fun Fact:** The Spitzer Space Telescope, an infrared observatory, has provided some of the most detailed images of star-forming regions, revealing the intricate structures within molecular clouds.

<img src="https://raw.githubusercontent.com/teaghan/astronomy-12/main/Unit3/figures/infrared.png" alt="Infrared" width="1000" style="display: block; margin-left: auto; margin-right: auto;">

**2. Radio Telescopes:**
Radio telescopes offer another window into the hidden process of star formation. These telescopes are designed to detect the faint radio waves emitted by molecules in cold, dense regions of space. One of the most important molecules detected in these regions is carbon monoxide (CO). By mapping the distribution of CO and other molecules, astronomers can identify the locations of molecular clouds, where star formation is actively occurring. Radio observations can also reveal the movement of gas within these clouds, helping to trace the dynamics of the star formation process.

> **Did You Know?** The Atacama Large Millimeter/submillimeter Array (ALMA) in Chile is one of the most powerful radio telescopes in the world, capable of capturing detailed images of star-forming regions in distant galaxies.

By combining these observational techniques, astronomers have been able to build a detailed picture of how stars form, even in the most obscured regions of the galaxy. These observations not only confirm theoretical models but also provide new insights that continue to push the boundaries of our understanding of the universe.

---

### Check Your Understanding

1. **Gravitational Collapse Trigger**: Explain how a supernova shockwave can trigger the gravitational collapse of a molecular cloud and initiate the process of star formation.
<br><br>

2. **Angular Momentum**: What roles does the conservation of angular momentum play in the various stages of star formation?
<br><br>

3. **Infrared Astronomy**: Why is infrared light particularly useful for observing star formation? What does it reveal that visible light cannot?
<br><br>

4. **Radio Telescopes and Molecular Clouds**: Explain how radio telescopes help astronomers detect molecular clouds and understand the star formation process. What specific molecules are often observed?
<br><br>

5. **Angular Momentum and Accretion Disks**: A collapsing molecular cloud forms an accretion disk around a protostar due to the conservation of angular momentum. If the initial radius of the disk is $500 \, \text{AU}$ with a rotation speed of $0.01 \, \text{rad/s}$, and the disk contracts to $50 \, \text{AU}$, calculate the final rotation speed of the disk.
<br><br>

6. **Planet Formation**: Describe the process by which planetesimals form within the accretion disk around a young star. How do these planetesimals eventually form planets?
<br><br>

7. **Moment of Inertia Calculation**: Calculate the moment of inertia for a solid sphere with a mass of $5 \times 10^{30} \, \text{kg}$ and a radius of $7 \times 10^{10} \, \text{m}$. 
<br><br>

---

## Resources

- **Astronomy (2016)**. Andrew Fraknoi, David Morrison, and Sidney C. Wolff.
- **Foundations of Astrophysics (2010)**. Barbara Ryden and Bradley M. Peterson.


# 3_2_atoms_particles.md
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---

# 3.2 Atoms and Particles

Stars are the glowing furnaces of the universe, converting simple particles into complex elements through nuclear fusion. But to truly understand how stars work, we first need to explore the building blocks of matter itself—atoms and their subatomic components. In this lesson, we’ll journey into the heart of the atom, uncover the forces that hold it together, and examine the role that particles play in the lives of stars.

<div class="alert alert-block alert-success">
  <h4>Video</h4>
  <p>Watch <a href="https://www.youtube.com/watch?v=VI32SlCynmo" target="_blank">this video</a> to gain a visual understanding of atomic structures.</p>
</div>

---

## Introduction to Atomic Structure

Atoms are the fundamental units of matter, making up everything in the universe, from the air we breathe to the stars we observe in the night sky. But what exactly are atoms composed of?

### **Protons, Neutrons, and Electrons**

Atoms consist of three primary subatomic particles: protons, neutrons, and electrons. These particles are responsible for the identity and behavior of atoms, including their role in stellar processes like nuclear fusion.

- **Protons**: Positively charged particles found in the nucleus of an atom. The number of protons determines the element's identity.
- **Neutrons**: Neutral particles also located in the nucleus. They provide stability to the nucleus by balancing the repulsive forces between protons.
- **Electrons**: Negatively charged particles that orbit the nucleus. Electrons are responsible for chemical bonds and the behavior of atoms in different states of matter.

### **Table of Subatomic Particles**

| **Particle**  | **Mass (kg)**                 | **Charge**      | **Location**      |
|---------------|-------------------------------|-----------------|-------------------|
| Proton        | $1.6726 \times 10^{-27}$       | +1 (positive)   | Nucleus           |
| Neutron       | $1.6749 \times 10^{-27}$       | 0 (neutral)     | Nucleus           |
| Electron      | $9.1094 \times 10^{-31}$       | -1 (negative)   | Orbiting nucleus  |

### **Atomic Number and Mass Number**

- **Atomic Number**: The number of protons in the nucleus of an atom, which defines the element.
- **Mass Number**: The total number of protons and neutrons in an atom’s nucleus.

For example, carbon has an atomic number of 6 and a typical mass number of 12, meaning it has 6 protons and 6 neutrons.

<img src="https://raw.githubusercontent.com/teaghan/astronomy-12/main/Unit3/figures/atomic_number.png" alt="atomic number" width="1000" style="display: block; margin-left: auto; margin-right: auto;">

### **Isotopes**

Isotopes are variations of an element that have the same number of protons but different numbers of neutrons. Some isotopes are stable, while others are radioactive and can decay over time. In stars, isotopes play a critical role in nuclear fusion processes, where atoms combine to form heavier elements, releasing vast amounts of energy.

> **Did You Know?** The Sun primarily fuses hydrogen isotopes (protons) to form helium, providing the energy that powers our solar system.

<img src="https://raw.githubusercontent.com/teaghan/astronomy-12/main/Unit3/figures/isotopes.png" alt="H isotopes" width="1000" style="display: block; margin-left: auto; margin-right: auto;">

---

## Neutrinos and Photons

Now, let’s explore two additional key particles—neutrinos and photons—that play critical roles in both atomic and stellar processes.

### **Neutrinos**

Neutrinos are incredibly light, neutral particles that interact very weakly with matter. Produced in vast quantities during nuclear reactions in stars, neutrinos can pass through entire planets without interacting with any matter. These elusive particles are critical in nuclear fusion processes in stars and play a key role in supernovae explosions, carrying away much of the energy released.

> **Fun Fact**: Every second, trillions of neutrinos from the Sun pass through your body unnoticed!

<img src="https://raw.githubusercontent.com/teaghan/astronomy-12/main/Unit3/figures/neutrino_detector.png" alt="Neutrino Detector" width="1000" style="display: block; margin-left: auto; margin-right: auto;">

### **Photons**

Photons are the basic units of light and all other forms of electromagnetic radiation. They are massless particles that travel at the speed of light and carry energy. In stars, photons are produced during nuclear fusion reactions and are responsible for the light and heat that stars emit.

#### **The Nature of Light: Wave-Particle Duality**

Light exhibits both wave-like and particle-like properties. As a wave, light can be described by its wavelength and frequency. As a particle, light is made up of photons. This wave-particle duality is a fundamental concept in quantum physics and helps explain the behavior of light in different situations, such as when it passes through a prism or interacts with matter.

> **Fun Fact**: Photons from the Sun take hundreds of thousands of years to escape from the Sun’s core to its surface, but only 8 minutes to travel the 150 million kilometers to Earth.

### **Table of Electromagnetic Radiation**

| **Type of Radiation** | **Wavelength Range (nm)** | **Radiated by Objects at This Temperature** | **Typical Sources**                                |
|-----------------------|---------------------------|---------------------------------------------|---------------------------------------------------|
| Gamma rays             | Less than 0.01            | More than $10^8$ K                          | Produced in nuclear reactions; require very high-energy processes |
| X-rays                 | 0.01–20                   | $10^6$–$10^8$ K                             | Gas in clusters of galaxies, supernova remnants, solar corona      |
| Ultraviolet            | 20–400                    | $10^4$–$10^6$ K                             | Supernova remnants, very hot stars                                |
| Visible                | 400–700                   | $10^3$–$10^4$ K                             | Stars                                                        |
| Infrared               | $10^3$–$10^6$             | 10–$10^3$ K                                 | Cool clouds of dust and gas, planets, moons                    |
| Microwave              | $10^6$–$10^9$             | Less than 10 K                              | Active galaxies, pulsars, cosmic background radiation         |
| Radio                  | More than $10^9$          | Less than 10 K                              | Supernova remnants, pulsars, cold gas                        |

---

## Forces within Atoms

Atoms are held together through the interaction of the four fundamental forces. Let’s explore these forces and their roles in both atomic structure and stellar processes.

### **The Electromagnetic Force**

The electromagnetic force is responsible for the attraction between negatively charged electrons and positively charged protons, keeping electrons in orbit around the nucleus. This force also governs the behavior of charged particles in stellar environments, such as plasma. Additionally, the electromagnetic force is responsible for the repulsion of like charges, such as the repulsive force between two positively charged protons within an atom's nucleus.

### **The Strong Nuclear Force**

The strong nuclear force is what holds protons and neutrons together in the nucleus, despite the repulsive electromagnetic force between the positively charged protons. This force is incredibly powerful but acts only at very short distances within the nucleus.

### **The Weak Nuclear Force**

The weak nuclear force is involved in processes like beta decay, where a neutron can transform into a proton, emitting a neutrino and an electron. This force plays a significant role in the nuclear reactions that occur in stars, particularly during the fusion of lighter elements into heavier ones.

### **Gravity**

While gravity is the weakest of the four fundamental forces on the atomic scale, it plays a crucial role in the formation and life cycles of stars. Gravity is the force that causes molecular clouds to collapse, initiating the star formation process, and it governs the structure of stars throughout their lifetimes.

---

## Radioactive Decay

Radioactive decay is a natural process in which unstable atomic nuclei lose energy by emitting radiation. This process transforms atoms into different elements or isotopes over time. In stellar environments, radioactive decay plays a crucial role in the production of energy and the synthesis of new elements, contributing to the life cycles of stars and the evolution of the universe.

### **Types of Radioactive Decay**

- **Alpha Decay**: In this type of decay, the nucleus emits an alpha particle (two protons and two neutrons), reducing the mass number of the atom.
- **Beta Decay**: During beta decay, a neutron is converted into a proton, and an electron (beta particle) and a neutrino are emitted.
- **Gamma Decay**: Gamma decay occurs when the nucleus releases energy in the form of gamma rays, without changing the number of protons or neutrons.

### **Half-Life and Its Significance**

The half-life of a radioactive isotope is the time it takes for half of the atoms in a sample to decay. Half-lives can range from fractions of a second to billions of years. In stars, radioactive decay processes contribute to the production of energy and the creation of new elements.

> **Example**: Uranium-238, with a half-life of 4.5 billion years, plays a key role in the heat generation inside planets like Earth.

<img src="https://raw.githubusercontent.com/teaghan/astronomy-12/main/Unit3/figures/radioactive_decay.png" alt="Radioactive Decay" width="1000" style="display: block; margin-left: auto; margin-right: auto;">

---
Here’s an enhanced version of the "Particles in Stellar Environments" section:

---

## Particles in Stellar Environments

The universe's most powerful laboratories are stars, where temperatures and pressures soar to unimaginable extremes. These harsh environments cause particles to behave in ways that defy our everyday experiences on Earth. 

### **High-Energy Environments in Stars**

At the heart of a star, the temperature can exceed millions of degrees Celsius—hot enough to strip atoms of their electrons. This creates a plasma, an incredibly energetic state of matter composed of freely moving ions and electrons. Plasma is often described as a "soup" of charged particles, and in this environment, particles collide at incredible speeds, facilitating nuclear fusion reactions that power the star.

As these particles interact, they generate intense magnetic fields and release vast amounts of energy in the form of light and heat. The movement of charged particles in this plasma also generates phenomena such as solar flares, which can burst out from the star's surface with explosive force, affecting entire planetary systems.

> **Fun Fact**: Plasma, the most common state of matter in the universe, makes up 99% of all visible matter, including stars, lightning, and even neon signs.

### **Stellar Winds and Supernovae**

Stars are not just passive energy emitters—they are also sources of powerful particle flows that shape the universe around them.

- **Stellar Winds**: Many stars, particularly massive ones, emit continuous streams of charged particles known as stellar winds. These winds are composed mainly of protons and electrons and can extend far beyond the star, influencing the formation of planets, asteroids, and even entire star systems. Stellar winds can erode the outer layers of a dying star, contributing to the creation of stunning nebulae—vast clouds of gas and dust that may eventually collapse to form new stars.

   Stellar winds are also responsible for the breathtaking beauty of planetary nebulae and the majestic structures seen in star-forming regions like the Eagle Nebula's "Pillars of Creation."

<img src="https://raw.githubusercontent.com/teaghan/astronomy-12/main/Unit3/figures/stellar_winds.png" alt="Stellar Winds" width="1000" style="display: block; margin-left: auto; margin-right: auto;">

- **Supernovae**: The death throes of a massive star culminate in a supernova—a colossal explosion that outshines entire galaxies. During a supernova, a star ejects enormous quantities of material into space at speeds approaching the speed of light. These high-energy particles, accelerated by the explosion, become cosmic rays that travel through the galaxy, bombarding planets and influencing their atmospheres.

   Supernovae are also nature’s forge for the heaviest elements, including iron, gold, and uranium. The very atoms that make up your body, the Earth, and everything around you were once created in the fiery death of a star. Without supernovae, the universe would lack the heavy elements necessary for life.

---

## Check Your Understanding

1. **Atomic Structure**: Explain how the atomic number and mass number define the identity and characteristics of an element. How do these properties influence the formation of isotopes in stellar environments?
<br><br>

2. **Subatomic Particles**: Compare and contrast the roles of protons, neutrons, and electrons in atomic structure. How do their respective masses and charges contribute to the stability and reactivity of an atom?
<br><br>

3. **Forces within Atoms**: Describe the strong nuclear force and explain why it is crucial in holding the nucleus together, despite the repulsive electromagnetic force between protons. How does this force compare to gravity on an atomic scale?
<br><br>

3. **Neutrinos**: Why are neutrinos so important in the study of stars, and why do you think they are so difficult to detect?
<br><br>

4. **Photons**: What are photons, and how do they contribute to the energy emitted by stars?
<br><br>

5. **Electromagnetic Radiation**: Examine the different types of electromagnetic radiation (gamma rays, X-rays, ultraviolet, visible light, etc.) and their relationship to the temperature and behavior of astronomical objects. How do these radiation types help astronomers study the universe?
<br><br>

---

## Resources

- **Astronomy (2016)**. Andrew Fraknoi, David Morrison, and Sidney C. Wolff.
- **Foundations of Astrophysics (2010)**. Barbara Ryden and Bradley M. Peterson.


# 3_3_nuclear_fusion.md
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# 3.3 Nuclear Fusion in Stars

The Sun provides warmth and light that is essential for life on Earth. But have you ever wondered what powers this immense ball of fire? The answer lies in nuclear fusion, a process that occurs deep within the core of the Sun and other stars, where lighter atomic nuclei combine to form heavier nuclei, releasing vast amounts of energy in the process. In this lesson, we'll explore how nuclear fusion works, the different types of fusion reactions in stars, and how these reactions shape the life cycles of stars and contribute to the formation of elements in the universe.

<div class="alert alert-block alert-success">
  <h4>Video</h4>
  <p>Watch <a href="https://www.youtube.com/watch?v=W8cX0YbRLFo" target="_blank">this video</a> for an excellent overview of nuclear fusion and its role in powering stars.</p>
</div>

---

## What is Nuclear Fusion?

Nuclear fusion is the process by which two light atomic nuclei combine to form a heavier nucleus, releasing energy in the process. This reaction is the primary energy source for stars, including our Sun. 

**Conditions for Fusion**: 
For nuclear fusion to occur, the nuclei must overcome their natural repulsion due to their positive charges. This requires extremely high temperatures (millions of degrees) and pressures, which are found in the cores of stars. The immense gravitational force in a star's core compresses the gas to such an extent that the temperature rises high enough to allow fusion to occur.

> **Did You Know?**
> The Sun's core temperature is around 15 million degrees Celsius, a condition necessary for hydrogen nuclei to fuse into helium.

---

## The Proton-Proton Chain Reaction

The proton-proton (p-p) chain is the dominant fusion process in stars the size of the Sun. It involves a series of reactions where hydrogen nuclei (protons) combine to form helium, releasing energy at each step.

**Step-by-Step Process**:

1. **Step 1: Formation of Deuterium**
   - Two protons collide at high speeds. Normally, they would repel each other due to their positive charges, but under the extreme conditions in a star's core, one proton undergoes beta decay and converts into a neutron. This forms Deuterium (Hydrogen-2), consisting of one proton and one neutron.
   - **Reaction:** The first step involves the fusion of two protons ($^1\text{H}$) to form deuterium ($^2\text{H}$), releasing a positron ($e^+$) and a neutrino ($\nu_e$).
> $$ ^1\text{H} + ^1\text{H} \rightarrow ^2\text{H} + e^+ + \nu_e $$

>> Here, the superscripts indicate the total number of protons and neutrons in the nucleus.
   - **Time Scale**: This step is the slowest, taking about a billion years on average for a proton in the Sun.

<img src="https://raw.githubusercontent.com/teaghan/astronomy-12/main/Unit3/figures/pp1.png" alt="PP Chain Step 1" width="1000" style="display: block; margin-left: auto; margin-right: auto;">

2. **Step 2: Formation of Helium-3**
   - The newly formed Deuterium nucleus collides with another proton, resulting in the formation of Helium-3 (two protons and one neutron).
   - **Reaction:** In the second step, the deuterium nucleus ($^2\text{H}$) produced in the first step fuses with another proton to form helium-3 ($^3\text{He}$), releasing a gamma-ray photon.
> $$ ^2\text{H} + ^1\text{H} \rightarrow ^3\text{He} + \gamma $$
   
   - **Time Scale**: This reaction happens quickly, in about 4 seconds.

<img src="https://raw.githubusercontent.com/teaghan/astronomy-12/main/Unit3/figures/pp2.png" alt="PP Chain Step 2" width="1000" style="display: block; margin-left: auto; margin-right: auto;">

3. **Step 3: Formation of Helium-4**
   - Finally, two Helium-3 nuclei collide to form Helium-4 (two protons and two neutrons) and release two protons.
   - **Reaction:** The third step involves the fusion of two helium-3 nuclei ($^3\text{He}$) to form helium-4 ($^4\text{He}$) and two protons. 
> $$ ^3\text{He} + ^3\text{He} \rightarrow ^4\text{He} + ^1\text{H} + ^1\text{H} $$
   
   - **Time Scale**: This step takes about 400 years for Helium-3 nuclei to collide in the Sun's core.


> **Note:**
>
>  The third step requires two helium-3 nuclei to start; the first two steps must happen twice before the third step can occur.


<img src="https://raw.githubusercontent.com/teaghan/astronomy-12/main/Unit3/figures/pp3.png" alt="PP Chain Step 3" width="1000" style="display: block; margin-left: auto; margin-right: auto;">

<div class="alert alert-block alert-info">
<h4>Timing:</h4> Although the first step in this chain of reactions is very unlikely to occur at any given moment and typically takes a very long time for a single proton to undergo fusion, the sheer number of protons in the Sun ensures that these reactions are happening constantly. Even though each individual reaction might take billions of years to occur, with countless protons in the Sun’s core, millions of these reactions are happening every second. Despite the slow probability for any single proton, the continuous, overlapping activity of countless atoms makes fusion a steady and powerful energy source.
</div>

> **Proton-Proton Chain Summary**:
> 
> The proton-proton chain converts four hydrogen nuclei into one helium nucleus, releasing energy in the form of light and heat. This energy is what powers the Sun and provides the warmth and light we receive on Earth.

