6.5: Relativistic Polytrope of Index 3

In Chapter 2, we remarked that the equation of state for a totally relativistic degenerate gas was a polytrope of index 3. In addition, we noted that an object dominated by radiation pressure would also be a polytrope of index n = 3. In the first category we find the extreme white dwarfs, those nearing the Chandrasekhar degeneracy limit. In the second category we find stars of very great mass where, from the β* theorem, we can expect the total pressure to be very nearly that of the pressure from photons. It is somewhat curious that such different types of stars should have their structures given by the same equilibrium model. However, both types are dominated by relativistic (in the sense of the special theory of relativity) particles, and this aspect of the gas is characterized by a polytrope of index n = 3.

Our approach to the study of these objects will be a little different from our previous discussions of stellar structure. Rather than concentrate on the internal properties and physics of these objects, we consider only their global properties, such as mass, radius, and internal energy. This will be sufficient to understand their stability and evolutionary history. The ideal vehicle for such an investigation is the Virial theorem.

a. Virial Theorem for Relativistic Stars

The Virial theorem for relativistic particles differs somewhat from that derived in Chapter 1. The effect of special relativity on the "mass" or momentum of such particles increases the gravitational energy required to confine the particles as the internal energy increases (see Collins⁸). Thus, for stable configurations, instead of \[2 \mathbf{T}+\boldsymbol{\Omega}=0\label{6.4.1}\]

we get \[\mathbf{T}+\boldsymbol{\Omega}=0\label{6.4.2}\]

which specifies the total energy of the configuration as \[\mathrm{E}=\mathbf{T}+\boldsymbol{\Omega}=0\label{6.4.3}\]

This is sometimes called the binding energy because it is the energy required to disperse the configuration throughout space. Thus polytropes of index n = 3 are neutrally stable since it would take no work at all to disperse them and as such these polytropes represent a limiting condition that can never be reached. To investigate the fate of objects approaching such a limit, it is necessary to look at the behavior of those conditions that lead to small departures from the limit. One of those conditions is the distortion of the metric of space caused by the matter-energy itself and so well described by the general theory of relativity.

Phenomenologically, we may view the effects of general relativity as increasing the effective "force of gravity". Thus, as we approach the limiting state of the relativistic polytrope, we would expect the effects of general relativity to cause the configuration to become unstable to collapse. So it is general relativity which sets the limit for the masses of white dwarfs, not the Pauli Exclusion Principle, just as general relativity set the limit for the masses of neutron stars. We could also expect such an effect for super-massive stars dominated by photon pressure.

To quantify these effects, we shall have to appeal to the Virial theorem in a non-Euclidean metric. Rather than re-derive the Boltzmann transport equation for the Schwarzschild metric, we obtain the relativistic Euler-Lagrange equations of hydrodynamic flow and take the appropriate spatial moments. We skip directly to the result of Fricke⁹. \[\begin{aligned}
\frac{1}{2} \frac{d^2 I_r}{d t^2}= & 3 \int_V P d V+\Omega \\
& +\frac{G}{c^2}\left[\int_V \frac{M(r)\left(P+\rho \dot{r}^2\right)}{r} d V-3 \int_V \frac{G M^2(r) \rho}{r^2} d V\right]
\end{aligned}\label{6.4.4}\]

Here \(\mathrm{I_r}\) is the moment of inertia defined about the center of the Schwarzschild metric. The effects of general relativity are largely contained in the third term in brackets which is multiplied by G/c² and contains the additions to the potential energy of the kinetic energy of the gas particles (as represented by the pressure) and the kinetic energy of mass motions of the configuration (as represented by \(\dot{\mathrm{r}}^2 \rho\)). The physical interpretation of the second term in the brackets is more obscure. For want of a better description, it can be viewed as a self-interaction term arising from the nonlinear nature of the general theory of relativity. Except for the relativistic term, equation \ref{6.4.4} is very similar to its Newtonian counterpart in Chapter 1 [equation \ref{1.2.34}]. The effect of the internal energy is included in the term \(3 \int_{\mathrm{V}} \mathrm{PdV}\). Since we will be considering stars that are near equilibrium, we take the total kinetic energy of mass motions to be zero. The \(\dot{\mathrm{r}}^2 \rho\) term was included in the relativistic term to emphasize its relativistic role.

