Introduction

In a previous paper [#!LiquidShells!#] we established the exact static solution of the Einstein field equations for the case of a spherically symmetric shell of liquid fluid located between two arbitrary radial positions $r_{1}$ and $r_{2}$ of the Schwarzschild system of coordinates. In this paper we will give the complete solution for a similar problem, that of a spherically symmetric distribution of a gaseous fluid that satisfies the equation of state of a polytrope. We will see that for most sets of values of the physical parameters of the system the Einstein field equations coupled with the polytropic equation of state automatically imply the existence of certain radii $r_{1}$ and $r_{2}$ where the energy density of the gas becomes exactly zero, thus giving rise to two gas-vacuum interfaces. These two values of the radial variable are not imposed by hand, but are a consequence of the equations describing the dynamics of the system. There are no geometrical free parameters, all the free parameters of the system are those describing the state and properties of the matter.

This puts us in a position, in the current problem, and in a very natural way, which is very similar to the one we had in [#!LiquidShells!#], with a shell of fluid matter surrounding an internal vacuum region and surrounded by an external vacuum region. In these two vacuum regions the solutions of the field equations are known exactly, and were in fact derived and discussed in detail in [#!LiquidShells!#]. Consequently, all that was said in that paper regarding the inner and outer vacuum regions is valid here without any change. However, our current problem in this paper is far less academic in nature, being much closer to the astrophysical applications. The family of solution that we find here can be considered a generalization of the particular family of solutions originally found by Tooper [#!tooper!#].

Results similar to the ones we present here were obtained for the case of neutron stars by Ni [#!NiNeutrStars!#], including the automatic generation of the inner and outer matter-vacuum interfaces. However, the crucial consideration of the interface boundary conditions was missing from that analysis, thus leading to incomplete results. The discussion of the interface boundary conditions was subsequently introduced by Neslušan [#!NeslusanNeutrStars!#,#!NeslusanReview!#], thus completing the analysis of the case of the neutron stars. Just as in [#!LiquidShells!#] and in the present work, the discussion of the interface boundary conditions led, also in that case, to an inner vacuum region containing a singularity at the origin and a gravitational field leading matter and energy away from the origin.

There is a connection between the purely numerical results presented in [#!NiNeutrStars!#] and [#!NeslusanNeutrStars!#] and the ones we present here, which are partly analytical and partly numerical. This is so because the equation of state for a neutron star can be approximated by a polytrope under certain particular conditions. On the other hand, however, the results we present here are not limited to cold neutron stars or some other particular type of object, but can be applied as well to normal stars of any type and size, at any temperature range, such as main sequence stars, red giants and white dwarfs, including configurations with two or more layers, with a different behavior of the matter in each layer.

In regard to the solution within the region containing the polytropic matter, we will present a solution which is partially analytical and partially numerical. We will reduce, by analytic means, all the quantities and functions involved to a single real function that is the solution of a second-order ordinary differential equation. We will show analytically that this function has a very simple behavior, which can be rigorously established without any recourse to numerical means. This function, which we will denote by $\beta(r)$, is the single element of the whole system that eventually has to determined in detail numerically, but its most important properties are analytically established beforehand.

The two radial positions $r_{1}$ and $r_{2}$ where the energy density becomes zero are soft singular points of the function $\beta(r)$, where by “soft” we mean that the function is not analytic, but also does not diverge to infinity at those points. Besides, the positions of these points are not known beforehand, since they are a consequence of the dynamics of the system. Due to all this, it is not really possible to integrate the differential equation numerically from these positions, which constitute the boundary of the region of space containing the matter, into the interior of the matter region. We will therefore have to develop an alternative way to solve this kind of differential problem numerically, since it is significantly different from a typical boundary value problem.

This paper is organized as follows. In Section 2 we gill give the full statement of the problem and describe the resolution method; in Section 3 we will obtain analytically the main properties of the solutions; in Section 4 we will present a few examples of complete numerical solutions; in Section 5 we will analyze and comment on the numerical results obtained; and in Section 6 we will present our conclusions.