4 Results on Am stars

4.1 HD 43478 (= V406 Aur = Renson 11540)

This star was classified A3-F2-F5 by Osawa (1965) and Ap Si Sr by Bertaud & Floquet (1974). As kindly pointed by Renson (1994, personal communication to PN), Babcock (1958) had already found it double-lined, but did not give any period. Interestingly, Babcock listed this star as probably magnetic, and mentioned the profile of the K line as peculiar; as Babel (1994) showed, cool magnetic Ap stars have a peculiar profile of the CaII K line which betrays a stratification of calcium in the star's atmosphere. Perhaps this star should be indeed classified Ap after all. We secured 56 points (see Tables 9, 10 and Fig. 7) and obtained the orbital elements listed in Table 1.

  

Figure 7: Radial-velocity curve of HD 43478. The period is $5.464086 \pm 0.000011$ days. Notice that the zero phase corresponds here to the quadrature (epoch given in Table 1) but not to the primary eclipse (Eq. 3) which would fall here at phase 0.75, when the more massive component passes in front of the less massive one

  

Figure 8: Lightcurves of HD 43478 for the [V] magnitude and the [U-B] and [B-V] colour indices of Geneva photometry, plotted according to the ephemeris given in Eq. (3). Notice the lack of colour change during eclipses. The best fit to the [V] curve is shown (see Table 4)

The system is especially interesting, because we discovered eclipses, which allow to determine the orbital inclination (North & Nicolet 1994). Unfortunately, the eclipses are shallow, as shown if Fig. 8 where the lightcurve is plotted according to the ephemeris:

$$\begin{eqnarray} {\rm HJD(Min I)} = 2\,446\,774.790 &+ 5.464086\,E \\ \pm 0.003.& \nonumber\end{eqnarray}$$ (3)

In addition, the number of measurements is small, due to the unfavourable period (close to 5.5 days) and we had a relatively small number of good nights at the Jungfraujoch station on the critical dates. The descending branch of the primary minimum was observed during a mediocre, partial night where only five standards could be measured; nevertheless, the scatter around the fitted lightcurve is fairly good. The two minima have about the same depth and are separated by exactly 0.5 in phase, confirming the circularity of the orbit. On the other hand, the [U-B] and [B-V] curves remain flat during both eclipses, showing that both components have similar effective temperatures. In spite of the small number of points, we analized the lightcurve with the EBOP16 programme (Etzel 1980, 1991). The Geneva and $uvby\beta$ photometric indices and parameters give very consistent $T_{\rm eff}$ and $\log g$ values through the calibration of Künzli et al. (1997) for the Geneva system, and through the calibration of Moon & Dworetsky (1985) for the $uvby\beta$ system (Table 3). For the $uvby\beta$ system, we applied the correction recommended by Dworetsky & Moon (1986) to the $\log g$ value for the Am stars.

  

Table 3: Physical parameters of HD 43478 according to its colours in the $uvby\beta$ and Geneva photometric systems. Both components are assumed to be identical. Note that $E(B2-V1)=1.146\ E(b-y)$. The errors quoted for the physical parameters determined with Geneva photometry are propagated from typical errors on the colour indices but do not include possible systematic errors related with the calibration itself. The reddening E(B2-V1)=0.089 corresponds to E(B-V)=0.1 suggested by the maps of Lucke (1978) and is mentioned only to illustrate its effect on the physical parameters; the adopted colour excess E(B2-V1)=0.045 is obtained from E(b-y) = 0.039, which results from the calibrated $uvby\beta$ colours

From the values of $T_{\rm eff}$ and $\log g$, we interpolated the linear limb-darkening coefficient u from the tables of Van Hamme (1993). With the available photometric data, it is impossible to fit simultaneously all the interesting parameters, namely the central surface brightness of the secondary $J_{\rm s}$, the radius $r_{\rm p}$ of the primary, the ratio k of the radii and the orbital inclination i. This is a well-known difficulty for all systems (even well detached ones) where both components are nearly identical, even when the eclipses are deep. Another type of data has to be used to constrain the ratio of radii, because the latter may be changed from e.g. 0.6 to 1.4, without any change in the rms scatter of the residuals. We do not have detailed spectroscopic informations, but the CORAVEL data allow to have a rough guess of the k ratio in the two following ways:

1.

