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

• $dx=vdt,\quad v=Fdt$,

where

• $F(x)=-\frac{x}{\vert x\vert^3}$         (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 $x$ at the distance $\vert x\vert$ 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 $F$ between two bodies with mass $M_1$ and $M_2$ at distance $r$, is given by

• $F=G\frac{M_1M_2}{r^2}$,

where $G$ 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.

Galileo's telescope