Evidence of the Galactic outer ring R1R2' from young open clusters and OB-associations

3.3 MB·

by  A. M. Melnik

Evidence of the Galactic outer ring R R′ from young open
1 2
clusters and OB-associations
A. M. Melnik1 • P. Rautiainen2 •
E. V. Glushkova1 • A. K. Dambis1
6
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2

n
a
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] Abstract 1 Introduction
A
ThedistributionofyoungopenclustersintheGalac-
G tic plane within 3 kpc from the Sun suggests the exis- Open clusters are compact groups of stars born inside
′
h. tence of the outer ring R1R2 in the Galaxy. The op- one giant molecular cloud during a short time interval.
p timum value of the solar position angle with respect Young open clusters are gravitationally bound objects
- to the major axis of the bar, θb, providing the best in distinction from OB-associations, which are loose
o
agreement between the distribution of open clusters groups of O and B-type stars. Such differences be-
r
◦
st and model particles is θb = 35±10 . The kinemati- tweenyoungclustersandOB-associationsarebasedon
a cal features obtained for young open clusters and OB- the comparison of their mass with the velocity disper-
[ associations with negative Galactocentric radial veloc- sionsinsidethem. Thefactthatallstarsinsideanopen
1 ity VR indicate the solar location near the descending cluster have nearly the same age gives researchers the
v segment of the outer ring R2. opportunitytofittheclustermainsequenceandcolour-
2
colour diagrams to model grids derived from zero age
8 Keywords Galaxy: structure; Galaxy: kinematics
2 main sequence (ZAMS) and a set of isochrones corre-
and dynamics; Galaxy: open clusters and associations;
1 sponding to different abundances. The result of fitting
galaxies: spirals
0 is the determination of many important physical char-
.
1 acteristicsofclusters,suchasheliocentricdistance,age,
0 A.M.Melnik and metallicity (Kholopov 1980; Mermilliod 1981).
6 P.Rautiainen Young open clusters indicate the positions of giant
1
: E.V.Glushkova molecularcloudsbutunlike gaseousobjects,openclus-
v
A.K.Dambis ters allow their distances to be determined quite pre-
i
X 1Sternberg Astronomical Institute, Lomonosov Moscow State cisely with an accuracy of ∼ 5% as far as we ignore
University,Universitetskijpr. 13,Moscow119991, Russia
r possible errors in the zero point of the adopted ZAMS
a 2Department of Astronomy and Space Physics, University of
(Dambis 1999). So the concentration of young open
Oulu,P.O.Box3000,FI-90014Oulunyliopisto,Finland
clustersinsomecomplexessuggeststhepresenceofgas
e-mail: [email protected]
thereandtracesthepositionsofspiralarmsandGalac-
tic rings.
We suppose that the Galaxy contains a two-com-
′
ponent outer ring R R made up of two elliptical
1 2
gaseous rings stretched perpendicularly to each other
and located near the solar circle (Fig. 1). The sign of
′
apostrophemeansthe pseudoringR –incomplete ring
2
made up of two tightly wound spiral arms. The idea
that the Galaxy contains outer rings was first put for-
ward by Kalnajs (1991).
Twomainclassesofouterringsandpseudoringshave
′
been identified: rings R (pseudorings R ) elongated
1 1
perpendicular to the bar and rings R (pseudorings
2
2
Y
R
2
R
1
X
Fig. 1 Distributionofyoungopenclusters(blackcircles)from thecatalogbyDias et al.(2002)andmodelparticles(gray
circles)intheGalacticplane. Onlyclusterswithlogage<8.00andlocatedwithin0.5kpc(|z|<0.5kpc)fromtheGalactic
plane are considered. The positions of model particles (gas and OB particles) correspond to the position angle of the Sun
with respect to the bar of θb = 45◦. The X-axis points in the direction of Galactic rotation and the Y-axis is directed
away from the Galactic center. One tick intervalalong theX- and Y-axis corresponds to 1 kpc. The Sun is located at (0,
7.5 kpc). The arrows show the positions of the outer rings R1 and R2. Of the two rings, R1 is located a bit closer to the
Galactic centerthan the R2.
3
( a ) Y ( b ) Y
III II
Perseus
LS
X Cygnus X
R2
Carina
Sagit.
