3. The optical light curve
There is now a general consensus that optical emission in persistent and transient (in outburst) low-mass X-ray binaries is due to reprocessing of X-rays by the outer disc (see e.g. van Paradijs & McClintock 1995). This is also the case for GRO J1655-40, since a simple estimate of the expected optical flux from a non-irradiated disc which reproduces the observed X-ray emission falls short of the observed optical flux by roughly an order of magnitude (see Fig. 2). Moreover, as we pointed out in Sect. 2, the optical light-curve is entirely consistent with the decline from maximum during a typical FRED-type outburst in an irradiated disc. Longer UV and optical decline time scales ( days, as compared to days for the X-ray flux) are simply due to the fact that as the irradiating flux declines, the outer disc edge becomes cooler and the peak of the emission moves into the optical band, thus compensating for the decrease in the total emission from the outer disc.
However, this simple picture clearly cannot explain the behaviour of the optical flux at times later than days after the onset of X-ray outburst. At this time, X-ray flux begins to increase, while the optical and UV emission continues to decline. Here we argue how the scenario for the X-ray emission of GRO J1655-40 described in Sect. 2 can reconcile these observations with the X-ray reprocessing origin of the optical emission.
As shown by Dubus et al. (1999b) the outer regions of a planar accretion disc cannot intercept the X-rays emitted by a point source located at the midplane. Therefore, in order for the outer disc to be irradiated, it must be warped (the other possible way for the outer disc to see the X-rays - an extended irradiating source - fails to explain why the outer disc is effectively geometrically thick while the vertical equilibrium implies very thin discs).
The origin and propagation of warps in accretion discs is still an open question (see e.g. Pringle 1999). Here we consider two possible regimes: one with low viscosity, when the warp propagation relies on sound waves; and another with high viscosity, when the warp evolution is driven by diffusion. As we shall see, in both cases the warp amplitude in GRO J1655-40 is likely to decay during an outburst.
If the viscosity is low, i.e. if , where H is the half-thickness of the disc, the warp can propagate as a non-dispersive wave at approximately the speed of sound (Papaloizou & Lin 1995). This regime can be relevant if the outer disc regions are not affected by the propagation of the heat front during the outburst. In the standard DIM the heat-front passage changes the viscosity parameter from a low, cold () to a high, hot () value. Therefore, if the heat front does not reach the outer disc it is conceivable (since we don't know the physical mechanism supposedly responsible for the change of ) that in the outer regions we would still have , while (see e.g. Dubus et al. 1999b). In such a case the increased stream of mass transfered from the secondary deposits matter moving in the orbital plane at . Since various parts of the disc communicate efficiently through sound waves, this new component will exert a torque on the outer disc reducing the warp on a time scale of a few forced precession periods, where the precession period is days (Larwood 1998).
On the other hand, the whole disc could be in a high state during an outburst. Since in a standard accretion disc in the outer regions, and the warp propagation is driven by viscous processes. The relevant viscosity is the one corresponding to the vertical shear. The ratio of this kinematic viscosity coefficient to the standard (radial) one is approximately , for (Papaloizou & Pringle 1983), and therefore the warp damping time is . At the outer disc edge of GRO J1655-40, is then about 100 days for , in very good agreement with our scenario. Note that in this regime the increase of mass transfer would not affect the warp.
Pringle (1996) found that warp can be radiation driven. In such a case the warp's viscous decay could be prevented by irradiation. For this to happen, the growth rate of the radiative instability must be shorter than the viscous damping time . This condition can be written as (e.g. Wijers & Pringle 1999):
where is the accretion efficiency and is the critical ratio of the radiative growth to the viscous damping times. One can see that for GRO J1655-40 the inequality above can be satisfied only for values higher than the ones usually assumed in the DIM. Unless such values are assumed a warp will be viscously damped on a time-scale estimated above.
Whatever the mechanism of warp decay, it would result in the reduction of the irradiating flux intercepted by the disc, and a consequent decrease in the observed optical flux from the system.