### Energy Generation in the Proton-Proton (p-p) Chain

In the proton-proton (p-p) chain, four hydrogen nuclei (protons) are ultimately fused to form a single helium-4 nucleus. The overall reaction can be represented as:

> $$ ^1\text{H} + ^1\text{H} + ^1\text{H} + ^1\text{H} \rightarrow ^4\text{He} + 2e^+ + 2\nu_e + 2\gamma + \text{energy} $$

To determine the amount of energy released during this process, we can calculate the difference in mass between the initial hydrogen atoms and the final helium atom. The mass of a hydrogen atom is 1.007825 atomic mass units (u), and the mass of a helium-4 atom is 4.00268 u. Here, we use the full atomic mass, including electrons, because the process involves the creation and annihilation of positrons and electrons.

The mass of the initial four hydrogen atoms is:

> $$ 4 \times 1.007825 \, \text{u} = 4.03130 \, \text{u} $$

The mass of the final helium atom is:

> $$ 4.00268 \, \text{u} $$

The difference in mass, which is converted to energy, is:

> $$ 4.03130 \, \text{u} - 4.00268 \, \text{u} = 0.02862 \, \text{u} $$

<div class="alert alert-block alert-info">
<h4>p-p Chain Mass loss:</h4> The mass loss of one proton-proton (p-p) chain reaction is 0.02862 u, which represents about 0.71% of the initial mass. Therefore, when 1 kilogram of hydrogen is converted into helium, approximately 0.0071 kilograms of mass is converted into energy.
</div>

Using Einstein's mass-energy equivalence formula, $E = mc^2$, where $c$ is the speed of light ($3 \times 10^8$ meters per second), the energy released by this mass conversion is:

> $$ E = 0.0071 \, \text{kg} \times \left(3 \times 10^8 \, \text{m/s}\right)^2 = 6.4 \times 10^{14} \, \text{J} $$

This amount of energy, released from the fusion of just 1 kilogram of hydrogen, is enough to supply all the electricity used in the United States for approximately two weeks.

The Sun, which shines with a luminosity of $4 \times 10^{26}$ watts, must fuse about 600 million tons of hydrogen into helium every second to maintain its brightness. Of this, around 4 million tons of mass are converted directly into energy. Despite the enormous quantities involved, the Sun has such a vast reserve of hydrogen that it can continue this fusion process for billions of years, ensuring its place as our solar system’s energy source for a long time to come.

---

## The CNO Cycle

In stars more massive than the Sun, the Carbon-Nitrogen-Oxygen (CNO) cycle dominates as the primary fusion process. While it still converts hydrogen into helium, it uses carbon, nitrogen, and oxygen as catalysts.

**Overview**:
- The CNO cycle involves the fusion of hydrogen nuclei with carbon, nitrogen, and oxygen, leading to the formation of helium. These heavier elements act as catalysts and are not consumed in the reaction.
- **Energy Released**: The CNO cycle releases more energy per cycle than the proton-proton chain, which is why it is more prevalent in hotter, more massive stars.

**Comparison with Proton-Proton Chain**:
- **Temperature Dependence**: The CNO cycle requires a higher core temperature than the proton-proton chain, which is why it is more efficient in more massive stars.
- **Role in Star Types**: The proton-proton chain is dominant in stars with a mass similar to or smaller than the Sun, while the CNO cycle is dominant in larger stars.

<div class="alert alert-block alert-info">
<h4>Did You Know?</h4> In stars with more than 1.3 times the mass of the Sun, the CNO cycle is the primary energy source.
</div>

---
## Energy Transport in Stars

The energy produced by nuclear fusion in the core of a star must journey to the star's surface before it can be radiated into space as light and heat. This energy travels through various layers of the star, each with unique properties that influence how energy is transferred.

#### Radiative Zone: 
   - In the radiative zone, energy is transported outward by radiation. Here, photons are absorbed and re-emitted by particles, slowly making their way from the hot core to cooler outer regions. This process is incredibly slow—each photon may take thousands of years to reach the next layer because it constantly interacts with particles, zigzagging through the dense material of the star. The radiative zone acts like a dense forest, where light must weave its way through countless obstacles.
   - The radiative zone is more prominent in stars like the Sun, where the energy from the core is primarily transferred by this slow but steady process.

#### Convection Zone: 
   - Above the radiative zone lies the convection zone, where the mode of energy transfer changes dramatically. In this region, the energy causes hot plasma to rise toward the star's surface, cool down, and then sink back toward the interior to be reheated—a process known as convection. This is similar to the way boiling water circulates in a pot. The convective zone is turbulent and dynamic, with massive currents of hot gas churning beneath the star's surface.
   - Convection is particularly important in cooler stars, where the outer layers are less efficient at radiating energy. In these stars, the convective zone can extend much deeper, playing a crucial role in energy transport.

<img src="https://raw.githubusercontent.com/teaghan/astronomy-12/main/Unit3/figures/layers.png" alt="Star Layers" width="1000" style="display: block; margin-left: auto; margin-right: auto;">

<div class="alert alert-block alert-info">
<h4>Interesting Side Note</h4> Despite the fact that light travels at an incredible speed—about 300,000 kilometers per second in a vacuum—the energy produced in the Sun's core can take thousands of years to reach the surface. This is because photons are continuously absorbed and re-emitted in the radiative zone, making their journey to the surface a slow and winding path. By the time a photon reaches the surface, its journey might have taken several million years!
</div>

<img src="https://raw.githubusercontent.com/teaghan/astronomy-12/main/Unit3/figures/photon_path.png" alt="Photon Path" width="1000" style="display: block; margin-left: auto; margin-right: auto;">

---

## Nuclear Fusion and Star Life Cycles

Nuclear fusion is not only the engine that powers stars; it also shapes their entire life cycle, from their stable years on the main sequence to their dramatic ends.

**The Role of Fusion in Stellar Evolution**:
- **Main Sequence Stars**:
   - Stars begin their lives on the main sequence, where they spend the majority of their existence fusing hydrogen into helium in their cores. This fusion provides the energy that counteracts gravity, keeping the star stable. The balance between the outward pressure from fusion and the inward pull of gravity is what maintains a star's equilibrium.
   - During this phase, a star's mass determines its brightness and color. More massive stars burn hotter and brighter but have shorter lifespans, while less massive stars burn cooler and dimmer, living much longer.

<img src="https://raw.githubusercontent.com/teaghan/astronomy-12/main/Unit3/figures/equilibrium.png" alt="Hydrostatic equilibrium" width="1000" style="display: block; margin-left: auto; margin-right: auto;">

- **Post-Main Sequence**:
   - As stars exhaust their hydrogen fuel, the core contracts and heats up, leading to the fusion of helium and other heavier elements. This marks the beginning of the post-main sequence phase. For stars like the Sun, this transition turns them into red giants, where their outer layers expand and cool, giving the star a reddish appearance.
   - More massive stars continue to fuse heavier elements in their cores, moving through stages of carbon, oxygen, and even silicon fusion. Each new stage of fusion occurs at increasingly higher temperatures and lasts for shorter periods of time.

**Helium Burning and Beyond**:
- **Red Giants and Supergiants**:
   - In stars like the Sun, once hydrogen is depleted, helium fusion begins, leading to the formation of carbon and oxygen. This helium burning phase causes the star to expand into a red giant. Eventually, the outer layers may be ejected into space, creating a planetary nebula, while the core remains as a white dwarf.
   - In more massive stars, after helium is used up, the core contracts further, and fusion of heavier elements like carbon and oxygen begins. These stars become supergiants, with a layered structure resembling an onion, where each layer is fusing different elements.

- **The Final Act—Supernovae and Stellar Remnants**:
   - When massive stars reach the point where their cores are made of iron, fusion can no longer produce energy. This leads to a catastrophic core collapse, resulting in a supernova explosion. This explosion scatters heavy elements into space, contributing to the cosmic material that forms new stars and planets.
   - The remnants of the explosion depend on the mass of the core:
     - **White Dwarf**: For smaller stars, the remnant is a white dwarf, which slowly cools over time.
     - **Neutron Star**: In more massive stars, the core collapse leads to the formation of a neutron star, an incredibly dense object composed mostly of neutrons.
     - **Black Hole**: If the star is massive enough, the core continues collapsing to form a black hole, an object with gravity so strong that not even light can escape.

---

## Check Your Understanding

1. **Proton-Proton Chain**: Describe the steps involved in the proton-proton chain reaction in stars like the Sun. Why is the first step the slowest?
<br><br>

2. **Energy Transport**: Explain how energy is transported from the core of the Sun to its surface. Why does this process take so long?
<br><br>

3. **Stellar Nucleosynthesis**: How does nucleosynthesis in stars contribute to the formation of elements in the universe? Why are supernovae important for element distribution?
<br><br>

4. **Energy Released in Fusion**: Calculate the energy released when a proton combines with a deuterium nucleus to form helium-3 ($^3\text{He}$). Use the following atomic masses: $^1\text{H} = 1.007825 \, \text{u}$, $^2\text{H} = 2.014102 \, \text{u}$, and $^3\text{He} = 3.016029 \, \text{u}$. Apply Einstein’s equation $E = mc^2$, where $c = 3 \times 10^8 \, \text{m/s}$, and note that 1 u = $1.66054 \times 10^{-27}$ kg.
<br><br>

5. **Energy Conversion Over Time**: The Sun converts $4 \times 10^9$ kg of mass to energy every second. How many years would it take the Sun to convert a mass equal to the mass of Earth ($6 \times 10^{24}$ kg) to energy?
<br><br>

6. **Total Energy Potential of the Sun**: Assume that the Sun's mass is 75% hydrogen and that all of this mass could be converted to energy according to Einstein’s equation $E = mc^2$. The mass of the Sun is $2 \times 10^{30}$ kg. How much total energy could the Sun generate? Express your answer in joules.
<br><br>

7. **Total Energy from Solar Hydrogen**: If all the hydrogen atoms in the Sun were converted into helium, how much total energy would be produced? Estimate the total number of hydrogen atoms in the Sun, considering that hydrogen constitutes 75% of the Sun’s mass. (**Hint**: The mass of one hydrogen atom is approximately $1.67 \times 10^{-27}$ kg.)
<br><br>

8. **Sun's Mass Reduction Over Time**: The Sun converts 4 million tons of matter into energy every second. How long will it take for the Sun to reduce its mass by 1%? Compare your result with the Sun’s current age of approximately 4.6 billion years. The mass of the Sun is $2 \times 10^{30}$ kg.
<br><br>

---

## Resources

- **Astronomy (2016)**. Andrew Fraknoi, David Morrison, and Sidney C. Wolff.
- **Foundations of Astrophysics (2010)**. Barbara Ryden and Bradley M. Peterson.


# 3_4_chemical_composition.md
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layout: embed_default
---

# 3.4 The Chemical Composition of Stars

Elements are the fundamental building blocks of matter. Everything we see around us—from the air we breathe to the stars in the night sky—is made up of elements. But where do these elements come from? The answer lies within the stars. 

In this lesson, we will explore how stars produce elements and how these elements evolve over time.

<div class="alert alert-block alert-success">
  <h4>Video</h4>
  <p>
Watch <a href="https://www.youtube.com/watch?v=fzTRHJanq58" target="_blank">this video</a> for an excellent introduction to the origin of elements.</p>
</div>

---

## Nucleosynthesis and the Origin of Elements

Nucleosynthesis is the process by which new atomic nuclei are created within stars. The story begins shortly after the Big Bang, approximately 13.7 billion years ago, when the universe was filled with hydrogen and a little bit of helium—the first two elements on the periodic table.

As the universe expanded, gravity caused hydrogen to clump together, forming the first stars. These stars, through nuclear fusion, began the process of creating new elements. The fusion processes that occur within stars are responsible for producing almost all the elements in the periodic table.

### Hydrogen Burning and the Proton-Proton Chain

<div class="alert alert-block alert-info">
<h4>Note:</h4> Remember that the term "burning" in astrophysics refers to nuclear fusion, not chemical combustion.
</div>

We already learned about the simplest and most common fusion process in stars, which is hydrogen burning. This occurs through the proton-proton chain. This process dominates in stars like our Sun, where four hydrogen nuclei (protons) combine to form a single helium nucleus. 

This chain reaction not only produces helium but also releases a significant amount of energy, which powers the star and balances the inward pull of gravity. This process can continue for billions of years, as in the case of our Sun, which is currently in the middle of its hydrogen-burning phase.

### Helium Burning and the Triple-Alpha Process

As stars evolve and exhaust their hydrogen fuel, the core contracts and heats up, enabling the fusion of helium. This process is known as helium burning, and it occurs through the triple-alpha process.

The triple-alpha process can be described as follows:

1. **Formation of Beryllium-8:**
   - Two helium-4 nuclei (alpha particles) collide to form beryllium-8. The reaction can be represented as:
> $$ \text{He}^4 + \text{He}^4 \rightarrow \text{Be}^8 $$

   - Beryllium-8 ($\text{Be}^8$) is highly unstable and typically decays back into two helium nuclei within a fraction of a second. However, under the extreme conditions in the core of a star, if another helium-4 nucleus collides with the beryllium-8 before it decays, the next step occurs.

2. **Formation of Carbon-12:**
   - The unstable beryllium-8 nucleus captures another helium-4 nucleus to form carbon-12. The reaction is:
> $$ \text{Be}^8 + \text{He}^4 \rightarrow \text{C}^{12} + \gamma $$
   - This reaction releases a gamma-ray photon ($\gamma$), which carries away the excess energy from the fusion process.

This sequence of reactions is the primary way that carbon, an essential element for life, is produced in stars. The creation of carbon and oxygen through the triple-alpha process marks a critical stage in the chemical evolution of the universe, as these elements are necessary for the formation of complex molecules and ultimately, life as we know it.

<img src="https://raw.githubusercontent.com/teaghan/astronomy-12/main/Unit3/figures/lowmass_layers.png" alt="Low mass layers" width="1000" style="display: block; margin-left: auto; margin-right: auto;">

### Advanced Nuclear Burning Stages

In more massive stars, after helium is exhausted, the core contracts further, and temperatures rise high enough to fuse heavier elements. These advanced nuclear burning stages include carbon burning, neon burning, oxygen burning, and silicon burning.

Each of these stages produces increasingly heavier elements, with silicon burning ultimately leading to the production of iron. Here’s a brief overview of these processes:

- **Carbon Burning:** Produces neon, sodium, and magnesium.
- **Neon Burning:** Produces oxygen and magnesium.
- **Oxygen Burning:** Produces silicon, sulfur, phosphorus, and magnesium.
- **Silicon Burning:** Produces elements up to iron.

<img src="https://raw.githubusercontent.com/teaghan/astronomy-12/main/Unit3/figures/highmass_layers.png" alt="High mass layers" width="1000" style="display: block; margin-left: auto; margin-right: auto;">

Iron represents the end of the line for fusion in stars. Unlike lighter elements, fusing iron does not release energy; instead, it consumes energy. This is because iron has the most tightly bound nucleus, making further fusion energetically unfavorable. Once a star’s core is predominantly iron, it has reached a critical stage in its life cycle.

<div class="alert alert-block alert-info">
<b>Note:</b> Iron is the heaviest element that can be formed through fusion in a star’s core. The process stops here because fusing iron requires more energy than it releases. This lack of energy production leads to the core's collapse, setting the stage for a supernova explosion and the creation of even heavier elements.
</div>

<div class="alert alert-block alert-warning">
<b>Example:</b> Calculate the energy released during the fusion of two Carbon-12 nuclei to form Magnesium-24 in the carbon burning stage.
</div>

> Given:
> 
> > Mass of a Carbon-12 nucleus: $ m_{\text{C}} = 12.000 \, \text{u} $
> > 
> > Mass of a Magnesium-24 nucleus: $ m_{\text{Mg}} = 23.985 \, \text{u} $
> 
> Solution:
> 
> > The fusion of two Carbon-12 nuclei forms one Magnesium-24 nucleus:
> > 
> > $$ \text{C}^{12} + \text{C}^{12} \rightarrow \text{Mg}^{24} $$
> > 
> > Calculate the mass difference between the initial and final nuclei:
> > 
> > $$ \Delta m = (2 \times 12.000 \, \text{u}) - 23.985 \, \text{u} $$
> > 
> > $$ \Delta m = 24.000 \, \text{u} - 23.985 \, \text{u} = 0.015 \, \text{u} $$
> > 
> > Convert this mass difference to kilograms using $1 \, \text{u} = 1.66054 \times 10^{-27} \, \text{kg}$:
> > 
> > $$ \Delta m = 0.015 \, \text{u} \times 1.66054 \times 10^{-27} \, \text{kg/u} $$
> > 
> > $$ \Delta m \approx 2.49 \times 10^{-29} \, \text{kg} $$
> > 
> > Now, apply Einstein’s mass-energy equivalence formula: $E = \Delta mc^2$
> > 
> > $$ E = (2.49 \times 10^{-29} \, \text{kg}) \times (3 \times 10^8 \, \text{m/s})^2 $$
> > 
> > $$ E \approx 2.24 \times 10^{-12} \, \text{J} $$
> 
> Therefore, the energy released during the fusion of two Carbon-12 nuclei to form Magnesium-24 is approximately $2.24 \times 10^{-12}$ Joules.

### Supernovae and the Creation of Heavy Elements

Imagine a colossal star, thousands of times more massive than our Sun, reaching the end of its life. After millions of years of burning through its nuclear fuel, the star’s core has fused lighter elements into progressively heavier ones, culminating in iron. At this point, the star faces a fundamental problem: iron is the dead end of nuclear fusion, which means the star can no longer generate the pressure needed to counteract the relentless pull of gravity.

As gravity takes over, the core of the star collapses in a matter of seconds. This rapid implosion is so violent that it triggers a catastrophic explosion known as a supernova. In this single, fleeting moment, the star unleashes more energy than our Sun will produce over its entire 10-billion-year lifespan. The explosion is so bright that it can outshine entire galaxies and is visible across vast distances in the universe.

But a supernova is more than just a spectacular light show; it’s a cosmic forge, where the universe’s heaviest elements are born. 

<img src="https://raw.githubusercontent.com/teaghan/astronomy-12/main/Unit3/figures/SN2014.png" alt="SN2014" width="1000" style="display: block; margin-left: auto; margin-right: auto;">


### The r-Process: A Frenzy of Element Creation

During a supernova, the conditions become extraordinarily extreme—temperatures soar to billions of degrees, and the density of matter skyrockets. These conditions are perfect for a process known as rapid neutron capture, or the r-process. Here’s how it works:

As the star's core collapses, an immense number of neutrons are suddenly available. In the blink of an eye, atomic nuclei start capturing these neutrons at a furious pace, much faster than they can decay. This rapid neutron bombardment leads to the formation of highly unstable, neutron-rich isotopes. 

These isotopes are so unstable that they immediately undergo a series of beta decays, where neutrons convert into protons, transforming the element into a new, heavier one. This chain reaction continues, building up nuclei that are far heavier than iron—elements like gold, platinum, uranium, and thorium, which cannot be formed through fusion in any other stellar process.

<div class="alert alert-block alert-info">
<h4>Did You Know?</h4> The gold in your jewelry and the uranium in nuclear reactors were formed in the violent explosion of a supernova billions of years ago!
</div>

### The Legacy of a Supernova: Seeding the Cosmos

But the story doesn’t end with the creation of these heavy elements. The tremendous energy of the supernova not only creates these elements but also flings them out into space, scattering them across the galaxy. This interstellar debris, rich with newly minted heavy elements, mixes with the surrounding gas and dust, enriching the interstellar medium.

