Measurement of the Epicycle Frequency in the Galactic Disc and Initial Velocities of Open Clusters

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.

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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 catalogue 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. Like in the case of a circular motion, the fects our results. We finally adopted v 5.0 km s− , since this value perturbed velocity rotates with the same frequency of the perturbed corresponds to a minimum of the rms= deviation of the adjusted displacement. The velocity components are: points in the determination of the epicycle frequency. However, this dξ choice is not critical for our results; almost the same values of epicy- ξ ′ b κ cos(κt φ); (4) 1 = dt = + cle frequencies were obtained with v in the range 3–8 km s− .If 1 we adopt an artificially incorrect value for v, like 10 or 20 km s− , the present-day velocity distribution becomes −clearly incorrect, 1 Available at the web page http://astro.iag.usp.br/-wilton. either very concentrated around 180◦ or 0◦. ±

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In this work, we do not take into account the velocities perpendic- guiding centre. The correction on γ can be written as a function of γ ; ular to the Galactic plane; only the velocity components projected the result is δγ ( V /r κ) sin(γ ). We also performed numerical on the plane are considered. calculations of δγ=−taking| | into account the ellipticity of the epicycle; the resulting δγ is not very different from that of the circular case. The errors that are introduced by not performing the δγ correction 3.1 Corrections for differential rotation and reference frames are small. For instance, if the modulus of the velocity perturbation 1 After electing a rotation curve, we subtract from the observed LSR is 10 km s− , the amplitude of δγ is less than 4◦ (the average value velocity of a cluster the velocity expected from pure circular rotation is 0). For such a small correction, a first-order estimation of it is at the position of the cluster; what is left is the perturbation velocity. sufficient. This is not different from what DL called correction for differential rotation. Without perturbation, we expect to observe the U and V 3.1.2 Rotation curves velocity components of pure circular rotation minus the velocity For radius ranges (or rings) relatively distant from R , the choice components (0, V0) of the LSR. The perturbation velocity vector 0 is obtained from the (observed expected) velocity components. Of of the rotation curve is more relevant than for the solar ring. For course, the perturbation velocity− depends slightly on the choice of instance, if the true rotation velocity of the Galaxy in a ring is larger than the one that we adopt for the corrections, the perturbation the Galactic parameters R0, V0,(dV/dR)0 (the radius of the solar orbit, the rotation velocity of the LSR around the Galactic Centre velocities are undercorrected in the direction of η′ and the observed and the slope of the rotation curve, respectively). However, we are γ s will tend to be closer to zero than they should be. The effect dealing with second-order (or differential) effects; if we adopt a is similar to the one that we described previously, concerning the v rotation curve that is too low or too high, the LSR velocity and component of the solar velocity: undercorrections or overcorrections the expected cluster velocity are affected in a similar way, and the tend to destroy the ‘normal’ distribution of present-day angles of the computed perturbation velocity does not change too much. velocity vectors. Once the components of a perturbation velocity vector have been In order to check the effect of the differential rotation corrections obtained, the angle γ of the vector with respect to the local rotation on the derived epicycle frequency, we performed the calculations direction is computed, as illustrated in Fig. 2. In this way, the vector over a grid of rotation curves. We varied R0 from 7.0 to 8.5 kpc, in 1 1 is now described in the local reference frame of the cluster. The steps of 0.5 kpc, and V0 from 170 to 250 