After reading in Longair's(adsbygoogle = window.adsbygoogle || []).push({}); Galaxy Formationthrough the derivation of the virial theorem in the context of a dynamical system in equilibrium consisting of "point masses" interacting only through gravity, I proceeded to try to understand his comments on how the theorem can be applied in order to determine the mass of a galaxy. In these comments, he makes a claim:

The total kinetic energy T is equal to one half the total mass of the system M times the velocity dispersion <v^2>. Now, based on the definition of velocity dispersion I was able to find (in another book, mind you), it is the root mean square of the velocities:

[tex] \langle v^2 \rangle = \frac{1}{\sqrt{N}}(\sum_i{v_i^2})^{\frac{1}{2}} [/tex]

Even if I assume in this context that <v^2> is actually that *without* the power of 1/2 (so that things will sort of make sense), his claim still basically amounts to:

[tex] 2T = M\langle v^2 \rangle [/tex]

[tex] \sum_i m_iv_i^2 = (\sum_i m_i)(\frac{1}{N}\sum_i{v_i^2}) [/tex]

This claim is clearly false. So what's going on? With what justification can he say that the total kinetic energy is just one half the total mass times the velocity dispersion squared?

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# Virial Theorem in Stellar Dynamics

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