Kepler’s Second Law

Kepler’s second law is most commonly stated as the sweeping motion of the satellite maps out equal area of the section for equal time intervals.

Proof - Using Specific Angular Momentum

Recall from the specific angular momentum has the following relationship:

\[ h = r^2 \frac{\mathrm d \nu}{\mathrm d t} \]

The differential area of the ellipse’s sector is given by

TODO: Proof

\[\begin{split} \mathrm d A = \frac{1}{2} r^2 \mathrm d \nu\\ d \nu = \frac{2}{r^2} \mathrm d A \end{split}\]

Joining the two equations,

\[ \mathrm dA = \frac{h}{2} \mathrm dt \]

Because \(h\) is constant with respect to time, any differential sector area is equal for the same differential time.