A&A 405, 199-206 (2003)
DOI: 10.1051/0004-6361:20030514
V. Simon
Astronomical Institute, Academy of Sciences of the Czech Republic, 251 65 Ondrejov, Czech Republic
Received 15 May 2002 / Accepted 11 March 2003
Abstract
An analysis of the long-term X-ray activity of the ultra-compact binary
4U 1820-30 within the years 1996-2002, using the ASM/RXTE data, is
presented. The X-ray light curve displays complicated large-amplitude
variations, even on short time scales of a few days. Episodes of brief low
states (BLS), lasting for several days, are superposed on the general course
of the cycle, the length of which we determined to be
days by the method of the O-C residuals. We found no significant secular
trend in
;
this is consistent with the stable period found over
longer time intervals by Chou & Grindlay (2001) and is thus consistent with
the triple model. The statistical distribution of 1.5-12 keV intensity,
,
for all data is not far from Gaussian although some deviations
exist. The
curve, folded with the 172 day cycle and smoothed,
shows a rapid rise, double-peaked maximum, and a slow decline. The course of
the smoothed residuals of this curve, the so called
curve,
is more complicated than the
curve, with several minima and
maxima. Two sharp maxima of
coincide with the rising
branch and the dip after the maximum of the
curve. We have
shown that the scatter of the folded
curve is dominated by
the BLS. The activity is discussed in terms of several models. Thermal
instability of the disk can be excluded. The increases of the mass transfer
from the donor star are likely to explain the 172 day cycle of activity.
However, we argue that the highly asymmetric X-ray light curve of this cycle,
with the long and complicated profile of the decay branch, cannot be
explained purely by the influence of a third body via the Mazeh & Shaham
(1979) mechanism. Instead, a hybrid model in which the irradiation-driven
instability of the donor star governs the complicated profile of the decay
with the superposed BLS appears to be more viable.
Key words: stars: neutron - stars: binaries: close - stars: circumstellar matter - stars: individual: 4U 1820-30
4U 1820-30 is a strong and persistent X-ray source, located in the
globular cluster NGC 6624. Its orbital period of just 685 s (11 min) is one of the shortest among known cosmic binaries (Stella et al. 1987a).
The evolutionary model (Rappaport et al. 1987) speaks in favour of a partly degenerate He-rich donor star, transferring matter onto a neutron star. There remains a problem with the observed sign of the secular changes of the orbital period which is opposite to the predicted one, but this difference may be explained by the attracting force of the globular cluster (Chou & Grindlay 2001).
The long-term variations of the X-ray intensity of 4U 1820-30 appear to be cyclic with the cycle-length of about 176 days originally estimated by Priedhorsky & Terrell (1984). The most comprehensive long-term timing analysis was conducted by Chou & Grindlay (2001). The X-ray spectrum of 4U 1820-30 has been subject to a number of analyses, the most recent and comprehensive one being that by Bloser et al. (2000). They found that the track of 4U 1820-30 in the color-color diagram during the 171 day cycle is consistent with an atoll source. This suggests that the cycle is a manifestation of mass-transfer variations, and not a precessing disk. This finding is in accordance with the occurrence of X-ray bursts. They are observed only when 4U 1820-30 is near the state of the minimum X-ray intensity. The X-ray bursts become more frequent as the X-ray intensity rises and then they disappear completely (Stella et al. 1984).
The recent monitoring of 4U 1820-30 by ASM/RXTE enabled us to obtain a very dense and almost continuous coverage within 1996-2002. About 13 cycles have been covered. This data set is very suitable for the analysis of the time-averaged properties of the long-term activity of this unique object.
The recent activity of 4U 1820-30 within the years 1996-2002 was
observed by the All Sky Monitor (ASM) onboard Rossi X-ray Timing
Explorer (RXTE) (http://xte.mit.edu/). This monitor provides
long-term observations of an unprecedented quality for this object. The data
file contained the sum band intensities
(1.5-12 keV) and the
hardness ratios
,
.
Only one-day
means (1693) were used in order to increase the signal/noise ratio. This
binning is sufficient because our analysis concentrates on the long-term
activity of 4U 1820-30 and hence on features in the light curve on the
time scale of days and longer. Modified Julian Date, used by RXTE, was
transformed into more often used Julian Date. Hereafter it is used in the
form of JD - 2 400 000.
