Subsystemsof RR Lyrae Variable Stars in Our Galaxy .
T. V. Borkova and V. A. Marsakov
Astronomy Reports, Vol. 46, No. 6, 2002, pp. 460-473.
Abstract
We have used published, high-accuracy, ground-based and satellite proper-motion measurements, a compilation of radial velocities, and photometric distances to compute the spatial velocities and Galactic orbital elements for 174 RR Lyrae (ab) variable stars in the solar neighborhood. The computed orbital elements and published heavy-element abundances are used to study relationships between the chemical, spatial, and kinematic chacteristics of nearby RR Lyrae variables. We observe abrupt changes of the spatial and kinematic characteristics at the metallicity [Fe/H] 
-0.95, and also when the residual spatial velocities relative to the LSR cross the critical value Vres 290 km/s. This provides evidence that the general population of RR Lyrae stars is not uniform, and includes at least three subsystems occupying different volumes in the Galaxy. Based on the agreement between typical parameters for corresponding subsystems of RR Lyrae stars and globular clusters, we conclude that metal-rich stars and globular clusters belong to a rapidly rotating and fairly flat, thick-disk subsystem with a large negative vertical metallicity gradient. Objects with larger metal deficiencies can, in turn, be subdivided into two populations, but using different criteria for stars and clusters. We suggest that field stars with velocities below the critical value and clusters with extremely blueh orizontal branchs form a spherical, slowly rotating subsystem of the proto-disk hlo, which has a common origin with the thick disk; this subsystemh has small but non-zero radial and vertical metallicity gradients. The dimensions of this subsystem, estimated from the apogalactic radii of orbits of field stars, are approximately the same. Field stars displaying more rapid motion and clusters with redderh orizontal branches constitute the spheroidal subsystem of the accreted outer halo, which is approximately a factor of three larger in size than the first two subsystems. It has no metallicity gradients; most of its stars have eccentric orbits, many display retrograde motion in the Galaxy, and their ages are comparatively low, supporting the hypothesis that the objects in this subsystemhad an extragalactic origin
1. INTRODUCTION
The complex, multicomponent structure of the Galactic halo is being confirmed by more and more studies. In particular, the halo contains the thick disk subsystem, with a relatively high abundance of havy elements, and two low-metallicity components: the proto-disk halo subsystem, which shares a common origin with the disk, and a subsystem that presumably was formed from extragalactic fragments captured by th Galaxy. The presence of two populations with different histories in the low-metallicity halo was suggested in Hartwick (1987), where it was shown that two components were needed to model the dynamics of RR Lyrae variables with metallicities [Fe/H] < -1.0 : a spherical and a somewhat flattened component, with the latter dominating at Galactocentric distances less than the radius of th solar circle. The idea that there are two populations in th metal-poor halo was primarily developed during studies of globular clusters. This is the only Galactic population that is observable without considerable selection effects as far away as the Galactic center, and even beyond it, making it possible to establish the spatial distributions of objects in the proposed subsystems with reasonable reliability. In addition, globular clusters possess a feature (the structure of their horizontal branchs) that can be used to distinguish between metal-poor clusters of different subsystems. Halo clusters with redder horizontal branches for a given metallicity (i.e., with horizontal branchs showing a considerable number of stars on the low-temperature side of the Schwarzschild gap) are predominantly outside the solar circle. In addition, they exhibit a larger velocity dispersion, slower orbital motion (a significant number having retrograde orbits), and are, on average, younger than clusters with very blue hrizontal branchs, which are concentrated within the solar circle. The explanation suggested for these differences was that the subsystem of older clusters formed together with the Galaxy as a whole, whereas the clusters of the youngh halo subsystem formed from fragments captured by the Galaxy from intergalactic space at later stages of its evolution Zinn (1993).
