Applying the Virial Theorem to Stellar winds

In summary, the conversation is discussing the time-dependent Virial Theorem for an isotropic stellar wind that has reached a certain distance from the center of a star. The task is to write down the theorem and calculate its various terms, but the problem lies in getting the time-dependent versions of those terms. The poster is seeking help in understanding how to approach this problem and is advised to post their question in the appropriate homework forum.
  • #1
JimLad
2
0
Hello,

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) :rolleyes:
 
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  • #2
this was posted 1 year ago and still no reply! i have the exaxt same problem, how the hell do you do this?
 
  • #3
anthonyhollin said:
this was posted 1 year ago and still no reply! i have the exaxt same problem, how the hell do you do this?

You need to post your question in the homework forums, and show your work. Please feel free to start a new question in the appropriate homework forum, since this is such an old thread. Oh, and please do not copy the work above-- you need to show your own work!
 

1. What is the Virial Theorem?

The Virial Theorem is a fundamental principle in physics that relates the potential and kinetic energies of a system. It states that the average kinetic energy of a system is equal to half of its potential energy. This theorem is widely used in astrophysics to study the dynamics of stars and other celestial bodies.

2. How is the Virial Theorem applied to stellar winds?

The Virial Theorem can be used to understand the energy balance in stellar winds. By comparing the kinetic energy of the wind with the gravitational potential energy of the star, we can determine the strength and direction of the wind. This helps us to understand the processes that drive the winds and their impact on the evolution of stars.

3. What types of stars have strong stellar winds?

Massive and evolved stars, such as red giants and supergiants, are known to have strong stellar winds. These winds are caused by the intense radiation pressure from the hot stellar atmosphere, which drives the gas away from the star. Young, hot stars also have strong winds due to their high luminosity and strong magnetic fields.

4. How does the Virial Theorem help us to study the effects of stellar winds on star formation?

The Virial Theorem provides insights into the dynamics of stellar winds and how they shape the surrounding interstellar medium. By studying the energy balance and momentum transfer between stars and their winds, we can better understand the role of stellar winds in the formation and evolution of stars, as well as the formation of planetary systems.

5. Can the Virial Theorem be applied to other astrophysical systems?

Yes, the Virial Theorem has wide applicability in astrophysics and can be used to study the dynamics of various systems, such as galaxies, clusters of galaxies, and even the entire universe. It is a powerful tool for understanding the energy balance and evolution of these systems, and has been used extensively in research and observations in astronomy and astrophysics.

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