![]() |
Figure 1: The X-ray light curve of 4U 1820-30 over the years 1996-2002 according to the ASM/RXTE observations. One-day means of the sum band (1.5-12 keV) intensity are shown. They are connected by the lines for the densely covered intervals. The uncertainties quoted in the original file of RXTE measurements are marked. The triangles above and below the curve denote the moments of maxima and minima, respectively (Sect. 3.2). The fit by the moving averages for Q=17 days is also shown. The vertical bars mark the positions of the prominent and well covered episodes of brief low states, analyzed in detail in Sect. 3.2.2. See Sect. 3.1 for details. (This figure is available in color in electronic form at http://www.edpsciences.org) |
The whole sum band ASM/RXTE X-ray light curve, covering the years
1996-2002, is shown in Fig. 1. The points are connected by the
lines for the densely covered intervals to guide the eye and to resolve the
rapid variations.
varies between about 5 and 35 ct/s. This
curve represents about 13 cycles of the length reported by Priedhorsky &
Terrell (1984) and Chou & Grindlay (2001). The course of the variations is
complicated, although the basic pattern of the
170 day cycle can be
resolved. The error bars which represent the quoted uncertainty are marked.
It can be seen that the data often display appreciable fluctuations, even on
short time scales of a few days. The X-ray orbital modulation was reported
to be quite small, of the order of 3 percent (Stella et al. 1987a). This
suggests that the complicated variations in Fig. 1 are not caused
by the orbital modulation. Episodes of brief low states, hereafter referred
to as BLS and described in detail in Sect. 3.2.2, are superposed on the
general course of the
170 day cycle (mostly on its decay branch).
In order to show clearly the smooth profile of the variations through
the 170 day cycle, the X-ray light curve was fitted by the two-sided
moving averages with the filter half-width Q=17 days and a step of one day.
Q refers to the semi-interval within which the data were averaged. Q=17
days was chosen after several trials; it yields the fit which satisfies the
general course of the observed curve but, at the same time, it averages
through most BLSs.
The statistical distribution of
for all observed data from
Fig. 1 is displayed in Fig. 2a. It is not far from
being Gaussian although some deviations exist. The top of the histogram is
flatter than the Gaussian. There is also an asymmetry between the left-hand
and the right-hand side. The slope of the right-hand side appears steeper
than that of the left-hand side and falls below the Gaussian curve for
ct/s. The high and low levels of intensity of the
170 day cycle are just barely resolvable in the histogram for the
individual measurements of
in Fig. 2a. This means
that there are no peaks near the high and low limits. The statistical
distribution for the fit from Fig. 1 displays a somewhat different
pattern (Fig. 2b). It is broader than the Gaussian curve for
ct/s and more narrow outside this interval. This suggests
that the histogram in Fig. 2a is dominated by the rapid
fluctuations and BLS while Fig. 2b better reflects the
properties of the
170 day cycle.
The determination of the cycle-length in the ASM/RXTE data set was
carried out by the method of the O-C residuals. It makes use of the
residuals from some reference period and enables us to determine
and to analyze its variations. This method is not sensitive to the exact
length of the reference period and allows us to check the position of each
measured feature on the O-C diagram at each epoch.
Three features in the X-ray light curve allowed us a reasonably accurate
timing: the moment of the maximum intensity,
;
the moment of the
minimum intensity,
,
and the moment when the intensity is
crossing
ct/s on the rising branch of the cycle.
and
were determined by fitting a polynomial. The
timings could be determined for all the consecutive events and are marked in
Fig. 1.
The O-C curves were constructed for all three features. The fluctuations
of O-C were small in comparison with the length of .
There is
no prominent secular trend in the O-C curve of any feature. The value of
days was determined from the linear fit to the O-C curve of the rising branch of the cycle. This feature displayed the smallest
scatter of the O-C's - the standard deviation of
,
determined
from the O-C's, is 10 days and is caused by the intrinsic cycle-to-cycle
fluctuations of the profile of the
curve. We therefore used
this value of
for the further analysis. The ephemeris for the
maxima of intensity which will be used in the next sections was derived:
![]() |
Figure 2:
The statistical distribution of
![]() |
![]() |
Figure 3:
a) The X-ray sum band light curve of 4U 1820-30
folded with the 172 day cycle of activity. The phases were calculated
according to the ephemeris in Eq. (1). The ASM/RXTE
measurements within the years 1996-2002 are displayed. The folded
data were smoothed by the two-sided moving averages for the filter
half-widths
Q=0.05, 0.06, 0.08, 0.10, and 0.12 phase (see the
legend in b)). b) Smoothed residuals
![]() |
The 172 day cycle repeats accurately enough to enable an analysis of the
properties of the folded
curve. All data were therefore
folded according to Eq. (1). The resulting curve is shown in
Fig. 3a. The profile with a steep rise and slower decline can
readily be resolved. The scatter is substantial, especially at some phases.
