# Kepler's Laws of Planetary Motion

Written by Jerry Ratzlaff on . Posted in Classical Mechanics

## Kepler's first law

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

## 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.

### Kepler's third law formula

$$\large{ P^2 = a^2 }$$

Where:

$$\large{ P }$$ = planet's distance from the sum

$$\large{ a }$$ = semi-major axis of the planet's orbit

$$\large{ G }$$ = universal gravitational constant

$$\large{ t }$$ = satellite orbit period (time)

$$\large{ r }$$ = satellite mean orbital radius

$$\large{ m }$$ = planet's mass

Solve for:

$$\large{ t = \sqrt { \frac {4\; \pi^2 \;r^3} {G\;m} } }$$

$$\large{ r = \sqrt { \frac {t^2 \;G\; m} {4\; \pi^2} } }$$

$$\large{ m = \frac {4\; \pi^2\; r^3} {G\;t^2} }$$