# Planetary Systems

*I demonstrate by means of philosophy that the earth is round, and is inhabited on all sides; that it is insignificantly small, and is borne through the stars.*(Johannes Kepler)

The equations of motion for a planet (viewed as a pointlike particle) of unit mass orbiting a fixed Sun of unit mass centered at the origin, take the form

- ,

where

- (Newton’s Gravitation Law)

is the *gravitational force*. This is a force acting at distance, because the Sun at the origin acts at the point at the distance from the origin.

Note that Newton’s Gravitation Law is the famous *inverse square law of gravitation* stating that the magnitude of the gravitational force between two bodies with mass and at distance , is given by

- ,

where is the gravitational constant.

We shall prove below that Newton’s Gravitation Law is a consequence of the fact that the gravitational potential satisfies a certain differential equation named *Laplace’s equation*, and we shall uncover the assumptions leading to Laplace’s equation. We can this way motivate that the exponent in Newton’s Law is 2 and nothing else.

Crab nebulosa: A macroscopic particle system.}

Galileo presenting mathematical arguments to disbelieving Catholic priests.

# Watch

# To Think About

- What are Kepler’s Laws
- What is the simplest solution of a 2-body problem?

# To Read

- BS How to prove Kepler’s laws yourself.
- \hyperref[chaptersolarsystem]{BS Solar System}

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