Local Circular Velocity with Cepheids

Galactic Local Circular Velocity from Gaia DR1 Cepheids and Effects of Non-Axisymmetry

Daisuke Kawata, Jo Bovy, Noriyuki Matsunaga and Junichi Baba
Mullard Space Science Laboratory, University College London, Holmbury St. Mary, Dorking, Surrey, RH5 6NT, UK
Department of Astronomy and Astrophysics, University of Toronto, 50 St. George Street, Toronto, ON M5S 3H4, Canada
Department of Astronomy, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan
National Astronomical Observatory of Japan, Mitaka, Tokyo 181-8588, Japan
E-mail: d.kawata@ucl.ca.ukAlfred P. Sloan Fellow
Accepted XXX. Received YYY; in original form ZZZ
Abstract

We apply a simple axisymmetric disc model to 174 Galactic Cepheids, to measure the local circular velocity, its radial gradient and the other Galactic parameters, such as the Sun’s proper motion in the Galactic rest frame, the solar radius and the velocity dispersion of the tracer populations. Accurate measurements of the distance and velocities for the Cepheids are obtained by cross-matching the existing Cepheids catalogue with the Tycho-Gaia astrometric solution (TGAS) catalogue. Our axisymmetric disc model fit to the kinematic data of the Cepheids determines the local circular velocity at the Sun’s location of  km s and the Sun’s azimuthal and radial peculiar motions of  km s and  km s, respectively, with strong priors of the solar radius,  kpc, and angular rotation velocity of the Sun,  km s kpc. To understand what these measurements from the local stellar tracers mean in terms of the local and global Galactic parameters, we also applied the axisymmetric model to the -body/hydrodynamic simulation data of a Milky Way-like galaxy with a bar and spiral arms by sampling the young and old disc star particles within the distance of 3 kpc from the observer placed at  kpc, but at different azimuthal angles. We found that our axisymmetric model fit to the young stars recovers the local circular velocity, defined with the radial gravitational force, reasonably well at the different locations of the disc, even in the face of significant non-axisymmetry in the forces.

keywords:
Galaxy: fundamental parameters — Galaxy: solar neighbourhood — Galaxy: disc — Galaxy: kinematics and dynamics — stars: variables: Cepheids — methods: numerical —
pubyear: 2017pagerange: Galactic Local Circular Velocity from Gaia DR1 Cepheids and Effects of Non-AxisymmetryGalactic Local Circular Velocity from Gaia DR1 Cepheids and Effects of Non-Axisymmetry

1 Introduction

The circular velocity at the location of the Sun is a fundamental parameter in the determination of the distribution of the total mass in the Galaxy. However, the measurement of the circular velocity at the Sun is not a straightforward process, and the different measurements provided systematically different results (see Bland-Hawthorn & Gerhard, 2016, for a review). The Sun’s rotation velocity with respect to the Galactic rest frame, , is most reliably measured as the proper motion of Sgr A, assuming that Sgr A is fixed at the Galactic centre. Using Very Long Baseline Interferometry (VLBI), Reid & Brunthaler (2004) measured the proper motion of Sgr A with respect to the background quasar, and obtained  km s kpc. However, the distance to the Galactic centre, , and the difference between the circular velocity, , and the Sun’s rotation speed, , are remained to be challenging measurements. The various measurements of are converging to around  kpc (Bland-Hawthorn & Gerhard, 2016). However, there are still systematic differences between the measurements (de Grijs & Bono, 2015; Bland-Hawthorn & Gerhard, 2016). The difference between and are referred as the Sun’s peculiar rotation velocity, , and evaluated using different kinematic tracers combined with Galactic disc dynamical models.

The star forming regions or young stellar populations are often used for such measurement, because their mean rotation velocity is close to the circular velocity, i.e. asymmetric drift can be ignored, as the circular velocity curve is measured with the rotation curve from the gas terminal velocity from the radio observations (e.g. Sofue, 2017). For example, using the accurate astrometric measurements of about 100 high mass star forming regions, Reid et al. (2014) measured  km s with  kpc (see also Honma et al., 2012).

Classical Cepheids are also considered to be a good kinematic tracer for this purpose, because of their young age (20300 Myr, e.g. Bono et al., 2005) and accurately measured distance based on their well known relation between the absolute luminosity and the period of the variation (Inno et al., 2013). Feast & Whitelock (1997) combined the proper motion measured by Hipparccos and the photometric distance of 220 Galactic Cepheids, and measured  km s and the angular velocity of circular rotation at the Sun,  km s kpc with  kpc. Recently, Bobylev (2017) combined the distance and line-of-sight velocity in Mel’nik et al. (2015) with the proper motion from the first release Gaia data (Gaia DR1, Gaia Collaboration et al., 2016a, b; Lindegren et al., 2016) for 249 Cepheids, and obtained  km s and  km s with  kpc.

