A new model for gravitational potential perturbations in disks of spiral galaxies. An application to our Galaxy

1.1 MB·English

by  T. C. Junqueira

Astronomy&Astrophysicsmanuscriptno.arxivpaper (cid:13)c ESO2013
January14,2013
A new model for gravitational potential perturbations in disks of
spiral galaxies. An application to our Galaxy
T.C.Junqueira, J.R.D.Le´pine,C.A.S.BragaandD.A.Barros
InstitutodeAstronomia,Geof´ısicaeCieˆnciasAtmosfe´ricas,UniversidadedeSa˜oPaulo,CidadeUniversita´ria,Sa˜oPaulo,SP,Brazil
e-mail:[email protected]
Received;accepted
3
1
ABSTRACT
0
2
Aims.Weproposeanew,morerealisticdescriptionoftheperturbedgravitationalpotentialofspiralgalaxies,withspiralarmshaving

n Gaussian-shapedgrooveprofiles.Theaimistoreachaself-consistentdescriptionofthespiralstructure,thatis,oneinwhichaninitial
a potentialperturbationgenerates,bymeansofthestellarorbits,spiralarmswithaprofilesimilartothatoftheimposedperturbation.
J Self-consistencyisaconditionforhavinglong-livedstructures.
1 Methods.Usingthenewperturbedpotentialweinvestigatethestablestellarorbitsingalacticdisksforgalaxieswithnobarorwith
onlyaweakbar.ThemodelisappliedtoourGalaxybymakinguseoftheaxisymmetriccomponentofthepotentialcomputedfrom
1
theGalacticrotationcurve,inadditiontootherinputparameterssimilartothoseofourGalaxy.Theinfluenceofthebulgemasson

thestellarorbitsintheinnerregionsofadiskisalsoinvestigated.
]
A Results.The new description offers the advantage of easy control of the parameters of the Gaussian profile of its potential. We
computethedensitycontrastbetweenarmandinter-armregions.Wefindarangeofvaluesfortheperturbationamplitudefrom400to
G
800km2s−2 kpc−1,whichimpliesanapproximatemaximumratioofthetangentialforcetotheaxisymmetricforcebetween3%and
. 6%.Goodself-consistencyofarmshapesisobtainedbetweentheInnerLindbladresonance(ILR)andthe4:1resonance.Nearthe
h
4:1resonancetheresponsedensitystartstodeviatefromtheimposedlogarithmicspiralform.Thiscreatesbifurcationsthatappear
p
asshortarms.Thereforethedeviationfromaperfectlogarithmicspiralingalaxiescanbeunderstood asanaturaleffectofthe4:1
-
o resonance.Beyondthe4:1resonancewefindclosedorbitsthathavesimilaritieswiththearmsobservedinourGalaxy.Inregionsnear
r thecenter,elongatedstellarorbitsappearnaturally,inthepresenceofamassivebulge,withoutimposinganybar-shapedpotential,
st butonlyextendingthespiralperturbationalittleinwardoftheILR.Thissuggeststhatabarisformedwithahalf-size∼3kpcbya
a mechanismsimilartothatofthespiralarms.
[ Conclusions.The potential energy perturbation that we adopted represents an important step in the direction of self-consistency,

comparedtoprevioussinefunctiondescriptionsofthepotential.Inaddition,ourmodelproducesarealisticdescriptionofthespiral
2
structure,whichisabletoexplainseveraldetailsthatwerenotyetunderstood.
v
2 Keywords.Galaxies:spiral–Galaxies:structure–Galaxy:kinematicsanddynamics–Galaxy:disk–galaxies:bulges
1
3
3
1. Introduction totheclosedstellarorbits,wecangenerateregionswherethese
.
2 orbitsarecrowded.Suchanorganizationisachievedbyrotating
1 Despiteitshistoricalmerit,thespiralstructuretheory, proposed orbits of increasing size by a given angle, relative to previous
2 byLin&Shu(1964),haswell−knownlimitations.Forinstance, ones.Theseregionsofhighstellardensitieslooklikelogarithmic
1 Lin&Shuaddgasandstarintoasinglecomponent,whentheir
spirals and rotate with the required pattern speed to transform
: dynamicsdifferin responsetothesamepotential;stars,forex-
v the usually open orbits into closed ones. Furthermore, such an
ample,arenotaffectedbypressuregradients.Additionally,real
i organizationofstellar orbitscan be achievedvia galactic colli-
X arms are not tightly wound, as assumed in that theory, and the
sions(Gerber&Lamb1994).ThegreatadvantageoftheKalnajs
r physicalpropertiesofthedisk(densities,velocityperturbations) model in comparison to Toomre’s is that the former’s spiral
a donothaveasine-shapeddependenceinazimuth,asarguedin structureisnotdisruptedbydifferentialrotation;inotherwords,
thepresentpaper.Aroundthesametime,Toomre(1964),inan
itcanbelonglived.Itisimportanttonotethatthesearchforsta-
opposingpointofview,arguedthatsincethelargestfractionof
ble or quasi-stationary solutions has traditionally been the aim
matterin the disk is in the formof stars, one can, in a first ap-
ofdynamicalmodelsofdisks.Thisisstillalegitimateconcern,
proximation,neglectthegasandworkwithstellardynamics.He
giventhatthenear-infraredmorphologyofseveralgranddesign
devoted much of his attention to understandingwhy an excess
galaxies reveals the existence of an underlying two-arm struc-
of stars in a given region does not grow without limit, due to
ture constituted mainly of old stars (Grosbøl&Patsis 1998;
theexcessofgravitationalattraction.Regardless,hismodelfor
Block&Wainscoat 1991; Rix&Rieke 1993; Rix&Zaritsky
thearmsfocusedmoreonstellardynamics,consideringthemas
1994;Blocketal.1994).Also,Le´pineetal.(2011b)arguedthat,
regionswithstellarexcessofstars,thatrotatewithvelocitythat
basedonthemetallicitygradient,thepresentspiralstructureof
doesnotdivergefromthevelocityofthediskmatter.
