# Two Body Motion

by Phill Edwards

## Kepler's Laws

Kepler's laws of motion were derived from observation data. They accurately describe the motion of a planet around its sun assuming that the sun's mass is significantly larger than the planet's mass and that the effects of other planets and relativity are negligible. They quite accurately describe the orbit of the Earth around the Sun. They do not describe the orbit of the Moon around the Earth well as the Sun's gravity has a significant effect on the Moon's motion.

### Kepler's First Law

The orbit of every planet is an ellipse with the Sun at one of the two foci.

The planet's position is defined by the distance from the sun r and the angle θ measured anti-clockwise from the point of perihelion. The equation of the ellipse is:

$r=\frac{p}{1+e\mathrm{cos}\theta }$ (1)

Where e is the eccentricity of the ellipse and p is the length of the semi latus rectum.

### Kepler's Second Law

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

### Kepler's Third Law

The square of the orbital period of a planet is directly proportional to the cube of the semi-major axis of its orbit.

If the period is measured in years and the semi-major axis is measures in AU, then the constant of proportionality is unity for our solar system. Hence T2=a3.

The exact equation is:

$\frac{{T}^{2}}{{a}^{3}}=\frac{4{\pi }^{2}}{G\left(M+m\right)}$

### Newton's Derivation

Kepler's laws can be derived and improved upon using vector algebra and Newton's laws or motion and gravitation. In the equations bold letters are vectors and bold italic vectors are unit vector. The following discussion assumes a Newtonian intertial frame of reference and hence no relativistic effects will be considered.

#### Torque

Torque τ is the rotational analogue of force. Newton's second law states that the net force applied to an object is equal to its rate of change of linear momentum. A net torque applied to a body is equal to its rate of change of angular momentum.

A sun's gravity can only exert a force on a body in a radial direction. There can be no tangential component and therefore no torque can be applied. This means that the angular momentum of a body must be constant.

#### Kepler's Second Law

The rate of change of area for elliptical motion is:

$\frac{\mathrm{dA}}{\mathrm{dt}}=\frac{1}{2}|\mathbf{r}⨯\frac{d\mathbf{r}}{dt}|$

The angular momentum L is:

$L=m|\mathbf{r}⨯\frac{d\mathbf{r}}{dt}|$

Substitution gives:

$\frac{\mathrm{dA}}{\mathrm{dt}}=\frac{L}{\mathrm{2m}}$

As L and m are known to be constant, the rate of change of area must also be constant. This proves Kepler's second law.