THE STELLAR KINEMATICS IN THE SOLAR NEIGHBORHOOD FROM LAMOST DATA

LAMOST is a quasi-meridian reflecting Schmidt telescope with an effective aperture of about 4 m and a field of view of $5^\circ$, operated by the National Astronomical Observatories, Chinese Academy of Sciences. It is a powerful instrument to survey the sky with the capability of recording 4000 spectra simultaneously (Cui et al. 2012; Zhao et al. 2012). In its five-year survey plan, it will obtain a few million stellar spectra in about 20,000 square degrees in the northern hemisphere (Deng et al. 2012).

The LAMOST Survey has internally delivered the first data release (DR1), which contains 2,204,860 spectra with a resolution of ∼1800 covering a wavelength range of $3800\lesssim \lambda \lesssim 9100\;\mathring{\rm{A}}$. The catalog contains 1,085,404 stellar spectra with estimated stellar atmospheric parameters and line-of-sight velocities. The distances to stars with estimated effective temperature, surface gravity, and metallicity are determined from isochrone fitting by Carlin et al. (2015). They developed two techniques, chi-square and Bayesian, to estimate the absolute magnitude. The chi-square technique finds the best-fit absolute magnitude that reaches the minimum of the chi-square between the measured ${T}_{\mathrm{eff}}$ log g, [Fe/H] and the isochrone ones. The Bayesian technique, on the other hand, considers both the selection effect and the different stellar populations along different lines of sight. They verified that the accuracies of the distance estimates derived from both techniques are ∼20%. They also found that their method may underestimate the distance of the giant stars by about 20%, but for main-sequence stars with $4000\;\lt \;$ ${T}_{\mathrm{eff}}$ $\;\lt \;7000$ K no correlation is found with ${T}_{\mathrm{eff}}$, as presented in Figure 1. Compared with the Hipparcos data, they obtained fairly good distance estimates with scatter of only ∼17%. Considering that only the FGK main-sequence stars are used in this work, the accuracy of the distance is sufficient for the studies of the local stellar kinematics within about 1 kpc. For such nearby stars, no significant difference is found between the two techniques. Therefore, we adopt the distance estimates from the chi-square technique in this work.

Zoom In Zoom Out Reset image size

We select nearby FGK-type main-sequence stars to investigate the kinematics of the solar neighborhood. In order to select main-sequence stars, we define the selection criteria separately for stars with ${T}_{\mathrm{eff}}$ larger and smaller than 5250 K. For the stars with ${T}_{\mathrm{eff}}$ higher than 5250 K (displayed by the blue dots in Figure 2), the selection criteria are as follows:

• 1.  
  5250 < ${T}_{\mathrm{eff}}$ < 7000 K,
• 2.  
  $\mathrm{log}g\ \gt$ (3.75–4.5)/(7000–5000)*(${T}_{\mathrm{eff}}$–5000) + 4.5,
• 3.  
  (−${T}_{\mathrm{eff}}$/500 + 14.1)$\;\lt \;{M}_{J}\;\lt \;$(−${T}_{\mathrm{eff}}$/550 + 15.0),

where M_J is the estimated absolute magnitude in J band from Carlin et al. For stars with ${T}_{\mathrm{eff}}$ < 5250 K (presented by the red points in Figure 2), the criteria are as follows:

• 1.  
  4000 < ${T}_{\mathrm{eff}}$ < 5250 K,
• 2.  
  $\mathrm{log}g\ \gt$ 4.2,
• 3.  
  (−${T}_{\mathrm{eff}}$/700 + 11.0)$\;\lt \;{M}_{J}\;\lt \;$(−${T}_{\mathrm{eff}}$/750 + 12.0).

The criteria on $\mathrm{log}g$ and M_J are empirically set to ensure that the selected stars are in the main sequence. The criteria for higher ${T}_{\mathrm{eff}}$ remove the possible subgiant branch stars so that the rest of the warm main-sequence stars may better trace the young populations. The criteria for lower ${T}_{\mathrm{eff}}$ exclude the giant stars. We further select stars whose spectra have signal-to-noise ratio higher than 10 in the g band, which is the most sensitive portion of the spectrum to the stellar parameters. We also select the stars within

• 1.  
  $100\lt | z| \;\lt \;$ 500 pc,
• 2.  
  $7\lt {R}_{{GC}}\;\lt \;$ 9 kpc,

adopting the position of the Sun to be $(X,Y,Z)$ = (8, 0, 0) kpc. The stars within $100\;\mathrm{pc}$ are not well observed in the LAMOST survey because most of these stars are too bright. Therefore, we only select the stars with $| z| \gt 100$ pc. Finally, a total of 209,316 FGK-type main-sequence stars are selected after applying all above criteria. It is worthwhile to point out that this sample is significantly extended compared to the previous works (DB98; S12; Nordström et al. 2004; Siebert et al. 2008; Büdenbender et al. 2014, etc.). Therefore, the statistics based on such a large sample may be substantially improved, and the sample containing a wide range of effective temperatures allows us to map the detailed kinematics for variant spectral types. Figure 3 shows the distribution of the sample in the Cartesian coordinate system with respect to the Galactic center (the top left, top right, and bottom left panels) and the ${T}_{\mathrm{eff}}$ versus $| z|$ plane (the bottom right panel). The five vertical dashed lines in the bottom right panel divide the effective temperature into six bins, in which the stellar kinematics are determined separately in the later sections. Figure 4 presents the distribution of the sample in the l versus b plane. From these figures, it is clear that the stars are not uniformly distributed in the sky. This leads to a distortion of the measured velocity ellipsoid, which is discussed in Section 3.3.2. Figure 5 displays that most of the selected stars have a signal-to-noise ratio in g band larger than 10.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

The LAMOST catalog provides the radial velocity of each star with uncertainty of about $5\;\mathrm{km}\;{{\rm{s}}}^{-1}$ (Gao et al. 2014). In order to investigate the systematics of the radial velocities, we cross-identify the LAMOST data with the APOGEE data released in SDSS DR10 (Ahn et al. 2014) and find ∼1000 common stars with good parameter estimates and velocity scatter smaller than $0.3\;\mathrm{km}\;{{\rm{s}}}^{-1}$ (which removes most multiple or variable stars) from APOGEE and signal-to-noise ratio higher than 20 in the LAMOST spectra. The radial velocity derived from the LAMOST pipeline is slower by 5.7 $\;\mathrm{km}\;{{\rm{s}}}^{-1}$ compared with APOGEE, as shown in Figure 6. The reason for this offset is unclear and is worthy of further investigation in future works. In this work, we simply add an additional 5.7 $\;\mathrm{km}\;{{\rm{s}}}^{-1}$ to the LAMOST-derived radial velocity to match the other survey data. It is noted that the offset is weakly anticorrelated with ${T}_{\mathrm{eff}}$, but there is no significant correlation with [Fe/H] and log g. We further investigate whether the weak anticorrelation with ${T}_{\mathrm{eff}}$ changes our result in Section 5.4.

Zoom In Zoom Out Reset image size

Consider a star with the line-of-sight unit vector ${\boldsymbol{r}}$ and heliocentric velocity of ${\boldsymbol{v}}$. The observed line-of-sight velocity v_los can be written as

$$\begin{eqnarray}&&{v}_{\mathrm{los}}={\boldsymbol{r}}\cdot {\boldsymbol{v}}.\end{eqnarray} \tag{ 1 }$$

${\boldsymbol{r}}$ is determined by the Galactic coordinates l and b, and the three components of ${\boldsymbol{v}}$ in the galactocentric Cartesian coordinates are U, V, and W, respectively. Then Equation (1) can be expanded as

$$\begin{eqnarray}&&{v}_{\mathrm{los}}=U\mathrm{cos}l\mathrm{cos}b+V\mathrm{sin}l\mathrm{cos}b+W\mathrm{sin}b.\end{eqnarray} \tag{ 2 }$$

Given a group of stars at similar distances along the line of sight ${\boldsymbol{r}}$, the velocity dispersion projected onto the line of sight should be

$$\begin{eqnarray}&&{\sigma }_{\mathrm{los}}^{2}=\displaystyle \frac{{\displaystyle \sum }_{i=1}^{N}{({{\boldsymbol{r}}}_{i}\cdot ({{\boldsymbol{v}}}_{i}-\bar{{\boldsymbol{v}}}))}^{2}}{N},\end{eqnarray} \tag{ 3 }$$

where N is the number of the stars. This can be specified as

$$\begin{eqnarray}{\sigma }_{\mathrm{los}}^{2} & = & {\sigma }_{U}^{2}{\mathrm{cos}}^{2}l{\mathrm{cos}}^{2}b+{\sigma }_{V}^{2}{\mathrm{sin}}^{2}l{\mathrm{cos}}^{2}b+{\sigma }_{W}^{2}{\mathrm{sin}}^{2}b\\ & + & {\sigma }_{{UV}}^{2}\mathrm{sin}2l{\mathrm{cos}}^{2}b+{\sigma }_{{UW}}^{2}\mathrm{cos}l\mathrm{sin}2b\\ & + & {\sigma }_{{VW}}^{2}\mathrm{sin}l\mathrm{sin}2b,\end{eqnarray} \tag{ 4 }$$

where the dispersions ${\sigma }_{U}$, ${\sigma }_{V}$, and ${\sigma }_{W}$ specify the size and shape of the velocity ellipsoids, and the cross-terms ${\sigma }_{{UV}}$, ${\sigma }_{{UW}}$, and ${\sigma }_{{VW}}$ determine the orientation of the velocity ellipsoids.