Sco
R1
IV I
1 kpc
1 kpc
Fig. 2 (a) Distribution of young (logage<8.00) open clusters (black circles) from the catalog by Dias et al. (2002) and
model particles (grey points) in the Galactic plane zoomed in to a larger scale. The Sun is at the origin. The positions
of the model particles are drawn for θb = 45◦. The X-axis points in the direction of Galactic rotation and the Y-axis
is directed away from the Galactic center. (b) The distribution of young open clusters (circles colored blue in electronic
edition)andrichOB-associations(asteriskscoloredgreeninelectronicedition)intheGalacticplane. OnlyOB-associations
containing more than 30 members (Nt >30) in the catalogue by Blaha & Humphreys(1989) are shown. The locations of
the outer rings R1 and R2 are indicated by gray arches. The positions of the Sagittarius, Scorpio, Carina, Cygnus, Local
System (LS) and Perseus stellar-gas complexes are drawn by ellipses. The Sagittarius and Scorpio complexes are located
inthevicinityoftheringR1. ThePerseuscomplexandLocalSystemlieneartheringR2. TheCarinacomplexissituated
in-between the two outer rings, where they seem to fuse together. As for the Cygnus complex, its connection with some
global structure is unclear. Roman numerals show thenumbersof quadrants.
4
′
R ) elongated parallel to the bar. In addition, there The explanation of the kinematics of young ob-
2
′
is a combinedmorphologicaltype R R which exhibits jects in the Perseus stellar-gas complex (see its lo-
1 2
elements of both classes (Buta 1995; Buta & Combes cation in Fig. 2b) is a serious test for different con-
1996; Buta & Crocker 1991). Modelling shows that cepts of the Galactic spiral structure. The fact that
outerringsareusuallylocatedneartheOuterLindblad the velocities of young stars in the Perseus stellar-
resonance(OLR) of the bar (Schwarz1981; Byrd et al. gas complex are directed toward the Galactic center,
1994;Rautiainen & Salo1999,2000,andotherpapers). if interpreted in terms of the density-wave concept
Comeron et al. (2014) used the data from mid- (Lin et al. 1969), indicates that the trailing fragment
infrared survey (Spitzer Survey of Stellar Structure in of the Perseus arm must be located inside the corota-
Galaxies, Sheth et al. 2010) to find that the frequency tion circle (CR) (Burton & Bania 1974; Mel’nik et al.
of outer rings is 16% for all spiral galaxies located in- 2001; Mel’nik 2003; Sitnik 2003), and hence imposes
side 20 Mpc and over 40% for disk galaxies of early an upper limit for its pattern speed Ω < 25 km s−1
sp
morphological types (galaxies with large bulges). The kpc−1, which is inconsistent with the pattern speed of
above authors have also found that the frequency of the bar Ω =40–65 km s−1 kpc−1 mentioned above.
b
outer rings increases from 15±2% to 32±7% when The studies of Galactic spiral structure are usually
going through the family sequence from SA to SAB, basedontheclassicalmodeldevelopedbyGeorgelin & Georgelin
and decreases again to 20±2% for SB galaxies. (1976),whichincludesfourspiralarmswithapitchan-
◦
Note that the catalogue by Buta (1995) includes gle of ∼ 12 (see e.g. the review by Vall´ee 2013).
′
several tens of galaxies with rings R R . Here are The main achievement of this purely spiral model is
1 2
′
some examples of galaxies with the R R morphol- that it can explain the distribution of HII regions in
1 2
ogy that can be viewed as possible prototypes of the Galactic disk (Russeil 2003). This model became
the Milky Way: ESO 245-1, NGC 1079, NGC 1211, more physical after incorporation of the bar into it
NGC 3081, NGC 5101, NGC 5701, NGC 6782, and (Englmaier & Gerhard 1999). However, the bar and
NGC 7098. Their images can be found in de Vau- spiral arms connected with it rotate with the angular
couleurs Atlas of Galaxies by Buta et al. (2007) at speed Ω = 50–60 km s−1 kpc−1, and this model can-
b
http://bama.ua.edu/∼rbuta/devatlas/ notexplainthekinematicsofyoungstarsinthePerseus
There is extensive evidence for the existence of the complex. Bissantz et al.(2003)developedthe modelof
bar in the Galaxy derived on the basis of infra-red ob- Englmaier & Gerhard(1999) by adding a pair of spiral
servations(Blitz & Spergel1991;Benjamin et al.2005; armsrotatingslowerthanthebarwithΩ =20kms−1
sp
Cabrera-Laverset al. 2007; Gonz´alez-Ferna´ndez et al. kpc−1. However, it remains unclear what mechanism
2012;Churchwell et al.2009)andgaskinematicsinthe can sustain this slower spiral pattern in the disk.
