5 Statistics of Ap binaries

5.1 The sample of binaries

Our sample is composed of all Ap stars known as spectroscopic binaries. The catalogue of Renson (1991) gives us the Ap stars. However, some stars considered as Am by Renson, have been considered in this work as Ap (HD 56495, HD 73709 and HD 188854). Among these stars, the spectroscopic binaries have been selected from the following sources. Fourteen orbits were determined thanks to the CORAVEL scanner (this paper and North et al. 1998). Lloyd et al. (1995) determined a new orbit for $\theta \ Carinae$ (HD 93030) and Stickland et al. (1994) for HD 49798. Recently, Leone & Catanzaro (1999) have published the orbital elements of 7 additional CP stars, two of which are He-strong, two are He-weak, one is Ap Si, one Ap HgPt and one Ap SrCrEu. Wade et al. (2000) provided the orbital elements of HD 81009. Other data come from the Batten et al. (1989) and Renson (1991) catalogues. Among all Ap, we kept 78 stars with known period and eccentricity, 74 of them having a published mass function.

5.2 Eccentricities and periods

The statistical test of Lucy & Sweeney (1971) has been applied to all binaries with moderate eccentricity, in order to see how far the latter is significant. The eccentricity was put to zero whenever it was found insignificant. We can notice the effect of tidal interactions (Zahn 1977, 1989; Zahn & Bouchet 1989) on the orbits of Ap stars (see Fig. 8a). Indeed, all orbits with P less than a given value ( $=P_{\rm circ}$) are circularized. According to the third Kepler law, a short period implies a small orbit where tidal forces are strong. The period-eccentricity diagram for the Ap stars does not show a well marked transition from circular to eccentric orbits, in the sense that circular as well as eccentric systems exist in the whole range of orbital periods between $P_{\rm circ}=5$ days to a maximum of about 160 days. The wider circular orbits probably result from systems where the more massive companion once went through the red giant phase; the radius of the former primary was then large enough to circularize the orbit in a very short time. This is quite consistent with the synchronization limit for giant stars of 3 to 4  $M_{\odot}~(P\sim 150$ days, Mermilliod & Mayor 1996).

An upper envelope seems well-defined in the e vs. $\log P$ diagram, especially for $\log P \la 2$, although four points lie above it. The leftmost of these, with ($\log P$,e) = (0.69, 0.52) has a rather ill-defined orbit. Whether this envelope has any significance remains to be confirmed with a larger sample than presently available.

Figure 8: a) Diagram eccentricity versus period (in days) for the 78 Ap stars. The symbols are according to the type of Ap: $\triangle$ HgMn, $\blacktriangle$ He weak, $\square$ SrCrEu, $\blacksquare$ Si. b) Diagram eccentricity versus period (in days) for G dwarf stars (Duquennoy & Mayor 1991).

It is interesting to compare our $e/\log P$ diagram with that established by Debernardi (2002) for his large sample of Am stars. For these stars, the limit between circular and eccentric orbits is much steeper, many systems having $e\sim 0.7$ at P=10 d only. The difference might be due to the wider mass distribution of the Bp-Ap stars (roughly 2 to 5 $M_\odot$) compared to that of Am stars (1.5 to 2-2.5 $M_\odot$), $P_{\rm circ}$ being different for each mass. One might also speculate that the lack of very eccentric and short periods is linked with the formation process of Ap stars, which for some reason (e.g. pseudo-synchronization in the PMS phase, leading to excessive equatorial velocities?) forbid this region.

