Classical Cepheids in the Galactic outer ring R1R2'
A. M. Mel'nik, P. Rautiainen, L. N. Berdnikov, A. K. Dambis, A. S. Rastorguev
aa r X i v : . [ a s t r o - ph . GA ] N ov Astronomische Nachrichten, 26 August 2018
Classical Cepheids in the Galactic outer ring R R ′ A. M. Mel’nik ⋆ , P. Rautiainen , L. N. Berdnikov , , A. K. Dambis & A. S. Rastorguev Sternberg Astronomical Institute, Lomonosov Moscow State University, 13 Universitetskij Prosp., Moscow 119991,Russia Department of Physics/Astronomy Division, University of Oulu, P.O. Box 3000, FIN-90014 Oulun yliopisto, Finland Astronomy and Astrophysics Research division, Entoto Observatory and Research Center, P.O.Box 8412, Addis Ababa,EthiopiaReceived 2014, accepted 2014Published online 2014
Key words
Galaxy: structure – Galaxy: kinematics and dynamics – (stars: variables:) Cepheids – galaxies: spiralThe kinematics and distribution of classical Cepheids within ∼ kpc from the Sun suggest the existence of the outerring R R ′ in the Galaxy. The optimum value of the solar position angle with respect to the major axis of the bar, θ b ,providing the best agreement between the distribution of Cepheids and model particles is θ b = 37 ± ◦ . The kinematicalfeatures obtained for Cepheids with negative Galactocentric radial velocity V R are consistent with the solar location nearthe descending segment of the outer ring R . The sharp rise of extinction toward of the Galactic center can be explainedby the presence of the outer ring R near the Sun. Copyright line will be provided by the publisher
Classical Cepheids – F–K-type supergiants with ages from20 to 200 Myr (Efremov 2003, Bono et al. 2005) – are goodtracers of the Galactic spiral structure and regions of highgas density. Due to the period-luminosity relation the dis-tances to classical Cepheids can be determined with an ac-curacy of ∼ % (Berdnikov, Dambis & Vozyakova 2000;Sandage & Tammann 2006).Most of the recent studies of the spiral structure of theGalaxy (see e.g. the review by Vall´ee (2013) and referencestherein) have typically suggested a 2- or 4-armed spiral pat-tern with a pitch angle of nearly i = 6 ◦ or 12 ◦ , respectively.In our previous studies (Mel’nik & Rautiainen 2009; Rauti-ainen & Mel’nik 2010) we proposed an alternative to thesepurely spiral models – a two-component outer ring R R ′ .The main advantage of the 4-armed spiral model is thatit can explain the distribution of HII regions in the Galacticdisk (Georgelin & Georgelin 1976; Russeil 2003; and otherpapers) and the existence of so-called tangential directionsrelated to the maxima in the thermal radio continuum, HIand CO emission which are associated with the tangents tothe spiral arms (Englmaier & Gerhard 1999; Vall´ee 2008).The main shortcoming of this approach is the absenceof a dynamical mechanism to maintain the global spiralpattern for a long time period (more than a few disk ro-tations) (Toomre 1977; Athanassoula 1984). Generally, thebar could support the spiral pattern, but in this case the spi-ral pattern must rotate with the angular velocity of the bar( Ω s = Ω b ) (Englmaier & Gerhard 1999). The estimates of ⋆ Corresponding author. e-mail: [email protected] the corotation radius (CR) of the Galactic bar do not ex-ceed 5 kpc, which gives the lower limit for its angular ve-locity Ω b > km s − kpc − (Weiner & Sellwood 1999).However, this model cannot explain the kinematics of thePerseus region: the direction of the velocities of young starsin the Perseus region, if interpreted in terms of the density-wave concept (Lin, Yuan & Shu 1969), indicates that a frag-ment of a Perseus arm must be located inside the corota-tion circle (CR) (Burton & Bania 1974; Mel’nik, Dambis &Rastorguev 2001; Mel’nik 2003; Sitnik 2003) implying anupper limit Ω s < km s − kpc − for its pattern speed,which is inconsistent with the one mentioned above. At-tempts have been undertaken to overcome this contradictionby introducing an analytical spiral potential rotating slowerthan the bar (Bissantz, Englmaier & Gerhard 2003). Notethat numerical simulations show that galactic stellar diskscan develop modes that rotate slower than the bar (Sellwood& Sparke 1988; Masset & Tagger 1997; Rautiainen & Salo1999, 2000). However, it is questionable whether the ampli-tude of slow modes can be large enough to determine thekinematics at twice the radius of the CR of the Galactic bar.Another concept of the Galactic spiral structure is thatthe disk rather than global modes forms transient spi-ral arms. Sellwood (2000, 2010) advances the idea thatfresh and decaying instabilities are connected through reso-nances, which is based on some specific features in the an-gular momentum distribution of old stars in the solar vicin-ity. Baba et al. (2009) build a model of the Galaxy withtransient spiral arms, which can explain the large peculiarvelocities, 20–30 km s − , of maser sources in the Galacticdisk. Transient spiral arms must heat the galactic disk in fewdisk rotation periods (Sellwood & Carlberg 1984). We can Copyright line will be provided by the publisher
A.M. Melnik et al.: Classical Cepheids in the Galactic outer ring R R ′ Fig. 1
Galaxies NGC 1211 and NGC 5701 with outer ring morphology R R R is located a bit closer to the galactic center than the R . suggest that young massive stars born in such arms shouldalso acquire large peculiar velocities. However, young stel-lar objects (classical Cepheids, young open clusters, OB-associations) in the wide solar neighborhood move in nearlycircular orbits with average velocity deviations of 7–13 kms − from the rotation curve (Zabolotskikh, Rastorguev &Dambis 2002; Mel’nik & Dambis 2009; Bobylev & Baikova2012; and other papers).Models of the Galaxy with the outer ring R R ′ can re-produce well the radial and azimuthal components of theresidual velocities (after substracting the velocity due to therotation curve and the solar motion to the apex) of OB-associations in the Sagittarius and Perseus stellar-gas com-plexes identified by Efremov & Sitnik (1988). The radialvelocities of most OB-associations in the Perseus region aredirected toward the Galactic center and this indicates thepresence of the ring