THE THREE-DIMENSIONAL VELOCITY STRUCTURE OF THE THICK DISK FROM SPM4 AND RAVE DR2

In Figure 9, we show the run of velocities as a function of distance from the Sun (top), distance from the Galactic plane (middle), and distance from GC (bottom). Left, middle, and right rows show V_R, V_θ, and V_z, respectively. For V_R and V_z we also show a moving average to be compared with the expected value of 0 km s⁻¹. While V_z is consistent with an average of 0 km s⁻¹ for the entire d_sun and |z| ranges, V_R has a mean of 0 km s⁻¹ only for stars within ∼1 kpc from the Sun. The slight departure of V_z from 0 km s⁻¹ in the plot as a function of R_GC at the extremes of the R_GC range is due to small-number statistics at the edges, and thus is not significant.

For the entire sample, we obtain an average V_R = 9.2 ± 1.1 km s⁻¹, and V_z = −2.0 ± 0.7 km s⁻¹. While V_z is only marginally different from 0, V_R is significantly different from 0. If we split the sample into a nearby subsample with d_sun < 1.0 kpc and a distant subsample with d_sun ⩾ 1.0 kpc, we obtain (1) V^near_R = 3.0 ± 1.7 km s⁻¹ and V^near_z = −2.6 ± 1.0 km s⁻¹ for 1053 stars and (2) V^far_R = 11.1 ± 1.3 km s⁻¹ and V^far_z = −1.9 ± 0.9 km s⁻¹ for 3367 stars. Clearly, the distant subsample behaves differently than the local one. We have also split the entire sample of 4420 stars into subsamples above and below the plane to check if a kinematical asymmetry is responsible for this non-zero V_R. We obtain V^above_R = 14.1 ± 1.9 km s⁻¹, V^above_z = −1.6 ± 1.4 km s⁻¹ for 1225 stars above the plane and V^below_R = 7.3 ± 1.3, V^below_z = −2.8 ± 0.8 km s⁻¹ for 3195 stars below the plane. The non-zero V_R is significant in both samples. The below-the-plane sample, which may include in quadrant one part of the Larsen & Humphreys (1996) overdensity (see Section 3), is thus not solely responsible for the non-zero V_R.

To further explore the origin of this non-zero V_R, we have tested other values for the Solar peculiar motion and for the LSR's rotation. Using the Schönrich et al. (2010) Solar peculiar motion, (V^☉_R, V^☉_θ, V^☉_z) = (−11.10, 12.24, 7.25) km s⁻¹, we obtain V_R = 7.5 ± 1.1 km s⁻¹ and V_z = −2.0 ± 0.7 km s⁻¹ for the entire sample. Adopting the Schönrich et al. (2010) Solar peculiar motion and the rotation of the LSR, V⁰_θ = 250 km s⁻¹ (Reid et al. 2009), we obtain V_R = 5.0 ± 1.1 km s⁻¹ and V_z = −2.1 ± 0.7 km s⁻¹. Thus, V_R is still at ∼5 σ level different from 0.

Other possible sources for this unexpected result are of course systematics in the observed quantities, of which the most suspect are the proper motions and distances. First we test the distance, i.e., modify it in order to obtain an average V_R = 0.0 km s⁻¹. We found that all distances have to be a factor of 0.5 smaller in order to satisfy our requirement. This corresponds to a magnitude difference of 1.5, or the absolute magnitude of red clump stars has to be wrong by this amount, which seems unrealistic. Next, we vary the proper motions in one coordinate, e.g., R.A. (keeping the distance and proper motions in the other coordinate fixed), first by adding an offset to it, and then introducing a slope with magnitude. The offset mimics an incorrect absolute proper-motion correction, while the slope with magnitude mimics magnitude-dependent systematics present in the proper motions. We found that we need an offset of 8 mas yr⁻¹ in order to satisfy our requirement; in which case the averages are V_R = 2.7 ± 1.3 km s⁻¹ and V_z = 4.6 ± 0.8 km s⁻¹. A slope of 3 mas yr⁻¹ mag⁻¹ gives V_R = 4.5 ± 1.3 km s⁻¹ and V_z = 4.1 ± 0.9 km s⁻¹, neither being satisfactory. We repeat this test by varying the proper motion along decl. and keeping the other quantities fixed. Here, within the range explored, we find no satisfactory solution because as V_R nears low values, V_z becomes ∼5 σ different from zero.

