This pages review theoretical bounds and estimates relating the mass $ M $ to the radius $ R $ of neutron stars.

## Black Hole Limit Edit

If the mass of a neutron star become too large (or, alternatively, the radius becomes too small) then the neutron star would become a black hole. Hence, a neutron star must exceed Schwarzschild's radius

$ R > \frac{2 G M}{c^2} $

where $ G $ is the universal constant of gravity and $ c $ is the speed of light.

## Causality Bound Edit

A related bound can be obtained from the condition that the speed of sound should not surpass the speed of light.

$ \frac{dP}{d \rho} < c^2 $

It is assumed that matter is cold, so the derivative is both at constant temperature and constant entropy. At very high values of pressure and density the above condition translates to

$ P < \rho c^2 $

Plugging the limiting value $ P = \rho c^2 $ into the general relativistic hydrostatic equation together with the mass - density relation

$ \frac{dm}{dr} = 4 \pi r^2 \rho $

yields an ordinary differential equation for the mass as a function of the radius $ m(r) $

$ m''(r) = \frac{2 m'(R)\left(r(c^2 - G m'(r))- 3 G m(r)\right)}{r\left(c^2 r - 2 G m(r)\right)} $

One boundary condition is that the enclosed mass vanishes at zero radius $ m(0) = 0 $. In order to numerically integrate the results the slope at zero radius is also required. However, it turns out that regardless of the choice of initial slope the mass radius curve always converges to the same line, as can be seen in the figure. The slope can be obtained from the ODE assuming $ m''(r) = 0 $

$ r \approx 4\frac{G m}{c^2} $

Since this was obtained using a limiting expression, it translate to the following inequality

$ R > \frac{4 G M}{c^2} $

This condition is stronger (more restrictive) then the former. It should be noted that this derivation is not entirely rigorous, as the definition of a relativistic speed of sound is different from its classical counterpart. In the relativistic case, the speed of sound is the square root of the derivative of the pressure with respect to enthalpy, not the density.

## Rotation Edit

The rotation frequency $ \omega $ of neutron stars can be readily measured. Rotation imposes two limits on the mass and radius. First, the neutron star must rotate slower than its breakup speed.

$ \frac{G M}{R^2} > \omega^2 R $

$ M > \omega^2 R^3 / G $

Second, the radius cannot exceed that of the light cylinder

$ R < \frac{c}{\omega} $