Stars and the Virial Theorem

The Virial Theorem is a theorem in mechanics that relates the kinetic energy of particles in a system to the potential of the system. This can be useful when talking about stars, which balance both an immense gravitational collapsing force from their huge mass with the outward force from the internal particles heated from fusion at the star's core.



 The Virial Theorem gives us this:



So the magnitude of the gravitational potential energy of the star is equal to the average kinetic (ie, thermal) energy of the electrons in the star's core. Why electrons? This has to do with degeneracy pressure, which is the tendency for particles to try to not occupy the same space at the same time. Think about the core of a star: it's very very dense, with a lots of electrons flying around. Furthermore, the inter-particle spacing is given by the De Broglie wavelength:



The smaller the momentum of a particle, the greater the inter-particle distance as given by the De Broglie relation. And electrons, being very tiny, tend to have a very small momentum. When such electrons are packed too tightly, this gives rise to degeneracy pressure, which wants to push them outward, counteracting the inward pull of gravitation. For more on this, check David's blog. Also, incidentally, I think degeneracy pressure has a hand in the formation of neutron stars: if the degeneracy pressure is unable to hold the collapsing force of gravitation at bay, it will be unable to force electrons to their proper inter-particle spacings. So, instead, it drives the nuclear reaction of electron capture, wherein a proton and electron form a neutron and an electron neutrino. Overall, this results in an extremely dense object made up mostly of neutrons.

But anyway, I digress. Based on the Virial Theorem and Degeneracy pressure, what can we find out about stars? Think of Heisenberg's Uncertainty Principle:



This says that the product of the uncertainty in a particle's position and the uncertainty in it's momentum must be greater than or equal to Planck's constant divided by four pi. How does this relate to the Virial Theorem? Well, we know that the position of a particle should be within the De Broglie wavelength. If that's so, then we can write the number density N as



since the cube of lambda should correspond to the volume taken up by one electron. Now we can use the uncertainty equation to solve for the De Broglie wavelength, and thereby find N in terms of properties of the star:



In order to use the Virial Theorem, it would be nice to get lambda in terms of the kinetic energy of the electrons. We can do this by squaring both sides of the equation:



Now solve for lambda squared:



Now we can find N:



Awesome. So now we know the number density of electrons in the star. The next question is, how is this related to the star's mass and radius? Well, the mass and radius are related to each other like this:



where rho is the mass density. What is the mass density, you say? Well, I'm glad you asked:



where N is the number density as calculated above, m is the mass of an electron, and n is the number of electrons per atom in the star. for the sake of simplicity, let's assume that the star is made up purely of hydrogen, so n = 1:



OK, so where does this get us? Well, N is in terms of the kinetic energy, which from the Virial Theorem we relate to gravitational potential energy, which is in terms of R and M. So, by substituting that in, we can then solve for the relationship between R and M. Now, normally this relationship would be fraught with messy constants, making everything more complicated. But we don't really care about the constants (what are we, mathematicians?). So let's just ignore them, and find out generally how R scales with M:



So, without the constants like G getting in the way, we have



Which gives