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_o, while Tylenda et al. ([1991]) gave a value of log L = 2.79 (L = 617 L_o), 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, Omega₁, Omega₂ 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 (T₁=35000 K case). Within Model 2 the secondary star temperature scales linearly with T₁ (see Sect. 5.2). Therefore, the luminosities for the T₁=35000 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.