Tracing the Galactic Disk with the Kinematics of Gaia Cepheids

Classical Cepheids (CCs) are excellent tracers for understanding the structure of the Milky Way disk. The latest Gaia Data Release 3 provides a large number of line-of-sight velocity information for Galactic CCs, offering an opportunity for studying the kinematics of the Milky Way. We determine the 3D velocities of 2057 CCs relative to the Galactic center. From the projections of the 3D velocities onto the X–Y plane of the Galactic disk, we find that the V R and V ϕ velocities of the northern and southern warps (directions with highest amplitude) are different. This phenomenon may be related to warp precession or asymmetry in warp structure. By investigating the kinematic warp model, we find that the vertical velocities of CCs are more suitable for constraining the warp precession rate than the line-of-node angle. Our results suggest that CCs at 12–14 kpc are the best sample for determining the Galactic warp precession rate. Based on the spatial structure parameters of Cepheid warp from Chen et al., we determine a warp precession rate of ω = 4.9 ± 1.6 km s−1 kpc−1 at 13 kpc, which supports a low precession rate in the warp model. In the future, more kinematic information on CCs will help to constrain the structure and evolution of the Milky Way better.


Introduction
Exploring the structure and evolution history of the Milky Way is one of the most important topics in astronomy.Constructing geometric and kinematic models of the Milky Way is limited by our location in the Galaxy and the effects of interstellar extinction, so a complete picture cannot be constructed as easily as for other galaxies.The research on the structure of the Galactic disk has been broadly divided into radial and vertical directions.In the radial direction, since the 1950s, we have known that the Milky Way has a spiral arm structure through the distribution of local OB associations (Morgan et al. 1953).Since then, many works have been devoted to studying the spiral arms of the Milky Way's disk (Reid et al. 2014(Reid et al. , 2016;;Xu et al. 2016).Reid et al. (2019) updated the logarithmic model of the spiral arms, and identified the main spiral arms as the Norma-Outer Arm, Sagittarius-Carina Arm, Scutum-Centaurus Arm, and Perseus Arm, as well as an isolated segment, the Local Arm.In addition, the Galactic disk has strong asymmetry and rich structures in velocity space.In particular, the structure of arches and ridges can be seen in Galactocentric azimuthal velocity and radius space (Antoja et al. 2018;Ramos et al. 2018;Fragkoudi et al. 2019;Khanna et al. 2019;Wang et al. 2020a).However, the dynamical nature of these phenomena and their formation mechanism remain unclear.Existing explanations include resonances with the bar (Monari et al. 2019;Trick et al. 2019;Laporte et al. 2020;Trick et al. 2021), and/or processes with spiral arms (Monari et al. 2019;Barros et al. 2020), and/ or external perturbations such as perturbations by the Sagittarius dwarf galaxy (Binney & Schönrich 2018;Bland-Hawthorn et al. 2019;Khanna et al. 2019).
Looking at the vertical structure of the Galactic disk, observations of 21 cm neutral hydrogen (Kerr 1957;Levine et al. 2006;Voskes & Butler Burton 2006), dust (Freudenreich et al. 1994), stars (Wouterloot et al. 1990), and stellar kinematics (Poggio et al. 2018;Romero-Gómez et al. 2019;Cheng et al. 2020) all suggest that the Galactic disk is warped and flared.At present, the mechanism that causes the warp's formation is not well understood, and the relevant theoretical explanations can be divided into two categories.One is that the warp is the result of gravitational interactions (Drimmel et al. 2000), including interactions with other satellite galaxies (Kim et al. 2014;Laporte et al. 2019) and a misaligned dark matter halo (Widrow et al. 2014;Amôres et al. 2017).The other