In order to determine the mean course of
,
the folded data
were smoothed by the two-sided moving averages. Several values of Q were
used. These fits are superposed in Fig. 3a; it can readily be
seen that they rapidly converge to a common and well determined smooth curve
with the decrease of Q. Further decrease of Q does not add any details,
hence the fits shown here can be taken as the representative of the course
of the folded
curve. The smoothed curve shows especially
these features: (a) rapid rise within phases 0.75-1.0; (b) a
depression just after the maximum; (c) slow, roughly linear decline
within phases 0.16-0.6.
The residuals of the fit to
were smoothed by the moving
averages again. This yielded the
curve, shown in
Fig. 3b. The
curve is a measure of the degree
to which the intensity curve in a given phase reproduces for the respective
epochs. Notice that the
curve is more complicated than the
curve. A broad minimum at phase
0.7 coincides with the
minimum of the intensity curve. The global maximum of the
curve occurs at phase 0.15 and coincides with the shallow depression which
follows the maximum of the intensity curve. The second highest peak in the
curve at phase 0.85 coincides with the middle of the
rising branch of the
curve.
Figure 3c shows the smoothed course of the skewness
of the residuals of the smoothed
curve.
corresponds to the symmetric distribution of the residuals.
and
correspond to the asymmetric distributions with a tail towards
more negative or positive values, respectively. It can easily be seen from
Fig. 3c that vast majority of phases possess
.
Notice that the minimum of the
curve corresponds to the maximum of
the
curve.
The X-ray light curve of 4U 1820-30 is quite complicated, with rapid
variations on the time scale of a few days, superposed on the 172 day cycle.
Even some brief fadings from the maximum down to the minimum level of
(with an amplitude of
20 ct/s) with the decay branch lasting
about 10 days can be resolved. An inspection of the curves revealed that
often recovers back to the previous level after the fading.
These fadings and subsequent rises were abbreviated as BLS in Sect. 3.1.
The light curves of the respective BLSs were examined. It emerged that there appears to be some maximum rate of decline even for the smoothest decay branches. A test of the working hypothesis that this slope remains the same for the individual events was therefore made for the BLSs with relatively smooth decays, using the method of matching a template. The mean rate of decline 1.85 ct/s/day was determined.
The match of the rising branches of the BLSs was less certain. An examination of the profiles revealed that the course of the rise generally displays larger rapid fluctuations than the decline. The rising branch profile sometimes shows faster risetimes near the maximum.
The X-ray color behaviour during the 172 day cycle was analyzed using
the HR1's and HR2's. All the available HR1's and HR2's were plotted
versus
.
Both curves were found to contain several points which
displayed quite large errors in HR's. In order to lower the noise, we
carried out the following procedure. Firstly, all points which had the quoted
uncertainty of their HR's
were rejected. Secondly,
in order to resolve the mean course of the HR1 and HR2 curves, they were
fitted as a function of
by the code HEC13, written by Dr. P.
Harmanec and based on the method of Vondrák (1969 and 1977). This code
can fit a smooth curve to the data no matter what their course is. The fits
with the parameters
,
ct/s were proven
to satisfy the data. Finally, 5 percent of the most negative and 5 percent of
the most positive residuals of the fit were rejected; a new fit with the same
parameters as before was then repeated for the remaining data. This fit was
truncated at
ct/s because the amounts of data were too low
to enable a reliable fit above this limit.
It is obvious from Fig. 4a that a monotonic function for HR1 was obtained. A very slight wave at the upper part of the curve is not
considered to be significant. On the other hand, the course of HR2 is more
complicated (Fig. 4b). HR2 reaches a minimum at
ct/s and increases toward both larger and smaller
's.