R in the Galaxy, while the radial ve-locities in the Sagittarius region are directed away from theGalactic center suggesting the existence of the ring R . Thenearly zero azimuthal component of the residual velocityof most OB-associations in the Sagittarius region preciselyconstrains the solar position angle with respect to the barmajor axis, θ b = 45 ± ◦ . We considered models with ana-lytical bars and N-body simulations (Mel’nik & Rautiainen2009; Rautiainen & Mel’nik 2010).Models of a two-component outer ring are consistentwith the (l, V LSR ) diagram by Dame, Hartmann & Thad-deus (2001). These models can explain the position of theCarina arm with respect to the Sun and with respect to the bar. They can also explain the existence of some of the tan-gential directions corresponding to the emission maximanear the terminal velocity curves which, in this case, can beassociated with the tangents to the outer and inner rings. Ourmodel diagrams (l, V LSR ) reproduce the maxima in the di-rection of the Carina, Crux (Centaurus), Norma, and Sagit-tarius arms. Additionally, N-body model yields maxima inthe directions of the Scutum and 3-kpc arms (Mel’nik &Rautiainen 2011, 2013).The elliptic outer rings can be divided into the ascendingand descending segments: in the ascending segments Galac-tocentric distance R decreases with increasing azimuthalangle θ , which itself increases in the direction of Galacticrotation, whereas the dependence is reversed in the descend-ing segments. Ascending and descending segments of therings can be regarded as fragments of trailing and leadingspiral arms, respectively. Note that if considered as frag-ments of the spiral arms, the ascending segments of theouter ring R have the pitch angle of ∼ ◦ (Mel’nik &Rautiainen 2011).Two main classes of outer rings and pseudorings (in-complete rings made up of two tightly wound spiral arms)have been identified: rings R (pseudorings R ′ ) elongatedperpendicular to the bar and rings R (pseudorings R ′ )elongated parallel to the bar. In addition, there is a com-bined morphological type R R ′ which exhibits elementsof both classes (Buta 1995; Buta & Combes 1996; Buta& Crocker 1991; Comeron et al. 2013). Modelling showsthat outer rings are usually located near the OLR of the bar Copyright line will be provided by the publisher
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XY Sagittarius armPerseus armCarina arm l<180 o l>180 o Fig. 2
Distribution of classical Cepheids (black circles) and model particles (gray circles) in the Galactic plane. For model particles thesolar position angle is chosen to be θ b = 45 ◦ . The arrows indicate the positions of the Sagittarius, Carina, and Perseus arm-fragments.The X -axis points in the direction of Galactic rotation and the Y -axis is directed away from the Galactic center. One tick interval alongthe X - and Y -axis corresponds to 1 kpc. The Sun is at the origin. We can see a fork-like structure in the distribution of Cepheids: onthe left ( l > ◦ ) Cepheids strongly concentrate to the only one arm (the Carina arm), while on the right ( l < ◦ ) there are twoarm-fragments located near the Perseus and Sagittarius regions. (Schwarz 1981; Byrd et al. 1994; Rautiainen & Salo 1999,2000; and other papers). However, the ring R can be lo-cated further inwards, closer to the outer 4/1-resonance insome cases (Treuthardt et al. 2008).Buta and Combes (1996) have shown that for the galax-ies in the lower red-shift range the frequency of the outerrings is 10% of all types of spiral galaxies. However, it in-creases to 20% for the early-type sample. Note, however,that Comeron et al. (2014) obtained a larger frequency froman analysis of the data from mid-infrared survey (SpitzerSurvey of Stellar Structure in Galaxies, Sheth at al. 2010):16% for all spiral galaxies located inside 20 Mpc and over40% for disk galaxies of early morphological types (galax-ies with large bulges). They have also found that the fre-quency of outer rings increases from ± % to ± %when going through the family sequence from SA to SAB,and decreases again to ± % for SB galaxies. The catalogof Southern Ringed Galaxies by Buta (1995) gives the fol-lowing statistics of the main ring classes: 18% ( R ), 37%( R ′ ), under 1% ( R ), 35% ( R ′ ), and 9% ( R R ′ ). Com- eron et al. (2014) generally confirm these results: ± %of outer rings in barred galaxies are parallel to the bar and ± % are oriented perpendicular to it. The small frac-tion of galaxies with R R ′ rings may be due to the selectioneffects – the rings R are often weaker than rings R , more-over rings R are often appear more conspicuous when ob-served in the B-band. The completeness of the catalogs re-garding the R features is difficult to estimate. Generally,the frequency of galaxies with outer rings R R ′ amongbarred galaxies may be as high as few per cent. Note thatthe catalog by Buta (1995) includes several tens of galaxieswith rings R R ′ .Figure 1 shows two galaxies with R R Copyright line will be provided by the publisher
A.M. Melnik et al.: Classical Cepheids in the Galactic outer ring R R ′ lar to each other. Of the two rings, R is located a bitcloser to the galactic center than the R . Other examplesof galaxies with the R R ′ morphology that can also beviewed as possible prototypes of the Galaxy are ESO 245-1, NGC 1079, NGC 3081, NGC 5101, NGC 6782, andNGC 7098. Their images can be found in de VaucouleursAtlas of Galaxies by Buta, Corwin, Odewahn (2007) athttp://bama.ua.edu/ ∼ rbuta/devatlas/Schwarz (1981) associates two main types of outer ringswith two main families of periodic orbits existing near theOLR of the bar (Contopoulos & Papayannopoulos 1980).The main periodic orbits x (1) and x (2) (in terms of thenomenclature of Contopoulos & Grosbol 1989) are fol-lowed by numerous chaotic orbits, and this guidance en-ables elliptical rings to hold a lot of gas in their vicinity.The rings R are supported by x (2) -orbits lying inside theOLR and elongated perpendicular to the bar while the rings R are supported by x (1) -orbits located slightly outsidethe OLR and elongated along the bar. However, the roleof chaotic and periodic orbits appears to be different