3 Results on Ap stars

3.1 HD 8441 (= BD +42$^\circ$293 = Renson 2050)

This bright A2 Sr star was already known as an SB1 system. Its rotational period, known from its photometric variability, is 69.43 days (Rakosch & Fiedler 1978). Renson (1966) found an orbital period of 106.3 days and published the radial-velocity curve, but not the orbital parameters. A total of 107 measurements have been made over almost 5000 days (Table 6), which confirms the 106 days period (see Fig. 1). However, the residuals are larger than expected from the precision of the measurements, and follow a very clear trend (see Fig. 2). The presence of a third component is certain, although its period is so long that we could not cover even one cycle. The orbital parameters of the primary are given in Table 1. This is the second spectroscopic triple system known among Ap stars, after the SiMg star HD 201433 whose periods are much shorter (see the catalogue of Tokovinin 1997).

  

Figure 1: Radial-velocity curve of HD 8441. The period is $106.357 \pm 0.009$ days

  

Figure 2: Radial-velocity residuals vs. time for HD 8441

The projected rotational velocity estimated from the width of the autocorrelation dip (Benz & Mayor 1984) is given in Table 2, with the restriction that in principle, such a quantity can only represent an upper limit to the true $v\sin i$.Indeed, the magnetic field commonly present among Ap Sr stars broadens the lines through the Zeeman effect, so that $v\sin i$ will be overestimated if this effect is neglected. In this particular case, however, the estimated $v\sin i$ is quite compatible with the 69.43 days rotational period, assuming a radius $R\sim 3 ~ R_\odot$.

  

Table 2: Visual magnitude and $v\sin i$ of the programme stars

The Hipparcos parallax of this star is $4.91\pm 0.80$ mas (Perryman et al. 1997); this translates into a distance d=232 pc, after having applied a Lutz-Kelker correction (Lutz & Kelker 1973) $\Delta M=-0.28$ which takes into account the exponential decrease of stellar density in the direction perpendicular to the galactic plane. On the other hand, the visual absorption estimated from Geneva photometry is A_v = 0.13. Assuming a contribution of about 0.23 magnitudes of the companions to the visual magnitude of the system, the apparent magnitude of the primary alone is V=6.92, and finally we obtain an absolute magnitude $M_V=-0.03\pm 0.42$ for this component. Adopting $T_{\rm eff}=9200$ K (Adelman et al. 1995) and interpolating in the evolutionary tracks of Schaller et al. (1992) for a solar metallicity Z=0.018 (and for a moderate overshooting distance $d_{\rm over}/H_{\rm p} = 0.2$), one obtains ${\cal M}_1=2.76\pm 0.18~{\cal M}_\odot$, $\log g = 3.71\pm 0.12$ (g in cgs units) and $R = 3.86\pm 0.66 \,R_\odot$.Although the uncertainties are fairly large, the primary is evolved and on the verge of leaving the Main Sequence; it is satisfying that our small $\log g$ value agrees with the spectroscopic estimate of Adelman et al. (1995) who gave $\log g = 3.35 - 3.8$.

  

Figure 3: Radial-velocity curve of HD 137909 given by CORAVEL only. The period is $3831.50 \pm 7.94$ days

3.2 $\beta$ CrB (= HD 137909 = BD +29$^\circ$2670 = Renson 39200)

This is a well-known, prototype cool Ap star classified A9 SrEuCr. Its rotational period, known from photometric, spectroscopic and magnetic variations is 18.4868 days (Leroy 1995). It is known as a binary, by both spectroscopy and speckle interferometry. A radial-velocity curve was published by Kamper et al. (1990) together with an astrometric orbit based on speckle observations. These authors suspected that a third body might be present, on the basis of radial velocities taken at Lick Observatory between 1930 and 1943. The system was monitored with CORAVEL for a little more than one cycle, which is very long, and 78 measurements have been obtained (see Table 7). The $V_{\rm r}$ curve is shown in Fig. 3 and the spectroscopic orbit is given in Table 1. Thanks to the precision and the homogeneity of the data, our $V_{\rm r}$ curve is more precise than that based on the data taken at David Dunlap Observatory by Kamper et al. Combining our measurements with those published by Kamper et al. (1990), by Oetken & Orwert (1984) and by Neubauer (1944), we can refine the period to P = 3858.13 days, but the accuracy of the orbital elements is not improved, due to the scatter of the residuals, which is more than twice larger than for CORAVEL observations alone. In order to fit Neubauer's data to the others, we had to subtract a constant value (2 kms^-1) to them, which was also done by Kamper et al. (1990). The resulting radial-velocity curve is shown in Fig. 4 and the corresponding orbital elements are given in Table 1. The residuals are shown in Fig. 5. A fit of the CORAVEL radial velocities alone has also been done keeping the orbital period fixed to the above, refined value and its results are displayed in the last line of Table 1. The scatter of the O-C residuals is hardly increased and only the eccentricity and the amplitude change by more than one sigma with respect to the fit where P was adjusted; the change is probably due to the rather inhomogeneous phase coverage of the observations.