According to Equations (2) and (4), one is able to reconstruct the 3D mean velocity and velocity ellipsoid at a given spatial position from the line-of-sight velocities of a group of stars located at the same position. To do this, we construct a likelihood as follows:

$$\begin{eqnarray}&&L=\displaystyle \prod _{k=1}^{N}\{\displaystyle \frac{1}{\sqrt{{\sigma }_{\mathrm{model},k}^{2}+{\epsilon }_{k}^{2}}}\mathrm{exp}[-\displaystyle \frac{{({v}_{\mathrm{los},k}-{v}_{\mathrm{model},k})}^{2}}{2({\sigma }_{\mathrm{model},k}^{2}+{\epsilon }_{k}^{2})}]\},\end{eqnarray} \tag{ 5 }$$

where ${v}_{\mathrm{los},k}$ is the measured line-of-sight velocity of the kth star, ${\epsilon }_{\mathrm{los},k}$ is the measured error of the line-of-sight velocity, and ${v}_{\mathrm{model},k}$ and ${\sigma }_{\mathrm{model},k}$ are the predicted mean line-of-sight velocity and the projected velocity dispersion along the line of sight of the kth star, respectively. The following nine quantities, U, V, W, ${\sigma }_{U}$, ${\sigma }_{V}$, ${\sigma }_{W}$, ${\sigma }_{{UV}}$, ${\sigma }_{{UW}}$, and ${\sigma }_{{VW}}$, are free parameters in Equation (5). The best-fit values for them can be found by maximizing the likelihood L.

We use the Markov chain Monte Carlo (MCMC) simulation to estimate the posterior probabilities of the velocities and the velocity ellipsoids. The priors of the three mean velocities, U, V, and W, are evenly distributed in all real values, while the priors of the velocity ellipsoids are evenly distributed in the range of (0, $+\infty$). In this work, we run the MCMC simulation with ${\boldsymbol{e}}{mcee}$, which implements the affine-invariant ensemble sampler (Goodman & Weare 2010; Foreman-Mackey et al. 2012).

The velocity deprojection technique has been applied by DB98, who used the proper motions of Hipparcos to estimate the velocity and dispersions of stars with median distance of ∼70 pc around the Sun. Later, Fuchs et al. (2009) applied a similar method to the SDSS data, which extends to a few hundred parsecs away from the Sun. However, McMillan & Binney (2009) then pointed out that the deprojection technique may produce systematic bias in the cross-terms, especially in ${\sigma }_{{UW}}$, if σ is a function of R and z. Because our method is very similar to that DB98 used and our data are located between 100 and 500 pc in $| z|$, we may also suffer from similar biases. Furthermore, unlike the Hipparcos data, the highly uneven spatial distribution of the LAMOST data may lead to another systematic bias. Therefore, before applying this method to the LAMOST data, we need to carefully validate it and understand these issues.

3.3.1. Validation with GCS Data

Our first validation uses data from the Geneva-Copenhagen Survey (GCS; Nordström et al. 2004), which is a volume-complete set of F- and G-type stars within ∼40 pc, and provides three-dimensional velocity components for each individual star. This data set does not extend to a large volume; hence, it may not be severely affected by the systematic bias explained by McMillan & Binney (2009). Thus, we expect that the deprojection method may give estimates close to the true values.

A subsample of 3712 dwarf stars for the validation test is selected from GCS data with 5500 K < ${T}_{\mathrm{eff}}$ < 6000 K, $| U| \lt 200\;\mathrm{km}\;{{\rm{s}}}^{-1}$, $| V| \lt 400\;\mathrm{km}\;{{\rm{s}}}^{-1}$, and $| W| \lt 200\;\mathrm{km}\;{{\rm{s}}}^{-1}$. Outliers beyond 4σ in the line-of-sight velocity distribution are removed. Only the line-of-sight velocity of these data is used to derive their mean velocities and velocity ellipsoid according to Equation (5) via MCMC simulation. In order to reduce the effect of the uneven distribution on the sky, we separate the sky into small equal-area bins and randomly select roughly equal numbers of stars in each bin so that the samples are approximately evenly distributed on the whole sky. We repeat the arbitrary draw of the data in equal-area bins 40 times and estimate the velocity and its ellipsoid for each random drawing with MCMC. The final derived velocities and velocity dispersions are the averaged values over the 40 random draws. Uncertainties on the velocities and their dispersions are composed of two parts: (1) the internal error from the MCMC, and (2) the external error due to the selection of the sample. The total uncertainties of the derived values are the square root of the quadratic sum of the two parts. We compare the best-fit values with the true values directly measured from the U, V, and W of the individual stars (see Table 1) and find that the derived velocities and the ellipsoid are in good agreement with the true values. This confirms that the deprojection method can reconstruct the 3D kinematics for the stars within a very local volume.

Table 1.  Comparison of the Velocity Ellipsoids Derived from Different Methods for the GCS Data

	$\langle U\rangle$	$\langle V\rangle$	$\langle W\rangle$	$\langle {\sigma }_{U}\rangle$	$\langle {\sigma }_{V}\rangle$	$\langle {\sigma }_{W}\rangle$	$\langle {\sigma }_{{UV}}\rangle$	$\langle {\sigma }_{{UW}}\rangle$	$\langle {\sigma }_{{VW}}\rangle$
Direct	−9.86 ± 0.64	−23.49 ± 0.46	−7.57 ± 0.33	37.10 ± 0.58	25.69 ± 0.67	20.33 ± 0.37	11.04 ± 0.97	−6.37 ± 1.49	4.05 ± 2.39
Deprojection	−9.73 ± 3.01	−22.84 ± 0.74	−6.67 ± 1.22	35.40 ± 1.02	24.86 ± 0.95	18.97 ± 0.84	9.95 ± 2.04	−8.65 ± 5.79	3.17 ± 6.88

Note. The first line gives the results from the direct estimation, while the second row gives the results derived from the deprojection method.

Download table as:  ASCIITypeset image

3.3.2. Validation with the Mock Data

LAMOST data are neither volume complete nor completely cover the whole 4π sky area. This subsequently leads to large spatial distortion in the velocity deprojection method since many lines of sight are not observed. Moreover, the LAMOST data in this work spread from 100 to 500 pc in $| z|$, far beyond the volume of both GCS and Hipparcos data. The systematic bias due to the spatial variation of σ may be stronger in this larger volume, as pointed out by McMillan & Binney (2009). In order to investigate both sources of systematics, we introduce a second test, assigning mock line-of-sight velocities from predefined velocity ellipsoids to data with the same spatial position as actual LAMOST stars. The random draws of the line-of-sight velocities are based on a presumed Gaussian distribution function in galactocentric cylindrical coordinates with mean velocity components of [$\langle {v}_{R}\rangle$, $\langle {v}_{\phi }\rangle$, $\langle {v}_{z}\rangle$ ] = [0, −26, 0]$\;\mathrm{km}\;{{\rm{s}}}^{-1}$ and velocity dispersions of

$$\begin{eqnarray}{\boldsymbol{\sigma }}(R,z) & = & ({{\boldsymbol{\sigma }}}_{{R}_{0}}+20z,{{\boldsymbol{\sigma }}}_{{\phi }_{0}}+20z,{{\boldsymbol{\sigma }}}_{{z}_{0}}+30z)\;\mathrm{km}\;{{\rm{s}}}^{-1}\\ & \times & \mathrm{exp}[({R}_{0}-R)/{R}_{\sigma }],\end{eqnarray} \tag{ 6 }$$

where ${R}_{0}=8.0\;\mathrm{kpc}$ is the galactocentric radius of the Sun, ${R}_{\sigma }$ is set to 5 kpc, ${{\boldsymbol{\sigma }}}_{0}\equiv ({\sigma }_{{R}_{0}},{\sigma }_{{\phi }_{0}},{\sigma }_{{z}_{0}})=(45,32,24)\;\mathrm{km}\;{{\rm{s}}}^{-1}$ following McMillan & Binney (2009), and z is in $\;\mathrm{kpc}$. The cross-terms ${\sigma }_{R\phi }$, ${\sigma }_{{Rz}}$, and ${\sigma }_{\phi z}^{2}$ are set to 0. For each observed star, its stellar parameters, as well as the three-dimensional positions, are kept the same as the observed values, but its three-dimensional velocity vector is randomly drawn from the Gaussian distribution function. The mock line-of-sight velocity for each star is then given by Equation (2), presuming that $U\cong {-v}_{R}$, $V\cong {v}_{\phi }$, and $W\cong {v}_{z}$ in the solar neighborhood.