centralregion(Binney et al.1991;Englmaier & Gerhard Liszt (1985) criticizes the use of kinematical dis-
1999;Weiner & Sellwood1999). Thegeneralconsensus tances for tracing the Galactic spiral structure. He
is that the major axis of the bar is oriented in the di- shows that kinematical distances derived for HII re-
◦
rection θ = 15–45 in such a way that the end of the gions,HIandCOcloudscanbewrongduetokinematic-
b
bar closest to the Sun lies in quadrant I, where θ is distance ambiguity and velocity perturbation fromspi-
b
the position angle between the line connecting the Sun ralarms. Moreover,Adler & Roberts(1992)showthat
and the Galactic center and the direction of the major bright spots in the diagrams (l, V ) which are in-
LSR
axis of the bar. The semi-major axis of the Galactic terpreted as ”clouds” can consist of a chain of clouds
bar is supposed to lie in the range a = 3.5–5.0 kpc. extending over several kpc along the line of sight.
′
Assuming that its end is located close to its corota- Models of the Galaxy with the outer ring R R
1 2
tion radius (CR), i.e. we are dealing with a so-called reproduce well the radial and azimuthal components
fast bar (Debattista & Sellwood 2000), and that the of the residual velocities (observed velocities minus
rotation curve is flat, we can estimate the bar angu- the velocity due to the rotation curve and solar mo-
lar speed Ω , which appears to be constrained to the tion to the apex) of OB-associations in the Sagittar-
b
interval Ω = 40–65 km s−1 kpc−1. This means that ius (see its location in Fig. 2b) and Perseus com-
b
the OLR of the bar is located in the solar vicinity: plexes. The radial velocities of most OB-associations
|R −R |<1.5kpc. Studiesofthekinematicsofold in the Perseus stellar-gas complex are directed toward
OLR 0
disk stars in the nearest solar neighbourhood, r < 250 the Galactic center and this indicates the presence
pc, reveal the bimodal structure of the distribution of of the ring R in the Galaxy, while the radial ve-
2
(u, v)velocities,whichisalsointerpretedtobe aresult locities in the Sagittarius complex are directed away
of the solar location near the OLR of the bar (Dehnen from the Galactic center suggesting the existence of
2000; Fux 2001, and other papers). the ring R . The nearly zero azimuthal component
1
5
of the residual velocity of most OB-associations in the corotation, while most orbits in the bar are ordered
Sagittarius complex precisely constrains the solar po- (Contopoulos & Patsis 2006; Voglis et al. 2007). Not
sition angle with respect to the bar major axis, θ = only periodic orbits induced by the bar and regular
b
◦
45 ± 5 . We considered models with analytical bars orbits related to them, but also manifolds connected
and N-body simulations (Mel’nik & Rautiainen 2009; to the unstable Lagrangian points near the ends of
Rautiainen & Mel’nik 2010). the bar, may contribute to the formation of outer
The classical model of Galactic spiral structure rings and pseudorings (Romero-Go´mez et al. 2007;
can explain the existence of so-called tangential di- Harsoula & Kalapotharakos 2009; Athanassoula et al.
rections related to the maxima in the thermal radio 2010).
continuum as well as HI and CO emission, which The study of classical Cepheids from the catalogue
are associated with the tangents to the spiral arms by Berdnikov et al. (2000) revealed the existence of
(Englmaier & Gerhard 1999; Vall´ee 2008). Models of ”the tuning-fork-like” structure in the distribution of
a two-component outer ring can also explain the ex- Cepheids: at longitudes l > 180◦ (quadrants III and
istence of some of the tangential directions which, in IV) Cepheids concentrate strongly to the arm located
this case, can be associated with the tangents to the near the Carina complex (the Carina arm), while at
outer and inner rings. Our model diagrams (l ,VLSR) longitudes l < 180◦ (quadrants I and II) there are two
reproduce the maxima in the direction of the Carina, regionsofhighsurfacedensitylocatednearthePerseus
Crux(Centaurus),Norma,andSagittariusarms. Addi- andSagittariuscomplexes. Theterm”theCarinaarm”
tionally,N-body modelyields maximainthe directions wasusedtodesignatethepartoftheSagittarius-Carina
of the Scutum and 3-kpc arms (Mel’nik & Rautiainen arm (Fig. 11 in Georgelin & Georgelin 1976) that
2011, 2013). starts near the Carina complex and continues to larger
Pettitt at al.(2014)simulatedthe(l,VLSR)diagrams Galactocentricdistances. Ina morphologicalstudy the
for models with analytical bar. Their gas disks form Carina arm cannot be distinguished from the ascend-
thetwo-componentouterringsR1R2200–500Myrafter ing segment of the ring R2. This morphology suggests
the start of the simulation. The above authors found that outer rings R and R come closest to each other
1 2
observationstoagreebestwiththemodelwiththesolar
somewherenearthe Carinacomplex(seeitslocationin
◦
positionangleofθ ≈45 andthe barpatternspeedin
b Fig.2b). Wehavealsofoundsomekinematicalfeatures
the range of Ω =50–60 km s−1 kpc−1.