is nongravitational mechanisms, such as the accretion of interstellar matter (López-Corredoira et al. 2002a;Wang et al. 2020b) or interactions with intergalactic magnetic fields (Battaner et al. 1990;Guijarro et al. 2010).Knowledge of stellar kinematics helps to study the origin of the Galactic warp, such as interactions with satellite galaxies (Poggio et al. 2020), the torque of a misaligned nonspherical halo (Jiang & Binney 1999), or the torque of the inner disk (Chen et al. 2019).The Galactic warp precession rate was measured for the first time using 12 million giants from Gaia Data Release 2 (DR2) by Poggio et al. (2020).They gave a value of precession β = 10.86 ± 0.03 (stat.)± 3.20 (syst.)km s −1 kpc −1 , and supported the scenario that the warp is the result of ongoing encounters with satellite galaxies.A counterexample is given by Chrobáková & López-Corredoira (2021), who obtained the precession rate β = 4 ± 5 km s −1 kpc −1 , indicating that the warp precession rate is not that high.They argued that there is a systematic error in the precession rate calculated by Poggio et al. (2020).The precession of Milky Way's warp is still an unsolved problem.
Classical Cepheids (CCs) are a class of pulsating variable stars whose luminosity varies linearly with period.This relationship is called the period-luminosity relation (PLR), which was first discovered by Leavitt & Pickering (1912).A CC's absolute magnitude can be determined by the PLR.Combined with its observed apparent magnitude, its distance can be determined.Therefore, CCs are a standard candle and the PLR is an essential component of the cosmic distance ladder used to make precise measurements of the Hubble constant (Freedman et al. 2012;Riess et al. 2022).Meanwhile, CCs are young stars found mainly in the spiral arms, the disk, and open clusters of the Milky Way.CCs can be used to trace structures of the disk and spiral arms.Age information of CCs can be obtained from the CC period-age relation (Zhou & Chen 2021).Therefore, CCs are an important tool for modeling the structure and evolution history of the Galactic disk (Lemasle et al. 2013;Skowron et al. 2019aSkowron et al. , 2019b;;Chen et al. 2019;Lemasle et al. 2022).In the past five years, Chen et al. (2019) and Skowron et al. (2019a) demonstrated the intuitive 3D structure of the Galactic disk for the first time based on CCs.Among them, Chen et al. (2019) discovered that the line of nodes (LONs) of warp do not always lie in the same direction, but show a pattern of leading spirals.Skowron et al. (2019a) revealed the evolution of spiral arms.Besides, Minniti et al. (2021) used CCs to extend to the far side disk of the Milky Way, while Lemasle et al. (2022) used CCs to investigate the overdensity regions in spiral arms.
Nowadays, the Gaia Data Release 3 (DR3) provides mean line-of-sight velocity (RV) information for the Milky Way's CCs, creating new opportunities to understand the Milky Way (Gaia Collaboration et al. 2023a).A new resonance-like feature was discovered based on data from Gaia DR3 CCs (Drimmel et al. 2023;Semczuk et al. 2023).Therefore, the goal of this paper is to study the Galactic outer disk based on the kinematic information of Gaia CCs, including the rotation curve, the Galactic warp, and flare.In this paper, we present the data and data processing procedures, including the velocity calculation and criteria in Section 2. The analysis of the asymmetric structure of the Milky Way and the warp precession model is discussed in Section 3. A summary of this work is in Section 4.

Data
In this section, we focus on the processing of the original data from Gaia DR3 to obtain the 3D position and velocity parameters of the CCs in Galactic cylindrical coordinates.Based on velocity histograms and an error analysis, we exclude possible outliers in the velocities and obtain the final sample for subsequent analysis.

