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 Johannes Kepler:

The line joining the sun and a planet sweeps out equal areas in equal times. 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.

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

for some constant $h$.

Without loss of generality, 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:

 $\displaystyle \dfrac {\d A} {\d \theta}$ $=$ $\displaystyle \dfrac {r^2} 2$ $\displaystyle \leadsto \ \$ $\displaystyle \int \rd A$ $=$ $\displaystyle \dfrac 1 2 \int r^2 \rd \theta$ $\displaystyle$ $=$ $\displaystyle \dfrac 1 2 \int r^2 \dfrac {\d \theta} {\d t} \rd t$ $\displaystyle$ $=$ $\displaystyle \dfrac 1 2 \int h \rd t$ $\displaystyle \leadsto \ \$ $\displaystyle \int_{t_1}^{t_2} \rd A$ $=$ $\displaystyle \dfrac 1 2 \int_{t_1}^{t_2} h \rd t$ integrating between times $t_1$ and $t_2$ $\displaystyle \leadsto \ \$ $\displaystyle \map A {t_2} - \map A {t_1}$ $=$ $\displaystyle \dfrac h 2 \paren {t_2 - t_1}$

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 $p$.

$\blacksquare$

Source of Name

This entry was named for Johannes Kepler.

Historical Note

Kepler derived his three laws of planetary motion in the early $1600$s from a concentrated study over the course of $20$ years of the colossal wealth of observational data which had been made previously by Tycho Brahe of the behavior of the planets of the solar system, and in particular Mars.

The first two of these results he published in his gigantic work Astronomia Nova.

The third appears some ten years later in his Harmonices Mundi of $1619$.

It was Isaac Newton who managed to interpret these three laws and so work out what is now known as Newton's Law of Universal Gravitation, from which Kepler's laws can straightforwardly be derived.