It is important to note that the star forming regions are likely affected by the systematic velocity due to the environment, like the spiral arms which triggers star formation (e.g. Baba et al., 2009; Bovy et al., 2009; McMillan & Binney, 2010). Chemin et al. (2015) demonstrate that the rotation velocity measured by the gas terminal velocity can be affected by the bar and spiral arms, which leads to incorrect measurement of the mean rotation velocity. The stars are also affected by the bar (e.g. Dehnen, 2000), spiral arms (e.g. Sellwood & Binney, 2002; De Simone et al., 2004; Grand et al., 2012a, b; Baba et al., 2013) and combination of bar and spiral arms (e.g. Quillen, 2003; Antoja et al., 2009; Minchev & Famaey, 2010). Trick et al. (2017) study how their measurements of the Galactic parameters with an advanced, action-based Galactic disc dynamical model are affected by the spiral arms (see also Chemin et al., 2016). They conclude that the dynamical models can be biased due to the influence of spiral arms, but if the sample of stars cover a large enough volume (with at least radius of 3 kpc), the dynamical model can recover the global Galactic parameters.

The Gaia collaboration will soon make their second data release (DR2), which will provide the accurately measured astrometry and radial velocity for stars covering this large enough volume around the location of the Sun. Interestingly, the cross-matched sample of the published Cepheids distance and radial velocity and the Gaia DR1 proper motion already cover a similar size of the volume, although there are only a small number of stars. To be ready for the Gaia DR2, we apply the simple axisymmetric dynamical model used in Bovy et al. (2012) to the classical Cepheids sample with the Gaia DR1 proper motion, and measure the Galactic parameters, including and .

We modify the axisymmetric kinematic model of Bovy et al. (2012) to fit both the line-of-sight velocity and the tangential velocity of stars, taking into account the observational errors of distance and velocities. This model takes into account the velocity dispersion and asymmetric drift, which are differences from Bobylev (2017). Hence, this model can also be applied for any kind of stellar tracer populations. We further discuss how the measured Galactic parameters are affected by the non-axisymmetric structures, using an -body/Smoothed Particle Hydrodynamics (SPH) simulation of a Milky Way-like disc galaxy with a bar and spiral arms. We demonstrate that young stars within 3 kpc of local volume from the observer is a suitable sample to recover the local circular velocity defined by the local gravitational force at the observer position, which can be different from the azimuthally averaged global circular velocity at the radius.

Section 2 shows our sample of Cepheids used for our axisymmetric disc model fit which is described in Section 3. The results of the axisymmetric disc model fit to the Cepheids data are shown in Section 4. Section 5 demonstrates that the axisymmetric model can recover the model parameters well for the mock data created with the axisymmetric disc with known parameter, as expected. Section 6 shows the results of applying the axisymmetric model to the -body/SPH simulation data of a Milky Way-like disc galaxy. A summary and discussion of this study are presented in Section 7.

2 Cepheids data

We used the same sample of classical Cepheids adopted in Baba et al. (2018). Our sample is taken from Genovali et al. (2014), where the distances to Cepheids are determined homogeneously by using near-infrared data. Errors in distance modulus are  mag for most of the Cepheids. We then cross-matched this sample with the Tycho-Gaia astrometric solution (TGAS; Michalik et al., 2015; Lindegren et al., 2016) catalogue to obtain the proper motion. We further cross-matched with a sample of Cepheids whose line-of-sight velocity, , are provided in Mel’nik et al. (2015). We use to describe the line-of-sight velocity in three dimension to distinguish from the line-of-sight velocity projected in the disc plane. For cross-matching these catalogue, we used TOPCAT (Taylor, 2005). The cross-match provided 206 Cepheids with known locations and 3D kinematics. We further limited the sample based on vertical position with respect to the Sun, kpc. To take into account the error, we define  kpc, where , , and are Galactic latitude, distance modulus and distance modulus error in magnitude, respectively. This limit was applied to eliminate clear outliers, although our sample shows a clear concentration around the Galactic plane, with more than 70 % being located within 100 pc, as expected for young stars like Cepheids.

To eliminate the data with a large velocity or distance uncertainty, we discarded the data whose uncertainty in velocity, , is larger than 20 km s, where , and are the uncertainties of the velocity measurements in the direction of Galactic longitude, , and latitude, , and heliocentric line-of-sight velocity, , respectively. and were evaluated by taking the standard deviation of the Monte-Carlo (MC) sampling of and computed from randomly selected right ascension (RA) and declination (Dec) proper motions and distance taken from the randomly selected distance modulus. The proper motions were sampled using the 2D Gaussian probability distribution with their measured mean, standard error and correlation between the RA and Dec proper motions. The distance modulus was taken from the Gaussian probability distribution with the measured distance modulus and their 1  uncertainty. For most of the Cepheids included, all of these three velocity errors are less than 4 km/s and are roughly comparable to each other. In addition, to avoid the results of our model fit being affected by a few Cepheids at a large distance with larger relative velocity with respect to the Sun, we also limited the sample to a distance of 4 kpc, beyond which the number of Cepheids drops dramatically. These selections left 174 Cepheids in our sample. The distribution of the Cepheid sample is shown in Fig. 1. The arrows in the left panel of the figure shows the velocities of Cepheid with respect to the Sun.

