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

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4. z -Probability and z -distribution

4.1. z -Probability in the orbit

We have investigated the probability with which one may find a given star at a particular z -distance. For that we calculated the orbits with equal time steps of 1 Myr. The statistics of these z -values gives the frequency [FORMULA] to find the star at a given z. Examples of such histograms are shown in Fig. 3.

[FIGURE] Fig. 3. Sample histograms showing the frequency [FORMULA] with which a star is found at a given height in z when considering fixed time intervals in the orbits calculated over 1 Gyr. The histogram intervals are z = 100 pc. The orbits themselves are included in Fig. 1. The present location of each star is indicated with [FORMULA] next to the z -axis

The frequency [FORMULA] has a relative minimum near [FORMULA] kpc while their maxima are away from the disk. This is to be expected. All stars will spend more time in the parts of the orbit away from the centre of the gravitational potential (where the star is far from the disk and the galactic centre) with a relatively small space velocity, compared to the larger z -velocity when the star is nearer to the centre of the gravitational potential. This means that on average a star spends more time away from the disk than 'in' the disk. However, selection effects may play a role (see Sect. 4.3.1).

4.2. z -Distribution from the entire sample

Working with a large sample of stars, we can add the histograms together resulting in the overall probability for such stars. This statistical distribution in z is shown in Fig. 4. In fact, the histogram shows the average spatial distribution of such stars, based on their actual kinematic behaviour.

[FIGURE] Fig. 4. Plot of the relative frequency [FORMULA], based on all stars of our sample, to find a star at a given z -distance in the Milky Way (histogram bins are 100 pc wide). Note that for [FORMULA] kpc the histogram is based on rapidly decreasing numbers of stars and the details of its shape there are thus not of significance. Panel a Overall distribution (linear scales). Panel b Enlarged plot (linear scales) of the frequency out to 1 kpc; note the rise and fall of N within this range. Panel c Logarithmic plot of ln [FORMULA] vs. z, showing that the distribution is consistent with an exponential one with a scale height of [FORMULA] kpc

The overall z -distribution for the stars is rather smooth. Fitting an exponential to the data one finds a scale height [FORMULA] = 0.97 kpc, based on the combination of the [FORMULA] and the [FORMULA] side of the histogram. The one-sided values are [FORMULA] =0.85 kpc and [FORMULA] =1.05 kpc, suggesting that the uncertainty in the derived overall scale height is of the order of 0.10 kpc.

The histogram is not identical to an exponential distribution in z (Fig. 4c). However, assuming an exponential and determining the slope of ln [FORMULA] allows to characterize the distribution with one number, which is of great help for the tests and comparisons to be performed.

One has to note that toward high z the distribution becomes biased to the very few stars reaching that far in z. In fact, overlooking all orbits, no star in the present sample reaches further than 6 kpc, while just 3 stars reach distances up to 3 to 6 kpc. The statistics in this z -range is therefore one of small numbers and cannot reliably be used for further numerical analysis. At the same time we concluded in Sect. 3.4, that the overall sample contains just very few stars (if any) stars going to Population II like z.

At small z the distribution found (Fig. 4 panel b) clearly differs from an exponential space distribution. The overall [FORMULA] shows a relative minimum near z = 0 kpc, i.e., it is more likely to find such stars at some distance away from the disk than in the disk. This finding has consequences for the concept of scale-height fitting from 'statistically complete samples' in a given direction, as used in several investigations. Looking back to studies of sdB star scale heights from stellar distances, it is clear that the distribution has to be sampled to well beyond 1 kpc in z to avoid problems with the relative minimum in the real spatial distribution.

We have calculated the orbits over 1 Gyr, although the phase life of sdB stars is about 108 yr. Doing the statistics for just that part of the orbit results in a scale height of 0.98 kpc, basically the same as our main value. Having used 1 Gyr apparently does not influence the scale height.

