Stationary Spherical Optically Thick Accretion into Black Holes

Citation

Flammang, Richard Alan (1982) Stationary Spherical Optically Thick Accretion into Black Holes. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/t73p-2e81. https://resolver.caltech.edu/CaltechTHESIS:08252017-133002176

Abstract

As its title indicates, this thesis treats the problem of stationary, spherical, optically thick accretion into black holes. By the phrase "optically thick" it is meant that (1) radiative energy transport can be adequately described by the diffusion approximation and (2) the photons are everywhere in local energy equilibrium (LTE) with the accreting gas particles.

In Chapter 1, a general set of equations governing time-independent spherical accretion into black holes is formulated. The equations are fully general relativistic and are applicable to optically thick regions, optically thin regions, and the transition regions which join them. The radiation is treated using frequency-integrated moments. The full, infinite series of moment equations is given, together with the limiting forms the equations take in the optically thick regime.

In Chapter 2, we present the mathematical theory of stationary spherical optically thick accretion. We analyze the integral curves of the differential equations describing the problem. We find a one-parameter family of critical points, where the inflow velocity equals the isothermal sound speed. Physical solutions must pass through one of these critical points. We obtain a complete set of boundary conditions which the solution must satisfy at the horizon of the black hole, and show that these, plus the requirement that the solution pass through a critical point, determine a unique solution to the problem. This analysis leads to a generalization of the well-known Bondi critical point constraint, which arises in the adiabatic accretion problem and which is effective at the point where the inflow velocity equals the adiabatic sound speed. We show that this point can be regarded as a "diffused critical point" in our problem. The analysis also yields a simple expression for the diffusive luminosity at radial infinity. Finally, we find a satisfying explanation for the rather peculiar critical point structure of this problem in an analysis of the characteristics and subcharacteristics present in the problem and in a "hierarchical" analysis of the waves which propagate along them.

In Chapter 3, we apply the theory of optically thick accretion developed in Chapter 2 to a wide range of physically different accretion regimes. Numerical solutions are presented and their physical properties are discussed. For solutions in which radiation pressure P_R dominates gas pressure P_G, but in which gas energy density (including its rest-mass) ρ_G dominates radiation energy density ρ_R, we pay particular attention to the adabaticity of the flow. Our quantitative results in this regime agree very well with Begelman's (1978) theory. We find the dimensionless number which governs the importance of heat diffusion in our problem and show that it reduces to the idea of "trapping of photons" and to the Péclet number in the appropriate limits. We find that solutions with P_R > P_G and ρ_R > ρ_G are always essentially adiabatic, owing in part to a relativistic suppression of heat flux which becomes important in this regime. The diffusive luminosity at infinity for these solutions is the Eddington limit of the black hole; with the adiabatic accretion rate, "efficiencies" of up to order unity are possible. We give preliminary consideration to the question of the stability of our solutions against convection and conclude that the Schwarzschild criterion is applicable, even for our non-static accretion flows. We show that solutions with P_R > P_G are everywhere stable against convection. On the other hand, solutions which start out at radial infinity with P_G > P_R are unstable to convection (if the adiabatic index of the gas γ_G exceeds 17/12) from radial infinity down to the point where P_R ~ P_G and the radiation-gas mixture has attained an adiabatic index of 17/12. The diffusive luminosity at infinity for these solutions is reduced from the Eddington limit of the black hole by the factor (γ_G - 1)4P_R∞/γ_GP_G∞; it is further reduced by the ratio of the electron scattering opacity to the actual opacity at infinity, if this differs from unity. In most cases, energy diffusion has a negligible effect on the accretion rate of these solutions.

Thesis Files