A common technique in stellar astrophysics is to perform a variational analysis of the Virial theorem as expressed by equation \ref{6.4.4}, but such a process is quite lengthy. Instead, we estimate the effects of general relativity by determining the magnitude of the relativistic terms as \(\gamma \rightarrow(4 / 3)\). Obviously if the left hand side of equation \ref{6.4.4} becomes negative, the star will begin to acceleratively contract and will become unstable. Thus we investigate the conditions where the star is just in equilibrium. Replacing \(3 \int_{\mathrm{V}} \mathrm{PdV}\) with its equivalent in terms of the internal energy [see equation \ref{5.4.2}], we get \[3(\gamma-1) U+\Omega=-\frac{G}{c^2} \int_V \frac{M(r) P}{r} d V+\frac{3 G^2}{c^2} \int_V \frac{M^2(r) \rho}{r^2} d V\label{6.4.5}\]

Now, since \(\gamma=4 / 3\) is a limiting condition, let \[\gamma=\frac{4}{3}+\epsilon \quad n=3(1-3 \epsilon)\label{6.4.6}\]

where ε is positive. We may use the variational relation between the internal and potential energies \[\delta U=-3(\gamma-1)^{-1} \delta \Omega\label{6.4.7}\]

(see Chandrasekhar¹⁰), and we get \[\operatorname{Lim}_{\gamma \rightarrow 4 / 3+\epsilon}[3(\gamma-1) U+\Omega]=(1+3 \epsilon)\left(U_0+\delta U\right)+\Omega_0+\delta \Omega=-3 \Omega_0 \epsilon\label{6.4.8}\]

Here the subscript ₀ denote the value of quantities when \(\gamma=4/3\), and, \(\mathrm{U}_0=-\Omega_0\) for that value of g so the Virial theorem becomes \[3 \epsilon \Omega_0=\frac{G}{c^2} \int_V \frac{M(r) P}{r} d V-\frac{3 G^2}{c^2} \int_V \frac{M^2(r) \rho}{r^2} d V\label{6.4.9}\]

We may now estimate the magnitude of the relativistic terms on the right-hand side as follows. Consider the first term where \[\frac{G}{c^2} \int_V \frac{M(r) P}{r} d V=\frac{G}{c^2} \overline{\left(\frac{M}{r}\right)}\left(\frac{-\Omega_0}{3}\right) \approx \frac{G M}{R c^2}\left(\frac{-\Omega_0}{3}\right)=-\frac{1}{6}\left(\frac{R_s}{R}\right) \Omega_0\label{6.4.10}\]

Here we have taken the pressure weighted mean of (M/R) to be M/R, and R_s is the Schwarzschild radius for the star. The second term can be dealt with in a similar manner, so \[\frac{3 G^2}{c^2} \int_V \frac{M^2(r) \rho}{r^2} d V=\frac{3 G^2}{c^2} \overline{\left(\frac{M}{r}\right)^2} M \approx \frac{3 G M^2}{2 R} \frac{R_s}{R}=-\frac{R_s}{R} \Omega_0\label{6.4.11}\]

Again, we have replaced the mean of M/R by M/R. Since the means of the two terms are not of precisely the same form, we expect this approach to yield only approximate results. Indeed, the central concentration of the polytrope will ensure that both terms are underestimates of the relativistic effects. Even worse, the mean-square of M(r)/r in equation \ref{6.4.11} will yield an even larger error than that of equation \ref{6.4.10}. Since the terms differ in sign, the combined effect could be quite large. However, we may be sure that the result will be a lower limit of the effects of general relativity, and the approximations do demonstrate the physical nature of the terms. With this large caveat, we shall proceed. Substituting into equation \ref{6.4.9}, we get \[\frac{R}{R_s} \approx \frac{5}{18 \epsilon}\label{6.4.12}\]

