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The electron degeneracy pressure (EDP) dominates in White dwarfs, where the electrons are compressed to very high densities. EDP is a purely quantum mechanical effect that arises due to Pauli's exclusion principle - the fact that no two fermions can occupy the same quantum state. It has nothing to do with electrostatic repulsion between electrons, and it is also not a thermal pressure (it exists, and in fact obtains a maximum at, $T=0$). This prevents further gravitational collapse of the star. The expression for the EDP depends on whether the electrons are non-relativistic or ultra-relativistic - as it can be seen that the expressions are independent of the electronic charge and the temperature: \begin{eqnarray} P_e = \begin{cases} \frac{(3\pi^2)^{2/3}}{5}\frac{\hbar^2}{m_e} n_e^{5/3}, & \mbox{if } ~\text{nonrelativistic} \\ \frac{(3\pi^2)^{1/3}}{4} \hbar c \, n_e^{4/3}, & \mbox{if} ~~\text{ultra-relativistic}. \end{cases} \end{eqnarray}

My question is the following.

On top of the EDP, should there not be another source of outward pressure that can potentially counteract the gravitational collapse - namely the electrostatic repulsion between the electrons? Intuitively, one would think that the electrostatic repulsion will prevent the electrons from coming too close to each other, and hence, also act against the inward gravity and help support the star against collapse. Is this effect small for some reason, and if so, why and how small? Can we compare how small the effect is (quantitatively) in comparison to the EDP?

Martin C.
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Solidification
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1 Answers1

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Electrostatic Coulomb effects actually reduce the pressure inside a white dwarf. This is because the electrons are (roughly) uniformly distributed, whilst the positive charges are concentrated in the nuclei. This makes the net Coulomb energy per nucleus negative and the gas more compressible.

It is a small correction - about 1.5 % of the relativistic degneracy pressure in carbon white dwarfs (it is proportional to $Z^{2/3} n_e^{4/3}$, where $Z$ is the atomic number of the nuclei and $n_e$ is the electron number density), but more important at lower densities. The small reduction in pressure at high densities slightly reduces the canonical Chandrasekhar limit by about $0.01M_\odot$. It also means that white dwarfs made of different chemical elements have slightly different mass-radius relationships even if the number of electrons per mass unit is identical (e.g. carbon vs oxygen vs neon).

An approximate treatment is discussed in chapter 2 of "Black Holes, White Dwarfs and Neutron Stars" by Shapiro & Teukolsky.

ProfRob
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