Metal-poor field stars show a similar multicomponent structure. In particular, a study of a large sample of stars in a deep survey in the direction of the North Galactic Pole demonstrated that stars further than 5 kpc from the plane of the disk tend to have retrograde orbits Majewski (1992). (The retrograde orbits clearly testify that the objects' origin was independent of that of the Galaxy, which exhibits prograde rotation.) Furthr evidence for the presence of objects with an extragalactic origin among field stars is the identification of objects with relatively young ages and low abundances of havy elements (so that they should have been older, according to abundance age indicators). The subsystem of accreted globular clusters was named the "young halo" for precisely this reason. On average, high-velocity field subdwarfs with highly eccentric orbits (e > 0.85) are younger than subdwarfs with similar metallicities but less eccentric orbits Carney (1996). The isochone ages of subdwarfs were derived in Carney (1996) based on Stromgren phtometry. Hanson et al. Hanson (1998), who also used abundances of [a/Fe]-elements as age indicators, likewise concluded that metal-poor red giants in retrograde orbits were relatively young. ([a/Fe] is known to be low for young objects formed from matter already enrichd in the ejecta of type Ia supernovae, whreas the higher relative abundances of a-elements in older stars are due to type II supernovae.) Unfortunately, no age indicators for individual RR Lyrae stars are available. However, observations of field RR Lyrae stars and blue hrizontal-branch stars were used in Laydan (1998) to estimate the number ratios of these objects in different directions from the Sun. Blue horizontal-branch giants dominate among stars close to the Galactic center and Galactic plane, whereas the numbers of variable stars and giants were approximately the same at greater distances. These stars have similar metallicities, suggesting that the inner, bluer population of the Galaxy is older than the outer population.
The recent papers Chiba, Yoshii (1998), Martin, Morrison (1998) present detailed studies of the kinematics of RR Lyrae stars in the solar neighborhod. The stars were assumed to form only two subsystems - a thick disk and a halo. In the current study, we attempt to identify "accreted hlo" stars among these RR Lyrae variables. We use the spatial velocities and computed elements of Galactic orbits as our main criteria for distinguishing the subsystems (taking into account the nearby location of the studied variables). We wish to investigate relationships between the physical, chemical, spatial, and kinematic chracteristics of RR Lyrae variables in each of the subsystems, determine the characteristic parameters of the subsystems, and compare them with the parameters of similar subsystems of globular clusters.
2. SOURCE DATA
The input catalog for our study was Laydan (1994), which presents metallicities for 302 RR Lyrae (ab) variables known up until 1985 and brigh ter th an apparent magnitude 13.5. In Laydan (1994),the abundances of heavy elements were derived from the ratio of the intensities of the CaII K line and Hd and Hb Balmer lines (an analog of th 
S index of Preston) and reduced to the [Fe/H] scale of Zinn & West (1984). Th typical uncertainty of an individual [Fe/H] value is 0.15 - 0.20 dex. We added several brigh stars that are missing from Laydan (1994), for which metallicities were computed in Laydan et. al (1996) from publishd S values. Chcks presented in Lambert et al. (1996) show that the metallicities of Laydan (1994) are in very good agreement with [Fe/H] values derived from Fe II lines using high-resolution spectra with high signal-to-noise ratios. Having in mind the large relative errors of the trigonometric parallaxes for distant objects, we chose to use the photometric distance scale of Laydan (1994) (assuming MV(RR)=0.15[Fe/H]+1.01). The radial velocities were also taken from Laydan (1994). Our final list contains 317 stars.
We created anothr sample for an analysis of the thee-dimensional motion of these stars as a function of metallicity, containing 124 stars in the solar neighborhod brigher than 12.5m from Chiba & Yoshii (1998), with proper motions taken from the Hipparcos catalog. This sample includes virtually all brigh RR Lyrae variables known prior to 1976 (such data were included in the satellite's input catalog). We supplemented our list with stars from Martin & Morrison (1998), which presents data for all nearby RR Lyrae stars of the northern sky brigher than 11m known in 1997 (a total of 130). The proper motions in Martin & Morrison (1998) are averages from seven sources with high accuracy ground-based observations, as well as from the Hipparcos catalog. Fifty stars from Martin & Morrison (1998) entered our sample.
