dif1d: provides analytical solution to 1d diffusion problem in a cylindrical geometry with inner sink.

This program computes the flux in one-dimensional diffusion problem in a cylindrical geometry, in application to studying the loss-cone problem of consumption of stars by a supermassive black hole. The evolution of the distribution function $ f(R,t) $ in the interval $ R_{min} <= R <= 1 $ is governed by the equation:

\[ \frac{ \partial f(R,t) }{ \partial t} = \frac{\partial}{\partial R} \left( D\, R\, \frac{\partial f(R,t)}{\partial R} \right) , \]

with initial conditions $ f(R,0) = 1 $, subject to the boundary conditions

\[ f(R_{min},t) = \alpha\, R_{min}\, \partial f/\partial R ; \quad \frac{\partial}{\partial R} f(1,t) = 0 . \]

The solution for the case $ \alpha=0 $ is given by [Milosavljevic&Merritt 2003] in terms of series of Bessel functions (MM03, Eqs.24-28). The generalization to arbitrary $ \alpha $ is done by introducing the effective value of $ R_0 = R_{min} \exp(-\alpha) $ at which the distribution function would be zero. The general form of the boundary condition interpolates between empty ( $ \alpha \ll 1 $) and full ( $ \alpha \gg 1 $) regimes.

This method is also used to estimate the capture rates in axisymmetric galactic nuclei, as described by an effective one-dimensional approximation of [Vasiliev&Merritt 2013]. In this case, the meaning of $ R_{min} $ is the effective capture boundary $ R_{eff} $, defined in terms of flattening coefficient $ R_{sep} $ (Eq.72 of VM13).

In addition, the draining of loss region is accounted for with an approximate method described in section 6 of [Vasiliev&Merritt 2013]: initially, the draining flux is equal to the full loss cone capture rate, and declines with time so that the total captured mass from the draining region approaches Rdrain. At the same time, the flux due to relaxation is gradually scaled up from 0 to the flux in the time-dependent solution with initially empty loss region.

The methods implemented in this code are based on the following papers: