Structure of Polytropic Stars in General Relativity
The inner structure of a star or a primordial interstellar cloud is a major topic in classical and relativistic physics. The impact that General Relativistic principles have on this structure has been the subject of many research papers. In this paper we consider within the context of General Relativity a prototype model for this problem by assuming that a star consists of polytropic gas. To justify this assumption we observe that stars undergo thermodynamically irreversible processes and emit heat and radiation to their surroundings. Due to the emission of this energy it is worthwhile to consider an idealized model in which the gas is polytropic. To find interior solutions to the Einstein equations of General Relativity in this setting we derive a single equation for the cumulative mass distribution of the star and use Tolman-Oppenheimer-Volkoff equation to derive formulas for the isentropic index and coefficient. Using these formulas we present analytic and numerical solutions for the polytropic structure of self-gravitating stars and examine their stability. We prove also that when the thermodynamics of a star as represented by the isentropic index and coefficient is known, the corresponding matter density within the star is uniquely determined.
Mass density pattern within a star is an important problem and has been the subject of intense ongoing research. Within the context of classical physics Euler-Poisson equations form the basis for this research [2, 3]. A special set of solutions to these equations for non-rotating spherically symmetric stars with mass-density and flow field is provided by the Lane-Emden functions. The generalization of these equations to include axi-symmetric rotations was by considered by Milne, Chandrasekhar[6, 7] and many others.
Another aspect of this problem relates to the emergence of density pattern within a primordial interstellar gas. This problem was considered first by Laplace in 1796 who conjectured that a primitive interstellar gas cloud may evolve under the influence of gravity to form a system of isolated rings which may in turn lead to the formation of planetary systems [18, 19, 20]. Such a system of rings around a protostar has been observed recently in the constellation Taurus.
It is obvious however that on physical grounds this problem should treated within the context of General relativity. The Einstein equations of General Relativity are highly nonlinear [4, 8] and their solution presents a challenge that has been addressed by many researchers [8, 9, 10]. An early solution of these equations is due to Schwarzschild for the field exterior to a spherical star . However, interior solutions (inside space occupied by matter) are especially difficult due to the fact that the energy-momentum tensor is not zero. Static solutions for this case were derived under idealized assumptions (such as constant density) by Tolman, Adler[17, 8], Buchdahl and were addressed more recently in the lecture series by Gourgoulhon and the review by Paschalidis and Stergioulas [31, 9, 11, 12, 13, 20] (these references contain a a lengthy list of publications on this topic). In addition various constraints were derived for the structure of a spherically symmetric body in static gravitational equilibrium [14, 15, 16, 17, 18]. Interior solutions in the presence of anisotropy and other geometries were considered also [21, 22, 23, 24]. An exhaustive list of references for exact solutions of the Einstein equations appears in [8, 9].
Due to the physical complexity of star interiors which involves several concurrent physical processes we consider in this paper an idealized model based on General Relativity in which the star (or the interstellar gas cloud) is polytropic and inquire about the mass density pattern within the star under this assumption. This model takes into account some of the thermodynamic processes within a star which have been ignored so far in the literature. To justify the imposed idealizations we observe that stars undergo thermodynamically irreversible processes and emit heat and radiation to their surroundings. Due the emission of this energy one can envision a situation in which the gas entropy within a star remains nearly constant.
For polytropic gas we have the following relationship between pressure and density
where is the isentropy index and is the isentropy coefficient. In the literature when the gas is considered to be isothermal. However, when (and only when) equals the ratio of specific heat at constant pressure or specific heat at constant volume, the gas is isentropic. For all other values of the gas is called polytropic. This marks finite heat exchanges within the fluid. However, one can consider a more general functional relationship between and where both and are dependent on . In this paper, however, we restrict ourselves and consider only functional relationships between and in which only one of these parameters is dependent on , viz. either where is constant or where is constant. These two position-dependent expressions for the isentropy relationship represent different physical properties of the gas.
The plan of the paper is as follows: In Section we review the basic theory and equations that govern mass distribution and the components of the metric tensor. In Section we derive an equation for the cumulative mass of the sphere as a function of and use the Tolman-Oppenheimer-Volkoff (TOV) equation to derive equations for the isentropy index and coefficient. We then prove that when these two parameters are predetermined the mass density within the star cannot be chosen arbitrarily. In Section we address the stability of a given mass distribution to small perturbations.In Section we present exact and numerical solutions for polytropic spheres with predetermined mass density distribution and determine their isentropy coefficients and stability. We summarize with some conclusions in Section .
