MNRAS 000, 1–14 (2015) Preprint 21 May 2019 Compiled using MNRAS LATEX style file v3.0 The spiral pattern rotation speed of the Galaxy and the corotation radius with GAIA DR2 W. S. Dias,1⋆ H. Monteiro,1, J. R. D. L´epine2 , and D. A. Barros3 1UNIFEI, Instituto de F´ısica e Qu´ımica, Universidade Federal de Itajub´a, Av. BPS 1303 Pinheirinho, 37500-903 Itajub´a, MG, Brazil 2Universidade de S˜ao Paulo, Instituto de Astronomia, Geof´ısica e Ciˆencias Atmosf´ericas, S˜ao Paulo - SP, Brazil 3Rua Sessenta e Trˆes, 125, Rio Doce, Olinda, 53090-393 Pernambuco, Brazil Accepted 2019 April 24. Received 2019 April 22; in original form 2019 March 20 ABSTRACT In this work we revisit the issue of the rotation speed of the spiral arms and the location of the corotation radius of our Galaxy. This research was performed using ho- mogeneous data set of young open clusters (age < 50 Myr) determined from Gaia DR2 data. The stellar astrometric membership were determined using proper motions and parallaxes, taking into account the full covariance matrix. The distance, age, reddening and metallicity of the clusters were determined by our non subjective multidimensional global optimization tool to fit theoretical isochrones to Gaia DR2 photometric data. The rotation speed of the arms is obtained from the relation between age and angu- lar distance of the birthplace of the clusters to the present-day position of the arms. Using the clusters belonging to the Sagittarius-Carina, Local and Perseus arms, and −1 adopting the Galactic parameters R0 = 8.3 kpc and V0 = 240 kms , we determine a pattern speed of 28.2 ± 2.1 kms−1 kpc−1, with no difference between the arms. This implies that the corotation radius is Rc =8.51 ± 0.64 kpc, close to the solar Galactic orbit (Rc/R0 =1.02 ± 0.07). Key words: Galaxy: kinematics and dynamics – structure 1 INTRODUCTION trailing at that radius. In one of the branches of the classical theory, the arms are small perturbations in the gravitational The Gaia mission recently released its second set of data potential of the disk, like gravitational potential grooves or (DR2) (Gaia Collaboration et al. 2018a) which provides pre- valleys, produced by the crowding of stellar orbits in some cise astrometric and photometric data for more than one regions of the disk. The orbits of the stars constituting the billion stars with magnitude G less than 21. Among other arms are closed ones, which repeat themselves after each scientific outcomes, a great improvement is expected in the turn (in the rotating frame of reference of the arms), and understanding of the structure and dynamics of spiral arms this guarantees a relatively long lifetime to the spiral struc- of the Milky Way, thanks to the new data on spiral arms ture (Pichardo et al. (2003),Junqueira et al. (2013)). tracers. Among these, a major role will be played by the arXiv:1905.08133v1 [astro-ph.GA] 20 May 2019 open clusters, due to their precise distance and ages that In spite of its merits, the classical model is being se- can be determined from the color magnitude diagrams. riously challenged. Another line of thinking considers that The Milky Way is a grand design galaxy, as revealed spiral arms are transient, appearing and disappearing in pe- by its large scale spiral structure (Georgelin & Georgelin riods of time of a few hundred million years (Sellwood 2010). (1976), Levine et al. (2006), Hou et al. (2009), Hou & Han In that view, Rc is not an important reference, as it changes (2014), Reid et al. (2014)). According to the classical the- frequently. Others consider multiple spiral structures with ory (see Shu (2016) for a review), the arms are located be- distinct rotation speed (Quillen & Minchev (2005) and tween the inner and outer Lindblad resonances. The radius Quillen et al. (2018)), which would mean that several Rc of corotation (Rc), where the velocity of rotation of stars coexist. Even among the researchers who adhere to the clas- coincides with the rotation velocity of the spiral arms (Ωp), sical view of the spiral structure, there has been no consen- is a fundamental parameter, which divides the Galactic disk sus concerning the location of the Rc. Mishurov & Zenina in two regions. The interstellar matter penetrates the arms (1999) and Dias & L´epine (2005) located it very close to in opposite directions, and the arms change from leading to the Solar orbit radius. However, Drimmel & Spergel (2001) located it at 6.7 kpc, Acharova et al. (2012) at 7 kpc, and Mongui´oet al. (2015) places it farther away than the ⋆ E-mail:[email protected] Perseus arm. Recently, Michtchenko et al. (2018) explained c 2015 The Authors 2 W. S. Dias