Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 347 / 3 / a

Solution

Vertically integrating the continuity equation and imposing a steady axisymmetric flow gives

\frac{1}{R} \frac{d}{d R} (R Σ u_{R}) = 0.

Thus the inward-positive mass accretion rate is

\overset{m}{˙} = - 2 π R Σ u_{R} = constant .

The zero-torque condition at

R_{ISCO}

gives the standard Shakura--Sunyaev thin disk relation

ν Σ = \frac{m ˙}{3 π} [1 - (\frac{R _{ISCO}}{R})^{1/2}] .

Combining this relation with mass conservation, or substituting it into the stated drift equation, yields

u_{R} = - \frac{3 ν}{2 R} [1 - (\frac{R _{ISCO}}{R})^{1/2}]^{- 1} .

Far outside the inner edge this reduces to

u_{R} ≃ - 3 ν / (2 R)

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