Derivation for Circular Orbits

For a satellite in circular orbit at radius r:

Centripetal acceleration = gravitational acceleration:

v_{0}^{2}/r = GM/$r^{2}$

v_{0} = √(GM/r)

Period: T = 2πr/v_{0} = 2πr/√(GM/r) = 2π√($r^{3}$/GM)

Squaring: $T^{2}$ = 4π^{2}$r^{3}$/GM

This is Kepler's Third Law with the constant k = 4π^{2}/GM.

Applications

Finding orbital period from radius: T = 2π√($r^{3}$/GM) = (2π/R)√($r^{3}$/g) × (1/√R)

Using GM = $gR^{2}$: T = (2π/R√g) × r^(3/2)

For Earth: T = r^(3/2) / (R√g) × 2π

Finding radius from period: r = (GM$T^{2}$/4π^{2})^(1/3)

For geostationary: r = (gR^{2}$$T^{2}/4π^{2})^(1/3) with T = 86,400 s

Comparing two orbits: $T_{1}$/$T_{2}$ = (r_{1}/r_{2})^(3/2)

Generalisation

Kepler's Third Law applies to:

• Natural planets around any star (constant = 4π^{2}/GM_star)
• Artificial satellites around any planet (constant = 4π^{2}/GM_planet)
• Moons around planets

The constant changes for different central bodies (different M).