Fitting the Lin Shu-type density-wave theory for our own Galaxy

Transcription

1 MNRAS 433, (2013) Advance Access publication 2013 June 18 doi: /mnras/stt923 Fitting the Lin Shu-type density-wave theory for our own Galaxy Evgeny Griv, 1 Chow-Choong Ngeow 2 and Ing-Guey Jiang 3 1 Department of Physics, Ben-Gurion University of the Negev, Beer-Sheva 84105, Israel 2 Graduate Institute of Astronomy, National Central University, Jhong-Li 32001, Taiwan 3 Department of Physics, National Tsing-Hua University, Hsin-Chu 30013, Taiwan Accepted 2013 May 23. Received 2013 May 22; in original form 2013 February 18 1 INTRODUCTION The most puzzling features of rapidly and differentially rotating disc galaxies are the ring and spiral structures seen on scales of 1 5 kpc. In our own Galaxy, spiral-arm indicators are well known: the arms concentrate the most massive gas clouds and very young stars and star clusters which have a rather small random velocity dispersion. At the present time, the origin of both the ring and spiral features in disc galaxies is far from being understood. In the spirit of Lin & Shu (1964), Lin, Yuan & Shu (1969), Yuan (1969) and Shu (1970), the vast majority of experts in the field regard regular visible spirals (and rings) in rotating galaxies as density waves. These waves do not remain stationary in time. It is assumed that the process of propagation of spiral waves is one of rotation of the waves around the galactic centre as a solid with a constant angular velocity (while the waves in the form of concentric rings propagate away from or towards the galactic centre) and a simultaneous growth in the wave amplitude. The alternating density enhancements (rings and spirals) and depletion zones (interarm regions) consist of different material at different times, and the point masses stream across the arms. See Rohlfs (1977) and Binney & Tremaine (2008) for reviews of the original Lin Shu wave theory. In memory of Professor Chi Yuan ( ). griv@bgu.ac.il (EG); cngeow@astro.ncu.edu.tw (C-CN); jiang@phys.nthu.edu.tw (I-GJ) ABSTRACT The presence of spiral structure in rapidly and differentially rotating disc galaxies is currently attributed to the phenomenon of unstable (that is to say, growing) Lin Shu compression-type waves, or density waves, rotating at a constant angular velocity around the system s centre. It is important that when a density-wave structure is present in a galaxy, the gravitational field of the spiral arms will systematically deflect the motion of gas and young stars away from their mean circular rotation, and point masses in such a model react by streaming motions that are of spiral shape too. We examine the kinematics of Milky Way s 233 Cepheid stars on the assumption that the system is subject to moderately growing spiral density waves by taking into account small-amplitude perturbations of the Galactic gravitational potential. Using Cepheid line-ofsight velocities, we propose new estimates of the parameters of solar motion and Galactic rotation corrected for the effects of density waves, the radial and azimuthal components of systematic stellar motion due to the spiral arms as well as the dynamical parameters of the waves. A basis is given for preferring the dominant one-armed spiral structure in the solar neighbourhood of the Galaxy. Key words: galaxies: kinematics and dynamics galaxies: spiral galaxies: structure. Lin & Shu (1964), Lin et al. (1969), Yuan (1969) and Shu (1970) stated clearly the concept of quasi-stationary density waves. Our present-day concept is somewhat different. This generalized wave theory was elaborated concurrently on the basis of the Lin and Shu original ideas by a number of authors, e.g. Lin & Lau (1979), Morozov, Torgashin & Fridman (1985), Montenegro, Yuan & Elmegreen (1999), Griv et al. (2000), Griv, Gedalin & Yuan (2002, 2006), Lou, Yuan & Fan (2001) and Griv & Gedalin (2012). Accordingly, a selfgravitating matter in a galaxy exhibits collective, gravitationally unstable modes of motions. Assuming the weakly inhomogeneous system, the behaviour of small perturbations of equilibrium parameters is sought in the form of a superposition of different Fourier harmonics corresponding to different self-excited growing modes of oscillation of the system under study, X 1 (R,t) = k R { X k (R, Z) [ e ik R iω kt ]}. (1) Here, we have introduced the coordinate system (R, ϕ, Z) with the velocity components V R =v R, V ϕ =R (R) + v ϕ and V Z = v Z,(R, ϕ, Z) are the cylindrical coordinates with the origin at the galactic centre and the axis of disc rotation is taken oriented along the Z-axis, X k (R, Z) is an amplitude, k is the real wave vector, ω k =Rω k + iiω k is some complex frequency of excited oscillations, suffixes k denote the kth Fourier component and t is the time. The real amplitude X k is a slowly varying function of space and time, the rapidly varying part of X 1 is absorbed in its phase C 2013 The Authors Published by Oxford University Press on behalf of the Royal Astronomical Society

2 2512 E. Griv, C.-C. Ngeow and I.-G. Jiang k R R 1, where k R is the radial wavenumber of the pattern. In the WKB method used throughout the theory, the radial wavenumber is presumed to be of the form k R (R) = A (R), where A is a large parameter and (R) is a smooth, slowly and monotonically varying function of R.In a local WKB approximation, we are actually exploring X k (R) = const/ R and k R = const (Shu 1970). The constant phase velocity of spiral density waves is p =Rω k /m called the pattern rotation speed, where m is the positive azimuthal mode number. Evidently X 1 is a periodic function of ϕ, and hence m must be an integer (= number of spiral arms for a given harmonic). Since p is a constant, independent of time or radius, each component will remain identical with time and therefore spirals do not wind up. A perturbation is considered to be a superposition of different oscillation modes, and the coexistence of several waves (both circular and spiral) is possible. Thus, the imaginary part of the wavefrequency Iω k corresponds to a growth or decay of the components in time, exp(iω k t), and Rω k /m corresponds to a rotation with constant angular velocity. Unstable modes with higher Iω k are more likely to achieve a higher amplitude and play more important roles. Since the components of the perturbation X 1 (R,t) with different m are separate, we may consider just one component at a time: X 1 =R[ X(R, Z)e ik RR imϕ iωt ]. A longitudinal Lin Shu density wave is associated with compression and decompression in the direction of travel, which is the same process as the ordinary sound waves in gases. The density waves are primarily collective modes in a galactic system. When Iω >0, the medium transfers its energy to the growing wave and oscillation buildup occurs. The solution in such a form represents in the plane a spiral wave with m arms or a circle (m = 0): the lines of constant RX 1 at a fixed moment of time are approximately given by the equation m(ϕ ϕ 0 ) = [k R (R) k R (R 0 )]. With ϕ increasing in the rotation direction, we have k R < 0 for trailing spiral patterns, which are almost always observed in galaxies, and k R > 0 for leading ones. With m = 0, we have the density waves in the form of concentric circles that propagate away from the centre when k R < 0ortowards the centre when k R > 0. This wave structure may be excited by real internal instabilities of gravity perturbations. Low-m (say, m < 4) waves are the most important because they are associated with large-scale phenomena (Lin et al. 1969; Rohlfs 1977). It is important that according to the theory the density wave ought to manifest