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Astron. Astrophys. 327, 577-586 (1997)

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3. Orbits, populations

3.1. The mass model for the galaxy

In order to calculate orbits a model for the gravitational potential of the Milky Way has to be adopted. We have based our study on the model potential by Allen & Santillan (1991). This model was particularly developed for use in an orbit program and has been applied to the orbits of nearby stars, metal poor stars, as well as globular cluster orbits (Allen & Santillan 1991, 1993, Scholz et al. 1996, Schuster & Allen 1997).

An alternative model of the kind which is suitable for numerical orbit integrations would, for instance, be the one of Dauphole & Colin (1995). However, previous investigations have shown that the results obtained with these different models agree as long as the orbits do not reach extreme distances from the galactic centre (Dauphole et al. 1996).

We used the Allen & Santillan model as implemented in an updated version of the computer program of Odenkirchen & Brosche (1992). In order to be consistent with the parameters of the model, our calculations of the stellar space velocities follow the current IAU values for the LSR [FORMULA] = 220 km s-1 and [FORMULA] kpc.

3.2. Calculated orbits

The observational data (Table 1) have been transformed into the positional ([FORMULA]) and velocity ([FORMULA]) coordinates in the galactic system (Table 3). Note that the X -axis points from the Sun toward the galactic centre with the zero point at the galactic centre.


Table 3. Stellar coordinates and orbital characteristics

The orbits were calculated over a total of 1 Gyr backwards. This time span is longer than the sdB evolutionary phase (assuming they are genuine horizontal-branch like stars), but we opted to use 1 Gyr to achieve good statistics for the stellar positions (see Sect. 4).

A selection of the orbits is shown in Fig. 1. Given are the meridional cuts, showing the motion of the star in projected galactocentric distance [FORMULA] and in distance to the plane of the Milky Way. The figure demonstrates the variety of orbits found.

[FIGURE] Fig. 1. For several stars the orbits are shown to demonstrate the variety in shape. The diagram shows the meridional cut, i.e., the plane through the rotation axis of the galaxy rotating along with the motion of the star. Plotted is the motion of the star in that plane in vertical distance z and galactocentric distance [FORMULA]. All orbits were calculated backward over 1 Gyr in steps of 1 Myr. For comparison we have added the orbit of the Sun

The orbits of the stars of our sample are rather well behaved. Most stars stay overall very close to the disk but 10 stars veer out to [FORMULA] kpc (PG 0918+029 reaches [FORMULA] kpc). Many of the stars have orbits reaching way in toward the centre of the Milky Way (7 stars to [FORMULA] kpc, the extreme is PG 0133+114 to [FORMULA] kpc). 8 stars move out to [FORMULA] kpc. A most notable result is that the orbits cover large portions of the galaxy (see also Fig. 5).

3.3. Disk and Halo orbits, populations

We have attempted to sort the stars according to 'halo'-like and 'disk'-like orbits. When trying to do so one has to consider the formation history of the stellar populations in our galaxy.

The stars of the globular clusters were among the first to be formed in the galaxy; they are called Population II stars. Relatively recently formed stars, like those of the open clusters, are part of the Population I. The very different morphology of the data point distribution in the respective colour-magnitude diagrams led Baade to the concept of these two populations. Star formation has most likely been a continuous process in our galaxy. This must mean that there is, in practice, a population continuum without sharp boundary between these populations, a fact exemplified in phrasings like old population I, old disk population, and the like.

In principle the sdB stars in the galaxy form a mix of stars of a large age range. Stars having started with about 2.5 M [FORMULA] evolve in about 0.3 Gyr to the horizontal-branch stage, whereas old stars having started as main-sequence star with [FORMULA] M [FORMULA] take some 12 to 15 Gyr to become the sdB star it is today. With a constant star formation rate in the galaxy we would expect that the sdB stars of today come from the past in proportion to the initial mass function. So we would expect that sdB star samples are dominated by the older ones.

The kinematics of the formation location may be reflected in the motion of today. Stars having formed in the halo still will have 'halo' orbits, while stars having formed in the galactic disk are expected to have orbits confined to the disk. However, dynamically interacting encounters with other stars during the full stellar life time will have had its effect on the kinematics too. For that reason one may expect that old stars of the disk have heated-up orbits.

We have calculated several parameters related to the orbits of the stars (see Table 3). [FORMULA], the velocity component parallel to the circular galactic rotation (cylindrical coordinates) shows a distribution with a maximum centered near the solar 220 km s-1 but with a rather large spread to [FORMULA] km s-1 (see Fig. 2a). For the angular momentum [FORMULA] the same holds, with a peak in the distribution at [FORMULA] kpc km s-1 in particular spreading to -700 kpc km s-1 (see Fig. 2b). Statistically one may expect a spread in each orthogonal velocity component of about 30 km s-1, based on (on average) errors in the radial velocity as well as in each component of the proper motion of about 30 km s-1. The observed spread in [FORMULA] (and [FORMULA]) is much larger than that and thus it is not due to noise in the input data. The stars with small [FORMULA] (small [FORMULA]) most likely are the older ones in the sample. Note that the peak [FORMULA] kpc km s-1 corresponds to [FORMULA] km s-1, less than the solar value.