To illustrate this argument we calculated a series of optical spectra from uniformly accreting thin discs with varying degree of irradiation. The value of the mass accretion rate was chosen to reproduce the black body component of X-ray emission (Sobczak et al. 1999). In our simple treatment here we specify neither the origin nor the structure of the warp. However, since we need some description of the photosphere of the disc above the orbital plane as a function of radius, we use the prescription,
chosen by analogy with formulae used in the literature, which (despite being based on an incorrect assumption about irradiated discs) seem to give a correct empirical description of the reprocessed X-ray flux (see Dubus et al. 1999b). All other properties of the warped disc, as far as irradiation is concerned, are described by , defined in Eq. (2). We use the prescription for irradiation of the outer disc by the inner disc edge (e.g. see Shakura & Sunyaev 1973; King & Ritter 1998), which combined with Eq. (8) above gives
where . Note that for the disc-disc irradiation geometry, the strength of irradiation is quadratic in , since it depends on the projections of both the emitting and irradiated annuli.
Of course the exact dependence of z on the disc radius is important in determining the shape of the optical spectrum. However, there are many other highly uncertain quantities in the calculation (e.g. radial profile of the albedo in the outer disc, angular distribution of the irradiating flux, details of radiative transfer in the atmospheres of irradiated discs) which all contribute significantly to the appearance of the disc in the optical band. In addition, we are using a flared planar disc `approximation' to describe a warped disc. Since all these uncertainties are hidden inside our parameter , the exact choice of is not very important. As far as we are concerned, Eqs. (8) and (9) simply describe the distribution of the irradiating flux with radius and do not result from some assumed vertical disc structure. Note especially that we do not assume that the irradiated disc is isothermal or adopt a particular value for the disc aspect ratio at the outer edge.
The resulting spectra computed for different values of are shown in Fig. 2. For comparison we also plotted the dereddened spectra of GRO J1655-40 observed (in order of decreasing flux) on May 14, June 8, June 20, June 30, and July 22, 1996 with HST /FOS red prism. (Note that these spectra correspond to the optical and UV data points shown in Fig. 1.) Since the contribution from the secondary is quite significant, especially at later times, we have subtracted from the data an estimate of the quiescent optical spectrum of GRO J1655-40, adjusted according to orbital phase (see Hynes et al. 1998 and Hynes 1999 for a fuller description of data processing). The broad hump centered at which remains after subtraction in the last two spectra has a similar spectrum to the secondary, and we suspect is probably due to a residual contribution from the companion. This suggests brightening of the secondary in outburst due to X-ray heating, supporting our irradiation scenario.
Though the model spectra are not intended as a formal fit to the data because of many simplifications in our calculation, the agreement between the five sets of spectra is fairly good. Fig. 2 shows qualitatively that by decreasing the degree of warping (as described by ), and therefore irradiation, of the outer disc, we can mimic the observed evolution of GRO J1655-40 in the optical band. Note how with decreasing the model optical spectra become softer, just as observed, even allowing for some residual contribution from the secondary in the data.
One should keep in mind that all model spectra shown in Fig. 2 were computed for a planar disc with and a fixed outer radius, given by Eq. (3). This means varying in our model corresponds simply to different values of the photospheric height at the outer radius. A more realistic description of the effects caused by enhanced mass transfer should, of course, include both changes in the disc profile (different functional form for ) as well as possible changes in the disc outer radius. We feel that as long as we use a flared planar disc to approximate the effects of the warp, the exact disc shape is beyond the scope of this paper. Similarly, it is difficult to calculate the change in the outer radius of the disc from first principles. However, observations and numerical simulations of dwarf nova discs (e.g. Smak 1984; 1999b) show that at the onset of the outburst expands by about (from the canonical to nearly of the Roche radius) for a constant mass transfer rate from the companion. By comparison, simulations with enhanced mass transfer (say by a factor of 30) simply decrease this value to , so the difference is too small to be constrained by the data.
© European Southern Observatory (ESO) 2000
Online publication: February 25, 2000