Differential Rotation of the Halo Traced by K-giant Stars

The stellar halo is one of the most important components in the Milky Way. Under the paradigm of the Λ cold dark matter model, the halo is formed through accretion and merging of satellites, and plenty of substructures are considered to be retained therein. So the halo has been recording information about its formation history. As a result, studies of the stellar halo can directly help us understand the formation of the Milky Way. However, the stellar halo is the most difficult component to study. It is relatively diffuse and of low density, and it can reach out to distances over 100 kpc (Bland-Hawthorn & Gerhard 2016). That makes it hard to obtain velocity information, i.e., proper motions and radial velocities. What is more, distance is also difficult to accurately determine, except for standard candles like RR Lyrae stars or blue horizontal branch stars (Xue et al. 2008; Hernitschek et al. 2018; Thomas et al. 2018). Thus the main obstacle to studying the properties of the stellar halo is obtaining a sufficient sample of tracers.

Thanks to the rapid development of large survey projects, e.g., the Sloan Digital Sky Survey (York et al. 2000), the Panoramic Survey Telescope and Rapid Response System (Bernard et al. 2016) and the Gaia mission (Gaia Collaboration et al. 2016, 2018a), many of those embedded substructures have been discovered in the halo, e.g., the Sagittarius Stream (Ibata et al. 1994), the GD-1 Stream (Grillmair & Dionatos 2006a, 2006b), and the ω-Cen Stream (Ibata et al. 2019). Those substructures, especially the thin cold streams, are helpful for studying the halo profile. Lux et al. (2013) introduced a method using the Markov chain Monte Carlo technique to constrain the halo shape with thin streams. The streams NGC 5466 and Pal 5 proved to be the best candidates from their orbit properties. Sanderson et al. (2015) studied how to constrain the halo profile with action distributions of streams. The results showed that, even for the simple case of a spherical potential, at least 20 streams with more than 100 member stars are required to ensure the potential is well constrained. Law & Majewski (2010) introduced a triaxial model which successfully reproduced the most prominent stream, the Sagittarius Stream. But there are still some points that are inconsistent with subsequent observations (see Dierickx & Loeb 2017 and references therein). Vera-Ciro and Helmi (2013) studied the halo shape using the Sagittarius Stream taking into account the effect of the Magellanic Clouds; their results suggested an oblate halo. From all of these studies, we find that the streams are powerful tracers to constrain the halo profile.

Many direct efforts, other than using tidal substructures, have also attempted to profile the halo. Valluri et al. (2012) showed that the halo shape can be probed using the orbital properties of individual halo stars, e.g., the action and frequency. Results from complementary simulations show that the disk plays an important role in the shape of the inner halo, making it oblate, but not the outer part. Using K-giant stars selected from data release 5 (DR5) of the Guoshoujing Telescope (Large Sky Area Multi-Object Fiber Spectroscopic Telescope; LAMOST), Xu et al. (2018) demonstrated a complicated halo, the profile being different for the inner and outer parts. Traced by the K-giant stars selected from the LAMOST DR5, the shape of the halo varies from oblate for the inner part to almost spherical for the outer part.

To determine the interaction between different components of the Milky Way, we require a deep analysis of their dynamics. The second data release of the Gaia mission (Gaia DR2) contains proper motions and parallaxes for more than 1.3 billion stars, and radial velocities for stars brighter than 14 in the G-band (Gaia Collaboration et al. 2018a). Accurate astrometric data greatly improve the study of the dynamics of the disk and halo (Belokurov et al. 2020). The phase spiral signature in the local volume was discovered by Gaia Collaboration et al. (2018b) and Antoja et al. (2018) for the first time, which indicates a possible interaction between the satellites of the Milky Way and the disk (Laporte et al. 2019; Y. Xu et al. 2020, in preparation). Spectroscopic surveys, including APOGEE and LAMOST, have made great progress in the study of the Milky Way. The combination of the APOGEE/LAMOST and Gaia data sets provides a unique opportunity to study the formation of the halo. Combining the astrometric data of Gaia DR2 and APOGEE, Helmi et al. (2018) revealed a major merger event in the local volume, named Gaia–Enceladus (also known as Gaia–Sausage; Belokurov et al. 2018). Using the combination of LAMOST and Gaia DR2, Tian et al. (2019, hereafter Paper I) measured a rotational velocity of ${V}_{T}=+{27}_{-5}^{+4}$ km s⁻¹ of the halo in the solar neighborhood using the K-giant sample.