Now all that remains to be done is to investigate how \(\gamma\rightarrow4/3\) in terms of the defining parameters of the star (M, L, R), and we will be able to estimate when the effects of general relativity become important.

b. Minimum Radius for White Dwarfs

We have indicated that the effects of general relativity should bring about the collapse of a white dwarf as it approaches the Chandrasekhar limiting mass. If we can characterize the approach of g to 4/3 in terms of the properties of the star, we will know how close to the limiting mass this occurs. As \(\gamma\rightarrow4/3\), the degeneracy parameter in the parametric degenerate equation of state approaches infinity. Carefully expanding f(x) of equation \ref{1.3.14} and determining its behavior as x → 4 we get \[\operatorname{Lim}_{x \rightarrow \infty} f(x)=2\left(x^4-x^2\right)\label{6.4.13}\]

From the polytropic equation of state \[\gamma=\frac{d(\ln P)}{d(\ln \rho)}\label{6.4.14}\]

Evaluating the right-hand side from the parametric equation of state [equation \ref{1.3.14}] and the result for f(x) given by equation \ref{6.4.13}, we can combine with the definition of e from equation \ref{6.4.6} to get \[\epsilon \approx \frac{2 x^{-2}}{3}\label{6.4.15}\]

If we neglect the effects of inverse beta decay in removing electrons from the gas, we can write the density in terms of the electron density and, with the aid of equation \ref{1.3.14}, in terms of the degeneracy parameter x. \[\rho=\frac{8 \pi c^3 m_e^3 \mu_e m_h x^3}{3 h^3}\label{6.4.16}\]

If we approximate the density by its mean value, we can solve for the average square degeneracy parameter for which we can expect collapse. \[\begin{aligned}
\bar{x}^2 & =\left(\frac{9 h^3}{32 \pi^2 c^3 m_e^3 \mu_e m_h}\right)^{2 / 3}\left(\frac{M}{R^3}\right)^{2 / 3} \\
& =\left(7.12 \times 10^6\right) \mu_e^{2 / 3}\left(\frac{M_{\odot}}{M}\right)^{4 / 3}\left(\frac{R_s}{R}\right)^2
\end{aligned}\label{6.4.17}\]

Combining this with equations \ref{6.4.15} and \ref{6.4.12}, we obtain an estimate for the manner in which the minimum stable radius of a white dwarf depends on mass as the limiting mass is approached. \[\frac{R}{R_s} \approx 144 \mu_e^{-2 / 9}\left(\frac{M_{\odot}}{M}\right)^{4 / 9}\label{6.4.18}\]

A more precise calculation involving a proper evaluation of the relativistic integrals and evaluation of the average internal degeneracy by Chandrasekhar and Trooper¹¹ yields a value of 246 Schwarzschild radii for the minimum radius of a white dwarf, instead of about 100 given by equation \ref{6.4.18}. We can then combine this with the mass-radius relation for white dwarfs to find the actual value of the mass for which the star will become unstable to general relativity. This is about 98 percent of the value given by the Chandrasekhar limit, so that for all practical purposes the degeneracy limit gives the appropriate value for the maximum mass of a white dwarf.

However, massive white dwarfs do not exist because general relativity brings about their collapse as the star approaches the Chandrasekhar limit. This point is far more dramatic in the case of neutron stars. Here the general relativistic terms bring about collapse long before the entire star becomes relativistically degenerate. A relativistically degenerate neutron gas has much more kinetic energy per gram than a relativistically degenerate electron gas, since a relativistic particle must have a kinetic energy greater than its rest energy, by definition. To contain such a gas, the gravitational forces must be correspondingly larger, which implies a greater importance for general relativity. Indeed, to confine a fully relativistically degenerate configuration, it would be necessary to restrict it to a volume essentially bounded by its Schwarzschild radius. This is not to say that the cores of neutron stars cannot be relativistically degenerate. Indeed they can, but the core is contained by the weight of the nonrelativistically degenerate layers above as well as its own self-gravity.

