Kepler's Second Law

Kepler's second law states: A line joining a planet and the Sun sweeps out equal areas during equal intervals of time.

Learning Objective

Apply Kepler's second law to describe planetary motion

Key Points

In a small time the planet sweeps out a small triangle having base line and height. The area of this triangle is given by $dA=\frac{1}{2} \cdot r \cdot rd \theta$. and so the constant areal velocity is $\frac{dA}{dt}=\frac{1}{2}r^{2}\frac{d \theta}{dt}$.
The period $P$ satisfies $\pi a b=P \cdot \frac{1}{2}r^{2} \dot \theta$. One can see that the product of $r^2$ and must be constant, so that when the planet is further from the Sun it travels at a slower rate and vise versa.
A planet travels fastest at perihelion and slowest at aphelion.

Terms

mean motion
An angle of $2\pi$ (radians) divided by the orbital period (of a celestial body in an elliptic orbit).
angular velocity
A vector quantity describing an object in circular motion; its magnitude is equal to the speed of the particle and the direction is perpendicular to the plane of its circular motion.

Full Text

Kepler's second law states:

A line joining a planet and the Sun sweeps out equal areas during equal intervals of time .

Understanding Kepler's 3 Laws and Orbits

In this video you will be introduced to Kepler's 3 laws and see how they are relevant to orbiting objects.

In a small time the planet sweeps out a small triangle having base line and height. The area of this triangle is given by:

$dA=\frac{1}{2} \cdot r \cdot rd \theta$

and so the constant areal velocity is:

$\displaystyle \frac{dA}{dt}=\frac{1}{2}r^{2}\frac{d \theta}{dt}$

Now as the first law states that the planet follows an ellipse, the planet is at different distances from the Sun at different parts in its orbit. So the planet has to move faster when it is closer to the Sun so that it sweeps equal areas in equal times.

The total area enclosed by the elliptical orbit is:

$A=\pi a b$

Therefore the period $P$ satisfies:

$\displaystyle \pi a b=P \cdot \frac{1}{2}r^{2} \dot \theta$ or $r^{2} \dot \theta = nab$

Where $\dot \theta = \frac{d \theta}{dt}$ is the angular velocity, (using Newton notation for differentiation), and $n=\frac{2 \pi}{P}$ is the mean motion of the planet around the Sun.

See below for an illustration of this effect. The planet traverses the distance between A and B, C and D, and E and F in equal times. When the planet is close to the Sun it has a larger velocity, making the base of the triangle larger, but the height of the triangle smaller, than when the planet is far from the Sun. One can see that the planet will travel fastest at perihelion and slowest at aphelion.

Kepler's Second Law

The shaded regions have equal areas. It takes equal times for m to go from A to B, from C to D, and from E to F. The mass m moves fastest when it is closest to M. Kepler's second law was originally devised for planets orbiting the Sun, but it has broader validity.

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