The mock sample is first split out into three slices in height, i.e., $100\;\mathrm{pc}\lt | z| \lt 300\;\mathrm{pc}$, $200\;\mathrm{pc}\lt | z| \lt 400\;\mathrm{pc}$, and $300\;\mathrm{pc}\lt | z| \lt 500\;\mathrm{pc}$. The overlap between neighboring slices can help to smooth the results. Then each $| z|$ slice is again split out into six bins in effective temperature with an interval of 500 K ranging from 4000 to 7000 K. Here we assume that the mean velocity components and velocity ellipsoids are the same for stars within the same ${T}_{\mathrm{eff}}$ bin in each $| z|$ slice. The results are displayed in Figure 7, with the solid lines representing the derived values and dashed lines for the true values. Because the velocity dispersions are functions of $| z|$ and stars with different effective temperatures have different spatial distributions, the true values are not strictly flat but show a slight bend with ${T}_{\mathrm{eff}}$. The reconstructed $\langle U\rangle$, $\langle V\rangle$, and $\langle W\rangle$ are consistent with the true values. However, neither the velocity dispersions nor the cross-terms are in agreement with the true values. Although McMillan & Binney (2009) showed similar deviation in the cross-terms, their test did not demonstrate the bias in the velocity dispersions. The inhomogeneous distribution in sky position of the LAMOST data sample (see Figure 3), which can induce even stronger geometric distortion, may be responsible for the deviation in the velocity dispersions.

Zoom In Zoom Out Reset image size

In general, the distortion in the velocity ellipsoid due to the unevenly distributed sample can be modeled as skew and tilt on the true velocity ellipsoid. Therefore, the distortion can be corrected by a skew and tilt model, which can be mathematically presented as matrix transformations.

The outline of the calibration is as follows. First, the mock stars are selected to be exactly located in the same spatial positions of the observed stars, combined with the simulated velocities based on a known velocity distribution function, which is only qualitatively similar to the true one. Because the mock data are selected with the identical spatial distribution as the observed data, the true velocity ellipsoid of the mock data would be affected by the same skew and tilt as the observations. Thus, we can measure the extents of the skew and tilt from the quantitive differences between the deprojection-derived velocity ellipsoids and their true values for the mock data.

In the rest of this section, we denote that all quantities with subscripts ${\cdot }_{o}$, ${\cdot }_{m}$, ${\cdot }_{t}$, and ${\cdot }_{c}$ represent the observed values, the values measured from the mock data, the true values, and the corrected values, respectively. First, we correct the velocity vector with a simple offset:

$$\begin{eqnarray}&&{{\boldsymbol{v}}}_{c}={{\boldsymbol{v}}}_{o}-\langle {\rm{\Delta }}{\boldsymbol{v}}\rangle ,\end{eqnarray} \tag{ 7 }$$

where $\langle {\rm{\Delta }}{\boldsymbol{v}}\rangle =\langle {{\boldsymbol{v}}}_{m}-{{\boldsymbol{v}}}_{t}\rangle$, which is derived by averaging over six simulations. Table 2 lists the mean offsets at different ${T}_{\mathrm{eff}}$ bins and $| z|$ slices. Most of the corrections are less than 1 $\;\mathrm{km}\;{{\rm{s}}}^{-1}$, implying that the distortion due to the unevenly distributed sample may not severely affect the derived mean velocities from the deprojection.

Table 2.  The Mean Offsets of Velocity to Calibrate the Geometric Distortion

Teff	$\langle N\rangle$	$\langle {\rm{\Delta }}U\rangle$	$\langle {\rm{\Delta }}V\rangle$	$\langle {\rm{\Delta }}W\rangle$
		100 $\;\lt \;| z| \lt 300$ pc
4250	13229	0.66 ± 0.68	0.22 ± 0.89	−0.09 ± 0.41
4750	19593	−0.10 ± 0.25	−0.23 ± 0.29	−0.23 ± 0.34
5250	25805	0.38 ± 0.38	−0.27 ± 0.40	−0.01 ± 0.32
5750	33453	−0.29 ± 0.20	0.15 ± 0.42	0.06 ± 0.37
6250	23565	0.36 ± 0.31	0.48 ± 0.30	−0.18 ± 0.53
6750	6739	0.24 ± 0.79	1.96 ± 1.29	−0.90 ± 1.44
		200 $\;\lt \;| z| \lt 400$ pc
4250	7600	2.44 ± 0.92	−0.22 ± 1.26	−0.31 ± 0.25
4750	16639	−0.26 ± 0.28	0.23 ± 0.73	0.05 ± 0.40
5250	27523	0.01 ± 0.32	0.10 ± 0.52	0.01 ± 0.23
5750	36303	−0.03 ± 0.27	0.27 ± 0.34	−0.05 ± 0.33
6250	18621	−0.22 ± 0.48	0.63 ± 0.63	−0.06 ± 0.46
6750	3436	−0.77 ± 0.92	2.13 ± 1.29	−0.15 ± 1.42
		300 $\;\lt \;| z| \lt 500$ pc
4250	3797	0.58 ± 2.11	0.04 ± 1.17	−0.30 ± 0.48
4750	11443	−0.16 ± 0.43	0.47 ± 0.76	−0.38 ± 0.51
5250	22503	−0.41 ± 0.50	0.04 ± 0.59	−0.01 ± 0.26
5750	33595	0.06 ± 0.27	0.20 ± 0.40	−0.40 ± 0.30
6250	13986	−0.24 ± 0.91	0.58 ± 0.44	−0.73 ± 0.29
6750	1566	−1.81 ± 2.24	1.36 ± 1.83	0.44 ± 2.09

Note. The $\langle {\rm{\Delta }}U\rangle$, $\langle {\rm{\Delta }}V\rangle$, and $\langle {\rm{\Delta }}W\rangle$ are the average offsets of the velocities for various ${T}_{\mathrm{eff}}$ bins at different $| z|$ slices.

Download table as:  ASCIITypeset image

Next, we consider the velocity ellipsoids. For simplicity, we assume that the stars are in a three-dimensional Gaussian distribution. It is worth noting that this assumption is not physically true, because the asymmetric drift of the stars in the Galactic disk skews the azimuthal velocity to the slower side (Binney & Tremaine 2008). An approximation of a Gaussian distribution may derive a slower peak of the azimuthal velocity distribution. However, the derived V and ${\sigma }_{V}$ based on the Gaussian approximation with the GCS data shown in Table 1 do not show obvious bias from the true values, implying that for the real observed data in the solar neighborhood, the Gaussian approximation may not contribute significant offsets in either V or ${\sigma }_{V}$. Therefore, we can still use a three-dimensional Gaussian profile to approximate the velocity distribution. A detailed investigation of the effect of the Gaussian approximation in the azimuthal velocity and its dispersion is discussed in Section 5.1.

We denote ${{\rm{}}\;{\rm{\Sigma }}}_{t}$ and ${{\rm{}}\;{\rm{\Sigma }}}_{m}$ as the true covariance matrices and the one derived with the mock data from the deprojection, respectively. Both can be rotated to align with the given axis by the following transform:

$$\begin{eqnarray}{{\boldsymbol{D}}}_{t} & = & {{\boldsymbol{V}}}_{t}^{-1}{{\rm{}}\;{\rm{\Sigma }}}_{t}{{\boldsymbol{V}}}_{t}\\ {{\boldsymbol{D}}}_{m} & = & {{\boldsymbol{V}}}_{m}^{-1}{{\rm{}}\;{\rm{\Sigma }}}_{m}{{\boldsymbol{V}}}_{m},\end{eqnarray} \tag{ 8 }$$

where ${{\boldsymbol{D}}}_{t}$ and ${{\boldsymbol{D}}}_{m}$ are diagonal matrices and ${{\boldsymbol{V}}}_{t}$ and ${{\boldsymbol{V}}}_{m}$ are the rotation transform matrices. The skew due to the inhomogeneity rescales the axis ratios and hence can be quantified by the division of the two diagonal matrices:

$$\begin{eqnarray}&&{{\boldsymbol{D}}}_{t}={\rm{}}\;{\rm{\Lambda }}{{\boldsymbol{D}}}_{m},\end{eqnarray} \tag{ 9 }$$

where ${\rm{}}\;{\rm{\Lambda }}$ is the scaled matrix. The tilt due to the inhomogeneity then can be quantified by the following transform:

$$\begin{eqnarray}&&{{\rm{}}\;{\rm{\Sigma }}}_{t}={{\boldsymbol{V}}}_{t}{\rm{}}\;{\rm{\Lambda }}{{\boldsymbol{V}}}_{m}^{-1}{{\rm{}}\;{\rm{\Sigma }}}_{m}{{\boldsymbol{V}}}_{m}{{\boldsymbol{V}}}_{t}^{-1}.\end{eqnarray} \tag{ 10 }$$

We define the correction matrices as ${\boldsymbol{A}}={{\boldsymbol{V}}}_{t}{\rm{}}\;{\rm{\Lambda }}{{\boldsymbol{V}}}_{m}^{-1}$ and ${\boldsymbol{B}}={{\boldsymbol{V}}}_{m}{{\boldsymbol{V}}}_{t}^{-1}$. Then, the observed velocity covariance ${{\rm{}}\;{\rm{\Sigma }}}_{o}$ can be calibrated by

$$\begin{eqnarray}&&{{\rm{}}\;{\rm{\Sigma }}}_{c}={\boldsymbol{A}}{{\rm{}}\;{\rm{\Sigma }}}_{o}{\boldsymbol{B}}.\end{eqnarray} \tag{ 11 }$$

The calibration matrices can be derived from the simulations with mock data combining the observed 3D positions with arbitrarily drawn velocities from given velocity ellipsoids. In practice, the random mock data may introduce some statistical fluctuations in ${\boldsymbol{A}}$ and ${\boldsymbol{B}}$. Therefore, we run six simulations to generate six sets of ${\boldsymbol{A}}$ and ${\boldsymbol{B}}$. The final calibrated velocity ellipsoid is averaged over the calibration by the six sets of transform matrices. Figure 8 shows the calibrated results, which are now consistent with their true values after the correction. It shows that not only the dispersions but also most of the cross-terms have been well reconstructed by the calibration. The calibration corrects the systematics raised from both the unevenly spatial distribution of the observed sample and the spatial variation of σ.

