 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2013/iii/paper-54/1/b/ii/solution

For no central accretion,

A = 0

and

\overset{ν}{ˉ} Σ = B r^{- 1/2}, v_{r} = 0, G = 3 π (G M_{*})^{1/2} B .

This nonaccreting constant-torque disk has a finite inner stress and normally a nonzero inner surface density. An external inner torque supplies the angular momentum transported to an outer sink, although mass does not flow. If both this zero-mass-flux condition and a zero inner torque are imposed,

B = 0

; there is no nontrivial positive-viscosity steady disk satisfying both.

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