In this section we will discuss the implications of the model calculations with the aim to decide between the two competing models. First, the formal acceptability of the fits is evaluated, while a discussion of the astrophysical aspects follows in Sect. 6.2.

6.1 Formal aspects of the model calculations

The prime requirement for a good fit is - apart from its astrophysical viability - that the O-C curve, i.e. the difference between the observations and the model light curve, should consist of randomly distributed data. In the lower frames of Figs. 1 and 2 the O-C curves of Models 1.1, 1.2 and 2.1, respectively, are shown.

It is obvious that the O-C curves of Model 1 show considerable structure, indicating a less than ideal model fit. The observed minimum is deeper than the calculated one, yielding positive O-C values. The opposite is true for the maximum; in particular in the case of Model 1.2 which predicts an unobserved shallow secondary minimum. But also at phases intermediate between maximum and minimum correlated residuals are clearly present.

In order to quantify the statistical deviations of the O-C values from a purely random distribution, a formal R-statistics analysis (Bruch 1999) was undertaken. For Model 1.1, R = 0.157 (for 651 data points). The corresponding R-probability (i.e. the statistical probability for completely random data scattered around 0 to have an even smaller value of R, or - somewhat loosely spoken - the probability for correlations to be present in the residuals) is P_R = 0.99998. Model 1.2 yields an R-probability even closer to 1. Thus, the formal analysis confirms the visual impression that from a purely statistical point of view Model 1 does not lead to a good fit to the observations.

This is different for Model 2.1. In this case the O-C curve does not show systematic deviations from a random distribution. The R-probability is P_R = 0.20 which is in excellent agreement with the hypothesis of the absence of correlated residuals. Therefore, statistically, Model 2.1 is completely satisfactory. The same holds true for Model 2.2. The R-probability remains close to P_R=0.20 for $0.19 \le q \le 5.7$ . Only very close to the borders of this range P_R starts to increase reaching a value of $P_R \approx 0.9$ at the edges.

These positive results open another way to determine confidence ranges which may be more realistic than the formal errors quoted in Table 4. We modified each parameter (keeping all other parameters fixed) until the R-probability for the residuals reaches P_R = 0.90, indicating a significant correlations between the residuals and thus systematic differences between data and fit. The corresponding "90% confidence ranges'' for the parameters are also quoted in Table 4. In some cases the parameters assumed unphysical values before P_R=0.90 was reached. Then, the physically sensible limit enters the table. This applies in particular to the upper boundary of the Roche-lobe filling factor. Larger values lead to an over-contact configuration (but note that in contrast to Model 1 this is not wholly impossible in the present case because the temperatures of the components are similar!). Although these confidence ranges might be somewhat more realistic than the formal errors they must still be regarded as lower limits because they do not take into account correlations between parameters: The effect of modifying one parameter could be neutralized by corresponding modifications of one or more other parameters, permitting wider parameter ranges.

This last point becomes particularly evident regarding the large range of permitted values for the mass ratio when a contact configuration is adopted. As was argued in Sect. 5.2 not the entire range of statistically acceptable values of q makes sense physically, but only values of $$q \mathrel{\mathchoice {\vcenter{\offinterlineskip\halign{\hfil $\displaystyle ...$ . To be definite, we limit the subsequent discussion to two particular values, namley q=1.0 (Model 2.2.1) and q=0.5 (Model 2.2.2). The corresponding confidence ranges for the free parameters in Model 2.2 are listed in Table 5.

**Table 5:** Parameter confidence ranges for Model 2.2.
	q = 1.0	q = 0.5	...
inclination i	64.8	...	66.6	66.4	...	68.1
temperature T₂	41800	...	48900	41040	...	48350

Finally, we note that the phase shift $\Delta \phi$ for both, Model 1 and Model 2, is well compatible with 0 within the confidence or error range. Therefore, there is no need to revise the zero-point of the ephemeris given in Sect. 4. The contribution of third light - 0 for Model 1, and expressed in Table 4 in fractions of the total system brightness at phase 0.25 for Model 2 - is vanishingly small, suggesting that the subtraction of nebular light was even more successful than could be expected. But since there is always the specter of parameter correlations we tried to find solutions with a larger (positive or negative) contribution of third light. However, these attempts did not meet success.

6.2 Astrophysical implications of the models

Perhaps the most significant result of the model calculations is the fact that (for both models) the primary as well as the secondary components of MT Ser are very close to filling their respective Roche lobes.

