Periodic Pattern in the Residual-Velocity Field of OB Associations
aa r X i v : . [ a s t r o - ph ] S e p Periodic Pattern in the Residual-Velocity Field of OBAssociations
A.M.Mel’nik ∗ , A.K.Dambis, and A.S.Rastorguev Sternberg Astronomical Institute, Moscow, Russia
Astronomy Letters, 2001, Vol. 27, pp. 521-533.
Abstract – An analysis of the residual-velocity field of OB associ-ations within 3 kpc of the Sun has revealed periodic variations in theradial residual velocities along the Galactic radius vector with a typicalscale length of λ = 2 . ± . f R = 7 ± − . The fact that the radial residual velocities of almost all OB-associations in rich stellar-gas complexes are directed toward the Galacticcenter suggests that the solar neighborhood under consideration is withinthe corotation radius. The azimuthal-velocity field exhibits a distinctperiodic pattern in the 0 < l < ◦ region, where the mean azimuthal-velocity amplitude is f θ = 6 ± − . There is no periodic pattern ofthe azimuthal-velocity field in the 180 < l < ◦ region. The locationsof the Cygnus arm, as well as the Perseus arm, inferred from an analysisof the radial- and azimuthal-velocity fields coincide. The periodic pat-terns of the residual-velocity fields of Cepheids and OB associations sharemany common features. Key words: star clusters and associations, stellardynamics, kinematics; Galaxy (Milky Way), spiral pattern.
1. INTRODUCTIONThe location of the spiral arms in our Galaxy and their influence on the kinemat-ics of gas and young stars is undoubtedly of great importance for understandingthe large-scale hydrodynamic processes and the evolution of stellar groupingsand that of the Galaxy as a whole. Only in our own Galaxy it is possible to de-rive the space velocity field of young stars and analyze the radial and azimuthalvelocity components simultaneously. However, even now the location of spiralarms in the Galaxy remains a subject of discussion. There are two main ap-proaches to the problem. The first one consists in identifying spiral arms fromregions of enhanced density of young objects in the galactic disk (Morgan etal. ∗ E-mail address for contacts: [email protected] etal. et al. etal. λ of pe-riodic velocity variations along the galactic radius-vector ignoring the nonzerospiral-arm pitch angle and assuming spiral-arm fragments to have the shapes ofcircular segments. The wavelength in question is to a first approximation equalto the interarm distance. We first applied this technique to the Cepheid velocityfield and found λ = 1 . ± . et al. m . We can then determine the mean pitch angle i of spiral arms interms of a model of regular spiral pattern using the simple relation tan i = λm πR of the density wave theory (Lin et al. λ = 1 . i = 5 ◦ .In the density-wave theory there is a fixed phase shift of π/ et al. < l < ◦ region.Unfortunately, our method allows us to determine the arm location onlyup to λ/
2, i.e., up to shifting arms into the interarm space. Moreover, it isimpossible to choose between the two solutions for the location of the spiralpattern based on kinematical data alone. For the final choice, we invoke the2dditional information about the location of starburst regions with respect tothe periodic pattern found in the velocity field studied. This information allowsus not only to choose the right spiral-pattern solution, but also to determinethe position of the region considered (within 3 kpc from the Sun) with respectto the corotation radius.It is impossible to analyze the spiral pattern of our Galaxy without theknowledge of reliable distances to the objects studied. The use of OB asso-ciations instead of individual stars allows, due to averaging, a more