Past exam of the physics B course of the University of Cambridge / 2021 / Paper 1 / Section B / B6 / a

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Solution

For the two-body approximation with a dominant central mass, Kepler's laws of planetary motion are:

Kepler's first law: a bound planetary orbit is an ellipse with the central attracting body at one focus.
Kepler's second law: the line from that focus to the planet sweeps equal areas in equal times, equivalent to constant areal velocity.
Kepler's third law: the squared orbital period is proportional to the cube of the semi-major axis, $T^{2} = 4 π^{2} a^{3} / (GM)$ when the orbiting mass is negligible.

The first law includes a circular Kepler orbit as a special case. The third-law constant is the same for bodies orbiting the same dominant mass.

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