Figure 1: Distribution of the Cepheids used for our model fit. Arrows in the left (right) panel indicate their velocity with respect to the solar motion in the Galactic rest frame before (after) subtracting the mean rotation of the best fit axisymmetric model.

3 Model

3.1 Axisymmetric disc kinematic model

Following Bovy et al. (2012), we compute the mean and dispersion of expected in an axisymmetric Galactic disc model in the Galactic rest frame, and derive the posterior probability for the parameters of the models by exploring likelihood of the model parameters with Markov-Chain Monte-Carlo (MCMC). We also include the component of the velocity in the Galactic longitude direction, , in the likelihood function, taking into account the covariance, because and are not independent of each other. The model is an axisymmetric and assumes a Gaussian velocity dispersion in the Galactic rotation and radial directions with no correlation, and zero mean radial velocity. We ignore the vertical motion or the thickness of the disc, but consider the distance and velocity projected in the disc plane. Hence, our axisymmetric model is considered to be a two-dimensional model in the disc plane. Therefore, note that the heliocentric velocities used from this section mean velocities projected in the disc plane, i.e. , where is the line-of-sight velocity in three dimension as used in the previous section.

In the axisymmetric disc with no mean radial velocity, we need to consider only the mean rotation velocity, , and velocity dispersion in the Galactic rotation, , and radial, , directions. The mean rotation velocity is calculated from asymmetric drift, , as , where is the circular velocity at the radius of . Following Bovy et al. (2012), we assume an exponential disc (stellar density ) and an exponentially declining radial velocity dispersion (), and then asymmetric drift can be calculated by

(1)

where , is azimuthal velocity dispersion (e.g. Binney & Tremaine, 2008) and is assumed to be constant for simplicity. As discussed in Bovy et al. (2012), we confirmed that the Galactic parameters we are interested in are not sensitive to the radial scale length of radial density profile, , or the radial scale length of radial velocity dispersion, . Here, we fix  kpc and  kpc. These values are inspired from the scale lengths seen for the star particles with the similar age range to Cepheids in our numerical simulation described in Section 6. The assumed is much larger than the conventional value for the Milky Way thin disc, e.g.  kpc suggested in Bland-Hawthorn & Gerhard (2016). However, the scale length for the young stars around the Sun is not well known, and could be different from that for the old stars. For example, Mackereth et al. (2017) analysed the stellar structure of the Galactic disc as a function of the age from the Apache Point Observatory Galactic Evolution Experiment (APOGEE) data, and found that the younger stars are having a flatter or even peaked radial profile around the solar radius. The scale length of radial velocity dispersion profile is also not well constrained either. From kinematics of old disc K giants, Lewis & Freeman (1989) showed  kpc. Huang et al. (2016) used red clump stars from APOGEE and the LAMOST Spectroscopic Survey of the Galactic Anti-centre and suggested  kpc. The scale length of radial velocity dispersion profile for the young stars could be different from these from relatively older stars. Hence, we use the above mentioned values obtained from our numerical simulation. As a result, our axisymmetric model can be described with , and . We also take into account the slope of the circular velocity, , at the solar radius of , when we calculate . As mentioned above, the velocity dispersion profile is assumed to be an exponential law. Therefore, for velocity dispersion, we consider the parameters of which is the radial velocity dispersion at , and is also a parameter.

In the Galactic rest frame, the mean rotation velocity at the position of the star, , can be projected to the line-of-sight velocity, , from the observer, i.e. the position of the Sun, as , where is Galactic longitude and is the angle between the line from the Galactic centre toward Sun and the one toward the position of the star, positive in the Galactic rotation direction. Following the same strategy, we can derive the projected longitudinal velocity from the rotation velocity of the axisymmetric model as .

The observational data provide the line-of-sight velocity, , and Galactic longitudinal velocity, , with respect to the Solar motion. Using the Solar radial velocity, (outward motion is positive) and rotation velocity, (clock-wise rotation is positive, ) in the Galactic rest frame, these velocities can be converted to the Galactic rest frame velocities as follows.

(2)
(3)

These can be compared with the expected velocity distribution of and from the model. Hence, fitting our axisymmetric disc model to the observed data, and for the tracer sample, requires to explore 7 model parameters, .

3.2 MCMC parameter probabilities

We consider the posterior probability to find the marginalised probability distribution function of our model parameters as follows.

where , describes the observational data, and corresponds to the model parameters. We run MCMC for . Likelihood function is described with

(5)

where

(6)

Here, and are the observed line-of-sight and longitudinal velocity in the Galactic rest frame for the observed star, , and and are the expected line-of-sight and longitudinal velocity in the Galactic rest frame from the axisymmetric model at the location of the star, . The covariance matrix, , is described as , where

(7)

and

(8)

To take into account the observational errors in the distance modulus, line-of-sight velocity and RA and Dec proper motion errors, we average each Cepheid’s likelihood over 1,000 Monte Carlo (MC) samples of the observed distance modulus, line-of-sight velocity and RA and Dec proper motions from Gaussian distributions with their observed values and errors. This can be described as

(9)

where is the likelihood for -th MC realisation for -th Cepheid, and . The correlation of the proper motion errors in the TGAS data are taken into account, using Cholesky decomposition.