reliabledistance scale to be constructed and more reliable velocities to be derived. OBassociations are sparse groupings of young stars (Ambartsumyan 1949). A com-parison of virial mass estimates of OB-associations with the masses estimated bymodeling their stellar content suggests that these groupings are gravitationallyunbound. Presently, there are several partitions of galactic OB-stars into asso-ciations, those of Blaha and Humphreys (1989), Garmany and Stencel (1992),and Mel’nik and Efremov (1995). All these partitions are based on the cat-alog of luminous stars of Blaha and Humphreys (1980). However, Garmanyand Stencel (1992) identified their OB-associations only in the 50 < l < ◦ region. Mel’nik and Efremov (1995) used cluster analysis technique to identifythe densest and most compact groups, the cores of OB-associations. However,these groupings contain twice as few stars than the associations of Blaha andHumphreys and, moreover, in dense regions, kinematical data are available fora smaller fraction of stars. We therefore consider the partition of OB-stars intoassociations suggested by Blaha and Humphreys (1989) to be more suitable forkinematical analyses. The sky-plane sizes of most of the OB-associations ofBlaha and Humpreys (1989) do not exceed 300 pc (except Cep OB1 and NGC2430), and the use of these objects for identifying periodic patterns with typicalscale lengths greater than 1 kpc is a quite correct procedure. The inclusion of alist OB-stars in the HIPPARCOS (1997) program allowed their space motionsto be analyzed for the first time (de Zeeuw et al. r BH of these associations;2. The catalog of classical Cepheids (Berdnikov 1987; Berdnikov et al. R = 7 . ± . et al. et al. et al. r = 0 . r BH , which is consis-tent with the short distance scale for Cepheids (Sitnik and Mel’nik 1996;Dambis et al. . The velocity of each OB association is based, onthe average, on 12 line-of-sight velocities and 11 proper motions of individualstars. We excluded the distant association Ara OB1B ( r = 2 . V z velocity component, which exceeds 20 km s − .3. AN APPROACH TO THE SOLUTIONIn the case of tightly wound spiral arms, the velocity field must exhibit variationsof the value and direction of residual stellar velocities (i.e., velocities correctedfor the Solar apex motion and galactic rotation) on Galactocentric distance.We now write the expressions for the perturbation of radial V R and azimuthal V θ components of residual velocities in the form of periodic functions of thelogarithm of Galactocentric distance R : V R = f R sin( 2 πR λ ln( RR ) + ϕ R ) , (1) V θ = f θ sin( 2 πR λ ln( RR ) + ϕ θ ) , (2)where f R and f θ are the amplitudes of variations of velocity components V R and V θ . Parameter λ (in kpc) characterizes the wavelength of the periodic velocityvariations along the galactic radius-vector. The angels ϕ R and ϕ θ determinethe phases of oscillations at the solar Galactocentric distance. Assuming thatgalactic arms have the shape of logarithmic spirals, we adopted a logarithmicdependence of the wave phase on Galactocentric distance, which degeneratesinto a linear function R ln ( R/R ) ≈ R − R if ( R − R ) /R is small.To demonstrate that the periodic pattern in the field of residual velocities isindependent of the adopted model of circular rotation, we found the parameters the catalog is available at http://lnfm1.sai.msu.ru/ ∼ anna/page3.html
4f the periodic pattern jointly with those of differential circular galactic rotationand the components of solar velocity. We inferred all these quantities from ajoint solution of Bottlinger equations (Kulikovski˘i 1985) for line-of-sight veloci-ties V r and velocity components V l ( V l = 4 .