In the subsequent discussion we will compare model predictions - considering Models 1.1, 2.2.1 and 2.2.2 - concerning the system luminosity and the distance to MT Ser to corresponding literature results, mainly based on studies of the planetary nebula. To do so, however, assumptions concerning the component masses are required. For all models, three mass values are considered: Two values which generously embrace the possible mass range which might be realized in MT Ser, and a plausible intermediate one. They will be referred to as the high (H) and low (L) mass limit, and the intermediate (I) mass, respectively. The adopted mass values are listed in Table 6.

In Model 1 we expect the primary component to be a hot sub-dwarf. It will probably evolve into a white dwarf without substantial further mass loss. Therefore, it must not have more than the Chandrasekhar mass (1.43 $M_\odot$ ), while the lower mass limit is given by the low mass cut-off of the white dwarf mass distribution which we place at 0.2 $M_\odot$ to be on the safe side. As intermediate mass a value of 0.6 $M_\odot$ is adopted, close to the peak of the white dwarf mass distribution. Together with the mass ratio of 0.92 (see Table 3) the component masses as listed in Table 6 result.

Table 6: Stellar parameters derived from model calculations.
Model 1.1 Model 2.2.1 Model 2.2.2

H I L H I L H I L

primary star mass ( $M_\odot$ ) 1.43 0.6 0.2 1.43 0.6 0.2 1.43 0.6 0.4

secondary star mass ( $M_\odot$ ) 1.31 0.55 0.18 1.43 0.6 0.2 0.72 0.30 0.2

component separation ( $R_\odot$ ) 1.38 1.03 0.71 2.22 1.66 1.15 2.02 1.51 1.32

primary star radius ( $R_\odot$ ) 0.62 0.46 0.32 1.00 0.75 0.52 1.05 0.78 0.68

secondary star radius ( $R_\odot$ ) 0.58 0.44 0.30 1.00 0.75 0.52 0.78 0.58 0.51

$\log g\, \rm {(cm\, s^{-2})}$ (primary star) 5.01 4.89 4.73 4.59 4.46 4.31 4.55 4.43 4.37

$\log g\, \rm {(cm\, s^{-2})}$ (secondary star) 5.02 4.90 4.73 4.59 4.46 4.31 4.51 4.38 4.32

system luminosity ( $10^3\,L_\odot$ ) ( $T_1=50\,000$ K) 2.14 1.20 0.57 9.42 5.28 2.51 8.37 4.68 3.58

system luminosity ( $10^3\,L_\odot$ ) ( $T_1=35\,000$ K) 0.44 0.25 0.12 2.24 1.31 0.61 2.00 1.13 0.86

M_B ( $T_1=50\,000$ K) 0.34 0.97 1.77 -1.38 -0.75 0.06 -1.22 -0.60 -0.30

M_B ( $T_1=35\,000$ K) 1.06 1.68 2.48 -0.80 -0.22 0.61 -0.66 -0.03 0.26

distance (kpc) ( $T_1=50\,000$ K; A_B=2.28) 5.9 4.9 3.0 13.0 9.7 6.7 12.0 9.0 7.8

distance (kpc) ( $T_1=50\,000$ K; A_B=1.09) 10.1 7.6 5.2 22.4 16.8 11.6 20.9 15.7 13.6

distance (kpc) ( $T_1=35\,000$ K; A_B=2.28) 4.2 3.2 2.2 9.9 7.6 5.2 9.3 6.9 6.1

distance (kpc) ( $T_1=35\,000$ K; A_B=1.09) 7.3 5.5 3.9 17.2 13.1 9.0 16.0 12.0 10.5