We found that is not well constrained by our observational Cepheids data (see Section 4). Hence, we introduced a Gaussian prior for as follows.

(10)

where we set kpc and kpc from Bland-Hawthorn & Gerhard (2016). We further use a Gaussian prior for the angular velocity of the Sun, , with km s kpc, because this is well constrained by the other observations (see Bland-Hawthorn & Gerhard, 2016, for a review).

We use emcee (Goodman & Weare, 2010; Foreman-Mackey et al., 2013) for the MCMC sampler and use 128 walkers and 1,000 chains per each walker. We use galpy (Bovy, 2015) for coordinate transformation.

Figure 2: Marginalised posterior probability distribution of our model parameters in our fiducial model. The number in the panel gives the correlation coefficient. In the panel of , the blue line shows equation (11), the best-fit linear relation.
Figure 3: Line-of-sight velocity, , (left) and Galactic longitudinal velocity, , (right) as a function of Galactic longitude for our Cepheids sample (solid circles) with the distance between 3 and 4 (top), 2 and 3 (2nd), 1 and 2 (3rd) and 0 and 1 kpc (bottom). The solid line in each panel show our best fit model results using the mean distance of the samples in each panel. The grey line in the bottom right panel shows from the Oort constant , and measured in Bovy (2017). The uncertainty is as small as the thickness of the grey line.
Figure 4: Marginalised posterior probability distribution of our model parameters in  kpc prior and no prior model. The number in the panel gives the correlation coefficient. In the panel of , the blue line shows equation (12), the best-fit linear relation. Note that the - and -axes ranges in each panel are different from Fig. 2.
Fiducial Bobylev (2017)
174 174 249

Gaussian prior of  kpc is applied, and no prior is used.
This is not a MCMC fit result, but the results in Bobylev (2017) who used a different method.
This is fixed.
This is calculated from  kpc,  km s kpc and  km s kpc.

Table 1: Results of the MCMC fit

4 Results for Cepheids in Gaia DR1

We applied our axisymmetric model fit described in Section 3 to our selected Cepheids sample showin in Fig. 1. The marginalised posterior probability distributions of the model parameters are shown in Fig. 2, and the mean and dispersion for each parameter probability distribution are summarised in Table 1. We call this model result ‘fiducial model result’. Later we will relax priors to study the sensitivity to our prior assumptions.

The fiducial model results are similar to Bobylev (2017) who applied a different method to similar Cepheids data. It is interesting that the radial velocity of the Sun measured from the Cepheids kinematics tends to provide a lower value ( km s in our fiducial model result) compared to the ones from kinematics of the local dwarf stars, e.g.  km s in Schönrich et al. (2010). In addition to the parameters explored in Bobylev (2017), our model fit provides velocity dispersion of and the ratio of the azimuthal and radial velocity dispersion, for the Cepheids sample.

Arrows in the right panel of Fig. 1 show the velocity of Cepheids in the Galactic rest frame after subtracting the rotation velocity from the best fit axisymmetric model. There is no obvious systematic residual velocity fields, which implies that the axisymmetric model is a reasonable model to describe the average velocity trend of our sample. As studied in Baba et al. (2018), the residual velocity fields are likely due to the spiral arms and provide the valuable information about the nature of the spiral arms. Because of the relatively large area covered by our Cepheids sample, our model fit is not so sensitive to the systematic velocity trend around the spiral arms (see also Trick et al., 2017). Fig. 3 compares the and from the best fit model at different distance with what is observed for our Cepheids sample. The figure shows that the best fit model describes both velocity components of the Cepheids reasonably well at the different distances. The grey line in the bottom right panel in Fig. 3 shows from the Oort constants, , and , measured in Bovy (2017) from the TGAS proper motion data of nearby dwarf stars with a typical distance of 230 pc. The grey line is broadly consistent with our samples within the distance of 1 kpc, and our best fit model.

Each panel of Fig. 2 also presents the correlation coefficient of the two parameters at the top-left corner in each panel. The rotation velocity of the Sun, , and the circular velocity at the location of the Sun, , are strongly correlated with each other. This indicates that the Cepheids data provide stronger constraints on the rotation of the Sun with respect to the circular velocity at the location of the Sun, , because the velocity dispersion of the Cepheids sample is small and asymmetric drift is negligible. In fact, our result provides very small asymmetric drift of  km s at . We obtained  km s in our fiducial model result as shown in Table 1. This is consistent with  km s in Schönrich et al. (2010) from the Hipparcos data of nearby (distance pc) dwarf stars.

