## 5. Time-dependent accretion in Keplerian disk## 5.1. Solutions to non-stationary Keplerian disk equationThe self-similar solutions of Eq. (6) were found by LS87. In
these solutions any physical characteristic of the disk, for instance,
the surface density , can be
presented in the form: , where the
scales and
depend on In a binary system the accretion picture has particular features. The main feature is the limitation of the outer radius. Thus, one cannot apply the LS87 solution at the third stage, that is, during the decay of accretion after the peak of the outburst. The spreading of the disk is to be confined by the tidal interactions. The tidal torque produced by a secondary star has strong radial dependence (Papaloizou & Pringle 1977). As Ichikawa & Osaki (1994) showed, the tidal effects are generally small in the accretion disk, except near to the tidal truncation radius, which is given by the last non-intersecting periodic particle orbit in the disk (Paczynski 1977). They concluded that once the disk expands to the tidal truncation radius, the tidal torques prevent the disk from expanding beyond the tidal radius. A class of solutions of Eq. (6), which this paper focuses on,
can be found on separating the variables Substitution of the product of two functions in Eq. (6) gives the time-dependent part of the solution:
The equation for is a non-linear differential equation of second order which is a particular case of the general Emden-Fowler equation (Zaycev & Polyanin 1996): the solution of which we seek as a polynomial Substituting into Eq. (22) we obtain for the second and the third term: and , are to be defined from the boundary conditions on . We consider the size of the disk to be maximum and invariant over
the period of outburst. As the drain of angular momentum occurs in a
narrow region near this truncation radius (Ichikawa &
Osaki 1994), we treat the region near this radius as the
-type channel, not considering the
details of the process. In other words, the smooth behaviour of
spatial factor
Corresponding and are displayed in Table 3.
Naturally, real accretion disks have finite value of , but still, in most cases, , that is equivalent to in our problem from the mathematical standpoint. Note that (21) implies a considerably The following subsections contain the explicit expressions for the physical characteristics of the disk. They are deduced from (7), (17), (18), and (21). We introduce for the mass of the central object the quantity . ## 5.2. Thomson opacity regime ( )Here the function and the values should be taken for the Thomson opacity regime. Then we have: is the effective optical thickness of the disk defined by the combined processes of scattering and absorption. We take approximately (cf. (19)): ## 5.3. Free-free opacity regime ( )Here the function This regime is characterized by lower temperature and density, and the optical thickness of the disk is defined by the processes of free-free absorption: . © European Southern Observatory (ESO) 2000 Online publication: March 28, 2000 |