We note that an offset of 8 mas yr⁻¹ in the absolute zero point of SPM4 is a huge and unrealistic value, since these systematics are expected to be of the order of 1–2 mas yr⁻¹ (T. M. Girard et al. 2011, in preparation). Likewise a slope of 3 mas yr⁻¹ mag⁻¹ is completely unacceptable for the SPM4 catalog, and in general for the SPM material and reductions. We remind the reader that our result for the absolute proper motion of cluster M 4 (Dinescu et al. 2004) obtained with SPM data is within 0.5 mas yr⁻¹ of the result obtained with Hubble Space Telescope (HST) data (Bedin et al. 2003; Kalirai et al. 2004). The ground-based result uses data in a much brighter magnitude range than the HST result: the first is tied to Hipparcos, the other to one QSO, and to galaxies. Thus, there is no reason to believe that systematics of ∼3 mas yr⁻¹ mag⁻¹ are present in the SPM data.

To obtain acceptable values for V_R and V_z, we need to have both proper-motion coordinates offset by 3 mas yr⁻¹, or both with slopes of 3 and 1 mas yr⁻¹ mag⁻¹. Systematic errors this large in the SPM4 catalog are unrealistic.

We turn now to an exploration of the literature regarding this issue. We consider only the most recent kinematical studies that include large samples of stars, with well-determined distances, proper motions, and radial velocities, and, when possible, homogeneous data. One recent study that uses SDSS data in stripe 82 is that by Smith et al. (2009). That study uses a sample of 1700 halo subdwarfs, where radial velocities are from SDSS spectra, and the proper motions are determined only from SDSS data in stripe 82 which has repeated observations over a period of seven years (Bramich et al. 2008). Their derived values are V_R = 8.9 ± 2.6 km s⁻¹ and V_z = −1.2 ± 1.6 km s⁻¹, for a sample spanning a heliocentric distance between 1 and 5 kpc. This is in excellent agreement with our result. Smith et al. (2009) admit that this result for V_R is significantly different from the nominal zero and suggest various causes for it such as kinematical substructure; systematic errors in proper motions, distances or radial velocities; the presence of binary stars.

Another study is that of ∼1200 metal-rich red giants at the south Galactic pole (SGP) done by Girard et al. (2006). This study used proper motions from an earlier version of the SPM catalog (namely, SPM3; Girard et al. 2004) and photometric distances estimated from 2MASS photometry. At the SGP, the proper motions are directly projected into U, V velocities. While the study focuses on the shear of the thick disk, they obtained an average velocity with respect to the Sun of U = 19.1 ± 2.7 km s⁻¹ at an average 2.2 kpc from the Galactic plane. Taking into account the Solar peculiar motion, their result is U = 9.1 ± 2.7 km s⁻¹, in agreement with ours and with Smith et al. (2009). We note that the Girard et al. (2006) study used different tracers (0.7 ⩽ (J − K) ⩽ 1.1) than our current study.

Another study of red clump giants at the NGP was made by Rybka & Yatsenko (2010). The red clump stars were selected from 2MASS in the traditional color range 0.5 ⩽ (J − K) ⩽ 0.7, and the proper motions come from various catalogs including Tycho2 and UCAC2. Using ∼1800 stars between 1 and 3 kpc from the Sun, they obtain an average velocity with respect to the LSR of U = 8.1 ± 1.8 km s⁻¹, in agreement with our result and the results mentioned above.

Two additional recent kinematical studies using SDSS DR7 data are those by C10 and B10. The C10 study focuses on "calibration" stars or a subsample of the SDSS data, while the B10 study uses all SDSS data. Both studies analyze dwarf stars (of the halo and disk), and use the same proper motions determined from USNO-B and SDSS positions (Munn et al. 2004, 2008). Radial velocities are obtained from SDSS spectra (see details in C10 and B10). Both studies have samples of tens of thousand of stars with heliocentric distances well outside the 1 kpc limit. Neither study gives the average V_R, V_z for the entire sample, thus we can judge a small positive offset only from the plots presented. Figure 6 in C10 shows histograms of each velocity component for various metallicity bins. It is apparent that for V_R the highest peak is always on the positive side of V_R for all metallicity bins, while this is not the case for V_z. In metallicity bins −1.6 to −1.2 and −2.0 to −1.6, this asymmetry is most apparent. However, for thick-disk stars selected in the metallicity range [Fe/H] = (−0.8, − 0.6), and distance from the Galactic plane |z| = (1, 2) kpc, C10 determine the average to be V_R = 2.5 ± 2.0 km s⁻¹.

B10 determine the average V_R and V_z for a subsample of 13,000 M dwarfs located within ∼1 kpc from the Sun and obtain values consistent with 0 km s⁻¹. This is consistent with our results for objects within 1 kpc from the Sun. However, in their Figure 7, where they show the run of the median V_R as a function of distance from the plane, there appears to be a small positive offset for stars between 1.5 and 3 kpc.

