Mon. Not. R. Astron. Soc. 386, 2081–2090 (2008) doi:10.1111/j.1365-2966.2008.13168.x Measurement of the epicycle frequency in the Galactic disc and initial velocities of open clusters J. R. D. Lepine,´ 1⋆ Wilton S. Dias2 and Yu. Mishurov3 1Instituto de Astronomia, Geof´ısica e Cienciasˆ Atmosfericas,´ Universidade de Sao˜ Paulo, Cidade Universitaria,´ Sao˜ Paulo SP, Brazil 2UNIFEI, Instituto de Cienciasˆ Exatas, Universidade Federal de Itajuba,´ Itajuba´ MG, Brazil 3South Federal University (Rostov State University), Rostov-on-Don, Russia Accepted 2008 February 28. Received 2008 February 27; in original form 2007 October 30 ABSTRACT A new method to measure the epicycle frequency κ in the Galactic disc is presented. We make use of the large data base on open clusters completed by our group to derive the observed velocity vector (amplitude and direction) of the clusters in the Galactic plane. In the epicycle approximation, this velocity is equal to the circular velocity given by the rotation curve, plus a residual or perturbation velocity, of which the direction rotates as a function of time with the frequency κ. Due to the non-random direction of the perturbation velocity at the birth time of the clusters, a plot of the present-day direction angle of this velocity as a function of the age of the clusters reveals systematic trends from which the epicycle frequency can be obtained. Our analysis considers that the Galactic potential is mainly axis-symmetric, or in other words, that the effect of the spiral arms on the Galactic orbits is small; in this sense, our results do not depend on any specific model of the spiral structure. The values of κ that we obtain provide constraints on the rotation velocity of the disc; in particular, V0 is found 1 to be 230 15 km s even if the short scale (R0 7.5 kpc) of the Galaxy is adopted. The ± − = measured κ at the solar radius is 43 5kms 1 kpc 1. The distribution of initial velocities of ± − − open clusters is discussed. Key words: Galaxy: kinematics and dynamics – open clusters and associations: general. long period of times. This property allowed us to develop a method 1 INTRODUCTION for a direct determination of the epicycle frequency. According to Open clusters are ideal objects to study the dynamics of the spi- the perturbation theory, if an open cluster was born with an initial ral structure of the Galaxy. A large number of them have known velocity different from that of the circular velocity given by the distances and space velocities (proper motions and radial veloci- rotation curve, it will follow an orbit that can be described by the ties) that are more precisely determined than those of individual epicycle approximation. The motion can be seen as the sum of stars. Another important parameter is their age which is obtained by the circular motion around the Galactic Centre, plus a small am- means of the Hertzsprung–Russell diagrams. In a recent work, Dias plitude rotation around the ‘guiding centre’ in the circular orbit. &Lepine´ (2005, hereafter DL) integrated backwards the orbits of a The epicycle frequency κ is a function of the galactic rotation ve- sample of open clusters to find their birthplaces as a function of time, locity and its derivative d/dR only (e.g. Binney & Tremaine and obtained the velocity of the spiral arms. In a similar manner, by 1987, hereafter BT87) as integrating the orbits, it is possible to determine the velocity (direc- 1 R d tion and amplitude) of the clusters at the time of their birth. Since the κ2 42 1 . (1) spiral arms are known to produce perturbations of the velocity field = + 2 dR with respect to circular rotation, it would not be surprising if the new Based on this equation, the epicycle frequency is easily derived from born clusters also present some systematic deviations from circular any observed rotation curve. However, since there are uncertainties motion. The study of initial velocities of clusters should possibly concerning the exact shape of the rotation curve, as several different put restrictions to the mechanisms that trigger star formation. curves are proposed in the literature, a direct measurement of the One result of our investigation is the discovery that part of the epicycle frequency, without passing through equation (1), is of great open clusters have initial velocities in one or more constant, rela- interest. Such a measurement, in turn, can help to select the best tively narrow angles, and that these preferred directions survive for Galactic parameters and rotation curve. We discuss in this work, in addition to the statistics of initial velocities of open clusters, the restrictions on the Galactic parameters that can be derived from the ⋆E-mail: [email protected] measured epicycle frequency. C C 2008 The Authors. Journal compilation 2008 RAS 2082 J. R. D. Lepine´ , W. S. Dias and Yu. Mishurov R Figure 1. Epicycle orbit around an equilibrium point rotating with circular velocity V around the Galactic Centre. Vp represents the perturbed velocity. Figure 2. The geometry of the velocity vectors. We adopt a usual convention with the Y-axis oriented along the circular rotation at the solar position, and 2 THE OPEN CLUSTERS CATALOGUE the X-axis in the Galactic radial direction, oriented towards external regions. 