Astron. Astrophys. 356, 363-372 (2000)
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
. For these most luminous parts of
the disk we take given by (20). The
overall emission of the disk is defined by the gravitational energy
release , where
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]](img154.gif)
As the temperature decreases, the law of decline switches to:
![[EQUATION]](img155.gif)
The mass of the disk can be derived by integrating
over its surface:
![[EQUATION]](img156.gif)
![[EQUATION]](img157.gif)
The constant is the same as in
the previous section. These solutions give an asymptotic law for the
disk bolometric luminosity variations. The value of
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
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 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 ). Imagine a situation
when one of these parameters, say ,
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
. Supposing that the mass of the
disk remains constant during this transition and taking into account
that the profile of does not
change, it can be seen from (7) that F changes discontinuously
and the luminosity jumps as
,
. The relation between the new and
the old F and D, obtained from (7), gives the new term
in (21):
![[EQUATION]](img166.gif)
Thus the increase in gives the
increase in D and, consequently,
which implies a steeper light curve
after the transition than before. The increase of
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]](img184.gif) |
Fig. 2. Bolometric luminosity and calculated for parameters: , , , . 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 . The inset illustrates the case of increase of 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
( ,
), where
and
(indexes 1, 2 denote different
opacity regimes) - two conditions allowing us naturally to define both
and
:
![[EQUATION]](img190.gif)
The right top indexes of and
indicate the opacity regimes. As the
profiles of are very close in these
two regimes (see Fig. 1), and parameters
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
( ,
, etc.) calculated in the two
solutions are sufficiently accurately equal.
At the time the free-free
absorption coefficient calculated
in the Thomson opacity regime and in the free-free opacity regime
takes the form:
![[EQUATION]](img195.gif)
The closeness of to
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
,
. Hereafter we substitute
with their typical values in a
self-consistent way. The transfer between Thomson and free-free
regimes begins at the moment ,
- arrow A at
in Fig. 2. We intersect the
curves at
![[EQUATION]](img202.gif)
what corresponds to and
for
(left small arrow). We call
"moment of transition". The
transition ends at the time when
the solutions match at - arrow
B at 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
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
(right small arrow) corresponds to
the other intersection of functions
, 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 decreases to the value
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
and
this happens at
:
. For investigation of the disk
evolution on larger time-scales see e.g. Cannizzo et al. (1995),
Cannizzo (1998), Kim et al. (1999).
© European Southern Observatory (ESO) 2000
Online publication: March 28, 2000
helpdesk.link@springer.de  |