Black Hole Accretion Discs

Abstract

This is an introduction to the models of accretion discs around black holes. After a presentation of the nonrelativistic equations describing the structure and evolution of geometrically thin accretion discs, we discuss their steady-state solutions and compare them to observation. Next, we describe in detail the thermal–viscous disc instability model and its application to dwarf novae for which it was designed and its X-ray irradiated disc version which explains the soft X-ray transients, i.e. outbursting black hole low-mass X-ray binaries. We then turn to the role of advection in accretion flows onto black holes illustrating its action and importance with a toy model describing both ADAFs and slim discs. We conclude with a presentation of the general-relativistic formalism describing accretion discs in the Kerr spacetime.

Appendix

Thermodynamical Relations

The equation of state can be expressed in the form:

$$\begin{aligned} P = P_r + {\mathcal{R} \over \mu _i } \rho T_i + {\mathcal{R} \over \mu _e } \rho T_e + {B^2 \over 24 \pi }, \end{aligned}$$

(1.228)

where $P_r$ is the radiation pressure, $\mathcal R$ is the gas constant, $\mu _i$ and $\mu _e$ are the mean molecular weights of ions and electrons respectively, $T_i$, and $T_e$ are ion and electron temperatures, a is the radiation constant (not to be confused with the dimensionless angular momentum a in the Kerr metric), and B is the intensity of a isotropically tangled magnetic field, includes the radiation, gas and magnetic pressures. The radiation pressure $P_r$, the gas pressure $P_g$ and the magnetic pressure $P_m$ correspond respectively to the first term, the second and third terms, and the last term in Eq. (1.228).

The mean molecular weights of ions and electrons can be well approximated by:

$$\begin{aligned} \mu _i \approx {4 \over 4X + Y}, \qquad \mu _e \approx {2 \over 1+X}, \end{aligned}$$

(1.229)

where X is the relative mass abundance of hydrogen and Y that of helium. We may define a temperature as

$$\begin{aligned} T=\mu \left( {T_i \over \mu _i} +{T_e \over \mu _e}\right) , \end{aligned}$$

(1.230)

where

$$\begin{aligned} \mu = \left( {1\over \mu _i} + {1\over \mu _e}\right) ^{-1} \approx {2 \over 1 + 3X + 1/2Y} \end{aligned}$$

(1.231)

is the mean molecular weight. In the case of a one-temperature gas ($T_i=T_e$), one has $T=T_i=T_e$. For an optically thick gas, $P_r=({4\sigma }/{3c}) T_r^4$.

For the frozen-in magnetic field pressure $P_m \sim B^2 \sim \rho ^{4/3}$, therefore, we may write the internal energy as

$$\begin{aligned} U = \frac{4\sigma }{\rho c} T_r^4 + {\mathcal{R} T \over \mu m_u (\gamma _g - 1) } + e_o \rho ^{1/3}, \end{aligned}$$

(1.232)

where $e_o$ is a constant ($P_m = 1/3 e_o \rho ^{4/3}$) and $\gamma _g$ is the ratio of the specific heats of the gas. We define

$$\begin{aligned} \beta = {P_g \over p}, \quad \beta _m = {P_g \over P_g + P_m}, \quad \beta ^*={4-\beta _m \over 3\beta _m} \beta . \end{aligned}$$

(1.233)

From Eqs. (1.228) and (1.232), one obtains the following formulae (see, e.g. “Cox and Giuli” 2004) for the specific heat at constant volume:

$$\begin{aligned} c_V = {\mathcal{R} \over \mu (\gamma _g - 1)} \left[ {12 (1 - \beta /\beta _m)(\gamma _g - 1)+\beta \over \beta }\right] = {4 - 3\beta ^* \over \varGamma _3 - 1} {P \over \rho T} \end{aligned}$$

(1.234)

and the adiabatic indices:

$$\begin{aligned} \varGamma _3 - 1 = {(4 - 3\beta ^*)(\gamma _g - 1) \over 12(1 - \beta /\beta _m)(\gamma _g -1) + \beta } \end{aligned}$$

(1.235)

$$\begin{aligned} \varGamma _1 = \beta ^* +(4 - 3\beta ^*)(\varGamma _3 - 1). \end{aligned}$$

(1.236)

The ratio of specific heats is $\gamma =c_p/c_V= \varGamma _1/\beta$. For $\beta =\beta _m$, we have $\varGamma _3=\gamma _g$ and $\varGamma _1=(4 - \beta )/3 + \beta (\gamma _g -1)$. For an equipartition magnetic field ($\beta =0.5$), one gets $\varGamma _1 = 1.5$ and for $\beta =0.95$, $\varGamma _1= 1.65$ (here, we have used $\gamma _g=5/3$). One expects $\beta _m \sim 0.5 - 1$. Since

$$\begin{aligned} T{dS \over dR}=c_V \left[ {d \ln T\over dR} - (\varGamma _3 -1) \left( {d \ln \varSigma \over dR} - {d \ln H\over dR }\right) \right] , \end{aligned}$$

(1.237)

the advective flux is written in the form:

$$\begin{aligned} Q^\mathrm{adv} = {\dot{M}\over 2\pi R^2} {P\over \rho } \xi _a \end{aligned}$$

(1.238)

where

$$\begin{aligned} \xi _a = -\left[ {4 - 3\beta ^* \over \varGamma _3 - 1} {d \ln T\over d\ln R} + (4 - 3\beta ^*) {d \ln \varSigma \over d \ln R}\right] . \end{aligned}$$

(1.239)

The term $\propto d \ln H/d \ln R$ has been neglected. Since no rigorous vertical averaging procedure exists, the presence or not of the $d \ln H/d \ln R$—type terms in this (and other) equation may be decided only by comparison with 2D calculations.

The formulae derived in this section are valid for the optically thin case $\tau =0$ if one assumes $\beta =\beta _m$.

Reference: Weiss, A., Hillebrandt, W., Thomas, H.-C., and Ritter, H. 2004, Cox and Giuli’s Principles of Stellar Structure, Cambridge, UK: Princeton Publishing Associates Ltd, 2004.