 | TABLE 1 Orbital elements for nearby RR Lyrae variables |
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Foreach star, we computed the spatial velocity components in cylindrical coordinates and orbital elements using the Galactic model of Allen & Santillan (1991), which includes a spherical bulge, disk, and extended massive halo. The model assumes the Galactocentric distance of the Sun to be R
=8.5 kpc and the Galactic rotation rate at the solar distance to be Vrot = 220 km/s. Table 1 presents the metallicities, Galactic azimuthal velocity components, full residual velocities relative to the local solar centroid [with (U, V, W) = (-10, 10, 6) km/s], and Galactic orbital elements for 174 RR Lyrae variables in our final sample. The orbital elements were computed by modeling five complete orbits around the Galactic center foreach star. The most informative quantities are Zmax, the maximum distance of the orbit from the Galactic center (the orbital apogalactic radius), and the eccentricity, e = (Ra -Rmin)/(Ra + Rmin),where Rmin is the minimum distance of the orbit from the Galactic center. To adequately estimate the chracteristic parameters of the proposed RR Lyrae subsystems, the sample must be representative of the objects in these subsystems.Our choice of stars based solely on their type of variability ensures an absence of kinematic selection effects. However, since light curves can be obtained only for the relatively bright variables,our initial sample already has insu ficient sampling depth to estimate the sizes of the oldest subsystems.Figure 1a presents the distributions of distance from the Sun for the stars in the initial sample.The solid line approximates the histogram with a function of the form 
to the class interval with the highest number (i.e.,for Rнабл < 1.75 kpc).The power-law index was found to be g = 1.4 ± 0.1.The shaded areas in the histogram show the distributions for stars in our final sample with known spatial velocities.These are brighter and, naturally,on average closer to the Sun. Here, the number of stars grows with distance from the Sun in accordance with the same law as for the initial sample, but this is valid within a smaller radius. Even allowing for the uncertainties,the resulting relation clearly differs from the quadratic law that is expected if the observed volume is uniformly populated.In other words, the sample is not complete. This is due to the observational selection criteria used in the Laydan (1994), which included only stars more than 10o from the Galactic plane in order to reduce the final uncertainty in [Fe/H]due to interstellar reddening. This is apparent in Fig. 1b, where we find virtually no stars near Zabs ± 250 pc. It also appears that proper motions have been measured for a larger percentage of stars in the northern hemisphere (Z > 0). Note that the subsystems of Galactic halo we are studying are large, and that the selection effects primarily affect subsys tems of the younger thin disk, so that our samples should be suffciently representative with respect to
the halo subsystems.
 | Fig. 1.Distributions of RR Lyrae stars in distance (a)from the Sun and (b) from the Galactic plane.The shaded part of the histogram corresponds to stars with known proper motions.The curve in the left histogram is a power-law approximation for the growth of the number of stars with distance in the intial sample for R < 1.75 kpc.  is the power-law exponent. The dip in the right distribution near Z± 250 pc is due to selection effects. |
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2. CRITERIA FOR DISTINGUISHING THE SUBSYSTEMS
Objects belonging to the thick disk subsystem
can be reliably distinguished based on their metallicity.In particular,the metallicity distribution of the globular clusters demonstrates an obvious depression near [Fe/H] -1.0, as well as abrupt changes in the velocity dispersions and distances from the Galactic center (cf., for instance, Borkova & Marsakov (2000)). Figure 2a shows the metallicity function for the RR Lyrae stars in our initial sample. It can also be described by two normal curves at a high confidence level. A maximum-likelihood test shows that the probability that rejection of the best-fit single-Gaussian curve in favor of a two-Gaussian fit would be erroneous is << 1%. The mean values and dispersions of the metallicities in the groups coincide with the corresponding parameters for globular clusters within the errors Borkova & Marsakov (2000). Like the globular clusters, the RR Lyrae stars in Fig. 2b demonstrate an abrupt change of the radial-velocity dispersions when the metallicity crosses a boundary value, equal, in this case, to [Fe/H]гр = -0.95. Abrupt changes in the dispersion and maximum distance of the stars from the Galactic plane are seen at the same metallicity in Fig. 2c.
 | Fig. 2. (a) Metallicity function and relations between (b) metallicity and radial velocity, (c) metallicity and maximum distance from the Galactic plane,and (d)metallicity and variability period for RR Lyrae variables in the initial sample. The curve in the first graph is the maximum-likelihood approximation of the distribution using a sum of two normal curves with the indicated means,dispersions,and total numbers. The dashed vertical lines in the second and the third panels are "eye estimates" of the location of the abrupt increase in the scatter of Vr and Zmax at [Fe/H ]=-0.95.. |
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Globular clusters also show an abrupt change of the dispersion of another intrinsic parameter near [Fe H] -1.0 - the morphological structure of the
horizontal branch, which is closely related to the mean period of the RR Lyrae variables in the cluster. In addition,the similarity of the period distributions for field and globular cluster RR Lyrae variables has been noted in several studies.For example, Carney (1999) presents corresponding histograms (his Figs.1.5 and 1.26)and discusses their bimodality and the similar positions of the two peaks. Figure 2d displays a metallicity-period diagram for the stars in our initial sample. However, the diagram shows no structural details apart from the well-known trend. Thus,we conclude that the variability periods of field RR Lyrae stars cannot be used as an additional criterion to identify objects belonging to the thick-disk subsystem.