Raw Sample
The European Space Agency's Gaia mission released the Gaia DR3 catalog in 2022 June.The celestial position, proper motion, parallax, and average photometry in the G, G Bp , and G Rp bands of each source were given in Gaia EDR3.In contrast, Gaia DR3 included newly determined mean RVs for more than 33 million stars, atmospheric parameters, and chemical abundances for about 32.2 million stars, and provided multiband time-series photometry for nearly 12 million variable stars (Eyer et al. 2023;Gaia Collaboration et al. 2023b).The gaiadr3.vari_cepheid catalog given by Gaia DR3 contains 15,021 different types of CCs, which are distributed in different galaxies such as Large Magellanic Cloud (LMC), Small Magellanic Cloud (SMC), Milky Way, M31, M33, etc.We obtained these CCs' astrometric parameters by crossmatching the gaiadr3.vari_cepheidcatalog with the gaiadr3.gaia_sourcecatalog from the Gaia archive.Ripepi et al. (2023) pointed out that there is a subsample of 799 CCs of all types that have timeseries RV curves, and the typical uncertainties in 〈RV〉 are about 1-1.5 km s −1 .To get a larger sample, Gaia Collaboration et al. (2023a) found that the average RV values of CCs calculated by the spectroscopic pipeline in the gaia_source catalog are in overall agreement with the RV curve fitting values in the vari_cepheid catalog, with a mean difference of 0.6 ± 6.0 km s −1 .They provided a catalog of a total of 3306 Galactic CCs, including a small number of CCs collected from the literature.We then crossmatched the previously obtained table of CCs with these 3306 CCs and obtained a total of 3038 CCs, of which 2057 have RV information.These subsamples with RVs serve as our initial Gaia sample, which includes 1364 fundamental CCs and 693 first-overtone CCs.The metallicities and errors for all CCs are from Gaia Collaboration et al. (2023a), which were estimated by the radial gradient of the Galactic disk metallicity determined by CCs: [Fe/H] = (−0.0527± 0.0022)R + (0.511 ± 0.022), rms = 0.11 dex (Ripepi et al. 2022).We now have a catalog that includes the spatial coordinates, proper motions, mean RVs, metallicities, periods, and mean intensity magnitudes in three bands for 2057 CCs.

Calculation of the Distance and Velocity of CCs
Each Gaia parallax has a zero-point offset, even after correction (Lindegren et al. 2021).Many studies have been devoted to the study of the zero-point deviation of the Gaia parallaxes (Groenewegen 2021;Ren et al. 2021;Zinn 2021).Gaia parallaxes can achieve high accuracy for objects within 1-2 kpc, and the parallax error is 10%-20% for stars within 4 kpc.But the relative errors of parallaxes are larger for objects at greater distances, which may affect the study of the outer disk of the Milky Way.Meanwhile, the improved accuracy of Gaia photometry (errors at the level of 1-10 mmag) and the Gaia-band period-Wesenheit relation of CCs provide opportunities for us to estimate more accurate distances to CCs.Therefore, we used the PLR-based distance instead of the Gaia parallax.We adopted the mean intensity magnitude of the three Gaia bands (〈G〉, 〈Bp〉, and 〈Rp〉).According to W G,Bp,Rp = 〈G〉 − 1.90(〈Bp〉 − 〈Rp〉) given by Ripepi et al. (2019), the Wesenheit magnitude of the Gaia bands was obtained, which reduces the effect of extinction.The Gaia-band absolute magnitude was obtained by the period-Wesenheit-metallicity relation (Ripepi et al. 2022): 5.988 0.018 3.176 0.044 log 1.0 0.520 0.090 Fe H . 1 For the first-overtone CCs, we converted their first-overtone periods (P 1O ) to the corresponding fundamental periods (P F ) through the equation of Feast & Catchpole (1997) 1 O , and then estimated their absolute Wesenheit magnitudes using Equation (1).Combined with the mean intensity magnitude of the three Gaia bands, the distance modulus of each CC is estimated by μ 0 = m W − M W . Then we used the following formula to convert the distance modulus to parallax in units of mas and distance in units of kpc: By comparison, we found that 98% of the stars have a distance deviation of less than 0.1 kpc.
To obtain 3D velocity components in Galactic cylindrical polar coordinates, we extracted the sample's proper motion and mean RV data (m m a d * v , , r ), where m a * = m d a cos .We converted (m a * , μ δ ) in units of mas yr −1 to tangential velocity in units of km s −1 by multiplying the stellar heliocentric distance d and 4.74047.We also multiplied it by the matrix conversion factor ¢ A G provided by the Gaia EDR3 online documentation to obtain velocities in Galactic coordinates.Then, we multiplied by the matrix of the normal triad A: The velocity of the heliocentric Cartesian coordinates is estimated by: The detailed velocity calculation equation, including the matrix information, was described in Section 3.
Based on the azimuth angle f, we calculated the 3D velocities in cylindrical polar coordinates.V R , V f , and V Z in Equation (6) are the radial velocity, the azimuthal velocity, and the vertical velocity, respectively.We now have the geometric and kinematic parameters needed to study the Galactic disk.