In this section we present a review of the basic theory, following chapter in .
The general form of the Einstein equations is
where and are respectively the contracted form of the Riemann tensor and the Ricci scalar,
is the matter stress-energy tensor, is Newton’s gravitational constant, is the speed of light in a vacuum and is the metric tensor.
The general expression for the stress-energy tensor is
where is the proper density of matter and is the four vector velocity of the flow.
In the following we shall assume that , and a metric tensor of the form
where , and are the spherical coordinates in 3-space.
When matter is static and takes the following form,
In the following we normalize to ; remains .
3 On the Structure of Isentropic Stars
In this section we consider Isenropic stars and derive general analytic expressions for , and .
3.1 General Equation for
Using the equations presented in the previous section one can derive a single equation for for a polytropic star where both and are functions of :
Differentiating this equation to obtain an expression for and substituting in (2.10) leads finally to the following general equation for :
This is a highly nonlinear equation but it simplifies considerably when is a constant or is an integer. A solution of this equation can then be used to compute the metric coefficients using (2.6) and (3.4). With this equation it is feasible to investigate the dependence of the mass distribution on the parameters and
In view of the difficulty of obtaining analytic solutions for (3.5) an alternative strategy should be used to investigate the structure of polytropic stars. Thus if we start with some analytic form of then we can use (2.5) to compute . With this data it is straightforward to derive differential equations for and using the TOV equation (2.9).
3.2 Equation for when is Constant
3.3 Equation for when is Constant
Following the same strategy as in the previous subsection we obtain a differential equation for
Thus in this setting (where is predetermined) one can use (3.6) or (3.7) to compute or by solving a first order differential equation. Alternatively, (3.6) and (3.7) can be converted to an equation for by using (2.5). We can then choose a functional form for either (and a constant value for in (3.6)) or (and a constant value for in (3.7)) to determine subject to proper boundary conditions. It follows then under the tenets of General Relativity the density of a polytropic star cannot be assigned arbitrarily. The same follows from (3.5) when the functional form and is predetermined.
We give several examples.
3.4 Equation for when and are Constant
Solving (3.7) algebraically for and substituting in (2.5), we obtain after some algebra a rather complicated equation for with and constant. Therefore we present only a special case of this equation in which both are constant.
Algebraically isolating and substituting in (2.5) we obtain the following equation for :
In particular, when and is normalized to (3.9) reduces to
A similar equation can be derived from (3.9) for with .
In Fig. 1 we present the solutions of these two cases ( and , each with ) by the red and blue dashed lines. The boundary conditions on are and . These boundary conditions are needed to avoid numerical singularities at and .
Similarly if we let (where is a constant) then for we obtain for in Fig. 1 the solid magenta and green lines, respectively.
Thus we demonstrate that in the context of general relativity the mass density of polytropic star with and constant cannot be assigned arbitrarily.
Using (3.6) and following the same steps described above we can obtain similar equations to the case where and is constant.
In this section we derive equations that determine the stability of polytropic stars using the two models that were discussed in (3.6) and (3.7). We then apply these results to the star model discussed in the previous section.
To implement this objective we introduce a perturbation to a star with initial cumulative mass distribution :
The star will then be considered stable if, for a perturbation with initial value , remains bounded and . It will be considered unstable otherwise. To derive the equation that satisfies we consider the two polytropic models separately.
For simplicity we treat here only the case . Following the same steps as in the previous subsection we obtain
where is the density which corresponds to .
5 Polytropic Gas Spheres and their Stability
In the present section we solve (2.5) through (2.8) for polytropic gas spheres. We present four solutions. The first is an analytic solution of these equations while the other two utilize numerical computations. We consider the stability of these solutions. The stability of the solution is calculated using (4.3).
5.1 Polytropic Sphere with Analytic Solution
For the present case we start by choosing a functional form for the density and then solve (2.5) for . Equation (2.6) becomes an algebraic equation for while (2.7) is a differential equation for . Substituting this result in (3.2), one can compute the isentropy coefficient (or isentropy index ).
The following illustrates this procedure and leads to an analytic solution for the metric coefficients.
