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.