6.4: Equations of Relativistic Stellar Structure and Their Solutions

In many respects the construction of stellar models for relativistic stars is easier than that for Newtonian models. The reasons can be found in the very conditions which make consideration of general relativity important. Except in the case of super-massive stars, when gravity has been able to compress matter to such an extent that general relativity is necessary to describe the metric of the space occupied by the star, all forms of energy generation which might provide opposition to gravity have ceased. Because of the high degree of compaction, the material generally has a high conductivity and is isothermal, so its cooling rate is limited only by the ability of the surface to radiate energy. In addition, the high density leads to equations of state in which the kinetic energy of the gas is relatively unimportant in determining the state of the gas. The pressure is determined by inter-nuclear forces and thus depends on only the density. In a way, the messy detailed physics of low-density gas, which depends on its chemical composition and internal energy, has been "squeezed" out of it and replaced by a simpler environment where gravity rules supreme. To be sure, the equation of state of nuclear matter is still an area of intense research interest. But progress in this area is limited as much by our inability to test the results of theoretical predictions as by the theoretical difficulties themselves.

a. A Comparison of Structure Equations

To see the sort of simplification that results from the effects of extreme gravity, let us compare the equations of stellar structure in the Newtonian limit, and the relativistic limit.

\[\label{6.3.1}\]

For relativistic stellar models, we need only solve equations \ref{6.3.1a} through \ref{6.3.1c} and \ref{6.3.1e} subject to certain boundary conditions. Combining equations \ref{6.3.1b} and \ref{6.3.1c}, we have just three equations in three unknowns − M(r), P, and ρ. Two of the equations are first-order differential equation requiring two constants of integration. One additional eigenvalue of the problem is required because we must specify the type (mass) of star we wish to make.

Thus, \[P(R)=0 \quad M(R)=M \quad M(0)=0\label{6.3.2}\]

For the eigenvalue, we might just as well have specified the central pressure for that would lead to a specific star and would make the problem an initial value problem. We can gain some insight into the effects of general relativity by looking at a concrete example.

b. A Simple Model

The reduction of the equation of state to the form \(P=\mathrm{P(\rho)}\) is reminiscent of the polytropic equation of state. For polytropes, the combination of the equation of state with hydrostatic equilibrium led to the Lane-Emden equation which specified the entire structure of the star subject to certain reasonable boundary conditions. To be sure, we could write a similar "relativistic" Lane-Emden equation for relativistic polytropes, but instead we take a different approach. Let us consider a situation where the constraint presented by the equation of state is replaced by a direct constraint on the density. While this does not result in a polytropic equation of state, it is illustrative and analytic, allowing for the solution to be obtained in closed form. Assume the density to be constant, so that \[\rho(r)=\rho_0=\text { constant }\label{6.3.3}\]

The first of the two remaining equations of stellar structure then has the direct solution \[M(r)=\frac{4 \pi r^3 \rho_0}{3}\label{6.3.4}\]

while the Oppenheimer-Volkoff equation of hydrostatic equilibrium becomes \[\frac{d P(r)}{d r}=-\frac{4 \pi G r \rho_0^2\left[1+P /\left(\rho_0 c^2\right)\right]\left[1+3 P /\left(\rho_0 c^2\right)\right]}{3\left[1-8 \pi G \rho_0 r^2 /\left(3 c^2\right)\right]}\label{6.3.5}\]

This equation has an analytic solution which can be obtained by direct, albeit somewhat messy, integration. We can facilitate the integration by introducing the variables \[y=\frac{P}{\rho_0} \quad \gamma=\frac{8 \pi G \rho_0}{3 c^2}=\frac{2 G M}{R^3 c^2}\label{6.3.6}\]

and rewrite the equation for hydrostatic equilibrium as \[\frac{d y}{d r}=-\frac{1}{2} \gamma c^2 \frac{\left(1+y / c^2\right)\left(1+3 y / c^2\right) r}{1-\gamma r^2}\label{6.3.7}\]

which is subject to the boundary condition \(\gamma(\mathrm{R})=0\). With zero as a value for the surface pressure, the solution of equation \ref{6.3.7} is \[y=c^2 \frac{\left(1-\gamma r^2\right)^{1 / 2}-\left(1-\gamma R^2\right)^{1 / 2}}{3\left(1-\gamma R^2\right)^{1 / 2}-\left(1-\gamma r^2\right)^{1 / 2}}\label{6.3.8}\]

in terms of physical variables this is \[P(r)=\rho_0 c^2 \frac{\left[1-2 G M r^2 /\left(R^3 c^2\right)\right]^{1 / 2}-\left[1-2 G M /\left(R c^2\right)\right]^{1 / 2}}{3\left[1-2 G M /\left(R c^2\right)\right]^{1 / 2}-\left[1-2 G M r^2 /\left(R^3 c^2\right)\right]^{1 / 2}}\label{6.3.9}\]

