COROTATION, STELLAR WANDERING, AND FINE STRUCTURE OF THE GALACTIC ABUNDANCE PATTERN J. R. D. Lépine, 1 I. A. Acharova, 2 and Yu. N.

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1 The Astrophysical Journal, 589: , 2003 May 20 # The American Astronomical Society. All rights reserved. Printed in U.S.A. COROTATION, STELLAR WANDERING, AND FINE STRUCTURE OF THE GALACTIC ABUNDANCE PATTERN J. R. D. Lépine, 1 I. A. Acharova, 2 and Yu. N. Mishurov 2 Received 2002 October 31; accepted 2003 January 30 ABSTRACT We consider the effects of the corotation resonance of the Galactic spiral structure on the stellar orbits. It is shown that because of resonant interaction with the spiral gravitation field, stars can wander in the radial direction over a large part of the Galactic disk, moving over distances 2 3 kpc in a short time, on the order of 1 billion years or even much less. This mechanism of radial stellar wandering is much faster than other stellar diffusion mechanisms that have been suggested in the literature. Corotation resonance also influences the Galactic distribution of heavy elements that are derived from old stellar-like objects. If at the initial time there is a simple linear distribution of metallicity in the disk, this is broken in 3 billion years. In the framework of the model for the spiral density wave pattern with the corotation resonance close to the solar position (supposed to be 8.5 kpc from the center), the bimodal abundance pattern with a gradient in the inner part of the Galaxy (R 7.5 kpc) and a plateau for R between 7.5 and about kpc forms under the influence of the corotation resonance. Subject headings: Galaxy: abundances Galaxy: evolution Galaxy: structure stellar dynamics 1. INTRODUCTION Chemical abundance is like a fossil record for Galactic history. However, to decode these records happens to be a very difficult task. A lot of factors influences the final Galactic radial distribution of heavy elements, so we should bear in mind that different objects can demonstrate a distinct chemical abundance pattern. Indeed, the youngest objects, H ii regions, OB stars and Cepheids, reflect the last stages of the chemical enrichment in the Galactic disk, whereas the older objects, such as planetary nebulae, inherit information about the distribution in a former epoch. Additional very important information may be given by some features in the abundance pattern, such as a minimum of metallicity at corotation. But because of dynamic smearing effects these distinctions can be rather subtle; hence, precise data and refined ideas about the Galaxy structure and evolution are needed to reconstruct the Galactic chemical history. Such observational data were derived only in the last decade (or even the last few years). In our previous paper (Mishurov et al. 2002, hereafter MLA) we investigated the effect of the corotation resonance on the chemical production and radial distribution of heavy elements in the Galactic disk. In the framework of the model for the spiral wave pattern with the corotation close to the Sun, the bimodal radial distribution of the metallicity revealed from Cepheids by Andrievsky et al. (2002a, 2002b) was explained. However, this is not the only effect of the corotation resonance on the Galactic radial abundance distribution: because of resonant interaction with the gravitational field of spiral arms near the corotation, the stars 1 Instituto de Astronomia, Geofísica e Ciências Atmosféricas, Universidade de São Paulo, Cidade Universitária, São Paulo, SP, Brazil; jacques@ iagusp.usp.br. 2 Department of Space Research, Rostov State University, 5, Zorge Street, Rostov-on-Don , Russia; kfk@phys.rsu.ru, mishurov@ phys.rsu.ru, mishurov@aaanet.ru; Isaac Newton Institute of Chile, Rostov-on-Don branch. 