km s− , in steps of 10 km s− . new frame has an axis in the local direction of rotation and the The method is very similar to that of DL for the determination of other in the local radial direction. The local frame of the clusters the pattern rotation speed; for each point of the grid, we make use is directly related to the Galactocentric polar coordinates, which of a rotation curve that passes at the corresponding (R0, V0) and fits is the most convenient system of reference to perform numerical the observed CO data taken from Clemens (1985); the same set of integration of Galactic orbits. Note that the motion of each cluster rotation curves of DL was used here. These are curves directly de- is studied individually (using numerical integration or the epicycle rived from observations with the rotation velocities recomputed for approximation); we are not interested in describing variations of each adopted (R0, V0). A simple and smooth analytical expression their relative positions, and anyway, we are dealing with clusters was fitted to each of these curves, in order to be able to compute the which have relative separations of several kpc. The local frame of corresponding ‘theoretical’ epicycle frequency. We refer to these each cluster is very close to the frame of its guiding centre; the small curves as CO-based curves or smooth curves. We also performed a correction required to place the velocity in the frame of reference number of experiments with constant velocity curves, or pure ‘flat’ of the guiding centre is next discussed. curves. Not all the points of this grid of rotation curves and radial ranges produce a valid measurement of the epicycle frequency. We later 3.1.1 The reference frame of the guiding centre discuss in more detail an example with a specific choice of R0,in order to clarify what we consider a valid measurement. The equations (2) to (5) are valid in the reference frame of the guiding centre of the epicycle motion, which does not coincide precisely 3.1.3 The modulus of the velocity perturbation with the local reference frame at present epoch. When a cluster is observed, we do not know a priory where is the guiding centre of The histogram of the modulus of the velocity vectors, or residual its epicycle motion. However, we suppose that the amplitude of the velocities after correction for differential rotation, is presented in 1 epicycle motion is small (as later discussed), so that the cluster is Fig. 3. The peak of the distribution is at about 10 km s− . In parts of never far from its guiding centre. Consequently, the Galactocentric the following study, we removed from the sample the clusters with 1 angle between the cluster and its guiding centre is small (of the order modulus larger than 20 km s− . The reason is that if the amplitude of 4◦; see below). This angle is the rotation angle required to pass of the perturbation is large, the simple epicycle approximation is from one reference frame to the other, or the error on γ due to the not valid. However, when we are dealing with numerical integration fact that we measure it in the local frame of the cluster instead of of the orbits which does not rely on the epicycle approximation, the frame of the guiding centre. this restriction is not needed. We also removed from the sample 1 A simple correction to γ can be made. This is easily understood the too small velocities ( V < 3kms− ), since the components of if we compare the epicycle motion to a uniform circular motion. the velocity vector would| be| of the same order of magnitude of the In this case, there is a π/2 phase difference between the directions measurement errors, and the error on the angle γ could be large. of the velocity vector and that of the position vector. We know the measured modulus V of the velocity vector; from equations (2) to | | 3.2 The case R0 7.5 kpc (4), we see that the modulus of the position vector is equal to that of = the velocity vector divided by κ. We therefore have both the angle Since the epicycle frequency depends on the Galactic radius equa- and the amplitude of the position of the cluster with respect to the tion (1), we must select the clusters in a narrow range of radius, to