The width (FWHM) of the autocorrelation dip can be translated in terms of $v\sin i$ through a proper calibration (Benz & Mayor 1984). Assuming there is no other cause of broadening than in normal stars (i.e. no Zeeman broadening, for instance), one obtains in this way the projected rotational velocities given in Table 2. If synchronism has taken place between spin and orbital periods, which appears highly probable given the rather evolved state of the system (low $\log g$)and the circular orbit (circularisation time is longer than synchronisation time according to tidal theories), then k is directly given by the ratio of the $v\sin i$ values, i.e. 0.73.

2.

The equivalent width W of the autocorrelation dip depends on effective temperature and metallicity of the star, but also on the amount of dilution of the stellar flux by the companion's flux. Assuming that both stars have the same effective temperature (as suggested by the flat [U-B] and [B-V] curves) and the same metallicity (a more adventurous assumption), the ratio W₂/W₁ gives directly the luminosity ratio L₂/L₁ and is equal to the square of the ratio of radii k². One obtains in this way k=0.79.

A larger weight has to be granted to the first method, so we adopt here k=0.75, keeping in mind that the uncertainty on this quantity remains considerable (20% or so). The final elements found with the EBOP16 programme are given in Table 4. They are rather approximate, but the inclination is relatively well determined and so are the masses too. It is necessary here to comment briefly on the definition of "primary'' and "secondary'' components, because it is not necessarily the same when radial velocities, respectively lightcurves are considered. From the radial-velocity standpoint, the primary evidently corresponds to the smaller amplitude K and to the more massive component. But, when interpreting the lightcurve, the EBOP code assumes that the deeper (or primary) eclipse corresponds to the secondary passing in front of the primary component. In this particular system, it is interesting to notice that the primary minimum corresponds to phase 0.75 of the $V_{\rm r}$ curve, where the less massive component lies behind the primary, not the reverse. Therefore, the adopted ratio of radii entered into the EBOP code should not be k = r₂/r₁ = 0.75, but $k = r_{\rm s}/r_{\rm p} = 1.333$, since we have to identify the dynamical primary (1) with the photometric secondary (s) and the dynamical secondary (2) with the photometric primary (p). Interestingly, this implies a larger surface brightness of the dynamical secondary than of the primary, i.e. a slightly larger effective temperature, a relatively rare occurence. The effective temperatures have been computed from an apparent $T_{\rm eff} = 6944$ K (average of Geneva and $uvby\beta$ estimates) which is assumed to result from a weighted average of the components' reciprocal temperatures: $\theta_{\rm eff}{\rm (apparent)}=0.7258=(L_1\theta_1+L_2\theta_2)/(L_1+L_2)$,and assuming $J_{\rm s}/J_{\rm p} = (T_{\rm eff s}/T_{\rm eff p})^4$.The bolometric luminosity has been computed assuming $M_{\rm bol\odot}=4.75$.

  

Table 4: Parameters of HD 43478 obtained from the [V] magnitude using the EBOP16 code and assuming the ratio of radii r₂/r₁ = 0.75. The indicated errors are the formal ones only and do not include the large uncertainty on k. Notice that the subscripts p and s refer to the photometric primary and secondary respectively (the "secondary'' being defined as the foreground star at Min. I) but correspond to the subscripts 2 and 1 (in this order), which correspond to the less and more massive star respectively

  

Table 5: Physical parameters of the components of HD 43478. The error on the masses includes a large, 20% uncertainty on k, which translates into a $\pm 0.27^{\rm o}$ uncertainty on i. The same is true of the radii, whose uncertainties are mutually anticorrelated since the sum of radii remains constant within 3% as k is varied

A summary of the physical parameters of the HD 43478 system is given in Table 5. The bolometric correction has been taken from Schmidt-Kaler (1982). The distance is shorter than indicated by the Hipparcos satellite, which gave $\pi = 3.87\pm 0.93$ mas or d = 258_-50⁺⁸² pc; however, the discrepancy cannot be considered significant since it remains largely within two sigmas. The distance deduced from the fundamental radii and photometric effective temperatures is almost twice more accurate than that given by Hipparcos (the error has been estimated using the usual propagation formula applied to the distance modulus). It is interesting that the fundamental $\log g$ value we find for the primary, which is quite reliable, is in excellent agreement with the value obtained from both Geneva and $uvby\beta$ colour indices.