210 wander over a large range of Galactic radius. This wandering should become apparent in the radial abundance pattern. The idea that stars diffuse along the Galactic radius was proposed by Wielen (1977). On the basis of chemical data Wielen et al. (1996, hereafter WFD) argued that the Sun was born about 2 kpc closer to the Galactic center and migrated during its life to the current distance, R =8.5 kpc. However, their theory for the stellar diffusion was a phenomenological one, since the basic mechanism for the gravitational perturbations of stellar orbits is not accurately known (WFD, p. 445). To find such mechanisms happened to be a difficult task (Lacey 1984; Fuchs 2001). In a recent paper Sellwood & Binney (2002, hereafter SB) showed that stars can move over a large distance because of resonant interaction with spiral density waves at corotation. By means of this mechanism they explained in particular the low age-metallicity correlation derived for old stars in the solar vicinity by Edvardsson et al. (1993). They also noted that this mechanism will flatten any radial metallicity gradient in the disk. These results represent an important contribution to the understanding of the chemical evolution of the disk; the physical mechanism for stellar migration proposed by SB seems to be efficient. Since the interaction of stars with spiral arms near corotation is a complex process, it is now necessary to fully explore theoretical and computational aspects and to verify to what extent the stellar migration process depends on the details of a specific model. In particular, SB performed unconstrained simulations in which a succession of spiral waves with different pattern velocity and corotation radii appears. In their simulations, the fact that the successive waves are uncorrelated produces migration distances that increase with time. However, the classical view of Lin & Shu s theory, according to which the spiral arms are long lived, is still widely accepted in the literature. In a series of papers, e.g., Lin & Bertin (1995), Shu et al. (2000), and Laughlin, Korchagin, & Adams (1997, 1998), the authors put forward some arguments that spiral density waves can be long lived. For instance, Amaral & Lépine (1997)

2 GALACTIC ABUNDANCE 211 determined the pattern rotation velocity from the ages of open clusters and suggested that the spiral waves are driven by the rotation of the bulge. This is an example of a constrained mechanism that would cause a constant pattern speed. According to various investigations, the present position of the Sun appears to be very close to the corotation resonance (Marochnik, Mishurov, & Suchkov 1972; Crézé & Mennessier 1973; Nelson & Matsuda 1977; Mishurov et al. 1997; Amaral & Lépine 1997; von Linden et al. 1998; Mishurov & Zenina 1999; Lépine, Mishurov, & Dedikov 2001; Fernández, Figueras, & Torra 2001, etc.). This is why we shall reconsider the problem of stellar wandering and the evolution of the metallicity distribution under the influence of the Galactic spiral arms in the framework of a model with corotation resonance close to the Sun. By means of direct numerical integration of the equations for the stellar motion perturbed by spiral density waves, we establish what happens with a star near the corotation resonance and how this will reflect on the chemical abundance pattern. Our model makes predictions concerning the metallicity distribution as a function of Galactic radius that are shown to agree with observations. We believe that the bimodal structure should also appear in external galaxies. 2. SPIRAL ARMS, COROTATION RESONANCE, AND STELLAR WANDERING To illustrate strong resonant influence of the gravitational field from spiral arms on the stellar motion near the corotation radius, let us consider the model task of the motion of a star in the Galactic plane perturbed by the gravitational field of spiral density waves. Note that Lynden-Bell & Kanaljs (1972) derived the effect of forcing at the Lindblad resonances but not at corotation. The equations of motions of the star are R ¼ R # _ S ; ð1þ dðr 2 _ #Þ dt ; ð2þ where 0 (R) is the unperturbed axisymmetric gravitational potential, S (R, #, t) is the corresponding spiral perturbation, (R, #) refers to the polar Galactocentric coordinate system, t is the time, and the prime denotes the derivative with respect to R. For the spiral gravitational perturbation S we adopt the representation in the form of a running wave from the density waves theory of Lin, Yuan, & Shu (1969): S ¼ A cos m cot i ln R # þ P t ; ð3þ R where A is the amplitude, i is the pitch angle of the arms, m is the number of arms, and P is the angular rotation velocity of the spiral pattern. Following Lin et al. (1969) we assume in