 Solution

ID: past-exam-of-the-astrophysics-course-of-the-university-of-oxford/2015/2/d/solution

Let

τ_{*} = P /∣ \dot{P} ∣ = 300

yr. With constant moment of inertia, the observed positive spin-up torque is

N_{*} = I \dot{Ω}_{*} = \frac{I Ω _{*}}{τ _{*}} = 4.84 \times 1 0^{28} N m .

Nearly Keplerian material at the inner disk carries specific angular momentum

ℓ = GM r_{Alf}

. Equating its inflow rate to the observed torque gives

\dot{M}_{acc} = \frac{I ( 2 π / P )}{τ _{*} GM r _{Alf}} = \frac{I ( 2 π / P )}{τ _{*} GM r _{co}} (\frac{r _{Alf}}{r _{co}})^{- 1/2} .

\dot{M}_{acc} = 2.23 (\frac{r _{Alf}}{r _{co}})^{- 1/2} \dot{M}_{Edd} ≃ 2 (\frac{r _{Alf}}{r _{co}})^{- 1/2} \dot{M}_{Edd} .

The printed coefficient is therefore recovered at its intended approximate precision. This is a material-torque model; magnetic torques, wind angular-momentum loss and a changing stellar moment of inertia are not included.

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