Three laws describing how planets orbit stars, derived empirically by Kepler (1609-1619) and later
explained by Newton's gravity.
1st Law
Orbits are ellipses with the star at one focus
2nd Law
Equal areas swept in equal times (faster near star)
3rd Law
T² ∝ a³ (period² proportional to distance³)
T² = (4π²/GM) × a³
For our Solar System: T²(yr) = a³(AU)Earth: a=1 AU → T=1
yr (by definition)Jupiter: a=5.2 AU → T=11.86 yr
Worked Example 1
Mars Orbital Period
Problem: Mars orbits at 1.524 AU from the Sun. Calculate its
orbital period.
Kepler's 3rd Law
T² = a³ = (1.524)³ = 3.540
T = √3.540 = 1.881 years = 687 days
Answer: 687 days (1.88 years). This matches observations
perfectly. Launch windows to Mars occur every ~26 months when Earth "catches up" to Mars.
2 Stellar Evolution - The Life Cycle of Stars
Stars are born in nebulae, spend most of their lives on the main sequence fusing hydrogen to helium, then
evolve off it when fuel runs out. Their fate depends almost entirely on their initial mass.
< 0.5 M☉
Red dwarf → white dwarf (trillions of years)
0.5–8 M☉
→ Red giant → planetary nebula → white dwarf
8–25 M☉
→ Supergiant → supernova → neutron star
> 25 M☉
→ Supergiant → supernova → black hole
Massive stars live fast, die
young: The Sun will live ~10 billion years. A star 10× more massive lives only ~20
million years - it burns fuel 10,000× faster despite having 10× more.
3 The Hertzsprung-Russell Diagram
The HR diagram plots stars by luminosity (brightness) vs. temperature (color). Most stars fall on the
"main sequence" - a diagonal band from hot/bright (upper-left) to cool/dim (lower-right). Giants and
white dwarfs occupy distinct regions.
L = 4πR²σT⁴ (Stefan-Boltzmann)
Luminosity depends on radius AND temperatureMain
sequence: L ∝ M^3.5 (mass-luminosity relation)Sun: L☉ = 3.85 × 10²⁶ W, T = 5,778
K
Worked Example 2
Stellar Luminosity - Sirius vs. Sun
Problem: Sirius A has T = 9,940 K and R = 1.71 R☉. The Sun has T
= 5,778 K. How many times more luminous is Sirius than the Sun?
Luminosity ratio from L ∝ R²T⁴
L_Sirius/L_Sun = (R_S/R_☉)² × (T_S/T_☉)⁴
= (1.71)² × (9940/5778)⁴
= 2.924 × (1.720)⁴ = 2.924 × 8.76
= 25.6× the Sun's luminosity
Answer: Sirius is ~25.6× more luminous than the Sun. Most of this
comes from its higher temperature (T⁴ contributes 8.76×) rather than its size (R² contributes only
2.9×). Temperature dominates luminosity.
4 Black Holes - When Gravity Wins
When a massive star's core collapses, if the remnant exceeds ~3 M☉ (Tolman-Oppenheimer-Volkoff limit), no
force can prevent total gravitational collapse. The result is a black hole - a region where gravity is
so strong that nothing, not even light, can escape.
Schwarzschild radius: r_s = 2GM/c²
Event horizon: the point of no returnSun → r_s = 3 km
(if compressed to a black hole)Earth → r_s = 8.9 mm
Worked Example 3
Schwarzschild Radius
Problem: Calculate the Schwarzschild radius for a 10 solar-mass
black hole. (M☉ = 1.989 × 10³⁰ kg, G = 6.674 × 10⁻¹¹)
r_s = 2GM/c²
r_s = 2(6.674×10⁻¹¹)(10 × 1.989×10³⁰) / (3×10⁸)²
= 2.654×10²¹ / 9×10¹⁶
r_s = 29,500 m ≈ 29.5 km
Answer: 29.5 km - a sphere roughly the size of a small city
containing 10 times the Sun's mass. Sagittarius A* (our galaxy's central black hole) has M ≈ 4 million
M☉ and r_s ≈ 12 million km (0.08 AU).
5 Cosmology - The Big Bang & Expanding Universe
The universe began 13.8 billion years ago from an extremely hot, dense state. Evidence: the cosmic
microwave background (CMB), Hubble's Law (galaxies are receding), and the abundance of light elements
(Big Bang nucleosynthesis).
Hubble's Law: v = H₀d
v = recession velocity, d = distanceH₀ ≈ 70 km/s/Mpc
(Hubble constant)Age ≈ 1/H₀ ≈ 14 billion years
6 Gravitational Waves
Ripples in spacetime caused by accelerating massive objects - predicted by Einstein in 1916, first
directly detected by LIGO on September 14, 2015. The source was two merging black holes (36 and 29 M☉)
1.3 billion light-years away.
Detection
LIGO uses laser interferometry over 4 km arms, detecting length changes of ~10⁻¹⁸ m - 1/1000th
the width of a proton. This is the most precise measurement ever made by humans.
Sources
Merging black holes, merging neutron stars, supernovae, and rotating asymmetric neutron stars
(pulsars). The 2017 neutron star merger was observed in both gravitational and electromagnetic
waves.
Worked Example 4
Hubble's Law - Galaxy Recession
Problem: A galaxy's spectrum is redshifted such that it appears
to be receding at 21,000 km/s. Using H₀ = 70 km/s/Mpc, how far away is it? How long ago did the light we
see leave that galaxy?
Hubble's Law
d = v/H₀ = 21,000/70 = 300 Mpc
= 300 × 3.26 million light-years
d ≈ 978 million light-years ≈ 1 billion ly
The light left ~1 billion years ago, when Earth was in the Proterozoic Eon and multicellular
life was just emerging.
Answer: ~1 billion light-years away. We're seeing this galaxy as
it was 1 billion years ago. Telescopes are literal time machines - the further we look in space, the
further back in time we see.