our model calculations the wave amplitude and the pitch angle are constant. Note that whereas the Galactic disk rotates differentially, i.e., the Galactic angular rotation velocity is a function of R (from the equilibrium equation 2 R = 0 0), the rotation velocity, P, of the spiral pattern is constant. In other words, spiral density waves rotate as a solid body. The distance R C at which both velocities coincide is called the corotation radius. It is obvious that a star near the corotation radius moves in a resonance with the wave. The effects of such resonant interactions are the aim of our investigation. The normal analytical approach to this problem is to assume that in the absence of spiral arms a star moves along a circle of radius R 0 at an equilibrium rotation velocity 0. Considering a small spiral perturbation, we can expand R and # in a series of powers of the spiral wave amplitude. If R 1 is the first-order perturbation in R, it is easy to show that close to corotation R 1 grows with time. The demonstration can be made in the manner followed by Lynden-Bell & Kalnajs (1972) for the Lindblad resonances. The usual effect of a resonance in any field of physics is growth in the amplitude of oscillation, because the driving force acts in phase with the natural oscillation during an interval of time. In the case of the corotation resonance, the effect is not much different. Without considering any perturbation, a star near corotation would stay at an almost constant relative position with respect to the spiral arms. If we now consider a perturbation produced by the spiral structure, in this fixed geometry the perturbation force will act on the star with a constant direction for a long time, and therefore, it will be able to accelerate the star. The analytical method that we mentioned above is not convenient to describe large deviations from the unperturbed state, so that we prefer to give numerical results Numerical Experiments To illustrate quantitatively the effects of the corotation resonance on stellar wandering, we give some results of direct numerical computations with equations (1) (3). The computations were performed by means of a fourth-order Runge-Kutta integration scheme with different values of the integration time step, and we checked that the result does not depend on it. The typical integration time step was 0.5 Myr although for control in many cases we computed the trajectories with much smaller steps. The typical deviation of the total energy of a particle at time 5 Gyr with respect to the initial one was on the order of The motion of a star depends on a set parameters: the rotation velocity of the Galactic disk, which is determined by the unperturbed gravitational potential (we used the rotation curve of Allen & Santillán 1991 with the standards 0 = 220 km s 1 and R = 8.5 kpc), the initial peculiar velocity, the initial Galactocentric distance R 0, the parameters of the spiral gravitational potential, and the initial wave phase of the particle 0, which determines the position of a star relative to the spiral arms (for different 0 the stellar trajectory will be quite different even if the other initial parameters are the same). The parameters describing the spiral perturbation are the wave amplitude A, the number of arms m, the pitch angle i, and the rotation velocity, P, of the pattern (in all cases the corotation radius R C is situated at the solar distance, i.e., P = ). Of course, stars with small peculiar velocities are more sensitive to small spiral perturbations, and a star with large peculiar velocity will not feel the spiral perturbation. For this reason, we adopted an initial peculiar velocity corresponding to the velocity of the Sun, U 0 = 10 km s 1 and V 0 =6kms 1. The amplitude of the spiral gravitational potential is a rather arbitrary value. In a series of papers Lin and his collaborators (see, e.g., Lin et al. 1969) proposed that the value of the amplitude of the spiral force at the position of the