738 [km s − kpc − (arcsec yr − ) − ] µ l r , where µ l is the proper-motion component along the galactic longitude) withallowance for perturbations induced by the density wave: V r = − ( − u ∗ cos l cos b + v sin l cos b + w sin b ) ++ R Ω ′ ( R − R ) sin l cos b ++0 . ′′ ( R − R ) sin l cos b −− f R sin( 2 πR λ ln( RR ) + ϕ R ) cos( l + θ ) cos b ++ f θ sin( 2 πR λ ln( RR ) + ϕ θ ) sin( l + θ ) cos b ; (3) V l = − ( u ∗ sin l + v cos b ) ++Ω ′ ( R − R )( R cos l − r cos b ) ++0 . ′′ ( R − R ) ( R cos l − r cos b ) − Ω r cos b ++ f R sin( 2 πR λ ln( RR ) + ϕ R ) sin( l + θ ) ++ f θ sin( 2 πR λ ln( RR ) + ϕ θ ) cos( l + θ ); (4)Here, θ is the azimuthal Galactocentric angle between the directions toward thestar and the Sun; Ω is the angular velocity of galactic rotation at the solarGalactocentric distance; Ω ′ and Ω ′′ are the first and second derivatives withrespect to Galactocentric distance taken at a distance of R ; u ∗ , v , and w are the solar velocity components relative to the centroid of OB-associations inthe directions of X , Y , and Z axes, respectively. The X -axis is directed awayfrom the Galactic center, the Y -axis is in the direction of galactic rotation,and the Z -axis points toward the North Galactic Pole. Velocity components u ∗ and v include the solar-velocity perturbation due to the spiral density wave.(In galactic astronomy the X -axis is traditionally directed toward the Galacticcenter and one should therefore compare − u ∗ and not u ∗ with the standardsolar apex). We adopted w = 7 km s − for the solar velocity component alongthe Z -coordinate (Kulikovski˘i, 1985; Rastorguev et al. ϕ R and ϕ θ , we rewrite the formulas for perturbations of velocity components V R and V θ as follows: V R = A R sin( 2 πR λ ln( RR )) + B R cos( 2 πR λ ln( RR )) (5) V θ = A θ sin( 2 πR λ ln( RR )) + B θ cos( 2 πR λ ln( RR )) (6)5he parameters f R , f θ , ϕ R and ϕ θ can then be found from the relations: f R = A R + B R ; f θ = A θ + B θ ; (7)tan( ϕ R ) = B R /A R ; tan( ϕ θ ) = B θ /A θ . (8)We computed the weight factors p in the equations for V r and V l as follows: p V r = ( σ + ε V r ) − / , (9) p V l = ( σ + (4738 ε µl r ) ) − / , (10)Here σ is the dispersion of residual velocities of OB associations with respectto the adopted model of motion (without allowance for the triaxial shape of thevelocity distribution); ε V r and ε µl are the standard errors of measured stellarline-of-sight velocities and proper motions, respectively. We determined thedispersion using iterations technique, which yielded σ = 6 . − .We then applied the least-squares technique to find a joint solution of thesystem of equations (3) and (4), which are linear with respect to the parameters u ∗ , v , Ω , Ω ′ , Ω ′′ , A R , B R , A θ , and B θ , with weight factors (9) and (10)and fixed λ (see p. 499 in the book by Press et al. (1987)). We estimate thewavelength λ by minimizing function χ ( λ ), which is equal to the sum of squaresof the normalized velocity residuals.4. RESULTS Figure 1 shows χ as a function of λ , based on a joint solution of the system ofequations (3) and (4) for line-of-sight and tangential velocities of OB associationslocated within 3 kpc from the Sun. χ takes its minimum value at λ = 2 . f R = 6 . ± . f θ = 1 . ± . − , respectively. Table 1 givesthe inferred values of all determined parameters: u ∗ , v , Ω ′ , Ω ′′ , Ω , λ , A R , B R , A θ , and B θ , as well as f R , f θ , ϕ R and ϕ θ computed using formulas (7) and (8).The table also gives the standard errors of the above parameters, the number N of equations used, and the rms residual σ .We then performed numerical simulations in order to estimate the standarderror of the resulting λ . To this end, we fixed the actual galactic coordinates ofOB-associations and simulated normally distributed random errors in the helio-centric distances of OB associations with standard deviations equal to 10% ofthe true distance, and then used formulas (3) and (4) to compute a theoreticalvelocity field with allowance for the perturbations due to the density wave (wetook all parameter values from Table 1). We then added to the theoretical ve-locities simulated