Key Formulas & Equations
Core Concepts
Stellar Evolution
Stars are born in nebulae (clouds of gas and dust). Gravity pulls material together, igniting hydrogen fusion in the core. Main sequence stars fuse H→He. Massive stars end as supernovae, leaving neutron stars or black holes. Low-mass stars become white dwarfs.
The Big Bang
The universe began ~13.8 billion years ago from an extremely hot, dense state. Evidence includes cosmic microwave background radiation, Hubble expansion, and abundance of light elements (H, He, Li). The universe continues to expand and cool.
Black Holes
Regions where gravity is so strong that nothing, not even light, can escape. Formed when massive stars collapse. The event horizon is the boundary of no return. Supermassive black holes (millions to billions of solar masses) reside at galaxy centers.
Dark Matter & Dark Energy
Dark matter (~27% of universe) does not emit light but exerts gravitational pull, inferred from galaxy rotation curves and gravitational lensing. Dark energy (~68%) drives the accelerating expansion of the universe. Only ~5% is normal (baryonic) matter.
Exoplanets
Planets orbiting stars other than the Sun. Detection methods: transit (planet passes in front of star, dimming light), radial velocity (star wobbles from gravitational pull), and direct imaging. Over 5,000 confirmed exoplanets as of 2024.
Spectroscopy
Analyzing the spectrum of light from celestial objects reveals their composition, temperature, velocity, and magnetic fields. Each element has a unique spectral fingerprint (emission/absorption lines). This is the primary tool for determining what stars and galaxies are made of.
Worked Examples
Example 1: Escape Velocity
Problem: Calculate the escape velocity from Earth. (M = 5.97×10²⁴ kg, r = 6.37×10⁶ m, G = 6.67×10⁻¹¹)
Step 1: v = √(2GM/r) = √(2 × 6.67×10⁻¹¹ × 5.97×10²⁴ / 6.37×10⁶)
Step 2: v = √(7.96×10¹⁴ / 6.37×10⁶) = √(1.25×10⁸)
Step 3: v ≈ 11,180 m/s ≈ 11.2 km/s
Answer: Earth's escape velocity is approximately 11.2 km/s.
Example 2: Parallax Distance
Problem: A star has a parallax of 0.5 arcseconds. How far away is it?
Step 1: d = 1/p = 1/0.5 = 2 parsecs
Step 2: Convert to light-years: 2 pc × 3.26 ly/pc = 6.52 ly
Answer: The star is 2 parsecs (6.52 light-years) away.
Example 3: Hubble's Law
Problem: A galaxy is 100 Mpc away. What is its recession velocity? (H₀ = 70 km/s/Mpc)
Step 1: v = H₀ × d
Step 2: v = 70 × 100 = 7,000 km/s
Answer: The galaxy is receding at 7,000 km/s.
Example 4: Stefan-Boltzmann
Problem: The Sun has T ≈ 5,778 K and R ≈ 6.96×10⁸ m. Find its luminosity. (σ = 5.67×10⁻⁸)
Step 1: L = 4πR²σT⁴
Step 2: L = 4π(6.96×10⁸)²(5.67×10⁻⁸)(5778)⁴
Step 3: L ≈ 3.83×10²⁶ W
Answer: Solar luminosity is approximately 3.83×10²⁶ Watts.
Practice Problems
Problem 1
What is the orbital period of a planet at 4 AU around a Sun-like star? (Use Kepler's Third Law)
Answer: T² = a³ → T² = 64 → T = 8 years
Problem 2
A galaxy recedes at 14,000 km/s. How far is it? (H₀ = 70 km/s/Mpc)
Answer: d = v/H₀ = 14,000/70 = 200 Mpc
Problem 3
What is the Schwarzschild radius of the Sun? (M = 2×10³⁰ kg)
Answer: r_s = 2GM/c² = 2(6.67×10⁻¹¹)(2×10³⁰)/(9×10¹⁶) ≈ 2,964 m ≈ 3 km
Problem 4
A star's apparent magnitude is 3, absolute magnitude is -1. What is its distance?
Answer: m - M = 5log₁₀(d/10) → 4 = 5log₁₀(d/10) → d = 10^(4/5+1) = 10^1.8 ≈ 63 pc
Problem 5
At what wavelength does a star with T = 3,000 K emit most strongly?
Answer: λ_max = b/T = 2.898×10⁻³/3000 ≈ 966 nm (infrared)
Problem 6
If a quasar's hydrogen line (656 nm) is observed at 1312 nm, what is z?
Answer: z = (1312-656)/656 = 1.0