Rotation curve of the Milky Way from Classical Cepheids
Flat rotation curves of spiral galaxies are considered as an evidence for dark matter, but the rotation curve of the Milky Way is difficult to measure. Various objects were used to track the rotation curve in the outer parts of the Galaxy, but most studies rely on incomplete kinematical information and inaccurate distances. Here, we use a sample of 835 Classical Cepheids with precise distances based on mid-infrared period-luminosity relations coupled with proper motions and radial velocities from Gaia to construct the accurate rotation curve of the Milky Way up to the distance of kpc from the Galactic center. We use a simple model of Galactic rotation to measure the distance of the Sun to the Galactic center kpc and the rotation speed of the Sun km s. The rotation curve at Galactocentric distances kpc is nearly flat with a small gradient of km s kpc. This is the most accurate Galactic rotation curve at distances kpc constructed so far.
Subject headings:Galaxy: kinematics and dynamics, Galaxy: fundamental parameters, Stars: kinematics and dynamics, Stars: variables: Cepheids
Flat rotation curves of spiral galaxies provide evidence for dark matter (Rubin et al., 1980) or even “new physics” (Milgrom, 1983), but the rotation curve of our Galaxy is notoriously difficult to measure, especially in the outer parts of the Milky Way. The most popular approach, the tangent-point method (e.g., Burton & Gordon 1978; Clemens 1985; Fich et al. 1989; Sofue et al. 2009; McClure-Griffiths & Dickey 2016), based on radio and mm observations of common molecules (Hi or CO), allows measuring the rotation curve within the solar orbit, although it is unreliable in the central regions of the Galaxy (Chemin et al., 2015). The rotation curve outside the solar orbit can be measured with known distances and velocities of some tracers: H ii regions (Fich et al., 1989; Brand & Blitz, 1993), Cepheids (Pont et al., 1994, 1997; Metzger et al., 1998), open clusters (Hron, 1987), or planetary nebulae (Durand et al., 1998), but the current uncertainties are considerable (see Figure 1 of Sofue et al. 2009), mostly because of poorly known distances. Such approach is prone to systematic errors, as usually only one component of the velocity vector (radial or tangential) is known and the circular rotation is assumed. As radial velocities and proper motions are measured relative to the Sun, both methods require independent information about the velocity of the Sun and distance to the Galactic center. See Bhattacharjee et al. (2014), Pato & Iocco (2017), Russeil et al. (2017), and references therein for recent data compilations.
A novel approach for constructing the Galactic rotation curve is presented by Reid et al. (2009, 2014) and Honma et al. (2012), who have measured accurate trigonometric parallaxes, proper motions and radial velocities of about 100 high-mass star-forming regions. They use the three-dimensional velocity information to calculate the rotation curve and to simultaneously estimate the velocity and location of the Sun. Their sample is relatively small and most of objects they analyze are located in the northern part of the Galactic disk, which may introduce some bias. The local rotation curve was published by the Gaia Collaboration et al. (2018b), who used the second Gaia data release (Gaia DR2) to study motions of nearby stars, but their parallaxes are accurate in the solar neighborhood, within kpc of the Sun.
Recently, Udalski et al. (2018; in preparation) presented the new OGLE Collection of Galactic Cepheids containing 1426 Classical Cepheids based on the survey of the Galactic plane carried out as part of the Optical Gravitational Lensing Experiment (OGLE). This data set more than doubled the number of known Galactic Cepheids. The survey covers over 2500 square degrees along the Galactic plane (, ) and probes the Galactic disk out to its expected boundary ( kpc from the Galactic center). That sample, supplemented with previously known all-sky Cepheids, was used by Skowron et al. (2018) to study the structure of the young Milky Way disk.
Here, we complement distances to Cepheids from Skowron et al. (2018) with the kinematical data (proper motions, radial velocities) to measure the three-dimensional velocities of Cepheids (Section 2). We use a simple model of Galactic rotation to measure the velocity of the Sun and its distance to the Galactic center (Section 3) and to construct the accurate rotation curve of the Milky Way up to the Galactocentric distance of 20 kpc (Section 4).
Skowron et al. (2018) measured accurate distances for 2190 Galactic Cepheids, using period-luminosity relations of Wang et al. (2018) and mid-infrared light curves, which virtually removes the effects of interstellar extinction. We cross-matched Skowron et al.’s catalog with Gaia DR2 (Gaia Collaboration et al., 2016, 2018a) and found that the full velocity information (proper motions and median radial velocities) is available for 835 objects.
We used distances measured by Skowron et al. (2018), as Gaia parallaxes are not sufficiently accurate for many objects from our sample. Additionally, Riess et al. (2018) and Groenewegen (2018) showed that Gaia parallaxes are systematically lower than accurate non-Gaia parallaxes of Classical Cepheids by mas and mas, respectively. The similar parallax zero-point offset, from to mas, was found for other tracers (see Groenewegen 2018 and references therein). We also measured the median offset of mas between Gaia and Skowron et al.’s (2018) parallaxes. Typical distance uncertainties are of a few per cent.
For the modeling, we removed obvious outliers and known Cepheids located in binary systems111http://www.konkoly.hu/CEP/intro.html (Szabados, 2003). We were left with 774 Cepheids. The radial velocities of Cepheids show variations with amplitudes up to 30 km s with the pulsation period (Joy, 1937; Stibbs, 1955). Radial velocities reported in the Gaia DR2 are median values of single-transit measurements. Cepheids from our sample were observed from 2 to 44 times with the median number of seven visits. Small number of single observations is usually reflected by large error bars, although in some cases, the uncertainties may be underestimated (if the measurements happened to be collected near the same pulsation phase). Thus, for the modeling, we added in quadrature km s to the reported radial velocity uncertainties.