Consider a sphere of radius (where ) with the density function
Observe that although is singular at the total mass of the sphere is finite.
Using (2.6) yields
For R=1 the solution is
At we have and the metric is singular at this point. This reflects the fact that the density function (5.4) has a singularity at (but the total mass of the sphere is finite). We observe that this singularity in at does not correspond to any of those classified by Arnold et al . This is due to the fact that none of the solutions presented in  has a periodic structure.
To determine the constants and we use the fact that at the value of should match the classic Schwarzschild exterior solution
and the pressure (see 2.8) is zero. These conditions lead to the following equations:
The solution of these equations is
Using (2.8) we obtain the following expression for the pressure
Assuming that we depict for this solution in Fig. 2.
Note that trying to model this result by a relationship of the form leads to discontinuities in the values of . For the differential equation for is
The solution of this equation is
and applying the boundary conditions on and the pressure at we find that
If we assume that the relationship between the pressure and the density is of the form then exhibits several local spikes in the range but is zero otherwise.
5.2 Spheres with Oscillatory Density Functions
Here we discuss several examples of spheres with oscillatory density functions and determine the appropriate polytropic index (or coefficient) that describes these spheres. We probe also for the stability of these mass configurations to small perturbations.
5.2.1 Infinite Sphere with Exponentially Decreasing Density
It follows from (2.6) (with ) that
Observe that although the sphere is assumed to be of infinite radius the density approaches zero exponentially as and the total mass of the sphere is finite.
Substituting these expressions in (3.6) with and and solving for we obtain Fig. 4 which exhibits a strong decline in the value of as the density decreases exponentially. If we substitute , and in (3.7 we obtain Fig. where has a steep negative gradient as .
5.2.2 Finite Sphere with Ring Structure
We consider a sphere of radius with density function
¿From (2.5) with we then have
where the total mass of the sphere is .
Fig. 7 depicts the solution of (3.6) for with , and . This figure exhibits a steep downward slope in the value of beyond due to the decrease in the density. Fig. 8 displays the solution of (3.7) for with and the same values for and . The spikes in the values of in this figure reflect the thermodynamics processes that are ongoing due to the oscillations in the density.
5.2.3 Infinite Sphere with Ring Structure
Consider a sphere of infinite radius with the density function
where , are constants and .
Solving (2.5) with the initial condition yields
Observe that although the sphere is assumed to be of infinite radius the density approaches zero exponentially as and the total mass of the sphere is finite.
If we interpret the density function (5.17) as one that corresponds to the density of a primordial gas cloud with ring structure then the results shown in Fig. 10 and Fig. 11 demonstrate that the thermodynamic activity within the cloud is reflected by the oscillatory behavior of and .
As to stability we found that when a polytropic model is used to describe the gas then it is stable only for for perturbations with . A plot of under this assumption is presented in Fig. 12. However when we assumed that the mass density in the cloud remained stable for perturbations satisfying and we obtained Fig. 13. This demonstrates that in this particular case the mass distribution (5.18) has a much larger basin of stability when the gas can be modeled by the relationship .
In this paper we considered the steady states of a spherical protostar or interstellar gas cloud where general relativistic considerations are taken into account. In addition we considered the gas to be polytropic, thereby removing the (implicit or explicit) assumption that it is isothermal. Two polytropic models for the gas were considered, the first in the form and the second in the form . Under these assumptions we were able to derive a single equation for the total mass of the sphere as a function of , from whose solution the corresponding metric coefficients can be computed in straightforward fashion. Using the TOV equation we derived equations for and . We proved that when either or are constants the mass density of the sphere cannot be chosen arbitrarily. We derived also an equation for stability of these configurations to perturbations in mass density.
Using several idealized models for the density within primordial gas clouds we were able to compute the appropriate polytropic coefficient and index and thus gain new insights about their thermodynamic structure. In particular we showed that the mass distribution of a gas cloud with ring structure can be stable to perturbations. The evolution of this ring structure in time (within the framework of General Relativity) will be investigated in a subsequent paper.
We conclude then that General Relativity can provide new and deeper insights about the actual structure of stars and primordial gas clouds and the emergence of density patterns within these objects.
To our best knowledge these solutions represent a new and different class of interior solutions to the Einstein equations which have not been explored in the literature.
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