The situation of both components in the HR diagram is shown in Fig. 9, together with Z=0.020 evolutionary tracks interpolated in those of Schaller et al. (1992) and isochrones with ages $\log t = 9.1$and 9.2. The primary is clearly at the end of its life on the Main Sequence, and the secondary is probably somewhat evolved too. In view of the shape of the isochrones, one can easily understand why the secondary is slightly hotter than the primary. Clearly, a more complete and precise lightcurve is needed, especially to give a more accurate, fundamental estimate of the radii and to assess thereby the validity of the assumption of synchronism.

  

Figure 9: HR and $\log g$ vs. $\log(T_{\rm eff})$ diagrams for both components of HD 43478. The position of the primary is fairly well defined, while that of the secondary is less reliable. The continuous lines are the ZAMS and evolutionary tracks interpolated for the measured masses, while the dotted and broken lines are the isochrones at $\log t = 9.1$ and $\log t = 9.2$ respectively

4.2 HD 96391 (= BD +72$^\circ$515 = Renson 27770)

  

Figure 10: Radial-velocity curve of HD 96391. The period is $4.915427 \pm 0.000008$ days

This star was classified A4-F0-F3 by Abt (1984). It is also an SB2 system with very similar companions. Unfortunately, we do not have Geneva photometry for that star, but Strömgren photometry done by Olsen (1983) and retrieved using the General Catalogue of Photometric Data (Mermilliod et al. 1997) gives $T_{\rm eff} = 7020$ K, $\log g = 3.85$through the calibration of Moon & Dworetsky (1985), and $\Delta$m₀ = -0.051, $R/R_\odot = 1.74$, M_V = 2.66, $M_{\rm bol} = 2.59$ and $\log (L/L_\odot) = 0.82$ through older calibrations included in Moon's (1985) code. We have 36 CORAVEL observations of this star (Tables 11 and 12); the orbital elements are listed in Table 1 and the $v\sin i$ value of each component is given in Table 2. The i angle remains unknown, since that star is not known as an eclipsing binary.

On the other hand, the Hipparcos parallax is $\pi = 6.81\pm 0.62$mas, which implies a distance d = 152 pc taking into account the Lutz-Kelker correction -0.07. Furthermore, from the reddening maps of Lucke (1978), a colour excess E(B-V) = 0.029 appears reasonable, so we adopt $A_V \sim 0.10$; to correct for the duplicity, the apparent visual magnitude is increased by 0.75 mag (so the result will relate to an average component), and one obtains M_V=1.82, $\log(L/L_\odot)=1.198\pm 0.089$, and by interpolation in theoretical evolutionary tracks, $\log g = 3.84\pm 0.08$dex, ${\cal M} = 1.819\pm 0.070~{\cal M}_\odot$ and $R = 2.69\pm 0.30~R_\odot$.The agreement of the $\log g$value obtained here with that given by the uvby photometry is excellent (the photometric luminosity is far off, but is obtained through an older calibration). Once again, this system appears close to the end of its life on the main sequence. It is now possible to estimate the orbital inclination i by comparing ${\cal M}\sin^3i \simeq 1.38~{\cal M}_\odot$ with ${\cal M}=1.82~{\cal M}_\odot$ and the result is $i\simeq 66\hbox{$^\circ$}$.This precludes eclipses, which would need an orbital inclination larger than $\sim 73\hbox{$^\circ$}$ to occur. The individual masses are about ${\cal M}_1 = 1.85~{\cal M}_\odot$ and ${\cal M}_2 = 1.73~{\cal M}_\odot$.The average equatorial velocity computed from the radius obtained above and from the assumption of synchronism is 27.7 kms^-1, which translates into $v\sin i\simeq 25$ kms^-1. This value may be compared with the observed $v\sin i$'s of both companions (Table 2), if the spin axes are perpendicular to the orbital plane: the observed values are smaller than those predicted by synchronism via the radius estimate, but the $11\%$ error on the latter is large enough to accomodate both results within two $\sigma$.The system is very probably synchronised.