According to the studies of Rodriguez-Gomez et al. (2017), the morphology for massive and dwarf galaxies significantly depends on their assembly history and spin, respectively. For Milky Way-like galaxies, the morphology depends on a combination of the two factors. To determine whether the halo shape is related to the spin variance, as in Paper I, we will also use K-giant stars to further study the rotation of the halo. The K-giant stars are perfect tracers for studying the dynamics of the halo; first, giant stars have higher luminosity, which is helpful in tracing distant volumes. Second, the K-giant stars are high in number, providing enough samples for statistics, especially for studies on the global properties of the stellar halo.

The LAMOST is a 4 m, quasi-meridian, reflecting Schmidt telescope. There are 4000 fibers, which make it efficient in obtaining spectra. DR5 provides radial velocities and metallicities for millions of stars with uncertainties of ∼5 km s⁻¹ and 0.1 dex respectively. The high efficiency allows LAMOST to obtain more than nine million spectra, the largest observation sample. Combined with Gaia DR2, it provides an unprecedented opportunity to study the Milky Way. More recently, DR7 includes low- and medium-resolution spectra⁶ with R = 1800 and 7500, respectively.

This paper is organized as follows. In Section 2, we briefly introduce the data set and the method. The results are shown in Section 3. Discussions are given in Section 4.

As in Paper I, we use the K-giant stars selected from LAMOST DR5 (Liu et al. 2014) to study the rotation velocities of the halo and thick disk. The LAMOST DR5 data set provides radial velocity and metallicity, with typical errors of ∼5 km s⁻¹ and 0.1 index, respectively. Following Paper I, we remove all stars with metallicity [Fe/H] $\gt -1$ to reduce contamination from the disk, especially the thin disk stars (see Figure 4 in Hayden et al. 2015). This proves to be efficient (as shown in Paper I). In this project, we address larger volumes, where the distances of the stars are no longer available from Gaia DR2. We adopt the distances provided by Carlin et al. (2015), which are estimated using a Bayesian approach by comparing the stellar parameters and a grid of stellar isochrones with typical relative errors ∼20%.

To avoid systematic offset of the distances and radial velocities from LAMOST DR5 (Tian et al. 2015; Schönrich & Aumer 2017; and Paper I), we use the common stars of LAMOST DR5 and Gaia DR2 to determine the offset and correct the values from the former. The distance is normalized by 0.805 and the radial velocity is corrected by adding ∼5.38 km s⁻¹. More details on distance and radial velocity corrections are described in Appendices A and B, respectively.

The proper motions of the K-giant star samples are obtained by cross-matching with Gaia DR2, and the positions and velocities are calculated using the Python package Galpy (Bovy 2015). The solar motion relative to the local standard of rest from Schönrich et al. (2010) is adopted: $({U}_{\odot },{V}_{\odot },{W}_{\odot })\,=(11.1,12.24,7.25)$ km s⁻¹.