Galaxyhasanageofatleastseveralbillionyears.
Kalnajs(1972)proposedaninnovativewayofunderstanding
thespiralstructure,whichwasfocusedsolelyonstellardynam- The Kalnajs model is the most promising one, as it gives
ics.Heshowedthatbyintroducingsomedegreeoforganization a correct understanding of the nature of the arms as regions
1
T.C.Junqueiraetal.:ThespiralstructureoftheGalaxy
where stellar orbits are crowded. However, Kalnajs’ model is curve and pattern speed (discussed, respectively, in Sects. 4.1
only a first approximation, since it makes use of stellar orbits and6.1).Thereforeitisworthconstructingamodelthatisade-
associated with an axisymmetric potential. In order to reach quateforcomparisonswith observationsandothermodels.We
self-consistency,one must introduce the effect of the perturba- areconsciousthatanumberofauthorsconsiderthatourGalaxy
tions of the potential on the stellar orbits. The perturbations hasastrongbar,sothatourmodelwouldnotbeapplicable,but
produce changes in the shapes of the orbits, which in turn inanycase,thecomparisonwithobservationsmaygiveusclues
change the regions where the orbits are crowded and gener- to this question. Othermodels that we have already mentioned
ate a new perturbation. If at the end of this cycle we obtain (A&L, Pichardo et al, Contopoulos & Grosbøl) use the same
approximately the same density distribution as at the end of hypothesis of spiral-dominated structure and are directly com-
the starting one, we say that the model is self-consistent, and parable.
that the structure will be long lived. Self-consistentmodels, or Theorganizationofthispaperisasfollows.InSect.2.1,we
at least approximately self-consistent models in terms of arm review the classical modelfor the spiral arms. In Sect. 2.2, we
shapes, were constructed by Contopoulos&Grosbøl (1986), presentour new modelfor the spiral arms discuss the required
Contopoulos&Grosbøl(1988)andPatsisetal.(1991)forlarge parameters.InSect.3,wededucetherelationbetweentheden-
spiralgalaxies,andbyAmaral&Lepine(1996)(hereafterA&L) sity contrast and the perturbation amplitude. We constrain the
andPichardoetal.(2003)forthespecialcaseofourGalaxy.In perturbationamplitudeto arangeofvaluesbymeansof obser-
the works of Contopoulos and collaborators and of A&L, the vationalevidenceforthedensitycontrast.InSect.4andsubsec-
classicalexpressionfortheperturbationΦ wasusedinaframe tions, we compute the axisymmetric potentialfor two different
1
ofreferencerotatingwithconstantvelocityΩ : rotation curves. First, we use a realistic rotation curve, which
p
isobtainedbyfittingthedataoftheGalaxy.Thesecondmodel
lnR usesa“flat”rotationcurvethatismodeledbytwoaxisymmetric
Φ1(R,ϕ)=ζ0Re−εsR cos mtani −mϕ!, (1) components(bulge+disk)withdifferentbulge-to-diskratiosin
ordertobetterunderstandtheeffectofthebulgecomponenton
withsomediversityintheamplitudeζ ,thebehavioroftheam- the stellar orbits. In Sect. 5.1, we presentthe equationsof mo-
0
plitudewithradius(hererepresentedby Re−εsR),thenumberof tionandtheintegrationscheme,andinSect.5.2,wederivethe
arms (m), and the pitch angle i. A property of this expression density response. In Sect. 6, we discuss the adopted corotation
is that if we fix the radius R, the variationof the potentialas a radius and the angular velocity of the spiral pattern. The main
function of the azimuthal angle is a sine function. In the work resultsarepresentedinthissection,discussedseparatelyfordif-
ofPichardoetal.(2003),theperturbingpotentialisrepresented ferent ranges of radii: between the Inner Lindblad Resonance
as a superposition of a large number of oblate spheroids, with (ILR)andthe4:1resonance,beyondthe4:1resonance,andthe
their centersdistributed along the locii of the spiral arms. This connectionbetweenthebulgeandtheinnerstellarorbits.InSect.
approachdoesnotgiveaccesstoasimpleanalyticalexpression 7,wegiveourconclusions.
fortheperturbation.