in thecenter region and on the galactic periphery: chaos is domi-nant outside the CR, while most orbits in the bar are ordered(Contopoulos & Patsis 2006; Voglis, Harsoula & Contopou-los 2007; Harsoula & Kalapotharakos 2009). Probably, it isnot only periodic orbits associated with the OLR that are re-sponsible for the formation of the outer rings/pseudorings.A concept has been proposed that the formation of outerrings as well as that of spiral arms is determined by man-ifolds associated with the Lagrangian points L1 and L2(Romero-G´omez et al. 2007; Athanassoula et al. 2010).The existence of the bar in the Galaxy is confirmedby numerous infra-red observations (Blitz & Spergel 1991;Benjamin et al. 2005; Cabrera-Lavers et al. 2007; Church-well et al. 2009, Gonz´alez-Fern´andez et al. 2012) and bygas kinematics in the central region (Binney et al. 1991; En-glmaier & Gerhard 1999; Weiner & Sellwood 1999). Thegeneral consensus is that the major axis of the bar is ori-ented in the direction θ b = 15 – ◦ in such a way that theend of the bar closest to the Sun lies in quadrant I. The semi-major axis of the Galactic bar is supposed to lie in the range a = 3 . – . kpc. Assuming that its end is located closeto its corotation radius, i.e. we are dealing with a so-calledfast bar (Debattista & Sellwood 2000), and that the rota-tion curve is flat, we can estimate the bar angular speed Ω b which appears to be constrained to the interval Ω b = 40 – km s − kpc − . This means that the OLR of the bar is lo-cated in the solar vicinity: | R OLR − R | < . kpc. Studiesof the kinematics of old disk stars in the nearest solar neigh-borhood, r < pc, reveal the bimodal structure of thedistribution of ( u , v ) velocities which is also interpreted as aresult of the solar location near the OLR of the bar (Kalnajs1991; Dehnen 2000; Fux 2001; and other papers).In this paper we show that the morphological and kine-matical features of the Cepheid sample considered are con-sistent with the presence of a ring R R ′ in the Galaxy.Section 2 describes the models and catalogues used; Sec- tion 3 considers the special features in the distribution ofCepheids; Section 4 studies the kinematics of Cepheids, andSection 5 presents the main results and their discussion. We used the catalogue of classical Cepheids by Berdnikovet al. (2000) which is continuously improved and updatedby incorporating new observations (Berdnikov et al. 2009a,2009b, 2011, 2014). The last version of the catalogue in-cludes the data for 674 Cepheids (Berdnikov, Dambis &Vozyakova 2014, in preparation). The procedure of deriv-ing distances is based on the K-band period-luminosity re-lation of Berdnikov, Vozyakova & Dambis (1996b) andinterstellar-extinction law derived in Berdnikov, Vozyakova& Dambis (1996a). The interstellar-extinction values areestimated using the B − V period-color relation of Dean,Warren, and Cousins (1978). Note the natural spread of theperiod-color relation does not introduce any substantial er-rors in the inferred distance values because the K -band ex-tinction is very small, A K = 0.274 E B − V . Our procedurein this case is essentially equivalent to using the V m λ We-senheit function with the deviations from the mean period-luminosity and period-color relations virtually cancellingeach other (Berdnikov et al. 1996a). Note the variations inthe distance scale up to 10% do not affect our conclusions.We use the simulation code developed by H. Salo (Salo1991; Salo & Laurikainen 2000) to construct two differ-ent types of models (models with analytical bars and N-body simulations) which reproduce the kinematics of OB-associations in the Perseus and Sagittarius regions. Amongmany models with outer rings, we chose model 3 from theseries of models with analytical bars (Mel’nik & Rautiainen2009) for comparison with observations. This model hasnearly flat rotation curve. The bar semi-axes are equal to a = 4 . kpc and b = 1 . kpc. The positions and velocitiesof × model particles (gas+OB) are considered at time T = 15 ( ∼ R = 7 . kpc (Rastorguev et al. 1994; Dambis, Mel’nik & Rastorguev1995; Glushkova et al. 1998; Nikiforov 2004; Feast etal. 2008; Groenewegen, Udalski & Bono 2008; Reid etal. 2009b; Dambis et al. 2013). As model 3 was adjustedfor R = 7 . kpc, we rescaled all distances for model par-ticles by a factor of k = 7 . / . . Note that the analysis ofmorphology and kinematics of stars located within 3 kpcfrom the Sun is practically independent of the choice of R in the range 7–9 kpc.Table 1, which is available in the online version of thepaper, gives the positions, line-of-sight velocities and propermotions of classical Cepheids. It ia also available at:http://lnfm1.sai.msu.ru/ ∼ anna/tables/tables _ Copyright line will be provided by the publisher
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It includes 674 classical Cepheids from the new re-lease of the catalog of classical Cepheids by Berdnikov etal. (2014, in preparation). For every classical Cepheid wegive its designation in the General Catalog of Variable Stars(GCVS) (Samus at al. 2007) or in the All Sky AutomatedSurvey (ASAS) (Pojmanski 2002), its type (see GCVS de-scription), fundamental period P F , intensity-mean V-bandmagnitude, J2000 equatorial coordinates α and δ , galac-tic coordinates l and b , and heliocentric distance r . Ta-ble 1 also gives the Cepheid line-of-sight velocities V r (theso-called γ -velocities, see Metzger, Caldwell & Schechter1992), their uncertainties ε V r and the references (1–6) tothe sources from which they are taken. The proper motionsof Cepheids were adopted from the new reduction of Hip-parcos data (ESA 1997) by van Leeuwen (2007). Table 1presents proper motions µ α and µ δ , their uncertainties ε µα and ε µδ and the corresponding Hipparcos catalog number n Hip .Table 2 lists the catalogs of line-of-sight velocities usedin our study. For each catalog it gives the correspondingreference number used in Table 1, the authors of the cata-log, the full reference to the paper/catalog, and the numberof velocities taken from each source. Some sources includea series of papers by one group of researches and/or dataavailable only online.