In order to study the variance of the rotational velocity of the halo at different heights to the disk plane, a sufficient sample for each volume is required. We first focus on the volumes with $6\lt R\lt 12\,\mathrm{kpc}$, $-30^\circ \lt \phi \lt 30^\circ$, and $-1\lt Z\lt 15\,\mathrm{kpc}$ in galactocentric cylindrical coordinates as shown in Figure 1. We adopt the solar location from Reid et al. (2014) with (R, ϕ, Z) = (8.3 kpc, 0°, 0 kpc). Here we ignore the distance from the Sun to the disk plane, which is too small to make any difference to our results. To constrain the uncertainties, only those stars with signal-to-noise ratio higher than 10, radial velocity errors ${\sigma }_{\mathrm{RV}}\lt 10$ km s⁻¹, and proper motion errors (${\sigma }_{{\mu }_{\alpha }}$, ${\sigma }_{{\mu }_{\delta }}$) smaller than 0.3 mas yr⁻¹ are used. This is labeled as the S-sample. A similar sample with different R range, $12\lt R\lt 20\,\mathrm{kpc}$, is selected for studying the outer part, labeled as the SO-sample in Figure 1.

Along the height to the disk plane, we divide each of the two samples into six sub-volumes to ensure each sub-volume contains enough stars. The information for each sub-volume is listed in Table 1.

Table 1.  Information for Each Subsample of Stars

	Selection	fH	VϕH	${\sigma }_{{V}_{\phi }}^{{\rm{H}}}$	fD	${V}_{\phi }^{{\rm{D}}}$	${\sigma }_{{V}_{\phi }}^{{\rm{D}}}$	fH3	${V}_{\phi }^{{{\rm{H}}}^{3}}$	${\sigma }_{{V}_{\phi }}^{{{\rm{H}}}^{3}}$	N
			(km s−1)	(km s−1)		(km s−1)	(km s−1)		(km s−1)	(km s−1)
S	−1 < Z < 1	0.70 + 0.04−0.04	37 + 7−7	75 + 4−4	0.30	185 + 5−5	41 + 3−3	⋯	⋯	⋯	2215
	1 < Z < 2	0.81 + 0.04−0.05	50 + 5−6	79 + 3−4	0.19	177 + 5−6	38 + 4−4	⋯	⋯	⋯	2268
	2 < Z < 4	0.12	−18 + 41−64	82 + 17−16	0.41 + 0.06−0.07	127 + 9−9	55 + 4−4	0.47 + 0.09−0.11	12 + 4−4	33 + 5−7	3415
	4 < Z < 6	0.77 + 0.05−0.09	40 + 6−4	73 + 3−3	⋯	⋯	⋯	0.23	6 + 7−7	18 + 16−12	1731
	6 < Z < 10	0.68 + 0.05−0.05	24 + 4−4	82 + 4−3	⋯	⋯	⋯	0.32	14 + 4−5	12 + 10−8	1327
	10 < Z < 15	1	10 + 4−4	63 + 3−3	⋯	⋯	⋯	⋯	⋯	⋯	496
SO	−1 < Z < 1	0.60 + 0.04−0.04	23 + 11−10	80 + 13−12	0.40	221 + 2−3	20 + 3−3	⋯	⋯	⋯	287
	1 < Z < 2	0.86 + 0.03−0.04	20 + 7−6	59 + 10−7	0.14	201 + 7−10	28 + 11−8	⋯	⋯	⋯	218
	2 < Z < 4	0.92 + 0.02−0.02	20 + 3−3	46 + 4−4	0.08	194 + 8−14	32 + 11−6	⋯	⋯	⋯	537
	4 < Z < 6	0.95 + 0.01−0.02	9 + 4−4	54 + 4−4	0.05	201 + 9−17	28 + 16−8	⋯	⋯	⋯	482
	6 < Z < 10	0.97 + 0.01−0.01	6 + 3−3	53 + 3−3	0.03	192 + 18−26	45 + 22−17	⋯	⋯	⋯	639
	10 < Z < 15	1	1 + 4−4	60 + 4−4	⋯	⋯	⋯	⋯	⋯	⋯	395

Note. The results, including fraction f, the rotational velocity $\langle {V}_{\phi }\rangle$ and dispersions ${\sigma }_{{V}_{\phi }}$, from the Bayesian method for each component are listed. Symbols H, D, and 3 denote the disk, halo and the third component, respectively. The last column lists the number of the K-giant stars in each sub-sample.