**Table 6:** Stellar parameters derived from model calculations.
	Model 1.1	Model 2.2.1	Model 2.2.2
	H	I	L	H	I	L	H	I	L
primary star mass ( $M_\odot$ )	1.43	0.6	0.2	1.43	0.6	0.2	1.43	0.6	0.4
secondary star mass ( $M_\odot$ )	1.31	0.55	0.18	1.43	0.6	0.2	0.72	0.30	0.2
component separation ( $R_\odot$ )	1.38	1.03	0.71	2.22	1.66	1.15	2.02	1.51	1.32
primary star radius ( $R_\odot$ )	0.62	0.46	0.32	1.00	0.75	0.52	1.05	0.78	0.68
secondary star radius ( $R_\odot$ )	0.58	0.44	0.30	1.00	0.75	0.52	0.78	0.58	0.51
$\log g\, \rm {(cm\, s^{-2})}$ (primary star)	5.01	4.89	4.73	4.59	4.46	4.31	4.55	4.43	4.37
$\log g\, \rm {(cm\, s^{-2})}$ (secondary star)	5.02	4.90	4.73	4.59	4.46	4.31	4.51	4.38	4.32
system luminosity ( $10^3\,L_\odot$ ) ( $T_1=50\,000$ K)	2.14	1.20	0.57	9.42	5.28	2.51	8.37	4.68	3.58
system luminosity ( $10^3\,L_\odot$ ) ( $T_1=35\,000$ K)	0.44	0.25	0.12	2.24	1.31	0.61	2.00	1.13	0.86
M_B ( $T_1=50\,000$ K)	0.34	0.97	1.77	-1.38	-0.75	0.06	-1.22	-0.60	-0.30
M_B ( $T_1=35\,000$ K)	1.06	1.68	2.48	-0.80	-0.22	0.61	-0.66	-0.03	0.26
distance (kpc) ( $T_1=50\,000$ K; A_B=2.28)	5.9	4.9	3.0	13.0	9.7	6.7	12.0	9.0	7.8
distance (kpc) ( $T_1=50\,000$ K; A_B=1.09)	10.1	7.6	5.2	22.4	16.8	11.6	20.9	15.7	13.6
distance (kpc) ( $T_1=35\,000$ K; A_B=2.28)	4.2	3.2	2.2	9.9	7.6	5.2	9.3	6.9	6.1
distance (kpc) ( $T_1=35\,000$ K; A_B=1.09)	7.3	5.5	3.9	17.2	13.1	9.0	16.0	12.0	10.5

Within Model 2 both components are hot sub-dwarfs which are expected to become white dwarfs. Therefore, for Model 2.2.1 (q=1) the above values of 1.43 $M_\odot$ , 0.6 $M_\odot$ , and 0.2 $M_\odot$ for the H, I, and L cases, respectively, are adopted for both stars, while for Model 2.2.2 (q=0.5) the additional contraint applies that none of the components may have a mass above or below the permitted range for a white dwarf. As intermediate mass case it is assumed that the primary component has a mass corresponding to the peak of the white dwarf mass distribution (0.6 $M_\odot$ ,).

6.2.1 System luminosity

The radii of the components in units of their separation a can be calculated from the Roche potentials at their surfaces as given by the best fit WD model. Both components are severely distorted by tidal effects. We take the stellar radii to be equal to the mean of the distance from their centres to the surfaces in the direction towards the companion star and in opposite direction. A more sophisticated definition of the mean radius is not warranted. Together with the known period, Kepler's third law furnishes the component separation a and then R₁ and R₂ in absolute units. The corresponding numbers are listed in Table 6. Finally, the radii and the temperatures - black body characteristics are assumed - of the two components yield the total luminosity of the system, also listed in Table 6.

The luminosities, based on T₁=50000 K, are in contradiction to values derived in the literature from observation of the planetary nebula. The Zanstra luminosity was determined by Shaw & Kaler (1989) to be <400 $L_\odot$ , while Tylenda et al. (1991) gave a value of $\log\,L$ =2.79 (L=617 $L_\odot$ ), only marginally consistent with the lower luminosity limit for Model 1.1. This discrepancy can be resolved to a large degree if our adopted temperatures are too high. In fact, the Zanstra temperature as measured by Shaw & Kaler (1989) is 35000 K, whereas Tylenda et al. (1991) found $\log T=4.51$ (T=32360K).

In order to investigate the effect of a lower temperature, a new WD solution for Model 1 was determined with T₁ fixed at 35000 K, leaving the more important parameters i, T₂, $\Omega_1$ , $\Omega_2$ and q free to vary, while keeping the atmospheric parameters fixed at their previously determined values. The WD fit converged on a solution (the statistical quality of which is as bad as that found in the high temperature case) with a somewhat higher secondary star temperature of 7940 K. Using the fitted values of q to calculate M₂ from the adopted value of M₁ for the H, I and L cases, and the surface potential to obtain the stellar radii, the system luminosity was recalculated as listed in Table 6 ( $T_1=35\,000$ K case). Within Model 2 the secondary star temperature scales linearly with T₁ (see Sect. 5.2). Therefore, the luminosities for the $T_1=35\,000$ K case were simply recalculated using the previously derived stellar radii and the lower temperature values.

It can be seen that in general the luminosities based on Model 1.1 are now even lower than quoted in the literature. Thus, allowing for the uncertainty of the primary star temperature (possibly somewhere between 50000 K and 35000 K) Model 1.1 is not in contradiction with independent measurements of the system luminosity. Concerning Model 2, the calculated luminosities are still too high. Only in the low mass case of Model 2.2.1 it is just within the range of the literature values.