et al. the moving groups observed in the solar neighborhood based much faster, as discussed in detail by Monteiro and Dias on a model which places Rc close to the Sun. (2019). The lack of a consensus on Rc shows the urgency of The sample with a complete vector (x, y, z, vx, vy, vz) of making use of the unprecedented volume and quality of initial conditions to orbit integration is composed by 80 open Gaia DR2 data to establish firm observational constraints clusters younger than 50 Myr. This final sample is relatively to it. The purpose of the present work is twofold. Firstly, small mainly because of our choice to use only open clusters we present a set of open clusters that has been analyzed us- with parameters (µαcosδ, µδ, radial velocity, distance, age) ing the Gaia DR2 catalogue. Secondly, we make use of this determined from the Gaia DR2 catalogue, and of the addi- sample of open clusters to perform a new determination of tional condition that the clusters had to be approved in a Rc. We adopt a method similar to that of Dias & L´epine visual inspection of the isochrone fit, as commented in the (2005), which consists in integrating the orbits of the clus- next section. ters back to their birthplaces. The clusters are believed to born in spiral arms, so that each cluster reveals the position of a spiral arm at a past time equal to its age. By comparing the past position of the arm with the present-day position, 3 MEAN ASTROMETRIC PARAMETERS the rotation angle of the spiral structure during that inter- AND MEMBERSHIP DETERMINATION val of time is obtained. This method is straightforward and has the advantage that it does not require any model nor In a 3D plot of the Gaia DR2 astrometric data (µαcosδ, µδ, any unknown parameter, except for the rotation curve of ̟) of the stars in the region of an open cluster, the members the Galaxy. As will be discussed, the rotation curve is suffi- of the cluster appear in a clump. From the statistical point ciently well known. of view, the closer to the center of the clump is a star the This paper is organized as follows: in the next Section greater the probability that it is a member of the cluster. we describe the sample of open clusters. Section 3 presents So the membership problem can be formulated in the data the method of membership determination. In Section 4 we space as a maximum likelihood one, based on the assumption comment on the method adopted for determining the dis- of normally distributed uncertainties in proper motion and tances and ages of the clusters. In Section 5 we discuss the parallax as: galactic rotation curves and the orbits of the open clusters. In Section 6 we present the approaches proposed to estimate 1 T −1 exp − 2 (X − µ) Σ (X − µ) the rotation speed of the spiral pattern and we present the f(X)= (1) p(2π)k|Σ| results. We summarize the main conclusions and comment on their consequences in Section 7. where X is the column vector (µαcosδ, µδ, ̟) composed of the proper motion components and the parallax, |Σ|≡ det Σ is the determinant of the covariance matrix Σ and µ the mean column vector. The covariance matrix incorporates all 2 THE SAMPLE OF YOUNG OPEN uncertainties and their correlations, which are all given in CLUSTERS the Gaia DR2 catalogue. We used the version 3.5 of the DAML02 catalogue Because of the large volume of data in Gaia DR2 and the (Dias et al. 2002) to select 441 clusters with ages lower than large number of clusters to be analyzed, the resolution of the 50 Myr. In addition we also analyzed 154 open clusters with equations above can take a considerable amount of compu- no age determined in DAML02, 31 clusters recently discov- tational time due to the matrix operations involved. So, we ered by Castro-Ginard et al. (2018) and 52 clusters discov- opted to estimate the membership in two steps, as performed ered by Cantat-Gaudin et al. (2018), looking for young open in Dias et al. (2018b) and Monteiro and Dias (2019). First clusters. As detailed in Section 4, we performed the isochrone we use the proper motion following the procedure described fit to the stars with membership greater than 0.51 in order in Dias et al. (2018a) to select the stars with membership to select the clusters younger than 50 Myr to compose our greater than 0.50 based on that model. Then we fit a 1D sample. parallax space assuming Gaussian distributions for the field For each cluster we searched for the stars in the Gaia as well as for the cluster stars. With the distribution param- DR2 catalogue, using the central coordinates and the ra- eters estimated (mean proper motion and parallax) used as dius taken from the DAML02 catalogue.