8 Discussion

8.1 Limb-darkening

Our adjusted values of x₁ (see Table 8) compare relatively well with the theoretical ones, in the sense that the difference hardly exceeds 0.1, or $15-20\%$ of the value (except for V where the difference reaches 0.15). Nevertheless, our values are systematically smaller than the theoretical ones (computed for a solar chemical composition). The discrepancies are larger than the formal errors indicated in Table 8, which may partly, but probably not entirely be explained by the fact that the errors listed should represent only lower limits to the true ones.

Figure 8 illustrates the dependence of x₁ on wavelength for a theoretical star with $T_{\rm eff}= 7770$ K and $\log g = 4.28$, for the passbands of Strömgren's uvby system and Johnson's UBV system (the values have been linearly interpolated in Table 2 of Van Hamme 1993). The fitted values for the Geneva U, B1, B, B2, V1, V, G passbands are shown for comparison.

  

Figure 8: Limb-darkening x1 coefficient of the primary as a function of wavelength. Points are theoretical heterochromatic values for the uvby (open dots) and UBV passbands (full dots), while crosses are for empirical values determined in this work for the passbands of the Geneva system. Notice that the theoretical points for the v and the V bands are superimposed

The theoretical dependence of x₁ on wavelength satisfies very well the observations from the U up to the B2 passband, except for a small, systematic vertical shift. For the V1 to G passbands, however, the observed x₁ parameter is much smaller than the theoretical one. Since the error bars are not larger than the symbols in Fig. 8 (except marginally for U), there is undoubtedly a significant discrepancy which remains to be explained. We obtained a simultaneous solution in the seven passbands, thus at least the relative values of x₁ should be reliable. But, even considering only differential values (e.g. x₁(V) - x₁(B)) a difference with the theoretical predictions does remain. Unfortunately, we did not see any recent discussion in the literature about the reliability of empirical limb-darkening coefficients in Algol-type systems.

8.2 Distance

From the values of V (Table 2), $M_{\rm bol}$ (Table 9), E[B2-V1] and [B2-V1]₀ (Sect. 4), we derive a distance of $270 \pm 12$pc for TZ Eri, by adopting a zero value for the bolometric correction BC of the primary, according to the BC-colour relation of Flower (1977).

8.3 Algol-type stars

Sarma et al. (1996) gave a detailed discussion on the evolutionary status of Algol components, on the basis of a comparison of their global parameters (masses, radii, luminosities, temperatures) with those of normal stars. As these authors noted:

1.

While primaries (mass gainers) of some semi-detached systems are overluminous, oversized and hotter for their masses, some others are underluminous, smaller and cooler.

2.

Except for a few cases, the primaries are lying either on the main sequence or near it.

3.

The secondaries (mass losers) have evolved off the main sequence (they were originally the more massive components).

Figures 9 and 10 present the Mass-Luminosity diagram ($\log L$ vs. $\log M$) and the HR diagram ($\log L$ vs. $\log T$) for the Algol systems in the lists by Sarma et al. (1996) and Maxted & Hilditch (1996) having a mass of the primary smaller than 4 $M_{\odot}$. The 3 systems with primaries off the main sequence (RZ Cnc, AR Mon, AW Peg) are not represented. The underluminous (black dots) and overluminous (black squares) have been separated in Fig. 10 by using the ZAMS (Schaller et al. 1992) to make the separation. We can see that:

1.

TZ Eri belongs to the group of the underluminous primaries.

2.

The mass of TZ Eri's primary is the smallest of the group of these underluminous primaries.

3.

The luminosity of TZ Eri's primary is that of a main sequence star of 1.70 $M_{\odot}$ located near the ZAMS (see Fig. 10), while the determined mass is 1.98 $M_{\odot}$ (see Table 9).

4.

With respect to a main sequence star of the same mass, TZ Eri's primary is underluminous by $\sim$0.3 in $\log L$ (or fainter by $\sim$0.75 in $M_{\rm bol}$) and cooler by $\sim$900 K in $T_{\rm eff}$.

5.

The mass transfer from the secondary to the primary has been important, taking into account the present masses of the components, i.e. 0.37 $M_{\odot}$ and 1.98 $M_{\odot}$. The minimum value of this transfered mass is given by:

$\Delta M = M_{1} - (M_{1} + M_{2})/2 \simeq 0.80\ M_{\odot}$.

  

Figure 9: Theoretical HR diagram for the components of Algol systems with very precisely determined physical parameters according to the lists by Sarma et al. (1996) and Maxted & Hilditch (1996), and to this paper for TZ Eri (see Sect. 8.3). Primaries are identified with filled symbols and secondaries with open symbols. TZ Eri is identified with plus symbols. Dots and squares refer to the systems having respectively under- and overluminous primaries with respect to their masses (see Fig. 10). The Zero-Age Main Sequence and 3 evolutionary tracks (3.0, 2.0 and 1.5 $M_{\odot}$) are drawn, according to Schaller et al. (1992)

  

Figure 10: Mass-Luminosity relation for the components of the same Algol systems as those in Fig. 9. The zero-age main sequence (ZAMS) and the Terminal-Age Main Sequence (TAMS) are drawn, according to Schaller et al. (1992). Primaries are identified with filled symbols and secondaries with open symbols. TZ Eri is identified with plus symbols. Dots and squares refer to the systems having respectively under- and overluminous primaries with respect to the ZAMS