normally distributed random errors with a standard deviation6 able 1. Parameters of the circular rotation law, periodic pat-tern, and solar-motion components inferred from an analysis ofthe line-of-sight velocities and proper motions of OB associations N u ∗ v Ω ′ Ω ′′ Ω λ km s − km s − km s − kpc − km s − kpc − km s − kpc − kpc132 -7.5 11.2 -5.0 1.5 30.2 2.0 ± . ± . ± . ± . ± . ± . A R B R A θ B θ f R f θ ϕ R ϕ θ σ km s − km s − km s − km s − km s − km s − deg deg km s − ◦ -33 ◦ ± . ± . ± . ± . ± . ± . ± ◦ ± ◦ of σ , which includes the contribution of observational errors ( σ = 1 /p V r and σ = 1 /p V l , see formulas (9) and (10)). We determined the wavelength λ foreach simulated velocity field and found the inferred λ values to be unbiased andto have a standard error of 0.2 kpc.We also explored the possibility of periodic patterns emerging accidentallyin the velocity field of OB associations due to chance deviations of individualvelocities from the circular rotation law. To this end, we simulated randomerrors in the velocities and heliocentric distances of OB associations and deter-mined λ , f R , and f θ for each simulated field. Numerical simulations showedthat 30% of all λ values fall within the wavelength interval 1 < λ < λ interval that random fluctuations of the fieldof circular velocities can be attributed to density-wave effects. The mean ampli-tudes f R and f θ are equal to 3 km s − , i.e., about twice the standard errors ofthe corresponding parameters inferred for the actual sample of OB associations(Table 1). However, the probability of a periodic pattern with an amplitude of f R ≥ . − and λ in the 1 < λ < P < f R = 6 . − emerging as a result of chance fluc-tuations in the velocities and heliocentric distances of OB associations can berejected at a confidence level of 1 − P > f θ ≥ . − and wavelengths λ in the 1 < λ < f θ = 1 . − can well be interpreted in terms of random fluctuations. A gravitational potential perturbation that propagates in a rotating disk at asupersonic speed produces a shock front, which affects the kinematics of gas andyoung stars born in this gas (Roberts 1969). The ages of OB associations donot exceed 5 × yr and, therefore, the motions of OB-associations must bedetermined mainly by the velocities of their parent molecular clouds (Sitnik etal. et al. u ∗ , v , Ω ′ , Ω ′′ , Ω , adopted fromTable 1. Also shown are circular arcs corresponding to the maximum meanradial velocity V R toward the Galactic center as defined by formula (1) and f R and ϕ R adopted from Table 1. Table 2 gives the following parameters for 59associations with known space velocities: radial ( V R ) and azimuthal ( V θ ) com-ponents of residual velocities; components V z of residual velocities along the z -coordinate; Galactocentric R and heliocentric r distances, and galactic coor-dinates l and b . To characterize the reliability of velocities and distances listedin Table 2, we also give the numbers n r and n l of association stars with knownline-of-sight velocities and proper motions, respectively, and also the number N of members of the OB associations used to determine distance r .Let us assume that the region studied is inside the corotation radius. Itthen follows, in view of the small value of the pitch angle, that the arcs shownin Fig. 2 should coincide with the shock front and must be located near theminima of gravitational potential minimum (Roberts 1969). Given a partitionof young galactic objects into stellar-gas complexes (Efremov and Sitnik 1988),one can identify the star-forming regions through which the arms drawn in Fig.2 pass. The arm located closer to the Galactic center passes in quadrant I nearthe OB associations of the Cygnus stellar-gas complex (Cyg OB3, OB1, OB8,and OB9) and in quadrant IV, through the OB associations and young clustersof the stellar-gas complex in the constellations of Carina, Crux, and Centaurus(Car OB1, OB2, Cru OB1, Cen OB1, Coll 228, Tr 16, Hogg 16, NGC 3766, andNGC 5606). Hereafter we refer to this arc as the Cygnus-Carina arm. Anotherarm, which is farther from the Galactic center, passes in quadrant II near the OBassociations of the