# Does uniform distribution of background matter affect the dynamics of a gravitational system?

Situation 1: A test particle of mass m moves around a big mass M in a Keplerian orbit. The orbital period is easily found, given certain initial conditions.

Situation 2: The same system of two bodies M and m, starts interacting under the same initial conditions as in the previous case. However, all of the space now is filled with a uniform distribution of background "dark matter" (density does not depend on position and is not very high, so that recourse to General Relativity is not needed). This background might affect m and M only gravitationally (i.e., there is no dissipation). Will the orbital period of m change?

I encounter a paradox here. Thinking on a primitive level, the mass inside the orbit is increased in the Situation 2 due to the background matter, so the motion must be different now. On the other hand, in the Situation 2 the gravitational potential in each point is changed only by a constant in comparison with Situation 1, so it cannot affect the dynamics, as this constant difference vanishes when taking the gradient to calculate forces.

• Is the space practically infinite? That is, is there a centre of this uniform dark matter distribution? Also, have you considered that the mass of the planet acts as a nucleation point for the dark matter to clump to? – Jim Dec 22 '14 at 17:16
• In fact, it would be interesting to consider both the purely abstract problem, where the space is infinite (and so is the background distribution), and a more "physical" situation, e.g., a planet orbiting a star in a region where there are no other gravitating bodies, and the dark matter is distributed in that region approximately uniformly, without distinct "centers". As for clumping of dark matter, let's take it to be practically unclumping (e.g., it's very "hot" dark matter). – ThisGuy Dec 22 '14 at 17:37
• Even hot dark matter falls into gravity wells. Perhaps "clump" was the wrong word. I mean it would attract all the dark matter around it and that matter would fall into its gravity well and, therefore, increase the mass of the well – Jim Dec 22 '14 at 17:40
• Case A: Infinite uniform dark matter cloud: This universe is a black hole. That, IMHO, constitutes a significant effect on the orbit of the test particle around the mass. – Jim Dec 22 '14 at 17:41
• Case B: Uniform dark matter cloud finite in extent: After finite time, the dark matter cloud falls into its own centre of gravity and creates a density gradient. The mass and the test particle are perturbed in their orbit depending on where this centre of gravity is and what the mass of the cloud is – Jim Dec 22 '14 at 17:43

The short answer is that a uniform matter distribution with Newtonian mechanics is an ill defined mathematical problem. The reason is equivalent to why $\int_{-\infty}^\infty |x|^{-1/2} dx$ is not defined: you only get a value with additional structure (e.g. a zero-point, or a rule for regularizing non-absolutely convergent series). While a uniform mass density cannot have a uniform gravitational field, you can find solutions that satisfy Gauss law, but they are non-uniform field solutions, that is, solution in which the uniform background does affect the dynamics (non-zero gravitational force).