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I am having a slight disagreement with my professor.

We can measure the mass of a distant star cluster from:

A.Its color

B.Its radius

C.Its metallicity

D.Smearing of lines in its total spectrum

My thinking was:

Law of Conservation of Energy

Kinetic Energy = Gravitational potential energy

1/2*m*v^2 =  G* m*M/r

v = sqrt(2GM/r)

So it is dependant on both velocity and radius.

Answer choice D would give you the velocity but I responded with answer choice B, radius.

After talking it over with my professor, she responded with:

If we think about it conceptually, the dominant parameter is the velocity. In other words, you can get a meaningful estimate of the cluster mass from the velocity dispersion without knowing the radius (because all clusters are the "same" size), but not vice versa.

So I'd say it still seems pretty clear that the best answer is the velocities.

Is she right or is the question worded poorly?

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    $\begingroup$ In the title you say "galaxy", but in the text you mention "cluster", which usually means either a stellar (globular) cluster, or a cluster of galaxies. If you mean "galaxy", then the problem with radius is that it's the virial radius, which is not readily measured, since it's dominated by dark matter. In that case, answer D is correct, since it gives you a good measure of the typical velocities, which can then be translated to M (since at a given redshift, there is a moderately tight relationship between M and r). $\endgroup$
    – pela
    Apr 3, 2019 at 11:30
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    $\begingroup$ The question title and main question body are about different tthings. $\endgroup$
    – ProfRob
    Apr 27, 2020 at 11:16

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I think she should have clarified the question by putting velocity and radius in the same choice.

By looking at her answer, it seems reasonable to think that velocity is more important. However without knowing the radius we cannot determine the mass of the stellar system. She argued this idea by saying that the size is approximately known for stellar systems. But I don't think that is a solid argument.

Theoritically we should know both R and V to calculate M.

Note: I did not understand why you wrote kinetic energy= gravitational force ? Thats not possible.

In stellar bounded system you should use virial theorem. By stating that $-2<E>=<U>$ and from there you get

$$M=v^2R/G$$

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