One motivation for this work is that it has become evident 2. Thespiralarms
that the traditional description of the potential perturbation in
terms of a sine function in the azimuthaldirection is not satis- 2.1.Theclassicalmodel
factory. It is simple to estimate the stellar density in a region
Thesurfacedensityofazero-thicknessdiskcanberepresented
wheretheorbitsarecrowdedandtocompareitwiththeaverage
mathematically as the sum of an axisymmetric or unperturbed
field,aswellastoperceivethattheperturbationsaremuchbet- surfacedensityΣ (R),andaperturbedsurfacedensity Σ (R,ϕ),
terdescribedasnarrowgroovesorchannelsinthegravitational 0 1
whichrepresentsthespiralpatterninaframethatrotatesatan-
potentialwithanapproximatelyGaussianprofile.Wepresentan gularspeed Ω .Theazimuthalcoordinateatthe rotatingframe
analyticaldescriptionoftheperturbingpotential,which isreal- ofreferenceispϕ = θ−Ω t, where θ isthe angleattheinertial
istic when compared to observations and produces better self- p
frame.Asusual,thephysicalsurfacedensityisgivenbythereal
consistency compared to other models. Strictly speaking, we
partofthefollowingequation(seeBinney&Tremaine2008):
give priority to the analysis of one aspect of self-consistency,
which corresponds to the shape of the arms or equivalently to Σ1(R,θ−Ωpt)=Σsei[m(θ−Ωpt)+fm(R)], (2)
thepositionofthedensitymaximaintheplaneofthegalaxyand
where f (R) is the shape function, which describes the spiral,
totheirprofiles.Weemploytheexpression“self-consistencyof m
andΣ isavaryingfunctionofradiusthatgivestheamplitudeof
armshape”todescribeourresults,althoughseveralauthorsthat s
thespiralpattern.FromΣ ,asdescribedbyEq.2,wederivethe
wementionedabovesimplyreferto“self-consistency”forsimi- 1
potentialΦ ,usingGauss’law,andobtain
laranalyses.Otheradvantagesrelatedtotheexpressionproposed d
inthisworkfortheperturbationare1)avoidingfluctuationsbe- Φ1(R,θ−Ωpt)=Φdei[m(θ−Ωpt)+fm(R)], (3)
tweenpositiveandnegativevaluesforthepotential(asproduced
byasinusoidalfunction)overthewholediskforthewholerange where
of parameters, 2) controlling the arm-interarm contrast and, 3) 2πGΣ
dealingofthethicknessofthespiralarms. Φd =− |k| s. (4)
The focus in this work is on normal galaxies in which the
structure is dominated by spiral arms, not by a bar. Naturally, In the above expression, k is the wavenumber and G the
many galaxies present a weak bar in their center, with an ex- Gravitationalconstant.UsingEqs.3and4,weget
tended region of spiral arms beyond the end of the bar. The 2πGΣ
presentmodelcanbeusefultointerprettheirstructure. Φ1(R,θ−Ωpt)=− |k| sei[m(θ−Ωpt)+fm(R)]. (5)
In addition to the motivations mentioned above, there has
beenconsiderableimprovementinrecentyearsintheknowledge Thisisawell−knownresult,foundbyLinetal.(1969)andde-
of the structure of ourGalaxy and, in particular, of its rotation rivedfromtheWKBtheory(seealsoAppendixB).
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T.C.Junqueiraetal.:ThespiralstructureoftheGalaxy
Fig.2.Mapoftheperturbingpotentialintheplaneofthegalaxy.
Fig.1. Azimuthal density profile at different radii for m = 2, ThecolorsrepresentthevaluesofΦ (R,ϕ)inarbitraryunits.The
1
i = 14o, and σ = 4.7 kpc. Here we set Σ = 1, since we only pictureonthe leftisthe potentialwith σ = 2.5kpc andonthe
wantto showhow thedensity profilevaries0swith the azimuthal rightσ=4.7kpc.Fori=14owehaveontheleftσ⊥ =0.6kpc
angle. andontherightσ⊥ =1.1kpc
2.2.Thenewpotentialforthespiralarms
Our Model
3 Classical Model
Beforewebegin,itisimportanttoclarifywhatweunderstandby
spiralarms.ThemodelweadoptisbasedontheideaofKalnajs
2
(1972) of stellar orbits of successively increasing radii in the
disk,organizedinsuchawaythattheygetclosetogetherinsome
regions,thuspresentingexcessesofstellardensity. 1
Traditionally, the perturbed surface density has been de- FR
scribed by Eq. 2. This approach, however, is not realistic. The 0
brightness profiles observed in galactic disks in circles around
the center are not sine functions, nor are the density profiles -1
obtained theoretically from the crowding of stellar orbits. The
surfacedensityweconsidertoberealisticisgivenbyadensity
-2
excessthatfollowsalogarithmicspiral,withaGaussianprofile
inthetransversaldirection.Thepotentialthatproducesthiskind 2 4 6 8 10 12
ofdensityhastheform R (kpc)
Φ1(R,ϕ,z)=−ζ0Re−σR22[1−cos(mϕ−fm(R))]−εsR−|kz|, (6) FTihge.3ci.rRclaedsiianlfreodrcreepdrueesetontththeesrpaidraiallafromrciendthueedtoiroeuctriopnotϕen=tia0l.,
while the crosses in green representthe radial force associated
whereζ istheperturbationamplitude,ε−1thescalelengthofthe
0 s withtheclassicalsinefunctionperturbation.Here,positiveval-
spiral, σthewidthofthe Gaussianprofilein thegalactocentric
azimuthaldirection, k = m the wavenumber,and f (R) the uesoftheforcemeanthatastarwouldbepushedoutwardand
Rtan(i) m negativesvaluesthat it would be pulled inward with respect to
shapefunction.Thescale-lengthofthediskandofthespiralare thegalacticcenter.Inthisfigure,thenumberofarmsis m = 2,
notnecessarily the same. This questionis discussed in Sect. 3. the pitch angle i = 14o, ε = 0.4 kpc−1, and forourmodelthe
Theshapefunctionis armwidthσ=4.7kpc s
m
f (R)= ln(R/R)+γ, (7)
m i
tan(i)
indirectionsofgalactocentriccircles(seedeductioninAppendix
where m is the number of arms, i is the pitch angle, R is the A).Becauseallthesecirclescrossthelogarithmicarmswiththe
i
pointwherethespiralcrossesthecoordinatex=0,andγisonly same angle i, the true width in a directionperpendicularto the
aphaseangle. armsisgivenbyσ⊥
SolvingPoisson’sequationusingEq.6,withtheassumption
ofzero-thicknessdiskandoftightlywoundspiralarms(TWA) σ⊥ =σsin(i). (10)
intheplanez=0wehave
Thisparameterallowsustoreproducearmsofdifferentwidths,
Σs =Σs0e−σR22[1−cos(mϕ−fm(R))], (8) asshFoigwunrein3Fpilgo.ts2t.heradialforceproducedbythespiralarmsin
where Σ is associated with the perturbation amplitude (see ourmodel(redcircles,whichareverysimilartoPichardoet al.