itself as a characteristic systematic change in the velocity field (Yuan 1969). The galactic material will exhibit azimuthal and radial systematic motions in addition to the basic circular rotation in a coherent structure like a density-wave pattern, and we are looking for these motions in the present study. In particular, the gas and stars move faster on the outer edge of the spiral arm and slower on the inner edge, and the denser (rarer) part of the arm is moving towards or away from the galactic centre. Yuan (1969) and Rohlfs (1977) have demonstrated the necessity of both the azimuthal and radial systematic motions in the framework of the densitywave theory and claimed that they can be used as an important test of the theory. Observations have indicated significant non-circular motions in many spiral galaxies (e.g. Fridman et al. 2005). In this work, from a sample of Cepheids as catalogued recently by Ngeow (2012), we determine both the geometrical and dynamical parameters of the spiral structure of the Galaxy and compare those results with the generalized stability theory. First, in accordance with the theoretical expectations, we are looking for timedependent modes of oscillations (equation 1). Secondly, we address the problems of the characteristic size of spirals and the spacing between them, and the velocity field of waves. Thirdly, we define the free parameters of the density-wave theory, namely the pattern speed p and the perturbations of the Galactic gravitational potential (R 0,Z 0 ) and surface density (R 0,Z 0 ) in the solar vicinity of the Galaxy (hereafter the subscript 0 denotes values corresponding to the position of the Sun). Crézé & Mennessier (1973) have already attempted to fit the Lin Shu theory to the observed velocities of nearby young stars, but evidently without success. The effect of the density waves upon the kinematics of young stars was subsequently considered by Byl & Ovenden (1978) and Mishurov, Pavlovskaya & Suchkov (1979), and later refined by Pavlovskaya & Suchkov (1980). The effect upon the kinematics of H II regions was then analysed by Grivnev (1981). The kinematics of various young Galactic objects was analysed on the assumption that the system is subject to density waves also by Mishurov & Zenina (1999), Fernández, Figueras & Torra (2001), Lépine, Mishurov & Dedikov (2001), Zabolotskikh, Rastorguev & Dambis (2002) and Bajkova & Bobylev (2012). The same problem was discussed by Siebert et al. (2012) considering the properties of the local spiral arms from RAVE data. Even though all these calculations demonstrate the presence in the solar neighbourhood of spiral density waves, they did not yield a reliable value for the number of spirals, which was taken to be m = 2 or, sometimes, m = 4. In contrast to all these studies, in our study, the calculations are performed for all possible m = 1, 2, 3 or 4. A basis is given for preferring the dominant m = 1 model. 2 PERTURBATION OF THE DENSITY AND THE VELOCITY FIELD Following Lin and Shu, if the waves are assumed to be moderately unstable ( Iω k /Rω k 1), in the plane the local gravitational potential of the Galaxy is taken in the form R0 (R, ϕ, t) = mean (R) + cos φ, (2) R with mean being the basic (mean) axisymmetric potential and R 0 /R cos φ the small non-axisymmetric perturbation, / mean 1 (e.g. Shu 1970). In equation (2), φ = φ 0 + m [ p t ϕ + (1/ tan p)ln(r/r 0 ) ] (3) is the phase of the wave, φ 0 is the phase of the wave at the Sun s location, ϕ is measured clockwise from the radius passing through the location of the Sun, p is the constant pitch angle of the spiral arms, λ = 2πR 0 tan p/m is the radial distance between arms, 2πR 0 /m is the angular distance between them and spirals with trailing arms have p < 0. As one can see, the geometrical form of a spiral arm is represented, as most other authors have done, by a single logarithmic function. [The shape of the arms of spiral galaxies is often logarithmic (Seigar & James 1998).] In the linear and tightly wound ( tan p 1) Lin Shu approximations, the perturbed surface density and the spiral components of the systematic motion are ( ) 1 = cos φ = kr 2 1 c2 s k2 R R0 mean cos φ, (4) κ 2 R κ2 ( ) v R = ṽ R cos φ = k R 1 c2 s k2 R R0 κ 2 R ω 2 ( ) v ϕ = ṽ ϕ sin φ = k R 1 c2 s k2 R R0 κ 2 κ 2 R 2 ω 2 ω 2 ω κ2 cos φ, (5) 1 sin φ, (6) κ2