Excluding Outliers
By visually inspecting a histogram of the velocity distributions of the 2057 Galactic CCs in the three directions, we found that some objects significantly deviate from the distribution of the overall sample.Therefore, we first excluded objects outside of the ranges of [−100, 100] km s −1 , and V f in [160, 300] km s −1 .After selection, there are 1967 CCs left in the sample, including both fundamental and first-overtone CCs.The velocity distributions of the remaining CCs are shown in Figure 1.The mean and median values of velocities in the different directions are also labeled in the figure.We found only four CCs with Galactocentric distances less than 4 kpc, and six CCs with Galactocentric distances greater than 19 kpc.These samples are too small to study the nature of the Galactic disk in these regions, so we excluded them, leaving 1957 CCs.
After that, we calculated the average velocities of the CCs in 1 kpc bins and plotted the variations of the 3D velocities with Galactocentric distance, as shown in Figure 2. It can be seen that some objects deviate from the mean velocity, so we estimated the standard deviation of the velocity in each bin and excluded these outliers by the 3σ clipping method.This procedure was performed for the V Z , V R , and V f velocities, and the remaining sample contained 1904 CCs.We noted that there are still very few stars with very large radial velocities at 4-6 kpc, so we excluded three CCs by the criterion of V R in [−65, 65] km s −1 .Figure 2 illustrates the distribution of the 3D velocities of the final 1901 CCs at different Galactocentric distances.We can see that the blue dots are evenly distributed around the mean velocities (black filled circles).We will analyze the velocity features in the next section.

Analysis of the Structure of the Galactic Disk
In this section we focus on the investigation of the different structures of the Milky Way using kinematic velocities of CCs.By analyzing the projection of the distribution of the velocities on the Galactic plane and the variation of the average velocities along the Galactocentric distance, we discuss the structure of the north-south asymmetry of the Galactic disk.We also discuss the warp model of the outer disk.