Zoom In Zoom Out Reset image size

The left panels of Figure 9 present the results of the three components of the average velocity ($\langle U\rangle$, $\langle V\rangle$, and $\langle W\rangle$).

The most obvious feature is that the three mean velocities shown in the left panels are all correlated with effective temperature for all $| z|$ bins. For the radial velocity $\langle U\rangle$ in the top left panel, at ${T}_{\mathrm{eff}}$ = 4250 K, $\langle U\rangle$ is around 10–12 $\;\mathrm{km}\;{{\rm{s}}}^{-1}$, which is at around 0 $\;\mathrm{km}\;{{\rm{s}}}^{-1}$ with respect to the local standard of rest (LSR; the black dashed line in the panel) adopting the solar motion of ${U}_{\odot }=11.1$ $\;\mathrm{km}\;{{\rm{s}}}^{-1}$ (Schönrich et al. 2010). In other ${T}_{\mathrm{eff}}$ bins, $\langle U\rangle$ for the stars within $100\lt | z| \lt 300$ pc are approximately flat at around $-9.07$ to $-8.74$ $\;\mathrm{km}\;{{\rm{s}}}^{-1}$. For the stars in $200\lt | z| \lt 400$ ($300\lt | z| \lt 500$), on the other hand, $\langle U\rangle$ increases from $-8.98\pm 1.47$ ($-9.37\pm 2.04$) $\;\mathrm{km}\;{{\rm{s}}}^{-1}$ at ${T}_{\mathrm{eff}}$ $\;=\;4750$ K to $-6.49\pm 1.54$ ($-7.24\pm 2.09$) $\;\mathrm{km}\;{{\rm{s}}}^{-1}$ at ${T}_{\mathrm{eff}}$ $\;=\;6750$ K. Although the change of $\langle U\rangle$ from cool to warm is small, the increasing trend is quite clear, especially when ${T}_{\mathrm{eff}}$ $\;\gt \;6000$ K. The direction of $\langle U\rangle$ is toward the Galactic center; thus, the top left panel of Figure 9 indicates that either the warmer stars tend to move inward in the Galaxy or the cooler stars tend to move outward.

For the azimuthal velocity $\langle V\rangle$ in the middle left panel, the stars show a clear break at ${T}_{\mathrm{eff}}$ $\;=\;6000$ K. The stars cooler than 6000 K are located at around −30 $\;\mathrm{km}\;{{\rm{s}}}^{-1}$, while those warmer than 6000 K go abruptly up to around −20 $\;\mathrm{km}\;{{\rm{s}}}^{-1}$. This is another version of Parenago's discontinuity (Parenago 1950) and has been rediscovered by DB98 at $B-V=0.61$, which corresponds to ${T}_{\mathrm{eff}}$ ∼ 6000 K according to Sekiguchi & Fukugita (2000). The cooler stars are, on average, older than the warmer stars, and hence most of them have had a longer time to experience scattering, which can increase the velocity dispersion and therefore increase the asymmetric drift (Jenkins 1992; Binney & Tremaine 2008). This leads to a slower azimuthal velocity for the older (cooler) stars. Another interesting feature is that the LAMOST data show an unusual gradient along $| z|$ for warmer stars. For the stars cooler than 6000 K, the azimuthal velocity decreases with $| z|$, while it turns out to increase with $| z|$ for the stars warmer than 6000 K. The former trend is natural since the stars at higher $| z|$ need more vertical energy and hence slightly lose their angular momenta and rotate more slowly, given that the stars are in equilibrium. However, it is difficult to understand why the warmer, or younger, stars show the trend turning around.

As did the other two velocity components, the vertical velocity $\langle W\rangle$ also shows a correlation with effective temperature. Compared to the zero velocity with respect to the LSR (shown as the black dashed horizontal line at 7.25 $\;\mathrm{km}\;{{\rm{s}}}^{-1}$ Schönrich et al. 2010), the stars with ${T}_{\mathrm{eff}}$ $\;\lt \;5500$ K roughly stay around the zero point. However, the warmer stars with ${T}_{\mathrm{eff}}$ $\;\gt \;5500$ K show higher $\langle W\rangle$ than the zero point, i.e., they are moving toward the Galactic north pole up to $\langle W\rangle$ $\sim \;-4$ $\;\mathrm{km}\;{{\rm{s}}}^{-1}$, which is equivalent to +3 $\;\mathrm{km}\;{{\rm{s}}}^{-1}$ with respect to LSR. The stars located at larger $| z|$ seem to have a smaller $\langle W\rangle$ than those located in lower $| z|$, although the trend is not quite statistically significant.

The stars cooler than 6000 K over all $| z|$ bins are on average old. Since they do not show a significant gradient in $\langle V\rangle$, it is safe to assume that they are in equilibrium. Then, they can be used to determine the solar motion with respect to LSR. The solar motion in radial and vertical directions can be derived from

$$\begin{eqnarray}{U}_{\mathrm{LSR}} & = & -\langle U\rangle \\ {W}_{\mathrm{LSR}} & = & -\langle W\rangle .\end{eqnarray} \tag{ 12 }$$

Thus, we directly obtain ${U}_{\odot }=9.58\pm 2.39\;\mathrm{km}\;{{\rm{s}}}^{-1}$ and ${W}_{\odot }=7.01\pm 1.67\;\mathrm{km}\;{{\rm{s}}}^{-1}$ by averaging over the $\langle U\rangle$ and $\langle W\rangle$ listed in the third and fifth columns of Table 8 for stars below 6000 K in all $| z|$ slices.

In order to estimate ${V}_{\odot }$, the asymmetric drift is required and a dynamical model has to be introduced. We start from Equation (4.228) of Binney & Tremaine (2008), which gives the approximation of the asymmetric drift from the circular speed, velocity dispersions, and the stellar density:

$$\begin{eqnarray}{\bar{\upsilon }}_{s}-{V}_{\odot } & = & {\upsilon }_{a}\simeq \displaystyle \frac{\bar{{\upsilon }_{R}^{2}}}{2{\upsilon }_{c}}\\ & & [\displaystyle \frac{{\sigma }_{\phi }^{2}}{\bar{{\upsilon }_{R}^{2}}}-1-\displaystyle \frac{\partial \mathrm{ln}(\nu \bar{{\upsilon }_{R}^{2}})}{\partial \mathrm{ln}R}-\displaystyle \frac{R}{\bar{{\upsilon }_{R}^{2}}}\displaystyle \frac{\partial (\bar{{\upsilon }_{R}{\upsilon }_{z}})}{\partial z}],\end{eqnarray} \tag{ 13 }$$

where R is galactocentric radius, z is the height above the plane, ${\upsilon }_{c}$ is the circular speed, and ν is the number density of stars, while a bar indicates the average value. Because the data are very close to the Galactic midplane, the cross-term of $\bar{{v}_{R}{v}_{z}}$ does not significantly vary with $| z|$ (Büdenbender et al. 2014), and therefore the last term of the equation is very small and can be negligible. Assume that the stellar density ν and ${\sigma }_{U}$ are exponential functions of R:

$$\begin{eqnarray}&&\nu \sim \mathrm{exp}(-\displaystyle \frac{R}{{h}_{R}})\end{eqnarray} \tag{ 14 }$$

$$\begin{eqnarray}&&{\sigma }_{U}^{2}\sim \mathrm{exp}(-\displaystyle \frac{R}{L}),\end{eqnarray} \tag{ 15 }$$

where h_R and L are scale length for the stellar density and ${\sigma }_{U}$, respectively. Then Equation (4.228) can be rewritten as

$$\begin{eqnarray}&&{v}_{a}\cong \displaystyle \frac{1}{2{v}_{c}}[{\sigma }_{V}^{2}+(\displaystyle \frac{1}{{h}_{R}}+\displaystyle \frac{1}{L}-\displaystyle \frac{1}{R})R{\sigma }_{U}^{2}].\end{eqnarray} \tag{ 16 }$$

Therefore, the solar motion in V can be estimated from

$$\begin{eqnarray}&&{V}_{\odot }=-(\langle V\rangle +{v}_{a}).\end{eqnarray} \tag{ 17 }$$

Here we set ${h}_{R}=2.6$ kpc following Jurić et al. (2008) or ${h}_{R}=2.5$ kpc following Schönrich & Binney (2012) and Sharma et al. (2014). According to van der Kruit & Freeman (2011), L is equal to $2{h}_{R}$ given an isothermal disk. Although this relation has not been confirmed for the case of our Galaxy, it is a reasonable guess. As a comparison, the different case of ${h}_{R}=3.0$ kpc and $L=6.0$ kpc is also considered. The final results, including all three components, are listed in Table 3. Our estimates of the solar motion in three dimensions are in agreement with Schönrich et al. (2010) and Huang et al. (2015).

