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

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4. The effect of spray and energy transport on the light curve

Fig. 4 shows four different models with and without the effects of the spray and the energy transport (see Table 2). For comparison we set the inclination of all models equal to [FORMULA] as this is the best value for our final fit model d. The efficiency parameter [FORMULA] is changed only for model b to [FORMULA] to reproduce the total luminosity.


Table 2. The differences between the models

4.1. Model a

Model a consists of a disk not modified by the spray and a star without energy transport. This solution shows that the contribution of the disk itself is too small to reproduce the high luminosity around eclipse. At these phases ([FORMULA]) the non-illuminated surface of the star shows only a constant signal. In contrast to this the width of the simulated eclipse fits the data very well. In connection with the V-shaped eclipse, this indicates an extension of the eclipsed body comparable to the disk size. a hot spot located only in the vicinity of the impact would produce a narrower U-shaped eclipse like that of a point source. Thus the effect of the stream-disk interaction has to modify the whole disk rim as the spray might do.

Because of the thin disk, the shadow on the secondary is small. The resulting large irradiated area yields a high luminosity around [FORMULA] where we look directly to the bright front side. Due to the small projected area of a thin disk also the depth of the secondary minimum around [FORMULA] is weak. This disk barely covers the secondary.

4.2. Model b

In contrast, the disk together with the spray produces a deeper secondary minimum shown in model b (see Fig. 5 for illustration of the different orbital phases). Its irradiated area is smaller than in model a and the overall contribution of the star is less although we set the efficiency parameter [FORMULA].

[FIGURE] Fig. 5. This figure illustrates an orbit of the system (model d) at ten different phases. Left column [FORMULA], right column [FORMULA].

This model of a spray at the disk rim reproduces the observed data much better compared to model a. Only the width of the eclipse is too small although most optical light comes from the spray at the outer disk regions. This indicates that the disk is even larger or that the secondary contributes to the luminosity at these phases ([FORMULA]). For the latter possibility we now investigate the effect of a transport of energy on the secondary.

4.3. Model c

We set the energy transport width [FORMULA] in Eq. (2). This value also fits the averaged blue light curve of the X-ray binary Her X-1 which is a binary with an irradiated 2.2 [FORMULA] companion suggesting a comparable stellar structure as in CAL 87. The advantage of that system is that there the contribution of the accretion disk to the optical light curve is small because of its special geometrical structure and thus nearly the whole light can be assigned to the secondary.

In general we find, the larger the transport width B, the larger the range of suitable fits. Model c corresponds to model a, but energy transport of width [FORMULA] is included. Even at phase [FORMULA] the star is brighter due to energy transported to the back side of the star. Due to this the width of the minimum of the stellar contribution is smaller when compared to models a and b. The clear increase of the optical stellar light between phases [FORMULA] and [FORMULA] reduces the demands on the disk luminosity.

The stellar luminosity is larger than in model a even around [FORMULA] where we look directly to the irradiated stellar surface. The directly illuminated parts contribute less than in model a, but the heated regions in the shadow and behind the illumination horizon contribute more than before, especially because of the lower bolometric corrections for these lower temperatures (Eq. (5)).

This shift of energy into the wavelength range of the optical filter is even more evident in the comparison of the stars of model b and d, although [FORMULA] of model b is twice that of model d.

4.4. Model d

Model d includes energy transport on the secondary and the spray described in Sect. 3.3. It combines two striking features of the light curve: (1) the strong depth of the secondary minimum is obtained by the spray because its large projected area covers the secondary well, (2) a slightly better fit around eclipse compared to model b results from the energy transport on the secondary, which contributes to the light curve at these phases, reducing the strong demand on the disk. Additionally, because of the more realistic efficiency parameter [FORMULA] we prefer model d to model b.

The solution agrees with the predictions of Cowley et al. (1990) who suggest that there is no significant heating of the secondary. vdH described the heated side of the star to be at least three times brighter than the non-heated side. This is also reproduced in model d.

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© European Southern Observatory (ESO) 1997

Online publication: July 8, 1998