3 where mean (R) is the mean surface density, κ = 2 [1 + (R/2 )(d /dr)] 1/2 is the epicyclic frequency, ω = m( p )is the Doppler-shifted (in a rotating frame) wavefrequency, c 2 s k2 R /κ2 < 1, the values of the phase φ = 2πn and n = 0, ±1,... correspond to density maxima (potential minima) in the field of the spiral arms, i.e. to the centre of an arm, and R 0 /R is the amplitude of the perturbed gravitational potential. Equations (4) (6) describe the small perturbation of the surface density and the small perturbed velocities of the matter under the action of the gravity perturbation, 1 / mean 1and v R, v ϕ R. A special analysis of the solution close to corotation (ω = 0) and Lindblad (ω =±κ)resonances is required. The spiral wave pattern is known to extend over a principal part of a disc where ω <κ. Hence, in this part of a galaxy in the above formulae, the common denominator ω 2 κ2 is negative. 3 NUMERICAL METHOD Consider a star at distance r from the Sun and at Galactic longitude l and latitude b. In order to model the situation, we assume that in the zeroth-order approximation objects in the disc are moving in perfectly circular orbits about an axis through the Galactic Centre perpendicular to the Galactic plane. In accordance with equations (5) and (6), in the next approximation we divide the motion of any star into three parts: mean circular rotation around the centre ( 0 )R 0, systematic motion in the field of the waves v R, v ϕ and the solar motion with respect to the local standard of rest (LSR) of stars in question (Rohlfs 1977; Mihalas & Binney 1981). The motion of the star along our line of sight as observed from the position of the Sun (the first Oort equation) is then v los = {[ ] 2A + 0.5R 0 0 (R R 0 (R R0 )sinl ṽ R cos φ sin(l + ϕ) + ṽ ϕ sin φ cos(l + ϕ) + u 0 cos l v 0 sin l} cos b w 0 sin b, (7) where A = R 0 0 /2 is the Oort constant describing the shearing motion, the dot denotes a derivative with respect to R and (u 0, v 0, w 0 ) are the components of the solar motion relative to the LSR. This approximation is roughly valid up to a distance r of less than 4 kpc. Similar equations can be written for the transversal velocities of stars; however, in this work we restricted ourselves only to lineof-sight velocities. Considering the flat-component Cepheid stars for which the Z coordinates rarely exceed 200 pc, we ignore all derivatives of with respect to Z. To compute the quantities 0, 0, ṽ R, ṽ ϕ, u 0, v 0 and w 0, we define the χ 2 merit function χ 2 = 1 N ( ) vobs,i v 2 mod,i (8) N q v i=1 obs,i and determine the best-fitting parameters by its minimization. In equation (8), N is the number of observations, q is the number of parameters to be derived, v obs, i and v mod, i are the observed and modelled (equation 7) line-of-sight velocities, respectively and v obs, i are the corresponding observational errors. The latter can be assumed to be normally distributed with zero mean and the dispersion v disp. As is seen, the model depends linearly on the set of seven parameters 0, 0, ṽ R, ṽ ϕ, u 0, v 0 and w 0, which can be determined by the standard least-squares fitting for specified m, φ 0 and p (or λ). The model, therefore, depends non-linearly on the set of three unknowns m, φ 0 and p, and we are looking for a minimum of χ 2 for each fixed set of these