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

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6. Bolometric light curves of time-dependent standard accretion disk: power law

In order to calculate the luminosity of the disk, we assume the quasi-stationary accretion rate as it is at [FORMULA]. For these most luminous parts of the disk we take [FORMULA] given by (20). The overall emission of the disk is defined by the gravitational energy release [FORMULA], where [FORMULA] is the efficiency of the process.

At early t, when the Thomson scattering is dominant, we derive that the bolometric luminosity of the disk varies as follows:

[EQUATION]

As the temperature decreases, the law of decline switches to:

[EQUATION]

The mass of the disk can be derived by integrating [FORMULA] over its surface:

[EQUATION]

[EQUATION]

The constant [FORMULA] is the same as in the previous section. These solutions give an asymptotic law for the disk bolometric luminosity variations. The value of [FORMULA] will be obtained in Sect. 6.2 when we shall discuss the transition between the regimes of opacity.

We remark that the observed X-ray light curves can have different (most probably, steeper) law of decay. Indeed, the energy band of an X-ray detector usually covers the region harder 1 keV where the multi-color photon spectrum of the disk (having appropriate temperature) can have turnover from [FORMULA] power law into exponential fall. This turnover is expected to change its position due to variations in temperature of the disk after the burst. The narrower the observed band, the more different the observed curve could look like in comparison with the expected bolometric flux light curve. In Sect. 7 we discuss this subject in more detail.

6.1. Luminosity - accretion disk parameters dependence

It is essential to point out that in formulae (36), (37) the parameters [FORMULA] cannot be changed to describe how luminosity depends on them. Indeed, these expressions were found as a result of solution of differential Eq. (6) with the constant coefficient D (which depends on parameters of the disk, except [FORMULA]). Imagine a situation when one of these parameters, say [FORMULA], quickly increases. This will not result in the decrease of the luminosity as it might seem from (36) or (37). What will happen really is that the accretion will change to another solution (during the same regime of opacity), according to the new [FORMULA]. Supposing that the mass of the disk remains constant during this transition and taking into account that the profile of [FORMULA] does not change, it can be seen from (7) that F changes discontinuously and the luminosity [FORMULA] jumps as [FORMULA], [FORMULA]. The relation between the new and the old F and D, obtained from (7), gives the new term [FORMULA] in (21):

[EQUATION]

Thus the increase in [FORMULA] gives the increase in D and, consequently, [FORMULA] which implies a steeper light curve after the transition than before. The increase of [FORMULA] can be possibly provided by the enhanced role of convection in the accretion disk and will result in the brightening of the disk. This situation is displayed in the inset in Fig. 2. We note that the descending portion of the curve after the increase is uncertain if convection is involved since the disk structure modifies from that presented in Sect. 4.

[FIGURE] Fig. 2. Bolometric luminosity [FORMULA] and [FORMULA] calculated for parameters: [FORMULA], [FORMULA], [FORMULA], [FORMULA]. Shown are the solution in Thomson opacity regime (solid line) and in the free-free opacity regime (dashed line). Their bold parts represent the resulting light curve of the disk. Small arrows mark two intersections when [FORMULA]. The inset illustrates the case of increase of [FORMULA] from 0.3 to 1

6.2. Thomson opacity - free-free opacity transition

The temperature of the disk decreases with time, and eventually the free-free and free-bound opacity supersedes the Thomson one. It is possible to connect two regimes at the point ([FORMULA], [FORMULA]), where [FORMULA] and [FORMULA] (indexes 1, 2 denote different opacity regimes) - two conditions allowing us naturally to define both [FORMULA] and [FORMULA]:

[EQUATION]

The right top indexes of [FORMULA] and [FORMULA] indicate the opacity regimes. As the profiles of [FORMULA] are very close in these two regimes (see Fig. 1), and parameters [FORMULA] vary slightly with radius (being roughly constant in the region where the substantial mass of the disk is enclosed), the physical parameters of the disk ([FORMULA], [FORMULA], etc.) calculated in the two solutions are sufficiently accurately equal.

At the time [FORMULA] the free-free absorption coefficient [FORMULA] calculated in the Thomson opacity regime and in the free-free opacity regime takes the form:

[EQUATION]

The closeness of [FORMULA] to [FORMULA] cm2g-1 confirms the reliability of our calculations and yields the smoothness of the transition.

Fig. 2 represents the bolometric light curve of the disk for [FORMULA], [FORMULA]. Hereafter we substitute [FORMULA] with their typical values in a self-consistent way. The transfer between Thomson and free-free regimes begins at the moment [FORMULA], [FORMULA] - arrow A at [FORMULA] in Fig. 2. We intersect the curves at

[EQUATION]

what corresponds to [FORMULA] and [FORMULA] for [FORMULA] (left small arrow). We call [FORMULA] "moment of transition". The transition ends at the time [FORMULA] when the solutions match at [FORMULA] - arrow B at [FORMULA] in Fig. 2.

This picture is reliable and useful, even though it implies the existence of two separate regimes, which is evidently not quite true. Indeed, at any epoch the inner part of the disk would be scattering dominated, the lower the accretion rate, the smaller this part. Obtaining of an exact solution needs consideration of combined free-free and Thomson opacity of the gas.

There is some t which corresponds to the Eddington limit [FORMULA] erg s-1. This means that the real source evolution could be described in our model only at later t. Thus, generally speaking, the solution before this moment appears inapplicable. As seen in Fig. 2, the applicable part of the solution belongs almost entirely to the free-free opacity regime (the bold dashed line).

The second intersection of the curves in Fig. 2 at [FORMULA] (right small arrow) corresponds to the other intersection of functions [FORMULA], meanwhile the physical parameters of the disk calculated using formulae (26)-(35) are different. Thus the disk is at the same (free-free) opacity regime as before.

When [FORMULA] decreases to the value [FORMULA] K, the convection (which presumably appeares in the zones of partial ionization) starts to influence the disk's structure, and the diffusive type of radiation transfer, which we use, is no longer valid. For [FORMULA] and [FORMULA] this happens at [FORMULA] : [FORMULA]. For investigation of the disk evolution on larger time-scales see e.g. Cannizzo et al. (1995), Cannizzo (1998), Kim et al. (1999).

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© European Southern Observatory (ESO) 2000

Online publication: March 28, 2000
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