The scale height derived from [FORMULA] is based on all positions in all orbits. However, the orbits cover large portions of the galactic plane. As noted in Sect. 3.3, the change in galactic potential with projected galactocentric distance [FORMULA] leads to a 'thickening' of the orbit. We therefore have redone the statistics in three intervals of [FORMULA]. For [FORMULA] [FORMULA] [FORMULA] kpc we find [FORMULA] kpc, for [FORMULA] [FORMULA] [FORMULA] kpc (the solar vicinity) we find [FORMULA] kpc, for [FORMULA] [FORMULA] [FORMULA] kpc we find [FORMULA] kpc. We conclude that the value from the full sample using all parts of the orbits is somewhat biased toward the large [FORMULA] portions. The value for the scale height of the sdB stars in the solar vicinity is therefore [FORMULA] kpc.

Before rushing to conclusions we will test in the next subsection the robustness of the result against variations in the input parameters. It will be shown that small adjustments in the final value of the scale height are needed.

4.3. Discussion of sources of error

4.3.1. Selection effects?

One may be concerned that selection effects have played a role in arriving at our results. Let us consider the ways in which the sample came together.

First, for all stars distances and radial velocities must be available. The selection of the stars (from the PG) for the investigations of Papers II..VII was essentially random on the sky so that no preference for any direction in the galaxy is present. However, our data taking started generally with the brighter stars. It means that the stars are on average relatively close by. But, for each star this proximity is only at the present epoch and we therefore sample these individuals by chance (see Fig. 5).

On the other hand, for the PG one has not attempted to survey the low galactic latitude portion of the sky (since the PG was aimed at finding quasars) and it does not cover the southern sky. In all, the sample therefore lacks stars in some directions, as visible in Fig 5. It may therefore be possible, that our sample underrepresents stars with orbits staying always very close to the disk (like that of the Sun; see Fig. 1). Adding such stars might fill in the relative minimum in [FORMULA] at z =0 kpc.

[FIGURE] Fig. 5. Location of the stars in space, projected onto the meridional plane (z, [FORMULA]). Panel a The present location is shown. The absence of stars in the general direction of the plane of the Milky Way is apparent (limits of the PG catalog). Panel b Location of the stars 3 [FORMULA] yr ago (being half the time of the sdB evolutionary phase), showing that the stars observed near the Sun came from a large variety of positions in the galaxy

Secondly, good proper motions can be determined for stars which have ample first epoch data. Since fields of the old plates are normally not defined in terms of galactic coordinates but based on the equatorial system, the first epoch aspect does not introduce a galactic bias. This is, e.g., true for the stars lying in the Bordeaux Zone of the Carte du Ciel. Yet, the low limiting magnitude of available first epoch plates biases our sample to the brighter and thus nearer ones. Especially, the proper motions of the objects in the list of T+97 are based on the Astrographic Catalogue which is limited to stars brighter than [FORMULA] mag. Therefore, the T+94 list has only the brightest sdB stars. On the other hand, the proper motions determined using the POSS as first epoch data (Sect. 2.1) pose in principle not a significant limit in terms of brightness. The sample of the Lick stars is not defined in magnitude range, since K+87 selected stars of all magnitudes being of astrophysical relevance at that time, even as faint as [FORMULA] mag.

4.3.2. Robustness of the scale height value

In order to verify that our results do not depend in a critical way on the input parameters used, we have experimented with the input data for the orbit calculation. As indicated in Sec. 4.2, we will assume that the histogram [FORMULA] can be represented by an exponential, for easy comparison. We made 3 kinds of experiments.

[FORMULA] Dividing the sample in two parts

For the half sample with stars now at [FORMULA] the scale height from the total orbits was 1.30 kpc whereas for the stars now being in the interior galactic half the scale height came out at 0.72 kpc. The average is again [FORMULA] kpc, and the difference reflects the difference in scale height in relation with [FORMULA] [FORMULA].

For the half sample with stars now at [FORMULA] the full orbits gave the scale height 1.0 kpc, the stars now being at [FORMULA] led to a scale height of 1.13 kpc.

[FORMULA] Variation of radial velocity and proper motion

We added 30 km s-1 to all radial velocities (being the observational uncertainty), repeated the orbit calculation, and made the z -distribution statistics. In a second attempt we reduced all radial velocities by 30 km s-1. We found from these experiments the values [FORMULA] and [FORMULA] kpc. Both values are larger than our original one, suggesting that the added error makes the average result less reliable. The actual [FORMULA] gives the smallest scale height.

We also added 5 mas/yr both in [FORMULA] and [FORMULA] and redid our calculations. We now found [FORMULA] 1.04 kpc, within the uncertainty range of our original value.