6.2.2 The distance to MT Ser

In order to obtain a photometric parallax for MT Ser, the ratio of the total energy emitted per surface unit of black bodies with temperatures such as those of the MT Ser components to the energy emitted per wavelength unit at the central wavelength of the B band (4400 Å) was first calculated. The luminosities of the individual components were divided by this ratio and then the resulting values of the two stars were summed, yielding the energy emitted by the system per wavelength unit at 4400 Å. Together with the calibration constant of Bessel (1979) for the B band this yields the absolute B magnitudes as listed in Table 6 for the two cases $T_1=50\,000$ K and $T_1=35\,000$ K.

This energy corresponds to the mean apparent B magnitude of MT Ser which can be estimated from Figs. 1 and 2 of GB83. However, the light level is different at the two observations, B=15.63 on 1982, May 28, and B=15.77 during 1982, April. GB83 attribute this difference to different filter sets and diaphragm sizes used at the two occasions, and in consequence to different contributions from the surrounding planetary nebula. At a diameter of $18^{\prime\prime}_{\raisebox{.6ex}{\hspace{.12em}.}}4$ (Acker et al. 1992) the latter has a size typical for apertures used in photoelectric photometry. Therefore we assume that those observations yielding the brighter mean magnitude for MT Ser contains most of the planetary nebula. The flux ratio between the nebula and the central star derived in Sect. 3 translates into a magnitude difference of $0^{\raisebox{.3ex}{\scriptsize m}}_{\raisebox{.6ex}{\hspace{.17em}.}}83$ . Thus, the B magnitude of the central star alone is $16^{\raisebox{.3ex}{\scriptsize m}}_{\raisebox{.6ex}{\hspace{.17em}.}}46$ . This is in good agreement with Shaw & Kaler (1989) who found B=16.45 $\pm$ 0.11.

The last ingredient needed to determine a photometric parallax is the interstellar extinction. Shaw & Kaler (1989) derived the reddening constant c (the logarithmic extinction at H $\beta$ ) from the H $\alpha$ -to-H $\beta$ ratio of the planetary nebula as c=0.85. Interpolating between their values of the conversion factor between c and E(B-V), which is itself a function of c (see also Kaler & Lutz 1985), we get E(B-V)=0.56. Assuming the ratio between total to selective absorption to be R=3.1 and using the interstellar extinction curve of Cardelli et al. (1989) this yields an absorption in the B band of $A_B = 2^{\raisebox{.3ex}{\scriptsize m}}_{\raisebox{.6ex}{\hspace{.17em}.}}28$ . A value of c discrepant from that of Shaw & Kaler (1989) has been derived by Tylenda et al. (1991). They give three values (two based on the H $\alpha$ -to-H $\beta$ ratio and the other one on the radio-to-H $\beta$ ratio) with a mean of c=0.38 which translates into A_B=1.09. An independent estimate of the extinction by Green et al. (1984) is based on the comparison of the expected colours of a pure Rayleigh-Jeans spectrum and the observed colours of MT Ser. They find A_V=1.5, corresponding to A_B=2.0, close to the higher of the two values derived from nebular physics.

The resulting distances for the high and low temperature cases and the two absorption values are listed in Table 6. In the literature statistical distances for the planetary nebula are cited as 4.3 kpc (Abell 1966), 4.51 kpc (Cahn & Kaler 1971), 5.4 kpc (Maciel 1984), <3.76 kpc (Shaw & Kaler 1989), and 4.60 kpc (Cahn et al. 1992). This is consistent with the range of photometric parallaxes predicted from Model 1 in the high absorption case. In constrast, the parallaxes derived from Model 2 are only marginally consistent with the literature values in the low mass, low temperature, high aborption case.

These results are certainly a strong argument in favour of Model 1. However, it would be premature to rule out Model 2 based on the only marginal consistency of the predicted distance with measurements for the planetary nebula. It is well known that such measurements are notoriously difficult and unreliable as discussed e.g. by Cahn et al. (1992), resulting in significant systematic errors.

6.2.3 The surface gravity of the components

Green et al. (1984) estimated the surface gravity of MT Ser comparing the mean of their two spectra with synthetic spectra for hot high-gravity stars of Wesemael et al. (1984). They get $\log g = 6 \pm 1$ . With the component radii as given in Table 6 and the assumed masses surface gravities can be calculated for the various models. They are also quoted in Table 6. In the case of Model 1 the secondary can be disregarded in this context because its contribution to the total light is very small; therefore the observed surface gravity is only due to the primary.

The calculated values for $\log g$ are only marginally consistent with the observations, and more so for Model 1 than for Model 2. Several factors may contribute to this discrepancy:

(1) Green et al. (1984) based their estimate of $\log g$ on LTE model atmosphere calculations. They point out that such models overestimate gravities by factors of 2-4 as compared to more realistic NLTE models.