Over time, gravity will cause these enriched clouds of gas and dust to coalesce, eventually giving birth to new stars and planets. This means that the gold in your jewelry, the uranium in nuclear reactors, and the elements that make up the Earth and even your own body were all forged in the heart of a long-dead star, billions of years ago, and carried to our solar system by the remnants of a supernova.

<img src="https://raw.githubusercontent.com/teaghan/astronomy-12/main/Unit3/figures/SN1006.png" alt="SN1007" width="1000" style="display: block; margin-left: auto; margin-right: auto;">

<div class="alert alert-block alert-info">
<h4>Star Dust</h4> You are quite literally made of star stuff! The carbon in your cells, the oxygen you breathe, and the iron in your blood were all created in stars billions of years ago. Every element heavier than hydrogen was born in the nuclear furnaces of stars or during the explosive deaths of massive stars. So, when you look up at the night sky, remember that the universe that created the stars also created you.
</div>

### Case Study: Supernova 1987A

Supernova 1987A (SN 1987A) offers a rare look at the final stages of massive stars. Discovered on February 24, 1987, in the Large Magellanic Cloud, it was the closest supernova observed in nearly 400 years, just 160,000 light-years away. The progenitor star, Sanduleak -69° 202, was a blue supergiant about 20 times the mass of the Sun.

#### Stellar Evolution and the Supernova Event

Sanduleak -69° 202 lived most of its life fusing hydrogen into helium as a main-sequence star, with a luminosity 60,000 times that of the Sun. After exhausting its hydrogen, it expanded into a red supergiant and later reverted to a blue supergiant before collapsing in a type II supernova. The explosion was powerful enough to outshine its entire galaxy for weeks, ejecting the star’s outer layers and creating an expanding shell of gas and dust.

<img src="https://raw.githubusercontent.com/teaghan/astronomy-12/main/Unit3/figures/SN1987A.png" alt="SN1987A" width="1000" style="display: block; margin-left: auto; margin-right: auto;">


#### Observations and Legacy

Post-explosion, the Hubble Space Telescope revealed a glowing ring of gas around SN 1987A, likely expelled during the star's red supergiant phase. The collision of the supernova shockwave with this ring provided stunning visuals and valuable data. Additionally, the detection of neutrinos confirmed theoretical models of core-collapse supernovae, providing direct evidence of neutron star formation.

SN 1987A has greatly advanced our understanding of massive star life cycles, supernova mechanics, and nucleosynthesis—the formation of elements in stars. It remains a cornerstone in astrophysics, offering insights that continue to shape our knowledge of the cosmos.

<img src="https://raw.githubusercontent.com/teaghan/astronomy-12/main/Unit3/figures/SN1987A_Brightness.png" alt="SN1987A Brightness" width="1000" style="display: block; margin-left: auto; margin-right: auto;">

### Evolution of the Star That Exploded as SN 1987A

| **Phase**          | **Central Temperature (K)** | **Central Density (g/cm³)** | **Time Spent in This Phase**   |
|--------------------|-----------------------------|-----------------------------|--------------------------------|
| Hydrogen fusion    | $40 \times 10^6$            | 5                           | $8 \times 10^6$ years          |
| Helium fusion      | $190 \times 10^6$           | 970                         | $10^6$ years                   |
| Carbon fusion      | $870 \times 10^6$           | $170,000$                   | 2000 years                     |
| Neon fusion        | $1.6 \times 10^9$           | $3.0 \times 10^6$           | 6 months                       |
| Oxygen fusion      | $2.0 \times 10^9$           | $5.6 \times 10^6$           | 1 year                         |
| Silicon fusion     | $3.3 \times 10^9$           | $4.3 \times 10^7$           | Days                           |
| Core collapse      | $200 \times 10^9$           | $2 \times 10^{14}$          | Tenths of a second             |

---

## Chemical Abundances in Stars

### Evolution of Chemical Abundances and Star Populations

The chemical composition of stars has evolved significantly since the early universe, and this evolution is closely tied to the classification of stars into three distinct populations: Population I, II, and III stars. Each of these populations reflects a different stage in the history of the universe’s chemical enrichment.

#### **Population III Stars: The First Generation**
- **Characteristics:** Population III stars were the first stars to form in the universe, composed almost entirely of hydrogen and helium—the only elements produced during the Big Bang.
- **Significance:** These stars are believed to have been massive and short-lived, and while they have not been directly observed, their existence is inferred from theoretical models. Through their life cycles and eventual supernova explosions, Population III stars began the process of enriching the interstellar medium with heavier elements (metals).
- **Legacy:** The metals produced by Population III stars paved the way for the formation of subsequent generations of stars.

#### **Population II Stars: The Metal-Poor Stars**
- **Characteristics:** Formed from the material enriched by Population III stars, Population II stars contain small amounts of metals. These stars are typically found in the halo of our galaxy and in globular clusters, indicating their ancient origins.
- **Significance:** Population II stars represent an early stage of chemical enrichment, where the universe’s metallicity was still relatively low. Despite their low metal content, they played a crucial role in further enriching the interstellar medium.

#### **Population I Stars: The Metal-Rich Stars**
- **Characteristics:** Population I stars are the youngest and most metal-rich stars, including those found in the disk of our galaxy, such as our Sun. These stars formed from the heavily enriched material of previous generations of stars.
- **Significance:** The higher metallicity of Population I stars allows for the formation of more complex elements and compounds, contributing to the diversity of planetary systems and the potential for life.

### The Evolution of Metallicity Over Time
As time progressed, each generation of stars contributed to the increasing metallicity of the universe. Population III stars initiated the enrichment process, Population II stars continued it, and Population I stars now represent the most chemically evolved stars. This gradual increase in metal content has led to more complex chemical compositions in later star-forming regions, influencing everything from star formation processes to the potential for life in the universe.

Understanding the differences between these star populations not only helps astronomers trace the history of chemical enrichment in the universe but also provides insights into the conditions under which different generations of stars and planetary systems have formed.

<div class="alert alert-block alert-info">
<b>Reminder:</b> The chemical composition of stars changes over time, and these changes help astronomers understand the history and evolution of the universe.
</div>

### Check Your Understanding

1. Describe the significance of the triple-alpha process in the context of cosmic evolution. Why is this process crucial for the existence of life?
<br><br>

2. Differentiate between Population I, II, and III stars in terms of their metallicity and significance in the history of the universe.
<br><br>

3. Calculate the energy released when four protons (hydrogen nuclei) fuse to form a helium-4 nucleus in the proton-proton chain. The mass of a proton is $1.00728 \, \text{u}$, and the mass of a helium-4 nucleus is $4.00260 \, \text{u}$. Use $1 \, \text{u} = 1.66054 \times 10^{-27} \, \text{kg}$ and $c = 3 \times 10^8 \, \text{m/s}$.
<br><br>

4. Consider a massive star at the end of its life cycle, primarily composed of iron in its core. Discuss what will happen to this star and why iron represents the end of fusion processes in stars.
<br><br>

5. Arrange the following stars in order of their evolution:
   - A. A star with no nuclear reactions going on in the core, which is made primarily of carbon and oxygen.
   - B. A star of uniform composition from center to surface; it contains hydrogen but has no nuclear reactions going on in the core.
   - C. A star that is fusing hydrogen to form helium in its core.
   - D. A star that is fusing helium to carbon in the core and hydrogen to helium in a shell around the core.
   - E. A star that has no nuclear reactions going on in the core but is fusing hydrogen to form helium in a shell around the core.
<br><br>

6. Using the triple-alpha process, calculate the energy released when three helium-4 nuclei fuse to form one carbon-12 nucleus. The mass of a helium-4 nucleus is $4.00260 \, \text{u}$, and the mass of a carbon-12 nucleus is $12.000 \, \text{u}$.
<br><br>

7. The ring around SN 1987A (Figure 23.12) initially became illuminated when energetic photons from the supernova interacted with the material in the ring. The radius of the ring is approximately 0.75 light-year from the supernova location. How long after the supernova did the ring become illuminated?
<br><br>

8. A supernova can eject material at a velocity of 10,000 km/s. How long would it take a supernova remnant to expand to a radius of 1 AU? How long would it take to expand to a radius of 1 light-year? Assume that the expansion velocity remains constant and use the relationship: expansion time = distance / expansion velocity.
<br><br>

9. A supernova remnant was observed in 2007 to be expanding at a velocity of 14,000 km/s and had a radius of 6.5 light-years. Assuming a constant expansion velocity, in what year did this supernova occur?
<br><br>

## Resources

- **Astronomy (2016)**. Andrew Fraknoi, David Morrison, and Sidney C. Wolff.
- **Foundations of Astrophysics (2010)**. Barbara Ryden and Bradley M. Peterson.


# 3_5_life_cycle.md
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layout: embed_default
---

# 3.5 Life Cycle of Stars

Stars are the building blocks of galaxies, and understanding their life cycle provides crucial insights into the workings of the universe. From the brilliant birth of a star to its final breath as a stellar remnant, the evolution of stars is a complex and fascinating journey governed primarily by one key factor: **mass**. In this lesson, we will explore how the initial mass of a star dictates its life path, the stages stars go through, and the remnants they leave behind.

<div class="alert alert-block alert-success">
  <h4>Video</h4>
  <p>Watch <a href="https://www.youtube.com/watch?v=_EtlJCfaxdc" target="_blank">this video</a> for an introduction to stellar evolution and the Hertzsprung-Russell (H-R) Diagram. It provides visual aids and explanations that will help you understand the key concepts covered in this lesson. I recommend watching it at x1.5 speed!</p>
</div>

---

## Initial Mass and Its Impact on Stellar Evolution

The most important factor in determining a star’s evolution is its initial mass. The initial mass determines not only how long the star will live but also the different stages it will go through and its final fate. Stars are classified into three main categories based on their mass:

- **Low-mass stars** (up to 0.5 solar masses): These stars have long lifespans, typically tens of billions of years.
- **Intermediate-mass stars** (between 0.5 and 8 solar masses): These stars have moderate lifespans, ranging from hundreds of millions to a few billion years.
- **High-mass stars** (greater than 8 solar masses): These stars have short but dramatic lifespans, often only a few million years.

### Why Does Mass Matter?
The greater the mass of a star, the higher the core temperature and pressure. This means that high-mass stars can sustain more vigorous nuclear fusion, leading to shorter lifespans but more spectacular end-of-life events, such as supernovae. In contrast, low-mass stars burn their fuel more slowly, leading to longer lifespans and more peaceful ends as white dwarfs.

<div class="alert alert-block alert-warning">
<h4>Stellar Lifetimes</h4> A low-mass star like our Sun will live for approximately 10 billion years, slowly fusing hydrogen into helium. In contrast, a massive star 20 times the mass of the Sun might live for only 10 million years before exploding as a supernova.
</div>

---

## The Hertzsprung-Russell (H-R) Diagram as a Tool for Understanding Stellar Evolution

The **Hertzsprung-Russell (H-R) Diagram** is one of the most important tools in astronomy for understanding stellar evolution. The diagram plots stars according to their luminosity and surface temperature, revealing distinct patterns that correspond to different stages of stellar life.

### Key Regions of the H-R Diagram:
- **Main Sequence**: Where stars spend most of their lives, fusing hydrogen into helium. About 90% of stars fall on the main sequence, with their position determined by mass—more massive stars are hotter and more luminous, while low-mass stars are cooler and dimmer.
- **Red Giants and Supergiants**: Large, cool stars that have exhausted their hydrogen and moved on to fuse heavier elements. These stars occupy the upper-right region of the diagram, indicating high luminosity despite their relatively cool surface temperatures.
- **White Dwarfs**: Hot but dim remnants of low- and intermediate-mass stars, found in the lower-left region of the diagram. These stars are small but have very high surface temperatures.

<img src="https://raw.githubusercontent.com/teaghan/astronomy-12/main/Unit3/figures/supergiant.png" alt="Super Giant" width="1000" style="display: block; margin-left: auto; margin-right: auto;">

### Stellar Radii and the H-R Diagram:
The H-R Diagram also allows astronomers to infer stellar radii. If two stars have the same surface temperature but one is more luminous, the more luminous star must have a larger radius. This explains why supergiants can be cool but extremely luminous—they have enormous surface areas. Conversely, white dwarfs are hot but dim because of their small size.

<img src="https://raw.githubusercontent.com/teaghan/astronomy-12/main/Unit3/figures/hr_1.png" alt="HR Small" width="1000" style="display: block; margin-left: auto; margin-right: auto;">

### Distribution of Stars:
It's important to recognize that the distribution of stars on the H-R Diagram does not fully represent the true population of stars in the universe. While many stars we observe are on the main sequence, faint stars like red dwarfs are underrepresented due to their low brightness and distance from us. Understanding this helps astronomers make more accurate models of stellar populations.

<div class="alert alert-block alert-info">
<h4>The H-R Diagram</h4> The H-R Diagram is often compared to the periodic table in chemistry—just as the periodic table organizes elements by their properties, the H-R Diagram organizes stars by their temperature and luminosity. It also helps astronomers estimate stellar radii and understand the diverse evolutionary paths of stars.
</div>

<img src="https://raw.githubusercontent.com/teaghan/astronomy-12/main/Unit3/figures/hr_2.png" alt="HR Large" width="1000" style="display: block; margin-left: auto; margin-right: auto;">

---

## Main Sequence Evolution

Stars spend the majority of their lives on the **main sequence**, where they fuse hydrogen into helium in their cores. This is the most stable and longest phase of a star’s life. The position of a star on the main sequence depends on its mass: more massive stars are hotter and more luminous, placing them higher on the Hertzsprung-Russell (H-R) Diagram.

The main sequence phase ends when the star exhausts the hydrogen in its core. What happens next depends on the star’s mass.

### Characteristics of Main-Sequence Stars

| **Spectral Type** | **Mass (Sun = 1)** | **Luminosity (Sun = 1)** | **Temperature** | **Radius (Sun = 1)** |
|:-----------------:|:------------------:|:------------------------:|:---------------:|:--------------------:|
| O5                | 40                 | 7 × 10<sup>5</sup>        | 40,000 K        | 18                   |
| B0                | 16                 | 2.7 × 10<sup>5</sup>      | 28,000 K        | 7                    |
| A0                | 3.3                | 55                        | 10,000 K        | 2.5                  |
| F0                | 1.7                | 5                         | 7500 K          | 1.4                  |
| G0                | 1.1                | 1.4                       | 6000 K          | 1.1                  |
| K0                | 0.8                | 0.35                      | 5000 K          | 0.8                  |
| M0                | 0.4                | 0.05                      | 3500 K          | 0.6                  |

<div class="alert alert-block alert-info">
<h4>Tip:</h4> On the H-R Diagram, stars on the main sequence form a diagonal line from the top left (hot, luminous stars) to the bottom right (cool, dim stars). Our Sun is a middle-aged main sequence star.
</div>

---

## Advanced Stages of Stellar Evolution

Once a star exhausts the hydrogen in its core, it enters the advanced stages of evolution. For low- and intermediate-mass stars, this means burning helium and possibly heavier elements, while high-mass stars go through a series of rapid fusion stages.

### Helium and Heavier Element Fusion
- **Helium Burning**: After hydrogen is depleted, the core contracts and heats up, allowing helium to fuse into carbon and oxygen in a process called the **triple-alpha process**.
- **Carbon, Oxygen, and Silicon Burning**: In high-mass stars, the core continues to contract and heat up, enabling the fusion of heavier elements like carbon, oxygen, and silicon. These stages are much shorter and more energetic than hydrogen fusion.

Eventually, high-mass stars create iron in their cores, which cannot undergo fusion to release energy. This leads to a catastrophic core collapse, resulting in a supernova explosion.

<div class="alert alert-block alert-info">
<h4>High Mass Stars</h4> A high-mass star may burn carbon in its core for only a few thousand years, while it spends millions of years fusing hydrogen on the main sequence.
</div>

---

## Post-Main Sequence Evolution

When a star exhausts the hydrogen in its core, it leaves the main sequence—the longest and most stable phase of its life. What happens next depends on the star’s mass, and the journey can take dramatically different paths. The post-main sequence evolution is where the true diversity of stellar life stories begins to unfold.

- **Low-Mass Stars**: For a star like our Sun, after billions of years of stable hydrogen fusion, the core runs out of fuel, and the star can no longer hold itself together. The outer layers begin to expand and cool, turning the star into a **red giant**—an enormous, bloated version of its former self. During this phase, the star is shedding its outer layers in a stellar wind, leaving behind a hot, exposed core. Over time, the core cools and contracts, becoming a dense, Earth-sized object known as a **white dwarf**. The star's brilliance fades, but it will continue to glow faintly for billions of years as it slowly radiates away its remaining heat.

- **Intermediate-Mass Stars**: Stars with slightly more mass follow a similar path but with a bit more flair. These stars also become red giants, but as they expel their outer layers, they often create beautiful, glowing shells of gas known as **planetary nebulae**. These nebulae are lit up by the ultraviolet radiation from the hot core, creating some of the most stunning images in astronomy. Like their smaller cousins, these stars end up as white dwarfs, quietly cooling for the rest of eternity.

- **High-Mass Stars**: The fate of high-mass stars, however, is far more dramatic. These stellar giants evolve into **supergiants**, with their cores burning through heavier and heavier elements—helium, carbon, oxygen, and so on—until they reach iron. Iron is a dead end for fusion, and without the energy from fusion to support it, the core collapses under its own gravity in a fraction of a second. This collapse triggers a **supernova**, one of the most powerful explosions in the universe. For a brief moment, the star shines brighter than an entire galaxy. What remains after this cosmic cataclysm is a dense core—either a **neutron star** or, if the star was massive enough, a **black hole**.

---

## The Evolution of Stellar Remnants

The death of a star is not the end of its story. The remnants left behind after a star’s dramatic final act continue to exist, and each type of remnant tells a different tale of the star it once was.

1. **White Dwarfs**: These remnants of low- and intermediate-mass stars are among the densest objects in the universe. Imagine compressing the mass of the Sun into a sphere the size of Earth—this is a white dwarf. The gravity at the surface of a white dwarf is so intense that a teaspoon of its material would weigh several tons on Earth. Though no longer capable of nuclear fusion, white dwarfs continue to glow faintly as they slowly cool over billions of years. Eventually, after a time far longer than the current age of the universe, they will cool to become cold, dark **black dwarfs**—essentially stellar embers that no longer emit light.

<img src="https://raw.githubusercontent.com/teaghan/astronomy-12/main/Unit3/figures/white_dwarf.png" alt="White Dwarf" width="1000" style="display: block; margin-left: auto; margin-right: auto;">

2. **Neutron Stars**: When a high-mass star goes supernova, the core that remains can be crushed into an incredibly dense object called a neutron star. Imagine a star more massive than the Sun compressed into a sphere just 20 kilometers across. The densities involved are so extreme that a sugar-cube-sized amount of neutron star material would weigh as much as a mountain! Neutron stars also have some fascinating properties: many rotate extremely rapidly, and as they spin, they emit beams of radiation from their magnetic poles. When these beams sweep past Earth, we detect them as pulses of light, which is why these neutron stars are known as **pulsars**.

