Astron. Astrophys. 317, 689-693 (1997)

## 3. Estimating the sdB kinematics and population type

Normally, to derive the kinematics and hence population type of a group of stars from proper motion or radial velocity data, and for which the absolute magnitude is unknown or uncertain, statistical parallax methods are used. Statistical parallax usually relies on a good distribution of the stars around the sky, which is certainly not the case for these small samples. Below we introduce a Monte Carlo technique which circumvents this problem, by simulating what we would expect to observe in small samples of stars drawn from one of four representative populations in the Galaxy.

We begin by analysing the stars for which we have radial velocities, since the kinematic information is in this case independent of distance. Saffer (1991) has measured radial velocity (RV) data for 31 sdB, and for 11 of his sdB we have now measured proper motions. For the 31 sdB, the dispersion of the line-of-sight velocities is km s-1, while for the 11 sdB with proper motions as well, the dispersion is 9 km s-1. (We have corrected these velocity dispersions by subtracting quadratically the quoted measuring errors in the radial velocities).

Let us first assume that these stars belong to one of four representative populations in the Galaxy, either the young disk (age less than about a Gyr), the old disk, the thick disk or the halo (see Freeman 1989 for a review of the properties of these populations). The kinematic properties (velocity ellipsoid and asymmetric drift, ) of these four groups are well known, and are shown in Table 1. Consider now a small sample of N stars, drawn from one of these populations, at positions on the sky for ...N. We carry out a large number of Monte Carlo simulations in which N objects at these positions on the sky are drawn from one of the representative stellar populations above. A space motion directed along the usual Galactic coordinates ) is randomly selected for each of the N objects, from a Gaussian distribution with dispersions and mean from which line of sight velocities can be computed. The dispersion of the line of sight velocities is then calculated (for a large number of simulations to smooth out statistical variations) for each of the representative populations, and compared to this quantity for the real stars. Further, if the apparent magnitude for each star is known in the actual sample, then for a given assumed mean absolute magnitude for the stars a distance along each line of sight in the Monte Carlo simulations can be assigned, and hence the expected distribution of proper motions can be simulated as well, and compared to the observations.

Table 1. Model population velocity dispersions in km s-1. From Mihalas and Binney (1981).

Table 2 gives the results for 2 samples of sdB stars - one being for those 31 sdB stars that Saffer gives RV data for (2 stars have been excluded as their velocities are so great that they are clearly interlopers from the halo), and the other being those 11 sdB stars for which we also have proper motions (PG0250+189, PG0342+026, PG0749+658, PG0918+029, PG0919+272, PG1114+072, PG1224+671, PG1230+052, PG1325+101, PG1704+221, PG2204+034). From the table one can see that the observed line of sight velocity dispersion indicates that our sdB sample comes from a population which is midway between the old disk and the thick disk.

Table 2. Table of radial velocity dispersions (in km s-1) for two samples of sdB stars and the four model populations.

For 11 of the sdB we have RV and proper motions, so for these stars we can calculate the space velocities of each star for an assumed mean absolute magnitude of the sdB. If we again assume that the stars belong to one of the four population types above, then the expected distribution of space velocity can be simulated for our small sample, and compared to the actual observations. Of course, since the space velocities derived depend on the distance via the two observed proper motions but are independent of the observed radial velocity, we can recover an estimate of the sdB absolute magnitude in this way.

If we assume for simplicity no intrinsic scatter in the absolute magnitudes of the sdB, then the best match between the space velocities we derive for the 11 stars and the space velocities of a large number of 11-star samples drawn from the Monte-Carlo simulation is for with an error of . The sdB may actually have an intrinsic scatter in their absolute magnitudes, which means that using a mean absolute magnitude for each of the 11 stars in order to derive the space velocities will artificially increase the dispersion of their space velocities. We examine this in more detail in the next section.

The absolute V magnitudes of sdB have been established by Heber (1986) (), and a similar result () is derivable from the larger work by Saffer (1991) if Bolometric Corrections (from Kurucz (1993) atmospheric models) are used and absolute magnitudes calculated from the atmospheric parameters and the assumption of a 0.5 mass.

We find there is good agreement between the observed absolute magnitudes and the one derived above from kinematics alone - this is the first confirmation of its kind and lends support to the assumption of the 0.5 mass. This support was previously only present from the comparison of theoretical EHB evolution tracks to observed atmospheric parameters.

© European Southern Observatory (ESO) 1997

Online publication: July 8, 1998