Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 347 / 2

Steady mass conservation with inward-positive accretion rate givesWrite the specific angular momentum as $l=R^2\Omega$. Multiplying the azimuthal equation by $2\pi R$ and using the mass equation shows that the sum of advected and viscous angular-momentum flux is constant:The right-hand side implements the zero-torque inner boundary condition. For a Keplerian accretion disk, $l\propto R^{1/2}$ and $R^{3}d\Omega /dR=-(3/2)l$. ThereforeandFar outside the inner edge, $\Sigma \simeq \dot{m}/(3\pi \nu )$ and $u_R\simeq -3\nu /(2R)$.

The Shakura--Sunyaev thin disk model replaces poorly resolved turbulent angular-momentum transport by a stress proportional to pressure,Equivalently, an eddy viscosity has $\nu \sim v_{\mathrm{eddy}}{\ell }_{\mathrm{eddy}}$. Turbulent motions much faster than the sound speed $c_s$ would shock, while eddies much larger than the disk scale height $H$ would not fit within the disk. Writing $v_{\mathrm{eddy}}{\ell }_{\mathrm{eddy}}=\alpha c_{s}H$ therefore gives the alpha disk prescriptionwhere vertical hydrostatic balance gives $H\simeq c_s/{\Omega }_K$. The parameter $\alpha$ summarizes the correlation and efficiency of turbulent or magnetic stresses; it is not a molecular viscosity.

Let $\epsilon =H/R\ll 1$. Vertical hydrostatic balance gives $c_s\sim H{\Omega }_K$, whereas leading radial force balance gives $u_{\varphi }\sim R{\Omega }_K$. HenceUsing part (a) and $\nu =\alpha c_{s}H$ far from the inner edge,Thus subsonic radial drift in a thin disk coexists with highly supersonic orbital motion. Gas follows nearly circular, pressure-coherent orbits and loses angular momentum only slowly, completing of order $(\alpha {\epsilon }^2{)}^{-1}$ revolutions during its viscous inflow.

At fixed radiative efficiency, the Eddington ratio implies $\dot{m}\propto f_{\mathrm{Edd}}M$. Far outside the innermost stable circular orbit, part (a) gives $\Sigma \propto \dot{m}/\nu$. The supplied viscosity law therefore gives, in physical radius,The disk mass follows by radial integration:Since the Schwarzschild radius satisfies $R_S\propto M$, replacing $R$ by $(R/R_S)R_S$ contributes another factor $M^{5/4}$. Thusso

The alpha disk relation and Keplerian angular frequency giveCombining this with $\Omega \propto M^{1/2}{R}^{-3/2}$ and the surface-density scaling from part (d), the Toomre stability criterion becomesIt decreases strictly with radius, so it crosses unity at a unique self-gravitating radius of an accretion disk. Expressing radius in units of $R_S\propto M$ givesSolving $Q(R_{\mathrm{sg}})=1$ therefore yieldsand

Let $x_{\mathrm{ISCO}}=R_{\mathrm{ISCO}}/R_S$. Requiring a non-self-gravitating annulus outside the innermost stable circular orbit gives $R_{\mathrm{sg}}>R_{\mathrm{ISCO}}$, and equality definesFor a nonspinning hole, $x_{\mathrm{ISCO}}=3$, and with $C_2\simeq 10^5$, $\alpha \simeq 0.1$, and $f_{\mathrm{Edd}}\simeq 1$, this is approximately $8\times 10^9{M}_{\odot }$, conventionally quoted as order $10^{10}{M}_{\odot }$. Above this mass the disk would become self-gravitating essentially as soon as stable circular orbits begin, so the assumed smooth Shakura--Sunyaev thin disk cannot provide a broad luminous accretion region.

The corresponding Eddington luminosity is of order $10^{48}\mathrm{erg}{\mathrm{s}}^{-1}$, comparable to the upper envelope of quasar luminosities. This supports self-gravity as one contributor to the observed luminous-mass ceiling. It is not an absolute upper bound on black-hole mass: mergers, radiatively inefficient growth, nonstandard gas supply, spin-dependent inner radii, and fragmented or episodic accretion can all build a more massive hole without maintaining this particular steady thin disk.