5 Conclusion

In this paper, we have studied the structure and the stability of isothermal gas spheres in the framework of general relativity. We have found that relativistic isothermal spheres (like neutron cores) exhibit the same kind of behaviors as Newtonian isothermal spheres but with a different interpretation of the parameters. In this analogy, the critical energy and the critical temperature found by Antonov (1962) and Lynden-Bell & Wood (1968) for an isothermal gas are in some sense the classical equivalent of the limiting mass of neutron stars discovered by Oppenheimer & Volkoff (1939) (there also exists a limiting mass for classical isothermal gas spheres with given temperature and radius, see Eq. (187)). In addition, the spiral behavior of the temperature-energy curve for a classical gas has the same origin as the mass-radius diagram for neutron stars. This spiral behavior, as well as the damped oscillations of the mass-density profile, are general features of isothermal configurations.

Although our model can reproduce qualitatively the main properties of neutron stars, it is however limited in its applications since the isothermal core is surrounded by an artificial "box'' instead of a more physical envelope like in the studies of Oppenheimer & Volkoff (1939), Misner & Zapolsky (1964) and Meltzer & Thorne (1966). However, this simplification allows us to study the stability problem analytically (or with graphical constructions) without any further approximation. This is a useful complement to the more elaborate works of the previous authors who had to solve the pulsation equation numerically or with approximations. This is also complementary to the study of Yabushita (1974) who considered isothermal gas spheres surrounded by an envelope exerting a constant pressure. The boundary conditions are of importance since Yabushita came to the conclusion that the onset of instability occurs before the first mass peak. In contrast, our model gives a stability criterion consistent with that of Misner & Zapolsky (1964) and other works. Furthermore, if we assume that the energy density at the edge of the isothermal core has a value $\sim$10 $^{15}gc^{2}/{\rm cm}^{3}$ (a typical prediction of nuclear models), we can reproduce almost quantitatively the results usually reported for neutron stars.

It should be stressed that the analogy between neutron stars and isothermal spheres is a pure effect of general relativity. When gravity is treated in the Newtonian framework, the classical condition of hydrostatic equilibrium requires a relationship between the pressure and the mass density $\rho=mn$ which, for dense matter (degenerate and relativistic), is a power law with index $\gamma=4/3$(see Chandrasekhar 1942). However, when gravity is treated in the framework of general relativity, the Oppenheimer-Volkoff equations require a relationship between the pressure and the mass-energy density $\epsilon$ which, for dense matter, is linear. Therefore, the core of neutron stars (treated with general relativity) is "isothermal'' while the core of white dwarf stars approaching the limiting mass (treated in a Newtonian framework) is "polytropic'', although the same equation of state is used (that for a completely degenerate and ultra-relativistic ideal Fermi gas). This is the intrinsic reason why the Mass-Radius relation for neutron stars exhibits a spiral behavior (Meltzner & Thorne 1966) while the Mass-Radius relation for white dwarf stars is monotonous (Chandrasekhar 1942). These remarks may increase the interest of studying isothermal gas spheres both in Newtonian mechanics and in general relativity.

Acknowledgements

This work was initiated during my stay at the Institute for Theoretical Physics, Santa Barbara, during the program on Hydrodynamical and Astrophysical Turbulence (February-June 2000). This research was supported in part by the National Science Foundation under Grant No. PHY94-07194.