<div class="alert alert-block alert-warning">
<h4>Did you know?</h4> A neutron star has a density so extreme that a sugar-cube-sized amount of its material would weigh as much as a mountain.
</div>

3. **Black Holes**: For the most massive stars, even neutron stars are not dense enough to contain their cores after a supernova. The core collapses further, forming a **black hole**—a region of space where gravity is so strong that not even light can escape. These enigmatic objects warp space and time around them, and while we cannot see black holes directly, we can observe their effects on nearby matter. Gas and dust falling into a black hole are heated to extreme temperatures and emit X-rays, providing a glimpse of these mysterious cosmic objects. Black holes can continue to grow over time by swallowing up surrounding matter, merging with other black holes, or both, creating even more massive versions known as **supermassive black holes** found at the centers of galaxies.

<img src="https://raw.githubusercontent.com/teaghan/astronomy-12/main/Unit3/figures/blackhole.png" alt="Black Hole" width="1000" style="display: block; margin-left: auto; margin-right: auto;">

---


## Check Your Understanding

1. **Mass and Evolution**: How does the initial mass of a star determine its life cycle and final fate?
   <br><br>
2. **Fusion Processes**: Why do more massive stars undergo faster fusion processes, leading to shorter lifespans?
   <br><br>
3. **H-R Diagram**: How can the Hertzsprung-Russell Diagram be used to track the life cycle of a star?
   <br><br>
4. **Supernovae**: What conditions lead to a supernova explosion, and what determines whether the remnant will be a neutron star or a black hole?
   <br><br>
5. **Main Sequence Trends**: Describe how the mass, luminosity, surface temperature, and radius of main-sequence stars change in value going from the “bottom” to the “top” of the main sequence.
   <br><br>
6. **Spectral Data Analysis**: Review the spectral data for the five stars in the table below:

   | **Star** | **Spectrum**      |
   |:--------:|:-----------------:|
   | 1        | G, main sequence  |
   | 2        | K, giant          |
   | 3        | K, main sequence  |
   | 4        | O, main sequence  |
   | 5        | M, main sequence  |

   - Which is the hottest? Coolest? Most luminous? Least luminous? In each case, give your reasoning.
   <br><br>
7. **Type-M Star**: An astronomer discovers a type-M star with a large luminosity. How is this possible? What kind of star is it?
   <br><br>
8. **Betelgeuse Density**: Betelgeuse has a radius of 0.75 billion kilometers and a mass that is 15 times that of the Sun, how would its average density compare to that of the Sun? Use the definition of density, $ \text{density} = \frac{\text{mass}}{\text{volume}} $, where the volume is that of a sphere.
   <br><br>
9. **Main Sequence Evolution**: How do stars typically “move” through the main sequence band on an H–R diagram? Why?
   <br><br>

---

## Resources
- Astronomy (2016). Andrew Fraknoi, David Morrison, and Sidney C. Wolff.
- Foundations of Astrophysics (2010). Barbara Ryden, Bradley M. Peterson.


# 4_1_nature_of_light.md
---
layout: embed_default
---

# 4.1 The Nature of Light

Since ancient times, humans have looked up at the night sky, captivated by the light from distant stars and planets. Early civilizations used this light to navigate, mark the passage of time, and even develop myths about the heavens. But what exactly is light, and how does it carry information from these distant celestial objects to our eyes and telescopes?

Our journey to understand the nature of light began in the 17th century with thinkers like Isaac Newton and Christiaan Huygens. Newton believed light consisted of tiny particles, while Huygens proposed that it behaved like a wave. These ideas paved the way for later discoveries. In the 19th century, James Clerk Maxwell revolutionized our understanding by developing the theory of classical electromagnetism. He described light as a transverse wave made up of oscillating electric and magnetic fields—an invisible yet powerful force connecting us to the distant reaches of the universe.

<div class="alert alert-block alert-success">
  <h4>Video</h4>
  <p>Watch <a href="https://www.youtube.com/watch?v=pj_ya0e20vE" target="_blank">this video</a> for an excellent introduction to the concept of light. The video includes a discussion on how light behaves, providing a visual representation of the topics covered in this lesson.</p>
</div>

---

## Electromagnetic Waves

Imagine light as ripples in a pond, where each ripple represents a wave. The *amplitude* of a wave refers to its height, which determines the brightness of the light. The *wavelength* is the distance between successive peaks, and the *frequency* is the number of wave cycles that pass a given point per second. These properties define the behavior of light and help categorize different types of light, from radio waves to gamma rays.

Light waves are unique because, unlike sound or water waves, they do not require a medium like air or water to travel. This ability to travel through the vacuum of space is what allows light to carry information from distant stars and galaxies to Earth, making it crucial for astronomy.

<div class="alert alert-block alert-info">
<h4>Wave Properties</h4>
Light is an electromagnetic wave with key properties such as <i>amplitude</i>, <i>wavelength</i> (&#x3BB;), and <i>frequency</i> (&#x66;). The speed of light in a vacuum is approximately 3 &times; 10<sup>8</sup> meters per second (or 300,000 kilometers per second), the fastest known speed in the universe.
</div>

<img src="https://raw.githubusercontent.com/teaghan/astronomy-12/main/Unit4/figures/wave_properties.png" alt="Wave Properties" width="1000" style="display: block; margin-left: auto; margin-right: auto;">


### Relationship Between Wavelength and Frequency

A fundamental principle in wave physics is that the speed of a wave is the product of its wavelength and frequency. For light, this relationship is expressed by the equation:

> $$ c = \lambda f $$

where $c$ is the speed of light, $\lambda$ is the wavelength, and $f$ is the frequency. This equation shows that shorter wavelengths correspond to higher frequencies and vice versa.

<div class="alert alert-block alert-info">
<h4>Units of Frequency</h4>
<p>Frequency is a measure of how many wave cycles pass a given point in one second. The standard unit of frequency is the Hertz (Hz), named after the physicist Heinrich Hertz. One Hertz (1 Hz) corresponds to one cycle per second.</p>
<p>For example, a frequency of 1 Hz means that one wave cycle is completed every second. Higher frequencies correspond to more cycles per second:</p>
<ul>
  <li>1 kHz (kilohertz) = 1,000 cycles per second</li>
  <li>1 MHz (megahertz) = 1,000,000 cycles per second</li>
  <li>1 GHz (gigahertz) = 1,000,000,000 cycles per second</li>
</ul>
</div>


Consider the light from a distant star. Short wavelengths produce blue or violet light (high frequency), while longer wavelengths produce red light (low frequency). This relationship not only explains the colors we see but also helps astronomers identify and classify celestial objects based on the light they emit.

<div class="alert alert-block alert-warning">
<h4>Deriving the Wave Equation</h4>
</div>

> The equation for the relationship between the speed, wavelength, and frequency of a wave can be derived from basic principles of motion. The general formula for average speed is:
>
> $$ \text{average speed} = \frac{\text{distance}}{\text{time}} $$
> 
> For example, a car traveling at 100 km/h covers a distance of 100 km in 1 hour.
>
> Now, let’s apply this concept to an electromagnetic wave. The distance traveled by the wave during one cycle is equal to its wavelength, $\lambda$. The time it takes to complete one cycle is related to the frequency of the wave, $f$. Frequency is defined as the number of cycles per second, so the time for one cycle is $t = \frac{1}{f}$.
>
> For an electromagnetic wave traveling at the speed of light, $c$, the speed can be expressed as:
> 
> $$ c = \frac{\text{distance}}{\text{time}} = \frac{\lambda}{t}$$
>
> Substituting $t = \frac{1}{f}$ into the equation, we get:
>
> $$c = \lambda \times f$$
>
> This equation shows that the speed of light is the product of the wavelength and the frequency of the wave. In other words, light waves with shorter wavelengths must have higher frequencies to travel at the constant speed of light.

<div class="alert alert-block alert-warning">
<h4>Example: Using the Wave Equation</h4>
</div>

> **Problem**: Calculate the wavelength of visible light with a frequency of $5.66 \times 10^{14}$ Hz.
>
> **Solution**:
> $$\lambda = \frac{c}{f} = \frac{3.00 \times 10^8 \, \text{m/s}}{5.66 \times 10^{14} \, \text{Hz}} = 5.30 \times 10^{-7} \, \text{m} = 530 \, \text{nm}$$
>
>> This wavelength corresponds to yellow-green light in the visible spectrum, which is commonly seen in sunlight.

<div class="alert alert-block alert-info">
<h4>The colour of light</h4> Remember, the colour of light is determined by its wavelength. This principle is used in spectroscopy to determine what stars are made of by analyzing the light they emit.
</div>

---

### The Electromagnetic Spectrum

The electromagnetic spectrum represents the full range of electromagnetic radiation, spanning from the shortest, most energetic gamma rays to the longest, least energetic radio waves. While we’re most familiar with the narrow band of visible light, which ranges from 400 to 700 nanometers, this is only a tiny slice of the vast spectrum. The invisible radiation that surrounds us—infrared, ultraviolet, X-rays, and more—plays a crucial role in our understanding of the universe.

<div class="alert alert-block alert-info">
<h4>Electromagnetic Radiation</h4>
What makes the electromagnetic spectrum so fascinating is that all forms of electromagnetic radiation, no matter the wavelength or frequency, travel at the same speed: the speed of light (3 &times; 10<sup>8</sup> meters per second) in a vacuum. Yet, their vastly different wavelengths and frequencies allow them to interact with matter in diverse ways, revealing different aspects of the universe.
</div>

Each type of radiation reveals something different about the universe. For example, X-rays help us see into the hearts of galaxies where black holes reside, while infrared radiation allows us to peer into cool dust clouds where stars are born. At either end of the spectrum are Gamma and Radio waves:

- **Gamma rays** are the shortest, most energetic waves, often produced by nuclear reactions and cosmic phenomena such as supernovae and black holes. These high-energy waves can penetrate through dense matter, providing insight into the most violent processes in the cosmos.
- **Radio waves** are at the opposite end of the spectrum, with the longest wavelengths. Despite their low energy, radio waves are invaluable in astronomy, penetrating dust clouds that block visible light and allowing astronomers to study the hidden cores of galaxies and nebulae.

<div style="text-align: center;">
    <img src="https://raw.githubusercontent.com/teaghan/astronomy-12/main/Unit4/figures/em_spectrum.png" alt="EM Spectrum" width="1000">
</div>

To better understand the different types of electromagnetic radiation, the following table summarizes their wavelength ranges, the temperatures of objects that emit them, and their typical sources:

| **Type of Radiation** | **Wavelength Range (nm)** | **Radiated by Objects at This Temperature** | **Typical Sources** |
|-----------------------|---------------------------|---------------------------------------------|---------------------|
| **Gamma rays**         | Less than 0.01            | More than $10^8$ K                          | Produced in nuclear reactions; require very high-energy processes |
| **X-rays**             | 0.01–20                   | $10^6$–$10^8$ K                             | Gas in clusters of galaxies, supernova remnants, solar corona |
| **Ultraviolet**        | 20–400                    | $10^4$–$10^6$ K                             | Supernova remnants, very hot stars |
| **Visible**            | 400–700                   | $10^3$–$10^4$ K                             | Stars |
| **Infrared**           | $10^3$–$10^6$             | 10–$10^3$ K                                 | Cool clouds of dust and gas, planets, moons |
| **Microwave**          | $10^6$–$10^9$             | Less than 10 K                              | Active galaxies, pulsars, cosmic background radiation |
| **Radio**              | More than $10^9$          | Less than 10 K                              | Supernova remnants, pulsars, cold gas |

## Applications in Astronomy

The electromagnetic spectrum is more than just a collection of different types of radiation—it’s a window into the universe. Each portion of the spectrum allows astronomers to observe different aspects of the cosmos, revealing phenomena that are invisible to the human eye. By harnessing the full range of electromagnetic waves, astronomers can study everything from the birth of stars to the behavior of black holes.

- **Radio Waves**: With their long wavelengths, radio waves are capable of penetrating dense clouds of gas and dust that block visible light. This makes them invaluable for studying star-forming regions and the hidden cores of galaxies. Radio astronomy has led to groundbreaking discoveries, such as the cosmic microwave background radiation, the afterglow of the Big Bang. Radio telescopes like the Event Horizon Telescope have even achieved the remarkable feat of imaging the shadow of a black hole, a direct glimpse into one of the universe's most enigmatic objects.

- **Infrared Radiation**: Infrared light is emitted by cooler objects, such as brown dwarfs, planets, and clouds of gas and dust. This part of the spectrum is especially useful for observing regions of space that are obscured by dust in visible light, such as the nurseries where new stars are being born. Telescopes like the James Webb Space Telescope (JWST) are optimized for infrared observations and are expected to reveal previously hidden details of the early universe, as well as the atmospheres of exoplanets.

- **X-rays**: X-rays are emitted by some of the most extreme phenomena in the universe, including supernovae, neutron stars, and black holes. These high-energy waves allow astronomers to observe the violent and energetic processes that are invisible in other wavelengths. NASA’s Chandra X-ray Observatory, for instance, has captured images of supernova remnants, revealing shock waves and hot gas expanding into space. X-ray astronomy also plays a critical role in studying the environments around black holes, where matter is heated to millions of degrees before falling into the event horizon.

<div style="text-align: center;">
    <img src="https://raw.githubusercontent.com/teaghan/astronomy-12/main/Unit4/figures/xray_sky.png" alt="The Sky in X-Ray" width="1000">
</div>

---

## Photons and their Energy

Light is a fascinating phenomenon that exhibits both wave-like and particle-like properties, a duality that puzzled scientists for centuries. This concept, known as *wave-particle duality*, is central to our understanding of light and its behavior in the universe.

On the one hand, light behaves like a wave, capable of interference and diffraction. It travels through space in the form of electromagnetic waves, with properties such as wavelength and frequency. However, light also behaves like a particle, composed of discrete packets of energy called **photons**. Each photon carries a specific amount of energy, which is determined by the frequency of the light.

The relationship between the energy of a photon and its frequency is given by **Planck’s equation**:

> $$ E = hf $$

In this equation, $E$ represents the energy of the photon, $h$ is Planck’s constant ($6.626 \times 10^{-34}$ J·s), and $f$ is the frequency of the light. This equation reveals that higher-frequency light, such as ultraviolet or X-rays, has more energetic photons compared to lower-frequency light, such as infrared or radio waves.

Wave-particle duality plays a crucial role in many astronomical observations. For example, in the *photoelectric effect*, photons striking a material can release electrons from its surface. This effect, which supports the particle theory of light, is not only foundational in quantum physics but also has practical applications in technology, such as in solar cells that convert sunlight into electricity.

<div class="alert alert-block alert-warning">
<h4>Example: Calculating the Energy of a Photon</h4>
Calculate the energy of a photon of ultraviolet light with a frequency of 7.5 &times; 10<sup>14</sup> Hz.
</div>

> 1. **Identify the given values**: 
>     - Frequency, $f = 7.5 \times 10^{14}$ Hz
>     - Planck’s constant, $h = 6.626 \times 10^{-34}$ J·s
>
> 2. **Use Planck’s equation** to calculate the energy of the photon:
> $$E = hf$$
> Substituting the given values:
> $$E = (6.626 \times 10^{-34} \, \text{J}\cdot\text{s}) \times (7.5 \times 10^{14} \, \text{Hz})$$
> 3. **Calculate the result**:
> $$E = 4.97 \times 10^{-19} \, \text{J}$$
>
>> The energy of a photon of ultraviolet light with a frequency of $7.5 \times 10^{14}$ Hz is $4.97 \times 10^{-19}$ joules. This high-energy photon is capable of causing chemical reactions, such as those responsible for sunburn, and can be observed in astronomical phenomena such as the radiation from hot stars.

---

## Interaction of Light with Matter

Light interacts with matter in several important ways, each of which plays a crucial role in both everyday life and astronomical observations. When light encounters a surface or passes through a material, it can be reflected, refracted, or absorbed, depending on the properties of the material and the angle of incidence.

- **Reflection**: When light strikes a surface and bounces back, we call this reflection. A familiar example is light reflecting off a mirror, allowing us to see ourselves. Reflection also occurs in astronomical instruments, such as reflecting telescopes, which use mirrors to collect and focus light from distant stars and galaxies.

- **Refraction**: Refraction occurs when light passes from one medium into another, bending in the process. This bending happens because light changes speed as it moves from one material to another. A common example is the way a straw appears bent when partially submerged in water. Refraction is a fundamental principle in refracting telescopes, which use lenses to bend and focus light.

- **Absorption**: In absorption, light is taken in by matter and converted into other forms of energy, such as heat. This process is why dark surfaces, like asphalt, feel hot after being exposed to sunlight. In astronomy, the absorption of light by dust and gas in space can obscure celestial objects, requiring astronomers to use different wavelengths of light to see through these clouds.

Understanding these interactions is essential for designing telescopes and other astronomical instruments. Reflecting telescopes, for example, use large mirrors to collect faint light from distant objects, while refracting telescopes rely on lenses to bend light and focus it onto a detector. These principles allow astronomers to gather and analyze light from across the universe, providing insights into the nature of stars, galaxies, and other celestial phenomena.

<div class="alert alert-block alert-info">
<h4>Wavelength Absorption</h4>
Remember, different materials absorb different wavelengths of light. This is why some telescopes are specifically designed to observe certain types of radiation, like X-rays or infrared light. Since Earth's atmosphere absorbs much of this radiation, space-based observatories are required to capture these wavelengths and reveal what lies beyond the visible spectrum.
</div>

---

## Check Your Understanding

1. **Electromagnetic Spectrum**: What distinguishes one type of electromagnetic radiation from another? Describe the main categories (or bands) of the electromagnetic spectrum and provide an example of an astronomical object or phenomenon observed in each band.
<br><br>
2. **Light Interaction with Matter**: Explain why a glass prism disperses light into a spectrum of colors. How is this principle used in astronomical spectrometers to analyze the light from stars and galaxies?
<br><br>
3. **Observing in Different Wavelengths**: With what type of electromagnetic radiation would you observe:
   - **A.** A star with a temperature of 5800 K (like our Sun)?
   - **B.** A gas heated to a temperature of one million K, such as in the solar corona?
   - **C.** A cool planet or moon that radiates in the infrared?
<br><br>
4. **Radiation and Health**: Why is it dangerous to be exposed to X-rays but not (or at least much less) dangerous to be exposed to radio waves? How do astronomers safely observe X-rays from distant cosmic sources?
<br><br>
5. **Atmospheric Transparency**: Astronomers want to make maps of the sky showing sources of X-rays or gamma rays. Explain why these observations must be made from above Earth’s atmosphere. What type of telescopes are used for this purpose?
<br><br>
6. A radio telescope is tuned to observe a distant galaxy emitting radio waves at a frequency of 1420 MHz (this is the frequency of neutral hydrogen, a key tracer of galactic structure). What is the wavelength of this radiation?
<br><br>
7. You observe a star emitting red light with a wavelength of 656 nm (the H-alpha line, common in many stars). What is the frequency of this light?
<br><br>
8. A gamma-ray burst is detected with a photon frequency of $2.4 \times 10^{21}$ Hz. What is the energy of one photon from this gamma-ray burst?
<br><br>
9. A space telescope is observing an exoplanet in infrared light with a wavelength of 10 micrometers. What is the frequency of this radiation? How does this help astronomers study the temperature and composition of the exoplanet?
<br><br>
10. The Hubble Space Telescope captures an image of a supernova in ultraviolet light with a wavelength of 250 nm. Calculate the energy of a photon at this wavelength. Why is ultraviolet light particularly useful for studying hot, young stars?
<br><br>

---

## Resources

- Astronomy (2016). Andrew Fraknoi, David Morrison, and Sidney C. Wolff.


# 4_2_blackbody.md
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# 4.2 Blackbody Radiation

The color and intensity of a star’s light aren’t just beautiful to behold—they are cosmic clues, revealing the star’s temperature, energy output, and even its life cycle. This fascinating story of light is explained by **blackbody radiation**—a key concept in astrophysics that helps us decode the mysteries of the universe.