 | Fig. 3. Relations between the residual spatial velocityand other characteristics of the RRLyrae stars.The filled circles represent stars with [Fe/H ] > -0.95; the large open circles with error bars are mean values and dispersions of the corresponding parameters in narrow intervals of Vres. The vertical and slanted dashed lines in the first panel mark the separation distinguishing are as occupied by thick-disk objects according to Hanson et al. (1998) and Martin, Morrison (1998), respectively. An abrupt change in the dispersion near Vres 290 km/s (the vertical dashed lines) is apparent in all panels except the first. |
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Since this metallicity criterion for selecting thick-
disk objects is clearly oversimplified,several criteria have been proposed to add metal-poorer stars to the subsystem. All use restrictions on the orbital velocity, and are thus suitable only for stars with known proper motions. For example, Hanson et al. (1998) proposed assigning all stars with velocities relative to the local centroid of the Sun below 125 km/s to the
old disk, based purely on the fact that most metal-rich stars meet this criterion. We have plotted the corresponding vertical dashed line for the RR Lyrae stars in our final sample. The filled circles represent stars with [Fe/H ] > -0.95. Virtually all lie to the left of the dashed line. The same area is also occupied by a fair number of metal-poorer stars,which form the "low-metallicity tail" of the thick disk. Note, however, that several metal-rich stars are found to the right of
the line. If we increase the threshold velocity for the criterion to include these, the subsystem will include many stars located much higher above the Galactic plane than any of the metal-rich stars,increasing the characteristic size of the thick disk. To avoid this, a criterion also taking into account the star's location was suggested in Martin & Morrison (1998)(see the slanting dashed line in Fig. 3a). In this case, the subsystem's characteristic size is obviously preserved,but the |Zobs| distributions of stars with different metallicities remain different; the diagram shows that the highest density of metal-rich stars is reached at Zobs < 0.5 kpc (see also Fig. 4a below),the highest density of stars in the low-metallicity tail occurs around Zobs 0.8 kpc. Moreover, the number of RR Lyrae stars in the low-metallicity tail of the thick disk is as high as the total number of metal-rich stars. This contradicts results obtained for nearby field stars, for which thick-disk stars were distinguished from the younger thin disk population. In particular, the recent detailed study Prochaska et al. (2000) indicates that the low-metallicity tail of the thick disk essentially disappears near [Fe/H] -1.5. Apparently, if there is a low-metallicity tail of RR Lyrae stars, it must be identified using more complex criteria. The large uncertainties in the tangential velocities of distant stars, such as RR Lyrae variables, make it impossible to do this adequately. For this reason, we decided not to artificially assign any metal-poor stars to the thick-disk subsystem,and use only the primary criterion [Fe/H ] > -0.95 when estimating the characteristic parameters of this subsystem. In our view, the fact that the ages of metal-rich globular clusters of the thick disk are considerably lower, and essentially do not overlap the ages of proto-disk halo clusters Borkova & Marsakov (2000), is another argument against the existence of a low-metallicity tail, since this indicates that the time intervals for their formation were different.
It is much more dificult to identify objects that have an extragalactic origin; i.e., those belonging to the accreted halo.According to the hypothesis that the protogalaxy collapsed monotonically from the halo to the disk Eggen et al. (1962), stars genetically related to the Galaxy cannot have retrograde orbits. (Only the oldest halo stars may be exceptions, since they could have retrograde orbits due to the natural initial velocity dispersion of the protostellar clouds.) On the other hand,some stars formed from extragalactic fragments captured by the Galaxy should have prograde orbits. In any case, such stars should have fairly large residual spatial velocities relative to the local velocity centroid. Figure 3b displays the relation between this velocity and the azimuthal velocity component, Q, for the RR Lyrae stars in our sample. We can see that their is a transition from prograde to retrograde orbits about the Galactic center near Vres 290 km/s. We also observe an abrupt increase in the dispersion of the azimuthal velocity component at the same place (see the error bars in the diagram). Figure 3c displays a significant increase in the scatter of the stars in Zmax when crossing the same threshold residual spatial velocity. The abrupt change in the apogalactic radii of the stellar orbits is even more evident (Fig. 3d). Both the orbital eccentricities and their dispersion abruptly increase when crossing the same point (Fig. 3e),and also demonstrate different relations. First, the orbital eccentricities increase almost linearly with the residual velocities, reaching a maximum near the threshold velocity level. With further increase in Vres, the mean and scatter of the eccentricities do not change within the errors. It is interesting that the metal abundances also demonstrate different behavior in different areas of Fig. 3f. First, we observe a clear metallicity decrease with increasing velocity, after which the metal abundance remains virtually
unchanged, and there is an appreciable decrease in its scatter. For this reason, we adopt Vcr 290 km/s as the critical value for distinguishing stars of the outer halo (Vres > Vcr). We suggest that stars with lower residual velocities have a Galactic origin, and belong to the thick-disk and proto-disk halo subsystems. Apparently, this kinematic criterion is not entirely unambiguous: some stars of the proto-disk halo may have larger residual spatial velocities. Evidence for
this is provided, in particular, by the increase in the stellar density immediately to the right of Vcr in our diagrams. However, we decided (as for the thick disk) to retain a simple criterion,in order not to artificially confuse the situation.