b inthe distributionofCepheids,whichsuggestthe loca-
Elliptic outer rings can be divided into the ascend-
tionoftheSunnearthedescendingsegmentofthering
ing and descending segments: in the ascending seg-
R (Mel’nik et al. 2015).
2
mentsgalactocentricdistanceRdecreaseswithincreas-
Inthispaperwestudythedistributionandkinemat-
ing azimuthal angle θ, which itself increases in the di-
icsofyoungopenclustersandOB-associations. Section
rection of galactic rotation, whereas in the descending
2 describes the models and catalogues used; Section 3
segmentsdistanceR,onthecontrary,increaseswithin-
considers the morphological and kinematical features
creasing angle θ. Ascending and descending segments ′
that suggest the existence of R R ring in the Galaxy,
1 2
of the rings can be regarded as fragments of trailing
and Section 4 presents the main conclusions.
and leading spiral arms, respectively. Note that if con-
sidered as fragments of the spiral arms, the ascending
segments of the outer ring R have the pitch angle of
2 2 Catalogues and Models
◦
∼6 (Mel’nik & Rautiainen 2011).
Schwarz (1981) associates two main types of outer
There are several large catalogues of open clusters.
rings with two main families of periodic orbits existing
Dias et al. (2002) compiled a catalogue of the phys-
neartheOLRofthebar(Contopoulos & Papayannopoulos
ical and kinematic parameters of open clusters us-
1980). The main periodic orbits are followed by nu-
ing data reported by different authors. This cat-
merous chaotic orbits, and this guidance enables el-
alogue, which is updated continuously, is available
liptical rings to hold a lot of gas in their vicinity.
at http://www.astro.iag.usp.br/ocdb/ and presently
The rings R are supported by x (2)-orbits (using
1 1 lists 2167 clusters. Kharchenko et al. (2013) deter-
the nomenclature of Contopoulos & Grosbol 1989) ly-
mined physical, structural and kinematic parame-
ing inside the OLR and elongated perpendicular to
ters of 3006 Galactic clusters. Mermilliod (1992)
the bar, while the rings R are supported by x (1)-
2 1 created WEBDA database of stars in open clusters
orbits located slightly outside the OLR and elongated
(https://www.univie.ac.at/webda/), where positional,
along the bar. However, the role of chaotic and pe-
photometricandspectroscopicdataforindividualstars
riodic orbits appears to be different inside and out-
inclusterfieldsisstored. Mermilliod & Paunzen(2003)
side the CR of the bar: chaos is dominant outside
6
analysed these data and derived the astrophysical pa- r (Sitnik & Mel’nik 1996). The kinematical data
BH
rameters (reddening, distance and age) of 573 open wereadoptedfromthe catalogueby Mel’nik & Dambis
clusters. (2009). The ages of OB-associations are supposed
In the last decade many embedded clusters (stel- to be less than 30 Myr (Humphreys & McElroy 1984;
lar groups recently born and still containing a lot of Bressan et al. 2012).