We use a simple model of circular rotation of the Milky Way. For each Cepheid, with known Galactic coordinates (longitude and latitude ) and heliocentric distance , we calculate the expected radial and tangential velocities and compare them with observations.
Our model has the following free parameters: – distance of the Sun to the Galactic center, – mean noncircular motion of the source in a Cartesian Galactocentric frame, – solar motion with respect to the local standard of rest (LSR), and one or two parameters that describe the shape of the rotation curve. We follow the notation from Appendix of Reid et al. (2009): is the velocity component toward the Galactic center, – along the Galactic rotation, – toward the North Galactic pole. We consider two analytical rotation curves: (model 1) and (model 2), where and are parameters and is the distance to the Galactic center.
The total velocity of the Cepheid is . Let be the angle between the Sun and the source as viewed from the Galactic center (see Figure 9 of Reid et al. 2009). We rotate the velocity vector through the angle and subtract the velocity of the Sun:
The radial velocity and tangential velocities in Galactic coordinates ( and ) can be calculated as follows:
Following Reid et al. (2014), we maximize the likelihood function:
where is the weighted residual for -th Cepheid. As we will show below, our simple model describes well the overall kinematical properties of our sample, but the scatter in residuals is large and it is significantly larger than some individual error bars (Figure 1). We use the likelihood function described by Equation (7) to minimize the influence of these “outlying” points (Sivia & Skilling, 2006).
The best-fit parameters are found by maximizing the likelihood function and are shown in Table 1. The uncertainties are estimated using the Markov chain Monte Carlo technique (Foreman-Mackey et al., 2013) and represent 68% confidence range of marginalized posterior distributions. As we found that the velocity of the Sun with respect to the LSR is poorly constrained by the data, we used the following Gaussian priors: km s, km s, and km s (Schönrich et al., 2010). We assumed uniform priors on other parameters.
|Parameter||Model 1||Model 2|
|(km s kpc)||(fixed)|
Note. Model 1: flat rotation curve , model 2: linear rotation curve .
To assess how distance uncertainties influence the final parameters, we carried out Monte Carlo simulations. For each Cepheid, we drew a new distance from the normal distribution and repeated our modeling procedure. We conducted 100 trials, in which we found the additional spread of , , and of 2.5 km s, 0.11 km s kpc, and 0.11 kpc, respectively. We add these quantities in quadrature to the uncertainties from Table 1, finding km s, km s kpc, and kpc.
Residuals from the best-fit models are shown in Figure 1, separately for radial, azimuthal, and vertical velocity components. Error bars of many individual objects are much lower than the scatter ( km s, km s, km s), likely because of peculiar (noncircular) motion of stars. Some Cepheids may be unrecognized members of binary systems.
Both and are in good agreement with previous determinations. Reid et al. (2014) found km s and kpc based on parallaxes and proper motions of high-mass star-forming regions. Reid & Brunthaler (2004) measured the proper motion of Sagittarius A* of mas/yr, which corresponds to km s for kpc. The angular velocity of circular rotation of the Sun ( km s kpc) in our model is consistent with Hipparcos ( km s kpc; Feast & Whitelock 1997) and Gaia ( km s kpc; Bovy 2017) measurements. Similarly, the distance to the Galactic center is consistent with recent determinations: kpc (Abuter et al., 2018), kpc (Chu et al., 2018), kpc (McMillan, 2017).
4. Galactic rotation curve
We use parameters (,) from Table 1, model 2 to construct the rotation curve of the Milky Way. We convert radial and tangential heliocentric velocities to the Galactocentric velocity by subtracting the motion of the Sun (equations (1–6)). The resulting rotation curve is shown in Figure 2, where we also plotted earlier data from Sofue et al. (2009), rescaled to new (,). Both data sets agree well up to a distance of 10–11 kpc from the Galactic center. Previous studies (e.g., Sofue et al. 2009; Reid et al. 2014; Russeil et al. 2017) found that the rotation curve outside 12 kpc is nearly constant or even rising (although its precise shape may depend on the choice of and ), but these data were affected by large uncertainties and small number of observations (Figure 2). Our rotation curve is nearly flat with a small gradient of km s kpc, contrary to some earlier claims that the rotation of Cepheids is Keplerian (Gnaciński, 2018).
Classical Cepheids are excellent tracers of the rotation curve in the outer parts of the Milky Way disk. Our rotation curve outside 12 kpc is more accurate than in any previous studies (Sofue et al., 2009; Reid et al., 2014) and can be used to constrain the distribution of dark matter in the Milky Way. Currently, our sample includes only 127 Cepheids at Galactocentric distances greater than 12 kpc (out of nearly 600 Cepheids with kpc from Skowron et al. 2018). Future Gaia data releases, as well as a dedicated spectroscopic survey of Cepheids, can provide more accurate insight into rotation of the outer parts of the Milky Way disk.
P.M. acknowledges support from the Foundation for Polish Science (Program START). The OGLE project has received funding from the National Science Centre, Poland, grant MAESTRO 2014/14/A/ST9/00121 to A.U. I.S. is also supported by the Polish National Science Centre grant MAESTRO 2016/22/A/ST9/00009. 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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