Velocity Asymmetric Structure
CCs are important tracers of the structure of the Galactic disk due to their precise distances and wide distribution in the Milky Way.Chen et al. (2019) and Skowron et al. (2019b) used the 3D spatial distribution of CCs to visualize the geometric structure of the Galactic disk.Based on the newly obtained RV and proper motion data, an updated kinematic map of CCs is available, which provides new clues to the archeology of the Galactic disk.Velocity projections in the X-Y plane can be used to visualize the asymmetry of the northern and southern disks, and the statistical effect of the 3D velocities at different Galactocentric distances reveals the structure of the Milky Way.In this section, we analyze the velocity projection in the X-Y plane in Figure 3 and the velocity variations with Galactocentric distance in Figure 2 to explore the distribution of 3D velocities and the asymmetric structure of the Galactic disk.
In Figure 3, we project the velocities of 1901 CCs on the X-Y plane of the Galactic disk, with the positive Y-direction being the northern disk and the negative Y-direction being the southern disk.Overall, the first thing we see is that the 3D velocities have a clear asymmetry in the northern and southern outer disks.This phenomenon is discussed in the last part of this subsection.From the projection of the vertical velocities onto the Galactic disk in the top panel of Figure 3 we see that the V Z values are dominated by positive velocities (solid circles in orange) outside of R = 12 kpc, with a significant increase in velocity compared to the internal velocities.This trend is more significant in the left panel of the Figure 2: the average value of V Z in the region from 5 to 11 kpc is about 1.4 km s −1 , and then the average velocity gradually increases from 11 to 14 kpc.The variation of V Z with R in both figures indicates that the Milky Way is warped in the vertical direction, which is consistent with its previously inferred warped morphology.Meanwhile, in the left panel of Figure 2, we find that the vertical velocity dispersion gradually increases from 5.0-6.0 km s −1 to 8.0-9.0 km s −1 from the inner disk to the outer disk.This suggests that the Galactic disk thickens vertically with increasing Galactocentric distance, which is consistent with the flare feature of the Galactic disk, reflected in thin and thick disks whose scale height is larger in the outer regions (Feast et al. 2014;Wang et al. 2018;Li et al. 2019;Chrobáková et al. 2022).The middle panel of Figure 3 shows the projection of V R in the X-Y plane.We see that the radial velocities are dominated by positive velocities (solid circles in orange) at about 10-13 kpc, indicating that there are more stars moving outward than inward in this region.This feature is more pronounced in the northern disk, indicating the asymmetry in the velocity structure between the northern and southern disks of the Milky Way.When we look at the middle panel of Figure 2, we see a peak shape at 10-14 kpc, dominated by positive radial velocities, with more stars moving outward than inward in this region, and the opposite in other regions.This is consistent with Figure 3. Within R < 10 kpc, the statistical average of the radial velocities is −1.9 km s −1 ; for R > 10 kpc, the average radial velocity of the stars starts to increase and reaches a maximum value of 8.3 km s −1 near 12 kpc, after which the average radial velocity decelerates and reverses motion again.Within 15 kpc, the maximum average values of the inward and outward velocities are 4.1 km s −1 and 8.In the right panel of Figure 2, we obtain the observed rotation curve of the outer Galactic disk.The results show that the rotation curve is generally flat with local fluctuations.The average rotational velocity decreases with a gradient of −3.81 km s −1 kpc −1 between 6 and 10 kpc, reaching a minimum of 229.66 km s −1 at 9.5 kpc.Then the rotation curve rises with a gradient of about 1.56 km s −1 kpc −1 until it reaches a maximum value of 235.88 km s −1 at 13.5 kpc, after which the rotational velocity gradually decreases.The shape of our rotation curve is consistent with Huang et al. (2016)  In the bottom panel of Figure 3, the rotational velocities present a clear asymmetry in the northern and southern warps in the directions with azimuthal angles around 90°and −90°.
Here, we define the northern and southern warps to directions around 90°and −90°.We analyze the rotation curves for eight different azimuthal angle ranges in Figure 4.In each range, the sample size is similar, and we calculated the average rotational velocity within each 1 kpc bin of Galactocentric distance.The bin with only one CC is not plotted because of the large error.From Figure 4, the most significant feature is the large difference between the rotation curves of the northern warp (red dots) and southern warp (blue dots).The northern warp has a lower rotational velocity and the southern warp has a higher rotational velocity compared to the mean rotational velocity at Galactocentric distances of 9-13 kpc.This difference is also found in the radial velocities, which are shown in Figure 5.The northern warp has positive radial velocities while the southern warp has negative radial velocities.
These velocity asymmetries may be due to warp precession, or to asymmetry in the structure of the northern and southern warps.Some previous works have found a north-south asymmetry around Galactocentric distances of 8 kpc that may be related to resonance of the bar, perturbation of the spiral arms (Monari et al. 2016), and/or interactions with the Sagittarius dwarf galaxy or the LMC (Gómez et al. 2013).
Based on the spiral arm potential of the perturbation distribution function, Monari et al. (2016) modeled the difference between the vertical velocities and radial velocities of stars on and between the spiral arms located at Galactocentric distances of 7-9 kpc.Based on tidal interactions with the Sagittarius dwarf galaxy, Gómez et al. (2013) modeled the asymmetry in the vertical direction locally at 8 kpc.The asymmetry in rotational velocities and radial velocities that we find here is mainly in the outer disk, which is more likely related to disk warp.
In contrast, the asymmetry in V Z is less obvious.In the warp model, the absolute value of the vertical velocities of the CCs is largest along the LONs and close to zero in directions with azimuthal angles of ±90°.The projection map of V Z agrees well with the warp model, and V Z is around zero in the northern and southern warps.