Table 3.   Solar Motion with Respect to the LSR

	$(\;\mathrm{km}\;{{\rm{s}}}^{-1})$
${U}_{\odot }$	9.58 ± 2.39
${V}_{\odot }$	10.52 ± 1.96 (hR = 2.6 kpc, L = 5.2 kpc)
	10.05 ± 1.98 (hR = 2.5 kpc, L = 5.0 kpc)
	13.09 ± 1.85 (hR = 3.0 kpc, L = 6.0 kpc)
${W}_{\odot }$	7.01 ± 1.67

Download table as:  ASCIITypeset image

The middle panels of Figure 9 show the velocity dispersions as functions of ${T}_{\mathrm{eff}}$ and $| z|$, and the right panels show the derived cross-terms. The velocity dispersions show clear trends along either ${T}_{\mathrm{eff}}$ or $| z|$. First, all three velocity dispersions show an abrupt drop at around ${T}_{\mathrm{eff}}$ ∼ 6000 K. This sudden drop has been seen in all $| z|$ bins. It is in agreement with the turnaround point in $\langle V\rangle$ shown in the middle left panel of Figure 9. The decreased velocity dispersions for stars with ${T}_{\mathrm{eff}}$ $\;\gt \;6000$ K again imply that the young stars are less affected by scattering and hence are kinematically cooler.

Second, the velocity dispersions show vertical gradients in most of the ${T}_{\mathrm{eff}}$ bins. Particularly, the vertical gradient of ${\sigma }_{U}$ is correlated with ${T}_{\mathrm{eff}}$, as shown in the second column of Table 4. Although the vertical gradients of ${\sigma }_{V}$ and ${\sigma }_{W}$ are mostly larger that those of ${\sigma }_{U}$, they do not show clear trends along ${T}_{\mathrm{eff}}$ (also see Table 4).

Table 4.   Vertical Gradient of the Velocity Dispersions at Each ${T}_{\mathrm{eff}}$ Bin

${T}_{\mathrm{eff}}$	$d{\sigma }_{U}/d| z|$	$d{\sigma }_{V}/d| z|$	$d{\sigma }_{W}/d| z|$
(K)	$(\mathrm{km}\;{{\rm{s}}}^{-1}\;{\mathrm{kpc}}^{-1})$	$(\mathrm{km}\;{{\rm{s}}}^{-1}\;{\mathrm{kpc}}^{-1})$	$(\mathrm{km}\;{{\rm{s}}}^{-1}\;{\mathrm{kpc}}^{-1})$
4250	−0.2 ± 1.4	54.5 ± 3.4	20.7 ± 0.7
4750	10.4 ± 1.7	22.9 ± 2.3	18.9 ± 1.0
5250	12.7 ± 2.1	14.3 ± 0.4	13.1 ± 0.5
5750	14.1 ± 2.9	30.7 ± 1.2	24.6 ± 1.1
6250	21.8 ± 0.4	29.4 ± 7.0	22.3 ± 2.2
6750	23.0 ± 3.4	5.8 ± 0.5	28.4 ± 8.3

Download table as:  ASCIITypeset image

The ratios of the velocity dispersions shown in Figure 10 indicate the shape of the velocity ellipsoids. Columns 12 and 13 in Table 8 also list the values of the two ratios. Both the ratios drop with $| z|$ in most ${T}_{\mathrm{eff}}$ bins. The only exception is at ${T}_{\mathrm{eff}}$ $\;=\;6750$ K, in which the ratios seem not correlated with $| z|$ (Figure 10).

Zoom In Zoom Out Reset image size

With derived cross-terms, we can also estimate the vertex deviation and the tilt angle at each ${T}_{\mathrm{eff}}$ bin and $| z|$ slice. The definition of the vertex deviation is

$$\begin{eqnarray}&&{l}_{v}\equiv \displaystyle \frac{1}{2}\mathrm{arctan}(\displaystyle \frac{2{\sigma }_{{UV}}^{2}}{{\sigma }_{U}^{2}-{\sigma }_{V}^{2}}),\end{eqnarray} \tag{ 18 }$$

and the definition of the tilt angle is

$$\begin{eqnarray}&&\alpha \equiv \displaystyle \frac{1}{2}\mathrm{arctan}(\displaystyle \frac{2{\sigma }_{{UW}}^{2}}{{\sigma }_{U}^{2}-{\sigma }_{W}^{2}}).\end{eqnarray} \tag{ 19 }$$

The right panels of Figure 9 show the derived cross-terms of the velocity ellipsoids. Their values are listed in Table 8. Figure 11 shows the estimated l_v and α as functions of ${T}_{\mathrm{eff}}$ and $| z|$.

Zoom In Zoom Out Reset image size

We find that l_v varies with $| z|$. It is around ${0}^{\circ }$ for stars with ${T}_{\mathrm{eff}}$ $\;\lt \;6500$ K at $100\lt | z| \lt 300$ and $200\lt | z| \lt 400$ pc, and then it becomes positive at $300\lt | z| \lt 500$ pc. The vertex deviation for stars with ${T}_{\mathrm{eff}}$ $\;\gt \;6500$ K as an exception changes with $| z|$ in the opposite sense, i.e., it decreases with $| z|$ from zero to $-6\buildrel{\circ}\over{.} 7$, although the uncertainty is very large. On the other hand, the tilt angle does not show clear trends with either ${T}_{\mathrm{eff}}$ or $| z|$, which is different from Binney et al. (2014, hereafter B14), who found that the tilt angle changes for the hot dwarfs compared to the giants or cool dwarfs using RAVE (Kordopatis et al. 2013) data. The reason for the discrepancy is hard to investigate, because of the large difference of the volumes between the LAMOST and RAVE data and the difference in methodologies.

Although the volume of our sample at $100\lt | z| \lt 300$ pc does not exactly overlap with that of the sample of DB98, which only extends to 70 pc, we will compare our sample to that of DB98 under the assumption that the velocity dispersions do not largely change from $| z| \lt 100$ to 300 pc. The color index $B-V$ used by DB98 can be converted into effective temperature according to Sekiguchi & Fukugita (2000). Specifically, the data at around $B-V=0.412$ (the color index bin 4 in DB98) correspond to ${T}_{\mathrm{eff}}$ ∼ 6600 K, the data at around $B-V=0.525$ (bin 6 in DB98) correspond to ${T}_{\mathrm{eff}}$ ∼ 6200 K, the data at around $B-V=0.582$ (bin 7 in DB98) correspond to ${T}_{\mathrm{eff}}$ ∼ 5800 K, and the data at $B-V=0.719$ (bin 9 in DB98) correspond to ${T}_{\mathrm{eff}}$ ∼ 5200 K. Because DB98 did not directly show the dispersions along the U, V, and W directions, but showed the dispersions along the major axis of the velocity ellipsoids and the ratio of the major axis to the middle and minor axis, we also rotate the velocity ellipsoid shown in Table 8 and find the longest, middle, and shortest axes of the derived ellipsoids to compare with DB98. Table 5 shows the comparison of the largest velocity dispersion ${\sigma }_{1}$ and the two ratios of ${\sigma }_{1}/{\sigma }_{2}$ and ${\sigma }_{1}/{\sigma }_{3}$ between DB98 and this work. In most cases, the velocity dispersions with roughly the same ${T}_{\mathrm{eff}}$ are slightly larger in our data than those in DB98 by up to ∼3 $\;\mathrm{km}\;{{\rm{s}}}^{-1}$. This is reasonable since the height of the data in this work is slightly larger than those in DB98.

Table 5.  Comparison of the Velocity Dispersions between DB98 and This Work

${T}_{\mathrm{eff}}$	$B-V$	DB98				This Work ($100\lt | z| \lt 300$)
		${\sigma }_{1}$	${\sigma }_{1}/{\sigma }_{2}$	${\sigma }_{1}/{\sigma }_{3}$	lv	${\sigma }_{1}$	${\sigma }_{1}/{\sigma }_{2}$	${\sigma }_{1}/{\sigma }_{3}$	lv
(K)	(mag)	$(\;\mathrm{km}\;{{\rm{s}}}^{-1})$			(degree)	$(\;\mathrm{km}\;{{\rm{s}}}^{-1})$			(degree)
5250	0.719	${37.20}_{-0.93}^{+1.41}$	${1.44}_{-0.12}^{+0.12}$	${2.04}_{-0.16}^{+0.61}$	${13.1}_{-7.6}^{+6.7}$	40.81 ± 5.06	1.51 ± 0.29	1.76 ± 2.05	−1.63 ± 5.90
5750	0.582	${37.64}_{-0.94}^{+1.37}$	${1.61}_{-0.18}^{+0.07}$	${1.78}_{-0.04}^{+0.48}$	${10.2}_{-6.0}^{+5.6}$	39.97 ± 5.11	1.50 ± 0.33	1.86 ± 1.21	−1.80 ± 8.98
6250	0.525	${32.93}_{-0.75}^{+1.09}$	${1.51}_{-0.12}^{+0.13}$	${2.19}_{-0.19}^{+0.64}$	${1.9}_{-6.1}^{+6.0}$	32.97 ± 5.77	1.62 ± 0.40	1.91 ± 3.11	0.45 ± 6.62
6750	0.412	${26.26}_{-0.59}^{+0.80}$	${1.66}_{-0.15}^{+0.12}$	${2.16}_{-0.15}^{+0.52}$	${10.2}_{-5.4}^{+5.0}$	26.05 ± 5.47	1.32 ± 0.44	1.56 ± 2.92	−0.27 ± 9.63

Download table as:  ASCIITypeset image

It also shows that the ratio ${\sigma }_{1}/{\sigma }_{2}$ derived in this work is, on average, consistent with DB98, while ${\sigma }_{1}/{\sigma }_{3}$ is slightly smaller than DB98. In other words, compared with DB98, the velocity ellipsoid in this work is slightly broader in the z direction, which is probably because the LAMOST data extend over a larger range in $| z|$. Notice that the errors in axis ratios are larger in this work. These errors may be overestimated because they are propagated from the large errors in the cross-terms via the rotation transforms, rather than deriving directly from the observed data.