parameters. To emphasize it again, Crézé & Mennessier (1973), Grivnev (1981), Mishurov & Zenina (1999), Fernández et al. (2001), Lépine et al. (2001), Zabolotskikh Fitting the density-wave theory 2513 et al. (2002), Bajkova & Bobylev (2012) and Siebert et al. (2012) adopted the constant values m = 2 or 4, and the value of χ 2 was therefore minimized with respect to only two parameters: p and φ 0 [see, however, Byl & Ovenden (1978) and Mishurov et al. (1979); in Siebert et al. (2012) the Sun s peculiar velocity was also fixed]. We take the standard values R 0 = 8 kpc, 0 = 26 km s 1 kpc 1 and κ 0 = 36 km s 1 kpc 1. Note here some features of our model. First, we will consider the most probable combination of parameters m, φ 0 and p to be the one for which χ 2 reaches the global minimum. Secondly, χ 2 is periodic in φ 0 with a period of π while ṽ R and ṽ ϕ are periodic with periods of 2π (equations 4 7). From equation (4), we see that because in the denser part of the spiral arm ( π/2 φ +π/2, or 3π/2 φ +5π/2) 1 > 0, the amplitude R 0 /R is negative. For a trailing pattern (i.e. k R < 0) in equation (6), in the principal part of the Galaxy ( ω 2 <κ2 ), out of the two phase values φ 0 and φ 0 + π we should choose for the phase of the spiral structure at the Sun s position the value that would satisfy the following inequality: ṽ ϕ < 0 (9) [this is because in equation (6) k R /(ω 2 κ2 ) is negative]. The azimuthal systematic velocity v ϕ = ṽ ϕ sin φ is in the direction of Galactic rotation at the phase φ = π/2, i.e. at the outer edge of the spiral arm, and v ϕ is in the opposite direction at the phase φ =+π/2, i.e. at the inner edge of the arm. For a trailing pattern with p < ( p > ), the denser part of the spiral arm is moving towards (away from) the Galactic Centre and the rarer part in the interarm region is moving away from (towards) it (Rohlfs 1977; equation 5). 4 GALACTIC CEPHEID DATA Relatively young classical Cepheid stars remain very important in many fields of astrophysics and Galactic research. This is because of the famous period luminosity relation, Cepheids are objects with the most reliable and homogeneous distance scale. Distances to Cepheids now available are quite accurate enough: modern calculations of the period luminosity relation yield uncertainties of less than 5 per cent (Metzger, Caldwell & Schechter 1998). In addition, their line-of-sight velocities are also measured with a good accuracy, v disp 1kms 1. Apart from everything else, they have a relatively small random velocity spread (c s 15 km s 1 ), so that the influence of the density waves on their velocity field is particularly striking (Lin et al. 1969; Griv et al. 2002, 2006). We have applied our method to a new sample of 256 Cepheids from Ngeow s (2012) catalogue. The line-of-sight velocities for these Cepheids were taken from Metzger et al. (1998), while their distances were adopted from Ngeow (2012). These distances are derived based on the Wesenheit function, where the zero point is calibrated using Galactic Cepheids that possess accurate Hubble Space Telescope parallax measurements from Benedict et al. (2007). Good agreement was found when comparing distances derived from the Wesenheit function and other independent distance measurements, including Hipparcos parallaxes, Baade Wesselink-type distances and main-sequence-fitting distances. Further details can be found in Ngeow (2012). We restricted ourselves to the stars within the region of r < 4.1 kpc. Since in Metzger et al. (1998) catalogue the individual uncertainties v obs, i for all stars are not known, we assume v obs, i = v disp, so that in equation (8) the weighting coefficients are assumed to be equal. Fig. 1 shows locations of 233 Cepheids used in the present calculation projected on to the Galactic plane and along the Z coordinate.