Cepheid Kinematic Warp Model
Earlier in the paper, we mentioned that the shape of the Galactic disk in the vertical direction is warped, and that CCs are considered to be ideal probes for uncovering the warp formation and evolution (Skowron et al. 2019b;Chen et al. 2019;Dehnen et al. 2023).In this section, we further investigate the Cepheid warp using the kinematic information.
Models of Galactic warp have been studied extensively in a number of works based on different tracers such as pulsars (Yusifov 2004)  geometric model of warp can be expressed in Galactocentric cylindrical coordinates (R, f, Z) as: where Z w is the average height of the stars above the plane of the Galactic disk, and R w and f w are the onset radius of warp and the Galactic azimuth of the warp's LONs, respectively.The parameters a and b are the fitting coefficients, representing the warp amplitude and the power-law index of the increase of warp amplitude with Galactocentric distance, respectively.We describe the warp precession model by introducing the evolution of the Galactic azimuth of the LONs as f w (t) = f 0,w + ωt, where f w = f 0,w is the current LONs of the warp.This warp model is the simplest model for warp precession given by Poggio et al. (2020), who derived an expression of V Z based on the zeroth moment of the collisionless Boltzmann equation.Ignoring variation of the warp amplitude with time and assuming that ω does not depend on R, the expression for the warp velocity is: In this formula, f V is the mean azimuthal velocity and ω denotes the precession rate of the Galactic azimuth of the LONs.
To discuss the warp model, we first fixed the Galactocentric distance to eliminate the variation of warp amplitude with Galactocentric distance.Here we chose CCs with Galactocentric distances in the range of 12.5-13.5kpc, i.e., a total of 134 for further study.We calculated their average azimuth velocity to » f -V 236.81 km s 1 and put it into Equation (8) together with R = 13 kpc.The parameters of the warp geometry model are directly taken from the CC warp model (Chen et al. 2019) with R w = 9.26, f w = 17.4,a = 0.148, and b = 1.In this case, ω is the only free parameter in the kinematic warp model.At the same time, Chen et al. (2019) show that there is a clear correlation between a, b, and R w , but f w is weakly correlated with these three parameters.This suggests that the values of a, b, and R w in the warp model obtained from different studies may be different, while the values of the parameter f w should be more consistent, i.e., f w is an independent parameter of the geometric warp model.From the kinematic warp model, it is known that stars on the warp's LONs have a maximum (minimum) vertical velocity.Therefore, we try to constrain the warp's LONs parameter f w and the warp progression rate ω by the vertical velocities of the CCs.
We investigated the effects of f w and ω on the kinematic warp model in Figure 6.The upper panel of Figure 6 shows the curves corresponding to the different warp's LONs in V Z versus f space.We fixed the precession rate as ω = 9.86 km s −1 kpc −1 , which is from Poggio et al. (2020) based on the warp linear model provided by Chen et al. (2019).Our geometric parameters were taken from the linear model of Chen et al. (2019).Unfortunately, the kinematic warp model is insensitive to the warp's LONs as seen in the figure, which means that the observed velocity does not constrain the warp's LONs well.Therefore, it is better to adopt the LONs from the geometric warp.In the lower panel of Figure 6, we fixed the warp's LONs to 17°. 4 and used different ω to generate kinematic warp models.Here, a positive ω indicates that the direction coincides with the rotation direction of the Milky Way.From the bottom panel of Figure 6, we can see that the curves of the kinematic warp models vary significantly with precession rate, with higher precession rates corresponding to smaller vertical velocities.Therefore, the kinematic data of the CCs can well constrain the precession rate of the warp.
Subsequently, we further relaxed the radius parameter of the kinematic warp model to study the differences in the warp model at different Galactocentric distances.In Figure 7, we plot the variation of the kinematic warp model with precession rate at different Galactocentric distances.The parameters of the colored solid lines in Figure 7 are the same as in the lower panel of Figure 6, but for different Galactocentric distances.Since the kinematics of the CCs can well constrain the precession rate of the warp, we determined the optimal warp precession rates at each radius by least-squares fitting, as shown by the black dashed line in each panel of Figure 7.In the fitting process, the azimuth angle of the warp's LONs was set to 17°.4 and the other geometric parameters were consistent with the linear model of Chen et al. (2019).
The first three subplots in Figure 7 show that the kinematic warp model is insensitive to changes in the warp precession rate in the 9-12 kpc range.The errors are too large to obtain an accurate warp precession rate.The kinematic warp model becomes sensitive to the warp precession rate as the Galactocentric distance increases, and the response is especially pronounced at azimuths near the warp's LONs.However, from the two subplots at the bottom of Figure 7, we find that the number of CCs beyond 15 kpc is small, and the error due to incompleteness may dominate.Therefore, we propose using the results of the Galactic warp precession rate at 12-14 kpc.
Since Poggio et al. (2020) used a sample at 13 kpc to obtain the precession rate, we finally chose to take the warp precession rate ω = 4.9 ± 1.6 km s −1 kpc −1 at 13 kpc as our result for the sake of comparison.We have indicated the best-fit result at 13 kpc in Figure 8 with a blue line, and used black and green lines to indicate the warp precession model from Poggio et al. (2020) and the static warp model, respectively.Our warp precession rate is lower than the result of Poggio et al. (2020).The Galactic warp precession rate given by Poggio et al. (2020)  give a warp precession rate of 9.86 km s −1 kpc −1 , which is 3.1σ larger than our results.Our results also exclude the static warp model with ω = 0, since the difference is 3.1 σ.
In contrast, our precession rate agrees with the low warp precession rate of ω = 4 km s −1 kpc −1 estimated by Chrobáková & López-Corredoira (2021) with a difference of only 0.6σ.They used the whole sample of Gaia DR2, whose average age is much older than the CCs.Chrobáková & López-Corredoira (2021) suggested that Poggio et al. (2020) determined their warp precession based on both young population structure and old population kinematic information, leading to a systematic uncertainty in the precession rate.They found that younger populations have a larger warp amplitude and a higher vertical velocity, while older populations have the opposite.Therefore, using a combination of the low velocities of the older population and the large amplitude of the younger population yields a higher precession rate.In our study, we used both the geometric warp model and the kinematic warp model of CCs to determine the warp precession rate and our  results support a low warp precession rate.Currently, the number of Galactic CCs with velocity information has greatly increased, providing us with a new opportunity to measure independently the Galactic warp precession rate based on CCs.In the future, based on more accurate and larger samples, the warp precession rate can be measured more accurately.