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Figs 5(a)–(c) are plots of the angle between the perturbed velocity vector and the direction of circular rotation, as a function of age. The three plots show clusters selected in different ranges of Galac- tic radius. The range shown in Fig. 5b is the solar ring, 7.1 < R < 7.9 kpc. A line representing the function (6) has been fitted. In Figs 5(a) and 5(c), two lines represent the function (6), for a same value of κ, but different values of ϕ. When a line reaches the maxi- mum angle that can be measured (180◦), it starts again at 180◦.In Fig. 5(b), only one line was plotted, as it seems to be a dominant− one, fitting alone about 14 clusters, considering that a cluster is fitted if it is situated within 20◦, in the vertical direction, from the line. The points situated around± this line are reproduced in Fig. 6 without the folding at multiples of 180◦; two points slightly off the 20◦ limit ± were included. The rms deviation of the points from the line is 14◦. Figure 3. Histogram of the modulus of the residual velocity vectors of the It can be noted from Fig. 5(b) that for ages larger than 50 Myr, no open clusters in the Galactic plane, after subtracting the normal rotation points not belonging to the line are closer than about 130 (or 9σ ) velocity at that radius. ◦ from it. This justifies the possibility of ‘isolating’ a line to fit the theoretical function. In the present case, the fitted value of κ is 43 1 1 ± 5kms− kpc− ; this is the epicycle frequency at the solar radius, one of the main results reported here. The initial angle is 74◦,in the frame of the guiding centre, or a few degrees more in the local frame of the clusters. The histograms of the initial directions of the open clusters obtained by projecting the points in frames (a)–(c) on to the age 0 axis using lines like those shown in the figure (using κ of the corresponding= Galactic radius range) are shown in (d)–(f). This is equivalent to invert equation (6) to find ϕ for each point, with afixedκ, and then determine γ (t 0). The histograms of initial angles are in good agreement with=those obtained from a totally different method, the numerical integration of the orbits, given in Figs 5(g)–(i). The probability of the apparent alignment of points in Figs 5(a)–(c) being a chance coincidence is discussed in the next section. Figure 4. Observed position and perturbation velocity vectors of the clusters The fact that a reasonable number of points fall on a same line in the range of Galactic radius 7.1 < R < 7.9 kpc. The radius range is indicated (for instance, 12 clusters within 20◦ of the line that begins at 74◦) by full lines and the solar radius by a dotted line. In the text, we refer to this reveals that some process confines±the direction of the initial velocity region as the R 7.5 kpc ring. The Galactic Centre is not contained in the = of the clusters to a same angle with respect to the direction of circular figure. rotation, for periods of time of more than 100 Myr. We postpone for the moment the discussion on the nature of such process. be able to observe a well-defined rotation rate of the perturbation 3.3 Probabilities of alignment in a random process velocity. We must make a compromise, since too narrow Galactic radius ranges will contain only a few clusters. After a number of In the case illustrated in Fig. 5(b), the sample contains 55 clusters. experiments, we found that a radial extent of the order of 0.6–1 kpc The probability of 14 clusters being fitted by a theoretical line, as it is convenient. The radius range that contains the Sun (let us say, happens in this case, is small. The probability of a point falling on the range R0 0.4 kpc) is less affected by differential rotation than a given line which has a ‘thickness’ of 40◦ is 40/360 1/9. The the others; whate± ver the rotation curve we choose, the circular rota- probability of 14 points being on a given line, out of 55 experiments,= tion velocity of the clusters is the same of LSR rotation around the or 14 successes and 41 failures, is [55!/(14!41!)](1/9)14(8/9)41, 3 ≈ Galactic Centre. The positions and velocities of the clusters situated 1.5 10− . in this ring are shown in Fig. 4. In×the example of Fig. 5(a), with a sample of 31 points, two lines In order to illustrate the measurement of κ, we adopt a set of pa- have been fitted, with a total of 13 points on them. In a random 1 rameters R0 and V0, 7.5 kpc and 230 km s− , respectively. We later process, the probability of falling on one of the two lines is 2/9. If argue that the measured κ gives us a link between these two parame- 31 points are randomly placed on the diagram, the probability of ters. Recent work often adopts R0 7.5 kpc (Racine & Harris 1989, 13 points falling on any one of two given lines (13 successes and 18 = 13 18 3 Reid 1993, and many others). This shorter Galactic scale, com- failures) is [31!/(13!18!)](2/9) (7/9) 6 10− . pared to the IAU recommended 8.5 kpc scale, is supported by Very In the first example, one could argue that≈ the× problem cannot be Long Base Interferometry observations of H2O masers associated reduced to that of throwing 55 points on graph and computing the with the Galactic Centre. The distance to the Galactic Centre derived probability of 14 of them being within a given distance of a line, since from infrared photometry of bulge red clump stars (Nishiyama et al. the line was not previously present; we fitted their parameters so that 2006) is R0 7.52 0.10 kpc, while astrometric and spectroscopic they pass through a group of points that resulted to be aligned. This observations=of the±star S2 orbiting the massive black hole in the is equivalent to reduce the number of ‘successes’, for instance, we Galactic Centre taken at the European Southern Obseratory (ESO) can choose two points to define the phase and slope of the line. In this Very Large Telescope (Eisenhauer et al. 2005) give 7.94 0.42 kpc. case, we require only 12 successes and 43 failures; the probability is ±

