Kepler's Laws of Planetary Motion/Second Law

Physical Law
Kepler's second law of planetary motion is one of the three physical laws of celestial mechanics deduced by :
 * The line joining the sun and a planet sweeps out equal areas in equal times.


 * KeplersSecondLaw.png

Proof
Consider a planet $p$ of mass $m$ moving around the sun in the plane under the influence of the force $\mathbf F$ imparted by the gravitational field which the two bodies give rise to.

Let the position of $p$ at time $t$ be given in polar coordinates as $\polar {r, \theta}$.

By definition of the gravitational field, $\mathbf F$ is a central force.

From Derivative of Angular Component under Central Force:


 * $r^2 \dfrac {\d \theta} {\d t} = h$

for some constant $h$.

, assume that $h > 0$, which means that $p$ is travelling in the direction of positive $\theta$.

Let $\map A t$ be the area swept out by $\mathbf r$ in time $t$ relative to some fixed point of reference.

For a small angle $\delta \theta$, the area $\delta A$ can be approximated to the area of a sector of a circle.

Thus:
 * $\delta A = \dfrac {r^2 \delta \theta} 2$

and so in the limit:

That is, given a time interval $t_2 - t_1$, the area $\map A {t_2} - \map A {t_1}$ is the same, whatever the physical position of $p$.

Also see

 * Kepler's First Law of Planetary Motion
 * Kepler's Third Law of Planetary Motion