The distribution of Cepheids in the Galactic plane can re-veal regions of high gas density which can be associatedwith spiral arms or Galactic rings. Figure 2 shows the dis-tributions of classical Cepheids and model particles. We cannotice two specific features here. First, the distribution ofCepheids is reminiscent of ”a tuning fork”: at longitudes l > ◦ there is only one spiral arm (the Carina arm) to-ward which Cepheids concentrate strongly while at longi-tudes l < ◦ there are two regions with high surface den-sity located near the Perseus and Sagittarius regions. Sec-ond, the concentration of Cepheids drops sharply in the di-rection to the Galactic center while it decreases more grad-ually in other directions. These features will be studied fur-ther.We also provide Figure 3 to emphasize the fork-like structure in the distribution of Cepheids. Positions ofCepheids are represented in coordinates ( θ , ∆ R ), where θ is the Galactocentric angle and ∆ R = R − R is the differ-ence between the Galactocentric distances of a Cepheid andthe Sun. Cepheids can be seen to concentrate to the Carinaarm at the negative angles θ and to the Perseus and Sagittar-ius regions at the positive θ .The position of two outer rings in the distribution ofmodel particles can be approximated by two ellipses ori-ented perpendicular to each other. The outer ring R can berepresented by the ellipse with the semi-axes a = 6 . and b = 5 . kpc, while the outer ring R fits well the ellipsewith a = 8 . and b = 7 . kpc. These values correspondto the solar Galactocentric distance R = 7 . kpc. The ring R is stretched perpendicular to the bar and the ring R isaligned with the bar, hence the position of the Sun and thatof the sample of Cepheids with respect to the rings is deter-mined by the Galactocentric angle θ b of the Sun with respectto the major axis of the bar. We now assume that Cepheidsconcentrate to the outer rings to find the optimum angle θ b providing the best agreement between the position of therings and the distribution of Cepheids.Figure 4 shows three χ functions – the sum of normal-ized squared deviations of Cepheids from the outer rings(Press et al. 1987) – calculated for different values of theangle θ b . For each star we determined the minimum dis-tances to the two ellipses and then took the smallest of thetwo values. However, the χ function appears to be verysensitive to the completeness of the sample. This problemwill be studied further. Here we show the results obtainedfor three distance-limited Cepheid samples including starslocated within r max = 2 . , 3.0, and 3.5 kpc from the Sun.Table 3 lists the parameters of different samples: the num-ber N of Cepheids, the minimal value χ min , the standarddeviation σ of a Cepheid from the model distribution, andthe angle θ b corresponding to χ min . In this section we con-sider the parameters derived without any segregation overperiods, deferring the discussion of the values obtained forshort- and long-period Cepheids separately to section 3.2.The first three rows of Table 3 indicate that the χ functionsreach their minima at θ b = 50 ◦ , 37 ◦ , and 25 ◦ , respectively.The random errors of these estimates are of about ± ◦ . Notethat the different values of χ min represented in Table 3 arederived for different samples and cannot be compared witheach other. This also holds true for the values of σ .The position angle estimates θ b = 50 ± ◦ and ± ◦ derived for the samples within r max = 2 . and 3.0 kpc,respectively, agree well with the estimate θ b = 45 ± ◦ obtained from the kinematics of OB-associations (Mel’nik& Rautiainen 2009; Rautiainen & Mel’nik 2010). It must be”the fork-shaped” distribution of Cepheids that determinesthe angle θ b being close to 45 ◦ . At longitudes l > ◦ twoouter rings fuse together to form one spiral fragment – theCarina arm, whereas at longitudes l < ◦ two outer ringsare prominent, generating the fragments of the Sagittariusand Perseus arms.We can see that the estimates of θ b decrease with in-creasing r max (Table 3). This shift can be attributed to nu-merous stars scattered in the direction of the anti-center atdistances r > kpc (Figures 2,3). Their inclusion into thesample makes the outer ring R to be aligned with the lineconnecting the Sun and the Galactic center. As the ring R is aligned with the bar, the increase of r max must be accom-panied by the decrease in θ b .On the whole, the optimum position angle θ b providingthe best agreement between the distribution of Cepheids in-side ± . kpc and the model of the outer ring R R ′ can Copyright line will be provided by the publisher
A.M. Melnik et al.: Classical Cepheids in the Galactic outer ring R R ′ Table 2 Sources of Cepheid line-of-sight velocities
Ref. Authors year journal numberMetzger et al. 1991 ApJS, 76, 803Metzger, Caldwell & Schechter 1992 AJ, 103, 5291 Metzger, Caldwell & Schechter 1998 AJ, 115, 635 191Gorynya et al. 1992 Sov. Astron. Lett., 18, 316Gorynya et al. 1998 Astron. Lett., 24, 815Gorynya et al. 2002 VizieR On-line Data Catalog: III/2292 Gorynya & Rastorguev 2014 in preparation 86Pont, Mayor & Burki 1994 A&A, 285, 4153 Pont et al. 1997 A&A, 318, 416 364 Barbier-Brossat, Petit & Figon 1994 A&AS, 108, 603 75 Malaroda, Levato & Gallianiet 2006 VizieR On-line Data Catalog: III/249 26 Fernie et al. 1995 IBVS, 4148, 1 6
Table 3
Parameters of different samples of CepheidsPeriod r max N χ min σ θ b ± ◦ All 3.0 kpc 372 415.17 0.81 kpc ± ◦ periods 3.5 kpc 413 540.57 0.90 kpc ± ◦ P < d 2.5 kpc 242 235.88 0.71 kpc ± ◦ P < d 3.0 kpc 281 358.64 0.81 kpc ± ◦ P < d 3.5 kpc 326 521.97 0.91 kpc ± ◦ P > d 2.5 kpc 72 75.33 0.74 kpc ± ◦ P > d 3.0 kpc 91 106.80 0.78 kpc ± ◦ P > d 3.5 kpc 105 153.16 0.87 kpc ± ◦ be derived by averaging the θ b estimates listed in the firstthree rows of Table 3, yielding θ b = 37 ◦ . The average scat-ter of these estimates is ± ◦ . Assuming that two errors inthe determination of θ b , ± and ± , are independent, wecan estimate the combined error as ε = ε + ε , or ∼ ◦ . -30 -20 -10 0 10 20 30-3.0-2.0-1.0 0.0 1.0 2.0 3.0 θ o ∆ R Fig. 3
Fork-like structure in the distribution of classicalCepheids (black circles). Positions of Cepheids are representedin coordinates ( θ , ∆ R ), where θ is the Galactocentric angle and ∆ R = R − R is the difference between the Galactocentric dis-tances of a Cepheid and the Sun. The position of the Sun is shownby a cross. We can see the concentration of Cepheids to the Carinaarm at the negative angles θ and their concentration to the Perseusand Sagittarius regions at the positive angles θ . There is a wide-spread opinion that only classical Cepheidswith long periods are suitable for study of the Galactic struc-ture. Note that Cepheids with the periods