It is also important to analyze the color changes during the BLS. The
reason is that the amplitude of
of some BLSs is comparable to
the full range of variations during the 172 day cycle. It is therefore
important to know if the color changes during BLS are consistent with the
mean course of variations of all data. The following well covered segments
were used for the three BLS: JD 2 450 322-JD 2 450 353 (decay and rise); JD 2 451 837-JD 2 451 866 (minimum and rise); JD 2 452 000-JD 2 452 026 (decay
and beginning of rise). The arithmetic means for the intervals of
5 ct/s were calculated instead of fitting because of a small amount of
data for the BLS. We can conclude that the course of the color changes during
the BLS is consistent with the mean profile of the color variations of the
whole data set (Fig. 4).
![]() |
Figure 4:
The dependence of HR1 and HR2 on
![]() ![]() |
The 3D X-ray color-color diagram, displayed in Fig. 5, was
constructed from the fits to the data in Fig. 4. This plot
enables us to analyze the spectral changes in the softer X-ray passband than
that used by Bloser et al. (2000) and to compare our results with theirs.
Notice that the banana-like track can be resolved even in the 1.5-12 keV
band. The determination of the color-color track from the fits to the HR versus
tracks thus appears to be a viable approach how to
determine meaningful tracks even from the data in relatively closely spaced
channels and suffering from a higher noise.
![]() |
Figure 5:
The 3D X-ray color-color diagram for 4U 1820-30.
Notice the atoll track in the HR1-HR2 plane. For clarity,
the symbols and numbers denote the corresponding values of
![]() |
The dependence of both HR1 and HR2 on the phase of the 172 day
cycle was examined by the moving averages with Q=0.05 and 0.08 phase
(Fig. 6). Both ratios display a clear dependence on the phase,
with the profiles resembling the
curve from Fig. 3a.
The amplitude of the HR1 curve is larger than that of HR2. HR1 displays
a small dip at phase 0.2 while this feature is absent in HR2. All this
behaviour is in accordance with the profiles in Fig. 4 where
the dependence of HR1 on
is monotonic while two values of
can correspond to a given HR2. However, we note that the part
between phases 0.8 and 1.0 in Fig. 6 may be distorted. The
scatter caused by the cycle-to-cycle variations of the
curve in
this phase range leads to a large scatter of the hardness ratios for a given
.
The fit that averages through this phase region then may not
accurately reproduce the steep, rapidly changing profile of the HR's.
The HR versus
profiles were also analyzed separately for
the rising and decay branches of the 172 day cycle. The fits by HEC13 did not
reveal significant differences between the branches. We only note that the
loop which is prominent in the bottom part of the fit to the whole data set
in Fig. 4b is clearly defined just for the decay branch.
However, this can be explained by the large asymmetry of the X-ray light
curve (Fig. 3a) which yields a much larger amount of data on the
decay branch than on the rising branch and hence allows a much more reliable
fit for the decays.
![]() |
Figure 6: HR1 and HR2 of 4U 1820-30 folded with the 172 day cycle of activity and smoothed by the two-sided moving averages for Q=0.05 and 0.08 phase. The phases were calculated according to Eq. (1). We note that the part between phases 0.8 and 1.0 may be distorted because of the steeply changing profile. See Sect. 3.3 for details. (This figure is available in color in electronic form.) |
We have carried out an analysis of the long-term X-ray activity of the
ultra-compact binary 4U 1820-30 within 1996-2002, using the ASM/RXTE
data. Complicated large-amplitude variations, even on short time scales of a
few days, with episodes of BLSs that are superposed on the general course of
the long cycle, were clearly resolved. We determined the length of this cycle
days by the method of the O-C residuals. This value
is consistent with that obtained by Fourier transformation by Chou &
Grindlay (2001). The O-C method has also shown that the mean cycle-length
is stable, without any secular trend. The rising branch of the cycle appears
to be the most stable part of the cycle. Bloser et al. (2000) have shown that
the 172 day cycle is due to the mass-transfer variations onto the neutron
star. The good stability of this cycle implies that the injection of matter
from the donor, discussed below, repeats accurately.
Several mechanisms which can give rise to the 172 day cycle will be discussed.
Stella et al. (1987b) determined the luminosity at the minimum of the
172 day cycle
ergs s-1 for
the 1-30 keV passband from the EXOSAT observations. The ASM intensity of 4U 1820-30 at the minimum of the X-ray light curve differs
just a little for the individual epochs. We can then make a plausible
assumption that the above-mentioned value of
can be
representative also of the minima observed by ASM/RXTE. The mass transfer
rate onto the neutron star,
,
can be determined from the accretion
luminosity, LX, through
,
where
and
are the mass and radius of the
neutron star, respectively.
refers to the efficiency with which the
potential energy is converted to radiation. We then obtain
yr-1 for
ergs s-1 and
,
assuming a 1.4
and 10 km radius neutron star.
