Ellipsoidal variations dominated model

The light curve folded on Period P₂ (Fig. 2) displays two almost equally deep minima, both of which are quite symmetrical. Above a certain minimum level the slope suddenly becomes less steep. This is best seen close to phase 0.1 (or in binned versions of the light curve). The maxima are significantly broader than the minima.

The minima are very reminiscent of eclipses. Their almost equal depth suggests that both component of the binary system should have a very similar temperature. The out-of-eclipse variations (i.e. the maxima) should then be due to ellipsoidal variations of one or both stars.

The temperature of 50000 K of MT Ser measured by Green et al. ([1984]) is so high that optical observations always sample the Rayleigh-Jeans part of the spectral energy distribution. The temperatures of both components being very similar in the present model, it is then practically impossible to determine the temperatures of both components by WD model calculations. Only the ratio of the primary and secondary star temperature, T₁/T₂, can be derived in this way. Therefore we fixed T₁ to 50000 K. Tests with other values confirmed that the best fit model is independent of T₁ even if T₁ is varied within a wide range, as long as T₂ is permitted to vary accordingly. The ratio T₁/T₂ remains constant.

Choosing again mode 2 of the WD code the same parameters as in Model 1 were left free to vary. The SIMPLEX algorithm (Caceci & Cacheris [1984]) then converged on a very good solution (Model 2.1). The best fit light curve is shown as a solid line in Fig. 2. The corresponding parameter values together with their formal errors as derived from the WD differential corrections routine are summarized in Table 4.

Table 4: Best fit WD model parameters for Model 2.1

Parameter	Best fit		Formal	Confidence range
	value		error
i [degrees]	67.12	+-	0.36	66.19	...	67.95
T2 [K]	45590	+-	1570	42500	...	49000
(FFRL)1	0.990	+-	0.010	0.963	...	1
(FFRL)2	0.995	+-	0.005	0.976	...	1
q=M2/M1	1.982	+-	0.008	1.961	...	1.994
beta1	1.18	+-	1.26	0.17	...	2.27
beta2	1.13	+-	0.68	0.35	...	2.00
mu1	0.14	+-	0.40	0	...	0.53
mu2	0.15	+-	0.24	0	...	0.53
A1	0.75	+-	1.85	0	...	2.70
A2	0.85	+-	0.84	0.06	...	1.80
Delta Phi	0.0011	+-	0.0003	-0.0019	...	0.0045
L3	-0.0004			-0.0033	...	0.0035

As in the case of Model 1 some parameters are not well constrained, reflecting their small influence on the light curve. These are the gravity darkening exponents (beta₁, beta₂), the albedos (A₁, A₂), and the limb darkening coefficients (mu₁, mu₂). Even so it is satisfying, although possibly accidental, that in spite of the large formal errors the best fit values of the former two quantities determined for MT Ser are not very different from the mean values found empirically for comparable stars by Rafert & Twigg ([1980]). We recalculated the WD model fit with fixed values of beta₁=beta₂=1; A₁=A₂=1; mu₁=mu₂=0.18 (see Sect. 5.1. As expected, the resulting best fit values for the free parameters deviate only insignificantly (within the formal errors) from those quoted in Table 4.

As can be seen from Table 4 the best fit solution leads to a configuration in which the two components are almost in contact with each other. Within the formal errors (which may well underestimate the true errors as argued above and will further be elaborated upon below) a contact configuration is permitted.

Moreover, the large value of the mass ratio q~2 is somewhat disturbing. The physical implications of the currently discussed model are those of a close binary having just emerged from a common envelope phase (as suggested by the presence of the planetary nebula). If both components are not in contact with each other, this configuration and their similar temperature indicate that they should be in a similar evolutionary stage. This makes it difficult to explain a large difference in mass. If they are in physical and thus thermal contact they might have similar temperatures even if the masses were different. But then we would expect the less evolved and thus less massive star to be heated by the more massive component. It can then attain a similar but not a higher temperature than the latter star. This is in contrast to the derived mass ratio which suggests that the hotter component is the less massive one. It is true that a different scenario might be devised in which the more evolved star lost so much mass during the common envelope phase that it emerges as the less massive component. However, while this might lead to a situation with q slightly larger than 1, it appears difficult to achieve q~2 in this way. Note that for similar reasons it is difficult to believe in the reality of the mass ratio of q=1.4 ensuing from Model 1.3.

In view of this problem we searched for physically more plausible solutions. Since the components are practically in contact with each other we investigated models adopting WD mode 4 (primary star fills its Roche lobe), mode 5 (secondary star fills its Roche lobe) and mode 6 (both components fill their Roche lobe; contact configuration). The important parameters determining the shape of the light curve are the orbital inclination i, the secondary star temperature T₂, the mass ratio q and the surface potential Omega of the secondary (mode 4) or the primary (mode 5) component, respectively. The other components were fixed to the theoretical values discussed above or the best fit values found in the initial model calculations.

It turned out that in all modes good solutions could be achieved within a wide range of mass ratios. They are practically indistinguishable from the solid line in Fig. 2. Thus, q is not well constrained. Therefore, in additional model calculations the mass ratio was fixed to a number of different values, and only the other parameters were permitted to vary. The differences between the individual models were minute: For all values of the mass ratio, 1 and T₂ varied within a small range. In the case of modes 4 and 5 the component which does not fill its Roche lobe generally attains a filling factor of more than 0.99. For some mass ratios the additional constraint that none of the stars overfills its Roche lobe had to be imposed in order to avoid a formal fit solution with a filling factor >1, indicating that any deviations from a true contact configuration are not significant. In view of the similarity of the results obtained with WD modes 4, 5 and 6 we restrict the subsequent discussion to the calculations in mode 6 (contact configuration) for simplicity (Model 2.2).

Statistically acceptable solutions (as discussed in more detail in Sect. 6.1) were found 0.19 <= q <= 5.7. The best fit solutions for the corresponding values of T₂ and i are plotted as a function of q in Fig. 3.

 

Figure 3: Best fit values for the orbital inclination (solid line; left hand scale) and the secondary star temperature (broken line; right hand scale) as a function of the adopted mass ratio q = M₂/M₁ for Model 2, calculated assuming a contact configuration.