4 2514 E. Griv, C.-C. Ngeow and I.-G. Jiang Figure 1. The distribution of Cepheids investigated and projected on to: (a) the Galactic X Y planeand (b) the X Z plane. Cartesian X, Y and Z coordinates are shown and the axis of the Galactic rotation is taken oriented along the Z-axis, with the Sun at (0, 8, 0). In Figs 1(a) and (b), respectively, the solar circle (R 0 = 8 kpc) and the Galactic equatorial Z = 0 plane are indicated by a dashed line. The stars are not uniformly distributed in the plane (Fig. 1a). The natural radial gradient in the surface density of stars is clearly seen: the density is greater on one side of the solar circle than on the other. In the distribution along the Z coordinate (Fig. 1b), a north south asymmetry in the number density of objects is also seen. The Z-odd distribution of the Cepheids in Fig. 1(b) cannot be accounted for by the well-known fact that the Sun lies slightly 25 pc above the plane (Jurić et al. 2008). Widrow et al. (2012) have already presented evidence for such a large-scale 400 pc asymmetry in the number density and bulk velocity of solar neighbourhood main-sequence stars. The problem is far from a solution at the present time. 5 RESULTS OF CALCULATION In the first step the calculation of the parameters of solar motion, Galactic rotation and spiral structure for each pair of values p and φ 0 (and given m), we have solved the linear least-squares problem given by equation (7) and determined the chi-square estimator χ 2. The surface of the estimator χ 2 and cuts through the chi-square space versus phase φ 0 and pitch angle p for the best-fitting m = 1 model are presented in Fig. 2. The global minimum near φ 0 = 247 and p = 6 is clearly seen. The m = 2andm = 3 models show a somewhat larger chi-square values near φ 0 = 242, p = 13 and φ 0 = 239, p = 20 (Table 1). As our calculation also shows, for the chi-square values of the best-fitting multi-armed (m 4) solutions, the pitch angle is larger than 30 ; thus, tan p 1and the Lin Shu approximation of tightly wound perturbations used throughout the theory does fail. Following Siebert et al. (2012), we abandon the hypothesis of multi-armed modes in the following discussion. Additional secondary minima of χ 2 with respect to p are also apparent from our simulation (Fig. 2). Mishurov et al. (1979) were the first to detect the appearance of such local minima. See also Pavlovskaya & Suchkov (1980), Mishurov et al. (1997) and Siebert et al. (2012) for a discussion of the problem. Following Pavlovskaya & Suchkov (1980), we strongly believe that the local minima χ 2 (p) are not a chance artefact or a result of a mathematical structure of the kinematic model given by equation (7). Perhaps, it may be a result of a superposition of Fourier modes of oscillation (equation 1), and several spiral patterns may coexist in the Galaxy. This hypothesis as well as the problem of the errors in the parameters estimated deserve a separate investigation. In the second step, we have used equations (4) (6) to determine the dynamical parameters of the Galactic spiral structure: the angular velocity of the spiral pattern p, the amplitude of the potential 0 and the relative perturbation in stellar density 0 / mean in the solar neighbourhood of the Galaxy. Table 1 presents the quantities calculated in this manner, along with the geometrical parameters of the Galactic spiral structure, Galactic rotation and solar motion. These data suggest the following. (i) p = km s 1 kpc 1 sets the corotation circle (the Galactocentric distance R cor where the angular velocities of the wave pattern and of the material are about the same) inside of the solar circle (R cor = 5 6 kpc, depending on the Galactic curve assumed). The Sun (R 0 = 8 kpc) is located sufficiently far from the corotation circle and, therefore, the Lin Shu theory (equations 4 6) is suitable. (ii) The Galactic velocity field is disturbed slightly by the spiral arms, ṽ R and ṽ ϕ R 0 0 and the perturbation found in the gravitational field is small, 1 / mean 2 0 / 2 0 R The latter indicates that non-linear effects are weak, and many features of first-order solutions of linearized equations of motion given by equations (5) and (6) are therefore valid. (iii) Among several minima of the chi-square estimator, χ 2 for the m = 1 model is the lowest one. The estimator increases slightly with m. In addition, 0 (m = 1) > 0 (m = 2) > 0 (m = 3). These data do not contradict the idea of a superposition of several

