White Dwarf Temperature

White dwarfs have a mass comparable to that of the sun $M_{\odot}$, and a radius comparable to that of the earth $R_e$ . Their temperature therefore cannot be greater than their gravitational binding energy

$k T < \frac{G M_{\odot} m_e}{R_{\oplus}} \approx 100 eV$

where $m_e$ is the mass of the electron.

White dwarfs cool mainly by blackbody emission, so the temperature is determined by the condition that the cooling time be shorter than the age of the universe $t_H$

$\frac{N k T}{R_{\oplus}^2 \sigma T^4} < t_H \Rightarrow k T > \left( \frac{M_{\odot}}{m_p} \frac{h^3 c^2}{R_{\oplus}^2 t_H} \right)^{1/3} \approx 120 meV$

where $N = M_{\odot}/m_p$ is the number of protons in a white dwarf (i.e. the ratio between the total mass and the mass of a proton, and $\sigma \approx k^4/h^3/c^2$ is the Stefan Boltzmann constant.

This calculation assumes that the temperature is roughly uniform throughout the white dwarf, unlike main sequence stars where a temperature gradient is maintained through nuclear energy production. Also, the calculation ignores degeneracy because the protons are not degenerate.