(2) The surface gravity is a sensitive function of the absorption line width. Larger widths lead to higher gravities. In a close binary system such as MT Ser a bound rotation of the components can be expected. Considering the orbital inclination, the period, and the stellar radii, projected rotational velocities of 97-187 kms^-1 (Model 1) and 105-205 kms^-1 (Model 2) are calculated for the different model assumptions.

(3) The surface gravity quoted by Green et al. (1984) is based on the sum of two spectra. Possible line shifts due to orbital motion which may mimick a broader line are not considered. In the different cases of Model 1 the orbital inclination, the component separation, the period and the mass ratio yield a projected radial velocity amplitude of the primary star (the contribution of the secondary to the optical light is negligible) in the range 109-210 kms^-1. Within Model 2 both components have comparable luminosities [ $L_1/L_2 = (R_1^2/R_2^2) \times (T_1^4/T_2^4) \approx 0.69$ in the most favourable case]. Therefore, the observed spectrum will be a superposition of the spectra of both components. Hence even in a single spectrum the lines are broadened by orbital motion. Here, the expected radial velocity difference of the components is up to 237-452 kms^-1 in the different model cases.

(4) At least within Model 1 a significant reflection effect is present which will tend to reduce the depth of the absorption lines.

In order to investigate if the effects of rotation and orbital motion are able to broaden the observed spectral lines sufficiently to mimick the high surface gravity observed by Green et al. (1984) we calculated line profiles for H $\beta$ , assuming LTE and hydrostatic equilibrium (and neglecting radiation pressure) for a temperature of 40000 K (note that in this case the line depth is somewhat deeper than for higher temperatures but the shape of the lines is not much affected). Three profiles were calculated assuming $\log g$ =6; $V \sin i$ =0 (profile h), $\log g$ =5; $V \sin i$ =0 (profile l1) and $\log g$ =5; $V \sin i$ =200kms^-1 (profile l2).

As expected, profile h is significantly broader than profiles l1 and l2. The difference between the latter is basically confined to the line centres. The rotation decreases the line depth, but the flancs are hardly affected. Thus, a rotation of the star up to the maximum velocity expected in the case of MT Ser does not significantly broaden the line. Therefore, this effect is not likely to explain the discrepancy between observed and predicted surface gravities.

This is different when orbital motion is considered. We shifted profile l2 by $\pm2.7$ Å ( $166\, {\rm km\,s}^{-1}$ ) and added the results in order to simulate the combined profile due to two stars in orbital revolution. The results (as well as profile h) were convolved with a Gaussian in order to mimick the resolution of the MT Ser spectrum of Green et al. (1984). The final line shapes of the simulated binary line and the high gravity line differ in detail, but - disregarding the shallow outer parts of the line flancs and the depth relative to the continuum - are not so different as to be easily distinguished in a spectrum with a S/N ratio as low as observed by Green et al. (1984). This, together with the fact that the value of $\log g$ quoted by Green et al. (1984) may in reality be an upper limit, lets us conclude that the apparent disparity between the observed and the calculated surface gravity of MT Ser is not a matter of concern.

Although best fit values of the only model parameters which are directly a function of the surface gravity, namely the limb darkening coefficients, are well consistent with high gravity hot stars (Wade & Rucinski 1985), their confidence ranges are large enough so that no contradictions arise even if considerably lower gravities are assumed. This holds true also if the component temperatures are as low as those proposed by Tylenda et al. (1991).

6.2.4 The geometry of the system

From the shape of the planetary nebula in a high-resolution CCD-image Pollaco & Bell (1997) deduce an orbital inclination of 66 $^\circ$ for the central binary system. This holds true under the assumption that an elliptical structure seen close to the centre of the nebula is in fact circular and inclined to the line of sight, and if it has been ejected in the orbital plane. Comparing this to the results or our model calculations we find disagreement with Model 1 ( $i=42^\circ_{\raisebox{.6ex}{\hspace{.05em}.}}5$ for Model 1.1 and $i=48^\circ_{\raisebox{.6ex}{\hspace{.05em}.}}8$ for Model 1.2), whereas the agreement with Model 2 (Model 2.1: $i=67^\circ_{\raisebox{.6ex}{\hspace{.05em}.}}1$ ; Model 2.2.1: $i=65^\circ_{\raisebox{.6ex}{\hspace{.05em}.}}7$ ; Model 2.2.2: $i=67^\circ_{\raisebox{.6ex}{\hspace{.05em}.}}2$ ) is almost perfect.