Note again that the principal criterion distinguishing globular clusters of the young halo are their redder horizontal branches and, simultaneously, the lower mean periods of their RR Lyrae stars compared to clusters of the proto-disk halo. To test this circumstance for field stars, we computed the mean periods of stars in a narrow metallicity range (-1.7 < [Fe/H] < -1.2,) for both halo subsystems. Each sample contained about forty stars. Their mean periods were the same. Thus,the variability periods of field RR Lyrae stars cannot serve as an additional criterion to divide them into the subsystems of the metal-poor halo, similar to the case of the thick disk.
Let us now compare the properties of the resulting
subsystems.
4.PROPERTIES OF STARS IN THE SUBSYSTEMS.
Since the membership of stars in the thick disk
is determined by their metallicity,we are able to first consider a number of properties of this subsystem using the initial sample of RR Lyrae stars, then compare
these properties to the results obtained for the smaller number of stars that have known spatial velocities and orbital elements. The characteristic feature of
the thick disk is its rapid rotation, which results in considerable flattening toward the Galactic plane. In particular, the natural logarithmic scale height of the density of globular clusters in the thick disk is 1.0 kpc. As shown above, the sampling depth of our initial sample is considerably larger, so that we can derive the scale height (Zo) of the subsystem of metal-rich RR Lyrae stars from their observed positions.
 | Fig. 4. (a) Distribution of distance from the Galactic plane, (b) kinematic diagram, and (c) relation between metallicity and distance from the Galactic plane for RR Lyrae stars with [Fe /H ] > -0.95. The curve in the left panel approximates the distribution with an exponential law with scale height ZO. The solid line in the middle panel is a regression line, whose slope is determined by the subsystem's rotation rate; the open circles and dashed line correspond to stars with [Fe/H] > -0.4, and the filled circles and dashed line to stars with -0.95 < [Fe/H] < -0.4. The line in the right panel is a least-squares regression line for | Z | < 2 kpc, whose slope is determined by the metallicity gradient; the squares show distant stars excluded from our computations. |
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Figure 4a shows that the scale height of the thick disk at the solar distance from the Galactic center is even somewhat lower than for globular clusters that are close to the center.Assuming non-rigid rotation of the subsystem with a constant linear velocity, we can obtain a least-squares fit to this velocity based solely on the radial velocities and observed
positions of the studied stars (see Zinn (1985) for details of the method). Figure 4b presents the corresponding cos 
- VS kinematic diagram, where is the angle between the line of sight and the vector for rotation about the Z axis and VS is the star's radial velocity reduced to an observer at rest at the position of the Sun. The slope of the least-squares regression line determines the rotational velocity of the system of metal-rich stars, Vrot=220 ±14 km/s, whereas the scatter around it determines the dispersion of the residual velocities,
res = 64 ± 6 km/s; the correlation coeficient is quite high, r = 0.9 ± 0.1. Using the apparent positions of the stars, we can also follow variations of the metallicity with distance from the Galactic plane (Fig. 4c). The regression line derived for the majority of the stars (i.e., those closer than kpc)indicates a considerable negative vertical metallicity gradient, gradZ[Fe/H] = -0.17 ± 0.09 kpc-1, with r = 0.3 ± 0.1. This result is stable: when distant stars are included,the absolute value of the gradient decreases,but remains outside the errors. However, stars with Zobs that are that high above the plane (beyond 3) do not necessarily belong to the disk subsystem.