Appendixs0Bformoredetails)bytheequation 2003)comparedwiththeclassicalmodel(greencrosses,derived
fromEq.3) asa functionof Galactocentricdistance R. In both
ζ m R2 casestheperturbationamplitudewassettobeequaltoone,since
Σs0(R)= 2π0Gσ2|tani|e−εsR. (9) wejustwanttoseehowtheforceprofilevariesandarenotinter-
ested inabsolutevalues.Thefirst thingwe cannoticeinFig. 3
TheazimuthalprofilecorrespondingtoEq.8isillustratedin isthatfortheclassicalmodeltheradialforceisstrongerforthe
Fig.1.Weemphasizethatσisthehalf-widthofthespiralarms sameperturbationamplitude,mainlyininnerregions.Thus,the
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T.C.Junqueiraetal.:ThespiralstructureoftheGalaxy
perturbationamplitudeintheclassicalcasewouldbelowerthan wantthe amplitudeof the spiralto droptoo fast outwards)and
in ourmodel. This is due to the fact that a sinusoidal potential compute the value of σ using the density of the solar neigh-
0
varies between positive and negative values and that the force, borhood,Σ (R ) = 49 M pc−2 (see Binney&Tremaine 2008,
d 0 ⊙
proportionalto the variationsof potential, is strongerthan in a Table1.1),forR =7.5kpcandε−1 =2.5kpcwegetσ =984
0 d 0
case wherethe potentialgroundlevelis equalto zero.Another M pc−2.Then,usingthevaluesfromTable1forthenumberof
⊙
differencebetweenthetwomodelsoccursaround9kpc,where, armsandthepitchangle,Eq.13reducesto
in the classical model, the force becomespositive while in our
modeltheforceslowlytendstozero,butitisstillnegative.This Σ ζ R2
happensbecausea starin thatregionfeelsthe effectofthe last Σs0 =3.10−4 σ02 . (14)
piece of the arm, as we can see in Fig. 2 in the x direction, so d
thatitwouldbepulledinwards.
Sincewehaveused,uptonow,parametersofthespiralstruc-
ture that are based on studies of our Galaxy, we shall estimate
thecontrastfora galactocentricdistanceofabout5 kpc,which
3. Relationbetweendensitycontrastand
inourmodelisaboutmidwaybetweentheILR andcorotation,
perturbationamplitude
asdiscussedinSect.6.NumericallywehaveR≈ σ,sothatEq.
FollowingAntojaetal.(2011)wetakeasameasureoftheden- 14becomes
sitycontrast
Σ
s0 ≈3.10−4ζ . (15)
Σ Σ 0
A ∼ s0, (11) d
2 Σ
d Aplotofthedensitycontrastbasedonthisequationisshown
wimhuemreΣddenissitthyeoafxitshyemsmpiertarlicasrumrfsa.cTehdiesnrseitlyataionndΣiss0viasltihdeumndaxer- icnusFsiegd.4a.bCovoem,pwaritihngththisefivagluurees,owfeAfi2nfrdoma rtahnegleiteorfataumrep,laistuddiess-
the assumption of a mass-to-light ratio of the order of 1 when oftheperturbationbetween400to800km2 s−2 kpc−1,whichis
A is measuredin the infraredbands(Kent1992). Antojaetal. compatiblewiththerangeofdensitycontrastsofTable1.Taking
(22011)collectedintheliteraturetheestimationsofGalacticand anaveragevalueA2 =0.18givesusζ0 =600km2s−2kpc−1.
extragalacticdensitycontrasts,andfoundthemtobeintherange
0.13 ≤ A ≤ 0.23.Therefore,anaveragevaluewouldbeabout
2
A =0.18. 0.36
2
0.32
Table1.Adoptedspiralarmsproperties 0.28
0.24
Property Symbol Value Unit
Numberofarms m 2 - Σd 0.2
PHiatclfhwanidgtlhe σi 144.7o kp-c Σ / s0 0.16
Scalelength ε−1 2.5 kpc
s 0.12
Spiralpatternspeed Ω 23 kms−1kpc−1
p
Perturbationamplitude ζ0 600 km2s−2kpc−1 0.08
0.04
The maximumdensityof the spiralarmsis givenbyEq. 9.
0
We refer to the maximum at a certain radius, not a maximum 100 200 300 400 500 600 700 800 900 1000
ζ0 (km2 s-2 kpc-1)
overthewholedisk.Tosimplifytheanalysis,weassumethatthe
axisymmetric surface density has an exponentialbehavior(Eq. Fig.4.DensitycontrastΣ /Σ asafunctionofperturbationam-
12).Galacticdisksareoftenrepresentedbyexponentialdensity plitudefortheMilkWay.s0 d
laws and, depending on the value of ε , can explain relatively
d
flatrotationcurves.We knowthatsuchanexponentiallawwill
notgeneratean exactfit to the rotationcurve,givenbyEq. 17,
butitsimplifiestheanalysisandgivesusa goodhintaboutthe This is an estimate for R ≈ 5kpc. However as we can
valueoftheperturbationamplitude see from Eq. 14, the density contrast increases with radius.