Conclusions
Based on RV information provided by Gaia DR3, we investigate the Galactic disk structure through CCs.The samples were taken from the Galactic CCs catalogs provided by Gaia Collaboration et al. (2023a).We extracted objects with RV information and crossmatched them with the Gaia archive to obtain their astrometric and photometry data, producing an initial sample of 2057 CCs.We estimated photometric distances for the CCs based on the period-Wessenheitmetallicity relation (Ripepi et al. 2022), and calculated the 3D positions and 3D velocities of CCs in the Galactic cylindrical coordinate system.
From the variation of 3D velocities with Galactocentric distance, we find warp and flare features from the vertical velocities, a peak shape in the radial velocities at 10-14 kpc, and local fluctuations in the rotation curve.In the projection of the 3D velocity on the X-Y plane, we clearly see the northsouth asymmetry of the Galactic disk.To investigate this phenomenon further, we plot the rotational velocities and radial velocities versus Galactocentric distance for CCs in different azimuthal ranges.The results show that the southern warp and the northern warp have significantly different patterns of rotational velocities and radial velocities.These differences appear in the outer disk, and we propose that they may be related to warp precession or asymmetry of the warp structure.
We investigate a kinematic warp model using CCs and find that the kinematic warp model is insensitive to the parameters of the LONs, but sensitive to changes in the warp precession rate.The warp's LONs can be determined by geometric information, while the vertical velocities can be used to constrain the warp precession rate.Meanwhile, we find that the kinematic warp model is insensitive to the warp precession rate up to 12 kpc, and as the distance increases the kinematic warp model becomes more sensitive and the model response is largest near the LONs.However, the current sample size beyond 15 kpc is too small, so we propose using samples at 12-14 kpc to obtain the Galactic warp precession rate.Based on the warp linear model parameters of Chen et al. (2019), we determine the warp precession rate at different Galactocentric distances.For comparison with the literature, we use the result at 13 kpc, ω = 4.9 ± 1.6 km s −1 kpc −1 , as the final Galactic warp precession rate.Both Poggio et al. (2020)ʼs precession rate, ω = 9.86 km s −1 kpc −1 , and the static warp model, ω = 0, are outside the 3σ error range of our results, and we support a low warp precession rate as in Chrobáková & López-Corredoira (2021).Future Gaia data releases will provide a larger sample of CCs with RV curves.This will help to constrain the precession of the warp, thus providing more evidence to constrain the mechanisms of Galactic disk formation and evolution.website is https://www.cosmos.esa.int/gaia.The Gaia archive website is https://archives.esac.esa.int/gaia.
the distance d from each CC to the Sun.Combining the Galactic longitude and latitude (l, b) of each object, we projected the heliocentric distances d into 3D Cartesian coordinate (x, y, z) centered on the Sun by the equation: positive x-direction is toward the center of the Milky Way.Then we assumed the Sun is 8 kpc away from the Galactic center in order to use the warp model parameters ofChen et al. (2019), located in the midplane (Z = 0), to obtain the Galactocentric Cartesian coordinates (X, Y, Z) through coordinate translation, where X = −x + 8.In this way, we calculated the Galactocentric distance R and azimuth f of each CC in Galactic cylindrical polar coordinates through = f increases with the rotation direction of the Milky Way.To verify the validity of the distances, we converted the heliocentric distances from GaiaCollaboration et al. (2023a)  to Galactocentric distances.

