5 Light curve synthesis

In view of the ambiguity of the true orbital period of MT Ser two radically different models for MT Ser appear to be viable.

Model 1: Assuming the shorter of the two possible periods (P₁) to be correct, a reflection effect dominated model suggests itself, with the system consisting of a hot primary and a cooler secondary. The variations are caused by the phase dependent aspect of that side of the secondary star which is heated by the primary. Such a scenario leads to a single maximum and minimum per orbit. This is the model advocated by GB83.

Model 2: If P₂ is the correct period, two minima and maxima per orbit must be explained. This is best done by a model in which ellipsoidal variations dominate. Again, the system is considered to be composed of two components. However, in this case their relative temperatures or luminosities are not constrained a priori. The size of at least one and possibly both of the components is not small compared to its Roche lobe, leading to ellipsoidal deformations of its shape. The orbital variations are then due to the variable aspect of the deformed component(s), leading to two maxima and minima per orbit.

In the following, model calculations using the Wilson-Devinney code (Wilson & Devinney 1971; Wilson 1979) will be analyzed in an attempt to discriminate between these two models.

Figure 2: Top: Light curve of MT Ser folded on the alternative orbital period, $2 \times P = P_2$, (dots) and model light curve (solid line). Bottom: O-C curve between the observations and the model light curve; for details, see text.

   
5.1 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 0.8. 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 $\Omega_1$ and $\Omega_2$ of the components (see Kopal 1959), the mass ratio q = M₂/M₁, and the atmospheric constants (gravity darkening exponents $\beta_1$ and $\beta_2$, limb darkening coefficients $\mu_1$ and $\mu_2$, 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 ( $FF_{\rm RL}$) - 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.

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 $\beta = 0.32$ 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=1.0. The limb darkening coefficients were fixed to $\mu_1 = 0.18$ and $\mu_2 = 0.58$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.

Figure 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.

 

 

Table 3: Best fit WD model parameters for Model 1.

Parameter	Model 1.1	Model 1.2	Model 1.3
i [$^\circ$]	42.52 $\pm$1.73	52.42 $\pm$0.87	42.52$^\ast$
T2 [K]	7517 $\pm$2065	7942 $\pm$260	8866 $\pm$85
$(FF_{\rm RL})_1$	0.995 $\pm$0.051	0.951 $\pm$0.006	0.999 $\pm$0.007
$(FF_{\rm RL})_2$	0.973 $\pm$0.056	0.882 $\pm$0.008	0.971 $\pm$0.012
q = M2/M1	0.916 $\pm$0.014	1.143 $\pm$0.009	1.415 $\pm$0.008
$\beta_1$	0.97 $\pm$2.38	1.0$^\ast$	1.0$^\ast$
$\beta_2$	0.32 $\pm$81.49	1.0$^\ast$	1.0$^\ast$
$\mu_1$	0.15 $\pm$0.27	0.18$^\ast$	0.18$^\ast$
$\mu_2$	0.16 $\pm$0.49	0.58$^\ast$	0.58$^\ast$
A1	1.00 $\pm$39.70	1.0$^\ast$	1.0$^\ast$
A2	2.42 $\pm$8.65	1.0$^\ast$	1.0$^\ast$
$\Delta \phi$	-0.0006 $\pm$0.0006	-0.0002 $\pm$0.0006	-0.0013 $\pm$0.0007

$\parbox{15cm}{ $^\ast$\space Fixed parameter.}$

   
5.2 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/T₂remains 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 [$^\circ$]	67.12	$\pm$	0.36	66.19	...	67.95
T2 [K]	45590	$\pm$	1570	42500	...	49000
$(FF_{\rm RL})_1$	0.990	$\pm$	0.010	0.963	...	1
$(FF_{\rm RL})_2$	0.995	$\pm$	0.005	0.976	...	1
q (M2/M1)	1.982	$\pm$	0.008	1.961	...	1.994
$\beta_1$	1.18	$\pm$	1.26	0.17	...	2.27
$\beta_2$	1.13	$\pm$	0.68	0.35	...	2.00
$\mu_1$	0.14	$\pm$	0.40	0	...	0.53
$\mu_2$	0.15	$\pm$	0.24	0	...	0.53
A1	0.75	$\pm$	1.85	0	...	2.70
A2	0.85	$\pm$	0.84	0.06	...	1.80
$\Delta \phi$	0.0011	$\pm$	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 ($\beta_1$, $\beta_2$), the albedos (A₁, A₂), and the limb darkening coefficients ($\mu_1$, $\mu_2$). 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 $\beta_1$=$\beta_2$=1; A₁=A₂=1 and $\mu_1$=$\mu_2$=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 $q \approx 2$ 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 $q \approx 2$ 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, i 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 for values $0.19 \le q \le 5.7$. 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.