c. Minimum Radius for Super-massive Stars

Since the early 1960s, super-massive stars have piqued the interest of some. It was thought that such objects might provide the power source for quasars. While their existence might be ephemeral, if super-massive stars were formed in sufficient numbers, their great luminosity might provide a solution to one of the foremost problems of the second half of the twentieth century. However, truly massive stars are subject to the same sort of instability as we investigated for white dwarfs. Indeed, the problem was first discussed by W. Fowler^12,13 for the super-massive stars and later extended by Chandrasekhar and Trooper¹¹ to white dwarfs. Finally the problem was re-discussed by Fricke⁹ and the effect of the metric on the relativistic integrals was correctly included.

By now you should not be surprised that such an instability exists because we know that massive stars are dominated by radiation pressure and can be well represented by polytropes of index n = 3. For super-massive stars, the departure from being a perfect relativistic polytrope results from some of the total energy being provided by the kinetic energy of the gas particles. To quantify the instability, we may proceed as we did with the white dwarf analysis. If we write the equilibrium Virial theorem and split the \(3 \int_{\mathrm{V}} \mathrm{PdV}\) term into a sum of the gas pressure and the radiation pressure, then we get \[\begin{aligned}
0= & 3 \int_V \beta P d V+3 \int_V(1-\beta) P d V+\Omega+\frac{G}{c^2} \int_V \frac{M(r) P}{r} d V \\
& -\frac{3 G^2}{c^2} \int_V \frac{M^2(r) \rho}{r^2} d V
\end{aligned}\label{6.4.19}\]

However, the total energy of the configuration is \[\begin{aligned}
E= & \frac{3}{2} \int_V \beta P d V+3 \int_V(1-\beta) P d V+\Omega+\frac{3 G}{c^2} \int_V \frac{M(r) P}{r} d V \\
& -\frac{3 G^2}{2 c^2} \int_V \frac{M^2(r) \rho}{r^2} d V
\end{aligned}\label{6.4.20}\]

Subtracting equation \ref{6.4.19} from (6.4.20), we get \[E=-\frac{3}{2} \int_V \beta P d V+\frac{2 G}{c^2} \int_V \frac{M(r) P}{r} d V+\frac{3 G^2}{2 c^2} \int_V \frac{M^2(r) \rho}{r^2} d V \approx 0\label{6.4.21}\]

When the total energy of the configuration becomes zero, we will have reached its minimum stable radius. Making the same approximations for the relativistic integrals that were made for the white dwarf analysis, we get \[\frac{R_0}{R_s} \approx \frac{5}{3 \beta}\label{6.4.22}\]

From equation \ref{2.2.11} we saw that the central value of beta, β_c, was bounded by the mass, so that \[\beta_c \propto M^{-1 / 2}\label{6.4.23}\]

Using the constant of proportionality implied by equation \ref{2.2.11} and combining with equation \ref{6.4.22}, we get \[\frac{R_0}{R_s} \simeq 0.18\left(\frac{M}{M_{\odot}}\right)^{1 / 2}\label{6.4.24}\]

Thus a super-massive star of 10⁸M_⊙ will become unstable at about 1800 Schwarzschild radii or about 14 AU. In units of the Schwarzschild radius, this result is rather larger than that for white dwarfs. This can be qualitatively understood by considering the nature of the relativistic particles providing the majority of the internal pressure in the two cases. The energy of the typical photon providing the radiation pressure for a super-massive star is far less than the energy of a typical degenerate electron whose degenerate pressure provides the support in a white dwarf. Thus a weaker gravitational field will be required to confine the photons as compared to the electron. This implies that as the total energy approaches zero, the mass required to confine the photons can be spread out over a larger volume, when measured in units of the Schwarzschild radius, than is the case for the electrons. This argument implies that neutron stars should be much closer to their Schwarzschild radius in size, which is indeed the case.

Perhaps the most surprising aspect of both these analyses is that general relativity can make a significant difference for structures that are many hundreds of times the dimensions that we usually associate with general relativity.