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Figure 5. Top panel (a)–(c): angle γ between the perturbed velocity and the direction of circular rotation as a function of the age of the clusters. The lines represent the function (6) for different values of ϕ. The clusters were selected in different ranges of Galactic radius indicated in each frame. The ﬁtted values 1 1 of κ were 48, 43 and 32 km s− kpc− , respectively. The middle frames (d)–(f) are histograms of the directions of initial velocities obtained by projecting the points of frames (a)–(c) on the age 0 axes with lines like those shown in the corresponding frames. The bottom frames (g)–(i) are the same histograms = obtained by numerical integration of the orbits (Section 3.4).

of fitted lines is arbitrary; similar conclusions can be obtained with 1, 2 or 3 lines. We also verified that our conclusions do not depend strongly on the ‘thickness’ of the lines; if we decide that points at distances up to 15◦ or 25◦ from a line are fitted points, the number of fitted points varies, but the change in the probability of being on a line by chance tends to compensate the variation, in the calculation of probability. This simple analysis of probabilities tells us that we are not dealing with random alignments, but also warns us that we should not attempt to draw conclusions from cases with a small number of aligned points, like for instance less than half of the total number of points being on three candidate lines or less than one-third of Figure 6. Angle γ between the perturbed velocity and the direction of cir- the points being on two lines, within a tolerance of 20◦. When the cular rotation as a function of the age of the clusters, for the clusters situated number of clusters decrease, or when the Galactic parameters or ro- around the line shown in Fig. 5(b), with the 180◦ folding removed. The ad- tation curves are different from the correct ones, we are sometimes justed line gives κ 43 5kms 1 kpc 1 and ϕ 79 , which corresponds = ± − − = ◦ confronted with situations in which very different values of κ seem to an initial angle 74 equation (6). ◦ to fit the points with about the same quality of the fit. This limits the number of Galactic rings and the range of parameters that can still only about 1 per cent. In reality, the probability of the observed be explored. alignments of points being spurious can be discarded, since the fitted value of κ is always very close to the ones expected from theory, and 3.4 Numerical integration the initial phases are confirmed by another method. Furthermore, of the orbits and histograms of initial directions there is a continuous variation of κ from one ring to the other; for instance, the clusters in Figs 5(a) and 5(b) are not the same, since the The approximation described in the previous sections is useful to Galactic rings do not overlap but the fitted slope changes smoothly provide an estimate of the epicycle frequency without depending from one ring to the other. We note that the choice of the number strongly on the knowledge of the rotation curve (the rotation curve