Blackbody radiation allows astronomers to measure the temperature of distant stars, understand their energy distribution, and classify them. In this lesson, we will uncover the secrets of this phenomenon and explore how the light from stars guides us to a deeper understanding of the cosmos.

<div class="alert alert-block alert-success">
  <h4>Video</h4>
  <p>Watch <a href="https://www.youtube.com/watch?v=XdXAdwb7loE" target="_blank">this video</a> for an explanation of blackbody radiation and Wien's Law.</p>
</div>

---

## The Blackbody Spectrum

At the heart of blackbody radiation is the concept of a **blackbody**—an idealized object that perfectly absorbs all the radiation that falls on it and re-emits it based purely on its temperature. This re-emitted radiation forms a characteristic curve known as the **blackbody spectrum**.

<img src="https://raw.githubusercontent.com/teaghan/astronomy-12/main/Unit4/figures/blackbody_spectrum.png" alt="Blackbody Spectrum" width="1000" style="display: block; margin-left: auto; margin-right: auto;">

The blackbody spectrum is a graph that shows how the intensity of radiation emitted by an object varies across different wavelengths. There are several key features of this graph that provide valuable insights into the nature of the object emitting the radiation:

1. **Wavelength (x-axis):** The horizontal axis represents the wavelength of the emitted radiation, typically measured in nanometers (nm) or micrometers (µm). As you move from left to right on the graph, the wavelength increases. Shorter wavelengths correspond to higher energy radiation, such as ultraviolet and visible light, while longer wavelengths correspond to lower energy radiation, such as infrared.

2. **Intensity (y-axis):** The vertical axis represents the intensity or amount of radiation emitted at each wavelength. The height of the curve at a particular wavelength shows how much energy is being radiated at that wavelength. The higher the peak, the more radiation is emitted at that wavelength.

3. **The Peak of the Curve:** The peak of the blackbody spectrum indicates the wavelength at which the object emits the most radiation. Hotter objects have their peaks at shorter wavelengths (bluer light), while cooler objects have their peaks at longer wavelengths (redder light). This relationship is governed by **Wien's Displacement Law**, which links temperature to the peak wavelength.

4. **The Shape of the Curve:** The shape of the blackbody curve rises steeply on the shorter wavelength side (left) and falls off more gradually on the longer wavelength side (right). This shape tells us how energy is distributed across different wavelengths, with hotter objects concentrating more energy in shorter wavelengths.

5. **Area Under the Curve:** The total area under the blackbody curve represents the **total energy emitted by the object** across all wavelengths. This area is directly related to the object's temperature and is described by the **Stefan-Boltzmann Law**. Hotter objects emit more total energy, which results in a larger area under the curve. This explains why hotter stars are more luminous than cooler ones.

> **Interactive Exploration:** Curious to see how temperature changes affect the blackbody spectrum? Use the simulator below to experiment with different temperatures and observe how the peak wavelength, total intensity, and area under the curve change. This hands-on experience will help you understand why hot objects glow blue while cooler ones appear red.

<div style="width: 100%; height: 0; padding-bottom: 75%; position: relative;">
    <iframe src="https://phet.colorado.edu/sims/html/blackbody-spectrum/latest/blackbody-spectrum_en.html" 
            style="position: absolute; width: 100%; height: 100%; top: 0; left: 0;" 
            allowfullscreen>
    </iframe>
</div>

---

## Wien’s Displacement Law

Have you ever noticed how some stars twinkle red in the night sky, while others gleam blue or white? This dazzling array of colors isn’t just for show—it’s a direct result of **Wien’s Displacement Law**, a powerful tool that helps astronomers determine the temperature of stars based on the color of their light.

**Wien’s Displacement Law** describes the relationship between the temperature of a blackbody and the wavelength at which it emits the most radiation. Essentially, it tells us that hotter objects radiate more at shorter wavelengths (toward the blue end of the spectrum), while cooler objects radiate more at longer wavelengths (toward the red end of the spectrum). Mathematically, this relationship is expressed as:

> $$ \lambda_{\text{max}} = \frac{b}{T} $$

where:
- $\lambda_{\text{max}}$ is the wavelength of maximum emission,
- $T$ is the temperature in Kelvin,
- $b$ is Wien’s constant, approximately $2.898 \times 10^{-3} \, \text{m/K}$.

What does this mean for stars? 

- **Cooler stars**, like red dwarfs, have their peak emission in the red part of the spectrum, which is why they glow with a reddish hue.
- **Hotter stars**, like blue giants, emit more light at shorter wavelengths, making them appear blue or white.

Wien’s Displacement Law provides a cosmic thermometer, letting us gauge the temperature of stars billions of kilometers away simply by observing their light.

<div class="alert alert-block alert-warning">
  <h4>Example</h4>
  <p>If a star has a surface temperature of 5800 K (similar to the Sun), the peak wavelength of its radiation can be calculated as:</p>
</div>

> $$ \lambda_{\text{max}} = \frac{2.898 \times 10^{-3} \, \text{m/K}}{5800 \, \text{K}} \approx 500 \, \text{nm} $$
>
> This wavelength is in the visible light range, explaining why the Sun appears white or yellowish to our eyes.

<div class="alert alert-block alert-info">
<h4>Stellar Age and Color</h4>
The color of a star can also tell us about its stage in the stellar life cycle. Hot, young stars tend to be blue or white, while older, cooler stars often appear red as they near the end of their life cycle. However, this relationship between color and age is complex and also depends on the star’s mass.
</div>

---

## Stefan-Boltzmann Law

Have you ever wondered why some stars shine brighter than others, even though they might be cooler? The secret lies in the **Stefan-Boltzmann Law**, a powerful relationship between the temperature of a star and the total energy it radiates.

The Stefan-Boltzmann Law tells us that the energy radiated per unit surface area by a blackbody increases dramatically with temperature—specifically, with the **fourth power** of the temperature! This means that even a small increase in a star’s temperature results in a massive increase in the energy it emits. The law is expressed mathematically as:

> $$ F = \sigma T^4 $$

where:
- $F$ is the energy radiated per unit area,
- $T$ is the temperature in Kelvin,
- $\sigma$ is the Stefan-Boltzmann constant, approximately $5.67 \times 10^{-8} \, \text{W/m}^2 \cdot \text{K}^4$.

What does this mean for stars? Consider this: if one star is twice as hot as another, it doesn’t just emit twice the energy—it emits **16 times** more energy per unit area! This is because the energy output scales with $T^4$, meaning that small differences in temperature can lead to enormous differences in brightness.

But temperature isn't the only factor at play. The **luminosity** of a star, which is the total energy it emits, also depends on the star’s surface area. Large stars like red giants, even though they might be cooler, can shine just as brightly as smaller, hotter stars because they have much more surface area over which to radiate their energy.

### Calculating the Power of a Star

While energy flux tells us how much power a star emits per square meter, we would often like to know how much total power is emitted by the star. We can determine that by multiplying the energy flux by the number of square meters on the surface of the star. Since stars are mostly spherical, we can use the formula $4\pi R^2$ for the surface area, where $R$ is the radius of the star. 

The total power emitted by the star (which we call the star’s “absolute luminosity”) can be found by multiplying the formula for energy flux by the formula for the surface area:

> $$ L = 4\pi R^2 \sigma T^4 $$

<div class="alert alert-block alert-warning">
  <h4>Example</h4>
  <p>Two stars have the same size and are the same distance from us. Star A has a surface temperature of 6000 K, and Star B has a surface temperature twice as high, 12,000 K. How much more luminous is Star B compared to Star A?</p>
</div>

> Start by writing the formula for the luminosity of each

 star:
>
> $$ L_A = 4\pi R_A^2 \sigma T_A^4 $$  
> $$ L_B = 4\pi R_B^2 \sigma T_B^4 $$
>
> Now, take the ratio of the luminosity of Star B to Star A:
> $$ \frac{L_B}{L_A} = \frac{4\pi R_B^2 \sigma T_B^4}{4\pi R_A^2 \sigma T_A^4} = \frac{R_B^2 T_B^4}{R_A^2 T_A^4} $$
>
> Since the two stars are the same size, we can simplify by setting $R_A = R_B$:
>
> $$ \frac{L_B}{L_A} = \frac{T_B^4}{T_A^4} = \left(\frac{12,000 \, \text{K}}{6,000 \, \text{K}}\right)^4 = 2^4 = 16 $$
>
>> So Star B is 16 times more luminous than Star A!

The Stefan-Boltzmann Law gives us a deeper understanding of how stars of different sizes and temperatures contribute to the dazzling array of brightness we see in the night sky.

<div class="alert alert-block alert-info">
<h4>Size, Temperature, and Luminosity</h4>
The Stefan-Boltzmann Law explains why a cool red giant can appear just as luminous as a small, blazing-hot blue star. Size and temperature both play crucial roles in determining a star’s brightness.
</div>

---

## Color Indices and Stellar Classification

When you look up at the night sky and see stars shining in different colors, you are witnessing more than just a beautiful display. Those colors are cosmic clues, revealing the temperature of the stars and providing a key to their classification. Astronomers use these colors, expressed as **color indices**, to categorize stars and unlock insights into their properties and evolution.

The **color index** is a numerical value that compares a star’s brightness at different wavelengths of light. To do this, astronomers use filters that allow specific wavelengths to pass through, measuring the star’s brightness in those wavelengths. One commonly used set of filters measures stellar brightness in three key regions of the spectrum:
- **Ultraviolet (U)** at 360 nm,
- **Blue (B)** at 420 nm,
- **Visual (V)** at 540 nm, which corresponds to yellow light.

The difference in brightness between two of these filters is called a **color index**. The most commonly used index in astronomy is the **B–V index**, which is calculated by subtracting the star’s visual magnitude (V) from its blue magnitude (B). This index is directly related to the star’s temperature:
- **Hot stars** (like O-type stars) emit more light in the blue part of the spectrum, resulting in a **negative B–V index** and giving them a blue appearance.
- **Cool stars** (like M-type stars) emit more light in the red part of the spectrum, resulting in a **positive B–V index** and giving them a red appearance.

Remember the **Hertzsprung-Russell (HR) diagram** you studied earlier? It plots stars based on their luminosity and temperature, and the color index helps pinpoint where a star belongs on this diagram. The spectral classification system (O, B, A, F, G, K, M) is based on the star’s temperature, which correlates with its color:
- **O-type stars**: The hottest and most massive, glowing blue with surface temperatures exceeding 25,000 K. These stars sit at the upper left of the HR diagram, where high-temperature, high-luminosity stars are found (remember the temperature on the HR diagram is typically inverted).
- **M-type stars**: The coolest and smallest, glowing red with temperatures around 3000 K. These stars are located at the lower right of the HR diagram, representing low-temperature, low-luminosity stars.

<div class="alert alert-block alert-info">
<h4>Color Indices</h4>
Vega, a bright star in the summer sky, has a B–V index close to 0, indicating it is a hot, blue-white star with a surface temperature around 10,000 K. In contrast, Betelgeuse, a cool red giant, has a B–V index around +2, reflecting its much lower surface temperature of about 3000 K.
</div>

The **color index** is not just a tool for measuring temperature—it’s a vital part of understanding **stellar evolution**. By knowing a star's color and classification, astronomers can determine its life cycle stage, from the fusion-powered main sequence to its final phases as a white dwarf, neutron star, or black hole.

---

## Check Your Understanding

1. Why does the peak wavelength of radiation emitted by a star shift towards shorter wavelengths as the star’s temperature increases?
<br><br>
2. How can two stars of the same temperature have different luminosities?
<br><br>
3. A star has a surface temperature of $3000 \, \text{K}$. Calculate its peak emission wavelength using Wien’s Displacement Law.
<br><br>
4. If the emitted radiation from a red dwarf star has a wavelength of maximum power at 1200 nm, what is the temperature of this star, assuming it is a blackbody?
<br><br>
5. What is the temperature of a star whose maximum light is emitted at a wavelength of 290 nm?
<br><br>
6. If a star emits $1.42 \times 10^6 \, \text{W/m}^2$ of energy, what is its surface temperature?
<br><br>
7. Two stars with identical diameters are the same distance away. One has a temperature of 8700 K and the other has a temperature of 2900 K. Which is brighter? How much brighter is it?
<br><br>
8. If the emitted infrared radiation from Pluto has a wavelength of maximum intensity at 75,000 nm, what is the temperature of Pluto assuming it follows Wien’s law?
<br><br>
9. Two stars have the same luminosity, but Star A has a temperature of 4000 K, while Star B has a temperature of 8000 K. Based on their temperatures, calculate the ratio of the radius of Star A to the radius of Star B (i.e. find $\frac{R_A}{R_B}$).
<br><br>

---

## Resources

- Astronomy (2016). Andrew Fraknoi, David Morrison, and Sidney C. Wolff.


# 4_3_atomic_energies.md
---
layout: embed_default
---

# 4.3 Atomic Energies

Understanding atomic energies is crucial in astronomy because it allows us to decipher the light from stars and other celestial bodies. By analyzing the energies associated with atomic transitions, astronomers can determine the composition, temperature, and other properties of stars.

<div class="alert alert-block alert-success">
  <h4>Video</h4>
  <p>Watch <a href="https://www.youtube.com/watch?v=31CW4axauxc" target="_blank">this video</a> for an explanation of atomic energy levels and emission spectra.</p>
</div>

---

## Recap: The Quantum Nature of Light

Remember from our previous lesson that light behaves as both a particle and a wave. This dual nature is fundamental to understanding how light interacts with matter. Photons are the particle aspect of light, and each photon carries a specific amount of energy, determined by its frequency:

> $$ E = h \cdot f $$

where:
- $E$ is the energy of the photon,
- $h$ is Planck’s constant ($6.63 \times 10^{-34}$ J·s),
- $f$ is the frequency of the photon.

Photons can transfer energy to electrons in atoms, causing the electrons to jump between energy levels. This is the foundation of atomic spectroscopy.

---

## Electron Volts

The energy of photons and electrons is often measured in electron volts (eV), a unit of energy that is more convenient for the small scales involved in atomic physics.

- **1 eV** is defined as the energy gained by an electron when it is accelerated through an electric potential difference of 1 volt:
  > $$ 1 \, \text{eV} = 1.6 \times 10^{-19} \, \text{J} $$

<div class="alert alert-block alert-warning">
  <h4>Example:</h4> Convert 5 eV to joules.
</div>

> Given:
> 
> > $ 1 \, \text{eV} = 1.6 \times 10^{-19} \, \text{J}$
> 
> Solution:
> 
> > $$ 5 \, \text{eV} = 5 \times 1.6 \times 10^{-19} \, \text{J} = 8 \times 10^{-19} \, \text{J} $$

---

## Bohr Model and Atomic Energy Levels

The Bohr model of the atom was a significant advancement in understanding atomic structure. Niels Bohr proposed that electrons orbit the nucleus in specific, quantized energy levels. Electrons can only occupy these discrete orbits and not the space in between.

- **Ground State**: The lowest energy level an electron can occupy.
- **Excited States**: Higher energy levels that electrons can jump to when they absorb energy.

These energy levels are quantized, meaning only certain values of energy are allowed. This quantization explains why atoms emit or absorb light at specific wavelengths.

<img src="https://raw.githubusercontent.com/teaghan/astronomy-12/main/Unit4/figures/bohr_model.png" alt="Bohr Model" width="1000" style="display: block; margin-left: auto; margin-right: auto;">

---

## Electron Transitions and Photons

Electrons within an atom are confined to specific energy levels, much like steps on a ladder. They can’t exist between these levels, so to move from one energy level to another, they must either gain or lose energy. This energy comes in the form of photons—packets of light that carry energy.

When an electron absorbs a photon, it gains enough energy to move to a higher energy level—this is known as **excitation**. Conversely, when the electron falls back to a lower energy level, it releases energy in the form of a photon—a process called **de-excitation**.

The energy of the photon absorbed or emitted corresponds precisely to the difference between the two energy levels:

> $$ \Delta E = h \cdot f = \frac{hc}{\lambda} $$

Where:
- $\Delta E$ is the energy difference between the levels,
- $h$ is Planck’s constant ($6.63 \times 10^{-34}$ J·s),
- $f$ is the frequency of the photon,
- $\lambda$ is the wavelength of the photon,
- $c$ is the speed of light ($3 \times 10^8 \, \text{m/s}$).

This relationship is crucial in atomic spectroscopy, the study of how atoms absorb and emit light.

---

### Brief Introduction to Spectra

We will be exploring spectra in more detail in the following lessons. For now, let's start with a simple concept: when white light passes through a prism, it splits into its component colors, creating a **continuous spectrum**. White light is made up of many different colors, each corresponding to a specific wavelength of light. As the light passes through the prism, it bends by different amounts depending on the wavelength, spreading out into a spectrum that ranges from red to violet.

This continuous spectrum is produced by sources of light that emit all wavelengths, such as heated objects like stars. In a continuous spectrum, all visible wavelengths are present, blending smoothly from one color to the next.

<img src="https://raw.githubusercontent.com/teaghan/astronomy-12/main/Unit4/figures/prism.png" alt="White light through a prism" width="1000" style="display: block; margin-left: auto; margin-right: auto;">

An example of a continuous spectrum is shown below. Notice how there are no gaps in the colors—every wavelength of visible light is present. This type of spectrum is typical of objects like the Sun and other stars, where light is emitted across a broad range of wavelengths.

<img src="https://raw.githubusercontent.com/teaghan/astronomy-12/main/Unit4/figures/continuous_spectrum.png" alt="A Continuous Spectrum" width="1000" style="display: block; margin-left: auto; margin-right: auto;">

---

### Emission and Absorption Spectra

When electrons move between energy levels, they absorb or emit photons, which gives rise to two other types of spectra:

- **Emission Spectrum**: When electrons drop from higher to lower energy levels, they emit photons. These photons appear as bright lines at specific wavelengths, corresponding to the energy difference between the levels. Each element has a unique set of emission lines, similar to a fingerprint.
  
- **Absorption Spectrum**: When electrons absorb photons and jump to higher energy levels, certain wavelengths of light are absorbed from a continuous spectrum. These absorbed wavelengths appear as dark lines in the spectrum, where light is missing. The pattern of these absorption lines is also unique to each element.

Understanding emission and absorption spectra is essential in astronomy, as they allow scientists to determine the composition and properties of stars and other celestial objects by analyzing the light they emit or absorb.

---

## The Hydrogen Atom Spectrum

Hydrogen, the simplest and most abundant atom in the universe, has been pivotal in shaping our understanding of atomic energy levels. With just one proton and one electron, hydrogen serves as an ideal model for exploring how electrons behave within atoms. The spectral lines produced by hydrogen offer a clear demonstration of how energy transitions within atoms correspond to specific wavelengths of light.

<img src="https://raw.githubusercontent.com/teaghan/astronomy-12/main/Unit4/figures/H_atom.png" alt="H atom" width="1000" style="display: block; margin-left: auto; margin-right: auto;">

### Hydrogen's Spectral Series and Energy-Level Diagrams

When the electron in a hydrogen atom moves between different energy levels, it either absorbs or emits light. These transitions can be visualized using an energy-level diagram, where each energy level is represented as a horizontal line, and transitions between these levels are depicted as arrows.

<img src="https://raw.githubusercontent.com/teaghan/astronomy-12/main/Unit4/figures/energy_levels.png" alt="H atom Energy Levels" width="1000" style="display: block; margin-left: auto; margin-right: auto;">

The resulting spectral lines from these transitions can be grouped into specific series, each corresponding to the electron's final energy level. These series cover different regions of the electromagnetic spectrum:

- **Lyman Series**: These transitions end at the ground state ($n = 1$). The photons emitted or absorbed in this series fall within the ultraviolet region, making them invisible to the naked eye, but crucial for studying stars and distant galaxies.
  
