Vertical Structure of the Galactic Disk in the Solar Neighborhood
V.A.Marsakov and Yu.G.Shevelev
Astronomy Reports, Vol.39, No.5, pp.559-568
1995, Astron.Zh., V.72, p.630
Abstract
The vertical tructure of star subsytems of various ages and metallicities is studied on the basis of a complete sample of F-stars within 50 pc from the Sun. It was found that the Sun lies 9±2 pc above the plane of hte Galactic disk in the direction of North Galactic pole and that F0-F9 stars contribute 0.0035M
/pc3 to the density of the Galactic disk. We also developed a method of statistical reconstruction of the actual height distribution of various objects at the Solar circle, based on calculation of three-dimensional orbits of stars. We used this method to determine the scale height of the subsystem of F stars of the Galactic disk, Z0=160±10 pc. It was shown htat in the process of its evolution the disk subsystem as a whole flattens, hoever, at each instant of time the metal-rich stars ([Fe/H]
-0.13) concentrate much stronger toward the Galactic plane than the metal-poor stars. It is suggested that scale height of the metal-poor stars subsystem is due to the infall of the metall-poor intergalactic gas onto the disk.
1. INTRODUCTION
The parameters of the spatial structure of Galactic subsystems are usually found either from star counts at various distances or by counting the total number of stars within specified distances, with subsequent modelling of the vertical distribution by an exponential law. The main uncertainty in methods stems from the problem of accurate correction of the distances to distant objects for interstellar extinction. We considered it more secure to study the vertical tructure of the disk using a complete sample of stars of a specified spectral class within a limited volume in the Solar neighborhood with precisely measured distances and spatial velosities. The decision was determined by the fact that every star in the course of its orbital motion crosses the plane of the Galactic disk, which is not far from the Sun. Therefore having determined the precise position of the Galactic disk plane, it is possible to reconstruct the true height distribution for all Galactic subsystems at the Solar Galactocentric distance; these subsystems must be present in the Solar neighborhood in sufficient numbers to obtain statistically reliable results.
2. OBSERVATIONAL DATA
In this study, we used our semple of F2-G2 stars Marsakov & Shevelev (1995), for which matallicities, isochrone ages, photometric distances, and other parameters for about 5500 F stars within 80 pc from the Sun have been calculated based on homogeneous uvby data taken from catalog Hauck & Mermilliod (1985) and positions and proper motions taken from Oschenbein (1980). We also calculated the elements of Galactic orbits for about one third of the sample stars, for which we found published radial velocities. The resulting sample meets the main requirement; it is representative of the objects of the disk subsystem. The initial catalog Hauck & Mermilliod (1985), in fact, contains virtually all stars within the spectral type range in which we are interested in that are brighter than V
8.3m. According to the standard table from Crawford (1975), the latest stars in our sample with temperature index b-y=0.412 correspond to spectral type G2, with absolute magnitude MV=4.9m. This implies that the sample is complete to 
50 pc from the Sun. This is well illustrated in Fig. 1, which shows the distribution of stars by their observed distances. (Since we used the original sample from Hauck & Mermilliod (1985), we were able to the distribution up to 120 pc from the Sun rather than to 80 pc, as was limited in Marsakov & Shevelev (1995)due to the substantial interstellar extinction at these distance.) The solid curve on the histogram gives an approximation for the part at Robs<50 pc by a function of the form 
. We determined the coefficients 
and 
by a least squares linear regression method in the log n - log Robs coordinates. We found the power-law index =1.85±0.05, with correlation coefficient r=0.99. Hence, at the 3
confidence level, the relation obtained corresponds to a square law, which is expected in the case of a uniform distribution of stars within the volume studies. There is an evident increasing deficit of stars relative to the theoretical distribution at greater distance. (The dashed curve on the histogram shows the extrapolation of the relation derived up to 60 pc.) At still larger distances, in addition to the limited depth of the semple, interstellar extinction begin to play an important part, and the relative number of distant stars sharply decreases. Hence, our sample contains virtually all F stars within 50 pc from the Sun. It is also of importance that in the spectral range under study (F2-G2), there are both the oldest and the youngest Galactic disk stars Shevelev & Marsakov (1993), which makes it possible to compare the spatial and kinematic characteristics of populations of various ages.