C C 2008 The Authors. Journal compilation 2008 RAS, MNRAS 386, 2081–2090 2086 J. R. D. Lepine´ , W. S. Dias and Yu. Mishurov is used in a ‘differential’ manner to compute the initial velocities). in radial velocities, proper motions and ages contained in the cat- Furthermore, a rough idea of the distribution of initial velocity di- alogue, as well as the small effect of the simplified δγcorrection rections is obtained simply by looking at the starting angles of the (Section 3.1), is all taken into account in the error on the slope given fitted lines (at age 0). by the least-square fitting. We consider that our final errors on κ are = 1 1 To obtain the distribution of initial angles, a second method is 5kms− kpc− . to perform a numerical integration of the stellar orbits. For this ± Other effects like the possible perturbation of the orbits by the integration, a first step is to obtain the axis-symmetric potential of spiral structure (see e.g. Lepine,´ Acharova & Mishurov 2003) or the Galaxy by integrating the force V(R)2/R based on the adopted the effects of the local wiggles of the rotation curve (Section 3.8) rotation curve. In this way, the potential does not depend on any are not considered here as errors of measurements, but the cause assumption on the different components of the Galaxy (disc, bulge, of discrepancies between the measured κ and theoretical values etc.). The present-day position and velocity of a cluster define a expected from too smooth rotation curves. quadrivector q(t) containing the polar coordinates, the radial velocity and the angular momentum which are the initial conditions of the V integration. In the integration, at each time-step, q(t dt) is deduced 3.6 Detailed analysis and the derived value of 0 + from q(t) using a Runge–Kutta IV method. The value of dt is chosen We expect that if a rotation curve is a good approximation of the true so as to have about 360 points along a circle around the Galactic one, it will produce a match between the theoretical and observed Centre. The integration is performed backward in time for a total κ, not only near R0, but also at other Galactic radii as well. In Fig. 7, time equal to the age. we present κ as a function of radius, obtained from the same method If the adopted rotation curve is close to the true one, the two meth- illustrated in Figs 5(a)–(c). The observed epicycle frequencies do ods should produce similar initial velocity distributions. Therefore, not depart too much from those predicted (based on equation 1) by a the comparison between the results of the two methods is a tool 1 smooth rotation curve with V0 232 km s− . The data in Fig. 7 were to make a choice among different rotation curves. As examples of also fitted by the same rotation=curve modified by the addition of a 1 histograms of initial velocity directions, those obtained with the Gaussian minimum centred at 8.8 kpc and peak value 12 km s− , same samples used in Figs 5(a)–(c), are shown in Figs 5(d)–(f) – 1 − the corresponding value of V0 being 230 km s− . The existence of using the slope-fitting method – and (g)–(i), obtained by numeri- a minimum is further discussed in a next section. We do not intend cal integration. We expect to obtain similar but not precisely the to present a precise fitting of it, but only to illustrate that it could same distribution of initial velocities with the two methods, since be an explanation for an anomalous slope of κ. This minimum is the methods rely on different simplifying assumptions. For instance, perceptible in the rotation curve (Fig. 9). the numerical integration takes into account the Galactic radius of Interesting results are obtained by varying the rotation curves individual clusters, while the slope-fitting method considers that κ with a fixed R0. Fig. 8 presents the theoretical and observed epicycle is constant within a ring. Another difference is the correction for frequency as a function of the adopted V0, for a fixed R0 7.5 kpc. the frame of reference of the guiding centres (Section 3.1.1) in the What we call here the theoretical κ is the value obtained= through slope-fitting method. direct use of equation (1) with (R) given by the adopted rotation The agreement between the two methods is found to be quite curve; the observed κ is obtained from the slopes of lines in plots satisfactory when rotation curves with high values of V (about 0 similar to those shown in Fig. 5. For each V0, the same rotation curve 1 1 230 km s− kpc− ) are used, but become poor with small V0, like is used to determine the theoretical frequency and for the differential 180 km s 1 kpc 1. − − rotation corrections. Fig. 8 turns clear how the best value of V0 can be derived, and how the uncertainty on κ affects this V0 value. Let us first consider the case of pure flat rotation curves [V(R) 3.5 Possible sources of errors = V(R0) constant ]. For such curves, it can be derived from equa- A source of uncertainty in the value of κ that we obtain, in the = tion (1) that κ √2.AtR0, V0/R0; the dashed line in Fig. 8 examples illustrated in Fig. 5, is the dependence of κ on the definition = = gives κ as a function of V0 for flat curves with R0 7.5 kpc. This of the LSR. It may happen that we are not using the best set of 1 = 1 line cuts the line of observed κ at about 43 km s− kpc− and V0 1 = solar velocity components for the LSR corrections, in the choice 230 km s− . In other words, if the rotation curve were a flat one, our described in Section 3. It can be seen from Fig. 2 that if we add a correction to Vy (in the direction of rotation), the computed value of γ will change. A small change in the LSR definition produces a same δ Vy for all the clusters, but different angles γ are affected in a different way, so that the slope of the fitted lines in Fig. 5 may 1 change. The arguments which led us to adopt v 5kms− are = 1 explained in Section 3. We verified that typically if we add 1 km s− to the Vy component of each cluster, to mimic the effect of an error on the v component of the solar motion, the fitted γ is changed about 1 1 1kms− kpc− . The error on κ introduced by the uncertainty in v is 1 1 about 3kms− kpc− . It w±as mentioned in Section 3 that the ratio a/b is approximately 1.5, but the precise value depends on the adopted and κ. A number of tests showed that a small variation of this ratio (like for instance Figure 7. Epicycle frequency measured at different Galactic radii. The taking a/b 1.4) does not produce any change in the fitted values dashed line represents the theoretical κ corresponding to a smooth rotation = 1 of κ, so that it was not even necessary to use an iterative procedure curve with V0 232 km s and R0 7.5 kpc. The dotted line corresponds = − = to determine κ. In principle, when a line is fitted to the observed to the same rotation curve to which a Gaussian minimum centred at 8.8 kpc 1 points like in Fig. 6, the scattering of the points due to the errors has been added, resulting in V0 230 km s . = −

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Table 1. Values of V0 for which the observed and theoretical κ are equal (see Fig. 8), for different adopted R0 and different radii of the Galactic ring.

