# Gravity vs. Degeneracy Pressure

• cepheid

#### cepheid

Staff Emeritus
Gold Member
Okay, I noticed that my OP got 39 views but no responses, so let me change my strategy. Here is my question:

An object made out of degenerate matter (e.g. white dwarf) will collapse if more gravitational energy is lost in getting smaller than the energy that is gained due to electron degeneracy. So, the sum of these two energies will decrease. But total energy must be conserved. So, what happens to the rest of the GPE that is lost (the part that doesn't go into doing work to overcome degeneracy pressure during contraction)? Is it just converted to heat?

My OP is below if you want more details on what I'm asking about.

My prof. was giving a sort of heuristic outline of how the Chandrasekhar mass limit arises. He started with the basic "equilibrium equation" that the energy E = Ekin + Epot should be minimized. This is in the context of a "cold" object, where most of "Ekin" is due to electron degeneracy. In other words, there is no thermal pressure support. All of the support against gravity comes from degeneracy pressure. He then worked out how each of these energies varies with the volume (or radius) of the object, and showed that in the case of non-relativistic electrons, there is an equilibrium point (the sum of the energies E is minimized for some finite radius R), and in the relativistic case, there is no global minimum, the solution is unchecked collapse. I looked into this further, and I realized that another way to think about this is in terms of pressure balance. You can equate the "gravitational pressure" to the negative of the degeneracy pressure, where

Pgrav = -∂Epot/∂V

and

Pdeg = -∂Ekin/∂V

This equation of pressures results in

∂(Epot + Ekin) / ∂V = 0

showing that pressure balance does indeed occur at a minimum in the sum of the energies.

Here is my question: in the situation where collapse occurs, gravitational pressure exceeds degeneracy pressure, meaning that the rate of change of Epot and Ekin with radius (or volume) is such that more GPE is lost in contracting than the energy that is gained due to electron degeneracy. In other words, not all of the GPE that is lost is "used up" in doing work against degeneracy pressure. The sum E is actually reduced. But total energy must be conserved right? The sum E must not be the total energy of the system. So what happens to the rest of that GPE?

An obvious answer would seem to be: "it is converted into heat." Is that the case? If so, then I have another related question...

Last edited:
It's converted to kinetic energy from ordered inward directed motion in the first place. If the star settles down again, it would become heat.

Heh, inwardly directed motion. How did I miss that?

## What is gravity?

Gravity is a fundamental force of nature that causes objects with mass to attract each other. It is responsible for the motion of planets around the sun and the formation of galaxies.

## What is degeneracy pressure?

Degeneracy pressure is a quantum mechanical effect that arises when particles, such as electrons, are packed tightly together in a small space. It resists further compression and helps to stabilize the object against the force of gravity.

## How do gravity and degeneracy pressure compare?

Gravity is a force that pulls objects together, while degeneracy pressure is a force that pushes particles apart. In a star, gravity tries to collapse the star while degeneracy pressure counteracts this collapse and helps support the star's structure.

## Which force is stronger, gravity or degeneracy pressure?

It depends on the circumstances. In general, gravity is the stronger force and dominates in larger, more massive objects such as stars. However, in extremely dense objects, such as white dwarfs and neutron stars, degeneracy pressure can be strong enough to counteract the force of gravity.

## How do gravity and degeneracy pressure affect the life cycle of a star?

Gravity plays a crucial role in the formation of a star, as it brings together the gas and dust in a molecular cloud. As the star ages, gravity continues to pull inwards, but degeneracy pressure prevents the star from collapsing completely. When the star runs out of fuel, gravity can overcome degeneracy pressure, causing the star to collapse and potentially resulting in a supernova explosion.