 | Fig. 1. The distribution of F stars in our sample from the observed distances from the Sun. Solid curve gives a power-law approximation of the number of stars as a function of the distance within Robs=50 pc and the dashed line is its extrapolation up to 60 pc. The arrow indicates the upper boundary of the domain where the sample is complete. |
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3. THE POSITION OF THE GALACTIC DISK
Stars of different ages most likely have different concentration toward the Galactic plane. Since the space volume under study is very limited, we can use the observed distances only to separate out the flat test subsystem of stars. In our work Shevelev & Marsakov (1993)? we found that the smallest velocity dispersions are observed for the youngest stars. Therefore, we divided all stars of the sample into four subgroups: first into two nearly equal metallicity groups separated by [Fe/H]=-0.13 (which corresponds to the maximum of the metallicity distribution for disk F stars Shevelev & Marsakov (1993)), and then each of these groups into two age subgroups separated by t=3 billion years. We determined the age of all stars using the New Yale Isochrones Shevelev & Marsakov (1993). Since in our study we do not use specific individual star ages, we considered it possible to adopt formal isochrone ages for stars near the ZAMS, where isochrones are so close to each other that the error in the derived age becomes comparable (and even greater) than the age itself.
 | Fig. 2.The observed distributions of stars of different ages and metallisities in the Z coordinate relative to Sun: (a) young (t3 Gyr) metall-rich ([Fe/H]-0.13) stars; (b) young metal-poor ([Fe/H]<-0.13) stars; (c) old (t>3 Gyr) metal-rich stars, and (d) old metal-poor stars. Broken lines show five-point sliding-averadge smoothed trends within ±50 pc, and the dashed line shows an approximation of the histogram by a second-order polynomial over the interval from -50 to +30 pc. The arrows indicate the range over which the sample is complete. The double arrow indicates the position of the Galactic disk. |
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We determined photometric distances of stars using uvby and V data from Hauck & Mermilliod (1985) and Oschenbein (1980), respectively (see a more detaled discussion in Marsakov & Shevelev (1988)). The height distributions of stars above the Galactic plane Zobs were constructed within a cylinder of radius 40 pc, where the Z axis is directed toward the North Galactic pole. Figures 2a-2d show these distributions for the four groups of stars indicated above. The arrows on the histograms show the heigt range for which the observational selection effect discussed in Section 1 is completely absent. the smoothed trends within ±50 pc shown in the figures suggest that only the young metal-rich group shows clear structure, whereas all other groups demonstrate flat distributions (determination of the power-law index for a polynomial trend showed that a parabolic approximation can be rejected with high statistical significance in favor of a strainght line fit). It is clearly seen in the histogram for young metal-rich stars (Fig.2) that the maximum of the distribution is shifted relative to the Sun toward the South Galactic pole. To find the precise position of the maximum of the density distribution of stars, e fitted the distribution within the range from -50 to +50 pc by a parabola. We then refined the positiion of the density maximum by using a least squares technique to fit a parabola to the data within a narrower range of Zobs, symmetric with respect to the previously estimated value (dashed line in Fig.2). Our final result is Zmax=-9±2 pc. This is apparently the position of the Galactic disk plane. We show it by a double arrow in Fig.2a. Our final estimate agrees with the value quoted in Allen's handbook Allen's handbook (1977), but the uncertainty in our estimate is considerably less.(We estimated the error of Zmax by comparing th results obtained by fitting parabolas to the distributions within various Zobs ranges.)
