Reflection dominated model

A model light curve of a pure reflection effect has always an almost sinusoidal shape. Different choices of gravitational or limb darkening coefficients lead only to minute modifications. The deviations from a pure sinusoidal shape increase slightly with increasing orbital inclination in the sense that the maximum becomes narrower.

In contrast, the observed light curve of MT Ser has a maximum which is significantly broader than the minimum (see Fig 1). Moreover, it has not the flat-topped appearance as the light curves observed by GB83 which they used as an argument against a model dominated by ellipsoidal variations. But even in their observations, the flat top does not appear always [see their Fig. 1 (first maximum) and Fig. 3]. In view of the above remarks on the reflection effect, the broad maximum immediately suggests that a pure reflection effect is not able to explain the variations. Moreover, the light curve exhibits an abrupt change in slope close to phase 0.28, and another less obvious break close to phase 0.8². Thus, apart from the reflection effect, some other cause must contribute to the variations.

A formal two-component sine fit with periods fixed at P₁ and P₁/2 matches the observed light curve approximately. While the phases of minimum of both sine terms agree with phase 0 within their uncertainties, the amplitude of the P₁/2-term is 23% of that of the P₁ term. This suggests that a satisfactory description of the light curve might be achieved by allowing for a (modest) contribution of an ellipsoidal effect along with the reflection effect.

Such a hybrid model requires a trade off between several parameters in order to match the observed light curve: (1) the orbital inclination must be small enough so that no eclipses occur; (2) the secondary must be large enough in order to intercept enough radiation of the primary to be heated sufficiently, (3) The ellipsoidal variations must not become so large compared to the reflection effect as to cause a secondary minimum near phase 0.5 in the light curve; (4) the temperature difference between the components must be large enough to cause a significant temperature difference also between the illuminated and unilluminated hemispheres of the secondary, and thus to enable a reflection effect leading to variations with the observed amplitudes in the light curve; (5) if the ellipsoidal variations are due to the secondary, the primary must not be too luminous since otherwise small ellipsoidal variations would be drowned; this can be achieved by reducing either the temperature or the size of the primary; (6) if the ellipsoidal variations are due to the primary, it must fill a considerable fraction of its Roche lobe.

Having these constraints in mind, mode 2 of the Wilson Devinney code (suitable for detached binaries with no constraints on surface potentials) was used in combination with the SIMPLEX parameter optimization algorithm (Caceci & Cacheris [1984]) to find a parameter set which leads to an optimal agreement between model light curve and observations. A circular orbit and, for simplicity, black body radiation characteristics for both stars were assumed. The temperature of the primary, T₁, was fixed at 50000 K, following Green et al. ([1984]). In order not to rely on the uncertain parameter estimates of GB83 all other parameters which might influence the light curve were left free to vary. These are: the orbital inclination i, the temperature of the secondary T₂, the dimensionless surface potentials Omega₁ and Omega₂ of the components (see Kopal [1959]), the mass ratio q=M₂/M₁, and the atmospheric constants (gravity darkening exponents mu₁ and mu₂, limb darkening coefficients beta₁ and beta₂, and albedos A₁ and A₂) of both components. We also permitted a contribution of a constant light source (third light L₃; positive or negative) in order to allow for uncertainties in the subtraction of the nebular contribution to the light curve (see Sect. 3). However, in all calculations the best fit solution yielded L₃=0. Finally, a possible slight phase shift Delta Phi of the minimum with respect to the epoch given in Sect. 4 was taken into account.

The resulting best fit parameters lead to a configuration in which (only) the primary overfills its Roche-lobe. This is unphysical because in such a situation both (not just one) components should be larger than their respective Roche-lobes. Moreover, it is difficult to see how in the ensuing contact configuration a drastic temperature difference between the components could be maintained. Therefore, we imposed the additional constraint on the optimization algorithm that both components must remain within their Roche-lobes.