R0 Ring radius Best V0 κ 1 1 1 (kpc) (kpc) (km s− ) (km s− kpc− ) 7.0 5.7 195 50 7.0 7.0 230 47 7.5 6.2 220 48 7.5 7.5 230 43 7.5 9.0 230 35 8.0 6.7 225 48 8.0 8.0 230 40 8.5 7.3 210 42 8.5 8.5 245 42

Figure 8. The epicycle frequency as a function of V0 for different rotation curves, all based on R0 7.5 kpc. The full line with positive slope is the = theoretical prediction for the series of CO-based rotation curves with differ- R0 0.4 kpc. The study was also made for rings that do not coincide V κ ent values of 0 (see Section 3.1.2). The dashed line is the theoretical for with±R ; the width of the rings was always about 0.4 kpc. In all pure flat curves with different V . The full line with negative slope shows the 0 0 the cases, the figures look similar to Fig. 8 with a rising± theoretical observed values of κ, based on angle versus age fitting, like in Figs 5(a)–(c). 1 line and a decreasing line of ‘observed’ κ. The intersection of the The theoretical and observed κ intersect at about 230 km s− . two lines is considered as the best estimate of V0 for the given R0. The results are summarized in Table 1. It can be seen that the observed κ at R(ring) R does not vary = 0 much for the different adopted R0. This is an expected result since the samples of clusters are the same in these cases, as the positions of the clusters are given relative to the Sun. The fact that a same best estimate of V0 is obtained for R0 7.0, 7.5 and 8.0 kpc indicates that this value is a robust result. Furthermore,= the fact that for R0 7.5 and 8.0 kpc almost the same value of V0 is obtained from different= rings is an indication that the smooth rotation curves that we are using in these cases are close to the real ones.

3.8 A wiggled rotation curve and the Oort’s constants It is well accepted that the rotation curve presents irregularities and, in particular, a minimum beyond the solar circle (e.g. Merrifield Figure 9. The rotation curve of the Galaxy for R0 7.5 kpc. The line is a = 1992; Brand & Blitz 1993; Amaral et al. 1996; Honma & Sofue fit to the CO data, using the empirical expression V 240 exp[ (3.6/r)2] = − + 350 exp( r/3.55 0.10/r); the units are km 1 and kpc. A shallow Gaussian 1997; Olling & Merrifield 1998). For instance, Brand & Blitz found − − − minimum 12 exp [ [(r 8.8)/1.0]2] was added, in order to better fit the the minimum centred at about 1.15 R0. Although our method based − − − CO data and the epicycle frequency data (Fig. 7). Including this term, V0 on the epicycle frequency measurement is unable to produce a de- 1 = 230 km s− at R0. tailed description of the minimum, we propose the curve presented in Fig. 9, which contains a minor wiggle represented by a Gaussian minimum centred at 8.8 kpc with a half-width of 1 kpc and depth measurement of κ 43 5kms 1 kpc 1 would imply that V is 1 − − 0 12 km s− . Such a wiggle produces a reasonable fit of the epicycle 1 = ± 230 15 km s− . ± frequency (see Fig. 7), and of the Clemens (1985) data. We note Besides the pure flat (slope zero) curves, the other family of curves that Clemens did not use the same VLSR transformation that we that we consider is that of smooth curves, obtained by fitting the CO adopted in this work, and that the we applied corrections to his data data with different values of V (Section 3.1.2). The line representing 0 for the different R0 that are strictly valid only for the data inside the the theoretical κ for this family of curves also cuts the observed solar circle. Up to R0 the curve is still quite flat, so that our analysis 1 line at 230 km s− . The coincidence of the two ‘theoretical’ lines of the sample of clusters (corrections for differential rotation) near intersecting at 226 km s 1 is not surprising, since the smooth curve − R0 is not affected by the minimum. 1 for V0 about 230 km s− is close to a flat one (see Fig. 9). In the presence of wiggles in the rotation curve, the measured slope of the curve (and consequently the Oort’s constants) depends on the precise radius range of the sample of objects which is used. 3.7 Rotation curves with R 7.0, 8.0 and 8.5 kpc 0 = This could explain the contradiction between different measure- An analysis similar to that of the previous sections has been per- ments of the Oort’s constants in the literature. In a first-order ap- formed, adopting R 7.0, 8.0 and 8.5 kpc. For each adopted R , proximation, the parameters V and R are linked together through 0 = 0 0 0 families of smooth rotation curves with different values of V0 (as the Oort’s constants A and B, which are determined by observations explained in Section 3.1.2) were used to derive the theoretical κ [V0/R0 A–B, (dV/dR)R0 A B]. The accepted value of A = =− − 1 1 − at the radius R0. On the other hand, the angle of the observed ve- B seemed to be in the range 25–27 (the units are km s− kpc− in locity perturbation with respect to the direction of circular rotation what follows), in the past and even in some recent works (Feast has been plotted, using the samples of clusters in the radius range & Whitelock 1997; Kalirai et al. 2004). The value of V0 that we