Now we evaluate equation \ref{6.3.9} for the central pressure by letting r go to zero. Then \[P_c=\rho_0 c^2 \frac{1-\left[1-2 G M /\left(R c^2\right)\right]^{1 / 2}}{3\left[1-2 G M /\left(R c^2\right)\right]^{1 / 2}-1}\label{6.3.10}\]

As the central pressure rises, the star will shrink, reflecting the larger effects of gravity so that \[\operatorname{Lim}_{P_c \rightarrow \infty} R=\frac{9}{8} \frac{2 G M}{c^2}=\frac{9}{8} R_s\label{6.3.11}\]

where \(\mathrm{R_s}\) is the Schwarzschild radius. This implies that the smallest stable radius for such an object would be slightly larger than its Schwarzschild radius. A more reasonable limit on the central pressure would be to limit the speed of sound to be less than or equal to the speed of light. A sound speed in excess of the speed of light would suggest conditions where the gas would violate the principle of causality. Namely, sound waves could propagate signals faster than the velocity of light. Since \(\mathrm{P}/p_0\) is the square of the local sound speed, consider \[\operatorname{Lim}_{P_c \rightarrow c^2 \rho_0} R=\frac{4}{3} R_s\label{6.3.12}\]

This lower value for the central pressure yields a somewhat larger minimum radius. Since any reasonable equation of state will require that the density monotonically decrease outward and since causality will always dictate that the sound speed be less than the speed of light, we conclude that any stable star must have a radius R such that \[R \geq \frac{4}{3} R_s\label{6.3.13}\]

In reality, this is an extreme lower limit, and neutron stars tend to be rather larger and of the order of 4 or 5 Schwarzschild radii. Nevertheless, neutron stars still represent stellar configurations in which the general theory of relativity plays a dominant role.

c. Neutron Star Structure

The larger size of actual neutron stars, compared to the above limit, results from detailed consideration of the physics that specifies the actual equation of state. Although this is still an active area of research and is likely to be so for some time, we will consider the results of an early equation of state given by Salpeter^5,6. He shows that we can write a parametric equation of state in the following way: \[\begin{aligned}
P & =\frac{1}{3} K\left(\operatorname{Sinh} t-8 \operatorname{Sinh} \frac{t}{2}+3 t\right) \\
\rho & =K(\operatorname{Sinh} t-t)
\end{aligned}\label{6.3.14}\]

where \[\begin{aligned}
K & =\frac{\pi \mu_0^4 c^5}{4 h^3} \\
t & =4 \log \left\{\frac{\hat{p}}{\mu_0 c}+\left[1+\left(\frac{\hat{p}}{\mu_0 c}\right)^2\right]^{1 / 2}\right\}
\end{aligned}\label{6.3.15}\]

and \(\hat{\rho}\) is the maximum Fermi momentum and may depend weakly on the temperature. The relationship between the mass and central density is shown in Figure 6.1. If one includes the energy losses from neutrinos due to inverse beta decay, there exists a local maximum for the mass at around 1 solar mass.

Figure 6.1 shows the variation of the mass of a degenerate object central density. The large drop in the stable mass at a density of about 10¹⁴ gm/cm³ represents the transition from the electron degenerate equation of state to the neutron degenerate equation of state.

More recent modifications to the equation of state show a second maximum occurring at slightly more than 2 solar masses. Considerations of causality set an absolute upper limit for neutron stars at about 5M_⊙. So there exists a mass limit for neutron stars, as there does for white dwarfs, and it is probably about 2.5M_⊙. However, unlike the Chandrasekhar limit, this mass limit arises because of the effects of the general theory of relativity. As we shall see in the next section, this is also true for the mass limit of white dwarfs.

We have not said anything about the formidable problems posed by the formulation of an equation of state for material that is unavailable for experiment. To provide some insight into the types of complications presented by the equation of state, we show below, in Figure 6.2 the structure of a neutron star as deduced by Rudermann⁷.

The equation of state for the central regions of such a star still remains in doubt as the Fermi energy reaches the level for the formation of hyperons. Some people have speculated that one might reach densities sufficient to yield a "quark soup". Whatever the details of the equation of state, they matter less and less as one approaches the critical mass. Gravity begins to snuff out the importance of the local microphysics. By the time one reaches a configuration that has contracted within its Schwarzschild radius only the macroscopic properties of total mass, angular momentum, and charge can be detected by an outside observer (for more on this subject see Olive, 1991).

Figure 6.2 shows a section of the internal structure of a neutron star. The formation of crystal structure in the outer layers of the neutron star greatly complicates its equation of state. Its structure may be testable by observing the shape changes of rapidly rotating pulsars as revealed by discontinuous changes in their spin rates as they slow down.

Although this ultimate result occurs only when the object has reached the Schwarzschild radius, aspects of its approach are manifest in the insensitivity of the global structure to the equation of state as the limiting radius is approached. This has the happy result for astronomy that the mass limit for neutron stars can comfortably be set at around 2½ M_⊙ regardless of the vagaries of the equation of state. It has an unhappy consequence for physics in that neutron stars will prove a difficult laboratory for testing the details of the equation of state for high- density matter.