However, this is significantly different from  km s in Bovy et al. (2012) who applied the same axisymmetric model to the APOGEE survey data which covers up to the distance of 10 kpc (see also Bovy et al., 2015, who obtained similar values from the mean velocity field of the APOGEE red clump stars). They suggest that nearby stars covered by the Hipparcos stars have a systematic motion as large as 10 km s with respect to the local circular velocity, due to streaming motion caused by non-axisymmetric structures. Our sample covers the distance up to  kpc, although the majority of them are within 3 kpc. It is interesting to find that our results are still consistent with measured with the solar neighbourhood sample within the very small volume covered by the Hipparcos stars. However, the number of our sample is still small, and we do not think that our result is conclusive enough to test the scenario of the streaming motion of the solar neighbourhood sample. The streaming motion affecting the measurement of for the solar neighbourhood star sample should be able to be tested with more observational data, which will be available in the Gaia DR2.

Both and are also strongly correlated with . This means that our Cepheids sample constrain the angular rotation speed more strongly than the rotation velocity in the Galactic rest frame. This is not due to our strong prior of the angular velocity of the Sun, . We ran the axisymmetric disc model fit with less strict solar radius prior of  kpc without any prior of . The result is shown in Fig. 4 and Table 1. The result shows the similar correlation of and with . Hence, this strong correlation is not due to the prior in the fiducial model, but intrinsic to the relatively local sample of our Cepheids data. Table 1 shows that kpc prior model result prefers higher . This means that our Cepheids data poorly constrain the Galactocentric radius of the Sun.

In Figs. 2 and 4, the panels of show a blue line which is a linear regression line of these two parameters. This line indicates the relation of

(11)
(12)

This relation requires and 7.4 kpc to have the IAU standard value of  km s (Kerr & Lynden-Bell, 1986) for the fiducial and kpc models, respectively. These are much smaller than the IAU standard value of  kpc (Kerr & Lynden-Bell, 1986) and they are too small compared to the other independent measurements of (see de Grijs & Bono, 2016; Bland-Hawthorn & Gerhard, 2016, for a review). Therefore, our results prefer a faster circular velocity at the location of the Sun.

Figure 5: Marginalised posterior probability distribution of the MCMC fitting to the axisymmetric disc mock data. The lines indicate the input parameters.
Input Model
236
248
15.0
1.0
8.2
Table 2: Results of the MCMC fitting to an axisymmetric disc mock data

5 Axisymmetric disc mock data test

To validate our MCMC fitting algorithm, we made mock data using the position of the observed 174 Cepheids and assign the velocity expected at the location of the Cepheids from an axisymmetric disc model with the known input parameters. Then, we created a mock data set by randomly displacing the distance modulus, line-of-sight radial velocity and proper motions using the observed errors of each Cepheid at each location. We applied the axisymmetric model fit to the mock data, taking into account the observed error with MC sampling as described in Section 3. We used the same and priors as our fiducial model fit in Section 4.

Marginalised posterior probability distribution of the parameters are shown in Fig. 5 and the results are summarised in Table 2. Table 2 also shows the input parameters for creating the mock data. All parameters are recovered well within 1  uncertainty. The uncertainties of parameter values from the model fit to the mock data are also similar to the fiducial model results to the real observational data in Table 1. This validates that our method provides the expected quality of the results from the number and accuracy of the distances and velocities of our Cepheids sample. The strong correlation between , and are also seen in the mock data fitting result in Fig. 5. This also presents that this correlation is an intrinsic limitation to the constraining power of this sample of the data.

Figure 6: Map of the local circular velocity, defined in equation (13), at different position of the disc over-plotted with contours of the over-density of stars. The positions indicated with the empty star symbols are the location of the observers chosen for our mock data. The numbers are indicated the angles in Fig. 8.

6 Fitting the mock data from an -body/SPH Milky Way-like simulation

The Galactic disc is not purely axisymmetric. The motion of stars are affected by non-axisymmetric structures, such as the bar and spiral arms (e.g. Lynden-Bell & Kalnajs, 1972; Dehnen, 2000; Baba et al., 2009; Grand et al., 2012a, 2014; Di Matteo et al., 2013; Bovy et al., 2015). Gravitational forces in the disc plane are also affected by the non-axisymmetric structures. Therefore, the ‘local’ circular velocity can be defined by

(13)

where is the gravitational force in the radial direction toward the Galactic centre at the position of in the Galactic plane. The local circular velocity can be different at the different azimuthal angles even at the same Galactocentric radius (Chemin et al., 2015; Chemin et al., 2016; Trick et al., 2017).

In this section, we discuss what the local circular velocity obtained by our axisymmetric modelling for the Cepheids kinematics means, applying the axisymmetric model fit to the particle data from a numerically simulated disc galaxy similar to the Milky Way. We use the same snapshot data as the DYN model at  Gyr in Baba et al. (2018). This is a Milky Way-like disc simulation with an -body/SPH code, ASURA, with self-gravity, radiative cooling, star formation and stellar feedback (Saitoh et al., 2008; Saitoh & Makino, 2009, 2010). This particular snapshot shows the clear bar component and spiral arms, and Baba et al. (2018) presented that the motion of the young star particles around the Perseus-like spiral arm is similar to what is observed for the same Cepheids sample used in this paper.

