D.1.1—Kepler’s three laws of orbital motion

State the three laws

1. A planet follows an elliptical orbit with the star at one focus.
2. The line from the star to the planet sweeps out equal areas in equal time intervals.
3. For bodies orbiting the same central star, the square of the orbital period is proportional to the cube of the semi-major axis: $T^2\propto a^3$.

Interpret the geometry

The semi-major axis $a$ is half the longest diameter of the ellipse. In an elliptical orbit the planet is closer to the star at one focus-side end and farther away at the other. Equal swept areas mean the planet moves faster when it is closer to the star and slower when it is farther away.

Compare orbital systems

For two planets around the same star,
$$\frac{T_Y}{T_X}=\left(\frac{a_Y}{a_X}\right)^{3/2}$$
Use the semi-major axes, not automatically the instantaneous distance from the star. The third law comparison assumes the same central mass.

Common trap

The star is at a focus, not generally at the centre of the ellipse. Also, the second law refers to equal swept areas, not equal arc lengths or equal distances travelled.