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# 4.1 The electron density

Because we will be considering electron degeneracy, we begin by deriving a new way of expressing the electron density. The general equation for the number density n of some type of particle is , where ρ is the gas density, X is the mass fraction and m is the particle mass.

We can write this for electrons just as easily as for nucleons: . However, it is more convenient to write where Ye is the number of electrons per nucleon. This leads to the expression

Equation label:(13)

- What is the value of Ye in: (a) pure ionised hydrogen; (b) pure ionised helium?

- (a) Hydrogen has one electron and one nucleon (its proton), so Ye = 1; (b) helium has two electrons and four nucleons (two protons and two neutrons), so Ye = 0.5.

You can readily convince yourself that Ye = 0.5 for the dominant isotopes of carbon-12 and oxygen-16 also (since carbon-12 has 12 nucleons and 6 electrons, while oxygen-16 has 16 nucleons and 8 electrons). For white dwarfs, the amount of hydrogen is negligible – it has all been burnt to helium or on to carbon and/or oxygen – so Ye = 0.5 for white dwarfs also.

For an electron-degenerate gas, you know that the pressure is independent of temperature, and is given by a constant multiplied by some power of the electron density. Using Equation 13, we can therefore write:

- for non-relativistic gas, .
- for ultra-relativistic gas,

The so-called Clayton model for the internal structure of a star gives the core pressure required to support a star in terms of its core density as:

Equation label:(14)

Here G is Newton’s gravitational constant (). By saying that degenerate electrons provide the required internal pressure to support the star, that is, by equating this to the degenerate-electron pressure, it is possible to write these equations purely in terms of the mass and core density, which therefore allows us to write one variable in terms of the other, as the following activity demonstrates.

## Activity 4

1. By equating the core pressure in a star to the degenerate pressure of non-relativistic electrons, derive an expression for the core density in terms of its mass M and the number of electrons per nucleon, Ye. Leave physical constants unevaluated.

2. Using the relation between electron number density ne and gas density ρc in the core of the star, express the core electron density as a function of stellar mass.

### Answer

1. Equating the core pressure to the pressure of non-relativistic degenerate electrons , where , we have

   Collecting terms in ρ on the left-hand side, and all others on the right, we get

   Cubing this gives

   and substituting for KNR gives

   Consolidating the two numerical factors, we obtain

2. The electron number density in the core of the star is , so substituting in the core density from part (a) gives

Activity 4 shows that in the non-relativistic case, the density in the core of a star may be written as

Equation label:(15)

or, in terms of the degenerate-electron number density, as

Equation label:(16)
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