gas within their volumes) were detected from near- We use the simulation code developed by H. Salo
and mid-infrared surveys (see e. g. the reviews by (Salo 1991; Salo & Laurikainen 2000) to construct two
Glushkova 2013; Morales et al. 2013). However, dis- different types of models (models with analytical bars
tancestomostoftheseobjectsremanuncertainmainly andmodelsbasedonN-bodysimulations),whichrepro-
becauseofthevariableextinctionlawinthefieldofem- duce the kinematics of OB-associations in the Perseus
bedded clusters. The determination of their colour ex- and Sagittarius complexes. Among many models with
cesses requires detailed photometric and spectroscopic outer rings, we chose model 3 from the series of mod-
studies. For example, the estimates of the distance to els with analytical bars (Mel’nik & Rautiainen 2009)
the embedded cluster Westerlund 2 ranged from r = 2 to compare with observations. This model has nearly
to8kpc,beforeadetailedstudyofthisregionhasbeen flat rotation curve. The bar semi-axes are equal to
carriedout(Carraro et al.2013). Inthispaperwecon- a = 4.0 kpc and b = 1.3 kpc. The positions and ve-
sider only optically observed clusters. locities of 5·104 model particles (gas+OB) are consid-
For our study we have chosen the catalogue by ered at time T ≈ 1 Gyr from the start of the sim-
Dias et al. (2002), which provides the most reliable es- ulation. We scaled and turned this model with re-
timates of distances, ages and other parameters. Its spect to the Sun to achieve the best agreement be-
new version (3.4) contains 627 young clusters with the tweenthevelocitiesofmodelparticlesandthoseofOB-
ages less than 100 Myr. associations in five stellar-gas complexes identified by
Paunzen & Netopil (2006) established a list of 72 Efremov & Sitnik (1988).
”standard”openclusterscoveringawiderangeofages, We adopt a solar Galactocentric distance of R =
0
reddeningsanddistancesselectedonthebasisofsmall- 7.5 kpc (Rastorguev et al. 1994; Dambis et al. 1995;
est errors from the available parameters in the liter- Glushkova et al.1998;Nikiforov2004;Feast et al.2008;
ature. Their analysis is based on the averaged val- Groenewegen et al.2008;Reid at al.2009b;Dambis et al.
ues from widely different methods and authors. The 2013; Francis & Anderson 2014). As model 3 was ad-
authors then compared the derived mean values with justed for R = 7.1 kpc, we rescaled all distances for
0
theparametersofopenclusterspublishedbyDias et al. model particles by a factor of k = 7.5/7.1. Note that
(2002). They found that if one uses the parameters of the particular choice of R in the range 7-9 kpc has
0
the catalogue by Dias et al. (2002) then the expected practically no effect on the analysis of the morphology
errors are comparable with those derived by averag- and kinematics of stars located within 3 kpc from the
ing the independent values from the literature. They Sun.
concluded that ages, reddenings and distances in the
catalogbyDias et al.(2002)aregoodfor statisticalre-
search. 3 Results
We adopted the proper motions of open clus-
ters based on the Hipparcos catalogue (Hipparcos 3.1 Space distribution of young open clusters
1997) from the paper by Baumgardt et al. (2000),
and if they were absent there, from the catalogues The distributionofyoungopenclusters inthe Galactic
by Glushkova et al. (1996, 1997), which are available planecanrevealregionsofintensestarformation,which
at https://www.univie.ac.at/webda/elena.html. In the can be associated with spiral arms or Galactic rings.
latter lists the proper motions were derived from the Figure 1 shows the distribution of young open clusters
Four-Million Star Catalogue of positions and proper from the catalog by Dias et al. (2002) and model par-
motions (4M-catalogue,Volchkov et al.1992)andthen ticles in the Galactic plane. Only clusters with ages
reducedto the Hipparcossystem. We chosethese cata- lessthan100Myrandlocatedwithin0.5kpc(|z|<0.5
loguesofpropermotionsbecauseofthecarefulselection kpc) from the Galactic plane are considered. We can
of star cluster members. see the model location of the outer rings R and R
1 2
For kinematical study, we also use OB-associations calculated for the solar position angle with respect to
◦
from the list by Blaha & Humphreys (1989), which in- the bar major axis of θ =45 .
b
cludes 91 objects. Their heliocentric distances r Figure2ashowsthedistributionofyoungopenclus-
BH
were reduced to the short distance scale r = 0.8 · ters from the catalog by Dias et al. (2002) and model
7
particles in a larger scale. To avoid cluttering with within r <3.5 kpc of the Sun. We can see a minimum
otherobjects,wemadeanotherplot(Fig.2b),wherewe at θ = 35◦ here. The random error of this estimate
min
indicate the positions of rich OB-associations as well. is of about ±3◦. Table 1 lists the parameters of the
ThecatalogbyBlaha & Humphreys(1989)includes27 observed sample: the number N of clusters, the stan-
richOB-associationscontainingmorethan30members dard deviation σ of a cluster from the model position
(Nt > 30). Figure 2b also presents the positions of of the outer rings, and the angle θmin corresponding to
the Sagittarius, Scorpio, Carina, Cygnus, Local Sys- the minimum on the χ2 curve.