- **Balmer Series**: This series involves transitions to the second energy level ($n = 2$). The Balmer series is visible in the visible light spectrum and is perhaps the most well-known series, producing distinct colors such as the red H-alpha line and the blue-green H-beta line.

- **Paschen Series**: These transitions end at the third energy level ($n = 3$). The resulting spectral lines fall within the infrared region, invisible to the human eye but detectable using infrared telescopes.

## Energy Calculations for Atomic Transitions

To calculate the energy of photons associated with atomic transitions, we use the formula:

> $$ E = \frac{hc}{\lambda} $$

<div class="alert alert-block alert-warning">
  <h4>Example:</h4> A photon emitted during the transition from $n = 3$ to $n = 2$ in a hydrogen atom has an energy of 1.89 eV. Calculate the wavelength of light emitted from this transition.
</div>

> We can use the formula:
> 
> > $$ \lambda = \frac{hc}{E} $$
> 
> Where:
> - $h = 6.63 \times 10^{-34} \, \text{J}\cdot {s}$ (Planck's constant),
> - $c = 3 \times 10^8 \, \text{m/s}$ (speed of light),
> - $E = 3.02 \times 10^{-19} \, \text{J}$ (energy of the photon).
> 
> Substituting the values:
> 
> > $$ \lambda = \frac{6.63 \times 10^{-34} \, \text{J}\cdot {s} \times 3 \times 10^8 \, \text{m/s}}{3.02 \times 10^{-19} \, \text{J}} $$
> 
> > $$ \lambda \approx 6.59 \times 10^{-7} \, \text{m} = 659 \, \text{nm} $$
> 
> This wavelength corresponds to red light in the visible spectrum.

---

## Ionization Energies

Imagine trying to pull a magnet off your fridge. The stronger the magnet, the more effort it takes to pull it away. Similarly, in an atom, the closer and more tightly bound an electron is to the nucleus, the more energy is needed to remove it. This energy is called **ionization energy**—the energy required to strip an electron completely away from the atom.

Ionization energy plays a critical role in the life of stars. In the searing heat of a star, atoms are constantly bombarded with energy, and when the temperature is high enough, electrons are forcibly removed from atoms. This process of ionization reveals important clues about the physical conditions of celestial bodies.

<div class="alert alert-block alert-info">
<h4>Ionization Energy for Hydrogen</h4>
In hydrogen, the simplest atom, it takes exactly 13.6 eV to remove its single electron and ionize the atom completely. This might not seem like much energy, but in the context of a star, it tells us a lot about the environment in which this atom exists.
</div>

---

## Applications of Atomic Spectra in Astronomy

Atomic spectra are like a cosmic fingerprint, revealing hidden secrets about the universe. By examining the light from distant stars, astronomers can unlock a treasure trove of information about these celestial objects. Every element in a star emits or absorbs light at specific wavelengths, producing unique spectral lines that act like barcodes. These barcodes allow astronomers to decode the star's story.

- **Composition**: Just as you can identify a song by its melody, astronomers can identify elements in a star by its spectral lines. Each element produces a distinct set of lines, so by analyzing the star’s light, we can figure out what it's made of—whether it’s hydrogen, helium, or even heavier elements like carbon and iron.

- **Temperature**: The intensity of these spectral lines and the ionization states of the elements give us vital clues about the star's temperature. A hotter star might show more highly ionized atoms, indicating extreme heat in its atmosphere. Spectra help us measure this heat from light-years away!

- **Motion**: Stars are constantly in motion, and atomic spectra help us track that movement. Thanks to the Doppler effect, the positions of the spectral lines shift depending on whether the star is moving toward or away from us. This shift allows us to measure the speed and direction of stars as they journey through space.

Understanding atomic energies is at the heart of these astronomical analyses, helping us map out the composition, temperature, and motion of stars across the universe.

---

## Check Your Understanding

1. A photon has a wavelength of $500 \, \text{nm}$. Calculate the energy of this photon in electron volts (eV).
   <br><br>

2. Explain how emission lines and absorption lines are formed. In what sorts of cosmic objects would you expect to see each?
   <br><br>

3. Explain how electrons use light energy to move among energy levels within an atom.
   <br><br>

4. Which of the following transitions in a hydrogen atom would produce a photon with the shortest wavelength?
   1. $n = 3$ to $n = 2$
   2. $n = 4$ to $n = 2$
   3. $n = 5$ to $n = 2$
   4. $n = 6$ to $n = 2$
   <br><br>

5. An astronomer observes a star and detects several absorption lines in its spectrum. How can the astronomer use these lines to determine the star's composition? What information do these absorption lines provide?
   <br><br>

6. Calculate the wavelength of light emitted when an electron in a hydrogen atom transitions from $n = 4$ to $n = 2$. Use the given energy for this transition of $2.55 \, \text{eV}$.
   <br><br>

7. Explain why ionization energy is important for determining the temperature of a star. What does the ionization state of an element tell us about the conditions within the star?
   <br><br>

8. A photon has an energy of $2.5 \, \text{eV}$. Calculate its wavelength in nanometers.
   <br><br>

--- 

## Resources

- Astronomy (2016). Andrew Fraknoi, David Morrison, and Sidney C. Wolff.


# 4_4_spectroscopy_1.md
---
layout: embed_default
---

# 4.4 Spectroscopy I: Principles of Spectroscopy

In this lesson, we will explore the different types of spectra—continuous, emission, and absorption—and how they reveal vital information about celestial objects. Spectroscopy is one of the most powerful tools in astronomy, allowing us to decode the physical conditions, chemical composition, and motions of stars, planets, and galaxies. By connecting this to previous lessons on blackbody radiation, atomic structure, and energy transitions, we'll see how light interacts with matter to produce unique patterns of spectral lines.

<div class="alert alert-block alert-success">
  <h4>Video</h4>
  <p>Watch <a href="https://www.youtube.com/watch?v=_1mpHBAXh1c" target="_blank">this video</a> for a general discussion on the importance of spectroscopy in astronomy.</p>
</div>

---

## Types of Spectra

In previous lessons, we explored the nature of light, blackbody radiation, and atomic energy levels. These concepts help explain the three main types of spectra that light can produce: continuous, emission, and absorption. Understanding these spectra is key to learning how light interacts with matter in the universe.

<div class="alert alert-block alert-info">
<h4>What is a Spectrum?</h4>
A <strong>spectrum</strong> is the range of different colors or wavelengths of light that an object emits, absorbs, or reflects. When light is passed through a prism, it spreads out into a rainbow of colors. Each part of the rainbow corresponds to a different wavelength of light, and by analyzing how much light is present at different wavelengths, astronomers can learn a lot about the objects that produced the light, such as stars or galaxies.
</div>

<img src="https://raw.githubusercontent.com/teaghan/astronomy-12/main/Unit4/figures/prism.png" alt="Prism" width="1000" style="display: block; margin-left: auto; margin-right: auto;">

---

### Continuous Spectrum

A **continuous spectrum** is when there is some amount of light present at all possible wavelengths (or colors), without any gaps. This type of spectrum is produced by hot, dense objects, such as the core of a star or a filament in a light bulb. When you look at a continuous spectrum, you see a complete rainbow of colors, blending from one to the next without interruption.

For example, in **Lesson 4.2**, we learned about blackbody radiation. Any object that has a temperature emits thermal radiation. The spectrum of this radiation is continuous, meaning it contains all wavelengths of light, and the exact shape of the spectrum depends on the object’s temperature. Stars, for instance, emit continuous spectra because their hot, dense interiors act like blackbodies.

A continuous spectrum is produced by moving charged particles, like electrons, in a dense environment. For example, in stars, the high-density plasma in the core scatters photons (light particles), spreading the light across all wavelengths. This creates a continuous range of colors with no missing parts.

The **Stefan-Boltzmann Law** explains how much energy a star emits based on its temperature. This law tells us that hotter stars shine more brightly and emit more energy across all wavelengths:

> $$ F = \sigma T^4 $$

Where:
- $F$ is the total energy emitted per unit area of the star's surface (in watts per square meter),
- $\sigma$ is the Stefan-Boltzmann constant, which is $5.67 \times 10^{-8} \, \text{W/m}^2 \cdot \text{K}^4$,
- $T$ is the temperature of the star in kelvins (K).

The **Wien's Displacement Law** helps us understand why hotter objects look bluer and cooler objects look redder. It shows that the peak wavelength of the light emitted by an object shifts depending on its temperature. Hotter objects emit more light at shorter (bluer) wavelengths, while cooler objects emit more light at longer (redder) wavelengths:

> $$ \lambda_{\text{max}} = \frac{b}{T} $$

Where:
- $\lambda_{\text{max}}$ is the peak wavelength of the emitted radiation (in meters),
- $b$ is Wien's displacement constant, which is $2.898 \times 10^{-3} \, \text{m} \cdot \text{K}$,
- $T$ is the temperature of the object in kelvins (K).

This is why the color of a star can tell us a lot about its temperature!

<img src="https://raw.githubusercontent.com/teaghan/astronomy-12/main/Unit4/figures/continuous_spectrum.png" alt="Continuous Spectrum" width="1000" style="display: block; margin-left: auto; margin-right: auto;">

---

### Emission Spectrum

An **emission spectrum** shows bright lines at specific wavelengths against a dark background. This occurs when a hot, low-density gas emits light as its electrons transition between energy levels. Each element has unique energy levels, so the emitted light forms a specific set of bright lines, creating a "spectral fingerprint."

When an electron in an atom gains energy, it moves to a higher energy level. As it falls back to a lower level, it releases energy as a photon. The energy of this photon corresponds to the difference between the two levels:

> $$ \Delta E = h \cdot f $$

where:
- $\Delta E$ is the energy difference between the levels,
- $h$ is Planck’s constant,
- $f$ is the frequency of the emitted photon.

The wavelength of the photon is related to its frequency by:

> $$ c = \lambda \cdot f $$

where $c$ is the speed of light and $\lambda$ is the wavelength.

For example, in the **hydrogen atom**, when an electron falls from the $n = 3$ to the $n = 2$ level, it emits red light with a wavelength of 656 nm, known as the **H-alpha line**. This emission line helps astronomers identify hydrogen in stars and galaxies. Other transitions, such as from $n = 4$ to $n = 2$, produce different colors.

Emission spectra are commonly observed in **emission nebulae** like the Orion Nebula, where the hot gas emits bright lines at specific wavelengths, revealing the chemical composition of the gas.

<img src="https://raw.githubusercontent.com/teaghan/astronomy-12/main/Unit4/figures/element_spectra.png" alt="element_spectra" width="1000" style="display: block; margin-left: auto; margin-right: auto;">

<div class="alert alert-block alert-info">
<h4>What is an Angstrom?</h4>
In the image above, the wavelengths on the horizontal axis are given in Angstroms (Å). An <strong>Angstrom (Å)</strong> is a unit of length commonly used to measure the wavelengths of light and other electromagnetic radiation. One angstrom is equal to \(1 \times 10^{-10}\) meters, or 0.1 nanometers. The angstrom is particularly useful in spectroscopy because many spectral lines, such as those of hydrogen, fall within the range of hundreds to thousands of angstroms. For example, the famous H-alpha line in the hydrogen spectrum has a wavelength of 6563 Å.
</div>

---

### Absorption Spectrum

An **absorption spectrum** shows dark lines superimposed on a continuous spectrum. This happens when light from a hot, dense object (like the core of a star) passes through a cooler, less dense gas. The cooler gas absorbs light at specific wavelengths, leaving dark lines in the spectrum.

In **Lesson 4.3**, we learned that electrons in atoms can absorb photons, causing them to jump to higher energy levels. For an electron to move to a higher energy level, the energy of the absorbed photon must exactly match the energy difference between the two levels. This absorption removes light at a specific wavelength from the continuous spectrum, creating dark lines in the absorption spectrum. 

We can relate the energy difference between the two levels to the wavelength of the absorbed light using the following equation:

> $$ \Delta E = h \cdot f = \frac{hc}{\lambda} $$

Where:
- $\Delta E$ is the energy difference between the two energy levels, in joules (J),
- $h$ is Planck’s constant, which is $6.626 \times 10^{-34} \, \text{J}\cdot\text{s}$, a fundamental constant that relates energy and frequency,
- $f$ is the frequency of the absorbed photon, in hertz (Hz),
- $c$ is the speed of light, which is $3.00 \times 10^8 \, \text{m/s}$,
- $\lambda$ is the wavelength of the absorbed photon, in meters (m).

Thus, by observing which wavelengths are missing from the spectrum, we can determine the energy levels involved in the electron transitions and identify the elements responsible for the absorption.

Absorption spectra are most commonly seen in **stellar atmospheres**. As light from the star’s core passes through the cooler outer layers, atoms in the atmosphere absorb light at certain wavelengths, creating absorption lines. These lines reveal the star's chemical composition.

One famous example is the **Fraunhofer lines** in the Sun’s spectrum, where dark lines indicate elements like hydrogen and calcium absorbing light at specific wavelengths.

<img src="https://raw.githubusercontent.com/teaghan/astronomy-12/main/Unit4/figures/sun_spetrum.png" alt="Sun Spectrum" width="1000" style="display: block; margin-left: auto; margin-right: auto;">

Modern astronomy represents absorption spectra as **graphs of wavelength vs. intensity**. In an absorption spectrum, most wavelengths of light maintain high intensity, but sharp dips occur at specific wavelengths where atoms absorb light. These dips correspond to the dark lines seen in the absorption spectrum and indicate the wavelengths at which light has been absorbed by the cooler gas.

<img src="https://raw.githubusercontent.com/teaghan/astronomy-12/main/Unit4/figures/spectrum_graph.png" alt="absorp_spectrum" width="1000" style="display: block; margin-left: auto; margin-right: auto;">

At these wavelengths, the intensity of the light decreases, forming distinct absorption lines. The positions of these lines allow astronomers to identify the elements present in the intervening gas, as each element absorbs light at specific wavelengths, leaving its "fingerprint" in the form of these absorption features.

<img src="https://raw.githubusercontent.com/teaghan/astronomy-12/main/Unit4/figures/absorp_spectrum.png" alt="absorp_spectrum" width="1000" style="display: block; margin-left: auto; margin-right: auto;">

---

## Kirchhoff’s Laws of Spectroscopy

In 1859, Gustav Kirchhoff summarized the observations of the different types of spectra into three laws that describe how spectra are produced under different physical conditions. These laws form the foundation of modern spectroscopy in astronomy.

### Kirchhoff’s First Law:
A **hot, dense object** (like a blackbody) emits a continuous spectrum. This law explains the emission from stars and other dense objects that radiate across a broad range of wavelengths without any gaps.

### Kirchhoff’s Second Law:
A **hot, low-density gas** emits an emission-line spectrum. This occurs because the gas's atoms emit light at specific wavelengths as electrons transition between discrete energy levels.

### Kirchhoff’s Third Law:
When light from a **hot, dense object** passes through a **cooler gas**, an absorption-line spectrum is produced. The cooler gas absorbs light at specific wavelengths, corresponding to the energy needed for electrons to jump to higher levels.

These laws help astronomers determine the conditions under which light is produced and modified as it travels through space. By applying these laws, astronomers can infer the temperature, chemical composition, and density of stars and interstellar clouds.

<img src="https://raw.githubusercontent.com/teaghan/astronomy-12/main/Unit4/figures/cont_emis_abs_spectra.png" alt="Three Types of Spectra" width="1000" style="display: block; margin-left: auto; margin-right: auto;">

---

## Spectral Line Formation in Stellar Atmospheres

In stars, the light we observe is affected by the outer layers of the star’s atmosphere. The core of the star emits a continuous spectrum, but this light passes through the cooler gases of the star’s outer atmosphere. As the light travels through these gases, atoms in the gas absorb specific wavelengths, causing absorption lines to appear in the spectrum.

In **Lesson 4.3**, we saw that each element has a unique set of energy levels, meaning that each element will absorb light at distinct wavelengths. By analyzing the absorption lines in a star’s spectrum, astronomers can determine the star's chemical composition, surface temperature, and other physical characteristics.

<img src="https://raw.githubusercontent.com/teaghan/astronomy-12/main/Unit4/figures/absorption-spectrum.png" alt="absorption-spectrum" width="1000" style="display: block; margin-left: auto; margin-right: auto;">

---

## Line Broadening and Astrophysical Implications

Spectral lines are not infinitely narrow. Several physical processes cause spectral lines to broaden, providing further insights into the conditions within stars, galaxies, and other astronomical objects.

### Doppler Broadening
The **Doppler effect** broadens spectral lines when atoms are moving toward or away from the observer. If the gas is in motion, atoms moving toward the observer cause the wavelength to shorten (blueshift), while atoms moving away cause the wavelength to lengthen (redshift). The faster the gas is moving, the broader the spectral lines.

<img src="https://raw.githubusercontent.com/teaghan/astronomy-12/main/Unit4/figures/Broadening_Spectrum.png" alt="absorp_spectrum" width="1000" style="display: block; margin-left: auto; margin-right: auto;">

### Pressure Broadening
**Pressure broadening** occurs in high-pressure environments, where collisions between atoms affect the energy levels of the electrons. In stars, the higher the pressure (usually in dense regions like the star’s core), the broader the spectral lines.

---

## Check Your Understanding

1. **Types of Spectra**: What key difference between the conditions of the gas in an emission nebula and the gas in a star's atmosphere leads to the formation of emission lines in one and absorption lines in the other?
<br><br>

2. **Wavelength of Absorption**: An electron absorbs a photon with an energy difference of $1.94 \times 10^{-18}$ J to jump from the $n = 1$ to the $n = 3$ energy level in a hydrogen atom. Calculate the wavelength of the absorbed photon. Use Planck’s constant $h = 6.626 \times 10^{-34} \, \text{J}\cdot \text{s}$ and the speed of light $c = 3.00 \times 10^8 \, \text{m/s}$.
<br><br>

3. **Color and Temperature**: A star is observed to emit its peak radiation at a wavelength of 450 nm. Using Wien’s Displacement Law, calculate the surface temperature of the star. ($b = 2.898 \times 10^{-3} \, \text{m}\cdot\text{K}$).
<br><br>

4. **Absorption vs. Emission Spectra**: Why do stars usually produce absorption spectra while nebulae produce emission lines?
<br><br>

5. **Identifying Elements**: A star’s absorption spectrum shows dark lines at wavelengths of 589 nm, 656 nm, and 486 nm. Using the table below, identify which elements are present in the star’s atmosphere and explain how the spectral lines act as "fingerprints."

| **Wavelength (nm)** | **Element**  |
|---------------------|--------------|
| 486                 | Hydrogen (H) |
| 517                 | Magnesium (Mg) |
| 589                 | Sodium (Na)  |
| 656                 | Hydrogen (H) |
| 670                 | Lithium (Li) |
| 706                 | Helium (He)  |

---

## Resources

- **Astronomy (2016)**. Andrew Fraknoi, David Morrison, and Sidney C. Wolff.
- **[The Atomic Spectrum](https://cosmosatyourdoorstep.com/2017/10/21/the-atomic-spectrum/)**


# 4_5_spectroscopy_2.md
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---

# 4.5 Spectroscopy II: Applications of Spectroscopy

In this lesson, we will dive deeper into the practical applications of spectroscopy and see how astronomers use these principles to unravel the mysteries of the cosmos. From identifying elements in stars to measuring the velocities of distant galaxies, spectroscopy provides an incredible window into the universe.

