Astron. Astrophys. 318, 73-80 (1997)

2. Observational and theoretical constraints

Looking to the light curve of CAL 87 its asymmetry will be noticed immediately. In this paper we want to follow the idea that the interaction of the accretion stream with the disk rim could be the origin of this: Looking to the basic constituents of a binary system, the compact star, its companion, the disk and the mass flow, only the effects of the hot spot can add an asymmetric component to the light curve. The high accretion rate suggested in the vdH model on the order of could lead to rather dramatic effects when this gas impinges the disk rim. In other X-ray binaries such sprays have already been considered.

The asymmetry of the light curves of eclipsing nova-like cataclysmic variables is also understood as resulting from the hot spot (e.g. Smak 1971, Smak 1994). In these systems the companion does not significantly contribute to the optical light and one observes only the disk and the hot spot. Their maximal optical light is observed just before the eclipse of the disk when we look directly towards the hot spot which is then located in front of the white dwarf (see Fig. 1 for the geometry which is the same for all close binaries with mass transfer). - In contrast, CAL 87 shows more optical luminosity after eclipse than before. In the model this is a natural consequence of the hot spot and the irradiation effect: The geometrical thick hot spot region acts as a large screen collecting a lot of radiation flux of the white dwarf. The emitted reprocessed optical light can be well observed at orbital phases where the hot spot is behind the white dwarf, that is after the eclipse (Fig. 1). Before eclipse we look against the non-irradiated outer rim of this area which covers the illuminated inner disk regions (Callanan et al. 1989, Cowley et al. 1990).

Fig. 1. A model of CAL 87 around eclipse.

The optical light curve of LMXB 4U 1822-37 shown by Mason et al. (1980) is similar to that of CAL 87. Also there the intensity drop at the phases before eclipse was connected to the interaction between accretion stream and disk. Hellier & Mason (1989) modelled the optical and X-ray light curves including a bulge around phase 0.8 or two bulges around phase 0.8 and 0.2 where they connect the 0.8 - bulge with the impact of the accretion stream on the disk (where their phase 0.8 corresponds to in our notation, see e.g. Fig. 3 below). In their simulation they added normalization parameters for the emission of the disk rim and the inner disk and also took for the illuminated secondary an optional constant plus sinusoidal component.

Fig. 2. The upper diagram shows parcticle trajectories projected in the orbital plane. The circle corresponds to the disk rim at 0.8 . In the lower part of the figure the corresponding values of in dependence on the azimuth are drawn.

Fig. 3. The upper panel shows the disk radius dependent on the azimuth where is the phase of impact. The maximal disk height of the hot spot is represented in the lower diagram.

Noting this similarity Callanan & Charles (1989) calculated a fit to the light curve of CAL 87 using a similar model to that used in determining the disk structure of LMXB 4U 1822-37 by Hellier & Mason (1989). The stellar masses in their LMXB model for CAL 87 are presumably different from those expected now (mass ratio in Hellier & Mason (1989), here). The phase dependence of the secondary's light described as a constant plus sinusoidal seems not to be adequate (compare the light curve in our Fig. 4, model a and b).

Fig. 4a and b. Shown are light curves of four simulations corresponding to model a and b (upper panel), c and d (lower panel). For a detailed description see Sect. 4. For each model the simulated optical light curve (solid lines) and the contribution of the star (dashed lines) and the disk (dotted lines) are drawn. The dots show the composite V light curve of CAL 87. The photometry is done between 1985 November and 1992 December (Schmidtke et al. 1993). A view to the systems at is shown below (1/4 resolution of the calculations).

Another fit to the optical light curve of CAL 87 was published by Khruzina & Cherepashchuk (1994) who included also a bulge at the rim of the accretion disk at the hot spot. In their model the compact object is a neutron star or a black hole and the mass ratio is on the order of 0.1.

For our analysis we first developed models for the hot spot region with a small vertical extension, without success. The total luminosity and the shape of the light curve (secondary minimum) did not nearly fit the observations. Instead of this, models with a thick structure at the disk rim allow to find a proper fit with small variations of the involved quantities (radius, height and temperature) in agreement with a similar result by Hellier and Mason (1980) for their simulations before.

This description is also supported by the low ratio suggesting that the central X-ray source is hidden from direct view similar to LMXBs, as supposed by Pakull et al. (1988). Additionally, there would be a deep eclipse in the X-ray light curve if the observed X-rays originated from the central white dwarf and not from scattering of surrounding gas as the observations suggest (Schmidtke et al. 1993, Kahabka et al. 1994). In order to cover the source the optically thick region has to have a vertical extension at each angle of azimuth of which is about 0.25 in our calculations. The maximal simulated opening angle of the gas will be 0.45.

We compare this to the scale height of the disk and of an X-ray induced corona where the temperature K corresponds to the X-ray flux obtained by the vdH model. Therefore, cold matter can not reach the required vertical height in form of a hydrostatic layering. Although the coronal scale height is of the right order it will not be optically thick to cover the central source continuously.

Therefore, the observed optical light has to be emitted by a cold and clumpy gas. This spray is embedded in the hot corona and both are in pressure equilibrium. All spray matter should then cover the central white dwarf all the time. Expelled in all directions the spray must move along free fall trajectories around the accretion disk. This can explain the large vertical height required to reproduce the observations mentioned above.

Although the hydrodynamic generation of such a two phase medium is not yet fully understood there are several conclusive arguments for their existence. Howarth & Wilson (1983) and Karitskaya et al. (1986) observed cold blobs ( K) with number densities of cm^-3 of the X-ray binary HZ Her/Her X-1 in optical and UV data. Bochkarev & Karitskaya (1989) suggested a physical model for these observed blobs. Schandl (1996) showed that the spray generated by a stream impinging onto a warped disk surface generates the preeclipse dips in the X-ray light curve of Her X-1. Frank et al. (1987) explained with such a two phase medium the optical light curves of dipping and coronal LMXBs. Also the observations of Cal 87 (Hutchings et al. 1995) showing gas in the temperature range between 25000 K and 29000 K around eclipse are consistent with such a spray.

Summarizing, the asymmetry in the optical light curve, the depth of the secondary minimum, the low ratio and the small decrease in X-rays during the deep optical eclipse may be understood by an optically thick spray of large vertical height which permanently covers the white dwarf.

© European Southern Observatory (ESO) 1997

Online publication: July 8, 1998
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