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

7. Observed light curves

As we mentioned in Sect. 6, the observed light curves can have a slope of decline which is different from that of the bolometric light curves due to particular spectral distribution. In this section we are going to illustrate this suggestion assuming the simplest spectral distribution of the disk emission.

To calculate the spectra, one can assume the quasi-stationary accretion rate in the inner parts of the disk because the variation is small there (, see Fig. 1). The outer parts of the disk, where accretion rate varies significantly, contributes to the low-frequency band of the spectrum. We discuss the X-ray band and the most luminous parts of the disk and, thus, we take given by (20).

Provided each ring in the disk emits as a black body, the temperature of the ring can be found as follows:

where  is the Stephan-Boltzmann constant. Then

In the last expression the stationary solution for is taken. The black-body approximation is satisfactory if . Then the outgoing spectrum is the sum of Planckian contributions of each ring of the disk and has the characteristic slope for photon energies , where is the maximum effective temperature of the disk (Lynden-Bell 1969). However, if the Thomson scattering on free electrons contributes substantially to the opacity, the outgoing spectrum is modified (Shakura & Sunyaev 1973). See e.g. Ross & Fabian (1996) for investigation of spectral forms of accretion disks in low-mass X-ray binaries.

The light curve is simulated by integrating at each t the spectral density

over the specific frequency range using (20) and (44), where  erg s is the Planck constant. The numerical factor in (45) corresponds to the luminosity outgoing from one side of the disk.

Explaining the observed faster-than-power decay of outbursts in soft X-ray transients, one must take into account the specificity of the energetic band of the detector. Naturally, the observed slope of the curve depends on width and location of the observing interval. The narrower this band, the more different the observed curve could look like in comparison with the expected bolometric light curve. Of course, this difference also reflects the spectral distribution of energy coming from the source.

We show here how the slope of the curve changes in the simplest case of multi-color black body disk spectrum according to which spectral range is observed. Following (45) we calculate and integrate it over three energy ranges: 3-6 keV, 1-20 keV and that one in which practically all energy is emitted. Fig. 3 shows the photon flux variations in two X-ray energy ranges (those of Ariel 5 and EXOSAT or Ginga observatories) and the bolometric flux variation for the face-on disk at an arbitrary distance of 1 kpc. The vertical line marks the time after which bolometric luminosity of the disk's one side is less than .

Fig. 3. The flux from one side of accretion disk at 1 kpc for parameters: , , , . The curves show the bolometric flux (upper curve), the 1-20 keV flux (middle curve) and the 3-6 keV flux (lower curve) during the Thomson opacity regime (solid parts) and the free-free opacity regime (dashed parts)

One can see an almost linear trend of the X-ray flux when bolometric luminosity is under the Eddington limit (to the right of the vertical line in Fig. 3), especially in intervals of . The decline becomes closer to the exponential one with time. The slope of the curve depends on , m, , and other parameters. For the same parameters as in Fig. 2, the e-folding time falls in the range 20-30 days for the lower curve (3-6 keV). For instance, smaller will result in less steep decline.

The natural explanation of such a result is the following: because the spectral shape of the disk emission has Wien-form (exponential fall-off) at the considered X-ray ranges, the law of variation of X-ray flux is roughly proportional to . In the free-free regime of opacity we have  (see Sect. 5.1). Consequently, the observed X-ray flux varies like , which is quite close to exponential behavior. We restrict ourselves to this brief and general discourse, as a detailed application of our model to observed sources is not a goal of this paper.

© European Southern Observatory (ESO) 2000

Online publication: March 28, 2000
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