2. The sample
2.1. Spectroscopic surface gravities
The sample has been built from two parts. First was considered the list of silicon stars for which North & Kroll (1989, hereafter NK89) give a spectroscopic estimate of based on the profile of the line. The estimate given in column 5 of their Table 1 was adopted, and corrected for a constant shift
which takes into account the systematic error displayed in their Fig. 16, although not exactly according to their Eq. 16 which would imply too large values for some stars. The intersection of this list with all stars having a known rotational period in the literature was then done, using an updated version of the database of Renson et al. (1991) which is a digital version of the catalogue of Renson (1991). Two stars in the list of NK89 which had no known period have been added, since their period has now been determined thanks to the Hipparcos mission (Perryman et al. 1997): they are HD 154856 ( days) and HD 161841 ( days). The original sample of NK89 was biased in favour of low photometric gravities, hence also of low spectroscopic (and hopefully real) gravities, since there is a loose correlation between them. Most stars of this sample are not very bright (), nor closeby enough that their parallax is significant, even for Hipparcos.
This sample contains 40 stars.
2.2. Hipparcos surface gravities
Second, the list of Si, SiCr or Cr stars with a Hipparcos parallax larger than 7 mas was defined and those with a known rotational period were retained. One star had no period in the literature but has a new one from Hipparcos (HD 74067, days). Their mass has been interpolated in theoretical evolutionary tracks (Schaller et al. 1992) from and . The effective temperature has been computed using the X and Y parameters of Geneva photometry calibrated by Künzli et al. (1997), and corrected according to the formula (Hauck & Künzli 1996) which replaces Eq. 1 of Hauck & North (1993) and where results from the calibration. The bolometric correction was interpolated in Table 6 of Lanz (1984) and corrected by plotted in his Fig. 4a. Contrary to the previous sample, this one is not biased regarding the distribution of the surface gravities, at least not a priori: it is a volume-limited sample which, although surely affected by a Malmquist-like bias, should be representative of field stars with a more or less uniform distribution of ages. Therefore, it contains a majority of stars which are rather close to the ZAMS in the HR diagram, just because stellar evolution is slower there than near the core-hydrogen exhaustion phase. If there is no a priori bias regarding the evolutionary state, one may say, nevertheless, that the distribution is biased towards high values, compared to a uniform distribution (which, then, would be strongly biased towards large ages).
This sample contains 56 stars.
2.2.1. Lutz-Kelker correction
The absolute magnitudes have been corrected for the Lutz-Kelker (1973) correction, but this correction was not applied in its original form which assumes a constant stellar density. Indeed, the distances involved are not negligible compared with the density scale height perpendicular to the galactic disk, so the following generalized formulae were adopted:
Let us recall that the correction on the absolute magnitude then reads:
where , is the scale height of the star density above the galactic plane tabulated by Allen (1976), is the true parallax and is the observed parallax affected by a gaussian error .
2.2.2. Visual absorption
The absolute magnitude also had to be corrected for the visual absorption, even though it remains negligible in most cases. Since Cramer (1982) found that the colour excess defined in the Geneva system was almost the same for Bp, Ap members of clusters as for normal B, A members - and with a smaller dispersion than - this colour excess was used, corrected using Cramer's relation where is the colour excess obtained using the intrinsic colours (of normal stars) of Cramer (1982). is then obtained through , since (Cramer 1994) and (Cramer 1984), and (Olson 1975).
2.2.3. Fundamental parameters from Hipparcos parallaxes
Once the effective temperature and bolometric correction are determined from photometry and the absolute magnitude from Hipparcos parallaxes as described above, it becomes possible to pinpoint the star on a theoretical HR diagram, the luminosity being obtained from
Then, we assume that Bp stars follow standard, solar-composition evolutionary tracks (since the chemical peculiarities are limited to superficial layers only) and the mass can be interpolated from and (using successive 3rd-degree splines in luminosity, and overall metallicity Z, with ) whenever there is a one-to-one relation between these quantities. The latter condition is not fulfilled near the core-hydrogen exhaustion phase, when increases, then decreases again, and in this domain we always assumed the star to lie on the lower, continuous branch of the evolutionary track, which also corresponds to the slowest evolution, hence to the higher probability. This assumption, if violated, will lead to a mass overestimate no larger than five percent.
The radius is directly obtained from
and the surface gravity from
The latter equation shows how strongly depends on , which remains a crucial quantity. The error on it was generally assumed to be 5 percent. The errors on the other quantities are estimated using the usual, linearized propagation formulae, but caring for the correlations between L, and M.
