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Mass-Radius relation

Using the TOV equations and an equation of state, a mass-radius curve can be found. The idea is that for the correct equation of state, every neutron star that could possibly exist would lie along that curve. This is one of the ways equations of state can be constrained by astronomical observations. To create these curves, one must solve the TOV equations for different central densities. For each central density, you numerically solve the mass and pressure equations until the pressure goes to zero, which is the outside of the star. Each solution gives a corresponding mass and radius for that central density.

Mass-radius curves determine what the maximum mass is for a given equation of state. Through most of the mass-radius curve, each radius corresponds to a unique mass value. At a certain point, the curve will reach a maximum and start going back down, leading to repeated mass values for different radii. This maximum point is what is known as the maximum mass. Beyond that mass, the star will no longer be stable, i.e. no longer be able to hold itself up against the force of gravity, and would collapse into a black hole. Since each equation of state leads to a different mass-radius curve, they also lead to a unique maximum mass value. The maximum mass value is unknown as long as the equation of state remains unknown.

This is very important when it comes to constraining the equation of state. Oppenheimer and Volkoff came up with the Tolman-Oppenheimer-Volkoff limit using a degenerate gas equation of state with the TOV equations that was ~0.7 Solar masses. Since the neutron stars that have been observed are more massive than that, that maximum mass was discarded. The most recent massive neutron star that was observed was PSR J0952-0607 which was 2.35 ± 0.17 solar masses. Any equation of state with a mass less than that would not predict that star and thus is much less likely to be correct.

An interesting phenomenon in this area of astrophysics relating to the maximum mass of neutron stars is what is called the "mass gap". The mass gap refers to a range of masses from roughly 2-5 solar masses where very few compact objects were observed. This range is based on the current assumed maximum mass of neutron stars (~2 solar masses) and the minimum black hole mass (~5 solar masses). Recently, some objects have been discovered that fall in that mass gap from gravitational wave detections. If the true maximum mass of neutron stars was known, it would help characterize compact objects in that mass range as either neutron stars or black holes.