Qualitatively, Ap stars behave roughly like normal G-dwarfs (Duquennoy & Mayor 1991) in the eccentricity-period diagram. There is also a lack of low eccentricities at long periods (here for $\log P > 2.0$), and the upper envelope is similar in both cases for $\log P > 1.0$. One difference is the presence of moderate eccentricities for periods shorter than 10 days among Ap binaries, and another is the complete lack of very short orbital periods ($P\leq 3$ days) among them, if one excepts HD 200405. The latter feature, already mentioned in the past (e.g. by GFH85) is especially striking because it does not occur for the Am binaries. The physical cause for it is probably that synchronization will take place rather early and force the components to rotate too fast to allow magnetic field and/or abundance anomalies to subsist. However, even an orbital period as short as one day will result in an equatorial velocity of only 152 km s^-1 for a 3 $R_{\odot}$ star, while single Bp or Ap stars rotating at this speed or even faster are known to exist (e.g. HD 60435, Bp SiMg, $P_{\rm rot}=0.4755$ d; North et al. 1988). The detection of one system with $P_{\rm orb}=1.635$ d further complicates the problem, even though it remains an exceptional case as yet.

The shorter $P_{\rm circ}$ value can be explained by the following effects:

• Ap stars are younger on average than G dwarfs, leaving less time for circularization;
• Ap stars have essentially radiative envelopes, on which tidal effects are less efficient: the circularization time increases much faster with the relative radius than for stars with convective envelopes: $t_{\rm circ} \propto (a/R)^{21/2}$ instead of $t_{\rm circ}\propto (a/R)^8$ (see e.g. Zahn 1977), a being the semi-major axis of the relative orbit and Rthe stellar radius;
• Therefore, it is essentially the secondary component which is responsible for the circularization of systems hosting an Ap star. Its radius being smaller than that of the Ap star, the circularization time is longer. Thus $=P_{\rm circ}$ is shorter in these systems than in those hosting G dwarfs.

Our results confirm those of GFH85. Many stars of the literature have imprecise orbital elements, making a detailed analysis difficult. We have not enough CORAVEL orbits of Ap stars yet to make a more precise statistics.

5.3 Mass function

The mass function contains the unknown orbital inclination i. Therefore, neither the individual masses nor the mass ratio can be calculated from it. However the orbital inclination of Ap stars can be assumed to be randomly oriented on the sky. Thus, we can compare the observed cumulative distribution of the mass functions (for our 74 stars) with a simulated distribution (see Fig. 9), where we assume random orbit orientations. We approximated the observed relative distribution of Ap masses (North 1993) by the function given below, then multiplied it by Salpeter's law ( ${\cal M}^{-2.35}$) in order to obtain the simulated distribution of the primary masses $f_{\rm Ap}$:

$$% f_{\rm Ap} {=} \left[ 0.3623 + 0.8764 \cdot {\cal M} + 8.48... ...ot {\rm e}^{-({\cal M} - 3.5)^2}\right] \cdot {\cal M}^{-2.35}$$ (9)

with a minimum mass of $1.5~{\cal M_\odot}$ and a maximum mass of $7~{\cal M_\odot}$.

Figure 9: a) Distributions of the masses of the secondary (left) and of the primary (right) used in simulation (100 000 dots). The dotted and dashed lines represent the distributions of the primary masses of 60 stars of the sample respectively with and without applying the Lutz-Kelker correction (1973). b) Observed and simulated cumulative distribution of the mass functions (including the He-weak and He-strong stars).

In order to check the simulated distribution of masses of primaries, we estimated directly the masses of 60 primaries in systems having Geneva and uvby photometric measurements, as well as Hipparcos parallaxes. The estimation was done by interpolation through the evolutionary tracks of Schaller et al. (1992), using the method described by North (1998). The mass estimate was complicated by the binary nature of the stars under study. In the case of SB2 systems, the V magnitude was increased by up to 0.75, depending on the luminosity ratio, and the adopted effective temperature was either the photometric value or a published spectroscopic value. In the case of SB1 systems, only a fixed, statistical correction $\Delta V=0.2$ was brought to the V magnitude, following North et al. (1997), which is valid for a magnitude difference of 1.7 between the components. The effective temperatures corresponding to the observed colours were increased by 3.5%, to take into account a cooler companion with the same magnitude difference (assuming both companions are on the main sequence). In general $T_{\rm eff}$ was obtained from Geneva photometry, using the reddening-free X and Y parameters for stars hotter than about 9700 K and the dereddened (B2-G) index for cooler stars. The recipe used is described by North (1998). Existing $uvby\beta$ photometry was used to check the validity of the $T_{\rm eff}$ estimates and the E(b-y) colour excess was used to estimate roughly the visual absorption through the relation $A_{\rm V}=4.3 E(b-y)$. The latter may be underestimated in some cases because the b-y index of Ap stars is bluer than that of normal stars with same effective temperature, but only exceptionally by more than about 0.1 mag. Reddening maps by Lucke (1978) where used for consistency checks.