stellar-gas complexes located in the constellations of Perseus,Casiopeia, and Cepheus (Per OB1, NGC 457, Cas OB8, OB7, OB6, OB5, OB4,OB2, OB1, and Cep OB1). In quadrant III neither stellar-gas complexes noreven simply rich OB associations can be found to lie along the extension of thisarc, which we refer to as the Perseus arm (Fig. 2). Note that Perseus-Cassiopeiaand Carina-Centaurus stellar-gas complexes are the richest ones in the sense ofthe number of luminous stars their associations contain (see, e.g., Table 2).If we assume that the solar neighborhood considered is located outside thecorotation radius, the shock front and the minimum of potential should coin-cide with the maximum velocity of streaming motions directed away from the8alactic center and maximum velocity of azimuthal streaming motions in thedirection of galactic rotation. The arms should then be shifted by λ/ However, things are not all that straightforward. An analysis of the data inTable 2 showed that about 30% of rich OB-associations (containing more than30 members with known photometric parameters,
N > V R directed away from the Galactic center.This is not a surprise, because star formation can also proceed in the interarmspace (Elmegreen and Wang 1987). In the solar neighborhood (Fig. 2) tworegions can be identified where most of OB associations have radial velocities V R directed away from the Galactic center. These are the associations of the Localsystem located in quadrants II and III (Vela OB2, Mon OB1, Coll 121, Ori OB1,and Per OB2) and the stellar-gas complex projected onto the constellations ofSagittarius, Scutim, and Serpens (Sgr OB1, OB7, OB4, Ser OB1, OB2, Sct OB2,and OB3) (Efremov and Sitnik 1988). These regions are located in the interarmspace of the pattern shown in Fig. 2. It is the alternation of star-formingregions with positive and negative radial residual velocities V R that determinesthe periodic pattern of the field of radial velocities of OB-associations.Within 3 kpc from the Sun a total of five star-forming regions can be iden-tified where almost all associations have the same direction of the radial com-ponent V R of residual velocity. The contours of these regions are shown in Fig.2. Table 3 gives for each such region its mean Galactocentric distance R , meanresidual velocities of associations V R and V θ ; the interval of coordinates l and r , and the names of OB-associations with known space velocities it contains.Table 3 shows a well-defined alternation of the directions of the mean radialvelocity V R of OB associations as a function of increasing Galactocentric dis-tance R . The periodic pattern is especially conspicuous in Fig. 3a, which showsthe variation of the radial component of residual velocity of OB assrciationsalong the Galactocentric distance. Radial velocities of OB associations in theCarina-Centaurus ( R = 6 . R = 6 . R = 8 .
4) complexes are directed mainly toward the Galactic center, whereasthose in the Sagittarius-Scutum complex ( R = 5 . R = 7 . able 3. Average residual velocities of OB-associations in the star-forming regionsRegion R , kpc V R , V θ , l , deg r , kpc Associationskm s − km s − Sagittarius 5.6 +11 ± − ± − ± ± − ± − ± ± ± − ± − ± inferred. l < ◦ and l > ◦ Regions
To study the specific features of the velocity field of OB associations, we ana-lyzed residual velocities of these objects as a function of Galactocentric distanceseparately for the two regions l < ◦ and l > ◦ (Fig. 4 and 5, respectively).Figure 4b shows the azimuthal velocity field of OB associations to exhibit a well-defined periodic pattern in the region l < ◦ , whereas no such pattern canbe seen in the field of velocity components V θ of the entire sample of OB as-sociations (Fig. 3b and Table 1). The mean amplitude of azimuthal velocityvariations in the region considered is as high as f θ = 5 . ± . − , i.e.,almost triple the value of f θ = 1 . ± . − inferred from the entire sam-ple of OB associations. One can see two well-defined minima at Galactocentricdistances R = 7 . R = 8 . R = 6 . − . R = 8 . − . able 4. Parameters of the periodic pattern in the ve-locity field of OB-associations located in different regionsRegion N λ , kpc f R f θ ϕ R ϕ θ σ km s − km s − deg deg km s − < l < ◦