Surprisingly, for the stars located at $100\lt | z| \lt 300$ pc, the vertex deviation is around zero for all ${T}_{\mathrm{eff}}$ bins. This is inconsistent with the results in DB98. First, it is noted that Sections 3.3.1 and 3.3.2 have shown that the cross-terms of the velocity ellipsoids are the easiest affected in the velocity deprojection method, as McMillan & Binney (2009) have stated. Both the methods in DB98 and in this work are based on the velocity deprojection. Therefore, both the results of the vertex deviation are possibly affected by systematic bias. Because DB98 deprojected the velocity components from the tangental velocities measured from proper motions, while we derive them from the line-of-sight velocities, the systematic bias in both works may go toward different directions. Second, even though we have calibrated the velocity ellipsoid, the corrected ${\sigma }_{{UW}}$ may still slightly bias from their true values (see the top right panels of Figure 8). As a consequence, it is quite hard to compare the vertex deviation between this work and DB98. This issue cannot be easily solved until the three-dimensional velocities of stars are provided.

B14 analyzed the kinematics of ∼400,000 RAVE stars and decomposed the sample into hot and cold dwarfs, red clump giants, and non-clump giants. For each of these classes, B14 provided an analytic model for the velocity ellipsoid at each point in the (R, z) plane. With the help of this model, we compared the velocity dispersions between B14 and this work. Table 6 shows the comparison of the largest velocity dispersion ${\sigma }_{1}$ and the two ratios of ${\sigma }_{1}/{\sigma }_{2}$ and ${\sigma }_{1}/{\sigma }_{3}$ between B14 and this work in the different layers. The ${\sigma }_{1}$ , ${\sigma }_{1}$/${\sigma }_{2}$, and ${\sigma }_{1}$/${\sigma }_{3}$ between the two works are in good agreement with each other within 1σ for most cases. However, for stars with ${T}_{\mathrm{eff}}$ $\;\gt \;6000$ K and $200\lt | z| \lt 400$ and those with ${T}_{\mathrm{eff}}$ $\;\lt \;6000$ and $300\lt | z| \lt 500$, the two works have differences by about 2σ.

Table 6.  Comparison of the Velocity Dispersions between B14 and This Work

Height	${T}_{\mathrm{eff}}$	B14			This Work
		${\sigma }_{1}$	${\sigma }_{1}/{\sigma }_{2}$	${\sigma }_{1}/{\sigma }_{3}$	${\sigma }_{1}$	${\sigma }_{1}/{\sigma }_{2}$	${\sigma }_{1}/{\sigma }_{3}$
(pc)	(K)	$(\;\mathrm{km}\;{{\rm{s}}}^{-1})$			$(\;\mathrm{km}\;{{\rm{s}}}^{-1})$
$100\lt | z| \lt 300$	>6000	30.47 ± 0.70	1.52 ± 0.11	1.95 ± 0.08	29.51 ± 4.89	1.47 ± 0.21	1.73 ± 0.25
	<6000	38.42 ± 2.30	1.52 ± 0.05	1.71 ± 0.05	40.71 ± 0.54	1.51 ± 0.01	1.80 ± 0.05
$200\lt | z| \lt 400$	>6000	32.70 ± 1.54	1.46 ± 0.19	1.80 ± 0.08	31.20 ± 5.53	1.51 ± 0.25	1.67 ± 0.09
	<6000	43.49 ± 2.77	1.57 ± 0.04	1.75 ± 0.01	42.14 ± 0.80	1.43 ± 0.09	1.70 ± 0.03
$300\lt | z| \lt 500$	>6000	36.57 ± 0.70	1.40 ± 0.18	1.69 ± 0.03	34.23 ± 4.78	1.42 ± 0.06	1.61 ± 0.16
	<6000	48.76 ± 2.50	1.63 ± 0.04	1.67 ± 0.07	42.83 ± 0.76	1.32 ± 0.15	1.62 ± 0.05

Download table as:  ASCIITypeset image

It is not easy to compare the dispersions with S12 since they separate the data into metallicity bins rather than temperature or color index. Moreover, most of the data in S12 are at heights larger than 500 pc, which does not overlap with our data. However, a qualitative comparison is still worthwhile since S12 directly used three velocity components to derive the velocity ellipsoids. First, we compare the velocity dispersions at $300\lt | z| \lt 500$ pc and ${T}_{\mathrm{eff}}$ $\;=\;6250$ K (see Table 8) with the data at $-0.5\lt$ [Fe/H] $\;\lt \;0.2$ and $\langle z\rangle =-0.69$ kpc in S12, which have ${\sigma }_{R}={38.8}_{-1.3}^{+1.2}$ $\;\mathrm{km}\;{{\rm{s}}}^{-1}$, ${\sigma }_{\phi }={27.7}_{-0.4}^{+0.4}$ $\;\mathrm{km}\;{{\rm{s}}}^{-1}$, and ${\sigma }_{z}={22.4}_{-0.7}^{+0.7}$ $\;\mathrm{km}\;{{\rm{s}}}^{-1}$, and find that they agree with each other. Then, we compare ${l}_{v}=9\buildrel{\circ}\over{.} 60\pm 2\buildrel{\circ}\over{.} 56$ with their result, which is $8\buildrel{\circ}\over{.} 3\pm 3\buildrel{\circ}\over{.} 4$. These two quantities are again consistent with each other. We also compare the tilt angle $\alpha =-1\buildrel{\circ}\over{.} 73\pm 0\buildrel{\circ}\over{.} 76$ with S12, which obtained $-5\buildrel{\circ}\over{.} 6\pm 2\buildrel{\circ}\over{.} 1.$ Although the directions of the tilt angles are same, the values have about $2\sigma$ difference. This is probably because the tilt angle increases very rapidly with $| z|$ (Büdenbender et al. 2014), and the data in S12 are located in larger $| z|$ than those in this work.

It is well known that the V distribution is not Gaussian but has a long tail to the slow side; hence, the Gaussian approximation in Equation (5) may not be correct. It is worthwhile to investigate the systematic bias in the derived velocity ellipsoids due to the Gaussian approximation.

We adopt the analytical form of the distribution function defined by Equation (15) of Cuddeford & Binney (2012), namely,

$$\begin{eqnarray}f(R,{v}_{R},{v}_{\phi }) & \sim & \mathrm{exp}(-\displaystyle \frac{{v}_{R}^{2}}{2{\sigma }_{d}^{2}}{e}^{{{yv}}_{\phi }/{v}_{c}})\\ & & \times \mathrm{exp}\{\displaystyle \frac{1}{2{\sigma }_{d}^{2}}[{v}_{\phi }^{2}-{v}_{c}^{2}\\ & & +2{v}_{c}^{2}\mathrm{ln}(\displaystyle \frac{{v}_{c}}{{v}_{\phi }})]{e}^{{{yv}}_{\phi }/{v}_{c}}\},\end{eqnarray} \tag{ 20 }$$

where R is the galactocentric radius, v_R the radial velocity, ${v}_{\phi }$ the azimuthal velocity, and $y\equiv 8/{R}_{d}$. The free parameters are R_d, ${\sigma }_{d}$, and v_c. We predefine three sets of the free parameters and generate three sets of mock data via Monte Carlo simulations. Then we use a Gaussian approximation to estimate the mean ${v}_{\phi }$ and ${\sigma }_{\phi }$. In all three simulations, we set R = 8 kpc, ${v}_{c}=220$ $\;\mathrm{km}\;{{\rm{s}}}^{-1}$, ${R}_{d}=2.5$ kpc. v_R follows a Gaussian distribution with zero mean and sigma of ${\sigma }_{U}={\sigma }_{d}\mathrm{exp}(-R/{R}_{d})$. We also add ${V}_{\odot }=12$ $\;\mathrm{km}\;{{\rm{s}}}^{-1}$ in the simulations. The only varying parameter is ${\sigma }_{d}$, which is chosen as 80, 110, and 140 $\;\mathrm{km}\;{{\rm{s}}}^{-1}$, respectively, for each simulation. We then compare with the directly estimated mean and standard deviation of ${v}_{\phi }$ without assuming a Gaussian profile. The ${V}_{\odot }$ is also calculated according to Equations (16) and (17) given L = 5 kpc. The comparison is listed in Table 7. This shows that $\langle V\rangle$ is underestimated by a larger amount as ${\sigma }_{d}$ increases. On the other hand, the velocity dispersion estimates from Gaussian fitting are also systematically underestimated by a few $\;\mathrm{km}\;{{\rm{s}}}^{-1}$. As a consequence, the derived ${V}_{\odot ,{\rm{G}}}$ from the Gaussian fitting is also slightly smaller than the true value.