The resulting distribution of masses is represented, with an appropriate scaling, in Fig. 9a, together with the simulated distribution of primary masses. The agreement is reasonably good, justifying the expression used above for the distribution of primary masses.

For the distribution of secondary masses, we use the distribution of Duquennoy & Mayor (1991) for the mass ratio $q=\frac{{\cal M}_{2}}{{\cal M}_{1}}$ of nearby G-dwarfs. We assume that the companions of the Ap stars are normal. The distribution of the mass ratio is a Gaussian:

$$% \xi(q) \propto {\rm e}^{-\frac{(q-\mu)^2}{2\sigma_{\rm q}^2}}$$ (10)

with $\mu = 0.23$ and $\sigma_{\rm q} = 0.42$. Knowing the mass of the primary and the mass ratio, we can determine the mass of the secondary. We assume a minimal companion mass of $0.08~{\cal M}_\odot$. Using the test of Kolmogorov-Smirnov (Breiman 1973), we find a confidence-level for the observed and simulated cumulative distributions of 92%. After elimination of the He-strong and He-weak stars, the KS test gives a confidence-level of 98%. In other words, there is only a 2% probability that the observed distribution differs from the simulated one. This indicates that spectroscopic binaries with an Ap primary have the same distribution of mass ratios as binaries with normal components (Duquennoy & Mayor 1991).

5.4 Percentage of cool Ap stars as members of spectroscopic binary systems

The sample used for estimating the binary percentage among cool Ap stars is only composed of the CORAVEL programme Ap stars described in the observation section, namely 119 stars. Including the stars HD 59435, HD 81009 and HD 137909 which have been published elsewhere and are not in Tables 1 and 2, the total number of programme stars is 122. However, 6 of them have only one measurement, so they are not relevant here; in addition, about 3 stars have very large errors on their RV values ( $\sigma > 3$ km s^-1), so they are not reliable. Thus there are 113 objects on which statistics can be done. We chose, as variability criterion, the probability $P(\chi^2)$ that the variations of velocity are only due to the internal dispersion. A star will be considered as double or intrinsically variable if $P(\chi^2)$ is less than 0.01 (Duquennoy & Mayor 1991). However, this test can not say anything about the nature of the variability.

For fast rotators, it is difficult to know whether the observed dispersion is just due to spots or betrays an orbital motion. Among the sample, 34 stars are assumed to be binaries, namely 30%. Nevertheless, this figure has to be corrected for detection biases; we have attempted to estimate the rate of detection through a simulation. For this, a sample of 1000 double stars was created. The mass distributions explained above are used again. The orbital elements $T_{\rm o}$, $\omega$ and i are selected from uniform distributions, while the eccentricity is fixed to zero for period less than 8 days and is distributed following a Gaussian with a mean equal to 0.31 and $\sigma=0.04$ (cases with negative eccentricity were dropped and replaced) for periods less than 1000 days and larger than 8 days, and following a distribution f(e) = 2 e for longer periods. The period is distributed according to a Gaussian distribution with a mean equal to $\overline{\log~(P)}=4.8$ and $\sigma_{\overline{\log~(P)}} = 2.3$(Duquennoy & Mayor 1991), where P is given in days. A cutoff at 2 and 5000 days was imposed. In a second step, the radial velocities of the created sample are computed at the epochs of observation of the real programme stars and with the real errors. Finally the $P(\chi^2)$ value of each star is computed. We obtain a simulated detection rate of 69%.

After correction, we find a rate of binaries among Ap stars of about 43% in excellent agreement with the one determined by GFH85 for the cool Ap stars of 44%. Our sample is however more homogeneous and reliable.