73 1.7 6.7 5.1 52 -15 6.2 ± . ± . ± . ± ± < l < ◦
56 1.4 6.9 6.1 -7 -26 6.0 ± . ± . ± . ± ± < l < ◦
59 2.4 8.1 4.4 7 251 6.2 ± . ± . ± . ± ± In the region l > ◦ the periodic pattern is represented by a single mini-mum and two maxima in the distribution of radial velocities V R (Fig. 5a). Theminimum at the Galactocentric distance R = 6 . − . R = 7 . l > ◦ (Fig. 5b).That is why merging the association samples from the two regions ( l < ◦ and l > ◦ ) washes out the periodic pattern (Fig. 3b), although the latter isclearly outlined by the associations of quadrants I and II (Fig. 4b).For a quantitative analysis, we inferred the parameters of the periodic pat-tern in the velocity field of OB associations in two regions: 0 < l < ◦ and180 < l < ◦ , by solving the system of equations (3) and (4) with weightfactors (9) and (10) and the parameters of circular rotation and solar-motioncomponents adopted from Table 1. Table 4 gives the following parameters ofthe periodic pattern inferred for the two regions: λ , f R , f θ , ϕ R and ϕ θ , theirstandard errors, the number N of equations used, and the mean residual σ .It is evident from Table 4 that in the region 0 < l < ◦ radial andazimuthal residual velocities have similar variation amplitudes equal to f R =6 . ± . f θ = 5 . ± . − , respectively. The phases of the variations ofthe radial and azimuthal velocity components ( ϕ R = 52 ±
15 and ϕ θ = − ± < l < ◦ ) break thealmost ideal periodic pattern outlined by the objects located in the Cygnus andPerseus arms and in the adjoining interarm space. Excluding from our samplethe OB-associations located in the region 0 < l < ◦ changes phase ϕ R signif-icantly. The parameters of the periodic pattern inferred in the 30 < l < ◦ sector for the objects located in the Cygnus and Perseus-Casseopeia arms andin the adjoining interarm space are also listed in Table 4. It is evident fromthis table that in the sector considered the phases of radial and azimuthal ve-locity oscillations agree with each other within the errors ( ϕ R = − ±
15 and11 θ = − ±
18, respectively). The reliably determined wavelength λ = 1 . ± . < l < ◦ is equal to the distance between the Cygnusand Perseus arms, or rather to that between the minima in the distributions ofboth radial and azimuthal residual velocities (Figs. 4a, 4b).In the region 180 < l < ◦ the mean amplitude of radial velocity oscilla-tions is equal to f R = 8 . ± . − . Here the wavelength is determined asthe distance between the maxima in the distribution of radial velocities, i.e., λ proves to be equal to the distance between the interarm-space objects (Fig. 5a).The wavelength inferred for this region, λ = 2 . ± . < l < ◦ ,with amplitudes equal to f R = 6 . f θ = 6 . − , respectively, are due toaccidental errors in the velocities and distances, to be rejected at a confidencelevel of 1 − P ≥ < l < ◦ is that of radial velocities (1 − P ≥ P = 15%) be interpretedin terms of random fluctuations.Figure 6a illustrates the specific features of the periodic pattern found inthis work. It also shows the field of residual velocities of OB-associations andthe circular arcs corresponding to the minima of residual radial ( V R ) (solid line)and azimuthal ( V θ ) (dashed line) velocities based on parameters λ , f R , f θ , ϕ R and ϕ θ , for regions 180 < l < ◦ and 30 < l < ◦ (Table 4). The arcs mustbe located in the vicinity of the lines of minimum gravitational potential. In theregion 30 < l < ◦ these lines determine the loci of the Cygnus and Perseus-Cassiopeia arms and in the region 180 < l < ◦ , that of the Carina-Centaurusarm. Numerical simulations showed the inferred Galactocentric distances ofarms and, correspondingly, the radii of arcs in Fig. 6a, to have standard errorsof 0 . − . < l < ◦ and the absence of such pattern in the region 180 < l < ◦ .Second, the agreement of Galactocentric distances of the Cygnus and Perseus-Cassiopeia arms as inferred from analyses of the fields of radial and azimuthalvelocities in the region 30 < l < ◦ . Third, a 0.3 kpc shift of the kinematicalpositions of the Carina arm ( R = 6 . ± . R = 6 . ± . et al. λ = 1 . ± able 5. Parameters of the periodic pattern in thevelocity field of Cepheids located in different regionsRegion N λ , kpc f R f θ ϕ R ϕ θ σ km s − km s − deg deg km s − < l < ◦