Figure 7: Distribution of all (left panel) and selected (right panel) star particles whose age is between 20 and 300 Myr. The x-y coordinate is the same as Fig 6, and the panels are centred at the assumed solar location, .
Figure 8: Local circular velocity (top), radial gradient of circular velocity (2nd), rotation velocity of the Sun (3rd), radial velocity of the Sun (4th), radial velocity dispersion (5th) and the ratio of squares of azimuthal and radial velocity dispersion (bottom) as a function of the azimuthal angle of the assumed observer as indicated in Fig. 6. Black filled circle with vertical error bars show the results from our model fit for the mock data of the young star particles with  Gyr (left) and the old star particles (right) at the positions highlighted in Fig. 6 and the horizontal error bar corresponds to the angle covered by the distance of 3 kpc. Red thick error bars indicate the fitting results for mock data with similar location to the observed Cepheids with the observed errors taken into account (see the main text for more detail). Grey solid lines indicate the true values measured at different azimuthal angles from the simulation model, and horizontal dashed lines indicate the mean of these values in the top and 2nd panels. In the 3rd panel, the horizontal solid grey line is the input parameter of the assumed observer’s rotation velocity. In the 4th, 5th and bottom panels, grey open circles with the horizontal error bars show the true values measured for the sampled star particles at different locations from the simulation data. In the 4th panel, the grey horizontal dotted line shows the assumed observer’s radial velocity, and the grey circles show the assumed observer’s radial motion plus the mean radial velocity of sampled particles around each assumed observer position.

Fig. 6 shows the local circular velocity at different positions around the disc and the contours show the over-density of the stars, which indicate the bar and spiral arms position. Fig. 6 shows that the local circular velocity changes significantly around the bar and spiral arms. Hence, the local circular velocity at different azimuthal positions can be different even at a fixed radius. The observer position used in Baba et al. (2018) corresponds to the position of kpc in Fig. 6, and we define that this is the angle of  deg. Open star symbols in Fig. 6 show the position at different at  kpc, and increase in anti-clockwise. The grey line in the top panels in Fig. 8 indicates the local circular velocity as a function of at  kpc. The horizontal grey dotted line shows the azimuthally averaged circular velocity at  kpc. We can see that the local circular velocity can be different by about 10 km s from the averaged circular velocity.

We created mock Cepheids data from the simulation snapshot by selecting the star particles whose age range is between 0.02 and 0.3 Gyr similar to the estimated age ranges of Cepheids (e.g. Bono et al., 2005). At  deg, these young star particles are distributed as shown in the left panel of Fig. 7. It is obvious that the spatial distribution of the young stars are not homogeneous. Instead, the particles are concentrated around where the spiral arms are, and there are several clumps with different sizes. We applied our axisymmetric model fit to these particle data, but because of this biased sampling around the spiral arm, the model fit could not recover the true local circular velocity. To mitigate this problem, we have selected 1,000 particles randomly, but as homogeneous as possible, which are shown in the right panel of Fig. 7. This is also motivated by the fact that our Cepheids sample are distributed reasonably homogeneously, as seen in Fig. 1. To select these particles, we randomly selected a location within the distance of 3 kpc from the observer position, and picked the closest particle from the randomly selected location after eliminating the particles already picked up. We have chosen the distance limit of 3 kpc, because the majority of our Cepheids data in Fig. 1 are within 3 kpc from the Sun, and we found that the larger distance limit breaks the axisymmetric assumption and provide worse results. We limited the number of particles to 1,000, because the particle resolution of the simulation allows us to pick the young star particles randomly and reasonably homogeneously up to around this number. We also assumed that the observer at  deg and  kpc was moving with  km s and  km s. In our axisymmetric model fit, we assumed  kpc and  kpc in equation (1), because this roughly matched the density and velocity dispersion profile. As mentioned above, we used the same and for our model fit in Section 4. We also applied the priors of  kpc and  km s kpc similar to what we did in Section 3.

The results of the axisymmetric model fit for these selected young particles around the assumed observer position of are shown at deg in the left panel of Fig. 8 (black filled circle with error bars). The top left panel shows that the true local at (grey solid line) is slightly higher than the mean circular velocity (grey dotted line), and our axisymmetric model fit (black filled circle) recovers this local circular velocity well. More remarkably, the left panel in the second row shows that the gradient of at is recovered very well, although the true local gradient shown with the solid grey line is significantly different from the azimuthally averaged (horizontal grey dotted line). The left panel in the third row demonstrates that the input rotation velocity of the observer shown with the horizontal grey solid line is also recovered well with our model fit at . The left panel in the fourth row shows the result of the radial velocity of the observer, , for which the true value is  km s (grey dotted line). Our model fit assumed that the mean radial velocity of the sample, , is zero, but if there is non-zero , the fitting result provide . Therefore, this panel shows the true  km s from the simulation data as an open grey circle with horizontal error bar. Our model fit result is shown with black filled circle with error bars. The mean radial velocity of  deg sample happened to be close to zero, and our model fit result recovered the assumed very well. The left panel in the fourth row shows the result of for the sampled star particles. The measured from the simulation data are shown with open grey circle with horizontal error bar. Our model fit result is shown with black filled circle with error bars (vertical error bars are not visible, because the measured uncertainty of from our model fit is very small), which reproduce the true velocity dispersion very well. The square of velocity dispersion ratio, , is also recovered well at  deg in the bottom left panel. Again, the true measured directly from the particle data are shown with grey open circles with horizontal error bar, and the model fit results are shown with black filled circles with error bars.