tem and Perseus stellar-gas complexes from the list by Figure 3b shows the χ2 functions computed for 10
Efremov & Sitnik (1988). We can see a tuning-fork- random samples also containing 564 objects and dis-
like structure in the distribution of young open clus- tributedalongtheheliocentricdistancer inaccordance
ters and OB-associations. At negative x-coordinates with a power law n(r) = r−1 that simulates the ef-
(on the left-hand side) mostofthe clusters concentrate fect of selection (see section 3.2). An example of a
to the only one arm (the Carina arm), while at posi- such a random sample is shown in Figure 4a. The χ2
tive x-coordinates (on the right-hand side) most of the functionscalculatedforrandomsamplesdemonstratea
clusters lie near the Perseus or the Sagittarius com- plateau in the 0 < θ ≤ 30◦ range and a steep rise in
b
plexes. The Carina stellar-gas complex is located in- the 30 < θ < 90◦ interval (Fig. 3b). A comparison
b
between the two outer rings, where they come closest of images shown in Figure 3a and Figure 3b indicates
to each other. The Sagittarius and Scorpio complexes that the minimum at θ = 35◦ exists only on the
min
lie near the ring R1, while the Perseus complex and curveobtainedfortheobservedobjectssuggestingthat
Local System, on the contrary, are situated near the it is not due to some specific best-fitting angles which
ring R2. In quadrant III, young objects within r <1.5 are, in principle, possible in the cases involving only a
kpc concentrate to the Sun, while more distant objects small region of the Galactic plane.
distribute nearly randomly over a large area. As for We made some modifications to the observed sam-
the Cygnus complex, its connection with some global
ple to show how the overdensities in the Perseus and
structure (outer ring or spiral arm) is unclear. How- Carina complexes influence the shape of the χ2 func-
ever, the kinematics of young objects in the Cygnus tion. Figure 5 shows the χ2 curves computed for the
complex is similar to that in the Perseus stellar-gas
observed sample of open clusters after the mirror re-
complex (Sitnik & Mel’nik 1996, 1999), and we there-
flection of some regionswith respect to the axis Y. All
fore tend to consider the Cygnus complex as a spur of χ2 functions were calculated for the same number of
the ring R , which can be rather lumpy.
2 objects N = 564. Sample designated as ”S1” was
obs
Using the model distribution, we can quite accu-
obtained from the observed distribution by changing
rately approximate the position of two outer rings by ◦
(l → 360 −l) for objects located in quadrants II and
two ellipses oriented perpendicular to each other. The
III.In sample”S1”the overdensityassociatedwiththe
outer ring R can be represented by the ellipse with
1 PerseuscomplexislocatedinquadrantIIIandthemin-
the semi-axes a1 = 6.3 and b1 = 5.8 kpc, while the imumofthe correspondingχ2 curvebecomesshallower
outer ring R fits well the ellipse with a = 8.5 and
2 2 incomparisonwiththatobtainedfortheobservedsam-
b = 7.6 kpc. These values correspond to the solar
2 ple. Thisflatteningoftheχ2 curveiscausedbythefact
Galactocentric distance R = 7.5 kpc. The ring R is
0 1 that the ring R reaches larger y-coordinates in quad-
2
stretched perpendicular to the bar and the ring R is
2 rant II than in quadrant III (Fig. 2). It is true for all
aligned with the bar, hence the position of the sample ◦
values of θ from the expected interval 15–45 . Hence
b
ofopenclusters withrespectto the ringsis determined
movingthePerseuscomplexintoquadrantIIIincreases
by the position angle θ of the Sun with respect to the
b deviations from the rings, and, consequently, increases
major axis of the bar. The outer rings do not touch
the corresponding values of χ2.
each other because gas particles located on the orbits
Sample ”S2” (Fig. 5) is obtained by changing (l →
which cross near the OLR were scattered due to colli- ◦
360 −l)forobjectsofquadrantsIVandI,whileobjects
sions during the formation of the outer rings. We now
ofquadrantsIandIIareleftattheiroriginalplaces. In
try to find the optimum angle θ providing the best
b
agreement between the positions of the open clusters
and the orientation of the outer rings.