Assuming the offset in V_R is real, we believe that, rather than having the entire Galaxy expand, it is more realistic to assume that stars in the solar neighborhood (within ∼1 kpc from the Sun) move inward, toward the GC compared to more distant stars that better represent a Galactic rest frame. If this is the case, then radial-velocity (i.e., line-of-sight velocity) studies at low latitude and toward the GC and anticenter should confirm/disprove this. In other words, bulge stars for instance should have an average radial velocity toward us, and stars at the anticenter (and beyond 1 kpc from the Sun) should have an average radial velocity away from us. Izumiura et al. (1995) present a study of SiO masers toward the Galactic bulge, and they find an average shift in radial velocity of −17.7 ± 7.6 km s⁻¹. Taking into account the Solar peculiar motion, this leaves a net radial velocity of ∼−7.7 ± 7.6 km s⁻¹. Other studies that observe various types of stars toward the bulge suffer from small sample size, large velocity dispersion of the bulge population, and the modeling of its rotation. One recent study that contains a significant number of stars (∼3300) is the radial-velocity study of M giants toward the bulge by Howard et al. (2008). This study does not detect any net offset from zero for fields located along the minor axis (their Figure 14); however, the scatter is rather large, of the order of 10 km s⁻¹. Their subsequent study (Howard et al. 2009) of a stripe along the major axis at b = −8° with ∼1200 red giants does show an offset of −9.1 ± 2.7 km s⁻¹ with respect to the LSR, which is intriguing and in agreement with our result. It is unclear if this could be due to the asymmetry of the fields sampled along the major axis of the bulge combined with the rotation of the bulge. Their radial-velocity histogram, however, has a Gaussian shape (their Figure 2).

As for radial-velocity studies toward the Anticenter, we note that of Metzger & Schechter (1994), who study 179 carbon stars at a distance of ∼6 kpc from the Sun. They find that these stars move with a mean of 6.6 ± 1.7 km s⁻¹ with respect to the LSR, thus radially outward. Therefore this sample of stars points yet again to the LSR's motion toward the GC, with an amount similar to our result.

Our investigation shows that local stars (lesser than ∼1 kpc from the Sun) do not exhibit a net V_R motion, and thus Hipparcos-based results, for instance, are unlikely to detect this. However, more distant samples with full three-dimensional velocities do detect it.

It is thus apparent that the entire local sample of stars has a net inward motion toward the GC, while the vertical motion remains zero. The first thought that comes to mind is that this is due to some noncircular motion induced by perturbations in the disk, such as spiral arms and/or bar. Indeed, a recent paper by Quillen et al. (2010) shows UV maps of various neighborhoods that sample the disk of an N-body simulated galaxy. These neighborhoods are placed at various radii from the GC and various angles from the bar. The maps show velocity clumps and arcs induced by the bar and spiral pattern that are offset from a mean of zero in velocity. Velocity offsets of these features can easily be as large as ∼50 km s⁻¹. Thus our offset of ∼12 km s⁻¹ for the entire local solar neighborhood is quite within the ranges predicted in Quillen et al. (2010).

In Figure 10, we show the run of each velocity component as a function of distance from the Galactic plane |z|; the averages in the top panel and the dispersions in the bottom panel. The data are grouped in bins of equal number of stars, here with 124 stars per bin. The data are restricted to a Galactocentric radius range of 7.5 < R_GC < 8.5 kpc. The average in each bin is calculated using probability plots (Hamaker 1978) trimmed at 10% on each side, and the uncertainty in the average is derived from the dispersion, also determined from probability plots. The bottom plot of Figure 10 shows the intrinsic velocity dispersions, i.e., corrected for velocity errors. Intrinsic velocity dispersions are calculated as follows. We start with an initial guess of the intrinsic dispersion. For each point, we calculate the square of the quadrature sum of this intrinsic dispersion and its velocity error, and we divide this number with the point's deviation from the mean. For all points this ratio should have a Gaussian distribution with a standard deviation of 1, if the intrinsic dispersion is correct. If not, the initial intrinsic dispersion is adjusted and the procedure is repeated until we satisfy the above condition.

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The top plot shows the variation of the rotation velocity V_θ as a function of |z|; of V_R, which is offset from 0 km s⁻¹ at z-distances larger than ∼0.7 kpc as discussed in the previous section; and of V_z, which is consistent with 0 km s⁻¹.

