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Astron. Astrophys. 331, 639-650 (1998)

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6. Light-curve modelling

6.1. Observational evidence

The observed light-curves refer to a broad wavelength range and to a fairly large time span: more than 2600 rotational periods between the first and the last observing runs. The fact that the light maxima are unequal (known as the O'Connell effect) precludes a complete explanation for the light variations based merely on the ellipticity of the stars and grazing eclipses. However, the analysis presented in Sect. 5and general considerations on the stability of the light-curve and on its colour dependence indicate which kind of models are excluded.


[TABLE]

Table 4. Differential Strömgren photometry in the sense TV Pic - HD 31 640. HJD are expressed relative to a zero point at HJD 2 440 000. [FORMULA] refers to Johnson's V magnitude as derived from our photometry. Remarks are coded as follows: 'c' = TV Pic measured relative to HD 31 587 only; '+' = longer series of observations (accuracy higher than average); '?' = observations not absolute (accuracy probably lower than average)


Several observational facts put strong constraints on the source of the additional light variability were it to be associated with the primary. Indeed, the spectrum emitted by the primary at either phase of quadrature is apparently not very different, and the colour variation in the light-curves is marginal (Fig. 6). If the variability is caused by a spot on the surface of the primary, then the spot must be small ([FORMULA] radius) and hotter than the larger part of the stellar surface. Only such spot can leave the [FORMULA] colour almost unchanged while also causing the difference between the light maxima observed in the passbands. The temperature contrast must therefore be smaller than those published for AG Vir (Bell et al. 1990 ) and similar systems. The parameter range for spots on the surface of the secondary is much less constrained because the influence of the spot on the total spectrum and on the colour variation is very diluted. However, such a spot cannot remain in a fixed position on the secondary's surface in view of the evidence for non-synchronous rotation and the fact that the light curve remained stable over the time span of the observations. There is no significant indication of any change in the light curve between November 1981 and January 1988. Undetected changes must be below the 0.01 mag level. Finally, the light curve does not show evidence of transient phenomena characteristic of strongly interactive systems.

[FIGURE] Fig. 6. Color changes of TV Pic, relative to HD 31 640, and fitted relations (Table 7). Ticks on the vertical axis indicate 5 millimag intervals

[TABLE]

Table 5. Differential Walraven photometry in the sense
TV Pic - HD 31 640. VB, BU, UW and BL stand for [FORMULA] 2.5 ([FORMULA]), etc. HJD are expressed relative to a zero point at HJD 2 440 000



[TABLE]

Table 6. Strömgren b photometry of TV Pic. HJD are expressed relative to a zero point at HJD 2 440 000


The light curve in Strömgren b is shown in Fig. 1. The colour variations (Fig. 6) are marginal and are dominated by a sinusoidal term; only in [FORMULA] and 2.5 ([FORMULA]) does the full range of variation exceed 10 mmag. The colours well above the Balmer jump all vary in phase with each other, and their amplitude is roughly proportional to the separation in wavelength between the filters used (Table 7) at a rate of 8 mmag full range per 1000 Å separation of the filters. The binary appears reddest when the primary is in front of the secondary, one tenth of the orbital period before the following quadrature; at the Balmer jump and shortwards of it, the phase of reddest colour is shifted about one quarter period in the direction of the preceeding quadrature. While a sinusoidal representation is useful for a first estimate of the amplitude and the phase relation, it is obvious from the physical background that the precise shape might be much more complex and the above statements should be interpreted accordingly. The evidence for the phase shifts and the small colour variability is presently not sufficiently detailed to warrant an attempt to model it.


[TABLE]

Table 7. Amplitude and phase of reddest colour for sinusoidal fits to the colours. The final fits were coupled by imposing the same phase relation within groups of colours. (Note that the sequence of effective wavelength of the passbands from UV to red is: [FORMULA]). The last column lists the rms of the fit to the observations