The best fit solution resulting under these conditions (Model 1.1) is only slightly different from the original solution. It is shown as a thin solid line in Fig. 1. The best fit parameters are listed in Table 3. Instead of the Roche potentials at the stellar surfaces the more directly interpretable Roche-lobe filling factor (FF_RL) - defined as the ratio of the distances from the stellar centre to the surface of the star and to the critical Roche surface, respectively, in the direction facing away from the companion star - is given in the table. The parameter errors were derived using the WD differential corrections routine. They are thus based on formal statistics but can probably only be regarded as lower limits to more realistic errors: while the internal errors represent the uncertainty related to the scattering of the observations and to the formal least squares solution, the possible existence of families of practically undistinguishable solutions may result in uncertainty intervals much larger than the formal ones. Regard, for example, the mass ratio which has a small formal error but is quite different for Models 1.1 and 1.3 (see below), although the resulting light curves are practically identical.

Table 3: Best fit WD model parameters for Model 1

Parameter	Model 1.1	Model 1.2	Model 1.3
i [degrees]	42.52 +-1.73	52.42 +-0.87	42.52*
T2 [K]	7517 +-2065	7942 +-260	8866 +-85
(FFRL)1	0.995 +-0.051	0.951 +-0.006	0.999 +-0.007
(FFRL)2	0.973 +-0.056	0.882 +-0.008	0.971 +-0.012
q=M2/2	0.916 +-0.014	1.143 +-0.009	1.415 +-0.008
beta1	0.97 +-2.38	1.0*	1.0*
beta2	0.32 +-81.49	1.0*	1.0*
mu1	0.15 +-0.27	0.18*	0.18*
mu2	0.16 +-0.49	0.58*	0.58*
A1	1.00 +-39.70	1.0*	1.0*
A2	2.42 +-8.65	1.0*	1.0*
Delta Phi	-0.0006 +-0.0006	-0.0002 +-0.0006	-0.0013 +-0.0007
*fixed parameter

As can be seen from Table 3 the errors of the atmospheric parameters beta, mu and A are exceedingly large, probably due to parameter correlations. Thus, it does not make much sense to handle them as independent parameters. Therefore, in another model run (Model 1.2) they were fixed to plausible values. For stars with radiative envelopes Rafert & Twigg ([1980]) found empirically a mean value of beta=0.96 for the gravity darkening exponent. Within the errors this is identical with the usually adopted theoretical value of beta=1 (Lucy [1967]). While the primary of MT Ser certainly has got a radiative envelope, this is less clear for the secondary. However, since the parameter optimization for Model 1.2 led to a somewhat hotter secondary than in the case of Model 1.1 (above the limit of the low beta stars; see Fig. 1 of Rafert & Twigg [1980]), beta=1.0 was adopted for both stars. Test calculation using beta=-.32 for the cool component (appropriate for stars with convective atmospheres; Lucy [1967]) yield almost indistinguishable light curves. Rafert & Twigg ([1980]) also determined empirically the albedos of stars in the temperature range of interest here and found a mean value of A=1.02. Since this is not significanly different from A=1 and since albedos larger than unity are difficult to explain, we fix the albedos of both components to A=1. The limb darkening coefficients were fixed to mu₁=0.18 and my₂=0.58 as calculated for stars of the respective temperatures and of intermediate surface accelerations (log g = 4.5) by Wade & Rucinski ([1985]). Leaving all other parameters free and imposing again the condition that none of the components overfills its Roche-lobe leads to a best fit model as shown by the thick solid line in Fig. 1. The corresponding parameter values and their errors are included in Table 3.

Fig. 1 shows that Model 1.2 predicts grazing eclipses, as evidenced by the V-shape of the calculated light curve at phase 0 and the slight dip around phase 0.5 (see insert in Fig. 1). While this improves the fit to the light curve minimum compared with Model 1.1, there is no trace of a secondary eclipse cutting into the maximum in the observations. Therefore, we calculated a third model (Model 1.3), keeping the atmospheric parameters fixed to those of Model 1.2 and the orbital inclination to the value found for Model 1.1. The resulting light curve is virtually indistinguishable from that of Model 1.1. The corresponding best fit parameter values are also included in Table 3.