C C 2008 The Authors. Journal compilation 2008 RAS, MNRAS 386, 2081–2090 2088 J. R. D. Lepine´ , W. S. Dias and Yu. Mishurov

this discussion that the value of V0 that we obtain, and our choice R0 7.5 kpc, is not in contradiction with the constraints of the Oort’=s constants. The existence of a relatively sharp (half-width about 1.0 kpc) minimum could possibly be related to the corotation resonance. A ‘vacuum’ of gas density tends to form at the corotation radius (e.g. Lepine,´ Mishurov & Dedikov 2001), and after a few billion year a ‘gap’ ring with a smaller stellar density forms.

3.9 Statistics of initial directions Typically, a ring with a width about 0.8 kpc contains two to four main directions of initial velocity perturbation seen as distinct peaks in the histograms like Figs 5(d)–(f). It was shown by DL that the clusters were born in spiral arms. A preliminary analysis of the position on the Galactic plane where open clusters were born indicates that at least two different directions can originate from a same position in the Galactic plane, at about a same epoch. This means that a same Figure 10. The present-day positions (asterisks) and birth time positions region of a spiral arm can produce star clusters going in different (triangles) of a group of clusters that share a same initial velocity direction directions. However, this kind of analysis is difficult to perform, (the sample of Fig. 6). since the samples become small when narrow bins ranges of ages and initial positions are considered. In Fig. 10, the present-day positions obtain with R 7.5 kpc might seem to be too large, since it cor- and birthplace positions of the clusters that share approximately the 0 = responds to V0/R0 30.7. However, Olling & Merrifield (1998) same initial velocity direction (the sample of Fig. 6) are shown. verified that the Oort’=s constants A and B differ significantly from Different rings may have different main directions, but some gen- the general V0/R0 dependence, in the solar neighbourhood. Olling eral tendencies can be observed. Fig. 11 presents the distribution of & Dehnen (2003) showed that the most reliable tracers of the ‘true’ initial velocities in the local (U, V) frame of each cluster, for differ- Oort’s constants are the red giants, and derived A B 33. Another ent ranges of radius, obtained by numerical integration. In frames determination of a high value of A B is that of−Back=er & Sramek (a)–(c), the same ranges of radius of Fig. 5 are shown for com- (1999), who measured the proper motion− of Sgr A, at the Galactic parison. We also considered larger Galactic radius intervals to ob- 1 Centre, equal to 6.18 0.19 mas yr− , equivalent to 29.2 0.9. This tain better statistics, in the frames (d)–(f). One region from 5.0 to is an interesting result± since, contrary to methods based±on Oort’s 7.0 kpc (frame d) is inside the corotation circle. The next one is the constants, it does not depend on local wiggles. Branham (2002) local region, from 7.0 to 8.3 kpc (frame e), which probably con- obtains 30.3, Fernandez,´ Figueras & Torra (2001), 30, tains the corotation radius (Rc/R0 1.06 0.08, according to DL). and Miyamoto= & Zhu (1998), 31.5, all of them from the=kine- Finally in frame (d), we show the=velocity±distribution for clusters matics of OB stars. Metzger, Cadwell= & Schechter (1998) obtained in the radius range 8.3–12 kpc, mostly beyond the corotation cir- 31 from Cepheid kinematics and Mendez´ et al. (1999) obtained cle. In each figure, one can see some preferential or non-random 31.7 from the Southern Proper Motion Program. We conclude from directions.

