Kepler’s Laws

Note

Law — Kepler’s laws of orbital motion

1 (orbits): the orbit is an ellipse with Earth at one focus (a circle is the special case with Earth at the centre).
2 (areas): the radius vector sweeps equal areas in equal times, so the satellite moves faster at perigee and slower at apogee.
3 (periods): the square of the period is proportional to the cube of the semi-major axis:

$$T^2=\frac{4\pi^2}{\mu}a^3,\qquad T=2\pi\sqrt{\frac{r^3}{\mu}},\qquad v=\sqrt{\frac{\mu}{r}}\ \text{(circular)},\quad \mu=GM.$$

Kepler’s laws: elliptical orbit with Earth at a focus and equal areas swept in equal times (left); the linear $T^2$–$a^3$ relation (right).

Here $T$ is the orbital period, $a$ the semi-major axis, $r$ the (circular) orbital radius and $\mu=GM=3.986e14\,m³/s²$ is Earth’s gravitational parameter.

Note

Example — Geostationary altitude from Kepler’s third law

For a geostationary satellite the period is one sidereal day, $T=86164\,s$. The orbital radius is

$$a=\left(\frac{\mu T^2}{4\pi^2}\right)^{1/3}\approx42164\,km.$$

Subtracting the Earth radius $R_E\approx6378\,km$,

$$h=a-R_E\approx\boxed{35786\,km}.$$

[!NOTE]

Key Point — Why Kepler matters

The laws give the orbital period, fix the GEO altitude, predict satellite position for tracking, and explain the perigee/apogee speed variation used in Molniya-type HEO coverage.