Kendalletal.(2011)andElmegreenetal.(2011)showedmany
Σd(R)=σ0e−εdR. (12) galaxies with a growth of density contrast with radius. If we
adopt ε = ε , the density contrast varies as the square of the
d s
Thereforethedensitycontrastis radius, which seems a little too fast in many cases. This prob-
lem can be attenuated in two ways: 1) the scale length of the
Σ ζ m R2 spiralandofthediskmaybeslightlydifferent(ε < ε ),which
s0 = 0 e−(εs−εd)R. (13) d s
Σ 2πGσ2|tani|σ will produce an exponential decrease, 2) the arms width may
d 0
increase with radius, producinga density contrast with moder-
The density contrast depends not only on the perturbation ateincrease,closertotheresultsofKendalletal.Foraspecific
amplitudeζ ,butalsoonσ,whichdescribesthearmswidth.In galaxy,itwouldbepossibletochecktheazimuthaldensitypro-
0
ourmodel,wecansetσasaconstantorafunctionofradius.If fileanditsvariationwithradiusandtoadjusttheparametersto
weassumethatthescalelengthisthesameforbothcomponents matchtheobservations.Onemeritofourmodelisthatitfacili-
(ε = ε , which is a reasonableassumptionbecause we do not tatessuchadjustments.
d s
4
T.C.Junqueiraetal.:ThespiralstructureoftheGalaxy
4. Theaxisymmetriccomponent
0
The main dynamical information of a spiral galaxy is derived
fromitsobservedrotationcurve.Infact,therotationcurvegives
-0.05
us the radial gradient of the potential, since we can compute
Φ0(R)byintegratingEq.16: 2)
-yr -0.1
∂Φ V2 (R) M
F =− 0 =− rot , (16) 2
∂R R Kpc -0.15
whereVrotistherotationcurve. ntial (
e -0.2
ot
4.1.TherotationcurveoftheGalaxy P
To contributeto a deepunderstandingof the spiral structureof -0.25
anygivengalaxy,wemustmaketheeffortofintroducingthereal
parameters,suchasthedimensionofthecomponents,therota-
-0.3
tioncurve,andthepatternrotationspeed.Althoughtheremaybe 0 2 4 6 8 10 12 14 16 18 20
controversiesconcerningsomeoftheseparameters,wechoseto R (kpc)
illustrate ourmodelof spiralperturbationby applyingit to our
Galaxy. However, we also used a simplified analytical expres- Fig.6.Axisymmetricalpotentialasafunctionofthegalactocen-
sion for the rotation curve, introduced in the next subsection, tricradiusR.
with the aim of reaching qualitatively a general understanding
oftheeffectofthethemassofthebulgeontheinnerorbits.
The“real”rotationcurveoftheGalaxythatwehaveadopted
(Fig. 5) is quite flat, except for a local minimum at about 8.8
kpc. The existence of this dip in the curve is revealed, for in- Thefirsttwocomponentscontaintermsin(1/R)and(1/R)2
stance, by the study of the epicycle frequency in the galactic inside the exponential function, which produce a decrease of
disk(Le´pineetal.2008).Thisisconfirmedinarecentworkby theircontributiontowardssmallradii.Thelast(Gaussian)term
Sofueetal. (2009), which makes use of different data sets, in- representsthelocaldip.Theinterpretationofasimilarcurve,ex-
cludingdataonmaserswithprecisedistancesmeasuredbypar- ceptforthedip,intermsofcomponentsoftheGalaxyisgivenby
allax,to trace the curve.The rotationcurvespresentedin these Le´pine&Leroy(2000)andisdifferentfromthatofSofueetal.
two papers are quite similar, except for minor scaling factors (2009).ThepatternspeedisdiscussedinSect.6.1.
due to the adoptedgalactic parameters, the solar distance from
Figure6showstheaxisymmetricpotentialthatresultsfrom
the Galactic centers R , and the solarvelocity V . The curveis
0 0 theintegrationofEq.16.Weusedthetrapeziumrulewithadap-
convenientlyfittedbytwoexponentialsandaGaussian
tivesteptosolvetheintegral,andwefixedthepotentialasequal
toΦ (100)= 0atR=100kpcinordertosetthearbitrarycon-
0
V (R)=398e−R/2.6−0.14/R+257e−R/65−(3.2/R)2 stant.
rot
−20e−[(R−8.8)/0.8]2. (17)
4.2.Amodelofaflatrotationcurvewithabulge
Many galaxies have a flat rotation curve, but some of them
300 presentapeakintheinnerpart.Forinstance,asseeninFig.5,a
peakforourGalaxyappearsataradiussmallerthan2kpc,with
250 amaximumaround300pc.Thenatureofthepeakisasubjectof
debate.Someauthorsconsiderthatitisnotduetorotation,like
200 Burton&Liszt(1993),whobelievethatthereisastrongvelocity
m/s) componentofgasoutflowfromthecentralregionsoftheGalaxy.