Note that the resulting high rate of rotation of the subsystem of metal-rich RR Lyrae stars, which coincides with the Galactic rotational velocity at the distance of the Sun within the errors, probably testifies to the presence of thin-disk stars. If we divide the sample into two groups, for example, at [Fe/H] = -0.4, the more metal-rich group will demonstrate a much higher rotational velocity in the kinematic diagram (Fig. 4b). Due to the spatial observational selection criteria used, our initial sample lacks stars closer to the Galactic plane than Zobs = ± 250 pc; since the scale height for the subsystem of old thin-disk stars is in this range Marsakov & Shevelev (1995), we cannot address this problem in more detail in the present study.
 | Fig. 5.Distribution of orbital elements for the RR Lyrae stars. Results for stars with [Fe/H ] > -0.95 (left column), [Fe/H ] < 0.95 and Vres < 290 km/s (middle column), and [Fe/H] < -0.95 and Vres > 290 km/s (right column) are shown separately. In the top row, we have plotted normal curves fitting the distributions. In the bottom row, the segmented lines are the reconstructed distributions of RR Lyrae stars in Z (see text for details), and the curves approximate the distributions with exponential laws (the corresponding scale heights are indicated). |
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 | TABLE 2Characteristics of the subsystems of RR Lyrae(ab) variables |
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The spatial velocities of the stars can be used to verify the above results for the metal-rich stars, and also to obtain estimates for a number of characteristics for both halo subsystems, if we can first reconstruct the stars' Galactic orbits. Figure 5 shows the distributions of the stars in our three subsystems in the elements of their Galactic orbits. The top row
presents histograms of the rotational velocities for stars of the thick disk,proto-disk halo, and outer accreted halo. All three distributions can be quite satisfactorily represented with normal laws (the curves in the histograms), whose maxima are separated nearly by the corresponding dispersion (Table 2). The good agreement between the mean rotational velocities and their dispersions for the metal-rich stars derived from radial velocities (Fig. 4b)and spatial velocities (Table 2) testifies to the good accuracy of the proper motions and the photometric distance scale used. The second row of histograms in Fig. 5 shows the corresponding distributions in orbital eccentricity. The characters of these histograms are obviously very different. The disk has virtually no stars with e 
0.5, all eccentricities are present in approximately equal numbers in the proto-disk halo,and stars with very eccentric orbits dominate in the outer halo (where almost two thirds of all stars have e > 0.8). The next row (Fig. 5g-i) presents the distributions in orbital inclination. As expected, the orbital inclinations of the disk stars are very low,not exceeding 15o. The stars of the two spherical subsystems can have any orbital inclination. In both cases, the number of stars strongly increases with decreasing inclination; however, this is true for stars with prograde orbits in the proto-disk halo, and for those with retrograde orbits in the outer halo. We must bear in mind that the deficit of stars with large orbital inclinations is largely due to the kinematic selection effect imposed on the sample. The vertical components of the spatial velocities of such stars in the solar neighborhood should be comparable
to the Galactic rotational velocity at this distance. Thus,the probability of their presence here is very low.
The fourth row of histograms (Fig. 5j-l) can be used to estimate the radial sizes of the subsystem, and the fifth row (Fig. 5m-o), their vertical sizes at the solar Galactocentric distance. Let us attempt to make a quantitative estimate of the outer sizes of the subsystems based on these distributions, using the standard rules of thumb for the behavior of an upper envelope, often applied in observational astronomy. For example,when selecting members of an open cluster, it is usual to reject the five brightest stars as possible field stars. Proceeding in this fashion, we simultaneously eliminate the largest uncertainties in the orbital elements and avoid possible errors in assigning some stars in our sample to a particular subsystem. Such estimates indicate that the outer sizes of the genetically related subsystems are comparable, whereas the outer halo is approximately a factor of three larger (Table 2). The sizes of the subsystems perpendicular to the Galactic plane differ
more dramatically: the half-thickness of the thick disk is smallest, that of the proto-disk halo is a factor of a few larger, and that of the outer halo is nearly an order of magnitude larger (Table 2).