 | TBLEVERTICAL STRUCTURE OF THE GALACTIC DISK. Velocity ellipsoid parameters, orbital elements, and number densities in the Galactic plane for four groups of F2-G2stars. |
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Our results confirmed the hypothesis that the group with the smallest velosity dispersion has the strongest concentration toward the Galactic plane. Consider now the kinematic characteristics of each group in more detail. The table gives velosity ellipsoid parameters calculated from tangential velocities (see Marsakov (1992) for a desription of the technique used), and the mean eccentricities of the Galactic orbits and maximum distances of the stars from the Galactic plane. We calculated the last two quantities for stars for which both tangential and radial velocity components are known; therefore the number of stars in each group is smaller than that given in the table. As will be demonstrated in Section 4, systematic variations of kinematic parameters with height above the Galactic disk are virtually absent in the Solar neighborhood, at least within 100 pc. Therefore, to obtain statistically reliable results, we used a sphere of radius 60 pc. It can be seen from the table that the ellipsoid semi-axes and orbital elements of the young metal-rich group differ markedly from those of other star groups, for which the parameters are nearly the same. Hence, the group of metal-rich stars, in addition to its stronger concentration toward the Galactic plane, demonstrates smaller dispersions of all spatial velocity components, a higher vertex inclination, a more elongated (farther from equilibrium) velocity ellipsoid, and more circular orbits with smaller oscillations about the plane of the disk.
4. THE DENSITY OF F STARS IN THE PLANE OF THE DISK
Since our sample is comlete to within 50, the Zobs distributions shown in Fig.2 make it possible to determine the number of F stars per unit volume of the Galactic disk. We used second-order aproximation polynomials to determine the number of stars in our groups within a cylinder of radius 40 pc with height equal to the width of the class interval at the Galactic plane (i.e., at distance Z=-9 pc from the Sun). We then divided the numbers obtained by the volume of this cylinder to derive the desired densities, which we give in the table for each group. The total density of F2-G2 stars (0.222(b-y)0.412) in the plane of the Galactic disk was n0=0.0040 pc-3. We then used a Salpeter mass function to extrapolate the distribution of sample stars using the temperature index (b-y) toward earlier spectral types to derive the density for the spectral range F0-F9 (0.180(b-y)0.370). We found that the ratio of the value of the new sample to that of the initial sample is 0.72, and therefore the density of F0-F9, stars in nF=0.0029 pc-3 and the corresponding mass density in the disk plane is 
F=0.0035M/ pc-3. This is a little higher than the result obtained by Kharadze et. al (1989) (where nF=0.002 pc-3) and the mass density presented in the handbook of Allen (1977) (F=0.003M pc-3). We consider this to be a result of underestimation of the number of F stars in earlier works due to interstellar extinction. However, we may not rule out the possibility that the Solar neighborhood has an enhanced local density of stars.
5. KINEMATIC AND SPATIAL INHOMOGENEITY IN THE NEAREST SOLAR NEIGHBORHOOD
In Fig.2a, a slight excess of stars at a height of 40-80 pc in the Northern hemisphere can be seen in the young metal-rich group. This effect is vertually absent in the other groups. The excess might have several origins. First, there could exist a real, very young clustering of genetically assoiated stars. In this case, this cluster should have a very small velocity dispersion, which could lead to the average kinematic parameters of stars at this height being different from the corresponding parameters of the remaining stars of the given group. Second, the excess could be due to weaker interstellar extinction in the Northern direction, since the Sun lies above the Galactic plane. In this case, the kinematics should not depend on Zobs. Third, the excess could be due to different quality of observations and survey depths in the Norhern and Southern hemisperes. In this case, there might appear a trend in the kinematic parameters calculated from photometric ditances and proper motions when crossing the Galactic plane.