  

Figure 4: Radial-velocity curve of HD 137909 including data published by Kamper et al. (1990), by Neubauer (1944) and by Oetken & Orwert (1984). A correction of $-2 {\rm\, km\,s^{-1}}$ has been added to the $V_{\rm r}$ values of Neubauer. The period is $3858.13 \pm 2.96$ days

  

Figure 5: Radial-velocity residuals vs time for HD 137909 for the whole sample

Thanks to the Hipparcos satellite, $\beta$ CrB has now a very precise parallax $\pi = 28.60 \pm 0.69$ mas which allows to compute the linear semi-major axis of the relative orbit from the angular semi-major axis obtained by speckle interferometry ($203.2 \pm 1.4$ mas). Since the inclination angle $i = 111.11\hbox{$^\circ$}\pm 0.46\hbox{$^\circ$}$ of the orbit is known from speckle interferometry and the quantity $a_1 \sin~i$ is known from CORAVEL measurements, the semi-major axis of the absolute orbit of the companion can be computed:

$$a_2=a-a_1=4.114 \pm 0.031~U.A. = 615.2 \ 10^6~{\rm km}$$ (1)

as well as the mass ratio:

$${{\cal M}_2\over{\cal M}_1}={a_1\over a_2}={0.727 \pm 0.033}.$$ (2)

Finally, one can obtain ${\cal M}_{2} = 1.356 \pm 0.073 ~{\cal M}_\odot$ from the mass function, as well as ${\cal M}_{1} = 1.87 \pm 0.13 ~{\cal M}_\odot$.Oetken & Orwert (1984) had found ${\cal M}_{1} = 1.82$ and ${\cal M}_{2} = 1.35$ using the same method but a pre-Hipparcos parallax of 31 mas. Our results, altough close to theirs, is more reliable. The radius of the primary, estimated from the Hipparcos parallax and from $T_{\rm eff}= 7750$ K (Faraggiana & Gerbaldi 1993), is $R = 3.03\pm 0.25 \,R_\odot$, which implies an equatorial velocity $v_{\rm eq}= 8.3\pm 0.7$ kms^-1. If the rotational equator of the star coincides with the orbital plane, then the projected rotational velocity is $v\sin i = 7.7\pm 0.7$ kms^-1, in excellent agreement with the value (which may be overestimated, however) listed in Table 2. In these estimates, we assumed a negligible interstellar absorption and adopted the difference $\Delta V = 1.7$ mag between the components of this speckle binary (Tokovinin 1985), so that the apparent visual magnitude of the primary component alone is 3.876 instead of 3.670 for the whole system (Rufener 1988).

The HR diagram is shown in Fig. 6. Strangely enough, the agreement between the observed location of $\beta$ CrB and the evolutionary track at the observed dynamical mass is very poor: both the primary and the secondary (if we rely on $\Delta V = 1.7$) appear overluminous compared to the evolutionary tracks drawn for their mass. Considered alone, the primary might well be at the very tip of the blue hook at the core-hydrogen exhaustion phase, which would reconcile within one $\sigma$ its observed and theoretical locations in the HR diagram. Its logarithmic age might then be 9.05 dex instead of 8.9. However, the secondary (indicated in Fig. 6 as a dot arbitrarily placed along the abcissa on the isochrone $\log t=8.9$) seems overluminous as well, making the puzzle more complicated but also more interesting, and certainly well worth further investigations. Unfortunately, it is not possible to test completely the position of the secondary because of its unknown colours. Such an information would be most interesting to test the idea of Hack et al. (1997) that the companion might be a $\lambda$ Boo star with $T_{\rm eff}\sim 8200$ K, although such a hypothesis appears difficult to maintain in view of Fig. 6.

  

Figure 6: HR diagram of $\beta$ CrB. The ZAMS, TAMS and evolutionary tracks interpolated for the observed masses are shown as solid lines, while those interpolated for the masses $\pm 1\ \sigma$ appear as broken lines. The dotted lines indicates the isochrones at $\log t=8.9$ and 9.05 (t in years)

The semi-major axis of the orbit of the binary's photocenter is given in the Hipparcos and Tycho Catalogues (Perryman et al. 1997). This allows an independant test of the magnitude difference $\Delta V$: assuming the photocenter to be defined by E₁ x₁ = E₂ x₂, where E₁ and E₂ are the respective brightnesses of the components in the $H_{\rm p}$ passband and x₁, x₂ the distances of the components to the photocenter such that x₁+x₂=a₁+a₂=a, one obtains $a_0= 1.77\pm 0.07$ au, on the basis of Tokovinin's $\Delta V = 1.7$ (one has a₀= a₁-x₁). This is in rough agreement (within three sigmas) with $a_0= 1.97\pm 0.05$ au given in the Hipparcos catalogue. Our estimate assumes a companion with $T_{\rm eff} = 7200$ K and takes into account the colour equation between $H{\rm p}$ and V (Vol. 2, p. 59 of Perryman et al. 1997) which leads to $\Delta H{\rm p} = 1.71$. Increasing $\Delta V$ by about 0.2 magnitudes would bring perfect agreement. Unfortunately, Tokovinin (1985) do not give any error estimate on $\Delta V$.