P < days ac-tually require more thorough study for classification. First,some of them are oscillating in the first overtone rather thanin the fundamental tone and this fact should be taken into ac- Copyright line will be provided by the publisher
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10 20 30 40 50 60 70 80350400450500550600650 r = 2.5 kpc max r = 3.0 kpc max r = 3.5 kpc max χ θ b ο Fig. 4
The χ functions calculated for different values of thesolar position angle θ b with respect to the major axis of the bar.Cepheids of all periods are included. Three distance-limited sam-ples of Cepheids confined by r max = 2 . , 3.0, and 3.5 kpc areconsidered. The χ functions reach their minima at θ b = 50 ◦ , ◦ , and ◦ , respectively. count when computing the distances. Second, short-periodclassical Cepheids can be confused with W Vir type vari-ables which are old stars of population II. However, clas-sical Cepheids have very specific features in light curves(Hertzsprung Progression) (Hertzsprung, 1926) which dis-tinguish them from Cepheids of other types.The subdivision of Cepheids into long- and short-periodgroups is a matter of convention. Usually, they are sepa-rated by the period of 10 days. However, we believe that thevalue of 8 days is more appropriate. Figure 5 shows the dis-tribution of fundamental periods of Cepheids. We can see aclear maximum in the distribution of Cepheids with periods P < days and almost a plateau distribution for Cepheidswith P > .Figure 6 shows the distribution of classical Cepheidswith short ( P < d) and long ( P > d) periods in theGalactic plane. It shows a conspicuous lack of long-periodCepheids in quadrant III. However, short- and long-periodCepheids generally show a similar behavior: they concen-trate to the Carina arm in quadrant IV, to the Sagittarius re-gion in quadrant I, and to the Perseus region in quadrant II.We repeat our study of the χ functions for short- andlong-period Cepheids separately. Table 3 presents the pa-rameters derived for Cepheids with periods P < and P > days in three regions: r max = 2 . , 3.0, and 3.5kpc. We can see that the angles θ b corresponding to χ min derived for the short- and long-period Cepheids in the sameregions coincide within the errors. On the whole, short- andlong-period Cepheids demonstrate the same tendency.
0 5 10 15 20 25 20 40 60 80100120 P, dN
Fig. 5
The number distribution of classical Cepheids from thecatalog by Berdnikov et al. (2000) over fundamental periods P . The average surface density and interstellar extinction ofCepheids in different directions can provide us with addi-tional information about the distribution of gas and dust inthe Galaxy. To study these characteristics, we subdivide thesample of Cepheids into four sectors: the directions to thecenter | l | < ◦ and anti-center | l − | < ◦ , those in thesense of Galactic rotation | l − | < ◦ and in the oppositesense | l − | < ◦ . For each sector we calculate the num-ber of Cepheids in annuli of width ∆ r = 1 kpc. The ratio ofthe number of Cepheids in a quarter of an annulus to its areagives us the average surface density n of Cepheids per kpc .This value depends on the sector and heliocentric distance r . We also calculate the average V -band extinction A V forCepheids in different annuli and in different sectors. Wecompute the A V value for each star as A V = 3 . E B − V (Berdnikov et al. 1996a), where E B − V is the color excessfrom the catalog by Berdnikov et al. (2000). Note that thevalues of A V were not used in the determination of Cepheiddistances which were derived from the K -band magnitudesand extinction A K (see also section 2).Figure 7 (a) and (b) shows the variations of n ( r ) and A V ( r ) along the heliocentric distance r calculated for thefour sectors. The surface density of Cepheids n in the direc-tion of the center ( | l | < ◦ ) can be seen to drop sharplywith increasing r : it decreases by a factor of ∼ fromthe distance range of 0–1 kpc to that of 2–3 kpc. For com-parison, this ratio amounts only 1.3–1.9 for the other threesectors. The surface density of Cepheids in the direction ofthe anti-center | l − | < ◦ is less than in other direc-tions within 4 kpc, for larger distances the estimates of n are Copyright line will be provided by the publisher
A.M. Melnik et al.: Classical Cepheids in the Galactic outer ring R R ′
1 2 3 4 5 6 7 8 5 10 15 20 25 30 35 r, kpcnkpc -2 ο ο ο ο (a) 6 8 10 12 14 16 18 20 20 40 60 80100 m-MN 0 ο ο ο ο (c) 1 2 3 4 5 6 7 81.02.03.0 r, kpcA Vm ο ο ο ο (b) 6 8 10 12 14 16 18 201.02.03.0 m-MA Vm ο ο ο ο (d) Fig. 7 (a) Variations of the average surface density n of Cepheids along the heliocentric distance r . (b) Variations of the averageextinction A V along the distance r . (c) Distribution of number of Cepheids N by the apparent distance modulus m − M , where N iscalculated in 2 m -wide intervals of distance modulus. (d) Growth of A V with the increasing m − M . Different lines show the valuescalculated for four sectors confined by the longitudes: l = 0 ± ◦ , l = 180 ± ◦ , l = 90 ± ◦ , and l = 270 ± ◦ . Frame(a) demonstrates the sharp drop of n ( r ) in the direction of the Galactic center | l | < ◦ (black solid line). Frame (c) shows that thedistribution of N in the direction of the Galactic center peaks at 13 m , whereas those for the other sectors peak at 15 m . The bars in frames(b) and (d) indicate the errors of A V which are shown only for the direction of the Galactic center and anti-center, in other sectors theyare less than 0.2 m . It is evident from frames (b) and (d) that extinction A V in the direction of Galactic center | l | < ◦ is greater than inother sectors. very uncertain. Extinction A V can also be seen to increasesharply toward the Galactic center ( | l | < ◦ ).Additionally, we consider the variations in the numberof Cepheids N and A V along the apparent distance mod-ulus m − M (Figure 7cd), where N and the average ex-tinction A V are calculated in 2 m -wide distance modulusbins. The distribution of N for the sector | l | < ◦ peaksat 13 m , whereas those for the other sectors peak at 15 m .It means that Cepheids can generally be discovered out to m − M = 15 m with the current instruments. Probably,the number of Cepheids N in the sector | l | < ◦ dropsconsiderably beyond 13 m due to their lower surface densitythere. However, the alternative explanation is that crowd- ing is much stronger in the direction of the Galactic center | l | < ◦ , preventing the discovery of Cepheids. Possibly,the survey VISTA variables in the Via Lactea (Saito et al.2012) would help solve this alternative.The bars in Figure 7 (b) and (d) indicate the errors inextinction A V which are shown only for the direction of theGalactic center | l | < ◦ and anti-center | l − | < ◦ . Inother sectors, the errors of A V are less than 0.2 m . We cansee that extinction A V in the direction of the Galactic center | l | < ◦ is greater than in other directions. Particularly, fordistance moduli m − M > m ( r > . kpc), the extinction A V in the direction | l | < ◦ is greater than in the oppositedirection | l − | < ◦ at a significance level of P > σ . Copyright line will be provided by the publisher