It is obviously not correct to compute the scale
height using Zmax, since all stars in a subsystem cannot simultaneously be located at the highest points of their orbits. To reconstruct the "real", instantaneous distributions of each subsystem from the values, we must "spread" each star over its orbit in proportion to the probability density of the star being located at each point of the orbit. This probability
density can easily be found from the computed bit of the star; the details of the procedure can be found in Marsakov & Shevelev (1995). The segmented lines in the bottom row of histograms (Figs. 5m-o)show the Z distributions for each subsystem reconstructed in this manner. In other words, this is how the stars will be distributed in height after some time if they are distributed randomly in their orbits. The solid curves in Figs. 5m-o are least-squares approximations to the reconstructed distributions using an exponential law, 
, where Z0 is the scale height (the corresponding Z0 values are indicated in Figs. 5m-o and Table 2). In this procedure, we did not take into account the first interval of Zmax for the segmented line corresponding to the thick disk,since the number
of objects in this interval is far too low due to the observational selection criteria. (In all cases, we excluded the five most distant stars when evaluating the
scale height.) Note the good agreement of the scale heights for the thick disk estimated from the observed positions and from the reconstructed distribution in
|Z|obs (Fig. 4a).
 | Fig. 6.Metallicities of the RR Lyrae stars as a function of apogalactic distance (a,b,c) and maximum distance from the Galactic plane (d,e,f). The straight lines are least-squares fits.The five most distant stars in each panel (in boxes) were excluded from our calculations. The correlation coefients and their uncertainties are indicated. |
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Let us now consider the metallicity gradients in
the subsystems. Figure 6 displays Ra - [Fe/H] and
Zmax - [Fe/H] diagrams for each of the subsystems. The straight lines are least-squares fits. Assuming that the stars are born near the apogalactic radii of their orbits, the slopes of these lines reflect the initial radial and vertical metallicity gradients for the subsystems. To increase the trustworthiness of the gradient estimates, we rejected the five most distant data points in each case. The resulting gradients are presented in Table 2, and the correlation coeficients in the corresponding diagrams. Only the vertical gradient in the thick disk appreciably exceeds the errors and has a high correlation coeficient. This gradient coincides within the errors with the value derived above (Fig. 4c)from the observed positions of the stars, testifying to the reliability of the result. The radial metallicity gradient in the disk is zero within
the errors. In the proto-disk halo, small gradients with similar magnitudes, only slightly exceeding their uncertainties, are found. The correlation coeficients given in Figs. 6b,e will become smaller if we add the five excluded distant stars, but will remain non-zero. Thus, the existence of gradients in the proto-disk halo is still an open question. In contrast, the complete absence of both gradients for the RR Lyrae stars of the accreted halo is beyond doubt.
5.DISCUSSION
The thick-disk objects can be reliably distinguished from halo objects due to the abrupt change of their spatial distribution and their velocity dispersion when crossing the threshold value [Fe/H] -1.0. Thick-disk globular clusters selected using the same criterion have Z0=1.0 ± 0.2 kpc, < [Fe/H] > =-0.56 ± 0.05, [Fe/H] = 0.28 ± 0.03, Vrot = 165 ± 38 km/s, res = 88 ± 15 km/s, gradR[Fe/H] = -0.01 ± 0.02 kpc-1, gradZ[Fe/H] = -0.16 ± 0.06 kpc-1. Comparing these to the parameters of the subsystems of RR Lyrae stars from Table 2, we find that the mean metallicities for the clusters and for our field stars nearly coincide. However, the metallicity function of the thick-disk globular clusters has a well defined depression in the number of clusters near the threshold metallicity, while the corresponding distribution for the field RR Lyrae stars does not show such a dip (There is even a local rise near [Fe/H] -0.85.) Though the rotational velocities of the subsystems are the same within the errors, the velocity for the RR Lyrae stars was ~ 35 km/s higher due to the contribution of thin-disk stars to the sample. The appreciable velocity dispersion of the disk globular clusters is probably the result of the large uncertainty in the computed cos values for clusters near the Galactic center. The corresponding gradients are essentially the same for the subsystems, whereas the scale height for the clusters is higher by approximately one-third. A natural explanation is that some of the metal-rich clusters near the Galactic
center probably belong to the bulge. The difference in the limiting radial sizes of the subsystems is more dificult to explain: this size is ~ 13 kpc for the field RR Lyrae stars,whereas almost no metal-rich globular clusters are observed beyond ~ 7 kpc (this may simply be due to the statistically small sample of globular clusters in the thick disk). Based on the coincidence of all but the last characteristic of the metal-rich subsystems of these different objects, we suggest that both the globular clusters and the field RR Lyrae stars with [Fe/H] -1.0 belong to the
same subsystem of the Galaxy - the thick disk.