5 Fitting the density-wave theory 2515 Figure 2. The surface of the chi-square estimator χ 2 as a function of the Sun s phase φ 0 and the pitch angle p, and cuts through the chi-square space versus φ 0 and p for the best-fitting m = 1 model. Table 1. Data on the geometrical and physical parameters of the Galactic spiral structure, Galactic rotation and solar motion derived from Cepheid kinematics. m χ 2 p φ 0 ṽ R ṽ ϕ p 0 0 / mean 0 u 0 v 0 w 0 ( ) ( ) (kms 1 ) (kms 1 ) (kms 1 kpc 1 ) (km 2 s 2 ) (kms 1 kpc 2 ) (kms 1 ) (kms 1 ) (kms 1 ) Fourier modes of oscillation developing in the Galaxy and the m = 1 mode is the main one. (iv) The phase φ 0 that we found is almost equal to 240.This means that the Sun is situated for all modes in the interarm region between the main inner Sagittarius and outer Perseus spiral-arm segments, closer to the Sagittarius one. (v) The pitch angle of the spiral pattern seems to be negative and relatively small. (vi) The wavelength of the spiral density wave in the solar neighbourhood (the distance between the centres of the Sagittarius and Perseus arms) λ local = 2πR 0 tan p/m 5.3 kpc. (vii) The spiral arms of the Galaxy represent rather small deviations of the surface density from a basic density distribution that is axisymmetric in the mean, 0 / mean < 1. Knowing λ local and φ 0, we can determine from these results the distance from the Sun to the Sagittarius and Perseus arm segments as outlined by the kinematics of Cepheids. The corresponding values for the m = 1 model are R inn 1.7 kpc and R out 3.6 kpc. 6 SUMMARY On the basis of the Lin Shu density-wave conception, we determined both the geometrical and dynamical parameters of the waves in the Galaxy from Cepheid kinematics. We made a determination of kinematical parameters of Cepheids, using a new sample of stars from Ngeow s (2012) catalogue. For the best-fitting m = 1 model, the sample of stars yields p 6, λ local 5.3 kpc, φ and 0 / mean The corotation radius R cor = 5 6 kpc lies inside of the solar circle; thus, p > 0, and the inner and outer Lindblad resonances are located at distances R ILR = 2 3 kpc and R OLR = kpc, respectively (adopting the flat rotation curve of the Galaxy between 4 and 14 kpc; Bovy et al. 2012). These results are consistent with the Lin Shu model of low-m, tightly wound, trailing and rapidly rotating spiral structure of our Galaxy. The spiral arms represent only small deviations of the equilibrium parameters from the basic axisymmetric state of the system. The Sun is located in the interarm region between the two main spiral-arm segments in Sagittarius and Perseus, on the outside of the Sagittarius one. We confirm Yuan s (1969, p. 885) conclusion that the Orion arm... is not a major arm.. In our model, this local arm segment is a secondary spiral arm, which is due to a secondary m = 2or even m = 3 Fourier harmonic of the Milky Way s oscillation. Such a secondary feature is often observed in the external galaxies. The distances from the Sun to the centres (the densest parts) of the inner Sagittarius and outer Perseus arms are R inn 1.7 kpc and R out 3.6 kpc. The data in Table 1 seem to indicate that the pattern speed p is similar for all three values of m. Interestingly, our theoretical model is consistent with this feature of our data, namely we have predicted that p does not depend on m and therefore each Fourier component of a gravity perturbation in an inhomogeneous system will rotate with the same angular velocity (see e.g. Griv et al for a discussion). To reiterate, our calculations do not contradict the idea of a superposition of several Fourier modes of oscillation developing in the plane of the Galaxy. We assume that, of the set of growing spiral modes (equation 1), the fastest mode for which the value of Iω k is a maximum is realized. In this work, the best-fitting model

