## Thursday, April 15, 2021

### [xyhswzxz] Fun with the 2-body problem

Rule 1: Any mass can have any number of satellite point masses placed in elliptic orbits around it.  The satellites do not affect each other, nor do they affect the central mass.  This corresponds to the 2-body problem when the satellite mass is much smaller than the central mass.  However, we do not constrain satellite mass to be much smaller; choose satellite masses however you wish.  As will be described below, this can result in physically unrealistic systems.

Rule 2: Any mass can be divided into a pair of point masses.  The motion of the center of mass stays the same.  The two masses orbit each other in elliptic orbits.  Satellites of the original mass continue unchanged, orbiting the center of mass.

Rule 3: Adjust the mass of a central mass (or its descendents after binary subdivision) and the velocities of its satellites so that paths remain the same, only sped up or slowed down.

Is this it for what can be modeled by ellipses?  Ellipses are nice because they are not chaotic as the three-body system is.  By solving Kepler's Equation with Newton's Method, one can compute positions for arbitrary times, even times in the far future, without the answer being sensitive to the initial values.

Repeatedly and recursively apply the above construction rules as many times as you wish.

One can easily create physically unrealistic systems using the rules.  Here are some examples of unrealistic systems:

• A satellite has mass greater than the body (or system) it orbits.
• A satellite orbits the center of mass of a binary pair, but the orbit of the satellite is inside the orbits of the binary pair.
• A mass X has satellite Y.  Y has satellite Z.  The orbit of Z around Y is larger than that of Y around X.

Consider not caring about these and other unrealistic possibilities because the point is not realism but the fun of combining ellipses upon ellipses, kind of like a spirograph.

Inspired by the "grav" hack in xscreensaver.