The results are displayed on Table 1.
Table 1. Fundamental parameters of the Si and He-weak stars derived from the Hipparcos parallaxes. The masses were obtained by interpolation in the evolutionary tracks of Schaller et al. (1992). Note that the errors are multiplied by a factor of 1000 for , 100 for Mass, and , and 10 for R. The rotational period from the literature (or from Hipparcos photometry in three cases, see text) is given in the last column. "LK" means "Lutz-Kelker correction" and is expressed in magnitudes.
2.3. Comparison between different sources of
A comparison between photometric and spectroscopic values was already shown by NK89. Fig. 1 shows how photometric and Hipparcos values compare, for Si and HgMn stars lying closer than 100 pc to the Sun. The diagrams look exactly the same as in the comparison of photometric vs. spectroscopic values, i.e. the Si stars are strongly scattered ( dex) while the HgMn stars follow the one-to-one relation much more closely ( dex), with the exception of HD 129174, a visual double which was excluded from the fit.
The nice behaviour of the HgMn stars in this diagram inspires confidence in the value of Hipparcos gravities.
The comparison between spectroscopic and Hipparcos gravities for Si stars is shown in Fig. 2, where all stars of the list of NK89 having Hipparcos parallaxes with are plotted (please note that some of them do not appear in Table 1 because they have mas). Unfortunately, only six objects fulfil this criterion; among them, four are on the equality line within the errors (at least within ), while two are clearly below.
The two outsiders are HD 147010 and HD 199728. Interestingly, these stars have the largest photometric amplitude, as shown in Table 2 where the peak-to-peak amplitude in Strömgren's u band (or Geneva band) is given with its source. This suggests that photometry overestimates in cases of extreme peculiarities 1, and is quite coherent with the fact that, in Table 1, some stars have values (determined from Hipparcos luminosities) around 4.5, which is about 0.2 dex more than the theoretical ZAMS value. This is probably due to an overestimate of their effective temperature. It seems that those Ap stars having a more or less fundamental value have on average less extreme peculiarities than those having a good rotational period (hence a large photometric amplitude) and considered here, so that the photometric calibration tends to overestimate for some of the latter. Nevertheless, no systematic correction will be made on in this sample, because the bias strongly depends on the individual stars.
Table 2. Peak-to-peak amplitudes of the 6 stars having both a spectroscopic and a Hipparcos value.
2.4. Hipparcos radii versus
In order to test the validity of the radii obtained using Hipparcos parallaxes, a comparison between the observed projected rotational velocities and equatorial velocities obtained from the formula of the oblique rotator model
Most stars fall below the equality line, as expected from ; therefore, the test appears rather successful, statistically speaking. However, seven of them are above, at least two of which simply because of the uncertainty on the determination (HD 126515 and HD 187474, with ). HD 199728 is only slightly above, but this may well be due to an underestimate of its radius linked with an overestimate of its effective temperature (see Subsect. 2.3 and Fig. 2 above). This is also the case of HD 24155, HD 142884 and HD 221006 (which have , 4.67 and 4.51 respectively, suggesting an overestimated radius), although the radius of the latter star would need to be strongly underestimated. The star HD 14392 (and possibly HD 221006 too) lies so high above the equality line that its rotational period may be questioned. Indeed, Pyper & Adelman (1985) proposed a period of 1.3102 days (following Winzer 1974), instead of 4.189 days proposed later by e.g. Adelman & Knox (1994). The photometric curves of HD 14392 are so scattered that the shorter period may be the right one after all; magnetic and spectroscopic observations should be done to settle the matter. The rotational period of HD 221006 has been found to lie around 2.31 days by Renson (1978) and this was confirmed by Manfroid & Mathys (1985) and by Leone et al. (1995). There seems to be no reason to question this value; therefore, we are left with two possibilities: either km s-1 (Uesugi & Fukuda 1970) is overestimated, or the radius is underestimated by more than 30 percent. This appears doubtful, since K has been estimated in a quasi-fundamental way (with the IR Flux Method) by Mégessier (1988) and is only 370 K lower than our photometric estimate: such a difference does not imply an increase of R by more than 3 percent.
Finally, HD 142884 has a reliable period and its radius must be underestimated by about 20 percent, as suggested by its very large (4.67). An independent estimate of its would be extremely welcome.
© European Southern Observatory (ESO) 1998
Online publication: May 12, 1998