Derivation of velocity dispersion from virial theorem?

GM/R, where T = kinetic energy and U = potential energy. The kinetic energy is given by K = (1/2)mv^2, where v is the velocity and m is the mass. Rearranging the equation, we get v^2 = GM/R. Therefore, the velocity dispersion, which is the average velocity of the particles, is proportional to the square root of the gravitational constant (G) and inversely proportional to the radius (R). In summary, the velocity dispersion can be derived from the virial theorem by rearranging the equation to find that it is proportional to the square root of the gravitational constant and inversely proportional to the radius.
  • #1
taylrl3
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Derivation of velocity dispersion from virial theorem??

Hey,

Im probably being a bit dim here but could anyone help me derive the velocity dispersion from the virial theorem. I've got 2K+U=0, K/m~sigma^2 and U/m~GM/R.

From rearranging I get a negative velocity^2? Or maybe its the gravitational force that's negative. Oh and a factor of 2. Is that just ignored??

My notes say the answer is sigma^2~GM/R

Just a bit confused, anyone got any clues?
 
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  • #2


The virial theorem says that 2T - |U| = 0, where U = -GM/R (the negative sign being important, since gravity is an attractive force).

2T = |U|
 

1. What is the virial theorem and how does it relate to velocity dispersion?

The virial theorem is a mathematical relationship between the kinetic and potential energy of a system. In the context of velocity dispersion, it states that the average kinetic energy of a group of particles is related to the gravitational potential energy of the system. This means that by measuring the velocities of particles in a system, we can determine the dispersion of their velocities, which is a measure of how spread out their velocities are.

2. How do we derive velocity dispersion from the virial theorem?

To derive velocity dispersion from the virial theorem, we start by assuming that the system is in equilibrium, meaning that the average kinetic energy is equal to the average potential energy. From there, we can use mathematical manipulations to solve for the velocity dispersion, which is given by the square root of the ratio of the gravitational potential energy to the kinetic energy.

3. What is the significance of velocity dispersion in astrophysics?

Velocity dispersion is an important measure in astrophysics as it can provide information about the dynamics and structure of a system. It can also be used to estimate the mass of a system, as the more massive a system is, the higher its velocity dispersion will be. Additionally, velocity dispersion can also reveal the presence of dark matter, which does not emit light but has a gravitational influence on the motion of particles.

4. Are there any limitations to using the virial theorem to derive velocity dispersion?

Yes, there are some limitations to using the virial theorem to derive velocity dispersion. For example, it assumes that the system is in equilibrium, which may not always be the case. It also assumes that the system is spherically symmetric, which may not be true for all systems. Additionally, the virial theorem only applies to systems where gravity is the dominant force, so it may not be applicable to all astrophysical systems.

5. How do we measure velocity dispersion in astronomical observations?

There are several methods for measuring velocity dispersion in astronomical observations. One common method is through spectroscopy, which measures the Doppler shifts in the wavelengths of light emitted by stars or galaxies. Another method is through the analysis of the broadening of spectral lines, which can be caused by the random motions of particles in a system. Other techniques, such as gravitational lensing, can also be used to indirectly measure velocity dispersion in systems such as galaxy clusters.

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