Because the local circular velocity depends on the azimuthal angle position even at a fixed radius as shown in the top left panel of Fig. 8, we applied the same exercise at the different azimuthal positions at  kpc, as indicated with open star symbols in Fig. 6. We sampled 1,000 young star particles with age between 20 and 300 Myr randomly and spatially homogeneously within the distance of 3 kpc from the observer after placing the observer position at different . Then, we applied our axisymmetric model fit to the sample, assuming the same observer’s rotation speed of  km s and radial motion of  km s. The results and comparison with the true values from the simulation data are shown in the left panels of Fig. 8. Interestingly, the local circular velocity, , is well recovered at the different . The measured local circular velocities are underestimated at and  deg, but considering the angle covered by the tracer sample, it is not much more than difference. The successful recovery of the local circular velocity at different in this complex simulation data with our simple axisymmetric model is encouraging. This means that applying an axisymmetric model to a relatively small region of the disc area can be valid to recover the local circular velocity.

The left panel in the second row shows that the radial gradient of circular velocity, , is also recovered reasonably well at  deg, while it is struggling at  deg. We found that the recovery of is worse, where changes non-monotonically, and shows significant oscillation within  kpc range in radius, because of the spiral arms. The rest of panels show the reasonable agreement between the true values and the measured values. However, and are significantly lower at  deg, where the spiral arm is affecting the velocity distribution. Still, the local circular velocity and the gradient is recovered well at  deg. This also indicates that our axisymmetric model fit provides a robust estimate of the local circular velocity than the other parameter values.

Interestingly, the Sun-like location, , recovers all the values very well. This location is between the two strong spiral arms, like Perseus and Scutum-Centarus arms in the Milky Way. The simple axisymmetric model works reasonably well at such location.

We made more comparable mock data to our Cepheids data by sampling the position and velocities of the young ( Age  Myr) star particles nearest to the observed position of 174 Cepheids used in Section 4 after placing the observer position at  deg and  kpc. We then applied our axisymmetric disc model to the mock data, taking into account the same observational errors as the Cepheids data closest to the corresponding particle. The results are shown with red open square with the error bars in the left panels of Fig. 8. The results show larger errors than the case of using all the particles with no error (black filled circle). The local circular velocity is slightly overestimated, which indicates a matter of caution for our Cepheid sample results. Still, our model fit results recover the true parameter values within uncertainty or slightly more. In fact, Fig. 1 shows that our Cepheids sample are distributed reasonably randomly, which must also help our good recovery of the true parameter values.

We have applied the same exercise for the 1,000 sample of stars selected randomly in the same way as did for the young star particles, but for star particles whose age is between 3.5 and 5 Gyr. Note that the simulation snapshot used is at  Gyr. In the initial condition, the formation time, , of the star particles are randomly assigned between and 0 Gyr, and the age was calculated by age  Gyr. Hence, these particles are the star particles already in the initial condition with the axisymmetric particle density distribution. Also, note that as shown later, the radial velocity dispersion of this old star sample is about 60 km s, which is as high as the thick disc population in the Milky Way. Therefore, the selected star particles do not mean the representative sample of stars with the similar age in the Milky Way, but they rather represent the older stars similar age to the thick disc stars in the Milky Way. Hence, we call this sample old stars simply.

The results are shown in the right panels of Fig. 8. Same as the left panels, the true values from the simulations are shown with grey lines or symbols, and our model fit results are shown with black filled circles with error bars. The model fit results show less variations with . For the local circular velocity, the model fit results for the old stars provide the similar local circular velocity to the average circular velocity or slightly higher than that. The results are not sensitive to the fluctuations of the local circular velocity with , because of their large velocity dispersion. On the other hand, the rotation velocity of the observer, , is well recovered at all . These results indicate that the old stars with large velocity dispersion are a good tracer to recover the azimuthally averaged properties of the disc, and not too sensitive to the local perturbation like spiral arms. However, the gradient of the circular velocity is completely off from the mean or local gradient values, except at  deg. This may not be surprising, because as shown in the top right panel of Fig. 8, the old stars’ kinematics are insensitive to the variation in the local circular velocity.

Even for the old stars, the mean radial velocity can be non-zero at some location of the disc, which are indicated from the difference between the grey open circle and the horizontal dashed line in the plot of . This means that unless is independently measured by the other method, the local velocity distribution fitting only provides and it is difficult to recover the true in the Galactic rest frame.