 | fig. 3.Velocity allipcoid parameters as functions of heigt above the Galactic disk for the complete sample of F-stars (left) and a young (t3 Gyr) metal-rich ([Fe/H]-0.13) group (reght). The curves give 1, 2, and 3 - the major, mean, and minor semi-axes of the ellipsoids, respectively; 2/1 is the velocity ellipsoid semi-axis ratio; l is the Solar apex galactic longitude; L1 is the longitude component of the vertex deviation; v, u, and w are the components of the space velocity of the group centroids with respect to the Sun - the axes are directed toward the Galactic center (u>), in the direction of the Galactic rotation (v), and toward the North Galactic pole (w). |
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Thus, to elucidete the nature of the excess of young metal-rich stars in the Northern hemisphere, e no investigate the relation of the velocity ellipcoid parameters to the currently observed height of stars above the Galactic plane. To make our conclusions more reliable, we compare the behavior of the velocity ellipsoid parameters of the young metal-rich group and of the entire star sample. We first consider the entire sample. We first consider hte entire sample within 100 pc and select from it stars within a cylinder of radius 60 pc. We then subdivide this subsample into six Zobs intervals separated by -45, -25, 0, +20, and +45 pc. The Galactic plane corresponds to the interval -25
0 pc. The left-hand column of the diagrams in Fig.3 show how the main parameters of the velocity ellipsoids of the entire star sample calculated from tangential velocities depend on Zobs. It can be seen that these parameters depend only weakly on Zobs alhtough there is some systematic trend, which however, is entirely within the statistical errors of the respective parameters. These errors are manifest in the figures as rondom variations between adjacent points. THe velocity allipsoids of the young metal-rich group (the right-hand column) which we subdivided into four subgroups of Zobs separated by -40, -10, and +20 pc, behave in a similar way. Fig.3 allows us to draw the following conclusions:
- Since we do not observe any significant differences between the behavior of the velocity ellipsoids of the metal-rich group and the entire sample, we can suggest that there relly exists a clustering of young stars, it is not kinematically distinguishable in any way.
- The fact that parameter variations with Zobs are not symmetrical suggest that the internal kinematics of groups within ±100. pc does not change ith increasing distance from the Galactic plane, and we can therefore use any volume to obtain unbiased estimates of the kinematical parametrs for various stellar groups.
- Small trends of parameters for both the stars of the entire sample and those of the young metal-rich group suggest some difference in the uvby systems used for observations in the Northern and Southern hemispheres. Further studies are required to make a final conclusion on the nature of the excess of young metal-rich stars in the Northern hemisphere.
6. THE THICKNESS OF THE DISK SUBSYSTEM
Among the stars that are now located in the Galactic plane, there are representatives of all Galactic subsystems. (It is evident that most of these stars are very young stars with almost circular orbits.) After a certain tame, the stars of any subsystem will randomly redistribute on their orbits and occupy the same volume (in the Z-coordinate) as thzt occupied now at a given Galactocentric distance by all other stars belonging to that subsustem. We shoed in Shavelev & Marsakov (1995) that almost all disk stars have boxlike orbits - i.e., after a lardge number of revolutions, they completely fill toroidal volumes with nearly rectangular cross sections (Fig.4). Therefore, we can use our complete sample of F stars within 50 pc from the Sun to reconstruct the true Z-distribution of all disk F stars at the Solar Galactocentric distance. Recall that among F2-G2 stars there are representatives of both the youngest (lying on the ZAMS) and the oldest (with turnoff point near G2) disc stars Sychkov et.al. (1989)
 | Fig. 4.The trajectory of a typical disk star (the Sun) in cylindrical coordinates (, Z). RP is the perigalactic radius and Ra, the apogalactic radius. Zmax is the maximum deviation of the star from the Galactic plane. THe dotted curve shows the position of the Galactic plane. The hatched horizontal band shows schematically the band in Z, where the probability of that location of the star was calculated. The vertical band shows the width of the column in which we reconstruct the star distribution in Z. |
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The catalog Marsakov & Shevelev (1995) gives Galactic orbital elements for stars with measured radial velocities. These orbits were calculated using a Galaxy model Allen & Santillan (1991) consisting of a spherical buldge, disk, and massive extended halo. In this model, the mass distribution of subsystems is chosen in such a way that the resulting rotation curve agrees with the observed curve over the Galactocentric distance range from 1 to 20 kpc and remains almost flat from 20 to 100 kpc/ The Galactocentric distance of theSun is taken to be R=8.5 kpc and the circular rotation velocity at the Solar distance to be VR=220 km/s. Unfortunately, the sample of stars with known space velocities and orbital elements is not complete; however, it nonetheless remains representative. This is evidenced by the abswnce of any limitations on the radialvelocity measurements (except the limiting magnitude) in catalog Oschenbein (1980), as well as the fact that the difference of velocity allipsoid parameters of F stars calculated from tangential and radial velocity components is not significant (see Table 1 in Marsakov (1992)).