Figure 1 .
Figure 1.3D velocity histogram of 1967 CCs in Galactic cylindrical polar coordinates.The mean and median values of the velocities in each direction are labeled.

Figure 2 .
Figure 2. 3D velocity distribution of 1901 CCs at different Galactocentric distances.We used the 3σ standard deviation of the mean in 1 kpc bins as a criterion to exclude outliers.The black solid dots give the average value of each 1 kpc bin.Error bars are standard deviations.

Figure 3 .
Figure 3. Projection of the CC 3D velocities in the X-Y plane.From top to bottom, the colors are coded by V Z , V R , and V f , where red represents larger velocities and blue represents smaller velocities.The position of the Galactic center and the Sun are marked with red plus signs, and the solid black circles represent Galactocentric distances of 8, 10, and 12 kpc.
and Gaia Collaboration et al. (2021), where Huang et al. (2016) used a sample of red clump giants and K giants, and Gaia Collaboration et al. (2021) used extremely young massive stars with ages younger than 0.2 Gyr.

Figure 4 .
Figure 4. Galactic rotation curves at different azimuths.We divided the azimuths into eight ranges as shown in the legend.The northern and southern disk asymmetries are significant with azimuthal angles of 45°< f < 90°and −90°< f < −45°, respectively.

Figure 5 .
Figure 5. Similar to Figure 4 but for radial velocities.
is 10.86 km s −1 kpc −1 , which is about 3.7σ larger than our result.Based on the linear warp model ofChen et al. (2019),Poggio et al. (2020)

Figure 6 .
Figure6.Effects of the warp's LONs f w and precession rate ω on the kinematic warp model.The blue dots show the distribution of 134 CCs with Galactocentric distances of 12.5-13.5kpc.The adopted warp geometric model is fromChen et al. (2019).In the upper panel, the warp precession rate is fixed at ω = 9.86 km s −1 kpc −1(Poggio et al. 2020), with different color curves representing different f w .In the lower panel, the warp's LONs f w is fixed to 17°. 4 and the different color curves represent different ω.

Figure 7 .
Figure 7. Warp precession rates at different Galactocentric distances.The black dashed line is the best-fit warp precession rate and the results are given in the legend.The different color curves are similar to the lower panel of Figure 6 but for different Galactocentric distances of the warp model.

Figure 8 .
Figure8.Determination of the precession rate based on CCs.The blue line represents the best kinematic warp model obtained by fitting the kinematics of the CCs (blue dots) with the least-squares method.The black line is the warp model with a higher precession rate ω = 9.86 km s −1 kpc −1 given byPoggio et al. (2020), and the green line is the static warp model with ω = 0 km s −1 kpc −1 .