From the results of this section, we conclude that the young stars are good tracer for the axisymmetric model fit to recover the local circular velocity at the different location of the Galactic disc. Cepheid variables provide the ideal sample of young stars within tightly constrained age range. However, their number density are not large and it would be difficult to increase the sample size by factor of 5 or more, as used in this section. Hence, we think that the young dwarf stars would be good tracer population for this purpose. The Gaia DR2 will provide the accurate distance and proper motion for many young dwarf stars. For example, the A-type dwarf stars are expected to be younger than 1 Gyr, and have small enough velocity dispersion for this application. Applying the axisymmetric disc model locally to the A-type dwarf stars in the Gaia DR2 data would be an alternative way of recovering the local circular velocity at the solar position and also at the different Galactocentric radii and the azimuthally different position from the Sun. Unfortunately, the line-of-sight velocity, , of the A-type stars will not be available in the Gaia DR2, because of their high effective temperature. Therefore, this requires to combine the Gaia DR2 parallax and proper motion measurements with the existing spectroscopic survey data or apply the antisymmetric model to only the longitudinal velocities, .

7 Summary and Discussion

We measured the local circular velocity and its radial gradient from 174 Galactic Cepheids whose accurate measurements of the distance and velocities are obtained by cross-matching the existing Cepheids catalogue with the Gaia DR1 TGAS catalogue. Our axisymmetric disc model fit also provides the Sun’s radial and azimuthal proper motions, and velocity dispersion of the sample.

Assuming strong priors of the solar radius,  kpc, and angular rotation velocity of the Sun,  km s kpc, we have obtained  km s,  km s,  km s,  km s, and  km s kpc. Here, is positive in the outward direction. With more conventional definition of the solar radial motion, , whose positive direction is inward, this is equivalent to  km s. These results are consistent with what are shown in Bobylev (2017) who used similar Cepheids data, but a different model fit. Our model provides and which are not fitted in Bobylev (2017).

Because this circular velocity is measured with modelling of the relatively local sample of the young disc stars, we call this the ‘local’ circular velocity in this paper, which is defined as , where is the radial gravitational force at the position of in the Galactic plane. Using an -body/SPH simulation data of a Milky Way-like disc galaxy and locating the observer at different azimuthal locations around the Galactic disc at a fixed radius similar to the solar radius, we demonstrate that our axisymmetric model fit to the sample of young stars within 3 kpc from the observer recovers the local circular velocity at the location of the observer well. This can be different from the azimuthally averaged circular velocity at the radius, because the bar and spiral arms lead to different radial force, , at the different azimuthal locations even at the same Galactocentric radius. The local circular velocity and the other Galactic parameters are recovered well, only when we used young star particles whose age is as young as classical Cepheids, and also we selected the particles randomly, but spatially homogeneously.

This is an encouraging result and suggests that the simple axisymmetric model approximation is valid for the sample of young stars within a small volume with 3 kpc in radius to measure the local circular velocity. The model fit result at the different location of the Galactic disc can provide the variation of the local circular velocities, which would be valuable information to understand the influence of the bar and spiral arm to the local gravitational field and stellar motions.

The final data release of the Gaia data will provide the accurate position and velocity for different kinds of stellar populations in a large volume of the Galactic disc. It is promising that advanced Galactic dynamical modelling (e.g. Bovy & Rix, 2013; Sanders & Binney, 2015; Trick et al., 2016) will uncover the azimuthally averaged circular velocity of the Galaxy and reveal the total mass distribution of the Galaxy. Also, -body based Galaxy modelling can take into account non-axisymmetric structures and it is expected to reveal the structures of the bar and possibly spiral arms (Hunt et al., 2013; Hunt & Kawata, 2014; Portail et al., 2017). We think that applying the simple axisymmetric model to the different region of the Galactic disc stars in the Gaia data would be a simpler alternative way of providing the local circular velocity and its radial gradient. The model is a simpler than the advanced methods listed above. However, a simpler model is often useful to validate a more complicated advanced model. Hence, we believe that this model would be also useful for studying the Galactic disc properties with the Gaia data. This paper also demonstrates that applying the model to the mock data from Milky Way-like simulated galaxies is important to understand the limits of dynamical modelling.

Acknowledgments

Kawata acknowledge the support of the UK’s Science & Technology Facilities Council (STFC Grant ST/N000811/1). Bovy received partial support from the Natural Sciences and Engineering Research Council of Canada. Bovy also received partial support from an Alfred P. Sloan Fellowship. Baba was supported by the Japan Society for the Promotion of Science (JSPS) Grant-in-Aid for Young Scientists (B) Grant Number 26800099. The -body/SPH simulation used in this paper were carried out on facilities of Center for Computational Astrophysics (CfCA), National Astronomical Observatory of Japan. This work also used the UCL facility Grace and the DiRAC Data Analytic system at the University of Cambridge, operated by the University of Cambridge High Performance Computing Service on behalf of the STFC DiRAC HPC Facility (www.dirac.ac.uk). This equipment was funded by BIS National E-infrastructure capital grant (ST/K001590/1), STFC capital grants ST/H008861/1 and ST/H00887X/1, and STFC DiRAC Operations grant ST/K00333X/1. DiRAC is part of the National E-Infrastructure. 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.

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