The light-curves and colour characteristics are very similar to the observations for TU Hor in the Geneva photometrical system (Waelkens 1982 ). The visual light-curve of TU Hor has two maxima differing by 0.05 mag, and a fairly sinusoidal colour variation of 6 mmag amplitude in the [FORMULA] index. This 'temperature' index is reddest about 0.1 period before the secondary maximum in visual light. Waelkens showed that the [FORMULA] light-curve can be transformed in a normal close-binary light-curve, with equal maxima separated by exactly half the orbital period, by subtracting a sinusoidal variation whose amplitude (30 mmag) and phase in [FORMULA] are compatible with the observed amplitude and phase in the [FORMULA] index (the variation in the index is ascribed to a 75 K variation between the stellar hemispheres visible at opposite quadratures). In our case, a slightly smaller sinusoidal correction in b (26 mmag) removes the O'Connell effect from the light-curve of TV Pic and brings the phase difference between the maxima to half the orbital period if the colour index is reddest [FORMULA] P before the secondary maximum in b. Such correction implies an amplitude of [FORMULA] mmag in [FORMULA] and [FORMULA] mmag in 2.5 [FORMULA]. The predicted phase for the 'temperature' index variations is compatible with the observed one, while the predicted amplitude is marginally larger than observed (Table 7).

6.2. Spot models

By using the Binary Maker code (Bradstreet 1993 ) we have investigated whether the light-curve can be reproduced by models with one temperature spot on either the primary or the secondary. We assumed that the light variability is mainly due to the ellipsoidal shape of the star and the occurrence of grazing eclipses. We had to position the (hypothetical) spot on the stellar surface so as to explain primarily the different luminosity seen at the quadrature phases; an appropriate starting point was in the orbital plane but [FORMULA] out of the inner Lagrangian point.

Fig. 1 shows that much of the light-curve asymmetry can be explained by such a simple model, and equally well with a spot on the primary as with a spot on the secondary. The models deviate nowhere more than 0.01 mag from a smooth spline representation of the data, and the parameters listed in Table 8 can be regarded as appropriate initial estimates for more powerful light-curve fitting codes. The most uncertain ones are the latitude of the spot and its extent (both affect the resulting temperature contrast). The primary has to fill its Roche lobe to reproduce the observed light-curve amplitude and the radius of the secondary cannot be varied appreciably without violating the spectroscopically-observed light ratio. TV Pic is a near-contact binary. The light-curve is fitted slightly better when the temperature of the secondary lies near the spectroscopically allowed maximum value. Fig. 7 shows the configuration at phase zero for both models. The lack of uniqueness (only the model including a single cool spot on the primary can be excluded) will presumably not be avoided by more powerful codes. Progress rather has to come from spectroscopic observations at more phases, in visual light and also at wavelengths where interacting matter may be more prominent (e.g. Lyman [FORMULA] emission, X-rays). We note that TV Pic is not detected as a bright source in the ROSAT survey (Voges et al. 1996 ); however, many of the nearby near-contact binaries are no bright sources as shown by Shaw et al. (1996 ). A more detailed analysis of X-ray data near TV Pic and TU Hor is necessary. Moreover, the slightly discordant MK classification of Houk (1978 ) may hint at a larger spectral variability over the orbital cycle than was recorded from comparing spectra taken at different quadratures. Clearly, such variation - if real - must be exploited and reproduced by future models.


[TABLE]

Table 8. Parameters for the two 'spot' models shown in Fig. 1. The longitude of the centre of the spot is measured from the inner Lagrangian point opposite to the sense of rotation. Co-latitude is measured from the 'upper' to the 'lower' pole. The phase of quadrature, [FORMULA], refers to the quadrature nearest to the secondary light maximum. The primary fills its Roche lobe. The radius given for the secondary is the 'back radius' (Wilson & Devinney 1971 ) in units of the distance between both components. See Table 9 for the stellar and orbital parameters


[FIGURE] Fig. 7. The configuration of the binary at phase zero (secondary light maximum) for the spotted models described in Table 8 (upper: spot on primary; lower: alternative model with spot on secondary)

6.3. Does the model need spots?

The spectroscopic evidence (Fig. 5) is that the spectrum of the secondary seems to have altered significantly between phase 0.5 (when it is red-shifted) and phase 0.0 (the lower light maximum). The puzzling aspect is the apparent doubling of the lines of the secondary at phase 0.0; the spectrum of the secondary itself is not visible at a velocity displacement that would mirror its red-shift at the other phase, nor do the two sets of split lines appear to be symmetrically placed around the velocity expected by comparison with its red-shifted displacement. One (admittedly generous) explanation is that the secondary is veiled by the presence of some sort of accreted material; such a model might account qualitatively for the fall in brightness and for the behaviour of the spectral lines. On the other hand, one would rather expect to find accreted material in the wake of the secondary (in the red-shifted situation) rather than ahead of it in its orbit. Further spectra are essential before we can make further progress in understanding the system better.

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© European Southern Observatory (ESO) 1998

Online publication: February 16, 1998
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