The bottom plot, which displays the velocity dispersions as a function of |z|, clearly shows the transition between the thin and thick disks. At |z| = 0.5 kpc, the sample is dominated by thin-disk red clump stars reflected in the low dispersion in all three components: $(\sigma _{V_R},\sigma _{V_{\theta }},\sigma _{V_z}) = (45, 30, 25)$ km s⁻¹. This dispersion increases rapidly between |z| = 0.5 and 1 kpc in all three components, to reach values of $(\sigma _{V_R},\sigma _{V_{\theta }},\sigma _{V_z}) \sim (70, 48, 38)$ km s⁻¹ at 1 kpc from the plane. This rapid increase reflects the mixture between the thin-disk population with its low velocity dispersion and the thick-disk population with a large dispersion. Before correcting for the contribution of the thin disk, we would like to discuss the results of three other studies. The continuous lines with corresponding colors for each velocity component show the dispersion dependence on |z| as determined by B10 for SDSS data for disk, dwarf stars. These stars were selected to be among the "blue stars," specifically with 0.2 < (g − r) < 0.6 and 0.7 < (u − g) < 2.0 (and two other criteria for combined colors, see B10), as well as to have photometrically determined metallicities [Fe/H] > − 0.9. Thus the metal-rich end is determined by the color cuts, and the sample is thought to represent the thick disk according to metallicity. In all three velocity components, the dispersions determined by B10 are substantially smaller than our values, except at |z| ∼ 0.5 kpc. In fact, the B10 values appear to reflect the kinematics of the thin disk rather than the thick disk. In the same plot, we show the run of velocity dispersions determined by Girard et al. (2006) at the SGP from the analysis of metal-rich, thick-disk giants. These values are more in line with what we obtain at |z|>1 kpc; however, our values have yet to be corrected for the contamination of the thin-disk contribution, a correction that will push the dispersion to higher values.

We add one more estimate of the dispersions, that determined by C10 from SDSS data for dwarf stars. Their thick-disk sample is selected to be in the metallicity range [Fe/H] = (−0.8, −0.6), where metallicities are determined from SDSS spectra. The dispersions are estimated for stars within |z| = 1–2 kpc. At an average |z| = 1.1 kpc, they obtain $(\sigma _{V_R},\sigma _{V_{\theta }},\sigma _{V_z}) = (53, 51, 35) \pm (2, 1, 1)$ km s⁻¹. These values are not corrected for observational errors in the velocities, which are assumed to be on the order of 5–6 km s⁻¹ (C10). We represent the C10 dispersions with a triangle symbol in our Figure 9. While their dispersions in the rotation and vertical component agree with ours, and are larger than the B10 values, the dispersion in the radial direction is substantially smaller than our value. It is also somewhat intriguing that the radial velocity dispersion is practically equal to the rotation velocity dispersion, clearly different from other thick-disk studies where the radial dispersion is larger by 20%–30%.

In light of the results summarized here, the B10 velocity dispersions seem unusually low, and it is worth exploring this a bit further. The C10 study used color cuts of "bluer" limits than B10, specifically, 0.0 < (g − r) < 0.6 and 0.6 < (u − g) < 1.2, and a metallicity range with a cutoff at the high end. Their dispersions are larger than those in B10. It is thus likely that the B10 sample of "disk" stars is significantly contaminated by thin-disk stars.

Next, we determine the velocity and velocity dispersion gradients, taking into account the contamination by thin-disk stars. As a first guess for the contribution of the thin disk in our sample, we use the density laws determined by Cabrera-Lavers et al. (2005, hereafter C05) from starcounts of red clump stars in 2MASS. The density laws are expressed as exponential functions of |z|, with given scale heights for the thin and thick disks, and a given normalization of thick-to-thin-disk stars. We show two parameterizations: (1) z_thin = 269 pc, z_thick = 1062 pc, and a local normalization of 8.6% thick-disk stars to thin disk, and (2) z_thin = 225 pc, z_thick = 1065 pc, and a local normalization of 11.4% thick-disk stars to thin disk. In Figure 11, we show the fraction of thin-disk stars (in %) as a function of |z| for the two descriptions. At ∼1 kpc, these descriptions indicate that the contribution of the thin disk is between 20% and 40% of the total stars. Our sample has a magnitude-selection imposed by the input RAVE catalog that affects this ratio that was derived for complete samples. Instead we choose to derive this ratio from our data by modeling the distribution of V_θ as the sum of two Gaussians: one representing the thin disk, the other the thick disk. The mean velocity of the thin disk is fixed at 220 km s⁻¹. We determine via χ² minimization of the V_θ distribution the velocity dispersions of the thin and thick disks, the fraction of thin-disk stars, and the mean velocity of the thick disk. We divide our sample into three |z| bins: (1) from 0.3 to 0.5 kpc, (2) from 0.5 to 1.0 kpc, and (3) from 1.0 to 1.5 kpc and apply this procedure for each bin. In Table 1 we show the best-fit results for each bin.