N header will be provided by the publisher 9
XY IIIIIIIV
Fig. 6
The distribution of classical Cepheids with short (smallcircles) and long (large circles) periods in the Galactic plane within3.5 kpc from the Sun. Periods are divided into two groups withrespect of P = 8 days. Roman numbers indicate quadrants. Wecan see the conspicuous lack of long-period Cepheids in quadrantIII. Generally, these particularities can be explained in termsof the model of the Galaxy with the outer rings located nearthe Sun. It is probable that the line of sight directed towardthe Galactic center intersects the outer ring R (Figure 2)which may contain a great amount of dust. Note that N-bodysimulations show that the ring R forms not only in the gassubsystem but also in the stellar component (Rautiainen &Salo 2000). We can expect that a high concentration of oldstars in the ring R can be accompanied by a great amountsof dust there. To estimate the radius of completeness of our sample,we need a hypothesis about the physical distribution ofCepheids. On the one hand, Cepheids should not be dis-tributed uniformly in the Galactic disk, they must concen-trate to spiral arms or to Galactic rings. On the other hand,Cepheids at larger distances are less likely to be discoveredthan those situated close to the Sun. Let us suppose that spi-ral arms or outer rings can increase the surface density ofCepheids not more than by a factor of two. That gives usa simple criterion: the sample is complete until the surfacedensity n(r) drops not more than to half its maximum value.When applied to the four sectors this criterion suggests thatour sample is complete out to 2 kpc in the direction of theGalactic center ( | l | < ◦ ), out to 3 kpc in two directions:that of the anti-center | l − | < ◦ and in the sense ofGalactic rotation | l − | < ◦ , and out to ∼ kpc in the sense opposite that of Galactic rotation | l − | < ◦ (Fig-ures 7a). The latter result may be due to the great interest ofobservers in the Carina arm and its extension.Another criterion of the completeness of the sample canbe formulated on the basis of variations in the number ofCepheids N along the visual distance modulus m − M (Fig-ure 7b). Suppose that the sample is complete until the num-ber of Cepheid N increases with increasing m − M . This es-tablishes the limits of m − M = 13 m for the sector in the di-rection of the Galactic center ( | l | < ◦ ) and m − M = 15 m for the three other sectors. To transform these apparent dis-tance moduli into distances, we must make an assump-tion about extinction. The variation of A V with distancemodulus (Figure 7d) shows that the average extinction at m − M = 13 m in the sector ( | l | < ◦ ) is nearly 2.0 m andit is also close to A V = 2 . m at m − M = 15 m in threeother sectors. So we can suppose that a Cepheid will be dis-covered if its true distance modulus is ( m − M ) = 11 . m in the sector ( | l | < ◦ ) and ( m − M ) = 13 . m in othersectors, i.e if it is located within r = 1 . kpc and r = 4 . kpc, respectively which is consistent with our previous esti-mates.Let us consider another criterion of completeness whichis based on the distribution of apparent magnitudes (Szaba-dos 2003, for example). Figure 9 shows the distribution ofapparent visual magnitudes m v of Cepheids from the cat-alog by Berdnikov et al. (2000). The number of Cepheidscan be seen to increase with m v till the value 12 m . Notethat the average apparent magnitudes m v of Cepheids in theheliocentric distance intervals 2–3 and 3–4 kpc are 11 and12 m , respectively. So again we can suppose that the catalogby Berdnikov et al. (2000) is nearly complete till ∼ kpc.To formulate more precisely, the incompleteness is gettingobvious at r > kpc. Before studying the non-circular motions, we must deter-mine the main parameters of the circular rotation of our ob-jects. Using the rotation curve derived from the same sampleof objects allows us to avoid systematical effects due to theeventual inconsistency of the distance scales of two differ-ent samples.We determine the parameters of the rotation curve basedonly on the Cepheids located within 3.5 kpc from the Sun( r < . kpc) and within 0.5 kpc ( | z | < . kpc) from theGalactic plane. This subsample includes 257 Cepheids withavailable accurate line-of-sight velocities and 217 Cepheidswith Hipparcos proper motions. The approach we use tosolve the Bottlinger equations was described in detail in ourearlier papers (Dambis et al. 1995; Mel’nik, Dambis & Ras-torguev 1999; Mel’nik & Dambis 2009), so we do not re-peat it here. We list the inferred parameters of the rotationcurve and solar motion in Table 4, where Ω is the angu-lar rotation velocity Ω( R ) at R = R ; Ω ′ and Ω ′′ are its Copyright line will be provided by the publisher R R ′ Table 4 Parameters of the rotation curve and the solar motion
Objects Ω Ω ′ Ω ′′ u v A σ Nkm s − km s − km s − km s − km s − km s − km s − kpc − kpc − kpc − kpc − Cepheids 28.8 -4.88 1.07 8.1 12.7 18.3 10.84 474all periods ± . ± . ± . ± . ± . ± . Cepheids 30.0 -4.90 0.92 8.3 12.1 18.4 11.19 323
P < d ± . ± . ± . ± . ± . ± . Cepheids 27.0 -4.85 1.31 7.7 14.2 18.2 10.04 151
