# 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:

## 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 $\left\langle{r, \theta}\right\rangle$.

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

From Derivative of Angular Component under Central Force:

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

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 $A \left({t}\right)$ 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:

\(\displaystyle \dfrac {\mathrm d A} {\mathrm d \theta}\) | \(=\) | \(\displaystyle \dfrac {r^2} 2\) | |||||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle \int \mathrm d A\) | \(=\) | \(\displaystyle \dfrac 1 2 \int r^2 \, \mathrm d \theta\) | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \dfrac 1 2 \int r^2 \dfrac {\mathrm d \theta} {\mathrm d t}\, \mathrm d t\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \dfrac 1 2 \int h \mathrm d t\) | |||||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle \int_{t_1}^{t_2} \mathrm d A\) | \(=\) | \(\displaystyle \dfrac 1 2 \int_{t_1}^{t_2} h \mathrm d t\) | integrating between times $t_1$ and $t_2$ | |||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle A \left({t_2}\right) - A \left({t_1}\right)\) | \(=\) | \(\displaystyle \dfrac h 2 \left({t_2 - t_1}\right)\) |

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

$\blacksquare$

## Also see

## 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.

## Sources

- 1937: Eric Temple Bell:
*Men of Mathematics*... (previous) ... (next): Chapter $\text{VI}$: On the Seashore - 1972: George F. Simmons:
*Differential Equations*... (previous) ... (next): $\S 3.21$: Newton's Law of Gravitation: $(10)$ - 1992: George F. Simmons:
*Calculus Gems*... (previous) ... (next): Chapter $\text {A}.10$: Kepler ($1571$ – $1630$) - 1992: George F. Simmons:
*Calculus Gems*... (previous) ... (next): Chapter $\text {B}.25$: Kepler's Laws and Newton's Law of Gravitation - 2008: Ian Stewart:
*Taming the Infinite*... (previous) ... (next): Chapter $8$: The System of the World