Solving Stellar Wind Virial Theorem for Exam

[anthonyhollin]

Homework Statement

Hi, i have 2 past exam questions that i need to be able to solve for my exam in 3 days time:

consider the portion of a stellar wind of mass density rho, pressure p, temperature T and velocity v between the radii r=R_i = 10R_* and r=R_o = 100R_* where R_* is the radius of the star. Assuming this portion has reached Virial Equilibrium, state the Virial theorem describing it.

{10 marks (=20%), so not just writing down the virial equation}

also, another similar question from the next year:

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.)

Homework Equations

time dependent virial theorem:
1/2 d^2I/dt^2 = 2T + W + 3Pi - closedIntegral(p r).ds

I = integral (rho v^2).dV is moment of inertia
T = kinetic energy = integral (1/2 rho v^2).dv
W = -integral(rho r . grad(Phi)).dV is the thermal energy, Phi = GM(r)/r
the surface integral accounts for outside pressure

The Attempt at a Solution

to be honest I've got got too far. For the fisrt question there is virial equalibrium, so d^I/dt^ = 0, but for the second question the next part is proving the wind cannot be in a state of equalibrium (however this course didnt go into much detail about magnetic field freezing in plasma, transporting angular momentum away, so i think (or hope) i can assume the wind doesn't rotate).

grad(Phi) = GM_*/r^2 since no wind self gravity and M_wind << M_*, and so W = GM_* integral rho / r dV where dV is 4 pi r^2 dr. My main problem is getting time depenmdent forms for this since i have the velocity v, and what to use for p, the isothermal equation of state p = rho Cs^2 where Cs is the sound speed, or should i use the form from assuming the corona is in hydrostatic equalibrium (which i know it isnt- hence the setllar wind), where
p(r) = P_0 exp{-L (1 - r_0/r)}
where P_0 = 2nkT by the ideal gas law where all gas is ionized H, so mu=1/2 and n=rho/m_p. L is the left over constants (G M_* m_p) /(2 k T r_0) which ~ ratio of a protons gravitational P.E to its K.E

sorry I am just not too sure where to go with this so i haven't done much. There really isn't anything in my notes (besides the virial equation and the isothermal EOS), so I am not missing out helpful stuff here

 

[astrorob]

It's been a while since I studied this, but I'd start by using the more basic version of the VT which is a little easier to understand (without that messy calculus!):

$$2(E_{kinetic} + E_{thermal}) + E_{gravitational} = 0$$​

Hints:

Kinetic energy

We'll need to know the mass of the region we're dealing with:

$$dM(r) = 4{\pi}r^{2}{\rho}(r)dr$$

Thermal energy:

The thermal energy of a monatomic gas is, as you probably know;

$${\frac{3}2}NkT$$

but for simplicity we can assume an ideal gas;

$$PV=NkT$$

A combination of these will yield an equation involving P and V. But of course, the V is not a considered variable in this case so you must convert that using the density and radius. The enclosed volume of the gas will be the volume of the higher radius sphere - the volume of the lower radius sphere.

Gravitational Energy

I'm sure you can work this bit out!