217 1.8 6.7 5.1 85 13 10.2 ± . ± . ± . ± ± < l < ◦
165 1.8 6.5 6.5 82 9 9.8 ± . ± . ± . ± ± < l < ◦
208 1.8 5.8 2.0 78 154 10.4 ± . ± . ± . ± ± . f R = 6 . ± . f θ = 2 . ± . − . It would be interesting to see whether the periodic pattern of the Cepheidvelocity field exhibits the same specific features as that of OB-associations.To answer this question, we inferred the parameters of the periodic patternof the Cepheid velocity field in three regions 0 < l < ◦ , 30 < l < ◦ , and180 < l < ◦ in the same way as we did it for OB-associations (Table 5).Note that the exclusion of the region 0 < l < ◦ has no effect on the inferredparameters of the periodic pattern in the Cepheid velocity field. We used theparameters of circular motion and solar-motion components that we inferredfrom the analysis of the entire sample of Cepheid located within 3 kpc from theSun [see Table in Mel’nik et al. (1999)]: u ∗ = − . ± . − ; v = 11 . ± . − ; Ω ′ = − . ± . − kpc − ; Ω ′′ = 1 . ± . − kpc − , andΩ = 29 ± − kpc − . These parameters agree within the errors with thecorresponding parameters of the velocity field of OB-associations (Table 1).It is evident from Table 5 that in the region 30 < l < ◦ both radialand azimuthal velocity fields contain a periodic component. The amplitudesof velocity variations are equal to f R = f θ = 6 . ± . − . Numericalsimulations show that the hypothesis about the accidental nature of the periodicvariations found in the fields of radial and azimuthal residual velocities in theregion 30 < l < ◦ can be rejected at a confidence level of 1 − P > < l < ◦ with an amplitude of f R = 5 . ± . − cannot be also due to randomfluctuations (1 − P > f θ = 2 . ± . − can well ( P = 25%) be due to randomfluctuations.Figure 6b shows the field of Cepheid residual velocities and, based on λ , f R , f θ , ϕ R , and ϕ θ from Table 5, the circular arcs corresponding to the minima ofmean radial ( V R ) (solid line) and azimuthal ( V θ ) (dashed line) velocities. Thesearcs determine the kinematical positions of the Cygnus, Perseus-Cassiopeia, andCarina-Centaurus arms. The radii of arcs in Fig. 6b are determined with anaccuracy of 0 . − . R = 6 . − . R = 8 . − . f θ ≈ ± − , whereas theazimuthal velocities of both populations show no periodic pattern in quadrantsIII and IV. However, the velocity field of Cepheids somewhat differs from thatOB associations. Thus Cepheids do not show the 0.3 kpc shift between thepositions of the Cygnus and Carina arms as inferred from radial velocities ofOB associations. The Galactocentric distances of the Cygnus arm as inferredfrom analyses of radial velocities of Cepheids ( R = 6 . ± . R = 6 . ± . V R along the galactic radius-vector with a typicalscale length of λ = 2 . ± . f R = 7 ± − .We revealed five kinematically distinct star-forming regions where almost allOB-associations have the same direction of radial residual velocity V R . The ra-dial velocities of OB associations in the Carina-Centaurus, Cygnus, and Perseus-Cassiopeia regions are directed mainly toward the Galactic center, whereas thoseof the Sagittarius-Scutum complex and of a part of the Local system, are di-rected away from the Galactic center. It is the alternation of star-forming regionswith positive and negative radial velocities V R that determines the periodic pat-tern of the radial velocity field of OB associations.The fact that rich Carina-Centaurus and Perseus-Casiopeia stellar-gas com-plexes lie in the vicinity of the minima in the distribution of radial velocities ofOB-associations indicates that the region considered is located inside the coro-tation radius. The enhanced density of high-luminosity stars in these regionscannot be due to observational selection, because the Carina-Centaurus andPerseus-Cassiopeia complexes are the most distant ones among those consideredin this paper (Fig. 2 and Table 3). Furthermore, these regions cover sky areasextending for several tens of degrees and at a heliocentric distance of 2 kpc themean extinction averaged over such large sectors depends little on the directionwithin the galactic plane. There is no doubt, the enhanced density of high-luminosity stars in the Carina-Centaurus and Perseus-Cassiopeia complexes isreal and not due to the extremely low extinction along the corresponding linesof sight.The fact that the Perseus-Cassiopeia complex ( R = 8 . | R c − R | > . p <