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Table 1. Thin-disk Fraction from the V_θ Distribution

$\overline{z}$	Nall	fthin	σthinθ	$\overline{V_{\theta }^{{\rm thick}}}$	σthickθ
(kpc)		(%)	(km s−1)	(km s−1)	(km s−1)
0.4	426	35 ± 42	19 ± 5	194 ± 17	29 ± 5
0.7	805	24 ± 6	18 ± 2	185 ± 03	35 ± 1
1.2	359	12 ± 2	16 ± 2	175 ± 02	50 ± 1

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The first z-bin is poorly constrained, due to the proximity of the rotation velocity of the thick disk to that of the thin disk. The next two z-bins have reasonable fits, with the middle one probably the best, due to the number of stars included. If we use the two C05 laws, but normalize such that the thin-to-thick disk ratio is set by the $\overline{z} = 0.7$ kpc result from Table 1, we obtain the dashed lines shown in Figure 11. This indicates that the thin-disk contamination is lower than that based directly on the C05 laws. Determinations from all three bins shown in Table 1 are also plotted in Figure 11. The most distant point from the plane is in reasonable agreement with the predicted values, while the nearest is simply a poor constraint. Since the two C05 laws are very similar with this new normalization at |z|> ∼ 0.5 kpc, we will adopt only one to correct the velocity dispersions, namely, the one with the higher scale height (dashed black line in Figure 11).

We correct the measured dispersions due to the contamination of the thin disk in the following way. The measured dispersion is

$$\begin{equation} \sigma _m^2 = a\times \sigma _a^2+b\times \sigma _b^2, \end{equation} \tag{ 1 }$$

where a is the fraction of thin-disk stars and σ_a is the dispersion of thin-disk stars, while b is the fraction of thick-disk stars and σ_b is the dispersion of thick-disk stars. We derive σ_b by adopting the fraction of thin- and thick-disk stars at each |z|from the relationship presented above, and adopting the following dispersions for the old, thin disk: $(\sigma _{V_R},\sigma _{V_{\theta }},\sigma _{V_z}) = (40, 24, 20)$ km s⁻¹ (Nordstrom et al. 2004). This correction is applicable if the mean velocities for thin and thick disks are the same, and thus can be applied in principle only to the V_R and V_z components. To understand the influence of thin-disk contamination on V_θ, we use Monte Carlo simulations to generate a population of thin- and thick-disk stars.

A set of 10,000 points are generated at a series of |z| values. Velocities are drawn from Gaussian distributions of given means and dispersions that resemble the thin and thick disks. The thick-disk rotation velocity varies linearly with |z|, with a gradient similar to that seen in observations. Velocities obtained this way are then smoothed with velocity errors, these being drawn from the observed distribution of errors in V_θ. The fraction of thin-disk stars as a function of |z| is that given by the C05 law, normalized with our data. We then calculate the average and the dispersion at each |z| value, applying the same procedure as with the observed data, i.e., probability plots for averages, and the intrinsic dispersion routine that eliminates measurement errors from the observed dispersions. We then compare these determined values with those from the input. We find that the dispersion correction is small (less than 2 km s⁻¹) for |z|>0.7 kpc and similar to the correction given by Equation (1). This is probably due to the fact that the thin-disk fraction is small at this distance from the plane (∼20% at |z| ∼ 0.8 kpc, see Figure 11). However, the average velocity is affected by the thin-disk contamination, well beyond |z| ∼ 0.7.

With this procedure, we determine the gradients of the velocity dispersions of the thick disk by applying Equation (1) and using only points with |z|>0.7 kpc, i.e., only seven bins, which are fit with a line. The fits are shown in Figure 12 (black lines) for each velocity component, together with our original data (gray symbols) and those corrected for thin-disk contamination (black symbols). The linear dependence determined by Girard et al. (2006) is shown with a red line; the thin-disk fraction is represented with a blue line; and the dashed line in the top plot represents one of the equilibrium models using the Jeans equation calculated in Girard et al. (2006) that best fit their data (namely, model 4, their Table 1 and Figure 8). The vertical dotted line indicates the limit |z| = 0.7 kpc. The results are summarized in Table 2.

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For both V_R and V_z our dispersion gradients are larger than those determined by Girard et al. (2006), which are 7.5 ± 3.1 km s⁻¹ kpc and 10.5 ± 3.3 km s⁻¹ kpc, respectively. This however does not represent disagreement, since the actual dependence is not linear. Girard et al. (2006) calculate several theoretical profiles which show that these dependencies are better approximated with quadratic functions. However they can be approximated with linear functions over limited |z| ranges. Toward low |z| values, the profiles become steeper: thus for the range in Girard et al. (2006) from 1 to 4 kpc, the dispersion gradient is shallower than ours where the range is from 0.7 to 2.5 kpc.