3 212 LE PINE, ACHAROVA, & MISHUROV Vol. 589 Fig. 1. Stellar radial wandering under the influence of the gravitation perturbation from spiral arms. Top, Dependence of R on t; bottom, dependence of the radial (solid line) and azimuthal (dotted line) components of peculiar velocities on t. The star starts at the corotation radius (R 0 = R C = R = 8.5 kpc), the number of spiral arms m = 2, and the pitch angle i = 7. Left, Initial wave phase 0 =90 ; right, 0 = 180. Sun, ma cot i F ¼ 2 R 2 ; ð4þ is But Roberts & Hausman (1984) give the value F 0.1, and Mishurov et al (1979) derived F If we use external galaxies to estimate the amplitude of the spiral force, we see that in M81 (this galaxy probably is similar to the Milky Way) at the distance corresponding to the solar one in our Galaxy, the amplitude is 0.1 (Visser 1980). A similar value can be derived for M100, based on the arminterarm contrast (Elmegreen, Elmegreen, & Seiden 1989). In the sample of Rix & Zaritsky (1995), the contrast for about half the galaxies is of order unity and leads to a spiral force of amplitude 0.2; so in our model experiments we adopt F = 0.1. Figures 1 and 2 illustrate the behavior of R and the radial and azimuthal peculiar velocities (U and V ) of the particle versus t for two-armed (m =2, i = 7 ) and four-armed (m =4,i = 12 ) patterns if the star starts from the solar Galactocentric distance. From these figures it is seen that a star that occurs at some moment of time near the corotation radius under the influence of the spiral gravitational field could indeed wander over a large range of Galactic radius, 2 3 kpc. However, unlike the diffusion of WFD this process is fast: the star can move over such a distance in a time of 1 Gyr or even much less. We would also like to emphasize that the peculiar velocities caused by the spiral perturbations are quite moderate and close to the typical observed stellar velocities. For comparison in Figure 3 we give the same results for a star that starts at R 0 = 7.5 kpc. It is seen that in some cases ( ) the wandering is more moderate than in the previous one. However, for other parameters ( 0 0 ) at some moment of time the star can undergo very fast radial drift. Of course this feature motivated a special study to check that it was not an artifact of the integration procedure. It is interesting to note that for a star starting at R =7 kpc, independently of 0 the radial wandering is essentially less than in the previous cases (see Fig. 4) and is close to the epicyclic amplitude; hence we can state that the effects of the corotation resonance on the stellar wandering terminates (in the inner part of the Galaxy) at about kpc. 3. COROTATION AND DEFORMATION OF THE ABUNDANCE PATTERN In this section we consider the evolution of the abundance pattern that can be expected for stellar-like objects. We examine the idea of SB about the flattening of the abundance distribution due to the corotation effects and show that because of the influence of the spiral arms on stellar motion near the corotation resonance the very popular linear radial abundance distribution is broken in a few billion years, and a bimodal structure forms. To demonstrate the net effect of the corotation resonance, let us consider the following task. At the initial moment of time t = 0, a number N of particles are distributed over a disk so that the surface concentration is l / exp( R/d ) with a radial scale d. In the absence of perturbations,

4 No. 1, 2003 GALACTIC ABUNDANCE 213 Fig. 2. Same as Fig. 1 but for m = 4 and i = 12. Left, 0 =90 ; right, 0 = 270. particles move in the Galactic plane rotating around the Galactic center with the corresponding equilibrium rotation velocity. Further, to each particle is assigned an abundance [X/H] according to the linear law: [X/H] = (R R ), where is the initial abundance gradient. Then we apply the perturbed gravitational potential of the spiral density waves and compute the perturbations in the motion of the particles. In our experiments we refer to stars that do not appreciably change their abundance with time; so in the diagram [X/H] versus R they move only in the horizontal direction, and it is easy to see what their original position was. The resulting abundance pattern computed with N = for the typical parameters, namely, the initial abundance gradient usually adopted for the whole Galaxy = 0.07 dex kpc 1, d = 3 kpc, and other parameters taken from the Fig. 3. Same as Fig. 1 but for R 0 = 7.5 kpc, m = 2, and i = 7. Left, 0 =0 ; right, 0 = 180.