Table 1 Parameters of thesample
Figure 3 shows the χ2 functions – the sums of nor-
malized squared deviations (Press et al. 1987) of open
Sample N σ θ
clusters from the outer rings – calculated for different obs min
valuesoftheangleθb. Figure3ashowstheχ2 curvede- 0<r<3.5 kpc 564 0.80 kpc 35±3◦
rived for the distribution of 564 young clusters located
8
χ2 ( a ) Observed χ2 ( b ) Random
800 800
700 700
600 600
500 500
400 400
ο θ ο θ
10 30 50 70 b 10 30 50 70 b
Fig. 3 The χ2 functions calculated for different values of the solar position angle θb with respect to the major axis of
the bar. (a) The χ2 function derived for the distribution of 564 young clusters from catalog by Dias et al. (2002) located
within r<3.5 kpc of the Sun. It has a minimum at θb =35±3◦. (b) The χ2 functions computed for 10 random samples
containing 564 objects and distributed in the Galactic plane in accordance with the power law n(r) ∼r−1 that simulates
the effect of selection. We show one such sample in Fig. 4a. The χ2 curves calculated for random samples demonstrate a
plateau at the 0<θb ≤30◦ interval followed by a steep rise in the 30<θb <90◦ interval. The dissimilarity of the curves
shown in ”a” and ”b” panels leads us to conclude that the minimum of the curve in panel ”a” is not due to some model
effects.
Y Y
( a ) ( b )
X X
1 kpc 1 kpc
Fig. 4 Examples of random samples generated to study selection effects. (a) Simulated objects are distributed in the
Galactic plane in accordance with the power law n(r)∼r−1. The sample contains 564 objects. (b) Simulated objects are
distributedneartheouterringsat theGaussian law with thestandard deviation of σr =0.8kpc. TheringR2 issupposed
to contain 64% of all objects. Only 20% of Nmod = 5000 objects are shown. All simulated objects are located within 3.5
kpcfromtheSun. TheX-axispointsinthedirectionofGalacticrotationandtheY-axisisdirectedawayfromtheGalactic
center. TheSun is at theorigin.
9
sample”S2”theoverdensityassociatedwiththeCarina
complexis locatedinquadrantIjust betweenthe rings
R and R . This transformation increases the devia-
1 2
tions from the rings for objects of the Carina complex
and causes the flattening the χ2 curve as well.
◦
Sample”S3”isaresultofthe(l→360 −l)transfor-
mation of coordinates of all objects (Fig. 5). Here the
overdensities associated with the Perseus and Carina χ2
complexes lie in quadrants III and I, respectively. We
can see that the corresponding χ2 curve is practically 800
◦
flat in the interval θ = 15–45 . The disappearance of
b
the minimum here is due to the tuning-fork-like struc-
ture in the distribution of the observed objects. The
700
mirror reflection creates the tuning-fork-like structure
pointed in the opposite direction (one segment lies at
S3
positive x-coordinatesand two segments are located at
negative x-coordinates), which is inconsistent with the S2
◦ 600
position of the outer rings obtained for θb =15–45 . S1
Figure6showsthehistogramofthedeviations(mini-
Obs
maldistances)dofyoungopenclustersfromthemodel
◦
positions of the outer rings calculated for θ = 35 .
b 500
Here we can clearly see the concentration of observed
clusterstothemodelpositionsoftheouterrings. Devi-
ationsdfromtheringsinthedirectionoftheincreasing
y-coordinates are considered positive and those in the
400
opposite direction are considered negative. The distri- 10 30 50 70ο θb
butionofdeviationscanbeapproximatedbytheGaus-
sianlawwithastandarddeviationofσ =0.8kpc. The Fig. 5 The χ2 functions computed for several samples of
objectsdesignatedas”Obs”(thethicksolidline),”S1”(the
excess of positive deviations at d > 1.5 kpc is due to
gray line),”S2” (the thin solid line), and ”S3” (the dashed
clusters located in the direction of the anticenter. The
line). The curve obtained for the observed sample of 564
fraction of clusters located in the vicinity of 1.5 kpc
young open clusters from the catalog by Dias et al. (2002)
(∼ 2σ) from the rings appears to be 95%, which is in (”Obs”) has the deepest minimum. The modified sample
good agreement with the Gaussian law. ”S1”isobtainedfromtheobservedonebythemirrorreflec-
We studied the effect of objects located in different tion of objects located in quadrants II and III with respect
regionsonthelocationoftheminimumoftheχ2 curve. to the axis Y, which is identical with the (l → 360◦ −l)
transformation. In sample ”S1” the overdensity associated
Removingtheclusterslocatedwithinthe0.5-kpcregion
◦ with thePerseus stellar-gas complex is located in quadrant
(r <0.5 kpc) decreases θ from 35 to 30 . This shift
min III.Sample”S2” isobtainedbychanging(l→360◦−l) for
appears due to the fact that the outer ring R passes
2 objectslocatedinquadrantsIVandI,whileobjectsofquad-
◦
throughthe solarvicinity atangleθ =90 . So nearby
b rants I and II are left at their original palaces. In sample
clusters ”favour” greater θb values. However, there is ”S2”theoverdensityassociated withtheCarinacomplexis
also an opposite effect: excluding distant clusters lo- located in quadrant I between the outer rigs R1 and R2.
