The size of a white dwarf

Adapted from IOAA 2024 (Rio de Janeiro), theory problem T4. We have restated it in our own words and kept the original data. The official solution gives 1.44 × 10⁶ m with rounder constants, and our constants give 1.43 × 10⁶ m. The solution was last checked on 23 October 2026.

A white dwarf is held up by the pressure of its degenerate electrons, $P=(3/8\pi {)}^{2/3}(h^2/5m_{\text{e}})n_{\text{e}}^{5/3}$, where $n_{\text{e}}$ is the number of electrons in a unit volume. Write $n_{\text{e}}$ in terms of the density $\rho$ using ${\mu }_{\text{e}}$, the number of nucleons for each electron, and take protons and neutrons to have the same mass $m_{\text{p}}$. The star is in hydrostatic equilibrium, $dP/dr=-GM(r)\rho (r)/r^2$. Assume that its density is uniform, that its central pressure obeys the same law, and that at the surface $dP/dr\approx -P_{\text{c}}/R$, where $P_{\text{c}}$ is the central pressure and $R$ the radius.

(a) The mass and radius are related by $R=a{M}^b$. Find $b$, and find $a$ in terms of physical constants and ${\mu }_{\text{e}}$. (6 points)

(b) Estimate the radius of a white dwarf of fully ionised carbon-12 with a mass of one solar mass. (4 points)

Answer(a) $b=-1/3$, and $a$ is as given in step 3. (b) The radius is $1.43\times {10}^6$ m, about 1400 km. Real white dwarfs of this mass are about four times larger, because the density is far from uniform and the electrons at the centre move close to the speed of light.