Table 7.  The Test Results of the Non-Gaussianity of V

${\sigma }_{d}$	$\langle V\rangle$	${\sigma }_{V}$	${V}_{\odot }$ a
$(\;\mathrm{km}\;{{\rm{s}}}^{-1})$
80	$-16.6\pm 2.5$	12.7 ± 0.5	14.0 ± 2.4
110	$-20.2\pm 2.7$	19.9 ± 1.1	15.2 ± 2.6
140	$-32.0\pm 14.1$	33.3 ± 5.6	22.7 ± 13.1
${\sigma }_{d}$	$\langle V{\rangle }_{G}$	${\sigma }_{V,{\rm{G}}}$	${V}_{\odot ,{\rm{G}}}$ b
$(\;\mathrm{km}\;{{\rm{s}}}^{-1})$
80	$-15.0\pm 2.3$	11.9 ± 0.4	12.5 ± 2.3
110	$-16.4\pm 2.2$	16.7 ± 0.5	11.6 ± 2.2
140	$-22.2\pm 10.9$	23.5 ± 4.4	14.2 ± 10.3

Notes.

^aThe mean and standard deviation of V are calculated directly from their definitions. The ${V}_{\odot }$ is calculated from the direct calculated standard deviation. No Gaussianity is assumed in these derived values. ^bThe mean and sigma values are calculated from fitting a Gaussian profile. And ${V}_{\odot ,{\rm{G}}}$ is calculated based on the Gaussian sigma ${\sigma }_{V,{\rm{G}}}$.

Download table as:  ASCIITypeset image

Table 8.   Derived Velocities and the Velocity Ellipsoids from the LAMOST Data

Teff	$\langle N\rangle$	$\langle U\rangle$	$\langle V\rangle$	$\langle W\rangle$	${\sigma }_{U}$	${\sigma }_{V}$	${\sigma }_{W}$	${\sigma }_{{UV}}$	${\sigma }_{{UW}}$	${\sigma }_{{VW}}$	${\sigma }_{U}/{\sigma }_{V}$	${\sigma }_{U}/{\sigma }_{W}$	lv	α
(K)		$(\;\mathrm{km}\;{{\rm{s}}}^{-1})$											(degree)	(degree)
100 $\;\lt \;| z| \;\lt \;$ 300
4250	13229	−10.41 ± 1.90	−26.09 ± 1.48	−7.26 ± 0.96	41.25 ± 1.84	27.16 ± 2.11	22.54 ± 1.10	−6.51 ± 10.23	−4.21 ± 7.61	−5.00 ± 15.45	1.52 ± 0.09	1.83 ± 0.07	−0.08 ± 7.41	−0.85 ± 3.92
4750	19593	−8.74 ± 1.49	−25.96 ± 1.52	−7.17 ± 0.92	40.77 ± 1.30	27.12 ± 2.14	23.43 ± 0.91	−5.96 ± 9.15	−5.40 ± 8.12	3.57 ± 7.98	1.50 ± 0.09	1.74 ± 0.05	−2.17 ± 8.12	−1.50 ± 4.05
5250	25805	−8.76 ± 1.41	−26.53 ± 1.68	−6.35 ± 1.23	40.79 ± 1.60	27.10 ± 1.55	23.17 ± 1.26	−5.16 ± 6.28	−4.83 ± 6.01	3.47 ± 8.18	1.51 ± 0.07	1.76 ± 0.07	−1.63 ± 5.90	−1.19 ± 8.58
5750	33453	−8.86 ± 1.40	−26.45 ± 1.62	−5.35 ± 1.19	39.86 ± 1.36	26.68 ± 2.00	21.80 ± 1.04	−5.29 ± 9.62	−9.70 ± 7.09	5.63 ± 5.14	1.49 ± 0.08	1.83 ± 0.06	−1.80 ± 8.98	−4.80 ± 3.86
6250	23565	−8.92 ± 1.50	−21.44 ± 0.93	−3.64 ± 1.16	32.97 ± 1.17	20.21 ± 1.99	17.36 ± 0.95	2.31 ± 6.00	−2.70 ± 12.51	−4.41 ± 6.65	1.63 ± 0.10	1.90 ± 0.07	0.45 ± 6.62	−0.53 ± 4.25
6750	6739	−9.07 ± 1.37	−23.17 ± 1.85	−4.57 ± 1.75	26.05 ± 1.05	19.68 ± 1.97	16.74 ± 1.86	−1.18 ± 9.26	1.79 ± 3.47	1.05 ± 7.94	1.32 ± 0.11	1.56 ± 0.12	−0.27 ± 9.63	0.46 ± 6.22
200 $\;\lt \;| z| \;\lt \;$ 400
4250	7600	−12.22 ± 2.19	−28.47 ± 2.00	−7.88 ± 1.30	40.79 ± 2.19	31.53 ± 2.86	24.83 ± 1.44	−2.11 ± 12.69	−11.21 ± 9.35	−2.24 ± 8.92	1.29 ± 0.11	1.64 ± 0.08	−0.68 ± 10.41	−6.75 ± 4.42
4750	16639	−8.98 ± 1.47	−29.05 ± 1.91	−6.90 ± 1.00	42.34 ± 2.07	28.69 ± 2.23	25.62 ± 1.09	−9.78 ± 11.28	−9.71 ± 3.75	2.56 ± 11.48	1.48 ± 0.09	1.65 ± 0.06	−5.56 ± 10.81	−4.71 ± 3.95
5250	27523	−8.20 ± 1.87	−28.68 ± 2.05	−6.77 ± 1.98	42.73 ± 1.50	28.40 ± 2.43	24.62 ± 1.05	3.65 ± 4.46	−4.40 ± 5.92	1.80 ± 7.40	1.50 ± 0.09	1.74 ± 0.06	0.76 ± 5.18	−0.91 ± 4.56
5750	36303	−8.08 ± 1.63	−27.97 ± 1.57	−5.73 ± 1.09	42.19 ± 1.47	30.13 ± 1.89	24.61 ± 1.33	8.79 ± 9.63	−7.39 ± 4.78	−2.15 ± 7.90	1.40 ± 0.07	1.71 ± 0.06	4.99 ± 7.87	−2.66 ± 5.12
6250	18621	−7.39 ± 1.41	−19.89 ± 1.29	−4.01 ± 0.95	35.02 ± 1.50	20.92 ± 2.14	20.29 ± 1.09	8.12 ± 3.20	−5.66 ± 6.71	−0.44 ± 5.57	1.67 ± 0.11	1.73 ± 0.07	4.73 ± 6.51	−2.25 ± 4.76
6750	3436	−6.49 ± 1.54	−20.47 ± 1.77	−5.28 ± 1.40	27.27 ± 1.39	20.42 ± 1.77	16.94 ± 1.34	−4.24 ± 10.13	3.21 ± 4.88	−1.41 ± 7.43	1.34 ± 0.10	1.61 ± 0.09	−3.16 ± 12.35	1.29 ± 7.38
300 $\;\lt \;| z| \;\lt \;$ 500
4250	3797	−10.58 ± 3.25	−35.84 ± 6.44	−7.49 ± 2.78	41.21 ± 10.82	38.06 ± 5.93	26.67 ± 6.46	9.62 ± 17.82	−11.10 ± 10.02	0.34 ± 9.78	1.08 ± 0.31	1.55 ± 0.36	27.20 ± 14.35	−7.01 ± 7.32
4750	11443	−9.37 ± 2.04	−32.45 ± 2.79	−6.33 ± 1.06	42.85 ± 2.12	31.70 ± 2.44	27.20 ± 5.43	5.96 ± 9.46	−7.46 ± 4.18	0.07 ± 9.94	1.35 ± 0.09	1.58 ± 0.21	1.80 ± 13.30	−2.90 ± 3.53
5250	22503	−9.03 ± 2.08	−30.24 ± 2.10	−6.98 ± 2.03	43.33 ± 2.80	29.96 ± 2.79	25.78 ± 1.56	8.68 ± 8.89	−6.24 ± 11.04	−4.10 ± 9.81	1.45 ± 0.11	1.68 ± 0.09	4.47 ± 8.44	−1.84 ± 6.28
5750	33595	−8.92 ± 1.62	−29.11 ± 1.35	−5.85 ± 1.39	42.68 ± 2.17	32.81 ± 2.67	26.71 ± 1.39	14.13 ± 7.54	−4.91 ± 2.45	−2.65 ± 8.72	1.30 ± 0.10	1.60 ± 0.07	14.10 ± 8.03	−1.25 ± 4.87
6250	13986	−8.40 ± 1.78	−18.99 ± 1.88	−4.64 ± 1.05	37.32 ± 2.01	26.09 ± 2.84	21.81 ± 1.32	11.13 ± 5.17	−5.26 ± 3.04	−3.20 ± 7.56	1.43 ± 0.12	1.71 ± 0.08	9.60 ± 8.41	−1.73 ± 5.93
6750	1566	−7.24 ± 2.09	−17.39 ± 1.81	−6.62 ± 1.91	30.64 ± 7.05	20.84 ± 7.12	22.42 ± 6.33	−7.45 ± 9.12	7.57 ± 11.32	2.71 ± 5.20	1.47 ± 0.41	1.37 ± 0.36	−6.71 ± 12.79	7.36 ± 10.43

Download table as:  ASCIITypeset image

Therefore, we can infer that the $\langle V\rangle$ and ${\sigma }_{V}$ from Equation (5) may be systematically underestimated. The same is true of the estimated ${V}_{\odot }$. However, it is worth noting that the velocity distribution function (20) is different than that of the real Galaxy. In reality, this underestimation may be very small and can be negligible. Indeed, the validation test with GCS data shows that the reconstructed V and ${\sigma }_{V}$ are only smaller than the true value within 1 $\;\mathrm{km}\;{{\rm{s}}}^{-1}$, as shown in Table 1. Therefore, the ${V}_{\odot }$ estimates may be only slightly smaller than the unknown true value. Going from low to high $| z|$, the disk becomes hotter and ${\sigma }_{d}$ increases. Then, the $\langle V\rangle$ is underestimated at higher $| z|$. However, this systematic underestimation likely occurs in all ${T}_{\mathrm{eff}}$ bins and consequently cannot be the reason for the opposite trend along $| z|$ for the stars with ${T}_{\mathrm{eff}}$ $\;\gt \;6000$ K.