We conclude that, given the size of the star sample, random errors in the input radial velocities and proper motions do not affect the value of the scale height in an essential way.

As a last test here we calculated the histogram based only on the orbits of the 21 stars which have absolute proper motions (from Table 2 and from K+87). In this case h comes out at 0.84 kpc and the remainder of the star orbits lead to a scale height well above 1 kpc. Possibly the C+94 and T+97 proper motions may be affected by additional systematic errors which lead to an increase in the z -distance in the orbits. This effect is similar to that of changing the radial velocities.

[FORMULA] Different distance scale

One of the input parameters is the stellar distance. Distance values have uncertainties of the order of 30%. We have not varied the input distances but tested the effects of distance errors in the following manner.

In the research on sdB stars a systematic difference exists between values of [FORMULA] and log g derived by some groups (Saffer et al.  1994) and by our group (Papers II,IV,VII). This difference leads to different distances for the stars (factors of 1.5 smaller distances from S+94 are not uncommon). In our orbit sample we have included stars investigated by both groups. We therefore divided our sample in two, one part using our distances and the other part using distances derived by S+94. For both groups the orbits were calculated and the z -distribution was determined. For the 32 stars from our data we find [FORMULA] = 1.07 kpc, while for the 17 stars with Saffer et al.  distances (there is some overlap) we find a scale height of 0.76 kpc. Changing the distance in a systematic way does make a difference (it changes also the tangential velocities).

[FORMULA] Different gravitational potential

The value of the scale height found is, of course, also a function of the nature of the potential model for the galaxy. If a smaller surface mass density is assumed, the vertical force will be smaller and consequently the scale height larger (see Allen & Santillan 1991). For the present study we will not explore these possibilities further.

4.4. Final scale height value and discussion

Overlooking all the tests, we conclude to the following for the z -distribution of the sdB stars. The all-orbit [FORMULA] is consistent with an exponential distribution with scale height [FORMULA] = 0.97 kpc. This scale height turned out to be biased somewhat to the z -values of stars reaching to large [FORMULA], because [FORMULA] in just the solar vicinity indicates that the base value is to be reduced to [FORMULA] =0.88 kpc. Subsets of the sample gave essentially the same scale height as the base value of 0.97 kpc, with the just noted exception of the division in inner galactic and outer galactic stars. Variations in the input velocities did not produce dramatic changes in the base value. Changing the distance of the stars in a systematic manner (tested by using the distances from Papers II,IV,VII versus those from S+94) lead to a difference in scale height of a factor 1.4. Taking the S+94 stars out of our data set means increasing the base value from 0.97 to 1.07 kpc

Combining these results from the tests we conclude that the sdB stars scale height [FORMULA] is best represented with the value of [FORMULA] kpc.

The scale height derived from our orbit data is much larger than the [FORMULA] pc derived for the sdB stars by Heber (1986) and in Paper II and IV. Clearly, such studies do not sample sdB stars to large enough distances, or they undersample the number of stars close by. It is well known that the value of a scale height is largely determined by the extreme points in such a distribution and both ends have a large risk of being unreliable given the small number of stars in those extreme bins. Also, the relative maximum near [FORMULA] pc in [FORMULA] of Fig. 4 makes clear that samples over limited distances can by definition not lead to a good characterisation of the true z -distribution.

In an analysis of the spatial distribution of sdB stars Villeneuve et al.  (1995) derived the stellar temperatures from photometry, the garavity by using a fixed relation between [FORMULA] and log g from Greenstein & Sargent (1974), and thus could calculate distances. Since they used just photometry, instead of going through a more detailed spectroscopic analysis, substantially larger distances could be reached. Villeneuve et al.  (1995) then found indications for scale heights ranging from 500 to 900 pc. That range tends more to what we have derived based on the orbit statistics.

The calculated mean asymmetric drift of -36 km s-1 points, together with the scale height of about 1 kpc, to a population of thick disc stars. These parameters are in remarkable agreement with the studies of Ojha et al. (1994) and Soubiran (1993). Those studies are based on investigations in limited fields (5o [FORMULA] 5o), whereas our study used a comparatively small number of objects but distributed over a large area of the sky.

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© European Southern Observatory (ESO) 1997

Online publication: April 6, 1998