Key ideas
Astronomy problems rely on a small set of scaling laws. Kepler’s third law links orbit size and period, parallax gives distance, Wien’s law gives temperature from color, and redshift gives recession speed.
- Kepler’s third law for bodies orbiting the Sun is T² = a³ with T in years and a in astronomical units.
- Parallax p in arcseconds gives the distance in parsecs, d = 1 ÷ p. One parsec is 3.26 light-years.
- Wien’s law gives the peak wavelength λmax = 2.898 × 10⁻³ m·K ÷ T. Luminosity scales as L ∝ R²T⁴.
- For small redshifts the recession speed is v = cz. Hubble’s law v = H₀d then gives distance, and light needs t = d ÷ c to cross it.
Key equations
T² = a³ (T in years, a in AU)d (pc) = 1 ÷ p (arcseconds)λmax = 2.898 × 10⁻³ m·K ÷ TL ∝ R² T⁴v = cz, v = H₀dv = √(GM ÷ r)
Worked example
A planet at a = 4 AU has T = √(4³) = 8 years. A star with p = 0.25″ is at d = 1 ÷ 0.25 = 4 pc = 4 × 3.26 = 13.0 light-years. A galaxy with z = 0.01 recedes at v = 3.00 × 10⁵ × 0.01 = 3000 km/s, which with H₀ = 70 km/s/Mpc is at d = 3000 ÷ 70 = 42.9 Mpc. A star with twice the Sun’s radius and twice its temperature is 2² × 2⁴ = 64 times as luminous.
Common mistakes
- Using kilometers or days in Kepler’s law without converting to AU and years.
- Multiplying by the parallax instead of dividing, which puts the star closer when it is farther.
- Forgetting that luminosity depends on the fourth power of temperature, not the first.
Practice
Test yourself with practice questions on this topic: Orbits and starlight