Orbits and Satellites

# Overview

Planets don't just magically orbit the sun. Johannes Kepler, throughout his lifetime, worked out the properties of planetary motion. This section encompasses the movement and interaction of planets, satellites, and other objects around bodies in space.

# Kepler's Laws

## The Law of Orbits

All planets move in elliptical orbits, with the Sun at one focus.

This law applies to other situations not including the sun because the basic principle is the same. In his time, all planets were believed to have perfectly circular orbits. This view supported Copernicus' theory that all the planets revolved around the sun. As the orbits get more and more eccentric (elongated), the foci will get further apart.

## The Law of Areas

A line that connects a planet to the Sun sweeps out equal areas in the plane of the planet's orbit in equal times; that is, the rate $\frac{dA}{dt}$ at which it sweeps out area $A$ is constant.

This just states that as the closer to the sun the body gets, the faster it moves. Inversely, the farther away the body is from the sun,the slower it moves. Kepler's second law applies to any orbit. This is called the law of areas because as the planet moves, the triangle it traverses over time has the same area. This picture, taken from this source, shows that as the satellite gets closer to the sun and speeds up, the area of the triangle is still the same as when it was slow.

## The Law of Periods

The square of the period of any planet is proportional to the cube of the semimajor axis of its orbit.

# Equations

## Escape Speed

$K + U = \frac{1}{2}mv^{2} + ( -\frac{GMm}{R}) = 0$

So solving for $v$

$v = \sqrt \frac {2GM}{R}$

Where
$G = 6.67 \times 10^{-11} \frac{N \cdot m^{2}}{kg^{2}}$ (Gravitational Constant)
$M$ is the mass of the large body
$R$ is the radius

## Law of Periods

$T^{2} = ( \frac{4\pi^{2}}{GM})r^{3}$

Where
$t$ is the period of the motion
$M$ is the mass of the large body
$r^{2}$ is the distance between them

## Kinetic Energy in a Satellite in Circular Orbit

$K = -\frac{U}{2}$

## Potential Energy in a Satellite in Circular Orbit

$U = -\frac{GMm}{r}$

## Total Mechanical Energy of a Satellite in Circular Orbit

$E = -\frac{GMm}{2r}$

## Total Mechanical Energy of a Satellite in Elliptical Orbit

$E = -\frac{GMm}{2a}$

Where
$a$ is the semimajor axis

Law of Periods Question

Escape Speed

# Special Cases

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• Example One
• Example Two

# Applications

• Planet Orbit Simulator
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