Past exam of the astrophysical fluid dynamics course of the University of Cambridge / 2014 / Paper 1 / 2 / b

An accretion disk forms when inward-moving gas retains enough angular momentum to orbit a compact object. In a geometrically thin Keplerian disk, radial force balance gives $\Omega \simeq \sqrt{GM/r^3}$ and vertical hydrostatic equilibrium gives a thickness $H\sim c_s/\Omega \ll r$. Matter cannot fall far inward while keeping its original specific angular momentum $r^2\Omega \propto r^{1/2}$: stresses must transport angular momentum outward.

The viscous torque in an accretion disk has outward magnitudewhere $\nu$ is the effective kinematic viscosity and $\Sigma$ the surface density. With constant inward accretion rate $\dot{M}>0$ and zero inner torque at $r_{\mathrm{in}}$, conservation of angular momentum givesThe associated standard thin-disk dissipation flux from each face isOrbital gravitational potential energy becomes heat and, in a radiatively efficient disk, escaping radiation. Integrating both faces gives the standard thin-disk luminosity $L=GM\dot{M}/(2r_{\mathrm{in}})$ under this boundary condition. A central stellar surface or an inner torque can release or redistribute additional energy.

The alpha disk prescription $\nu \sim \alpha c_{s}H$ summarizes an effective stress rather than a claim that molecular viscosity alone explains observed rates. In an ionized disk, magnetorotational instability can produce magnetic and fluid stresses. The inward evolution occurs on the viscous timescale $t_{\nu }\sim r^2/\nu$, much longer than an orbit for a thin disk with $\alpha \lesssim 1$. Hot inner annuli and cooler outer annuli naturally give a range of emitted temperatures; the blackbody approximation additionally requires sufficient optical depth.