P > d ± . ± . ± . ± . ± . ± . OB-associations 30.6 -4.73 1.43 7.7 11.6 17.7 7.16 132 ± . ± . ± . ± . ± . ± . first and second derivatives taken at R = R ; u and v are the components of the solar motion with respect to thecentroid of the sample in the direction toward the Galacticcenter and Galactic rotation, respectively; N is the numberof conditional equations. Table 4 also includes the parame-ters derived for short- and long-period Cepheids separatelyand those for OB-associations.Figure 8 (top panel) shows the rotation curve derivedfrom an analysis of the line-of-sight velocities and propermotions of Cepheids of all periods. For comparison, wealso show the rotation curve based on the data for OB-associations (Mel’nik & Dambis 2009). The parameters ofthe rotation curve derived for Cepheids and OB-associationsagree within their errors, see Table 4. Our Cepheid sampleyields Ω = 28 . ± . km s − kpc − which is also con-sistent with other studies of the Cepheid kinematics: Ω =27 . ± . (Feast & Whitelock 1997) and Ω = 27 . ± . (Bobylev & Baikova 2012). However, there is a systematicaldifference between the value of Ω obtained for Cepheidsand that inferred for OB-associations and maser sources, Ω = 31 ± km s − (Reid et al. 2009a, Mel’nik & Dambis2009; Bobylev & Baikova 2010). The linear rotation ve-locity at the solar Galactocentric distance estimated fromthe kinematics of Cepheids is systematically lower than thevalue derived from OB-associations. Moreover, the rotationcurve of Cepheids seems to be slightly descending, whereasthat of OB-associations is nearly flat within the 3 kpc neigh-borhood of the Sun. We are inclined to attribute these dif-ferences to the fact that the distances and proper motionsavailable for Cepheids are less accurate than those of OB-associations. Averaging the distances and proper motions ofstars within each OB-association may have given a consid-erable advantage. This problem requires further study.The lower panel of Figure 8 shows the scatter of indi-vidual azimuthal velocities of Cepheids with respect to therotation curve. The standard deviation of the projected ve-locities onto the Galactic plane from the rotation curve is σ = 10 . km s − for Cepheids of all periods.We also calculated the parameters of the rotation curvefor short- and long-period Cepheids (Table 4). It can be
5 6 7 8 9 10 50100150200250 4 V ckm s-1 R, kpcCepheidsOB-associations
5 6 7 8 9 10 50100150200250 4 V ckm s-1 R, kpcCepheids
Fig. 8
Top panel: the Galactic rotation curve derived from ananalysis of line-of-sight velocities and proper motions of Cepheids(thick line) and those of OB-associations (thin line). The positionof the Sun is shown by a circle. Lower panel: the scatter of indi-vidual azimuthal velocities of Cepheids with respect to the rotationcurve. It is built for Cepheids of all periods. seen that parameters determined for Cepheids with
P < and P > days are consistent within the errors exceptfor Ω , which is conspicuously smaller for long-periodCepheids. Note that the standard deviation of the velocitiesfrom the rotation curve equals σ = 11 . and 10.04 kms − for short- and long-period Cepheids, respectively. Thesmall difference between them suggests that both groups ofCepheids are suitable for study the Galactic structure. Copyright line will be provided by the publisher
N header will be provided by the publisher 11
0 2 4 6 8 10 12 14 16 18 20 40 60 80100120 m v N Fig. 9
The distribution apparent magnitudes m v of classicalCepheids. Residual velocities characterize non-circular motions inthe Galactic disk. We calculate the residual velocities forCepheids as the differences between the observed heliocen-tric velocities and the computed velocities due to the cir-cular rotation law and the adopted components of the so-lar motion defined by the parameters listed in Table 4 (firstrow). For model particles the residual velocities are deter-mined with respect to the model rotation curve. We con-sider the residual velocities in the radial V R and azimuthal V T directions. Positive radial residual velocities V R are di-rected away from the Galactic center while positive az-imuthal residual velocities V T are in the sense of Galacticrotation.Figure 10 shows the distribution of model particles withnegative and positive residual velocities V R and V T locatedwithin 3.5 kpc from the Sun. The left panel demonstratesthe distribution of radial residual velocities V R . Model par-ticles in the outer ring R (the one which is closer to theGalactic center) have positive velocities V R (black circles),whereas those in the ring R have negative velocities V R (gray circles). Particles located at the Galactic periphery( R > R + 2 kpc) have close to zero velocities V R (blackpoints). The right panel shows the distribution of azimuthalvelocity V T . Model particles in the ring R lying left of theSagittarius complex ( l ≈ ◦ , r ≈ . kpc) have mostlynegative V T velocities while those situated right of it havemostly positive V T velocities. It is not a chance coincidencebecause the position angle of the bar θ b was chosen in sucha way that model particles reproduce nearly zero velocities V T of OB-associations in the Sagittarius complex (Mel’nik& Rautiainen 2009). In the ring R we see the opposite ve- locity gradient: model particles located at the negative x-coordinates have mostly positive V T velocities (black cir-cles), whereas those situated at the positive x-coordinateshave mainly negative V T velocities (gray circles).Let us consider the distribution of Cepheids and modelparticles with negative ( V R < ) and positive ( V R > )radial residual velocities (Figure 11). The left panel showsthe distribution of objects with negative radial velocities V R which are supposed to belong to the outer ring R . Within r < . kpc from the Sun, the elliptic ring R can be rep-resented as a fragment of the spiral arm. Its pitch angle ap-pears to be i = 8 . ± . ◦ and i = 6 . ± . ◦ for Cepheidsand model particles, respectively. The positive value of thepitch angle i indicates that the spiral arm is leading and cor-responds to the solar position near the descending segmentof the outer ring R . Since the ring R is aligned with thebar, the location of the Sun near the descending segment ofthe outer ring R reflects the well-known fact that the bar’send closest to the Sun lies in quadrant I.Figure 11 (right panel) shows Cepheids and model parti-cles with positive radial residual velocities ( V R > ). Theseobjects are expected to concentrate to the ascending seg-ment of the outer ring R . Unfortunately, we see no goodagreement here. However, Cepheids do not occupy all the3-kpc solar neighborhood, they as well as model particleswith V R > are mostly located at negative y -coordinates.Figure 12 shows the dependence of azimuthal velocity V T on coordinate x for Cepheids and model particles withnegative radial velocities ( V R < ). The objects studied arelocated within 3 kpc of the Sun and are supposed to belongto the descending segment of the outer ring R . We consideronly Cepheids with small errors ( ε vl < km s − ) of thetangential velocity in the Galactic plane V l . The value of 10km s − corresponds to the average deviation of the