Figure 11. Distribution of initial perturbation velocities of open clusters. In frames (a)–(c), the clusters were selected in the same Galactic radius ranges of Figs 5(a)–(c). In frames (d)–(f), wider ranges (indicated in the frames) were investigated. The U, V velocities are in the local frame of each cluster, with U pointing opposite to the Galactic Centre.

C C 2008 The Authors. Journal compilation 2008 RAS, MNRAS 386, 2081–2090 Epicycle frequency and velocities of clusters 2089

A number of differences appear between the three regions. There the other uses the plots of the velocity direction versus age of the are only a few points in the second quadrant (γ between 90◦ and open clusters. We compared the results of the two methods using 180◦) in frame (d) (small Galactic radii), while in frame (f) (large different rotation curves based on different values of R0 and of V0. Galactic radii) it is the fourth quadrant which is almost empty. In If we fix for instance R0 and vary V0, we can find a best value of V0, frame (f), we see dominant directions at about 50◦ and 130◦ with for which the ‘observed’ and the ‘theoretical’ values of κ coincide. respect to the rotation direction. The direction of these initial veloc- If R0 7.5 kpc is adopted, using the sample of clusters in the = 1 ities can be transformed to the frame of reference of the spiral arms range 7.1 < R < 7.9 kpc, we find V 230 15 km s− . The 0 = ± by adding a vector in the counterclockwise rotation direction, which same value of V0 is obtained when we adopt R0 7.0 or 8.0 kpc. 1 = represents the velocity of the frame of reference of the cluster with Therefore, V 230 15 km s− is a robust result which does not 0 = ± respect to the arms (at distances larger than corotation, the Galac- depend on the precise choice of R0, in the 7–8 kpc range. 1 tic material enters into the arms in the counterclockwise direction). The rotation curve R0 7.5 kpc , V0 230 km s− , with a min- 1 = = If for instance we add a vector with modulus about 100 km s− in imum at 8.8 kpc, is found to be a satisfactory one, since it predicts 1 that direction (180◦) to a vector about 15 km s− with direction 50◦ ‘theoretical’ values of κ that are consistent with the observed ones, or 130◦, the resultant is a vector with direction about 170◦. Since not only at R0, but also at different Galactic radii. In addition, this the spiral arms have pitch angles typically between 7◦ and 15◦, this curve produces a good agreement between the distribution of ini- means that a large fraction of the clusters start their life travelling tial velocities of the clusters obtained by numerical integration of along the spiral arms. In frame (d), the dominant directions are not the orbits and by plots of velocity directions versus age. For this 1 1 so clear, but many points are situated in the directions around 60◦ curve, V /R is 30.7 km s− kpc− . As we discussed, this value is − 0 0 and 130◦. In this case, the correction is in the clockwise direction, supported by many recent measurements that result in V0/R0 close − 1 1 but the same conclusion that many clusters start moving along the to or larger than 30 km s− kpc− including a method that does not spiral arms is reached. This interesting property of the initial ve- depend on local irregularities of the rotation curve. The method to locities of the clusters certainly contributes to the narrow aspects link the values of V0 and R0 proposed in this paper is a new one, of the arms, since the clusters do not leave the arms in a short time which is something much needed in view of some conflicting results compared to the lifetime of the brightest stars. based on analyses of Oort’s constants that have been published in the last decade. 4 CONCLUSIONS ACKNOWLEDGMENTS The study of samples of open clusters situated in rings with width <1 kpc, reveals the presence of several sequences of 4 to 12 clusters The work was supported, in part, by the Sao Paulo State agency that have been formed with a same initial direction of the perturba- FAPESP (fellowship 03/12813-4) by the Conselho Nacional de De- tion velocity, as measured with respect to the direction of circular senvolvimento Cientifico e Tecnologico.´ We thank the referee for rotation. 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