7 Photometric solution

The photometric solution for TZ Eri eclipses was obtained with the Wilson-Devinney WD programme (Wilson & Devinney 1971; Wilson 1992), using the version revised in 1995. We used the WD programme in MODE 5, adapted for semi-detached systems , allowing a simultaneous computation on the seven Geneva photometric light curves, based on 393 measurements (see Sect. 3), and the radial velocity curves of both components (see Table 4).

  

Table 8:

Lobe filling of the cool loser component was assumed. The semi-major axis of the relative orbit was initially set to A = 10.2 $R_{\odot}$ as calculated from radial velocity results (see Sect. 5). Orbital eccentricity is fixed to zero and longitude of the periastron to $90 \hbox{$^\circ$}$. Primary star temperature was fixed to 7770 K as determined in Sect. 4. For both components, the stellar atmosphere models of Kurucz (1994) integrated by Nicolet (1998) through the Geneva photometry passbands (Rufener & Nicolet 1988) have been used.

The bolometric albedos for hot and cool components were taken at the theoretical value of 1.0 and 0.5 respectively (radiative and convective cases). Noise was set to 1 because scintillation can been neglected, the photon noise being dominant (see Bartholdi et al. 1984). The grid resolution values were taken as 20, 20, 20, 20 for N1, N2, N1L and N2L respectively (see WD programme). For each of the seven Geneva magnitudes, the passband mean wavelength $\lambda_{0}$ was the one calculated by Rufener & Nicolet (1988) as shown in Table 6. We assigned the $\lambda_{0}$ value for the B filter to the passband of the spectrographs used for the radial velocity measurements. Stellar rotation is assumed to be synchronized for both components. For both the primary and secondary components, a logarithmic limb-darkening law of the form:

$$I = I_0(1-x+x\cos\theta - y\cos\theta \ln(\cos\theta))$$ (6)

was assumed (Van Hamme 1993). For the secondary, both x and y parameters were fixed to their theoretical values, interpolated from Table 2 of Van Hamme (1993). Indeed, the secondary minimum is too shallow to allow the determination of x₂. For the primary, the y₁ parameter was fixed to its theoretical value, because it is not possible in practice to determine both x₁ and y₁ parameters since they are strongly correlated. The x₁ parameter, on the contrary, was left free and was fitted in the least-squares procedure, for each of the seven passbands. The adopted values of y₁, x₂ and y₂ are listed in Table 7, while the fitted values of x₁ are given in Table 8. The gravity darkening exponent g₂ is relevant to the secondary component and is defined by the expression (Wilson & Biermann 1976):

$$T_{\rm local} = T_{\rm r.p.}\left(\frac{a_{\rm local}}{a_{\rm r.p.}}\right)^{0.25 g}$$ (7)

where $T_{\rm local}$ is the local effective temperature, $T_{\rm r.p.}$is the temperature at a reference point which here is the component's pole, and $a_{\rm local}$, $a_{\rm r.p.}$ are the accelerations due to gravity at the same respective points.

The adjustable parameters are then semi-major axis A, inclination i, mass ratio q=M₂/M₁, cool star temperature T₂, hot and cool star luminosities L₁ and L₂ in each passband, cool star gravity darkening exponent g₂ (see Wilson & Biermann 1976), hot star limb darkening coefficient x₁ in each passband and potential $\Omega_{1}$ at the hot star surface. In reality, $\Omega$ is a non-dimentional parameter which is a linear function of the true potential $\Psi$ (Kopal 1959; Wilson & Devinney 1971). First, preliminary light curves were generated using the Light Curve sub-programme of WD until a satisfactory fit to the observed light curves was obtained. Second, the method recommended by Van Hamme & Wilson (1986) was applied for the determination of the temperature and luminosity of the cool component. In a first step, T₂ was determined with IPB option (see the WD programme) equal to 0, i.e. the luminosity is coupled to the temperature. In a second step, temperature and luminosity were de-coupled (IPB = 1), and it was possible to determine L₂. Finally, the following sets of parameters were adjusted: first, successively (T₂, $\Omega_{1}$, i), (L₁, L₂), (A, i), (x₁, g₂), and then simultaneously (A, i, $\Omega_{1}$, q, L₁, L₂), as recommended by the users of the WD programme. The solution was carried out to the point where the probable errors of all these parameters were smaller than the computed parameter corrections.

