### More about These Orbits...

**Planetary Motion with Different Force Laws**

We take a central force proportional to some power of the radius: force *F* = *Gr*^{n}

For example, for *n* = 1, the force is directly proportional to distance from the center. This is a two-dimensional harmonic oscillator, the orbits are closed ellipses, the attractive "sun" being at the center of the ellipse.

The inverse square law (our Solar System), *n* = -2, also gives elliptic orbits, but now the Sun is at one focus of the ellipse, not at the center. (There are also unbound parabolic and hyperbolic orbits.)

These two values,

*n*=1,-2 are the only ones giving closed orbits (except for special initial values, for example circular orbits are always possible, but not necessarily stable).

Moving slightly away from *n*= -2 gives a *precessing* elliptical orbit (try it!). This is the behaviour of Mercury's orbit. The precession is partially caused by other planetary attractions, but, crucially, there is a significant contribution from general relativity. This was historically very important: it convinced Einstein he was on the right track.

Orbits with *n* = -3 or less are unstable. You'll find they get pretty wild on getting below -2.7 or so .

What do you expect for *n* = 0? For *n* = 10? Check them out, and interpret.

*Code by David Bang*