Central Temperature and Density of Stars in Gravitational Equilibrium

Abstract

The relation between the central temperature and the central density is investigated for a polytropic star and a polytropic (or isothermal) stellar core, which are in gravitational equilibrium. The ranges of the temperature and density cover the cases of non-degenerate and degenerate, non-relativistic as well as relativistic, electrons. It is shown that, when the mass of the star or of the core is smaller than a certain critical value which depends slightly on the polytropic index N and is equal to Chandrasekhar's limiting mass for N =3, the central temperature takes a maximum value depending on the mass at the onset stage of the electron degeneracy. §I. Introduction and summary Throughout the life of a star, the central temperature and density changes to a considerable extent. The features of the over-all evolution of the star are determined mainly by how far the central temperature rises in its whole life and accordingly how far the synthesis of the chemical elements proceeds 1n the interior. For example, it is well known that a star smaller than about one-tenth of the solar mass is unable to undergo the hydrogen burning phase, since at a certain stage of its pre-main-sequence contraction electron degeneracy sets in and afterwards the central temperature has to decrease. 1 ), 2 ),B) On the other hand, if a star is massive enough, it will follow all the phases of nuclear burning, from hydrogen burning to the formation of iron, developing a shell structure composed of zones of different chemical compositions. The evolution of such a massive star has already been studied up to the central carbon-burning phase for a stellar mass of 16 M 0 . 4 ), 5 ) It has been found that, in the stages before the onset of carbon burning, electrons in the central region are relativistically degenerate if the energy loss by neutrino emission is taken into account. In order to investigate the over-all features of stellar evolution and their dependence on the mass, it will be necessary first to know the over-all varia­ tions in the central temperature and density which are allowed for stellar con­ figurations in gravitational equilibrium. Strictly speaking, these variations depend on the details c;>f the development of the shell structure. In this paper, however, an attempt is made to find an approximate relation which holds between the central temperature and density of a star or of a stellar core, which satisfies the following simplified conditions.