V (krot 150 rOicth,elirksecaotnrsiaidxeiarltbhualtgaenacimcoprodrintagnttodGepearhrtaurrde&froVmietarix(is1y9m86m),eits-
100
required to explain the peak. To perform a first approximation
analysis on the effects of the presenceof a bulge on the stellar
50
orbitsin theinnerregionsofa galaxy,we adoptthe hypothesis
thatthebulgeisaxisymmetric,asitseemstobe,atleastapproxi-
0
0 2 4 6 8 10 12 14 mately,inmostspiralgalaxies(Me´ndez-Abreuetal.2010).Also
R (kpc)
inaxisymmetricbulges,onecanfindellipticalorbitsintheircen-
Fig.5.RotationcurveoftheGalaxy.Theredlineistherotation tralregions.Then,evenaverysmallperturbationcouldgiverise
curvefitted usingClemens’data(blackdots),andthe blueline to an oval structure. Since in this part of the work we are not
isthepatternspeedΩ ×Rcorrespondingtoanangularvelocity interested in solving the specific case of our Galaxy, but want
p
Ω = 23kms−1 kpc−1. Thecorotationradiusisat8.4kpc.We to have a moregeneralidea of the effectof a bulge,we do not
p
adoptedR =7.5kpcandV =210kms−1. use the observedrotation curve of our Galaxy. Instead, we use
0 0
asimpleronefromwhichthepotentialisobtainedanalytically,
thusavoidingnumericalintegrations.Forthisstudyweadoptthe
model for the rotation curve given by Contopoulos&Grosbøl
5
T.C.Junqueiraetal.:ThespiralstructureoftheGalaxy
(1986), in which the presence of the peak is modeled by two Poincare´’ssurfaceofsection(Poincare´ 1892).ThePoincare´sec-
axisymmetriccomponents(bulge+disk): tionwasfixedwheretheorbitscrosstheaxis ϕ= 0o. Theperi-
odic orbits trapped around circular orbits are represented by a
Vrot(R)=Vmax fbǫbRexp(−ǫbR)+[1−exp(−ǫdR)]. (18) sequence of points in the phase-space (R,pr), lying on a curve
calledaninvariantcurve.
p
Hereǫ−1 andǫ−1 arethescalelengthforthebulgeanddisk Inpractice,whatwedoistocomputetheenergyofacircular
b d
components, respectively. The relative importance of the two orbitusingEq.19.Thusthisequationbecomes
componentsisgivenbythebulgefraction f .
b
We analyzed the inner stellar orbits obtained with rotation h(R)= J02 −Ω J +Φ (R). (24)
curveswith differentvaluesof f , thatis, with differentimpor- 2R2 p 0 0
b
tanceofthebulge.Theresultsarediscussedlater,inSect.6.
Thisexpressionhasnoperturbationterms.Theangularmo-
mentumpermassunitisgivenby J = RV ,andΦ (R)isthe
0 rot 0
5. Stellarorbits galacticpotential.Foreachradius,wehaveavalueofenergyh,
then we come back to the Eq. 19 with H = h. Thus, we have
5.1.Theintegrationscheme tofindtheinitialconditions(R,p ,J ,ϕ)thatwillclosetheorbit
r 1
Onceweknowthetwocomponentsofthepotential,Φ (R)and for a given h. The last variable can be fixed at ϕ = 0, and the
Φ (R,ϕ), we can derive the equations of motion gove0rned by angularmomentumJ1isfoundsolvingEq.19foragivenRand
1 p . Therefore, we only have to deal with the pair of variables
them.Figure7illustratestheanglesandangularvelocitiesused r
(R,p ).Thepairthatgivesusastableperiodicorbitisfoundat
intheequations.WhentheGalacticplaneisrepresented,themo- r
thecenteroftheislandinthePoincare´section.
tion of stars andthe patternspeed are shown clockwise,which
The families of periodic orbits that we obtained (starting
correspondstonegativevaluesofangularvelocities.
fromenergiesequivalenttocircularorbitsspaced0.2kpcapart,
from3upto8kpc)areshowninFig.8.Theinnerpartfrom2up
to3kpcwasavoidedhere;wediscussthisregioninSect.6.4.
5.2.Responsedensity
Theresponsedensityiscomputedbyconsideringtheconserva-
tion offlux betweentwo successive orbitsin the perturbedand
unperturbed cases (Contopoulos&Grosbøl 1986). In our case
weconsideraseriesofcircularorbitsspacedat0.2kpc.Wecan
imagineineachorbitalargenumberofstarsequallyspacedin
Fig.7.SchemeoftheGalacticplane.Theyellowdotrepresents the unperturbed case. The area S of an angular sector, which
astarrotatingwithspeedθ˙inaframeofreference(dashedline) is between two neighboring orbits and two successive stars in
corotatingwiththespiralarmsattheangularvelocityΩ . eachorbit,istransformedintoS′whentheperturbationisintro-
p
duced.Thetwoareascontainthesamemass,sothatthedensity
is proportional to 1/S′. There are two reasons for the change
indistancebetweenneighboringstars.Intheradialdirection,it
TheHamiltonianforsuchasystemisgivenby is because the orbitsapproachorrecede one fromthe other. In
the azimuthal direction, it is because the stars travel at differ-
H = 1 p 2+ J12 −Ω J +Φ (R)+Φ (R,ϕ), (19) ent angular velocities along the orbit. Due to the conservation
2" r R2 # p 1 0 1 of angular momentum, the stars move more slowly when they
areatlargergalacticradius,andgetclosertooneanother, since
wherep and J arethelinearandangularmomentumpermass thefluxisconserved.Thisiseasytotakeintoaccount,sincethe
r 1
unit,respectively.ItisimportanttonotethatJ ismeasuredwith spacingbetweenstarsis∆ϕ∝ϕ˙−1.Theresponsedensitycanbe
1
respecttotheinertialframe.Theequationsofmotionare writtenas
Σ 2πR ∆R ∆t
dR = pr, (20) Σresp = cTR∆cR∆ϕc , (25)
dt
dp J2 ∂Φ (R) ∂Φ (R,ϕ) whereΣc isthedensity,Rc istheradius,and∆Rc isthespacing
r = 1 − 0 − 1 , (21) between two successive orbits, all of them in the unperturbed
dt R3 ∂R ∂R
case, while the same quantities in the denominatorrefer to the
perturbedsituation.Thequantity∆tisthetimespentbyastarto
dϕ J
= 1 −Ω , (22) movefromϕtoθ+∆ϕ,whileT istheperiod.Inpractice,what
dt R2 p
wedoistodividethecirclesinNsectors,thusmakingitpossible
dJ ∂Φ (R,ϕ) tofindthepositionsRandthetimeforeachangle,definedbyi∆θ
1 =− 1 . (23) (i=1..N).Thereforewecanfindthetime∆tspentbythestarto
dt ∂ϕ
movebetweentwoadjacentsectors.
