Is the gravitational binding energy formula different for stars and galaxies?

[kmarinas86]

From http://www.astro.cornell.edu/academics/courses/astro201/vt.htm :

Conditions:
Stable
Self-gravitating
Spherical distributions
Equal mass objects

$KE=\frac{1}{2}M_{tot}v^2$
$PE\simeq-\frac{1}{2}G\frac{M_{tot}^2}{R_{tot}}$
$KE\simeq-\frac{1}{2}PE$
$M_{tot}\simeq 2\frac{R_{tot}v^2}{G}$

http://www.physics.uq.edu.au/people/ross/phys2080/nuc/virial.htm

"When an ideal self gravitating system contract, half of the gravitational binding energy goes into thermal motion (heat) and the other half goes into radiation which is lost into space."

From above:

$KE\simeq-\frac{1}{2}PE$

Given the quote just above:

$Gravitational\ binding\ energy\simeq-PE$

$Gravitational\ binding\ energy\simeq\frac{1}{2}G\frac{M_{tot}^2}{R_{tot}}$

But, this is not right for a star. For a star, it is:

$Gravitational\ binding\ energy=G\frac{M_{tot}^2}{R_{tot}}$

This would mean that one fourth of the gravitational binding energy goes into thermal energy, or one half of of the gravitational potential energy.

Something is not jibing, but what is it?

_________

 

[Gokul43201]

$Gravitational\ binding\ energy\simeq\frac{1}{2}G\frac{M_{tot}^2}{R_{tot}}$

But, this is not right for a star. For a star, it is:

$Gravitational\ binding\ energy=G\frac{M_{tot}^2}{R_{tot}}$

Well, at least one of those two expressions must be wrong (I think they both are). Where did you get them from? Shouldn't there be a 3/5 factor for the GPE of a uniform, spherical object?

 

[kmarinas86]

Well, at least one of those two expressions must be wrong (I think they both are). Where did you get them from? Shouldn't there be a 3/5 factor for the GPE of a uniform, spherical object?

Yes. But for a star its different $GM^2/r$.

http://en.wikipedia.org/wiki/Gravitational_binding_energy

For a galaxy, I'm not sure.

The fact that there are different fractions used makes me wary. Anyone have the full list?