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

## 5. Fundamental parameters for the binary and its components

Since the two maxima in the light curve of TV Pic are not the same, we cannot derive the physical elements (radii and masses) of the component stars in the classical way. One solution is to construct a specific model for the light curve by including a spot on one or both stars. It is therefore instructive to investigate first the constraints placed on that model by the observations, before additional degrees of freedom are introduced.

The fact that the spectra were not taken exactly at quadrature implies that the velocity semi-amplitudes and are slightly larger than the observed velocity differences. We assume a sinusoidal velocity curve, and apply a correction of 1.01; hence, we estimate km s-1 and km s-1 (see Sect. 4.1for the meaning of the errors), implying a mass ratio and a mass function

where P is the orbital period in days, and the secondary's mass. Thus, and . The Roche lobe geometry may be computed from Eggleton (1983):

where is the radius of a sphere with the same volume as the primary Roche lobe, and A is the binary separation. We find and, using the inverse of q, .

In view of the large light amplitude due to ellipticity and of the absence of deep eclipses, it follows, conservatively, that . This is because the ellipticity of the system cannot cause a sufficiently large light amplitude at smaller inclinations, while the (grazing) eclipse of the primary becomes much to deep for inclinations exceeding the upper limit given above. Moreover, the width of the shoulders of the light curve shows that the primary must be nearly filling its Roche lobe; a 5% smaller radius produces unacceptably broad shoulders. These bounds to the inclination also set bounds to the separation A between the components, through the relation

and on their masses (Table 3). If the period is given in days and the mass in solar masses, this formula will yield the separation in solar radii. Even before carrying out a more precise determination of the inclination by detailed light curve modelling, we are finding that the masses of the stars are too small for (near-) main-sequence stars of the inferred spectral types. If we anticipate the inclination of - derived in Sect. 6.2, we deduce a dynamical estimate of for the primary - between the spectroscopic and the photometric estimates - and for the secondary; the value of 3.5 obtained spectroscopically for the latter is very uncertain.

Table 3. Preliminary solutions for masses, separation, and Roche-lobe radii using the mass ratios derived from different assumptions

When a star fills its Roche lobe, synchronous rotation requires

If the star is smaller, the right-hand side of this equation gives an upper limit to the required apparent rotational velocity. We thus derive km s-1 and km s-1. However, the widths of the lines in the secondary indicate significantly faster rotation (112 km s-1), so either the secondary is not in synchronous rotation, or rotation is not the dominant source of line broadening.

If we assume that the primary fills its Roche lobe and is rotating synchronously, then from the above equation we obtain the slightly smaller mass ratio q  = 0.273 0.014. The uncertainty does not include that associated with derivation of rotational line broadening from the simple line-broadened model, as noted in Sect. 3. The latter is estimated by Collins & Truax (1995 ) to be about 10% for single stars, so  0.31 appears compatible with the radial-velocity amplitudes as well as with the primary's line broadening. The possible solutions presented in Table 3 and the rotational broadening of the primary suggest strongly that TV Pic is a near-contact binary (see Shaw 1994 and references therein).

© European Southern Observatory (ESO) 1998

Online publication: February 16, 1998
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