Gravitational instability of finite isothermal spheres in general relativity. Analogy with neutron stars

We investigate the effects of relativity on the gravitational instability of finite isothermal gaseous spheres. In the first part of the paper, we treat the gravitational field within the framework of Newtonian mechanics but we use a relativistic equation of state in the condition of hydrostatic equilibrium. In the second part of the paper, we study the full general relativistic problem for a gas described by an equation of state $p=qϵ$ such that the pressure is proportional to the energy density (``isothermal’’ distribution). For $q=1\mathrm{/}3$, this equation of state describes the core of neutron stars. The mass-density diagram displays some damped oscillations and there exists a critical value of mass-energy above which no equilibrium state is possible. We show analytically that the mass peaks are associated with new modes of instability. These results are strikingly similar to those obtained by Antonov [Vest. Leningr. Gos. Univ. 7, 135 (1962)] and Lynden-Bell & Wood (1968) for a classical isothermal gas. Our study completes the analogy between isothermal spheres and neutron stars investigated by Yabushita [MNRAS 167, 95 (1974)].