 | Fig. 5.Reconstruction of the Z distribution for F disk stars: (a) the distribution of the maximum deviation from the plane of the Galactic disk (Zmax) for orbits of stars in the Solar neighborhood that are a 60x60x60 pc3 cubic volume; (b) the weighted histogram (a) reflecting the distribution of Zmax for stars at the Solar Galactocentric distance in a column with a 60x60 pc2 square base; and (c), the final reconstructed distribution of Z for stars of the Galactic disk at the Solar Galactocentric distance. The solid curve at the bottom show an approximation of the distribution by an exponential law, and the numbers give the scale height and its error. |
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Thus, we wish to reconstruct the Z-distribution of all F stars currently within a cylinder perpendicular to the plane of the Galactic disk. We first reconstruct the distribution of the maximum deviation from the Galactic plane (Zmax) for orbits of stars in catalog Marsakov & Shevelev (1995) that are now within a cubic valume of edge length 60 pc centered on the Galactic plane at the Solar Galactocentric distance (shown in Fig.5a) The southern face of the cube is 40 pc and the northern face 20 pc from the Sun. (This shape of the valume, within which our sample is complete was selected in order to simplify the following calculations.) A stars moving along its orbit around the Galactic center, crosses the Galactic plane several times (Fig.4). The greater the vertical velocity component at Z=0, the greater the maximum deviation of the star's orbit from the plane of the Galactic disk, and the lower the probability of finding the star near the Galactic plane. Therefore, there is a selection effect in the histigram of Zmax, which is manifested as a deficit of stars with high Zmax. To allow for this selection effect and reconstruct the distribution of Zmax for all stars at the Solar Galactocentric distance, we assigned the i-th stars a weight equal to the proportion of the time spent in a cylinder of height Zmax(tcyl) to the time spent in the selected cube (tcube): pi=tcyl/tcube. The weight is equal to unity for stars with orbits lying entirely within the specified band (i.e., with |Z|max30 pc). Numerical simulations have shown that for stars with boxlike orbits (like the orbits of most disk stars Marsakov & Shevelev (1994)) pi is equal to t/t3 within 5%, where t is the total time of observation of the star and t3 is the time spent by the star within the -30 псZ+30 pc zone. Fig.4 shows the meridional selection of a stellar orbit (i.e., a section of the torus completely filled by the stellar orbit after a sufficiently large number of revolutions) and the cylinder in which we reconstruct the Z-distribution (the horizontal and vertical bands, respectively). The "weight" of each disk star within a vertical column 60x60xZmax i pc3 is pi. After performing the same operation for all stars of the sample and constructing a histogram of the weighted Zmax distribution, we derived the Zmax distribution for all dis stars that are now within the vertical column with the base specified above (without allowing for the fact that some stars lack VR measurements). When calculating the weights, we naturally assumed the Galactic disk to be quasi-static, and the time required for stars that are now in the Solar neighborhood to becom