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I somewhere see that in a white dwarf, we have this equation:

$\rho = 2 N_e * m_p / V = 2 n_e * m_p$

where $N_e$ is the number of particles (in this case e, electrons) and $n_e$ is the number density. and $m_p$ mass is mass of proton. My problem is with the 2 in this equation. why $2 * n_e$ and not just $n_e$?

Also about Electric Potential Energy between a Ion and electron, somewhere said:

$ E = (Z e * 1e) * n_e^{1/3} / (4 \pi \epsilon_0) $

I don't understand the $n_e$ term. Is the above really a equation or is it an approximation? I think it can't be an equation because $ne \neq 1/r^3$.

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The 2 in your equation is the average number of mass units per electron. e.g. totally ionised carbon has 6 electrons and 12 mass units. In your formula, the mass of a proton is used as a mass unit (close enough in this case).

In your second equation $n_e^{-1/3}$ is an approximation for the inter-particle separation. This is an approximation and there is a small numerical factor that would need to be introduced in an accurate treatment.

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