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In Paper I, a Bayesian model including three Gaussian components was used for the local volume: the halo, the thick disk, and a possible retrogradely rotating component. In this paper, we adopt the same method and include the halo, the thick disk and a possible additional component. The Gaussian distribution of each component can be written as follows:

$$\begin{eqnarray}\begin{array}{rcl}{p}_{i}({V}_{\phi }^{(k)}| {f}_{i},{V}_{\phi ,i},{\sigma }_{i},{\epsilon }^{(k)}) & = & \displaystyle \frac{{f}_{i}}{\sqrt{2\pi ({\sigma }_{i}^{2}+{\epsilon }_{k}^{2})}}\\ & & \times \exp \left(-\displaystyle \frac{{\left({V}_{\phi }^{k}-{V}_{\phi ,i}\right)}^{2}}{2({\sigma }_{i}^{2}+{\epsilon }_{k}^{2})}\right),\end{array}\end{eqnarray} \tag{ 1 }$$

where ${V}_{\phi ,i}$ and ${\sigma }_{i}$ are the rotational velocity and its dispersion of the ith component, f_i is the number fraction of the ith component, and ${V}_{\phi }^{k}$ and ${\epsilon }_{k}$ are the rotational velocity and its uncertainty of the kth star in cylindrical coordinates. Considering that the errors in the rotational velocity will be larger for distant stars, which may affect the results, here we use the probability calculation with uncertainties included, as shown in Equation (1). To determine the rotational velocities of the three components, the Python package emcee is applied to the samples with different space selections as listed in Table 1. An affline solver is used with 50 walkers and 6000 iterations in total, including 3000 burn-in iterations. To find the best-fit parameters for the components, we test models with different numbers of components, e.g., 1, 2, or 3 according to the results given in Paper I. The median values are adopted for each of the parameters. The lower and upper uncertainties are determined from the differences between the median value and the 16% and 84% values. A typical result for the parameter determination is showed in Appendix C. Due to the large contribution from the disk, the third component has a low fraction in the lower volumes; in this paper, we have much fewer stars in each volume than in Paper I.

We find that the model with the disk and halo components is better for volumes with $Z\lt 2\,\mathrm{kpc}$. As the contamination of the disk becomes lower for higher volume, i.e., $Z\gt 4\,\mathrm{kpc}$, the model with the halo and retrogradely rotating components is better, rather than the halo and the disk. In general, the disk contributes very few stars ($\sim 8 \%$) in those higher-Z volumes (Z = 5 kpc) around R = 9 kpc (Wang et al. 2018). As a result, the contributions of the halo and the additional contribution will rise. In Table 1, for the transition volume with $2\lt Z\lt 4\,\mathrm{kpc}$ in the S-sample, the disk contribution becomes weaker and the third component (the Gaia–Enceladus–Sausage (GES) as described in the following section) increases. Then we constrain the parameters of a model with all three components. This is not done for the volume with similar height in the SO-sample because of the low number of samples and the low contribution of the GES component (see below). The median values are adopted as the best-fit parameters for each volume. The results for those space volumes are listed in Table 1. The markers H, D, and 3 denote the halo, the disk, and the third component, respectively.

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Figure 2 presents the distributions of the rotational velocity and its dispersion.

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3.1. Rotational Velocity Distribution of the Halo

Focusing on the halo component, which is the main target in this paper, we find that the halo in the S-sample (the blue symbols in Figure 2) is progradely rotating with rotational velocity dispersion around 75 km s⁻¹. The rotational velocity generally decreases with height to the disk plane. This is clearer in the left panel in Figure 3. The black solid line shows the linear fitting results, with rotational velocity uncertainties taken into account. The fitting results show that the decreasing rate is −3.04 ± 0.63 km s⁻¹ kpc⁻¹. The intercept of the line represents the rotational velocity of the halo at the disk plane, of 49 ± 5.34 km s⁻¹. The volume with $2\lt Z\lt 4\,\mathrm{kpc}$ shows an exception of the variance. The uncertainties in the parameters are very large, because the fraction ($\sim 12 \%$) of the halo is too low to be well constrained.