We reveal that the kinematic features for the stars with ${T}_{\mathrm{eff}}$ $\;\gt \;6000$ K are significantly different from those for the cooler stars. In this section, we first give a theoretical estimation of the age–${T}_{\mathrm{eff}}$ relation and then discuss the possible reasons why the kinematic features are so different for the stars with ${T}_{\mathrm{eff}}$ $\;\gt \;6000$ K based on the age distribution in the next section.

In order to derive the age–${T}_{\mathrm{eff}}$ relation, we first need to know the star formation history of the Milky Way. We adopt the star formation history shown in Figure 1 of Schönrich & Binney (2009), which contains a star formation peak at an age of about 11 Gyr and a star formation rate decreasing with time (see the top panel of Figure 12). Second, we use synthetic isochrones (Marigo et al. 2008) to select the range of ${T}_{\mathrm{eff}}$ for the main-sequence stars at a given age. We also adopt the initial mass function derived by Chabrier (2003) to assign the weight for the main-sequence stars at different temperatures. Finally, we set up the stellar distribution in the age versus ${T}_{\mathrm{eff}}$ plane, as shown in the bottom panel of Figure 12. The blue thick line shows the mean age at different ${T}_{\mathrm{eff}}$. It shows that for the stars with $6500\;\lt \;$ ${T}_{\mathrm{eff}}$ $\;\lt \;7000$ K the mean age is between 2 and 3 Gyr and there are almost no stars with age larger than 4 Gyr. For the stars with $6000\;\lt \;$ ${T}_{\mathrm{eff}}$ $\;\lt \;6500$ K the average age varies between 3 and 6 Gyr. For the stars with ${T}_{\mathrm{eff}}$ $\;\lt \;6000$ K the average age is around 8 Gyr, with a very broad range covering all ages. However, since the star formation rate has a peak at around 11 Gyr, the stars with age older than 8 Gyr dominate this region. The ${T}_{\mathrm{eff}}$ of the abrupt change in age is perfectly consistent with that of the sudden change in the mean velocity and velocity dispersions shown in the left and middle panels of Figure 9. This implies that the Parenago discontinuity occurs at around 6 Gyr. This value is about 3 Gyr earlier than the result by Quillen & Garnett (2001).

Zoom In Zoom Out Reset image size

Similar asymmetric motion is also reported in other works. Carlin et al. (2013) found that both the vertical and radial velocities are not at around zero with the LAMOST data. Although the data analyzed by them are located in a larger volume than the data we use in this work, they are qualitatively consistent with the kinematic features of the young stars in our samples. Williams et al. (2013) also found similar trends in the solar vicinity with the red clump stars observed by RAVE (Siebert et al. 2011). Particularly, they found a faster ${v}_{\phi }$ (see their Figure 10); a negative v_R (see their Figure 12), which is equivalent to a net positive U in the Cartesian coordinates; and slightly positive v_Z (see their Figure 13) at the position close to the Galactic midplane. Although the quantitative comparison is quite difficult since the spectral types of the stars in Williams et al. (2013) are completely different and the spatial sampling is also different, the orientations and the values of the offsets in three-dimensional velocity are quite similar to our results. In general, red clump stars are thought to be quite young, and Girardi & Salaris (2001) argued that the peak of the age of nearby red clump stars is only around 1 Gyr. Then, it is not surprising that similar asymmetric motion is found in the red clump samples and the younger stars from our data set.

There are possibly three channels to explain the unusual asymmetric motions for the young stars. We discuss these three scenarios here separately.

First, if the young stars are still not completely relaxed, they may keep the peculiar motion of the molecular cloud in which they are born. Assume that the young stars are formed in a giant molecular cloud containing a total of 10³–10⁴ newly formed stars within a scale of ∼200 pc; then the relaxation time for the group of stars is between 0.5 and 4 Gyr. As shown in Figure 12, for the youngest stars with ${T}_{\mathrm{eff}}$ $\;\gt \;6500$ K even the maximum age is only ∼2 Gyr. Therefore, it is possible that these young stars are still not completely relaxed and still display some kinematic features of their birthplace.

Second, it is likely that the young stars are perturbed by the nonaxisymmetric structure in the disk, e.g., the central rotating bar, spiral arms, etc. Some studies have argued that the perturbation must affect both young and old stars (e.g., Famaey et al. 2005). However, in general, orbits of stars in nearly circular orbits and very close to the Galactic midplane can be approximated by three harmonic oscillations in azimuthal, radial, and vertical orientations. Subsequently, the relatively fixed frequencies of the oscillations make the stars more easily respond to the resonance induced by the rotating bar or spiral arms. Since young stars are mostly in near-circular orbits and concentrated in the midplane, they are more easily perturbed by nonaxisymmetric structures than the old stars. However, the bar and spiral structures are mostly in the Galactic midplane and thus may not affect the vertical motion of the young stars (though for some mechanisms by which the bar and spiral arms can vertically perturb the disk see Debattista 2014; Faure et al. 2014). Therefore, it is hard to explain why the young stars also show a net positive value in W.

Finally, the disk may have been perturbed by a merging event. Indeed, Widrow et al. (2012) and Yanny & Gardner (2013) found that the stellar vertical density shows wave-like features, which could be the density wave excited by a minor merger (Gómez et al. 2013).

Certainly, none of the scenarios discussed above can be easily ruled out. Further observational and theoretical works are required to better address this challenging issue.

An interesting question raised here is why previous studies of local kinematics based on the Hipparcos data do not show the asymmetric motion. This is quite hard to answer. The different volume of the Hipparcos may be one reason. More detailed investigations with the future Gaia data including accurate distance estimates and 3D velocities (Perryman et al. 2001) may be needed to complete the picture of local disk kinematics.

Although Figure 1 does not show any dependence of the distance estimates on ${T}_{\mathrm{eff}}$, some extreme tests, e.g., the distance is only over- or underestimated for the warmer stars but unchanged for the cooler ones, are very helpful to verify whether the kinematic results are robust. The left and middle panels of Figure 13 show the mean velocities derived from the distances with artificial 20% over- and underestimation only for the stars with ${T}_{\mathrm{eff}}$ $\;\gt \;6000$ K, respectively. Essentially, the asymmetric motions of the young stars still exist in both $\langle U\rangle$ and $\langle W\rangle$ with slightly shifted values. And Parenago's discontinuity and the turnover phenomenon in $\langle V\rangle$ are still substantial. We also do not find any systematic bias due to the over- or underestimation of the distance in the velocity ellipsoids.

Zoom In Zoom Out Reset image size

Figure 6 presents a weak anticorrelation between the offset of the radial velocities and ${T}_{\mathrm{eff}}$. This drives us to further investigate whether the results are stable when the offset varies with ${T}_{\mathrm{eff}}$. We then fit the offset of the radial velocity with a linear function of ${T}_{\mathrm{eff}}$ and correct the radial velocities for the individual stars with the ${T}_{\mathrm{eff}}$-independent offset. We find that this leads to slight shifts in $\langle U\rangle$, $\langle V\rangle$, and $\langle W\rangle$, as shown in the right panels of Figure 13. Although the difference in $\langle U\rangle$ between the young and old stars has vanished, the differences in $\langle V\rangle$ and $\langle W\rangle$ are even stronger. It may also shift the derived local motion of the Sun. For ${U}_{\odot }$, it may decrease by ∼1 $\;\mathrm{km}\;{{\rm{s}}}^{-1}$, and for ${W}_{\odot }$, it may increase by ∼1 $\;\mathrm{km}\;{{\rm{s}}}^{-1}$. These shifts are within 1σ uncertainties according to Table 3. We also notice that Gao et al. (2015) found that the offset is $3.8\;\mathrm{km}\;{{\rm{s}}}^{-1}$ compared to the LAMOST data with the PASTEL catalog (Soubiran et al. 2010), implying that the calibration of the radial velocity may also depend on different calibrators. We performed another test with the offset of $3.8\;\mathrm{km}\;{{\rm{s}}}^{-1}$ and found that the mean velocities shift by $\sim 1\;\mathrm{km}\;{{\rm{s}}}^{-1}$ for all stars and the systematic differences between the young and old stars are unchanged. Therefore, in order to maintain simplification, we adopt the correction of the radial velocity with a constant $5.7\;\mathrm{km}\;{{\rm{s}}}^{-1}$ for all stars in this work. According to the tests with various corrections, we conclude that the differences in $\langle V\rangle$ and $\langle W\rangle$ between the young and old stars are real, while the mild difference in $\langle U\rangle$ shown in Figure 9 may be insignificant.