The mass transfer rate,
,
driven purely by the
gravitational radiation, can be determined from Eq. (1) of King et al.
(2002):
![]() |
(2) |
Thermal instability of the accretion disk is quite unlikely to be the
cause of the 172 day cycle and the BLS. The radius of the disk in 4U 1820-30
is
cm if we assume
.
Applying the models for He, C, and O disks by Menou et al. (2002), we always
obtain
,
below which the thermal instability can occur,
several times lower than
,
even for the extreme values of
and of the viscosity parameter
.
Moreover, the disk is strongly irradiated by the neutron star in 4U 1820-30,
which further suppresses the instability.
The variations of the mass outflow from the donor,
,
appear to be the most plausible explanation for the 172 day cycle. Two
promising triggering mechanisms can be assumed. (a) The tidal influence
of a more distant third body via the Mazeh & Shaham (1979) mechanism (Chou
& Grindlay 2001). (b) The irradiation-induced instability (IDI) of
the donor, invoked by the irradiation of the donor by the neutron star and
the disk corona.
The highly asymmetric profile of the X-ray light curve of the 172 day
cycle (Fig. 3a) suggests a highly non-sinusoidal course of
changes of .
A rather symmetric burst of
is
expected in the case of the Mazeh & Shaham (1979) mechanism. In this
mechanism, a third body orbiting the binary with period P3 induces a
long-term modulation of the eccentricity of the inner binary with a period
.
Thus for
min, and the constant
(dependent on the relative inclination and mass ratio of the
third body), the long-term modulation of 172 d implies P3=1.1 d (Chou &
Grindlay 2001). The observed X-ray intensity depends on the mass transfer
rate in the inner disk region,
.
The initial profile of the
burst of
will be modified by propagating through the
disk. The model by Livio & Verbunt (1988) shows that even an abrupt increase
of
will give rise to a gradual and slower increase
of
(and hence also
), occurring on the
viscous time scale. If we set
yr-1, then we can determine the viscous time of the disk to
be
and 2 days for
and 0.5, respectively,
making use of Eq. (4) of Esin et al. (2000). The typical observed duration
of the rises, about 20 days, is significantly longer than
,
unless
.
In the framework of the model by Bath et al.
(1986), this occurs when the profile of the burst is slower than
;
is then governed by the changes of
.
days is consistent with the decay time scale
of the BLS, but not with the much longer duration of the decay branch of the
172 day cycle. The circularization time of the close pair will be much longer
than 172 days (Mazeh & Shaham 1979). This is in accordance with the fact
that the minima of the 172 cycle display no secular trend in intensity.
The color behaviour during BLS is consistent with that of the whole data
set (Fig. 4) - this suggests that real large variations of
occur during BLS. BLS cannot be invoked by the action of the third body via the mechanism by Bailyn (1987). The reason is that BLSs
occur irregularly, on time scales significantly longer than the time scales
appropriate for 4U 1820-30 (which would be on the order of the orbital
period of the third body
1.1 day, suggested by Chou & Grindlay
2001). All these lines of evidence suggest that even in the framework of the
triple star model the decay branch of the X-ray light curve must be prolonged
and modulated by additional mechanism(s), possibly by a superposed IDI in a
hybrid model, as outlined below.
The irradiative flux on the surface of the donor is
.
It is then obvious that the donor in 4U 1820-30 will be
very strongly irradiated because of its very small
cm. The IDI will be influenced by the variable shielding of the donor by the
disk outer rim; the latitudes up to about 20
can be shielded during
the maximum of
.
The irradiated parts can be heated up to about 105 K (Arons & King 1993). There are several models for IDI (e.g.
Hameury et al. 1986; Gontikakis & Hameury 1993; Wu et al. 1995). They agree
that in some cases a hysteresis curve for
versus the
lobe-filling factor is possible. The donor then can switch between high and
low
.
