# White Dwarf Radii: Minimizing Total Energy

• quasar987
In summary, the reasoning behind the minimization of the energy in a white dwarf is that the gravitational potential energy and the degeneracy pressure of the electrons are balanced.
quasar987
Homework Helper
Gold Member
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 ?

Astronomy news on Phys.org
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.

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.

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.

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

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

Last edited:
Kurdt said:
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.

quasar987 said:
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.

quasar987 said:
$$E=\frac{A}{R^2}-\frac{B}{R}$$

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.

Last edited:
SpaceTiger said:
[...]
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?

## What is a white dwarf?

A white dwarf is a type of star that is at the end of its life cycle. It has exhausted all of its nuclear fuel and has collapsed to a very small size, similar to the size of Earth.

## How do white dwarfs minimize total energy?

White dwarfs minimize total energy by balancing the forces of gravity and electron degeneracy pressure. The electrons in a white dwarf are packed tightly together, creating a high pressure that counteracts the force of gravity. This balance keeps the star from collapsing further and minimizes its total energy.

## What is the relationship between white dwarf radii and mass?

The radius of a white dwarf is inversely proportional to its mass. This means that the more massive a white dwarf is, the smaller its radius will be. This is due to the strong force of gravity compressing the star as it loses energy.

## How do white dwarf radii compare to other types of stars?

White dwarf radii are much smaller than other types of stars. They typically have a radius of only a few thousand kilometers, whereas a star like the sun has a radius of about 700,000 kilometers. This is because white dwarfs have much less mass and are able to be compressed to a smaller size.

## Can white dwarf radii change over time?

Yes, white dwarf radii can change over time. As they continue to lose energy, their radii will decrease even further. However, this process occurs very slowly and can take billions of years to significantly change the star's size.

Replies
1
Views
1K
Replies
49
Views
3K
Replies
9
Views
2K
Replies
5
Views
2K
Replies
9
Views
3K
Replies
7
Views
5K
Replies
23
Views
2K
Replies
4
Views
3K
Replies
7
Views
5K
Replies
3
Views
2K