Let us nowturn to the thick disk's "low-metallicity tail". Of course, the abrupt end of the metallicity distribution at [Fe/H] = -1.0 seems artificial. The maximum-likelihood method also &wuot;requires&wuot; amuch higher metallicity dispersion, as can be seen in a comparison of the data in Fig. 2 and Table 2. Therefore, it is natural to wish to smooth the metallicity function by including in the subsystem metal-poorer stars that do not reach large distances from the Galactic plane and have large orbital velocities. This does not cause significant changes in any of the characteristic spatial or kinematic parameters of the subsystem. However, adding metal-poor, low-stars to the sample completely removes the vertical metallicity gradient in the disk. This is the reason for the contradictory results concerning this gradient obtained in different studies, and for disagreements about models for the formation and evolution of this subsystem. The presence of a vertical gradient would
provide evidence for a slow, dissipative collapse as the mechanism for the formation of the thick disk, whereas the absence of a vertical gradient would rule out this process.We are inclined to believe that the stars in the "low-metallicity tail" of the disk (if there are any) probably formed from low-metallicity gas and dust fragments weakly interacting with the bulk of the parent proto-disk cloud, where star formation (and hence enrichment in heavy elements) was suppressed for a long time. In this case, low-metallicity disk stars can be located in a thin layer of the Galaxy, considerably reducing the observed vertical metallicity gradient among genetically related stars of the subsystem.
Let us now consider the halo stars. As earlier, we will compare the characteristics of the metal-poor RR Lyrae stars derived in this study to the parameters of corresponding subsystems of globular clusters from Borkova & Marsakov (2000), since only these objects were distinguished based on an intrinsic,physical parameter rather than interrelated spatial and kinematic criteria. The metallicity distributions of corresponding halo subsystems differ somewhat. In particular, the mean metallicity of the proto-disk halo derived from globular clusters is lower than that of the outer halo. The metallicity dispersion is also lower. The field stars show the opposite pattern (Table 2). In all cases, however, the differences are comparable to the formally computed uncertainties, indicating that any conclusions about differences between these parameters have low statistical significance. The gradients in the proto-disk halo are completely consistent. In the accreted halo, both gradients are absent for the field RR Lyrae stars but non-zero for the clusters. However, both gradients for the globular clusters are due exclusively to metal-richer objects close to the Galactic center (R 7 kpc), and objects that far from the Sun do not enter our sample of field RR Lyrae stars.We have identi fied the halo subsystems based on spatial velocity, and differences between the proto-disk halo and accreted halo in any kinematic parameter should be more prominent. Indeed, while the orbital velocities for the globular clusters of the proto-disk halo and the accreted halo are 77 ± 33 km/s and -23 ± 54 km/s, so that the difference between them is ~ 100km/s, this difference is approximately 40% higher for the RR Lyrae stars (Table 2).The velocity-dispersion estimates for the subsystems of globular clusters (129 ± 19 km/s and 140 ± 18 km/s) are obviously overestimated due to the large distance uncertainties and, as a result, they
are much higher than the values for the field RR Lyrae stars. The mean eccentricities in the proto-disk halo subsystens are the same, whereas, in the outer halo, the eccentricities are, on average, higher for the field stars, as expected (note that orbital eccentricities are known only for a small number of clusters, and with large uncertainties). The radial size of the proto-disk halo was approximately a factor of 1.5 larger for the field stars than for the clusters, whereas the two scale
heights were the same within the errors. Recall that we can estimate the radial sizes of RR Lyrae subsystems only from their maximum distances from the Galactic center, which leads to appreciable overestimation of these sizes. The radial and vertical sizes of the outer accreted halo subsystem of field stars are naturally the largest, and are in reasonable agreement with the corresponding sizes for the subsystem of
globular clusters. Note that, in order to obtain correct estimates of sizes of Galactic subsystems based on data for nearby stars, it is necessary to take into
account the kinematic selection effects, which lead to adeficiency of stars with large R a and max in the solar neighborhood.
Thus, the generally good agreement between the
characteristics of corresponding subsystems of field RR Lyrae stars and globular clusters distinguished using different criteria shows that both populations are not uniform. Both the clusters and field stars belong to at least three Galactic subsystems: the thick disk and genetically related inner proto-disk halo, and the outer accreted halo. The collected results indicate that this subsystem is characterized by a large size, an absence of appreciable metallicity gradients, predominantly large orbital eccentricities, a large number of objects in retrograde orbits,and younger ages for its objects, supporting the hypothesis that this subsys-
tem had an extragalactic origin.
ACKNOWLEDGMENTS
The authors are grateful to the anonymous referee, who found several inconsistencies in the manuscript. This study was supported by the Russian Foundation for Basic Research (projects 00-02-17689 and 01-02-06449).
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