Finally, we estimate the gradient of V_θ as a function of |z|. Our simulations show that the average V_θ is affected by the thin-disk contamination out to |z| ∼ 1.1 kpc, where the bias is about the size of the error in the average velocity (i.e., ∼5 km s⁻¹). Only three bins in our data set are beyond this |z| value. Thus, based on our simulations, we derive the bias in the average velocity at each |z| bin. We apply this bias to all of our data bins, and then fit a line to obtain a gradient ∂V_θ/∂z = −25.2 ± 2.1 km s⁻¹ kpc, with V_θ|_z=0 = 200.8 ± 2.1 km s⁻¹. Since we are uncertain of the thin-disk fraction at low |z| values, and therefore of the correction to be applied, we provide yet another estimate of the gradient. We use only the outermost three data points from the binned data, and two points obtained from the two-Gaussian fit of V_θ distribution, those at |z| = 0.7 and 1.2 kpc (see Table 1). The gradient obtained this way is ∂V_θ/∂z = −24.4 ± 1.6 km s⁻¹ kpc, and V_θ|_z=0 = 202.3 ± 2.4 km s⁻¹, and in agreement with the previous determination. We also note that the gradient obtained using all bins, and no bias correction for thin-disk contamination is ∂V_R/∂z = −30.9 ± 2.1 km s⁻¹ kpc, indicating that the contamination tends to steepen the gradient. Our determination is less steep than that determined by Girard et al. (2006) of −30.3 ± 3.2 km s⁻¹ kpc; however, it is within 1σ error. The C10 determination of −36 ± 1 km s⁻¹ kpc is significantly different from ours.

We also calculate V_R and V_z as given by the average over bins at |z|>0.7 kpc, to avoid the issue associated with a local sample of stars as discussed in Section 4. These values are $\overline{V}_R = 12.7\pm 3.1$ km s⁻¹ and $\overline{V}_z = -0.7\pm 1.9$ km s⁻¹.

To study the dependence with R_GC, we first trim our sample to include stars at |z|>1.0 kpc. By doing so we avoid the contamination of the thin disk, while still having a representative number of stars. We then define three layers with 1.0 < |z| < 1.5 kpc (1115 stars), 1.5 < |z| < 2.0 kpc (401 stars), and 2.0 < |z| < 2.5 kpc (190 stars). In each of these layers, we calculate the velocity averages and intrinsic dispersions in four equal-number R_GC bins using the same procedures as in Section 5.1. The velocity average and dispersions as a function of R_GC are shown in Figure 13, for each velocity component. Each z-layer is represented with a different symbol: filled circles for 1.0 < |z| < 1.5 kpc, open circles for 1.5 < |z| < 2.0 kpc, and open triangles for 2.0 < |z| < 2.5 kpc. Here, velocity dispersions are not corrected for thin-disk contamination, which is very small and essentially similar in each z-layer.

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Inspection of Figure 13 shows that velocity averages do not show significant gradients with R_GC, except V_θ which decreases mildly with decreasing distance form GC. As for dispersions, the only significant gradient with R_GC is that for $\sigma _{V_z}$. In Table 3, we list the gradients for each velocity component, and each z-layer. These were obtained using linear fits to the data. Note that the middle layer spans the largest R_GC range.

Table 3. Velocity and Velocity Dispersion Gradients with R_GC

z-range	N	∂VR/∂RGC	∂Vθ/∂RGC	∂Vz/∂RGC		$\partial \sigma _{V_R} / \partial R_{\rm GC}$	$\partial \sigma _{V_{\theta }} / \partial R_{\rm GC}$	$\partial \sigma _{V_z} / \partial R_{\rm GC}$
(kpc)		(km s−1 kpc−1)				(km s−1 kpc−1)
1.0–1.5	1112	−1.0 ± 2.1	6.0 ± 1.4	1.3 ± 1.8		−1.6 ± 3.2	−1.7 ± 4.2	−7.5 ± 1.3
1.5–2.0	400	−0.5 ± 5.0	13.0 ± 4.5	6.6 ± 4.6		−1.6 ± 6.3	−1.0 ± 4.3	−8.3 ± 4.7
2.0–2.5	188	−2.5 ± 4.5	12.0 ± 8.1	2.1 ± 1.2		−3.4 ± 2.9	−6.0 ± 3.5	−8.4 ± 3.1

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The gradient $\partial \sigma _{V_z} / \partial R_{\rm GC}$ is related to the underlying gravitational potential and provides a dynamical estimate of the thin-disk scale length to be discussed in Section 7.

We derive the profile of $\sigma _{V_z}$ as a function of R_GC for a relaxed thick-disk population in equilibrium within the underlying static gravitational potential of the Galaxy. We use the Jeans equation

$$\begin{equation} \frac{\partial (\rho \overline{V_R V_z})}{\partial R} + \frac{\partial (\rho \overline{V_z^2})}{\partial z} + \frac{\rho \overline{V_R V_z}}{R} = - \rho \frac{\partial \Phi _{{\rm tot}}}{\partial z}. \end{equation} \tag{ 7 }$$