Since the photometric temperature of the primary star is determined with an uncertainty of $\pm 100$ K (see Sect. 4), it is interesting to calculate the solution also for the two T₁ values 7670 K and 7870 K. The results are presented in Table 8. The differences are very small, except of course on T₂.

It is especially noteworthy that, for T₁= 7770 K, the value of T₂ ($4563 \pm 2$ K) is very close to the value obtained in Sect. 4 on the basis of the photometric calibrations. The mass ratio $q=M_{2}/M_{1} = 0.1865 \pm 0.0003$in Table 8 is a little smaller than the value derived from the radial velocity analysis ($K_{1}/K_{2} = 0.193 \pm 0.013$, see Table 5), but still well within the uncertainty of the spectroscopic determination. A similar difference can be noted on the semi-major axis of the relative orbit A = a₁+a₂ ($10.16 \pm 0.22$$R_{\odot}$ in Table 5 and $10.57 \pm 0.16$ $R_{\odot}$ in Table 8). These differences are not surprising. We have to recall that the radial velocity curve for the secondary component is based on only one measurement. The constraints imposed by the photometric measurements of the eclipses produced an improvement of the orbital and physical parameters of TZ Eri.

The small difference between the photometric and spectroscopic values of q could indicate that the lobe filling by the secondary is not complete. If this were true, the real value of the uncertainty on q would be larger than the photometric value (0.0003), may be as large as the spectroscopic value (0.013). Nevertheless, test solutions have been obtained with the cool star slightly detached from its Roche lobe, and the results are less satisfactory than those obtained in MODE 5 of the WD code: as the cool star became farther detached from the lobe, solution errors grew. This confirms the semi-detached status of the system.

The uncertainties on the derived parameters are of three types:

- Intrinsic, i.e. resulting from the mathematical analysis of the light and radial velocity curves (e.g. $\pm$ 2 K on T₂, see Table 8). Note that the intrinsic errors on R_1,2 for A fixed are an order of magnitude smaller than the values in Table 9.

- Strongly correlated with the determination of A. For example, in Table 9, the errors on M_1,2, R_1,2 and $\log g_{1,2}$almost entirely come from the uncertainty on A, and are pair wise strongly correlated (as well as with A).

- Depending on the photometric determination of $T_{\rm eff_1}$. For example, the values of $M_{\rm bol_{1,2}}$ strongly depend on the adopted value for $T_{\rm eff_1}$.

The light curves for the seven filters U, B₁, B, B₂, V₁, V and G are shown in Figs. 4 and 5. The quality of the fits is clearly extremely good. Figures 6 and 7 present two views of TZ Eri. Figure 6 is a "classical'' representation of the two components in the equatorial plane. The 3-dimentional representation of the potential (Fig. 7) allows a better understanding of the meaning of the Lagrange points and of the path for the flow of the material from the secondary when it fills its Roche lobe.

  

Figure 4: Light curves of TZ Eri in 6 of the 7 Geneva photometric passbands (U, B1, B, B2, V1, G). The light curve in V is given in Fig. 5

  

Figure 5: The light curve of TZ Eri in magnitude V, with an enlargement of the primary eclipse

  

Figure 6: Schematic view of the system in the equatorial plane (coordinates in $R_{\odot}$). Some equipotential lines and the Lagrange points are indicated. The secondary star fills its Roche lobe

  

Figure 7: Schematic view of the gravity potential $\Omega$ (see Sect. 7 for the definition) of the TZ Eri system. The horizontal coordinates are in $R_{\odot}$

   Table 9: Computed parameters of the system TZ Eri from the Wilson-Devinney programme. The parameters are, for the hot primary (1) and the cool secondary (2) components: the mass M, the mean radius R, the surface gravity $\log g$, the bolometric magnitude $M_{\rm bol}$ and the various radii (in units of semi-major axis) of the deformed components, i.e. pole (perpendicular to the orbital plane), point (in the direction of the other component), side (in the orbital plane, in the direction perpendicular to the direction of the other component) and back (in the direction opposite to the other component). The uncertainty on $M_{\rm bol}$ is calculated with an uncertainty on $T_{\rm eff}$ of $\pm$ 100 K