<div class="alert alert-block alert-success">
  <h4>Video</h4>
  <p>Watch <a href="https://www.youtube.com/watch?v=-mQ41yA6LaA" target="_blank">this video</a> for a detailed explanation of the Doppler effect and its significance in astronomy.</p>
</div>

---

## Identifying Elements in Stars

### Spectral Lines as Cosmic Fingerprints

Every element in the universe has a unique set of spectral lines, much like fingerprints. These lines are created by the transitions of electrons between energy levels in atoms. When light from a star passes through its atmosphere, certain wavelengths are absorbed by the atoms in the atmosphere. This absorption produces dark lines in the star's spectrum, known as **absorption lines**. These lines allow us to determine the elements present in the star, as each element absorbs light at specific wavelengths.

<img src="https://raw.githubusercontent.com/teaghan/astronomy-12/main/Unit4/figures/cont_emiss_absorp.png" alt="Elemental Spectra" width="1000" style="display: block; margin-left: auto; margin-right: auto;">

### Element Identification

Astronomers use spectroscopy to identify the chemical composition of stars by comparing observed spectral lines to known atomic spectra. For example, the Balmer series of hydrogen, which produces lines in the visible part of the spectrum, is often seen in stellar spectra. The positions of these lines can be matched with laboratory data to confirm the presence of hydrogen.

Other elements like helium, calcium, sodium, and iron also have characteristic spectral lines. By analyzing these lines, astronomers can determine not only the presence of elements but also their relative abundance. This information is crucial in understanding the life cycles of stars, as stars of different ages and masses show varying compositions.

<img src="https://raw.githubusercontent.com/teaghan/astronomy-12/main/Unit4/figures/abs_emiss_spectra.png" alt="Elemental Spectra" width="1000" style="display: block; margin-left: auto; margin-right: auto;">

### Application in Stellar Spectra

For example, if a star's spectrum shows strong hydrogen lines, it suggests that hydrogen is abundant in the star’s atmosphere, which is typical for young stars. On the other hand, older stars or stars near the end of their life may show strong lines of heavier elements like carbon or iron, indicating that nuclear fusion has progressed to the production of heavier elements.

To give you a practical understanding, here are some common spectral lines astronomers look for:
- **Hydrogen (H-alpha line)**: Found at 656.3 nm, a prominent red line in the spectra of many stars.
- **Helium (He I and He II lines)**: Visible in the spectra of young, hot stars.
- **Calcium (Ca II H and K lines)**: Common in the spectra of cooler stars.
  
Each of these lines provides clues about the temperature, composition, and age of a star.

<img src="https://raw.githubusercontent.com/teaghan/astronomy-12/main/Unit4/figures/spectral_lines.png" alt="Elemental Lines" width="1000" style="display: block; margin-left: auto; margin-right: auto;">

---

## Measuring Stellar Velocities Using Doppler Shifts

### The Doppler Effect in Astronomy

The **Doppler effect** is a phenomenon where the wavelength of light changes depending on the relative motion between the source of the light and the observer. In astronomy, this is used to measure the radial velocity of stars and galaxies—whether they are moving toward or away from us.

<img src="https://raw.githubusercontent.com/teaghan/astronomy-12/main/Unit4/figures/doppler_galaxy.png" alt="doppler galaxy" width="1000" style="display: block; margin-left: auto; margin-right: auto;">

When a star or galaxy moves **toward** us, the wavelengths of light are compressed, causing the spectral lines to shift toward the **blue** end of the spectrum, known as **blueshift**. Conversely, when the object is moving **away** from us, the wavelengths are stretched, shifting the lines toward the **red** end of the spectrum, known as **redshift**.

<img src="https://raw.githubusercontent.com/teaghan/astronomy-12/main/Unit4/figures/doppler_shift.png" alt="doppler shift" width="1000" style="display: block; margin-left: auto; margin-right: auto;">

### Radial Velocity Measurements

By measuring the shift in the position of spectral lines, astronomers can calculate the velocity of a star or galaxy along the line of sight, known as the **radial velocity**. The larger the shift, the faster the object is moving. This technique is invaluable for studying the motion of stars within galaxies, the expansion of the universe, and the motion of galaxies relative to one another.

The Doppler formula used to calculate the radial velocity is:

> $$ v = \frac{\lambda - \lambda_0}{\lambda_0} \cdot c $$

Where:
- $\lambda$ is the observed wavelength,
- $\lambda_0$ is the rest wavelength (the original wavelength when the object is not moving relative to the observer),
- $v$ is the velocity of the object (positive for receding, negative for approaching),
- $c$ is the speed of light, $3.00 \times 10^8 \, \text{m/s}$.

If $v$ is positive, the object is moving **away** from us (redshift). If $v$ is negative, the object is moving **toward** us (blueshift).

<div class="alert alert-block alert-warning">
<h4>Example: Calculating the Radial Velocity of a Star</h4>
</div>

> **Problem**: A spectral line that is normally observed at 500 nm appears at 502 nm in the spectrum of a distant star. Calculate the star's radial velocity and determine whether it is moving toward or away from us.
>
> **Solution**:
> The observed wavelength $\lambda = 502 \, \text{nm}$ and the rest wavelength $\lambda_0 = 500 \, \text{nm}$. Using the Doppler formula:
>
> $$ v = \frac{\lambda - \lambda_0}{\lambda_0} \cdot c = \frac{502 \, \text{nm} - 500 \, \text{nm}}{500 \, \text{nm}} \cdot 3.00 \times 10^8 \, \text{m/s} $$
>
> $$ v = \frac{2}{500} \cdot 3.00 \times 10^8 \, \text{m/s} $$
>
> $$ v = 1.2 \times 10^6 \, \text{m/s} $$
>
> Since $v$ is positive, the star is moving **away** from us at a velocity of $1.2 \times 10^6 \, \text{m/s}$ (redshift).

In the image below, we can compare the spectrum of Arcturus to that of the Sun. Notice how the absorption lines in the spectrum of Arcturus appear shifted compared to those in the solar spectrum. This shift indicates that Arcturus is moving relative to the Earth. 

By measuring the displacement of these absorption lines, astronomers can determine Arcturus’s radial velocity. In this case, if the lines are shifted toward the red end of the spectrum, it means that Arcturus is moving away from us (redshift). If they were shifted toward the blue end, Arcturus would be moving toward us (blueshift). 

This comparison of the spectral lines provides a powerful tool for understanding the motion of stars and the dynamics of stellar systems.

<img src="https://raw.githubusercontent.com/teaghan/astronomy-12/main/Unit4/figures/arcturus_sun_spectra.png" alt="Arcturus and the Sun" width="1000" style="display: block; margin-left: auto; margin-right: auto;">

### Exoplanet Detection Using Radial Velocity

One of the most exciting applications of the Doppler effect is in the detection of **exoplanets**—planets that orbit stars other than the Sun. As a planet orbits a star, the gravitational pull of the planet causes the star to "wobble." This wobble causes tiny shifts in the star's spectral lines as the star moves toward and away from us. By measuring these shifts, astronomers can infer the presence of planets, even if they are too faint to be observed directly.

The radial velocity method has been one of the most successful techniques in detecting exoplanets, providing evidence of planets in orbit around stars far beyond our solar system.

<img src="https://raw.githubusercontent.com/teaghan/astronomy-12/main/Unit4/figures/doppler_planet.png" alt="Doppler Planet transit" width="1000" style="display: block; margin-left: auto; margin-right: auto;">

---

## Spectral Line Catalogs and Practical Spectroscopy

### Line Lists and Databases

Spectral lines are the foundation of spectroscopic analysis, but they are only useful if we can identify them correctly. For this, astronomers rely on **line lists** and **catalogs**, which are databases of spectral lines corresponding to different elements and ions. One well-known resource is the **NIST Atomic Spectra Database**, which provides detailed information about the wavelengths of spectral lines, their strengths, and the conditions under which they occur.

When astronomers observe a spectrum from a star or galaxy, they compare the observed spectral lines to these databases to identify the elements present. This process involves:
- Matching the wavelength of the observed lines to the database.
- Checking the intensity and pattern of the lines to confirm the element.
- Determining if any Doppler shift has occurred due to the star's motion.

### Real-World Analysis

In practical spectroscopy, astronomers use a variety of telescopes equipped with spectrometers to capture light from distant stars, galaxies, and nebulae. These instruments spread the light into its component wavelengths, producing a spectrum. The data is then processed using software that compares the observed spectral lines with known line lists to identify the elements and infer conditions such as temperature, density, and velocity.

Spectroscopy is performed at both ground-based observatories, like the **Keck Observatory** in Hawaii, and space-based telescopes, such as the **Hubble Space Telescope**. Space telescopes are particularly useful for observing in the ultraviolet and infrared parts of the spectrum, which are absorbed by Earth’s atmosphere.

<img src="https://raw.githubusercontent.com/teaghan/astronomy-12/main/Unit4/figures/galaxy_spectrum.png" alt="Galaxy Spectrum" width="1000" style="display: block; margin-left: auto; margin-right: auto;">

---

## Check Your Understanding

1. **Spectral Lines and Element Identification**: Describe how astronomers use spectral lines to identify the elements present in a star’s atmosphere. Why is each element associated with a unique set of spectral lines?
<br><br>

2. **Doppler Effect**: Explain the Doppler effect and how it helps astronomers measure the velocity of a star. How would you differentiate between a star moving toward us and one moving away?
<br><br>

3. **Radial Velocity Measurement**: A star’s spectrum shows a hydrogen line that is normally observed at 656.3 nm but appears at 657.1 nm. Calculate the radial velocity of the star. Is the star moving toward or away from us? (Use the speed of light, $c = 3.00 \times 10^8 \, \text{m/s}$)
<br><br>

4. **Exoplanet Detection**: How does the radial velocity method work in detecting exoplanets? Why does a star “wobble” due to an orbiting planet?
<br><br>

5. **Doppler Shift Formula**: A spectral line from a distant galaxy appears at 400.5 nm, while its rest wavelength is 399.0 nm. Calculate the galaxy's velocity relative to Earth using the Doppler shift formula. Is the galaxy moving toward or away from us?
<br><br>

6. **Velocity of Stars in a Galaxy**: How can measuring the radial velocities of stars within a galaxy help astronomers understand the structure and rotation of the galaxy? Provide an example of what might be observed.
<br><br>

7. **Spectral Line Catalogs**: Why are spectral line catalogs and databases essential for spectroscopic analysis? How do astronomers use these tools to match observed spectral lines with known elements?
<br><br>

8. **Wavelength and Frequency Relationship**: A star emits a spectral line with a frequency of $5.00 \times 10^{14} \, \text{Hz}$. Calculate the wavelength of this line in nanometers. (Use the speed of light, $c = 3.00 \times 10^8 \, \text{m/s}$)
<br><br>

9. **Doppler Effect and Expanding Universe**: How does the Doppler effect provide evidence for the expanding universe? Explain how observations of redshifted galaxies support the Big Bang theory. (This one might require some research).
<br><br>

10. **Stellar Composition Over Time**: Stars of different ages often show different spectral lines. Why do older stars exhibit strong lines of heavier elements like carbon and iron, while younger stars show more hydrogen and helium lines?
<br><br>

---

## Resources

- Astronomy (2016). Andrew Fraknoi, David Morrison, and Sidney C. Wolff.
- [NIST Atomic Spectra Database](https://www.nist.gov/pml/atomic-spectra-database)


# 5_1_early_universe.md
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# 5.1 The Big Bang and the Early Universe

https://www.youtube.com/watch?v=0L_MvPLspdg

Ignore the mentioning of the HEisenberg uncertainty principle!


# 5_2_cosmology.md
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# 6.2 Cosmology


# 5_3_types_of_galaxies.md
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# 6.3 Future of the Universe


# 5_4_black_holes.md
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# 6.4 The Expanding Universe


# 5_5_dark_matter_and_energy.md
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# 6.5 Dark Matter and Dark Energy



# Unit 1 Assignment
Astronomy Assignment: Celestial Explorers Name:	____________________________	Embark	on	a	journey	through	the	night	sky	using	the	Stellarium	tool	to	identify	celestial	objects,	record	their	characteristics,	and	explore	the	vast	distances	of	the	universe.	This	assignment	will	help	you	become	familiar	with	key	astronomical	concepts	while	enjoying	the	beauty	of	the	cosmos.	Resources •	Stellarium	Web	-	An	online	planetarium	for	observing	the	night	sky.	Part 1: Getting Started with Stellarium 1. Set	Your	Location:	Open	Stellarium	and	set	your	location	to	your	current	city	or	a	nearby	location.	This	will	help	you	see	the	night	sky	as	it	appears	from	your	specific	spot	on	Earth.	2. Familiarize	Yourself:	Spend	some	time	exploring	the	Stellarium	interface.	Learn	how	to	navigate	the	sky,	zoom	in	on	objects,	and	access	information	about	them.	Part 2: Identifying Celestial Objects Choose	a	clear	night	when	the	stars	are	visible,	and	find	an	open	space	to	get	the	widest	view	possible.	Use	the	Stellarium	tool	to	help	you	locate	visible	objects	in	the	sky.	The North Star (Polaris) 1. Find	Polaris:	Use	Stellarium	to	help	locate	the	North	Star,	Polaris.		2. Record	the	following	information	from	Stellarium:				-	Right	Ascension	(RA):	____________________________				-	Declination	(Dec):	____________________________				-	Distance	from	Earth	(light-years):	____________________________		3. Calculate	Light	Travel	Time:	Using	the	distance,	calculate	how	many	years	into	the	past	you	are	seeing	when	you	observe	Polaris.						Light	Travel	Time:	____________________________	Visible Planets 4. Identify	Visible	Planets:	Use	Stellarium	to	help	you	identify	at	least	two	planets	visible	in	the	night	sky.	5. Record	the	following	information:	a. Planet	1:						Name:	____________________________						RA:	____________________________						Dec:	____________________________						Distance	from	Earth	(astronomical	units):	____________________						Light	Travel	Time:	___________________________	b. Planet	2:						Name:	____________________________						RA:	____________________________						Dec:	____________________________						Distance	from	Earth	(astronomical	units):	____________________						Light	Travel	Time:	____________________________	 Bright Stars 6. Identify	Bright	Stars:	Use	Stellarium	to	help	you	identify	at	least	two	bright	stars	in	the	night	sky.	7. Record	the	following	information:	a. Star	1:						-	Name:	____________________________						-	RA:	____________________________						-	Dec:	____________________________						-	Distance	from	Earth	(light-years):	____________________________						-	Light	Travel	Time:	____________________________	b. Star	2:						-	Name:	____________________________						-	RA:	____________________________						-	Dec:	____________________________						-	Distance	from	Earth	(light-years):	____________________________						-	Light	Travel	Time:	____________________________						Constellations 8. Constellation	Exploration:	Choose	one	constellation	and	identify	its	main	stars.				-	Constellation	Name:	____________________________				-	Main	Stars:	List	their	names,	RA,	Dec,	distance,	and	light	travel	time.							Part 3: Cosmic Reflections 	1. Write	a	short	reflection	on	what	you	found	most	interesting	or	surprising	about	your	observations.	How	does	understanding	the	light	travel	time	change	your	perception	of	the	stars	and	planets	you	observed?						2. What	do	you	notice	about	the	distances	of	the	stars	in	your	chosen	constellation?	Are	these	stars	actually	close	to	each	other	in	space?						


