Calculating Consistency of a Point Mass System w/ Dark Matter

[hilbert2]

Suppose we have a point mass system that consists of $m + n$ particles. There are $m$ normal visible point masses and $n$ invisible "dark matter" point masses. The point masses interact gravitationally with a $1/r$ potential.

Now when someone observes the motion of the visible masses, he will notice that they are not consistent with Newton's laws of motion (of course, because the invisible masses perturb the trajectories of the visible masses).

The question is, can such an observer calculate how many dark matter point masses there must be in the observed system to make the system self-consistent, and what are their masses and trajectories? In what cases a unique solution exists to this problem?

 

[hilbert2]

I probably didn't express this clearly enough... (English isn't my native language) I have seen that the observed rotational motion of galaxies is explained with a hypothesis that there is a "halo" of dark matter around all galaxies. How does one deduce the required density distribution of dark matter from the observed motion of visible matter? Here "visible matter" is something that interacts electromagnetically and not only gravitationally.

http://en.wikipedia.org/wiki/Dark_matter_halo

 

[Nugatory]

The question is, can such an observer calculate how many dark matter point masses there must be in the observed system to make the system self-consistent, and what are their masses and trajectories? In what cases a unique solution exists to this problem?

This technique has been used to show the existence of astronomical bodies that aren't directly visible - for example, we infer the existence of a dark companion when we observe a bright star making otherwise unexplained motions. It's harder in a more complex many-body problem, of course, but in principle it's still possible.

The solutions need not be unique in any practical sense, because two bodies of mass M close enough to one another and distant enough from the other objects will be observationally indistinguishable from one body of mass 2M and located at their center of mass.

 

[dauto]

I probably didn't express this clearly enough... (English isn't my native language) I have seen that the observed rotational motion of galaxies is explained with a hypothesis that there is a "halo" of dark matter around all galaxies. How does one deduce the required density distribution of dark matter from the observed motion of visible matter? Here "visible matter" is something that interacts electromagnetically and not only gravitationally.

http://en.wikipedia.org/wiki/Dark_matter_halo

That question is much easier to answer (The previous question is very interesting but not trivial at all).

It is very easy. Find out the force required to keep a star in circular motion around the center of the galaxy using F = mv²/r.

Compare that with the force produced by the gravitational force due to the visible mass.

Assume the difference between the two numbers is due to a spherically symmetric distribution of dark matter.

Compare that with the gravitational force produced by such a spherical mass distribution in order to figure its mass.

Voila!