5. Time-dependent accretion in Keplerian disk
5.1. Solutions to non-stationary Keplerian disk equation
The 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 t in a particular way, and is a universal function of one self-similar variable (Zeldovich & Raizer 1967). The solutions represent three stages of the non-stationary accretion on an object. The first stage is the formation of the accretion disk from some finite torus around an object. The second stage is the developing of the quasi-stationary regime of accretion, and the third - the decay of accretion when the external boundary of the disk is spreading away to infinity. LS87 obtained the self-similar solutions of type II for the first two stages and the self-similar solution of type I (Zeldovich & Raizer 1967) for the final stage (when there is conservation of the total angular momentum of the disk).
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 h and t. We seek the solution in the form , where , . From Eq. (4), substituting , it follows that
D is the constant defined by the vertical structure of the disk (Sect. 4, Eq. (18)); is a negative separation constant which can be found from boundary conditions on ; is an integration constant. From here on we set for the Thomson opacity regime. We calculate a value of for the free-free opacity regime in Sect. 6.2. Expression (21) represents asymptotic law for after-peak evolution of a real source.
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
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 f in the moment of viscous forces F (which increases as in the inner parts of the disk, then flattens, reaches the maximum and drops down near due to tidal torque) is analytically treated as increasing, flattening, and reaching maximum at , which is the end of the disk (this profile is shown in Fig. 1). Thus we propose the boundary conditions as follows:
Corresponding and are displayed in Table 3.
Table 3. Summary of parameters in solutions for two opacity regimes for the Keplerian disk
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 steeper time dependence than the solution by LS87 does. The latter yields the accretion rate as a function of if , and if . In our case this dependence is if , and if . This difference is due to the non-conservation of angular momentum in the disk in our case.
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 ( )
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 f and the values should be taken for the free-free opacity regime. The following formulae contain the constant appeared in expression (21). It was neglected in the previous subsection; here accounts for the possibility of time shifts between the solutions in the two opacity regimes. We have:
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