Cepheidvelocity from rotation curve ( σ in Table 4). The error ε vl isdetermined by the error of the proper motion in the Galacticplane µ l and the distance r , ε vl = 4 . rε µ l , where propermotion is in mas yr − and the distance is in kpc. We usethe linear law to describe the V T - x relation: V T = ax + b .The slope a for Cepheids and model particles appears to be a = − . ± . and a = − . ± . , respectively. Bothvalues of a are negative, and this trend for Cepheids has asignificance level of P > σ .Generally, the kinematical features obtained forCepheids with negative radial velocities V R are consistentwith the solar location near the descending segment of theouter ring R . We use the data from the catalog by Berdnikov etal. (2000) to study the distribution and kinematics of clas-sical Cepheids in terms of the model of the Galactic ring R R ′ (Mel’nik & Rautiainen 2009). The best agreementbetween the distribution of Cepheids located within r max =2 . , 3.0, and 3.5 kpc from the Sun and the model of the outer Copyright line will be provided by the publisher R R ′ XY VR > +3 km/sVR < -3 km/s|VR| < 3 km/s V R Model particles R R VT > +3 km/sVT < -3 km/s|VT| < 3 km/s V T Model particles R R Fig. 10
Distribution of negative and positive residual velocities V R (left panel) and V T (right panel) calculated for model particleslocated in the solar neighborhood of 3.5 kpc. The radial velocities V R are subdivided into three groups: negative ( V R < − km s − ),positive ( V R > +3 km s − ) and close to zero ( | V R | < km s − ). The same was done for the azimuthal velocities V T . The solar positionangle θ b for model particles is chosen to be θ b = 45 ◦ . The arrows show the locations of the fragments of the outer rings R and R : thering R is closer to the center than the ring R . The X -axis is directed in the sense of the Galactic rotation, the Y -axis is directed awayfrom the Galactic center. The Sun is in the origin. One tick interval along the X - and Y -axis corresponds to 1 kpc. XY V < 0 R XY V > 0 R Fig. 11
Distribution of Cepheids (black circles) and model particles (gray circles) with negative ( V R < ) and positive ( V R > )radial residual velocities in the Galactic plane. Left: objects with V R < are supposed to belong to the descending segment of the outerring R . Right: objects with the positive radial residual velocities ( V R > ) are mostly located in quadrants I and IV. The X -axis isdirected in the sense of the Galactic rotation, the Y -axis is directed away from the Galactic center. The Sun is at the origin. One tickinterval along the X - and Y -axis corresponds to 1 kpc. Copyright line will be provided by the publisher
N header will be provided by the publisher 13 -3 -2 -1 0 1 2-20-10 0 10 20 x, kpcV T km s -1 Fig. 12
Dependence of azimuthal velocity V T on coordinate x for the Cepheids (black circles) and model particles (gray circles)with negative radial residual velocities ( V R < ). These objectsare supposed to belong to the descending segment of the outer ring R . Both Cepheids and model particles demonstrate a decrease in V T with increasing x . ring R R ′ is obtained if the position angle of the Sun withrespect to the bar major axis is θ b = 50 ± ◦ , ± ◦ , and ± ◦ , respectively. Averaging these values gives the finalestimate of θ b = 37 ± ◦ . It is ”the fork-like” structurein the distribution of Cepheids that determines the angle θ b being close to 45 ◦ : at longitudes l > ◦ two outer ringsfuse together to form one spiral fragment – the Carina arm,whereas at longitudes l < ◦ two outer rings run sepa-rately producing the Sagittarius and Perseus arm-fragments.To study the surface density n and extinction A V ofCepheids in different directions, we subdivide the sampleinto four sectors and calculate the average values of n and A V at different heliocentric distances r . The surface den-sity n of Cepheids appears to drop sharply in the directionof the Galactic center ( | l | < ◦ ). Furthermore, extinction A V grows most rapidly just in this direction. These featurescan be due to the presence of the ring R located in the di-rection of the Galactic center at 1–2 kpc from the Sun.Our analysis of variations in the surface density ofCepheids along distance gives us the estimate of the com-pleteness radius of the sample which appears to lie in the2-4 kpc interval for the different sectors. A similar resultwas obtained from an analysis of distance moduli and vi-sual magnitudes. We adopt ± kpc as the average value.Probably, within 3 kpc from the Sun, the apparent distribu-tion of known Cepheids reflects their physical distributionrather than instrumental-related difficulties in their discov-ery and study. The parameters of the rotation curve derived fromthe samples of Cepheids and OB-associations (Mel’nik &Dambis 2009) are consistent within the errors (Table 4).We study the distribution of Cepheids and model par-ticles with negative radial residual velocities ( V R < ),which inside 3 kpc of the Sun must belong to the outerring R . The selected Cepheids and model particles demon-strate similar distribution in the Galactic plane: both sam-ples concentrate to the fragment of the leading spiral armwith the pitch angle of i = 8 . ± . ◦ and i = 6 . ± . ◦ ,respectively. A similar leading fragment was found in thedistribution of OB-associations with negative radial resid-ual velocities ( V R < ) (Mel’nik 2005). The appearance ofthe leading fragment suggests that the Sun is located nearthe descending segment of the ring R . Moreover, selectedCepheids and model particles exhibit similar variations ofazimuthal velocity V T in the direction of Galactic rotation(the x coordinate).All this morphological and kinematical evidence sug-gests the existence of a ring R R ′ in the Galaxy. N-bodysimulations show that the descending segments of the outerrings R often include clumps and spurs (Rautiainen & Salo2000). Probably, the Galactic outer ring R is not homo-geneous in the solar neighborhood. The local Cygnus arm( l = 70 – ◦ , r = 1 – kpc) and the observed fragment ofthe Perseus arm ( l = 100 – ◦ , r = 1 . – . kpc) can beassociated with some of these model spurs. Such spurs usu-ally have a larger pitch angle than large-scale patterns likesegments of elliptical rings or global spiral arms. The natureof the fragmentation is not clear, it can be of purely hydro-dynamical origin (Dobbs & Bonnell 2006) or be associatedwith slow modes forming in the stellar population of thedisk (Rautiainen & Salo 1999, 2000). Acknowledgements.
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