White Dwarf Radii: Minimizing Total Energy

[quasar987]

In the simplistic treatement of white dwarfs we covered in my thermo class, we obtained an expression for the total energy of the star, namely

$$E=\frac{A}{R^2}-\frac{B}{R}$$

where A and B are constants. Then, we said that the actual radius of the dwarf is the radius that minimizes the energy.

On what principles is this reasoning based ?

 

[Danger]

I can't address your question directly. All that I can say is that the minumum radius is that at which it must collapse into a neutron star.

 

[Jheriko]

I would guess that one of these terms represents the energy from gravity trying to compress the star and the other represents the energy from the degeneracy pressure of the electrons. Which is which and where they come from, I do not know.

 

[Kurdt]

All systems try to reach a state where the energy is minimised. Look at electrons in an atom, they always head to the ground state. Of course if the white dwarf is stable then it is bound to be in its minimal energy state anyway otherwise it would have to collapse further.

EDIT: Just to add the place where I see this the most is in quantum mechanics. I'm not sure how common it is in thermodynamics.

Obviously you know how to minimise it. Differentiate wrt R and set to zero then rearrange for R.

$$0=\frac{-2A}{R^3} +\frac{B}{R^2}$$

$$R=\frac{2A}{B}$$

 

[quasar987]

All systems try to reach a state where the energy is minimised.

So is this an axiom of physics or what? I have a feeling that this holds in classical mechanics too and that it's not an axiom but rather a property that can be proved from more fundamental assumptions.

 

[Kurdt]

So is this an axiom of physics or what? I have a feeling that this holds in classical mechanics too and that it's not an axiom but rather a property that can be proved from more fundamental assumptions.

Interesting point. I've never really seen this set out specifically in texts I've come across. I've attempted answering this several times but what I come up with is more confusing than anything else so I'll leave it to somebody more skilled than me or else give me a while to word myself clearly.

 

[SpaceTiger]

$$E=\frac{A}{R^2}-\frac{B}{R}$$

The second term is the gravitational potential energy. In a uniform density star,

$$B=\frac{3GM^2}{5}$$

The first term is the thermal energy. In a degenerate gas, this will be proportional to the Fermi energy:

$$E_F=\frac{\hbar^2}{2m}(3\pi^2n)^{2/3}$$

In a uniform density star,

$$E_T \propto n^{2/3} \propto \frac{M^{2/3}}{R^2}$$

Minimizing the total energy is effectively equivalent to balancing the forces of gravity and degeneracy. To get a feel for this, consider the one-dimensional Newtonian expression:

$$F=\frac{\partial V}{\partial x}$$

When the net force is zero, the energy is either minimized or maximized. In the latter case, however, the equilibrium is unstable, so we usually consider only energy minimization.

 

[quasar987]

[...]
Minimizing the total energy is effectively equivalent to balancing the forces of gravity and degeneracy. To get a feel for this, consider the one-dimensional Newtonian expression:
[...]

Hey ST, thanks for the reply. What I'm looking for however, is for more than "a feel". I'm interested in the exact justification for this. Don't be afraid to use technical terms, I'll sort out what it all means myself.

I was thinking it was something like "R is a generalized coordinate and the system is in equilibrium when the generalized force dE/dR vanishes."

Anything close?