# Unit 2 Assignment
Investigating Galaxy Data
Name:
Objective
The purpose of this assignment is to apply your knowledge of circular motion and gravity to analyze
galaxy data and determine if there are any discrepancies between our observations and expectations.
1What is a Model?
A model is a simplified representation of a system or phenomenon that helps us understand and predict
its behavior. In science, models are essential tools for testing hypotheses and validating theories against
observations.
Models are constructed based on known laws and principles and include key variables and relationships
that define the system being studied. For example, in astronomy, we use models to describe the motion of
planets, stars, and galaxies by applying Newton’s laws of motion and gravitation.
A good model should:
•Capture the essential features of the system.
•Make accurate predictions that can be tested against experimental or observational data.
•Be as simple as possible while still being accurate.
However, no model is perfect. Models often make assumptions and simplifications that may not hold
true in all situations. For instance, when modeling the orbits of planets, we might assume they are perfect
circles, even though they are slightly elliptical in reality. These simplifications are necessary to make the
problem manageable but can lead to discrepancies between the model’s predictions and actual observa-
tions.
In this assignment, we will create a model for the rotational velocity of stars in a galaxy and compare
it to observational data. By examining any discrepancies, we can explore potential reasons why our model
might not perfectly match reality, leading to new insights and improvements in our understanding.
12Gravity Review
Newton’s Second Law states that the force on a mass, m, is related to its acceleration, a, through the
equation:
F=ma (1)
Additionally, the gravitational force between two masses, m1andm2, is given by:
FG=Gm 1m2
r2(2)
where ris the distance between the two objects, and G= 6.6743 ×10−11m3·kg−1·s−2is the gravita-
tional constant.
For example, if we use m1=M⊙= 1.989×1030kg (the mass of the Sun) and place a planet with
the same mass as the Earth ( m2=ME= 5.972×1024kg) at various distances from the Sun, we can
see how the gravitational force between them decreases as the distance increases. In other words, we can
use Equation 2 to model the force between the Sun and a planet with the same mass as Earth for different
distances. This model is shown in Figure 1.
Sample calculation for r= 1.25×1011m:
FG=GM⊙ME
r2=(6.6743 ×10−11)(1.989×1030)(5.972×1024)
(1.25×1011)2N
FG= 5.074×1022N
2Figure 1: Modeling the force between the Sun and Earth.
3Circular Motion
When an object undergoes circular motion with a radius, r, and a constant speed, v, it is actually accel-
erating towards the center of the circle. This acceleration is given by:
a=v2
r(3)
We can relate this equation to Newton’s Second Law (Equation 1) to find the relationship between the
centripetal force and the orbital velocity. According to Newton’s Second Law:
F=ma and a=v2
r
Therefore, the centripetal force, Fc, is:
Fc=mv2
r(4)
Let’s apply this to the Earth-Sun system. The Earth is located approximately rE= 1.475×1011m from
the Sun. Based on the time it takes to complete one orbit (365.25 days), we can determine the Earth’s
3orbital speed, which is vmeas= 29,780m/s. Using this orbital speed, we can calculate the centripetal force
on Earth and compare it to the theoretical gravitational force using our model with r= 1.475×1011m.
Calculating the “measured” force between the Sun and Earth:
Fmeas=ME·v2
meas
rE=(5.972×1024)(29,780)2
(1.475×1011)N
Fmeas= 3.591×1022N
Figure 2: The force between the Sun and Earth.
The calculated force is quite close to the theoretical value!
To quantify this closeness, we use the percentage error formula:
%error =|Fmeas−Fmodel|
Fmodel×100% (5)
4Calculating the percentage error:
%error =|3.591×1022−3.644×1022|
3.644×1022×100%
%error = 1.45%
While this error is quite small, we can hypothesize why our model does not perfectly match the mea-
surement. For instance:
•The Earth’s orbit is not perfectly circular, so Fc=mv2
ris not an exact model.
•Large planets, like Jupiter, also have a substantial gravitational interaction with Earth, so FG=Gm 1m2
r2
is not exact either.
This example should give you some guidance on how to:
1.Create a model of a physical phenomenon.
2.Compare your measurements to that model.
3.Use this comparison to hypothesize what might be missing from your model.
54Galaxies
Similar to how the Earth orbits around the Sun, stars in a galaxy rotate around the center of the galaxy. A
rotation curve is a plot of the rotational (or orbital) velocity of stars in a galaxy as a function of their distance
from the center (radius).
Using photometric data of the luminous matter, we can create a rotation curve of a galaxy. This curve
is used to estimate the enclosed mass within a certain radius by equating the centripetal force to the gravi-
tational force. We assume that, like the Earth orbiting the Sun, stars have circular orbits. Therefore, we can
use the same relationship as before:
mv2
r=GmM enc(r)
r2(6)
where:
•v= rotational velocity
•G= gravitational constant
•m= mass of the star
•Menc(r)= enclosed mass as a function of radius (note: this is not the same as Menc×r)
•r= radius or distance from the center of the galaxy
Since stars are very massive objects and the distances we consider are also very large, we typically use
units of solar masses ( 1M⊙= 1.989×1030kg) and kiloparsecs ( 1kpc= 3.086×1019m). We can also
express the gravitational constant in these units, where G= 4.30×10−6kpc·km2/M⊙/s2.
Observations of galaxies show that the luminous matter is largely contained within the central “bulge.”
Therefore, at larger distances, if we assume that Menc=Mbulge, Equation 6 simplifies to:
mv2
r=GmM bulge
r2(7)
65Activity Part 1: Creating a Rotational Model
Your goal for this activity is to:
1.Use the relationship in Equation 7 to create a model for the rotational velocity of a star in a galaxy in
terms of the mass enclosed within that star’s orbit.
2.Compare this model to measured data from the provided data tables.
3.Discuss whether or not your model adequately represents the data.
5.1 Creating a model
a) Using Equation 7 and your algebra skills, develop an equation for vthat will serve as your model for
the rotation curve of your galaxy.
To do this, follow these steps:
1. Start with Equation 7:
mv2
r=GmM bulge
r2
2. Simplify the equation to solve for v. This will give you a model for the rotational velocity as a function
of radius rand mass Mbulge.
75.2 Galaxy data
NGC7814, also known as the “Little Sombrero,” is a spiral galaxy located approximately 40 million light-
years away in the constellation Pegasus. It is characterized by a prominent central bulge and a thin, flat
disk. The galaxy’s edge-on orientation provides a clear view of its structure, making it an excellent subject
for studying galactic dynamics and mass distribution.
Figure 3: NGC7814 - The Little Sombrero Galaxy
The table below provides some measurements obtained from NGC7814 of the rotational velocities ( v)
of stars at different distances from the center of the galaxy ( r).
Galaxy Name NGC7814
Mbulge 0.5912 ×1010M⊙
r (kpc) vmeas (km/s)
0.6300 265.0000
2.7900 233.0000
5.0300 228.0000
7.2600 227.0000
9.5000 223.0000
11.7400 216.0000
14.0100 214.0000
16.1800 214.0000
18.4400 214.0000
85.3 Plotting the data
For this activity, you can either use a graphing program of your choice or do the calculations and graphs
by hand.
a) Use the data table for NGC7814 to create a rotation curve plot (i.e., plot the measurements for v
against r) with appropriate axis labels and units.
b) On this same plot, include a curve that represents your model. To accomplish this, follow these steps:
1. Use your model equation to calculate the theoretical rotational velocity for various values of r.
2. Fill in the table below with your calculated velocities.
3. Plot these points on your graph and connect them with a smooth curve.
Show at least one sample calculation.
r (kpc) v (km/s)
5.3.1 Sample Calculations
95.3.2 Results
a) Calculate the percentage error between the measurements and your model for each star, then cal-
culate the average percentage error. Show at least one sample calculation.
Sample percentage error calculation:
%error =|vmeas −vmodel |
vmodel×100%
10b) Write down what you observe about the agreement between the measurements and your model.
What can you say about the measured rotational velocities at large distances from the galaxy’s center?
c) Does your model appear to accurately represent the measurements? Use evidence from your com-
parisons to justify your answer.
d) What do you think might be missing from your model? (Hint: what assumptions did we make?)
116Enclosed Mass
Previously, we modeled the rotational velocity of a star orbiting its galaxy using the equation:
v=√
GM enc
r(8)
where:
•v= rotational velocity
•G= 4.30×10−6kpc·km2/M⊙/s2(the gravitational constant)
•Menc= enclosed mass as a function of radius
•r= radius or distance from the center of the galaxy
However, when we assumed that the enclosed mass was simply the mass of the bulge of the galaxy,
the model did not accurately represent the experimental data. This discrepancy leads us to explore how
we can more accurately estimate the enclosed mass.
In reality, multiple sources contribute to the mass of a galaxy. Some of the main sources include:
1. The central black hole
2. The bulge of the galaxy
3. The main stellar disk
4. The surrounding gas cloud
Our goal is to model the enclosed mass by including these contributions and compare this model to
the measured enclosed mass obtained from the radial velocity curve of the galaxy.
12Figure 4: The structure of a galaxy.
To begin, our model will include these four contributions as follows:
Menc(r) =MBH+Mbulge +Mdisk+Mgas(r) (9)
where:
•MBH= mass of the central black hole
•Mbulge = mass of the galaxy bulge
•Mdisk= mass of the main stellar disk
•Mgas(r)= mass of the gas cloud as a function of radius
The measurements we will be analyzing are from stars outside of the bulge and main stellar disk. There-
fore, Mbulge andMdiskcan be determined from their average spherical densities, ρbulge andρdisk, using
the formula:
Mass =Volume ×Density =4
3πr3ρ (10)
Equation 10 can also be used for Mgas, but note that this mass will continue to increase as we move
further from the center of the galaxy.
137Activity Part 2: Creating a Model
Your goal for this activity is to:
1.Use the relationship in Equation 9 to create a model for the enclosed mass within a given radius of
the galaxy.
2.Compare this model to measured data from the provided data tables.
3.Discuss any discrepancies between the model and the measured data.
Table 1: Components of the enclosed mass for NGC 7814.
Galaxy Name NGC7814
MBH 1.00×108M⊙
ρbulge 1.524×108M⊙/kpc3
rbulge 2.10kpc
ρdisk 5.592×107M⊙/kpc3
rdisk 6.05kpc
ρgas 7.329×104M⊙/kpc3
Table 2: Rotational curve data for NGC 7814.
r (kpc) v (km/s)
6.14 228
7.26 227
8.37 226
9.50 223
10.65 219
11.74 216
12.82 214
14.01 214
15.09 214
16.18 214
17.36 214
18.44 214
19.53 214
147.1 Creating and Plotting the Model
a) Expand Equation 9 to create a model forMencin terms of r,MBH,ρbulge,rbulge,ρdisk,rdisk, and
ρgas. You can follow the steps below:
1.Start with Equation 9:
Menc(r) =MBH+Mbulge +Mdisk+Mgas(r)
2.Use the provided values in Table 1 to express Mbulge,Mdisk, and Mgas(r)in terms of their densities
and radii. Remember that the mass of a spherical region is given by:
Mass =Volume ×Density =4
3πr3ρ
3.Substitute these expressions into Equation 9 to get a full model for Menc(r).
15b) Use your model and the data in Tables 1 and 2 to determine the enclosed mass within each star’s
orbit.
Show at least one sample calculation for the model value of the enclosed mass ( Menc(r)).
c) To obtain the measured enclosed mass from the rotational curve data, rearrange Equation 8 and use
the measured velocities in Table 2.
Show at least one sample calculation for the measured enclosed mass.
Table 3: Enclosed Mass Data for NGC7814
r (kpc) Menc(model, M⊙)Menc(measured, M⊙)
6.14
7.26
8.37
9.50
10.65
11.74
12.82
14.01
15.09
16.18
17.36
18.44
19.53
d) In the graph area on the following page (or with your graphing software), plot both the model and
measured enclosed mass vs. the radius. Include appropriate axis labels and legends.
7.1.1 Sample Calculations
167.1.2 Results
a) Atr= 19.53kpc, how much enclosed mass is missing from your model compared to the measured
value?
17b) Write down what you observe about the trend of the model. What about the measurements?
c) Does your model appear to accurately represent the measurements? Use evidence from your com-
parisons to justify your answer.
d) What do you think might be missing from your model? Be creative!!
188Self-Assessment
For each statement, circle the answer that best represents how you feel about today’s lesson.
a) After today’s lesson, I have an understanding of what a “model” is.
Strongly Agree | Agree | Disagree
b) After today’s lesson, my confidence in my ability to compare measurements to a model .
Improved | Did not improve
19


# Unit 3 Assignment
StellarEvolution
Name:
Stars are the fundamental building blocks of the universe, serving as the engines that fuel galaxies and
the laboratories where elements are forged. From the Sun, our closest star, to the distant giants that illumi-
nate the night sky, stars come in a wide variety of sizes, temperatures, and luminosities. In this assignment,
you will become a stellar detective, using the tools and concepts you’ve learned to unravel the mysteries
of a select group of stars.
KeyConceptsandFormulas
LuminosityandTemperature
The luminosity of a star is the total energy it emits per second. The Stefan-Boltzmann law relates a star’s
luminosity ( L) to its surface area and temperature:
L= 4πR2σT4
Where:
•Lis the luminosity,
•Ris the radius of the star,
•Tis the surface temperature,
•σis the Stefan-Boltzmann constant.
This equation shows that a star’s luminosity depends on both its size and temperature. Hotter and
larger stars emit more energy, making them more luminous.
TheInverseSquareLawandApparentBrightness
The apparent brightness or flux ( F) of a star as seen from Earth is related to its luminosity and distance
(r) from Earth:
F=L
4πr2
Distant stars appear dimmer than those that are closer, even if they have the same luminosity. By
knowing a star’s apparent brightness and distance, you can calculate its true luminosity.
1StellarEvolutionandtheHRDiagram
Stars evolve over time, following a life cycle that depends on their mass. The Hertzsprung-Russell (HR)
diagram plots stars according to their luminosity and temperature, revealing patterns that correspond to
different stages of stellar evolution. Most stars spend the majority of their lives on the mainsequence ,
fusing hydrogen into helium in their cores.
Mass-LuminosityRelationshipforMain-SequenceStars
The luminosity of main-sequence stars is approximately proportional to the mass raised to the power
of 3.5:
L
L⊙=(M
M⊙)3.5
This equation shows that more massive stars are more luminous but have shorter lifespans.
NucleosynthesisandEnergyProduction
Stars generate energy through nuclear fusion, where lighter elements, like hydrogen, are fused into
heavier elements, like helium, in the core. As stars evolve, they can fuse heavier elements, creating the
elements found throughout the universe.
2Part1:DataAnalysisandHRDiagramPlotting
Star # Temperature (K) Luminosity (L
L⊙)Radius (R
R⊙)Absolute
Magnitude ( Mv) Star Type Spectral Class
1 9030 45.0 2.63 1.450 2A
2 7720 7.92 1.34 2.440 2F
3 16500 0.013 0.014 11.890 1B
4 14245 231000 42 -6.120 3O
5 3575 123000 45 -6.780 3M
6 11900 0.00067 0.00898 11.380 1B
7 16390 1278 5.68 -3.320 2B
8 39000 204000 10.6 -4.700 2O
9 5300 0.59 0.91 5.490 2F
10 19400 10920 6.03 -3.080 2B
11 5936 1.357 1.106 4.460 2F
12 18290 0.0013 0.00934 12.780 1B
13 30000 28840 6.3 -4.200 2B
14 12010 0.00078 0.00920 12.130 1B
15 9373 424520 24 -5.990 3O
16 15680 0.00122 0.01140 11.920 1B
17 8250 9.25 1.93 -0.980 2F
18 25000 0.056 0.00840 10.580 1B
19 17383 342900 30 -6.090 3O
20 34190 198200 6.390 -4.570 2O
21 8924 0.00028 0.00879 14.870 1A
22 11250 672 6.98 -2.300 2A
23 10574 0.00014 0.00920 12.020 1F
24 29560 188000 6.02 -4.010 2B
25 4077 0.085 0.79500 6.228 2K
1.PlottheStarsontheHRDiagram
•Use the provided data to plot each star on an HR diagram. The x-axis represents temperature (loga-
rithmic scale), and the y-axis represents luminosity (logarithmic scale).
•Use the luminosity and temperature to determine the position of each star on the diagram.
32.IdentifyStarTypes
Each star is categorized as Type 1, 2, or 3. Use the HR diagram, the stars’ properties, and your knowledge
on stellar evolution to determine what these types represent.
3.ExplainStarClassification
After plotting the stars and identifying their types, write a brief explanation of how you determined the
classification for each star type. Reference the position of each star on the HR diagram and connect their
properties to stellar evolution.
4Part2:ConceptualQuestions
1.EvolutionaryPaths
Consider Star #11 from the table. Explain its likely evolutionary path. Justify your prediction based on
the star’s mass, luminosity, and evolutionary stage.
Hint: For main-sequence stars, luminosity is related to mass by the formulaL
L⊙=(
M
M⊙)3.5
.
2.EnergyProductionandFusion
Describe the energy production process (i.e., the specific fusion processes) occurring inside stars at
different stages of their life cycles.
3.FusionandLuminosity
How does the fusion process inside a star relate to its luminosity and temperature? Why do stars be-
come more luminous as they evolve off the main sequence?
54.ConnectingMass,Luminosity,andLifespan
Reflect on the relationship between a star’s mass, luminosity, and lifespan. Why do stars with higher
masses have shorter lifespans despite having more fuel?
Hint: Recall that the luminosity is the rate of energy output and that L∝M3.5.
Part3:CalculationQuestions
1.SurfaceFlux
Star 5 (M-type star) and Star 7 (B-type star) have very different temperatures and radii. Star 5 is much
cooler but significantly larger, while Star 7 is much hotter but smaller. Calculate the surface flux (energy
per square meter per second) for both stars using the Stefan-Boltzmann law: F=σT4, where Tis the
temperature and σ= 5.67×10−8W/m2/K4. Use the calculated flux and the given radii to explain how
each star achieves its luminosity. Does the difference in temperature fully account for the difference in
luminosity, or does the size of the star play a crucial role?
62.EstimatingtheLifetimeofStar7
In lesson 3.3, we found that the mass loss of one proton-proton (p-p) chain reaction is 0.02862 u, which
represents about 0.71% of the initial mass. Therefore, when 1 kilogram of hydrogen is converted into he-
lium, approximately 0.0071 kilograms of mass is converted into energy.
•Models of massive stars like Star 7 indicate that only about 10% of the total hydrogen in the star
will participate in nuclear reactions, as it is only the hydrogen in the central regions that is at a high
enough temperature.
•Use this information to estimate the lifetime of Star 7.
Given:
•Luminosity of Star 7: 1,278L⊙
•Luminosity of the Sun: L⊙= 3.8×1026watts
•Mass of the Sun: M⊙= 2×1030kg
(i)CalculatethemassofStar7 using the mass-luminosity relationship:
L
L⊙=(M
M⊙)3.5
Solve for Min terms of LandL⊙, and then calculate the mass of Star 7 in kilograms.
(ii)Determinethetotalenergyavailablefromhydrogenfusion in Star 7’s core. Assume that 10% of
Star 7’s mass participates in nuclear fusion. Calculate the total energy released from this mass of
hydrogen using the fact that 0.0071 kg of mass is converted into energy for every 1 kg of hydrogen.
Hint: E= ∆m·c2
(iii)EstimatethelifetimeofStar7 . Use the total energy available from fusion and the luminosity of Star
7 to estimate the lifetime in seconds, then convert this value to years.
Hint: Think about the units: luminosity is energy per unit time (1 watt = 1 Joule/s) and you want to get
units of time!
7


# Unit 4 Assignment
Unit 4 Assignment: Stellar Spectroscopy Analysis
Name:
Introduction
In this assignment, you will act as an astronomer analyzing real stellar spectra. This two-part activity will
guide you through determining the temperatures of stars using Wien’s Law and identifying the chemical
composition of stars by matching their absorption lines to known elements. You will use real astronomical
data, making your work similar to what professional astronomers do!
• InPart1 , you will calculate the temperatures of 10 stars from their spectra.
• InPart2 , you will compare the absorption lines of 5 stars to elemental lines to determine the stars’
chemical composition.
Part1:DeterminingStellarTemperaturesUsingWien’sLaw
Stars emit light across a range of wavelengths, and the peak wavelength of their emission can help us de-
termine their temperature. Using Wien’s Displacement Law, we can calculate a star’s temperature based
on the wavelength at which the spectrum peaks.
T=b
λmax
Where:
•Tis the temperature in Kelvin (K)
•b= 2.898×10−3m·K, which is Wien’s constant
•λmaxis the peak wavelength in meters
1ExampleCalculation
YourTask
You will find 10 stellar spectra labeled A–J in the Data section at the end of this document. Identify the peak
wavelength for each spectrum and use Wien’s Law to calculate the temperature of each star.
Hint: Pay close attention to the wavelength units!
StarPeakWavelength(nm) PeakWavelength(m) Temperature(K)
A
B
C
D
E
F
G
H
I
J
2AnalysisQuestionsPart1
1.IdentifyingTrends: After calculating the temperature of all 10 stars, what general trend do you ob-
serve between the temperature and the shape of the spectrum? (i.e. How do cooler stars differ from
hotter stars in terms of their emitted light?)
2.Wien’sLawApplication: Calculate the approximate wavelength of peak emission for a star with a
temperature of 7500 K. Which part of the spectrum (ultraviolet, visible, infrared) is the peak emission?
3.VisibleLight: Based on what you know about blackbody radiation and stellar spectra, why do you
think humans evolved to see light in the wavelength range of 380 to 700 nanometers?
4.TemperatureandLuminosity: Two stars have the same temperature, but Star B has a radius that is
four times larger than Star C. How would the luminosity (total energy emitted) of Star B compare to
Star C? Use your understanding of temperature, luminosity, and star radius to explain your answer.
3Part2:IdentifyingElementsinStarsUsingAbsorptionLines
Each element produces a unique set of spectral lines, acting like a “fingerprint” that helps us identify what
elements are present in a star. By comparing the absorption lines from stars to known element spectra, we
can determine the chemical composition of the stars.
ElementSpectra
In the Data section, you will find the emission spectra of seven elements commonly found in stars: Hydro-
gen, Helium, Neon, Sodium, Lithium, Magnesium, and Iron. These spectra will be used to identify elements
in the stars’ absorption spectra.
YourTask
In the Data section, you will also find the absorption line spectra of five stars labeled Star 1–5. Compare the
absorption lines of each star to the element spectra and identify which elements are present in each star.
Recordtheelementspresentineachstarbycheckingtheboxesinthetablebelow.
Element Star1Star2Star3Star4Star5
Hydrogen
Helium
Neon
Sodium
Lithium
Magnesium
Iron
AnalysisQuestionsPart2
1.SpectroscopyImportance: Why is spectroscopy an important tool for studying stars that are light-
years away?
2.CommonElements: Are there any elements present in all five stars? Why might some elements be
found in every star, while others are only found in some stars?
43.ComparingStars: Do any of the stars contain all seven elements? If so, what can you infer about
these stars compared to others that contain fewer elements?
4.ElementalFingerprints: Why is it important for astronomers to study and understand the emission
spectra of individual elements?
5.AbsorptionLineStrength: Consider two stars with similar temperatures, but Star A has stronger
absorption lines for sodium than Star B. What could explain the difference in the strength of these
lines? (The image below is included to remind you what absorption lines; a stronger absorption line
would reflect a deeper dip in the intensity.)
56.MetallicityandStarFormation: If a star has a high abundance of heavy elements (metals), what can
this tell you about the star’s generation? How does this information relate to the star’s formation
history?
6Data:StellarSpectraA–J
7891011Data:ElementEmissionLine
12Data:StellarAbsorptionLines
13