The mean density of the donor in 4U 1820-30 about
3200 g cm-3 is relatively low for a white dwarf; this star need not be
highly degenerate, with a degenerate core and a non-degenerate envelope
(Rappaport et al. 1987). This outer layer may be able to react to the outer
irradiative flux by expansion. Currently, there is no model of the response
of such a kind of star to the irradiation. Following King (1989), we can
infer that the conditions in 4U 1820-30 are such that the intense hard X-ray
emission from the neutron star is able to reach deep into the outer layer of
the donor and that the ratio of the irradiative and intrinsic flux of the
donor is high - this speaks in favour of the feasibility of IDI model in
4U 1820-30. The propagation of the burst of
in IDI through the disk will be similar to that described above. Although the highly
asymmetric profile of the X-ray light curve of the 172 day cycle may be
attributed to a long-lasting burst of
,
comparable to the
duration of the decay branch (the course of
is governed by
the decay of
in this case (Bath et al. 1986)), the
complicated course of the BLS needs a superposition of an additional
mechanism.
Alternatively, a hybrid model, in which the influence of the third body
triggers a burst of
,
can be offered. IDI can then be
superposed on the changes invoked by the third body and solve the problem of
the long and complicated profile of the decay branch of the 172 day cycle. In
this respect, the BLS, or sometimes just a shallow depression, which often
appears on the top of the X-ray light curve and whose depth is highly
variable for the individual epochs (Fig. 3b) deserves a special
attention. We offer a possible scenario for its formation in which IDI plays
a role. The height of the outer rim is expected to vary by more than 1/3
during the 172 day cycle because of variable
.
Due to a
large increase of the inflowing matter during the burst of
,
the vertical thickness of the disk, z0, increases. The shielding
of the L1 point therefore increases - this in turn leads to a temporary
decrease of
and hence to a decrease of
on
.
The corresponding decrease of z0 and consequent increase
of irradiation of the donor again lead to an additional mass outflow from the
donor. The time scale of this process will also be controlled by the delay
caused by the time needed to transport the heat from the higher latitudes of
the donor to the vicinity of the L1 point.
In the light of the above-mentioned lines of evidence, we argue that the
complicated profile of the 172 cycle is governed by a superposition of
several mechanisms. The course of
with the phase of the cycle,
with the prevailing
(Fig. 3c), suggests that the
BLSs dominate the course of the X-ray intensity curve during the cycle. Both
the profile and the rate of decline of these BLSs remain quite similar for
the respective events with relatively smooth curves. The smaller slope of
the more complicated declines can be interpreted by the superposed rapid
(day-to-day) fluctuations. The rises from the BLSs are more complicated but
some events are symmetric with the rate of decline very similar to that of
the rise. This suggests that the rises are controlled by
in a
similar way as the decays, as argued above. Also the X-ray color variations
are helpful in the identification of the underlying mechanism. They
display the atoll track in the 1.5-12 keV band during the 172 day cycle
(Fig. 5). The atoll track in 4U 1820-30 was previously resolved
over a wider passband (2-16 keV) by Bloser et al. (2000). Here we have shown
that such a track can be resolved also in the softer and more narrow
passband. Figure 5 shows that both HR1 and HR2 increase for
most of the length of the track. Bloser's et al. (2000) data reveal that
their hard color decreaseswhile their soft color increases for most of the
length of the track. These differences may be interpreted as due to (in part)
the relative differences in the sensitivities of the ASM vs. the PCA for
the bands chosen. We also resolved the color behaviour during the BLS
(Fig. 4) (no episode of a deep BLS was covered by the
observations by Bloser et al. 2000). The color changes during BLS are
consistent with those of the whole data set. This suggests that real large
variations of
occur during BLS and that the variations of
the disk- magnetosphere boundary (see Bloser et al. 2000) are the same as
during the 172 day cycle. The long time scale of the BLS, on the order of
several days, is comparable to
and is much longer than the
orbital time of the disk. Also the amplitude of some BLSs is very high, even
comparable to the full range of
during the whole 172 day cycle.
All this suggests that the whole disk is involved in BLS. Real variations of
,
modified by the viscous processes in the disk, appear
to be the most promising explanation.
Acknowledgements
This research has made use of NASA's Astrophysics Data System Abstract Service and the observations provided by the ASM/RXTE team. I thank Dr. Hudec for reading the manuscript and for his comments. I am indebted to Dr. Harmanec for providing me with the program HEC13. The support by the post-doctoral grant 205/00/P013 of the Grant Agency of the Czech Republic and the project ESA PRODEX INTEGRAL 14527 is acknowledged.