6 2516 E. Griv, C.-C. Ngeow and I.-G. Jiang is obtained for a dominant one-armed spiral mode. Even preliminary, the calculations thus yield the most reliable value for the number of spiral arms in the solar vicinity of the Galaxy, which is m = 1. This is the main new result. Note that the so-called lopsidedness, or the m = 1 asymmetry, is often seen in the distribution of stars and gas in the outer discs of many galaxies (e.g. van Eymeren et al. 2011). The data obtained here will be submitted to statistical analysis and search the literature for observations that support (or deny!) our results in the following publications of the series. ACKNOWLEDGEMENTS This work was begun while the first author was visiting the Academia Sinica in Taiwan in 2011, thanks to a visitor grant of the Theoretical Institute for Advanced Research in Astrophysics. EG is grateful to Wen-Ping Chen and Ronald Taam for the hospitality they and their colleagues extended to EG both in Jhong-Li and Taipei. CCN thanks the funding from National Science Council of Taiwan under the contract NSC M MY3. The ideas of Michael Gedalin, Edward Liverts and Irena Zlatopolsky concerning the manuscript are appreciated. The work was funded, in part, by the Israel Science Foundation, the Israeli Ministry of Immigrant Absorption in the framework of the programme KAMEA and the National Science Council of Taiwan. REFERENCES Bajkova A. T., Bobylev V. V., 2012, Astron. Lett., 38, 549 Benedict G. F. et al., 2007, AJ, 133, 1810 Binney J., Tremaine S., 2008, Galactic Dynamics, 2nd edn. Princeton Univ. Press, Princeton, NJ Bovy J. et al., 2012, ApJ, 759, 131 Byl J., Ovenden M. W., 1978, ApJ, 225, 496 Crézé M., Mennessier M. O., 1973, A&A, 27, 281 Fernández D., Figueras F., Torra J., 2001, A&A, 372, 833 Fridman A. M., Afanasiev V. L., Dodonov S. N., Khoruzhii O. V., Moiseev A. V., Silchenko O. K., Zasov A. V., 2005, A&A, 430, 67 Griv E., Gedalin M., 2012, MNRAS, 422, 600 Griv E., Gedalin M., Eichler D., Yuan C., 2000, Phys. Rev. Lett., 84, 4280 Griv E., Gedalin M., Yuan C., 2002, A&A, 383, 338 Griv E., Gedalin M., Yuan C., 2006, Adv. Space Res., 38, 47 Grivnev E. M., 1981, Sov. Astron. Lett., 7, 303 Jurić M. et al., 2008, ApJ, 673, 864 Lépine J. R. D., Mishurov Yu. N., Dedikov S. Yu., 2001, ApJ, 546, 234 Lin C. C., Lau Y. Y., 1979, SIAM Stud. Appl. Math., 60, 97 Lin C. C., Shu F. H., 1964, ApJ, 140, 646 Lin C. C., Yuan C., Shu F. H., 1969, ApJ, 155, 721 (erratum, 156, 797) Lou Y.-Q., Yuan C., Fan Z., 2001, ApJ, 552, 189 Metzger M. R., Caldwell J. A. R., Schechter P. L., 1998, AJ, 115, 635 Mihalas D., Binney J., 1981, Galactic Astronomy, 2nd edn. Freeman & Co., San Francisco Mishurov Yu. N., Zenina I. A., 1999, Astron. Rep., 43, 487 Mishurov Yu. N., Pavlovskaya E. D., Suchkov A. A., 1979, SvA, 23, 147 Mishurov Yu. N., Zenina I. A., Dambis A. K., Melnik A. M., Rastorguev A. S., 1997, A&A, 323, 775 Montenegro L. E., Yuan C., Elmegreen B. G., 1999, ApJ, 520, 592 Morozov A. G., Torgashin Yu. M., Fridman A. M., 1985, Sov. Astron. Lett., 11, 94 Ngeow C.-C., 2012, ApJ, 747, 50 Pavlovskaya E. D., Suchkov A. A., 1980, SvA, 24, 164 Rohlfs K., 1977, Lecture Notes in Physics, Vol. 69, Lectures on Density Wave Theory. Springer, Berlin Seigar M. S., James P. A., 1998, MNRAS, 299, 685 Shu F. H., 1970, ApJ, 160, 99 Siebert A. et al., 2012, MNRAS, 425, 2335 van Eymeren J., Jütte E., Jog C. J., Stein Y., Dettmar R.-J., 2011, A&A, 530, 30 Widrow L. M., Gardner S., Yanny B., Dodelson S., Chen H.-Y., 2012, ApJ, 750, L4 Yuan C., 1969, ApJ, 158, 871 Zabolotskikh M. V., Rastorguev A. S., Dambis A. K., 2002, Astron. Lett., 28, 454 This paper has been typeset from a TEX/LATEX file prepared by the author.