Here, R and z are Galactocentric cylindrical coordinates, ρ(R, z) is the volume density of the thin-disk stars, and Φ_tot(R, z) is the gravitational potential. Also, the velocity dispersions and cross term are

$$\begin{equation} \overline{V_R^2} = \sigma ^2_{V_R}, \overline{V_z^2} = \sigma ^2_{V_z}, \overline{V_R V_z} = \sigma ^2_{V_{Rz}}. \end{equation} \tag{ 8 }$$

We adopt for the cross term the expression derived by Binney & Tremaine (1987, p 199) in terms of the limiting case of spherical alignment:

$$\begin{equation} \overline{V_R V_z} = (\overline{V_R^2} - \overline{V_z^2}) \frac{z}{R} = \overline{V_z^2} \beta \frac{z}{R}, \end{equation} \tag{ 9 }$$

where β is a dimensionless number between 0 and 1, with 0 for cylindrical alignment of the velocity ellipsoid and 1 for spherical alignment. The density distribution of the thick-disk population is represented by exponential functions in radial and vertical directions:

$$\begin{equation} \rho (R,z) = \rho _0 \exp \left(- \frac{R}{R_{{\rm thick}}} - \frac{z}{z_{{\rm thick}}} \right), \end{equation} \tag{ 10 }$$

where R_thick and z_thick are the scale length and scale height of the thick disk. Using the formulations of Equations (8) and (9) in Equation (7), we obtain

$$\begin{equation} \overline{V_z^2(R,z)} = \frac{1}{\frac{z \beta }{R R_{{\rm thick}}} + \frac{1}{z_{{\rm thick}}}} \Bigg (\frac{\partial \Phi _{{\rm tot}}}{\partial z} - \frac{z \beta }{R} \frac{\partial \overline{V_z^2}}{\partial R} \Bigg). \end{equation} \tag{ 11 }$$

The Galactic potential separated by component is

$$\begin{equation} \Phi _{{\rm tot}}= \Phi _{{\rm bulge}} + \Phi _{{\rm disk}} + \Phi _{{\rm halo}}. \end{equation} \tag{ 12 }$$

The bulge potential adopted from the formulation of Johnston et al. (1995) is

$$\begin{equation} \Phi _{{\rm bulge}} = - \frac{G M_b}{r+c}, \end{equation} \tag{ 13 }$$

where $r = \sqrt{R^2+z^2}$, c = 0.7 kpc, G is the gravitational constant, and M_b = 3.4 × 10¹⁰M_☉ is the mass of the bulge. The disk potential, representing here the thin disk, is expressed with Bessel functions as in Girard et al. (2006). This potential, with an exponential surface density of the form Σ_thin(R) = Σ₀exp(−R/R_thin), is

$$\begin{equation} \Phi _{{\rm disk}} = -2\pi G \Sigma _0 R_{{\rm thin}}^2 \times \int _0^{\infty } \frac{J_0(kR) \exp (-k|z|) dk}{(1+k^2R_{{\rm thin}}^2)^{3/2}}, \end{equation} \tag{ 14 }$$

where R_thin is the scale length of the thin disk. The halo potential is a Plummer model given by

$$\begin{equation} \Phi _{{\rm halo}} = - \frac{G M_h}{(r^2+a^2)^{1/2}}, \end{equation} \tag{ 15 }$$

where a = 6.3 kpc is the core radius and M_h is the halo mass. The halo mass is expressed in terms of the rotational velocity of the LSR, V_c as

$$\begin{equation} V_c^2 = \frac{G M_h \it R_{\odot }^2}{(\it R_{\odot }^2+a^2)^{3/2}} + V_{{\rm disk}}^2(\it R_{\odot },0) + V_{{\rm bulge}}^2(\it R_{\odot },0), \end{equation} \tag{ 16 }$$

where V_disk and V_bulge represent the rotation associated with the thin-disk gravitational potential and the bulge potential, respectively. These can be expressed as

$$\begin{eqnarray} V_{{\rm disk}}^2(R,z) &=& R\frac{\partial \Phi _{{\rm disk}}}{\partial R} = 2 \pi G \Sigma _0 R_{{\rm thin}}^2 R\nonumber\\ && \times \int _0^{\infty } \frac{J_1(kR) \exp (-k|z|) k dk}{(1+k^2R_{{\rm thin}}^2)^{3/2}} \end{eqnarray} \tag{ 17 }$$

$$\begin{equation} V_{{\rm disk}}^2(\it R_{\odot },0) = 4 \pi G \Sigma _0 R_{{\rm thin}} x^2 \times [I_0 (x) K_0(x) - I_1(x) K_1(x)], \end{equation} \tag{ 18 }$$

where x = R_☉/2R_thin (see Freeman 1970; Binney & Tremaine 1987) and

$$\begin{equation} V_{{\rm bulge}}^2(R,z) = R \frac{\partial \Phi _{{\rm bulge}}}{\partial R} = \frac{G M_b R^2}{r(r+c)^2} \end{equation} \tag{ 19 }$$

$$\begin{equation} V_{{\rm bulge}}^2(\it R_{\odot },0) = \frac{G M_b R_{\odot }}{(R+c)^2}. \end{equation} \tag{ 20 }$$

In what follows, we will use Equation (11) to parameterize the dependence of $\overline{V_z^2}$ as a function of R, for a given z, where the term $\partial \overline{V_z^2} / \partial R$ is measured from our data.

