A problem from Binney and Tremaine

[krishna mohan]

Hi...

I have been pondering over this problem for sometime...
This is problem no 4-9 in Binney and Tremaine's Galactic Dynamics first edition...

In a singular isothermal sphere with an isotropic dispersion tensor, the RMS speed is $$\sqrt{\frac{3}{2}}v_c$$, where $$v_c$$ is the circular speed. In a system with the same mass distribution, but with all stars on randomly oriented circular orbits, the RMS speed is $$v_c$$.
Thus, the two systems have identical density distributions but different amounts of kinetic energy per star.
How is this consistent with the virial theorem?

Can anyone suggest anything?

 

[krishna mohan]

For some reason, am not able to edit this post.
Please see the new post on the same topic.

 

[harcel]

The virial theorem is about the system as a whole, when it is in virial equilibrium. The v_c that you quote here is not a unique number for every star. In case of an isothermal sphere it is, as the mass distribution it has is such that the velocity is constant, as a function of radius. For all other mass distributions, v_c is a function of r, and the mass enclosed within r (M(r)): v_c^2 = M(r)/r So for any given mass distribution other than isothermal, the circular velocity is not constant. If all stars move on circular orbits, they will all have v_c as their velocity, but the value of v_c depends on the radii of all other orbits.