Questions on accretion

1. Assume that a black hole grows by accretion at the Eddington rate (i.e. the accretion rate is such that the accretion luminosity is the Eddington luminosity). Derive an expression for the mass of the black hole as a function of time. You should find that there is a characteristic timescale on which the mass grows (known as the Salpeter time). What is it?

2. One important result for thin disks is the disk spectrum. Assuming each annulus of the disk radiates as a blackbody, and using the result \(T_{\rm eff}\propto r^{-3/4}\), what should the disk spectrum look like? Sketch \(F_\nu\) against \(\nu\).

3. The local thermal timescale in a thin disk can be defined as

\[ t_\mathrm{therm}={\Sigma c_PT\over 2\sigma T_\mathrm{eff}^4}, \]

where \(c_P\) is the specific heat capacity (\(\approx k_B/m_p\)). Using the results above for the thin disk, show that \(t_\mathrm{therm}\approx \alpha^{-1}\Omega^{-1}\). Show that \(\Omega^{-1}<t_\mathrm{therm}<t_\mathrm{visc}\) for a thin disk, consistent with the assumptions underlying the thin disk model.

4. Calculate the orbital period of a Keplerian orbit at \(r=r_M\). This is a lower limit to the spin period of the accreting star because if the star (and its magnetosphere) rotate faster than the Keplerian frequency at \(r_M\), the accreting matter does not have enough angular momentum to attach to the magnetic field. (In this case, the matter is flung away in what is known as the “propeller” regime).