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In the process of revising for an exam I have, I am having difficulty with this question.

"Consider an isotropic stellar wind of mass density rho, pressure p, temperature T and velocity v that has reached to a distance r=R_w from the centre of a star. The star has mass M* and radius R*. Write down the time-dependent Virial Theorem describing the wind between the spherical surfaces r=R* and r=R_w. Assume the gravitational acceleration of the material in the wind is dominated by the mass of the star (ie. you can neglect the self gravity of the gas in the wind)."

Given the full Virial theorem is: 1/2 (d^2 I / dt^2) = W + 2T + 3Pi - closedintegral(dS r p)

where W = gravitational potential energy, T = kinetic energy , Pi = thermal energy, the closed integral is to account for an outside pressure for the bit of the system we are looking at, and I is moment of inertia.

The stellar wind won't be rotating, so I = 0. So then I tried to calculate the each of the terms W, T, Pi and the surface integral term, but I havn't got them in time dependent form. An example would be W:

W=integral {dV rho r grad(Psi)}

where Psi is the gravitational potential = -GM(r)/r

So for the region R*< r < Rw, M(r) = M* since we can neglect the self gravity of the stellar wind. Hence:

grad{Psi}=GM*/r^2

W= GM* integral {dV rho / r}

Now dV=4pi r^2 dr and I could sub in for rho and integrate, but this would only be spatially dependent. How do I get the time dependent version? It's a similar case for the other terms - I have only got them in terms of spatial quantities.

Thanks for any help, hope that was clear(ish)

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# Applying the Virial Theorem to Stellar winds

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