cated in the direction of the anticenter (y > 2.5 kpc) Sample ”S3” is made byapplying the (l→360◦−l) trans-
fromtheobservedsampleincreasesθ from35to40◦. formation to all objects. Here the overdensities associated
min
withthePerseusandCarinacomplexesarelocatedinquad-
These clusters (y > 2.5 kpc) are located far from both
◦ rantsIII and I,respectively. Wecan see that theχ2 curves
outer rings and therefore ”favour” the θ = 0 value
b calculated successively for the samples ”Obs”, ”S1”, ”S2”,
in the case of which the ring R crosses the Y-axis at
2 and ”S3” haveincreasingly shallow minima.
maximum Galactocentric distance. We hence tend re-
◦
gard the uncertainty of ±5 as a random error of our
method.
We madeanattemptto simulatethe influence ofse-
lection effects by assigning to every model object the
probability P of its detection. Generally, the detection
of a cluster depends on a lot of things: the richness
10
andangularsizeofacluster,thenumberofresolvedin-
dividual members and their visual brightness, the sur-
facedensityoffieldstars,andtheamountofextinction
along the line of sight (Morales et al. 2013).
Let us suppose that the probability of detection of
a cluster is determined mainly by the brightness of its
stars. Then the probability of cluster detection is a
function of its apparent distance modulus DM which
depends on the heliocentric distance to the cluster r
and the extinction A toward it:
V
DM =5lgr+10+A , (1)
N V
where r is in kpc.
120
The probability P of detection of some objects is
usually assumed to be equal to unity within some re-
100 gion of parameters and to be exponentially decreasing
function beyond it. We can thus write the probability
P(DM) in the following way:
80
1 if DM <DM
60 0

P(DM)=

e−(DM−DM0)/s0 else,
40 
(2)
where the scale factor s and zero point DM are de-
20 0 0
terminedbyfittingbetweenthedistributionsofthedis-
tance moduli DM of observed and model clusters.
To simulate the sample of clusters we adopted the
-3 -2 -1 0 1 2 3 d, kpc
value of the solar position angle θ and scattered
tru
Fig.6 Distributionofthedeviations(minimaldistances)d Nmod = 5000 model objects with respect to the outer
ofyoungopenclustersfromthemodelpositionsoftheouter rings R1 and R2 in accordance with the Gaussian law
rings calculated for θb = 35◦. It can be approximated by with the standard deviation of σr =0.8 kpc within 3.5
theGaussianlawwithastandarddeviationofσr =0.8kpc kpc of the Sun, as it is shown in Figure 4b.
(thesolidline). Deviationsfromtheringsinthedirectionof Note that among 564 young open clusters from the
increasingy-coordinatesareconsideredpositiveandthosein
catalog by Dias et al. (2002) located within r < 3.5
theoppositedirectionareconsiderednegative. Theexcessof
kpc, 408 objects ( 70%) appears to lie within 0.8 kpc
positive deviationsat d>1.5 kpcis dueto clusterslocated
from one of the two outer rings, of those 262 (64%)
in the direction of the anticenter. The fraction of clusters
located within 1.5 kpc (∼ 2σ) of the rings appears to be are located in the vicinity of the ring R2. We therefore
95%, as expected in thecase of theGaussian law. distributed simulated objects among two rings placing
64% of all objects in the ring R .
2
To calculate the distance modulus for a model ob-
ject we must assign to it some value of the extinction
A . Thathasbeendoneinaccordancewiththeextinc-
V
tionofobservedyoung(logage<8.00)clusterslocated
in the nearby region and derived from their colour ex-
cess AV = 3.1EB−V (Cardelli et al. 1989). For each
model objects situated at point (x, y) we selected ob-
served young clusters from the catalog by Dias et al.
(2002) located within the radius of r =0.25 kpc from
e
the point (x, y), and calculated their average value