Sinceweareinterestedinthepositionofmaximumresponse
The equations above were integrated using the 6th order
densityforeachorbit,Eq.25reducesto
implicit Runge-Kutta-Gauss (RKG) (see Sanz-Serna&Calvo
1994,formoredetails),withafixedtimestep dt = 10−1 Myrs. ∆t
To find the stable periodic orbits, we applied the method of Σresp ∝ R∆R∆θ. (26)
6
T.C.Junqueiraetal.:ThespiralstructureoftheGalaxy
AsthevaluesofΣ ,R ,∆R ,andT donotchangeinagiven
c c c
circle,theycanbeignoredinEq.25.Thusthepositionofmax-
imum density is the sector where Eq. 26 reaches its maximum
value.
Weareinterestedinthepositionofmaximumresponseden-
sityforeachorbit.Thepositionofthedensitymaximaisshown
inFig.8.
10
5
c)
kp 0
Y (
-5 Fig.9.Thisfigureshowsaplotoftheobservedarmstructureof
theGalaxyinthesolarneighbourhood.TheSunisatthecenter
of the figure, and the distances on the two axes are relative to
theSun.TheGalacticcenterisdownwardandoutsidethefigure.
-10-10 -5 0 5 10 Different kinds of objects have been added to better trace the
X (kpc) structure, all of them shown with a same symbol (gray dots):
Cepheids,openclusters,CSsources(seetextfordetails).Curve
Fig.8.Hereweshowaseriesofclosedorbitsintheplaneofthe
a, in brown,represents a stellar orbitat the 4:1 resonance, and
GalaxyobtainedusingtherotationcurveofEq.17.Thespirals
curvebanorbitatthe6:1resonance.Curvecisnotidentifiedin
indicatedin green representthe imposedperturbation;theyco-
termsofresonances,sinceitisalinearfittotheobservedpoints.
incidewiththeresponsedensity(reddots)upto4:1resonance.
Theotherlines(twoinred,oneingreen)arelogarithmicspirals
Theredcircleisthecorotationradius(R =8.4kpc).Theper-
cor fittedtothedata.
turbationamplitudeisζ = 600km2 s−2 kpc−1.Theyellowdot
0
showstheSunposition.
clusters back to their birthplaces. As shown by Dias&Le´pine
(2005), the result, R = (1.06±0.08)R , does not have any
CR sun
strongdependenceon the adoptedrotation curve.A seconddi-
rect observation is the position of the ring-shaped HI void at
6. Results corotation(Amoˆresetal. 2009), anda thirdone thepositionof
the square-shapedspiralarmassociated with the 4:1resonance
6.1. AmodelforourGalaxy
(Le´pineetal.2011a).Theexistenceofa gapinthedistribution
Throughoutthiswork,wehaveadoptedtheparametersthatap- ofopenclustersandofCepheids,alongwithastepinthemetal-
pearedtoustocorrespondtoourGalaxy,likethosepresentedin licity distribution, is naturally explained in terms of the coro-
Table1,andtherotationcurve.Inthenextsections,wepresent tation resonance (Le´pineetal. 2011b). What we call a direct
bothgeneralresults,whicharevalidinprincipleforotherspiral methodisonethatdoesnotinvolveanycomplexmodelwithun-
galaxies,andspecificresults,whichdependontheexactchoice certainparametersandthereforegeneratesrobustresults.Thisis
ofparameters.Oneofthemostimportantchoiceswhenweapply notthecaseofn-bodysimulations,orchemicalevolutionmod-
amodeltoagalaxyisthatofthepatternspeed,orequivalently, els.Based ontheresultsabove,weadoptRCR = 8.4±0.2kpc;
iftherotationcurveisknown,ofthecorotationradius. accordingly,withEq.17,wefindΩp =23kms−1kpc−1.