Another parameter of relevance is the eccentricity of the orbit defined as [FORMULA], with [FORMULA] and [FORMULA] the apo- and perigalactic distances of the stars (as in Allen et al.  1991). The average for our stars is 0.24 but a notable number has [FORMULA] (note that the orbit of the Sun has [FORMULA] in the above definition). The orbits of stars with small [FORMULA] have, of course, large eccentricities (see Fig. 2b).

The orbits show the effect of diminishing gravitational potential with galactocentric distance: each star moves to larger z when at larger [FORMULA] than at smaller projected galactocentric distance. In order to properly quantify the maximum height the star reaches outside the disk we have defined the 'normalised z -extent', [FORMULA]. The value of this parameter for each orbit is given in Table 3. The average for our sample is 0.16. A large value of nze is an indication for a halo-like orbit.

The parameters ecc and nze are plotted together in Fig. 2c. In general one would expect that orbits very different from that of the Sun would be those of old stars. We therefore suspect that the stars with orbits with approximately either [FORMULA] kpc km s-1, or [FORMULA], or [FORMULA], in general be considered to be the older ones.

Can we identify individual stars as old ones based on their orbit? PG 0918+029 reaches to [FORMULA] kpc, the largest z -value in the sample, while PG 1519+640 has a very elongated orbit reaching [FORMULA] kpc. These two stars do not have, however, proper motions based on an extragalactic reference. PG 0212+148 reaches to [FORMULA] kpc (the z -extent of this orbit is much more limited than the one given by Colin et al., essentially due to a more accurate proper motion). PG 0142+148 has an orbit with [FORMULA] covering the range of [FORMULA] kpc. These orbits may be the halo like ones, but none of the stars of our sample exhibits clear halo orbit characteristics.

Finally, the average kinematic properties of our sample may also be of relevance for characterising the sdB star population. We have calculated the mean asymmetric drift which turns out to be [FORMULA] km s-1. The dispersion in the values of the kinematical parameters is [FORMULA] km s-1, [FORMULA] km s-1, and [FORMULA] km s-1. These values are in very good agreement with the kinematical properties of thick disc stars (Ojha et al.  1994, and references therein).

[FIGURE] Fig. 2. Several parameters of the stellar orbits are plotted (for a discussion see Sect. 3.3). Panel a shows the histogram of the values of the present velocity component [FORMULA]. Panel b shows the orbit parameters eccentricity, [FORMULA], and angular momentum, [FORMULA]. Note the large number of stars with highly eccentric orbits as well as the stars with Sun like orbit parameters (see Sect. 3.3). Panel c shows the values for the eccentricity, ecc, together with the normalised z extent, [FORMULA], or the halo-ness of the orbit. In all panels the value for the Sun is indicated as [FORMULA]

3.4. Discussion of the results

We conclude that our sample does not contain stars which can be uniquely identified as old and thus as Population II stars. Either the sample is still too small or over time all orbits have been modified to general disk like orbits. However, several stars have orbits indicative of larger age, identified in Fig.  2b and Fig. 2c as those whose orbital parameters are very different from the Suns orbit parameters.

The orbits found are generally well behaved and there are no orbits indicating chaotic behaviour. Schuster & Allen (1997) have investigated a large sample of metal poor high-velocity stars. These do show chaotic orbits, but Schuster & Allen conclude this shows up predominantly in stars whose orbits reach to galactocentric distances [FORMULA] [FORMULA] kpc. Our sdB stars stay all further out. The Schuster & Allen star orbits also reach to much larger z -values than those of our sdB stars.

The relatively large values of [FORMULA] (and of [FORMULA]) for our orbits suggest that, as a sample, the sdB stars move in the galaxy not too dissimilar from the LSR. In fact, the average properties of the orbits are quite different from those of the metal poor high-velocity stars studied by Allen et al.  (1991) and Schuster & Allen (1997). This difference may be an indication for a difference in origin. The sdB stars either are not very metal poor and not very old while the metal poor high-velocity stars are much older. However, if those metal poor stars are older than the sdB stars, one wonders where the horizontal-branch like stars emerging from such an old population have gone. Alternatively one could speculate about a completely different origin for the sdB stars (see e.g. Paper V). More stars have to be investigated to clarify these questions.

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© European Southern Observatory (ESO) 1997

Online publication: April 6, 1998