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Unlike in the S-sample, the rotational velocity dispersion is no longer flat in the SO-sample, but first decreases and then remains around a lower value of 60 km s⁻¹. This is consistent with the variance of the rotational velocity dispersion showed by Bird et al. (2019).

Similar to the S-sample, the SO-sample shows a decreasing trend of the rotational velocity of the halo. As shown in the right panel of Figure 3, the decreasing rate of the rotational velocity is lower than that of the S-sample, of −1.89 ± 0.37 km s⁻¹ kpc⁻¹. The intercept of the fitting line is of 22.40 ± 2.66 km s⁻¹. This is much lower than that of the S-sample, meaning the rotational velocity of the halo close to the disk plane is larger for the inner part.

3.2. Contribution of the GES

The locations with $Z\gt 2\,\mathrm{kpc}$ show a second halo component with low rotational velocity dispersion and close-to-zero rotational velocity (the green symbols in the top panels). This is consistent with the properties of the GES (Belokurov et al. 2018; Helmi et al. 2018; Myeong et al. 2019). It should be noted that the GES component is not recognized in the lower volumes. That does not mean there is no contribution of the GES component in those volumes; the main reason is that the fraction of the GES member stars is too low to be well constrained at lower-Z locations.

According to the distribution of the components, we calculate the probabilities of the components for each star. Figures 4 and 5 show the distributions of the stars in each volume color-coded by the probability of the disk component. Because the disk component almost vanishes in the higher volumes in the S-sample, where there are only the halo and GES components, we present the probability of the GES in the subsample with $4\lt Z\lt 6\,\mathrm{kpc}$ (the top left panel in Figure 4).

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The GES is included in the Bayesian model as an independent component, because it has significantly different dynamical information from the halo (Helmi et al. 2018), smaller rotational velocity and dispersion, and larger energy. As shown in Figures 4 and 5, the probabilities for the GES member stars are around 50% at most. That means it is quite difficult to select a pure sample to study its chemical information. In Figure 6, the metallicity distributions of the stars with probability higher or lower than 0.4 in the S-sample with $4\lt Z\lt 6\,\mathrm{kpc}$ are shown. We find that the distribution with higher probabilities is still different from that of Helmi et al. (2018), because of the high contamination ($\gt 50 \%$).

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What should be noted is that the GES component is more significant in the top right panel in Figure 4, where the stars are located with $4\lt Z\lt 6\,\mathrm{kpc}$, but this is not clear for the outer volumes with $12\lt R\lt 20\,\mathrm{kpc}$ in Figure 5. It is not clear that the missing GES is intrinsic or that the larger uncertainties of the distances make the distribution of the action J_ϕ more diffuse.

3.3. Rotational Velocity Distribution of The Disk

3.4. Disk Flare

4.1. Interaction between the Halo and The Disk

There are many mechanisms that can generate the differential rotation of the halo. One possible mechanism for the decreasing trend of the halo rotational velocity versus the height to the disk plane is the interaction between the halo and the disk. To check this scenario, we use the simulated galaxies from the TNG100 simulation (Marinacci et al. 2018; Nelson et al. 2018, 2019; Naiman et al. 2018; Pillepich et al. 2018a; Springel et al. 2018). The TNG100 simulation is a magnetohydrodynamic cosmological simulation, which contains 2 × 1820³ resolution elements in a cosmological  ${(110\mathrm{Mpc})}^{3}$ box. Compared with the original Illustris simulation (Vogelsberger et al. 2013; Torrey et al. 2014), the TNG simulation has adopted new physics models and improved the implementations of galactic winds, stellar evolution, chemical enrichment (Pillepich et al. 2018b), and active galactic nucleus feedback (Weinberger et al. 2017). Therefore, the TNG simulation can better reproduce many observed galaxy properties and scaling relations to different degrees. Galaxies in their host dark-matter halos were identified using the subfind halo finding algorithm (Dolag et al. 2009). In Figure 7, we show the velocity ${v}_{\phi }$ and velocity dispersion ${\sigma }_{{v}_{\phi }}$ for eight galaxies from the TNG simulation with different axis ratios. It is seen that there are clear velocity gradients in ${v}_{\phi }$ for stars in panel (a). These four galaxies have an oblate disk, with axis ratio c/a (minor axis over major axis) lower than 0.5. For the other four galaxies as shown in panel (b), the stellar systems are nearly spherical or triaxial, with $c/a\gt 0.5$, and there are no obvious velocity gradients in ${v}_{\phi }$ versus height to the disk.

