A large program has been underway for some years now, to determine the space velocities of nearby stars. the author was kindly lent some data from this project by Andersen & Nordström (private communication). For all but 33 of the stars left in the sample there are velocity data and calculated orbits (described in Fux 1997). These data are based on the same absolute magnitudes as the results presented here.
With these data it is possible to make an in-depth study of the scale height weighting problem. As theoretical models usually concern a cylinder around the sun and the observed sample is confined to the solar neighbourhood, some correction must be made. This is usually done by weighting the sample according to the average velocity in each velocity bin to correct for the different scale hights. According to Mayor et al. (1977) the U and V velocities does not produce a noticeable effect (meaning that the radial metallicity gradient is low), thus only velocities are usually considered.
From Fig. 6 it is clear that most of the stars have velocities below . There are 10 stars with velocities above . These stars have between 1 and 2 kpc, and could be either halo stars or the end of the disk velocity distribution. They do not have exceptional metallicities. The remaining stars are approximately gaussian distributed. A fitted exponential distribution would predict no stars with velocities above , which suggest that this might be a way to separate the possible halo stars from the sample.
Another result evident from this figure is that many stars have very low velocities. This can cause a problem when using conventional corrections such as weighting the stars by their velocities; When one star has a velocity of e.g. and another has a velocity of e.g. , this will cause the second star to have six times the weight of the first star, while in reality none of them spend much, if any, time more than 40 pc from the galactic plane. As seen in Fig. 6 there is only a very general trend for to depend on [Fe/H]. This makes it somewhat unreliable to use average values, but they will at least give the general trend. Averaging the velocities in 0.1 [Fe/H] bins gives Fig. 7. In the figure a linear fit is shown (with a dashed line). Note though, that it suffers from the bad statistics in the low metallicity part with 2-4 stars per bin, as opposed to in the more metal rich part.
Another possible correction is to use the f method from Sommer-Larsen (1991). Using this method would ignore the velocity data, which is not optimal as this discards useful information.
Another possibility is to use the calculated maximum height above the plane. The distribution of is shown in Fig. 8. There seem to be an exponential decay (a rough guesstimate suggests a scale height of around 300 pc). The important feature of the plot is that the maximum height of a considerable part of the sample is comparable to the distance limit (40 pc). Thus is not a good indicator of the time the star spends inside our volume, as some stars inside the sample will be near their and will therefore have low velocities. These low velocities (e.g. ) will give extremely low weights with a standard correction method.
The correction is done by assuming that the stars move vertically in a harmonic oscillator potential. A harmonic oscillator potential is a fair approximation, as long as z is small compared to the scale height of the disk, and this is true for most of the sample (it is not the method used to calculate , but for these purposes it is acceptable). This method also assumes that the Sun is located in the Galactic plane. While a more complete analysis would remove this assumption is is used as it greatly simplifies the problem. With these assumptions it is then easy to find the proportion of time spent within 40 pc of the Galactic plane knowing the maximum distance that the star reaches. Each star is then weighted with the reciprocal time:
Eq. 1 is used, if the star reaches further than 40 pc away from the Galactic plane. If it does not, its weight is set to one. For the 33 stars where no velocity data were available, the correction from the average values of was used, normalized to the weights from the method described above.
Unfortunately this method is also prone to distortions from a few stars. e.g. some stars gain weights of , compared to weight for a star that does not move more than 40 pc from the Galactic plane. This is of the same order of magnitude as the sample size and can be very disruptive. This problem is compounded by the fact that the harmonic oscillator approximation breaks down at large distances from the plane. Another problem is that with the large scatter in velocities and the very few stars in the low metallicity bins, this effect will primarily affect the bins with many stars, as the bins with few stars will probably not have any extreme stars. To rectify this problem all stars with velocities above were removed.
The different metallicity distributions that result from the different corrections are shown in Fig. 9. The most interesting point of this plot is that the distribution below [Fe/H] does not change much, although the Sommer-Larsen correction is a bit above the others. Another interesting point is the position of the main "bulge". The bulge shifts position with changing correction method. The difference between the correction and the truncated correction indicates that the cut-off at does remove some of the weighting, but as the distributions are comparable at lower metallicities, the cut-off does not adversely affect the distribution. As the distribution with cut-off is comparable to the other corrections, it will be used.
© European Southern Observatory (ESO) 2000
Online publication: December 5, 2000