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Astron. Astrophys. 356, 363-372 (2000)

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8. Viscous evolution of advective disk

As we know, the structure of an accretion disk in the vertical direction, the relation between the viscous tensor and the surface density in particular, defines the type of its temporal evolution. In advective disks, which are the low-radiative accretion flows, the relations between their characteristic physical parameters differ significantly from those in standard disks. In this section, we discuss the results of Sect. 2 as applied to the disks which radial structure was presented by Spruit et al. (1987) and Narayan & Yi (1994, 1995, hereafter NY).

The viscous stress and the surface density are related through the kinematic coefficient of turbulent viscosity. Integrating the component of viscous stress tensor one obtains (cf. (14) and (15)):

[EQUATION]

where [FORMULA] is the averaged kinematic coefficient of turbulent viscosity. Then the relation between [FORMULA] and [FORMULA] is given by

[EQUATION]

Recall that [FORMULA] is the real specific angular momentum and h is the Keplerian one. It can be seen that [FORMULA] and [FORMULA] define what class of solutions Eq. (5) will have.

If one adopts for the structure of advection-dominated accretion flow (ADAF) the self-similar solution by NY, it can be easily inferred that such disks exhibit the exponential with time behaviour. The solution of NY is given by:

[EQUATION]

Expressing [FORMULA] in the basic Eq. (5) in terms of F, we obtain from (47) and (48):

[EQUATION]

Solution (48) enables deriving the relation between [FORMULA] and [FORMULA] using the [FORMULA] prescription of viscosity:

[EQUATION]

where [FORMULA] is the isothermal sound speed. Thus [FORMULA] is a function of radius alone,

[EQUATION]

and Eq. (49) can be rewritten in the form:

[EQUATION]

Solution to (52) is sought as a product of two functions [FORMULA] and [FORMULA], with [FORMULA], [FORMULA] being some value of h:

[EQUATION]

[EQUATION]

The exponential temporal behaviour of NY flow is evident. Generally speaking, any disk possessing such properties of [FORMULA] as constancy in time would have such exponential behaviour because its evolution would be described by a linear equation (like (52)).

The question is, would the confined NY disk keep such properties or it would not. The fact is that NY solution describes the infinite disk. Either the boundary conditions destroy the linearity of (52) or just the characteristic decay time changes, this problem requires further accurate numerical investigation. For instance, Narayan et al. (1997) calculated numerically the global structure of stationary advection-dominated flow with consistent boundary conditions; they noted that although the self-similar solution (48) makes significant errors close to the boundaries, it gives the reasonable description of the overall properties of the flow.

Further we assume that exponential trend of solution persists. Generally speaking, the equation determining [FORMULA] will differ from (54). This difference may be not very significant. One can see that Eq. (54) is a particular case of (22) where [FORMULA] and [FORMULA] and, hence, the solution can be found according to (24) and (23). Besides, the solution of (54) can be found in terms of Airy functions (Bessel functions of order [FORMULA]).

The accretion rate evolves with time as follows (cf. (20) and (53)):

[EQUATION]

The value of accretion rate can be determined if an initial condition is imposed at some t. Mahadevan (1997) showed that ADAF luminosity [FORMULA] or [FORMULA] according to whether the electron heating is dominated by the Coulomb interactions or by the viscous friction. Subsequently, the luminosity has an exponential decay too.

We can estimate the characteristic time of evolution of such flow. It can be obtained from (55). Let us compare the diffusion time [FORMULA] with the corresponding orbital period [FORMULA]. Since [FORMULA], with (48) we have:

[EQUATION]

We use the expressions for [FORMULA] and [FORMULA] from NY. Here [FORMULA] is the ratio of specific heats; [FORMULA] measures the efficiency of radiative cooling. In the limit of no radiative cooling, we have [FORMULA] while in the opposite limit of very efficient cooling [FORMULA]. NY solution is degenerate if [FORMULA] because the angular velocity of the flow is zero in this case.

We can see that the time-dependent advection-dominated disk is quickly depleted if [FORMULA] is not small. For example, consider the light curves of X-ray novae which have the exponential decay time scales [FORMULA]. To obtain [FORMULA] of such order, [FORMULA] should be [FORMULA]. However, the advection-dominated solution ceases to exist if the accretion rate is greater than the critical value [FORMULA] (Narayan & Yi 1995, Mahadevan 1997), where [FORMULA]. Hence, [FORMULA] yields the critical accretion rate [FORMULA] g s[FORMULA].

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© European Southern Observatory (ESO) 2000

Online publication: March 28, 2000
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