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As shown in Wang et al. (2019), oblate galaxies have larger spin parameters than prolate and triaxial galaxies. In other words, the oblate system has the larger rotational velocity. Close to the oblate disk, the fast rotation disk dominates the rotation velocity. With increasing height, the halo begins to dominate the kinematics of the system, and the halo is more spherical or triaxial. Therefore, ${v}_{\phi }$ decreases with height to the disk. In other words, the decreasing trend of the halo versus the height to the disk is likely caused by the dynamical interaction between the disk and the halo. A stronger disk gives rise to a larger decreasing rate. From the comparison, we confirm that the halo must be oblate, which is consistent with the conclusion of Xu et al. (2018).

4.2. Interaction between the Halo and Bar

4.3. Halo Assembly History

4.4. Dichotomy of the Halo

4.5. Effect of the Distance Calculation

We use the K-giant stars from LAMOST DR5 to investigate the rotation information of the halo and the disk. We find that the rotational velocity of the halo decreases with increasing height to the disk plane. The dispersion of the halo is almost flat up to 15 kpc. The rotational velocity of the inner part decreases faster than that of the outer part, −2.75 and −1.88 km s⁻¹ kpc⁻¹ respectively. Analyzing all the possible mechanisms, we claim that the decreasing trend suggests an oblate halo profile, which is consistent with that proposed by Xu et al. (2018). This is possibly caused by the interaction between the halo and the disk component.

The signal of the merging event GES is clearly seen only in the volumes with height from 2 to 10 kpc and galactocentric distance between 6 and 12 kpc. The rotational velocity dispersion of the disk is larger in higher volumes. At the same time, the rotational velocity decreases with height. Our results also show a flaring disk, which can reach heights of 6–10 kpc with galactocentric distance from 12 to 20 kpc. Limited by the sample, we claim that the disk can reach at least 20 kpc.

In order to avoid contamination from different components, we use a Bayesian method to determine the rotational velocity for each component at different locations statistically. With the rotational velocity distribution of each component, this method can determine the probability for each star belonging to the different components. As shown in Figures 4 and 5, only the disk component can be clearly discerned; the GES and the halo are still difficult to separate. This will be improved in the future for selection of the two components using full phase space information. Purer samples will greatly aid chemical studies with spectral data sets from LAMOST (Liu et al. 2020), SEGUE (Yanny et al. 2009), and APOGEE (Majewski et al. 2017).

We thank Lia Athanassoula and Victor Debattista for helpful discussions. This work is supported by National Key R&D Program of China No. 2019YFA0405500. C.L. thanks the National Natural Science Foundation of China (NSFC) with grant No. 11835057. Y.W. thanks the National Natural Science Foundation of China (NSFC) with grant No. 11773034. X.-X.X. thanks NSFC for support under grants No. 11873052 and No. 11890694. The LAMOST FELLOWSHIP is supported by Special Funding for Advanced Users, budgeted and administrated by Center for Astronomical Mega-Science, Chinese Academy of Sciences (CAMS). This work is supported by Cultivation Project for LAMOST Scientific Payoff and Research Achievement of CAMS-CAS.