25 km s − kpc − .Our conclusion about the corotation radius R c being located in the outer14art of the Galaxy beyond the Perseus-Cassiopeia arm is in conflict with theconclusions of Mishurov and Zenina (1999a, 1999b) who argue that the Sunis located near the corotation radius | R − R c | < f R = 2 ± − ) andlarge value of azimuthal ( f θ = 8 ± − ) velocity perturbation amplitudesthey inferred (Mishurov and Zenina 1999b). Our analysis yielded a reversedamplitude proportion with f R = 7 ± f θ = 2 ± − both for Cepheid(Mel’nik et al. λ = 2 kpc and is consistent with our results.As for the periodic pattern in the field of azimuthal velocities of OB-associations,it is observed in the quadrants I and II and is absent in the quadrants III andIV. The mean amplitude of azimuthal velocity variations in the region of theCygnus and Perseus arms (30 < l < ◦ ) is as high as f θ = 6 ± − ,i.e., triple that of the entire sample of OB associations. Another specific fea-ture is the striking agreement between the positions of the Cygnus and Perseusarms as inferred from separate analyses of radial and azimuthal velocity fieldsof OB associations. This feature can be explained by shocks that develop whena density wave propagates through gas at a supersonic velocity.The periodic patterns in the residual velocity fields of Cepheids and OB-associations have very much in common: similar scale length of radial velocityvariations along the galactic radius-vector, λ = 2 ± . R = 6 . − . R = 8 . − . λ = 2kpc.The wavelength value that we inferred, λ = 2 kpc, seems to coincide withthat of the most unstable mode of galactic disk oscillations at the given Galacto-centric distance. The very existence of the spiral pattern suggests that the galac-tic disk is marginally unstable at the solar Galactocentric distance. Presently,the gaseous component of the galactic disk is considered to be instrumentalin maintaining such a marginal instability (Jog and Solomon 1984; Bertin andRomeo 1988; Bertin et al. − λ = 2kpc suggests that both components play important part in the dynamics of ourGalaxy. 15CKNOWLEDGMENTSWe are grateful to Yu.N.Efremov, A.V.Zasov, and A.V.Khoperskov for the dis-cussions, as well as for the useful remarks and advice.REFERENCESAmbartsumian, V.A. Astron. Zh. , 1949, v. 26, p. 3 (rus).Barbier-Brossat, M., Figon, P.
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The function χ ( λ ) for the joint solution of the system of equations (3) and (4)for the entire sample of OB associations YX -1 Figure 2:
The field of space velocities of OB-associations projected onto the galactic plane.The X -axis is directed away from the Galactic center, and the Y -axis is in the direction ofgalactic rotation. The Sun is at the origin. The circular arcs correspond to the maximumradial component V R of residual velocity toward the galactic center. One can see five star-forming regions where almost all OB-associations have the same direction of radial velocity V R .
10 0 10 20-20 V , k m s - R (3a) 0 < l < 360 o -10 0 10 20-20 V , k m s − θ (3b)-10 0 10 20-20 V , k m s - R (4a) 0 < l < 180 o -10 0 10 20-20 V , k m s − θ (4b)-10 0 10 20-20 V , k m s - R (5a) 180 < l < 360 o V , k m s − θ (5b) Figure 3: 4, 5.
Residual velocities V R and V θ of OB-associations as a function of Galacto-centric distance R . The whole sample and the regions 0 < l < ◦ and 180 < l < ◦ areconcidered. X 1 kpc30 km s -1 (a) YX 1 kpc30 km s -1 (b) Figure 6:
The field of residual velocities of (a) OB-associations and (b) Cepheids. Thecircular arcs correspond to the positions of the arm fragments as inferred from analyses ofradial (solid line) and azimuthal (dashed line) residual velocities in the regions 30 < l < ◦ and 180 < l < ◦ ..