Our theoretical $\overline{V_z^2}$ profiles are to be compared with data in one z layer, namely, 1.5 < |z| < 2.0 kpc, which includes ∼400 stars, has the largest span in R, and has no thin-disk contamination. The expression for $\overline{V_z^2}$ contains several parameters which have a range of possible values. These parameters are: β, ellipsoid alignment geometry; z_thick, the scale height of the thick disk; R_thick, the scale length of the thick disk; $\partial \overline{V_z^2} / \partial R$, the gradient of the velocity dispersion; Σ_thin, the surface density of the thin disk at the Sun's location; V_c, the rotation of the LSR; M_b, the bulge mass; c, the core radius of the bulge; a, the core radius of the halo; and finally, R_thin, the scale length of the thin disk. Parameter values are listed in Table 6 with the default value shown in bold.

Table 6. Model Parameters

Parameter	Values Explored
β	0, 1
zthick (kpc)	0.50, 0.75, 1.00
Rthick (kpc)	3.0, 4.1
$\partial \overline{V_z^2} / \partial R$ (km2 s−2 kpc−1)	−686, −759, −832
Σthin (M☉ pc−2)	42.0, 56.0
Vc (km s−1)	220, 250
Mb (1010M☉)	1.0, 3.4, 6.0
c (kpc)	0.4,0.7, 1.0
a (kpc)	5.0, 6.3, 7.0
Rthin (kpc)	1.5, 2.0, 2.5, 3.0

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The tests are made keeping all parameters fixed at default values and varying only one, to isolate its impact on the profile. If its influence is minimal, we eliminate it as a potential uncertainty in the final determination of the thin-disk scale length. The default parameters were chosen as follows: the thick-disk scale height and length are from De Jong et al. (2010), the surface density of the thin disk at the Sun's location is from Holmberg & Flynn (2004), the bulge parameters are from Johnston et al. (1995), and the halo core radius is estimated as in Girard et al. (2006). The value of $\partial \overline{V_z^2} / \partial R$ is from our measurements, while R_thin, which we will eventually fit, is initially set to the most common value found in the literature based on starcounts.

In Figure 15, we show the effect on the model profile of varying each parameter listed in Table 6, superposed on the observations. The default model is represented with a black line. Clearly, large changes in β, R_thick, $\partial \overline{V_z^2} / \partial R$, M_b, c, or a do not affect the profile at the level that observations can discriminate. We are thus left with four parameters, of which z_thick, Σ_thin, and V_c more or less show a net vertical shift in the profile when their values change, while R_thin causes the profile to change its shape, becoming steeper for shorter scale lengths. From the plots it is seen that the profiles are most sensitive to the parameters z_thick and R_thin. Thus we apply a χ² minimization for the fit of the observations with the model, to determine the most likely values of these two parameters. We do so separately for the four combinations of the test values of the second-most relevant parameters Σ_thin and V_c. The results are summarized in Table 7. Errors shown are formal errors derived from the fit, but are themselves highly uncertain due to the small number of bins. Formally, the values of the fitted parameters vary within 0.2 kpc for both the scale height of the thick disk and the scale length of the thin disk.

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We attempt to determine more realistic uncertainties in z_thick and R_thin by running Monte Carlo simulations of our binned data. We generate a set of 1000 representations of the four $\sigma _{V_{z}}$ data points, using Gaussian deviates from the observed values and with dispersion equal to each bin's uncertainty. Then, we run the χ² minimization routine with fixed Σ_thin = 56 M_☉ pc⁻² and V_c = 220 km s⁻¹ and determine a set of 1000 determinations of z_thick and R_thin. The resulting 1σ range is 0.1 kpc in z_thick and 0.4 kpc in R_thin. With these uncertainties, the solutions in Table 7 are equally valid, and we cannot distinguish between the two values for Σ_thin and V_c. We therefore adopt as our best determination z_thick = 0.7 ± 0.1 kpc, and R_thin = 2.0 ± 0.4 kpc. Our thick-disk scale height is in very good agreement with the determination by de Jong et al. (2010) of 0.75 ± 0.07 kpc from SDSS and with Girard et al. (2006) of 0.78 ± 0.05 kpc from 2MASS-selected red giants.

The thin-disk scale length is also in reasonable agreement with starcount determinations which range from roughly 2 to 3 kpc (Siegel et al. 2002; Cignoni et al. 2008 and references therein), with our value being on the low end of the range.