Guoshoujing Telescope (the Large Sky Area Multi-Object Fiber Spectroscopic Telescope; LAMOST) is a National Major Scientific Project built by the Chinese Academy of Sciences. Funding for the project has been provided by the National Development and Reform Commission. LAMOST is operated and managed by the National Astronomical Observatories, Chinese Academy of Sciences. This work has made use of data from the European Space Agency (ESA) mission Gaia (https://www.cosmos.esa.int/gaia), processed by the Gaia Data Processing and Analysis Consortium (DPAC, https://www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the Gaia Multilateral Agreement.

Software: astropy (Astropy Collaboration et al. 2013), Galpy (Bovy 2015), emcee (Foreman-Mackey et al. 2013).

To ensure the distances from Gaia DR2 and LAMOST DR5 are consistent, we first select the K-giant stars common to both according to the following criteria:

• 1.  
  $\omega /{\omega }_{\mathrm{err}}\gt 5$
• 2.  
  ${D}_{\mathrm{Gaia}}\lt 3$ kpc
• 3.  
  ${\sigma }_{{D}_{\mathrm{Gaia}}}/{D}_{\mathrm{Gaia}}\lt 0.1$
• 4.  
  ${\rm{S}}/{\rm{N}}\gt 10$.

The first criterion is used to select those stars whose parallaxes are well measured. The second and the third are used to constrain the distance D_Gaia smaller than 3 kpc with high accuracy, provided by Bailer-Jones et al. (2018). The last is used to constrain the data from LAMOST DR5 with high signal-to-noise ratio. After selection, we have 5560 K-giant stars with distance accurately measured by Gaia and reliable spectra from LAMOST DR5.

To compare the distances we define the difference ${\rm{\Delta }}=\tfrac{{D}_{L}-{D}_{\mathrm{Gaia}}}{{D}_{\mathrm{Gaia}}}$, where D_L is the distance provided by Liu et al. (2014), and D_Gaia is the distance from Bailer-Jones et al. (2018).

Figure A1 shows the distribution of the distance difference versus the Gaia distance D_Gaia in the left panel and its histogram distribution in the right panel. From the distribution we find that there is a system offset with Δ a constant around −0.2. To obtain the true value, we calculate the mean value of Δ and its dispersion, $\langle {\rm{\Delta }}\rangle$ and ${\sigma }_{{\rm{\Delta }}}$, and then select those stars within $1{\sigma }_{{\rm{\Delta }}}$. We iterate this step until the mean value does not change significantly, lower than 0.005. This threshold is chosen because with this value the distance system error will be lower than 0.15 kpc at 30 kpc. Finally, we obtain the value ${\rm{\Delta }}\sim$−0.195, as shown by the red line in both panels. In this case, the distance from LAMOST DR5 is corrected by dividing by 0.805. Figure A2 shows the corrected LAMOST distance distribution as a function of D_Gaia. The red line represents the 1:1 relation.

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The radial velocity provided by Gaia DR2 is only available for bright stars with $G\lt 14$. This is because the sample cannot trace distant volumes. In this paper we adopt the radial velocity from LAMOST DR5. Figure B1 shows the comparison between the radial velocities provided by LAMOST DR5 and Gaia DR2 of the common K-giant stars used in Paper I. We clearly find an offset of ∼5.38 km s⁻¹ and a dispersion 6.39 km s⁻¹. In this paper, we correct the radial velocity from LAMOST DR5 by +5.38 km s⁻¹.

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We adopt the median values for each parameter in the Bayesian method from emcee. The upper and lower uncertainties are determined as the difference between the median value and the 84% and 16% values. Figure C1 shows a corner distribution of the possible values for each parameter. The blue lines represent the median values for each parameter, while the dashed lines represent the 16%, 50%, and 84% values. Figure C2 shows the fitting results of the rotational velocity distribution. The red and green dashed lines represent the halo and disk components, respectively. The vertical cyan dashed line represents the rotational velocity 0.

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