### 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} = $\sqrt{GM/r}$
Period: T = 2πr/v_{0} = 2πr/$\sqrt{GM/r}$ = 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)
| r_{1}/r_{2} | $T_{1}$/$T_{2}$ |
|---|---|
| 1/4 | 1/8 |
| 1/2 | 1/(2√2) |
| 2 | 2√2 |
| 4 | 8 |
| 9 | 27 |
### 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).$
Part of ME-06 — Gravitation
Kepler's Third Law — Applications and Derivation
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