Asymptotic Behavior of Cold Superdense Stars

Abstract

The behavior of superdense ("neutron") stars at absolute zero has been studied. It is shown that, with certain assumptions about the equation of state, the mass and radius of such a star approach constants in an oscillatory fashion as the star's central density increases. The assumptions about the equation of state are: $\frac{\mathrm{dP}}{d\rho }>0\mathrm{}$ everywhere, and for sufficiently high-density $\rho$, the pressure divided by the density $\frac{P}{\rho }$ approaches a constant. These assumptions are physically reasonable, especially if one assumes a real speed of sound which is finite, but always less than the speed of light. The results of the paper show that there exists for such stars an infinite series of ranges of the central density, in which ranges $\frac{\mathrm{dM}}{d{\rho }_0}$ alternates in sign, where $M$ is the total star mass and ${\rho }_0$ is the central density. This indicates alternate local stability and instability; however, the total binding energy is positive for ${\rho }_0$ greater than ∼${10}^{16}$ g/${\mathrm{cm}}^3$, so that instability against large-scale deformation exists. A striking feature of the results of this paper is that their qualitative nature does not depend on whether or not the general relativistic form of the equations is used. The exact quantitative results do, of course, depend on the form of the equations, as well as on the exact equation of state used.