1 1 One can see that γ θ tan (Vx /Vy ), with θ sin (y/R). We make use of the New Catalogue of Optically visible Open Clus- = + − = − ters and Candidates of Dias et al. (2002).1 This catalogue updates the previous catalogues of Lynga¨ (1987) and Mermilliod (1995). dη η′ a κ sin(κt φ). (5) The present 2.7 version of the catalogue contains 1756 objects, of = dt =− + which 850 have published ages and distances, 889 have published The ratio of the velocity amplitudes is the same of the displacement proper motions (most of them determined by our group, Dias et al. amplitudes. The angle γ between the perturbed velocity vector and 2006 and references therein) and 359 have radial velocities. Re- the direction of circular rotation, taken as a reference (see Fig. 2), is cently, Paunzen & Netopil (2006) compared statistically our cata- logue with other data from the literature and reached a result which 1 ξ ′ 1 γ (t) tan− tan− [(b/a) tan(κt ϕ)]. (6) is in favour of the data from the catalogue. The list of objects that we = η′ = + used in this paper is easily reproduced from the file clustersgal.txt We introduced a new symbol, ϕ, for the phase because we want available at the site, ignoring the clusters that have an empty field to adopt the direction of circular rotation as the origin for γ ; the in any one of the columns corresponding to distance, age, proper relation between the two phases is ϕ φ π/2. In the figures, we motion in longitude and latitude, and radial velocity. We found 374 represent the Galactic rotation in the=clockwise+ direction, while γ objects having all the required data. increases in the counterclockwise direction. The situation γ 0◦ occurs when the cluster is at its minimum Galactic radius, along=the 3 THE METHOD epicycle trajectory (φ π/2, in equation 2). In the hypothetic case=−a b, γ would increase linearly with time, The main result of the perturbation theory of stellar orbits in a cen- with a slope κ. In the case= a/b 1.5, γ is a smooth increasing tral potential, applied to the Galactic disc, is that the motion of a function of time, with a slope that on= the average is equal to κ, since star (or of an open cluster) can be described as the sum of a circu- in a time T 2π/κ, γ varies 360◦. If all the clusters had a same lar rotation around the Galactic Centre, with angular velocity (R), starting angle,= in a plot of γ versus age, the slope of the fitted line plus an epicycle perturbation, as illustrated in Fig. 1. The epicycle would give directly the value of κ. perturbation is an oscillation with frequency κ around the unper- The velocity vector of a cluster with respect to the Sun is the sum turbed circular orbit, in both the radial direction and the direction of two components: the radial velocity and the velocity in the plane of circular rotation: of the sky, derived from the proper motion and distance. These data are given in the catalogue. We then convert the velocity to the Local r(t) r ξ(t), with ξ(t) b sin(κt φ); (2) = 0 + = + Standard of Rest (LSR). Different sets of solar velocity components η(t) (u, v, w) with respect to the LSR are recommended in the litera- θ(t) θ0 t , with η(t) a cos(κt φ) (3) ture, like the Standard Solar Motion, with (u, v, w) ( 10.4, 14.8, = + + r0 = + 1 = − 7.3) (in km s− , Mihalas & Binney 1981), ( 10.0, 5.2, 7.2) from (ξ and η represent the displacements in the two directions, b and the analysis of the Hipparcos data by Dehnen− & Binney (1998), a the amplitudes in units of length, and φ the phase). In the frame ( 9.7, 5.2, 6.7) also based on Hipparcos data by Bienayme´ (1999) of reference of the guiding centre of the epicycle motion, rotating or−( 9.9, 12.1, 7.5) according to Abad et al. (2003), among oth- − 1 with angular velocity , the perturbed displacement describes an ers. Since differences of a few km s− between the various systems ellipse, with the ratio of the amplitudes a/b 2 /κ (BT87), which occur only in the v component, we varied it to verify how it af- = 1 is approximately 1.5.