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

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3. The simulation

3.1. The secondary star

We describe the surface of the secondary star with a spherical grid of [FORMULA] resolution in the polar and the azimuthal angles. Further on, the grid points lie on the equipotential surface of the Roche lobe of the star.

The observation of the secondary star is difficult because of a star which is in the close projected vicinity of CAL 87. After subtracting the light of this star the remaining spectrum shows apparently no signature of the secondary star (Pakull et al. 1988 and private communication 1995). Therefore we assume a temperature for the non-irradiated secondary of about [FORMULA] = 8000 K as calculated for a main sequence star of 1.5 [FORMULA] with a radius of 9.7 1010 cm (= Roche radius) and 0.25 solar metallicity specific for the LMC (Schaller et al. 1992).

We describe the source of radiation, the white dwarf, as a point source. The irradiation of each surface element of the secondary depends on the angle of incidence [FORMULA] and one obtains for the temperature of the irradiated star


with [FORMULA] Stefan-Boltzmann constant, r distance between the white dwarf and the surface element. The parameter [FORMULA] is the efficiency of the reprocessing of the illuminating radiation from the white dwarf to thermal radiation, comparable to [FORMULA] where a is the albedo.

Depending on the geometry there is no irradiation of surface areas located behind the illumination horizon or in the shadow of the accretion disk (see Fig. 4 model a or b).

In some models below we include energy transport from irradiated to non-irradiated parts of the stellar surface. We took a simple description because the correct solution of this problem is complex and beyond the scope of this paper. We spread the luminosity [FORMULA] of each surface element with area dq weighted by a Gaussian kernel and integrate over the whole surface. The resulting luminosity of each surface element is


[FORMULA] is the angle between the normal vectors of the surface elements dq and [FORMULA] and the parameter B describes the angular width of the spreading. In our simulations we use [FORMULA]. The normalization constant [FORMULA] achieves the equality of the total luminosity of the star with and without energy transport:


We take a black body spectrum for the emitted radiation of the secondary star where [FORMULA] is the wavelength, h the Planck constant, c velocity of light and k the Boltzmann constant,


This radiation flux is folded with an optical filter function [FORMULA] (Allen 1973) to get the optical flux of each surface element


Finally the total observed optical flux is calculated using the angle [FORMULA] between the normal vector of the surface element and the direction to earth:


3.2. The accretion disk

This section describes the disk without any interaction with the accretion stream. The modifications due to the spray are the topic of the next section.

The disk height [FORMULA] (height of the photosphere above midplane) of the irradiated disk was determined from vertical structure computations for several values of [FORMULA], [FORMULA] and [FORMULA] consistently. Approximation by power law gives


The disk temperature is determined by frictional heating and irradiation, with [FORMULA] again angle of incidence


with G the constant of gravitation. The visual light of the black body radiation is calculated following Eqs. (4) - (6).

The disk size is taken as [FORMULA] ([FORMULA] Roche lobe radius) according to the models of Paczyski (1977) and Papaloizou & Pringle (1977).

3.3. The spray

We model the optically thick surface of the spray matter in form of an increase of the disk size, height and temperature. As observed by Hutchings et al. (1995) the temperature around eclipse is between 25000 K and 29000 K. Due to the illumination by the white dwarf, the corresponding surface temperature increases (see Eq. 8) and this increases the observed mean temperature of all surface elements. We assume the lower observed limit, 25000 K, for the temperature of the un-illuminated spray.

As a guide-line to the expected form of a spray Fig. 2 shows some trajectories of particles freely falling in the potential of the binary. They start at the inner Lagrangian point L1, hit the disk and spray around. The vertical height of the point of impact varies between [FORMULA]. The mean direction of the spray is tangential to the disk and has an opening angle of up to [FORMULA]. The particle velocities vary from [FORMULA] to [FORMULA]. We also assumed that the spray matter, when impacting on the disk again, produces a new but reduced spray. Thus we calculated the trajectories further-on in the following way: We reduced the vertical velocity by a factor of 2 and we reflected the trajectories. In the orbital plane the reflection angle was half the angle of incidence, relative to the tangent of the disk. At the second impact of the spray matter on the disk we stopped the calculations.

The figure shows that the extention of the spray increases strongly in radial and vertical directions just after the impact of the accretion stream. Afterwards the spray seems to be like a diffuse nebula moving around the disk (see also Bochkarev & Karitskaya 1989).

The spray moves not only along the disk rim but also towards smaller disk radii ([FORMULA]). But during our investigations we found that there is no real difference in the final light curve between solutions which also modify temperature and height at these inner disk regions. The reason is the high inclination of CAL 87 which makes the vertical extension much more relevant than the radial one. Therefore we restrict the modification of the disk shape to radii [FORMULA].

In Fig. 3 the radial and vertical modifications are shown in dependence on the azimuth [FORMULA] starting at the phase of impact. The radius increases up to [FORMULA] at about [FORMULA] and decreases at later phases to its original value. This model follows calculations of free fall trajectories of matter expelled at the hot spot. The maximal vertical extension of the disk of about [FORMULA] and the maximal radial extension are placed at about [FORMULA] after the impact.

The irradiation of the spray is calculated according to Eq. (1), the spectrum is taken as black body and the final optical flux calculation is performed in the same way as for the secondary star.

Numerically we describe the disk modified by the spray in polar coordinates with a resolution of [FORMULA] in azimuth using 200 concentric rings. Because the disk size varies with the azimuth some outer surface elements do not contribute.

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© European Southern Observatory (ESO) 1997

Online publication: July 8, 1998