6.6: Fate of Super-massive Stars

The relativistic polytrope can be used to set minimum sizes for both white dwarfs and very massive stars. However, super-massive stars are steady-state structures and will evolve, while white dwarfs are equilibrium structures and will remain stable unless they are changed by outside sources. Let us now see what can be said about the evolution of the super-massive stars.

a. Eddington Luminosity

Sir Arthur Stanley Eddington observed that radiation and gravitation both obey inverse-square laws and so there would be instances when the two forces could be in balance irrespective of distance. Thus there should exist a maximum luminosity for a star of a given mass, where the force of radiation on the surface material would exactly balance the force of gravity. If we balance the gravitational acceleration against the radiative pressure gradient [equation \ref{4.2.11}] for electron scattering, we can write \[\nabla P_r=-\frac{\sigma_e \rho L(r)}{4 \pi c R^2}=-\frac{G M \rho}{R^2}\label{6.5.1}\]

Therefore, any object that has a luminosity greater than \[L_{\mathrm{Edd}}=\frac{4 \pi G c}{\sigma_e} M \approx\left(1.3 \times 10^{38}\right)\left(\frac{M}{M_{\odot}}\right) \quad \mathrm{erg} / \mathrm{s}\label{6.5.2}\]

will be forced into instability by its own radiation pressure. This effectively provides a mass-luminosity relationship for super-massive stars since these radiation-dominated configurations will radiate near their limit.

b. Equilibrium Mass-Radius Relation

If we now assume that the star can reach a steady-state, that represents a near-equilibrium state on a dynamical time, then the energy production must equal the energy lost through the luminosity. Eugene Capriotti¹⁴ has evaluated the luminosity integral and gets \[L=\int_V \rho \epsilon d V \simeq\left(2.9 \times 10^{-67}\right)\left(\frac{M^{8.625}}{R^{16.25}}\right) \quad \mathrm{erg} / \mathrm{s}\label{6.5.3}\]

We can assume that massive stars will derive the nuclear energy needed to maintain their equilibrium from the CNO cycle, can evaluate e as indicated in Chapter 3 [equation \ref{3.3.19}], and can evaluate the central term of equation \ref{6.5.3} to obtain the approximate relation on the right. Assuming that the stars will indeed radiate at the Eddington luminosity, we can use equation \ref{6.5.2} to find \[R_e \approx\left(1.7 \times 10^{11}\right)\left(\frac{M}{M_{\odot}}\right)^{0.47} \mathrm{~cm}\label{6.5.4}\]

Thus we have a relation between the mass and radius for any super-massive star that would reach equilibrium through the production of nuclear energy. However, we have yet to show that the star can reach that equilibrium state.

c. Limiting Masses for Super-massive Stars

Let us add equations \ref{6.4.19} and \ref{6.4.20} taking care to express the relativistic integrals as dimensionless integrals by making use of the homology relations for pressure and density, and get for the total energy: \[\mathrm{E=-\frac{1}{2} \bar{\beta} \Omega+\frac{2 G^2 M^3}{R^2 c^2} \int_0^1\left[\frac{M(r) R}{M r}\right] \frac{P}{P_c} \frac{\rho_c}{\rho} \frac{d M(r)}{M}-\frac{9 G^2 M^3}{2 R^2 c^2} \int_0^1\left[\frac{M(r) R}{M r}\right]^2 \frac{d M(r)}{M}}\label{6.5.5}\]

We must be very careful in evaluating these integrals, for any polytrope in Euclidean space as the radial coordinate used to obtain those integrals is defined by the Schwarzschild metric (see Fricke⁹ p. 942). We must do so here, since we will not be content to find a crude result for the mass limits.

Figure 6.3 shows the variation of the binding energy in units of the rest energy of the sun as a function of the radius in units of the minimum stable radius. In is clear that a minimum (most negative) binding energy exists and that the minimum is a specific value for all super massive stars.

Replacing β by its limiting value given by the β* theorem and evaluating the relativistic integrals for a polytrope of index n = 3, we obtain \[E=-\frac{27 G M_{\odot}^2}{4 R_{\odot}}\left(\frac{M}{M_{\odot}}\right)^{3 / 2} \frac{R_{\odot}}{R}+5.07 \frac{G^2 M_{\odot}^3}{R_{\odot}^2 c^2}\left[\left(\frac{M}{M_{\odot}}\right)^{3 / 2} \frac{R_{\odot}}{R}\right]^2\label{6.5.6}\]

If we now seek the radial value for which E = 0, we get \[\begin{aligned}
R_0 & \approx 0.756\left(\frac{G M_{\odot}}{c^2}\right)\left(\frac{M}{M_{\odot}}\right)^{3 / 2}=\left(1.1 \times 10^5\right)\left(\frac{M}{M_{\odot}}\right)^{3 / 2} \mathrm{~cm} \\
& =0.37 R_s\left(\frac{M}{M_{\odot}}\right)^{1 / 2}
\end{aligned}\label{6.5.7}\]

