3 Properties of the wave-like variation

In this section we compare the observed parameters of the wave-like variation - its amplitude and profile - to those given by the reflection effect.

3.1 The amplitude of light curves

The largest amplitude, between $\sim$0.5 and $\sim$2.5mag, is observed in the U band (Fig. 1, Table 1). In the model of the reflection effect, it is given by the luminosity ratio of the illuminated to the non-illuminated giant's hemisphere. The upper limit ( $i=90^{\circ }$) of the magnitude difference between the two hemispheres, $\Delta m_{\rm max}$, can be expressed as

$$\Delta m_{\rm max} = - 2.5\log (1 + 2 L_{\rm RE}/L_{\rm g}),$$ (1)

where $L_{\rm g}$ is the luminosity of the giant and $L_{\rm RE}$ represents the portion of the hot component luminosity, $L_{\rm h}$, impacting the giant. If the separation between the stars, A, is significantly larger than the giant's radius, $R_{\rm g}$, then $L_{\rm RE} = (R_{\rm g}/2A)^{2}\times L_{\rm h}$, and Eq. (1) reads as

$$\Delta m_{\rm max} = - 2.5\log (1 + \beta /2),$$ (2)

where the parameter

$$\beta=\frac{R_{\rm g}^{2}}{A^{2}}\frac{L_{\rm h}}{L_{\rm g}}$$ (3)

measures the strength of the illuminating radiation field relative to that of the giant. So the amplitude of the LC caused by the reflection effect is determined only by the parameter ${\beta }$. Because of its importance in defining the investigated amplitude we also determined its uncertainties in the cases under discussion (Appendix A). Results are summarized in Table 1. We can see that the parameter ${\beta }$ determined from observations can produce a maximum magnitude difference $\Delta m_{\rm max} < 0.1$mag, which is far below the observed quantities. On the other hand, the reflection effect (Eq. 2) requires $\beta \ge 1 - 10$ to match the observed amplitudes.

Figure 2: Profiles of LCs given by the reflection function for inclination of the orbit $i=90^{\circ }$ and $45^{\circ }$ and limb darkening coefficient u=0.6. They are characterized by the parameter (Sect. 3.2)

However, this problem should be treated by solving the radiation transfer of a very hot light ( $T_{\star} \sim 10^{5}\,{\rm K}$) throughout a very cool atmosphere ( $T_{\rm eff} \sim 3\,000\,{\rm K}$) to learn how the reprocessed UV/EUV radiation contributes into the Balmer and Paschen continua. An exact approach to the reflection effect was outlined by Vaz & Nordlund (1985) and Nordlund & Vaz (1990) for similar effective temperatures of the component stars, but not for very cool stars, where the TiO absorption bands are important. Recently, Proga et al. (1996) treated this problem for symbiotic binary stars using a non-LTE photoionization model, but also without including molecules. According to their model, significant changes in the structure of the red giant atmosphere are expected for $\beta \gg$ 1 and negligible when $\beta \ll$ 1. Their calculations showed that the magnitude difference between the illuminated and non-illuminated hemisphere mag in the range of considered temperatures between 20000 and 200000K for ${\beta }$ = 10, 1, 0.1 and 0.01 (cf. their Fig. 6). Also according to this study, the theoretical differences in broadband magnitudes between the opposite hemispheres of an illuminated red giant are also very small, $\sim0.0 - 0.1$mag, far from those observed.

3.2 The shape of light curves

To characterize the shape of the observed LC we introduce a parameter a as

$$a = \frac{m(0) - m(0.25)}{\Delta m_{\rm max}},$$ (4)

where m(0) and m(0.25) are the magnitudes at the orbital phases 0 and 0.25, respectively, and $\Delta m_{\rm max}$ is the amplitude of the LC. The shape of the LC resembles a sinusoidal curve for a = 0.5, but a > 0.5 implies a broader maximum than minimum. We can recognize 3 types of LCs, characterized by (i) $a \sim 0.5$ (e.g. V1329Cyg, AGPeg, left panels of Fig. 1), (ii) 0.5 < a < 1 (e.g. CICyg, AXPer, mid panels of Fig. 1) and (iii) $a \approx 1$ (e.g. He2-467, EGAnd, right panels of Fig. 1). In some cases the parameter a depends on wavelength: (Table 1). Note that extreme values of $a \sim$1 indicate the possible presence of a secondary minimum in the LC.

Shaping of LCs caused by the reflection effect is determined by a reflection function. To construct such a LC we assume that the observed luminosity of the binary, $L(\epsilon)$, is given by the sum of a constant part of the system luminosity, $L_{\rm const}$, given mainly by stellar components, and a phase dependent variable light, $L_{\rm v}(\epsilon)$, given by the reflection function. Then we can write

$$L(\epsilon) = L_{\rm const} + L_{\rm v}(\epsilon),$$ (5)

where $\epsilon = \arccos(\sin i \cos\theta)$ is the angle between the line of sight and the direction to the hot component, the phase angle $\theta$ is determined by the position of the hot star on its relative orbit around the cool giant and i is the orbital inclination. The observed magnitude of the system then can be expressed as

$$m(\epsilon) = - 2.5\log\left[1+\frac{L_{\rm v}(\epsilon)}{L_{\rm const}} \right]+m_{\rm const}.$$ (6)

In our calculations we adopted the reflection function derived by Hadrava (1992). Figure 2 shows example of LCs given by Eq. (6). We can see that the reflection effect produces a strictly periodic modulation of the light along the orbital cycle characterized by the parameter . Therefore the observed profiles characterized with cannot be reproduced by the reflection effect. In addition, recently revealed systematic variation in the minima positions in the LCs of symbiotic binaries (Skopal 1998a) also argues against the location of the main source of the optical continuum on the giant's hemisphere.

We conclude that the observational characteristics of the LCs of symbiotic binaries - the large amplitude, the profile of minima and variation in their positions - cannot be reproduced by the reflection effect.