5 214 LE PINE, ACHAROVA, & MISHUROV Vol ) 3 and range from the Galactic interior up to R 4 5 kpc. However, to use them we give another representation of the data: we grouped the abundances by averaging them over R within 0.5 kpc bins and normalized them so that [X/H] = 0 for R = R. The corresponding distributions for oxygen and sulphur superposed on our theoretical results are given in Figures 6 and 7. From these figures one can see that the observational data confirm the bimodal-like structure. Fig. 4. Same as Fig. 3 but for R 0 = 7 kpc previous section (the case for m = 2), is shown in Figure 5. Note that the above structure forms at t 3 Gyr and is maintained afterward. From this figure it is seen that a bimodal abundance structure forms under the influence of spiral density waves: whereas inside the Galactocentric distance R 7.5 kpc the gradient does not noticeably change, between 7.5 and kpc the plateau-like structure forms. This distortion of the abundance pattern is obviously due to the influence of the corotation resonance and stellar wandering (recall that in our model R C = R = 8.5 kpc). For a comparison of our results with observations, the data of Maciel & Quireza (1999) for type II planetary nebulae were used. They have ages 4 6 Gyr (Maciel & Köppen 4. DISCUSSION In the present paper we illustrate the ideas of SB about the effects of the corotation resonance on the stellar radial wandering and the evolution of the radial Galactic abundance distribution. It was shown that because of strong resonant interaction with spiral arms at the corotation radius stars wander over a large part of the Galactic disk. They can move in a radial direction for a distance 2 3 kpc in a period of 1 Gyr or even less. This is a much faster mechanism than the stellar diffusion of WFD. Stellar wandering due to resonant interaction with spiral arms at corotation will be reflected in the radial abundance structure: if the initial distribution is a linear-like function a very popular point of view after a few billion years it will be transformed into a bimodal pattern with a plateau in the middle of the Galaxy (between 7.5 and kpc) and a steep gradient in the interior region (R 7.5 kpc) with a fracture at R kpc. The plateau forms in the vicinity of the corotation radius. Of course, this mechanism for the bimodal abundance structure reaches its maximum effect in a model in which the corotation radius does not vary appreciably over 3 Gyr; otherwise, some smoothing would occur. This hypothesis, adopted in the present paper, is different 3 We would like to outline that Twarog, Ashman, & Anthony-Twarog 1997 perhaps were the first to recognize a plateau-like (or step-like) feature in the radial Galactic abundance distribution. However, the open clusters in their sample are predominantly young: about a half of them are younger than 1 Gyr. Hence, they can simply keep the plateau-like distribution from the interstellar medium (see MLA). Fig. 5. Evolution of the abundance radial distribution under the influence of the spiral arms, showing t = 0(solid line) and t = 5 Gyr (open circles). Note the formation of the bimodal structure: the steep gradient for R 7.5 kpc and the plateau between 7.5 and kpc.

6 No. 1, 2003 GALACTIC ABUNDANCE 215 Fig. 6. Comparison of the resulting radial abundance distribution arising under the perturbations from spiral arms (open circles) with the data for planetary nebulae from Maciel & Quireza (1999) for oxygen. The bimodal structure is well seen in both the theoretical and observational data. from that of SB. On the other hand, the hypothesis of SB of varying corotation radius has in principle the advantage of being able to produce migrations over a larger range of radii. Possibly in the future very detailed observations of the metallicity distribution of stars will make it possible to choose between these two different views of the history of variations of the corotation radius. The above mechanism for the formation of the plateau structure of metallicity differs from the one proposed in our previous paper (see MLA). Indeed in that paper we considered the chemical enrichment process in the interstellar medium due to the effects of corotation and diffusion. This result must be applied only to the young objects such as Cepheids, H ii regions, etc., that reveal the present value of the metallicity in the interstellar medium. In the present paper we did not consider any process of chemical production. We inspected the dynamical effects only on the Galactic radial abundance pattern. This result can be applied to sufficiently old stellar (or stellar-like) objects. From our results it follows that the simple initial Galactic abundance distribution in a form of a linear function of Galactocentric distance is broken after a few (3 5) billion years and a bimodal structure forms. Of course, if the initial abundance distribution is already similar to a bimodal distribution as we argued in MLA, this structure will be conserved; so we can state that there are two distinct mechanisms for the bimodal structure formation, which affect objects of different ages. Since the calculations of stellar orbits were made in two dimensions in the Galactic plane, we cannot say at the moment if the corotation mechanism could also contribute to the increase in the vertical scale height of the stellar populations with their mean age. This work was partially supported by the Russian Federal Program Integration (grant B 0087/2122). Fig. 7. Comparison of the theoretical distribution with the data for sulphur of Maciel & Quireza (1999)

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