Comparing this result with equation \ref{6.4.22}, we see that our crude approximation of the relativistic integrals was low by about an order of magnitude. Equation \ref{6.5.6} is quadratic in M^3/2/R and will become positive at small R. Figure 6.3 shows the dependence of the binding energy on the radius. Differentiation of equation \ref{6.5.6} shows that the greatest (most negative) binding energy will occur at \[R_m=2 R_0\label{6.5.8}\]

and this corresponds to an energy of \[E_m=-2.25 M_{\odot} c^2\label{6.5.9}\]

This energy is a constant because of the quadratic nature of the energy equation. The relativistic terms simply vary as the next power of \(\mathrm{[M^{3/2}/R]}\) compared to the Newtonian terms. Hence the minimum will depend only on physical constants.

A star that is contracting toward its equilibrium position may reach equilibrium for any radial value that is greater than, or equal to R_m, providing an energy source exists to replace the energy lost to space. We have already found the equilibrium radius for energy produced by the CNO cycle [equation \ref{6.5.4}]. Combining that with the minimum energy radius, we find \[\frac{M}{M_{\odot}} \leq 5.2 \times 10^5\label{6.5.10}\]

Thus any star with a mass less than about half a million solar masses can come to equilibrium burning hydrogen via the CNO cycle, albeit with a short lifetime. More massive stars are destined to continue to contract. Of course, more massive stars will produce nuclear energy at an ever-increasing rate as their central temperatures rise. However, the rate of energy production cannot increase without bound. This is suggested by the declining exponents of the temperature dependence shown in Table 3.4. The nuclear reactions that involve β decay set a limit on how fast the CNO cycle can run, and β decay is independent of temperature. So there is a maximum rate at which energy can be produced by the CNO cycle operating in these stars.

If the nuclear energy produced is sufficient to bring the total energy above the binding energy curve, the star will explode. However, should the energy not be produced at a rate sufficient to catch the binding energy that is rising due to the relativistic collapse, the star will continue an unrestrained collapse to the Schwarzschild radius and become a black hole. Which scenario is played out will depend on the star's mass. For these stars, the temperature gradient will be above the adiabatic gradient, so convection will exist. However, the only energy transportable by convection is the kinetic energy of the gas, which is an insignificant fraction of the internal energy. Therefore, unlike normal main sequence stars, although it is present, convection will be a very inefficient vehicle for the transport of energy. This is why the star remains with a structure of a polytrope of index n = 3 in the presence of convection. The pressure support that determines the density distribution comes entirely from radiation and is not governed by the mode of energy transport. We saw a similar situation for degenerate white dwarfs. The equation of state indicated that their structure would be that of a polytrope of index n= 1.5 (for nonrelativistic degeneracy) and yet the star would be isothermal due to the long mean free path of the degenerate electrons. However, the structure is not that of an isothermal sphere since the pressure support came almost entirely from the degenerate electron gas and is largely independent of the energy and temperature distribution of the ions.

The star will radiate at the Eddington luminosity, and that will set the time scale for collapse. Remember that the total energy of these stars is small compared to the gravitational energy. So most of the energy derived from gravitational contraction must go into supporting the star, and very little is available to supply the Eddington luminosity. This can be seen from the relativistic Virial theorem [equation \ref{6.4.2}], which indicates that any change in the gravitational energy is taken up by the kinetic energy. Relativistic particles (in this case, photons) are much more difficult to bind by gravitation than ordinary matter; thus little of the gravitational energy resulting from collapse will be available to let the star shine. The collapse will proceed very quickly on a time scale that is much nearer to the dynamical time scale than the Kelvin-Helmholtz time scale. The onset of nuclear reactions will slow the collapse, but will not stop it for the massive stars.

A dynamical analysis by Appenzeller and Fricke^15,16 (see also Fricke⁹) shows that stars more massive than about 7.5 × 10⁵M_⊙ will undergo collapse to a black hole. Here the collapse proceeds so quickly and the gravity is so powerful that the nuclear reactions, being limited by β decay at the resulting high temperatures, do not have the time to produce sufficient energy to arrest the collapse. For less massive stars, this is not the case. Stars in the narrow range of 5×10⁵M_⊙ # M # 7.5×10⁵M_⊙ will undergo explosive nuclear energy generation resulting in the probable destruction of the star.

Nothing has been said about the role of chemical composition in the evolution of these stars. Clearly, if there is no carbon present, the CNO cycle is not available for the stabilization of the star. Model calculations show that the triple-a process cannot stop the collapse. For stars with low metal abundance, only the proton-proton cycle is available as an energy source. This has the effect of lowering the value of the maximum stable mass. Surprisingly, there is no range at which an explosion occurs. If the star cannot stabilize before reaching R_m, it will continue in a state of unrestrained gravitational collapse to a black hole. Thus, it seems unlikely that stars more massive than about a half million solar masses could exist. In addition, it seems unlikely that black holes exist with masses greater than a few solar masses and less than half a million solar masses. If they do, they must form by accretion and not as a single entity.