Minimum Neutron Star Mass

This entry is largely based on Burrows and Ostriker 2014.

Let us consider a white dwarf that collapses into a neutron star. In this process the object gains gravitational potential energy, but has to "pay" an energy fee for pushing the nucleons close together. This energy fee per nucleon is roughly (see entry on the strong force)

$u \approx m_{\pi} c^2 \left(m_{\pi}/m_n\right)^2$

where $m_{\pi}$ is the pion mass, $c$ is the speed of light and $m_n$ is the average mass of a nucleon. The gravitational binding energy scales as the mass squared $M^2$ while the nuclear binding energy is linear in the mass, and hence below a certain threshold the collapse is energetically dis-favourable. We can estimate this mass by equating the gravitational energy with the nuclear energy

$\frac{G M^2}{R} \approx u \frac{M}{m_n}$

If we assume nuclear density, i.e. that the average distance between nucleons is the pion Compton length $\lambda_{\pi}$, we can relate the radius to the mass

$R \approx \lambda_{\pi} \left(\frac{M}{m_n}\right)$

Solving for the mass yields

$M \approx \frac{m_{\pi} m_p^3}{m_n^5}$

where $m_p \approx \sqrt{\hbar c/G}$ is the Planck mass. The value of this lower bound is around 5 Jupiter masses.