In the last few decades, many papers have pointed to a Inthepresentwork,wehavenotinvestigatedthespiralstruc-
corotation radius of the Galaxy situated close to the orbit turebeyondcorotation,ataskleftforfuturework.Aswediscuss
of the Sun (Marochniketal. 1972, Creze&Mennessier 1973, inthenextsections,thereisachangeinthenatureofthearmsat
Mishurov&Zenina 1999, among others). An additional argu- the4:1resonance.OnlybetweentheILRandthatresonanceare
ment is that the classical theory of spiral arms tells us that thearmsexplainedbythecrowdingofstellarorbits.
the arms exist between the Inner and Outer Lindblad reso- There are models of the spiral structure that are in conflict
nances and that the corotation falls roughly midway between withthecorotationradiusthatweadvocate.Forinstance,model
these resonances. The data from the literature collected by oftheinnerGalaxybyDehnen (2000)suggeststhattheradius
Scarano&Le´pine (2012) supportsthe theory,showingthat the oftheOuterLindbladResonance(OLR)oftheGalacticbarlies
corotationoccursaboutmidway between extremities of the re- inthevicinityoftheSun.Hismodeldoesnotbelongtothecate-
gion where the arms are seen. In our Galaxy, the spiral arms goryofdirectmethods,aswedefinedabove,sinceitisasimula-
extend from about3 kpc to 13 kpc (e.g., Russeil 2003), which tionandbynature,basedonmanyapproximationsanduncertain
wouldsituatethecorotationatabout8kpc.Recently,a fewdi- parameters.Theauthordoesnotclaimthathepresentsaprecise
rectmethodsallowedthedecreaseoftheuncertaintyonthecoro- determinationoftheOLRofthebarandsuggeststhatthisradius
tation radius to a few hundredparsecs. One exampleis the de- is≈6-9kpc.ArecentreviewofpatternspeedsintheGalaxyis
terminationofthepatternspeedbyintegratingtheorbitsofopen given by Gerhard (2011). He recognizesthat the pattern speed
7
T.C.Junqueiraetal.:ThespiralstructureoftheGalaxy
Fig.10. Rotation curvesand correspondinginnerorbitsfordifferentstrengthsof the bulge.At the top are the rotationcurvesfor
differentvaluesof f (seetext).Fromlefttoright,theparameter f is3.2,2.2,1.2,and0,respectively.Belowaretheorbitsassociated
b b
witheachrotationcurve.Thebluelineisthebulgecontribution,thegreenlineisthediskcontribution,andtheredlineisthebulge
+diskcontribution.
of the spiral structure is about 25 km s−1 kpc−1, but favors the concentratedaround24kms−1kpc−1(Scarano&Le´pine2012),
ideathatthebarrotatesatahighervelocity.Thereviewdoesnot inotherwords,theyarenotarbitrary.
mention the work of Ibata&Gilmore (1995), who determined We wouldliketo callattentiontoan interestingeffect.One
therotationspeedofthebulge,whichwasfoundtobethesame can see that the maximum response density starts to be out of
thatweadoptforthespiralarms.Inastudyofthemorphology phasewithrespecttothelogarithmicspiralperturbationnearthe
of bars in spiral galaxies,Elmegreen&Elmegreen(1985) con- 4:1 resonance. This suggests that the deviation from a perfect
cluded that in late-type spirals the bar may extend only to the logarithmicspiralingalaxiescanbeunderstoodnaturally asan
ILR.Thisisthehypothesisthatweadopttocompareourmodel effectofthe4:1resonance.Aconsequenceofthisistheappear-
with the Galaxy, based on the fact that the solar neighborhood ance of bifurcations. They happen because when the response
containsspiralarmsonly. densitystartstobeoutofphasewiththeimposeddensity,partof
thematterremainsfollowingtheimposeddensity.Thisresultsin
aweakerpieceofarm.Ingalaxieswithastructuresimilartothat
6.2.ThestructurebetweentheILRandthe4:1resonance
oftheMilkyWay,theappearanceofbifurcationsispossiblyas-
sociatedwiththe4:1resonance,ratherthanwiththecorotation.
Many studies of self-consistency have been per-
This is not a new result, since it was already discovered from
formed (Contopoulos&Grosbøl 1988; Patsisetal. 1991;
a theoretical approach, for instance, by Patsisetal. (1994) and
Amaral&Lepine 1997; Pichardoetal. 2003; Martosetal.
Patsisetal.(1997).Thisisalsoarobustresultbecauseitdoesnot
2004; Antojaetal. 2011). Contopoulos&Grosbøl (1986)
dependonthedetailsofagivenrotationcurve.Observationally,
already studied the response density and concludedthat strong
the workof Elmegreen&Elmegreen(1995) also states thatbi-
spiralsdo notexistbeyondthe 4:1innerresonance.Thisresult
furcationsstartattheendofthetwoprominentsymmetricarms.
wasconfirmedbytheotherworksandalsobythepresentwork.
InFig.8weshowthelociiofmaximumresponsedensityofour
model.Itcanbeseen thatthereis averygoodself-consistency 6.3.Beyondthe4:1resonance
of the shape of the spiral arms in the region inside the 4:1
resonance. This self-consistency is particularly satisfactory in In our model, there is a clear change in the nature of the arms
the case of our model, since a potential perturbation with a beyondthe4:1resonance.Therearenomorearmsproducedby
Gaussian profile generates a very similar response density and theconcentrationofstellarorbits,butonlydiscreteresonantor-
responsepotential.Thefactthattwowell−behavedlogarithmic bits. Howcoulda resonantorbitbehavelike a spiralarm?This
spiral arms extend about the 4:1 resonance is a robust result, mayhappenifstarsarecapturedwithinresonances,similarlyto
whichwasencounteredbythevariousworksmentionedabove, asteroidsintheSolarsystem,forwhichthetrappingmechanism
although they use a variety of axisymmetric potentials and wasdescribedbyGoldreich(1965).Forinstance,intheasteroid
perturbation potentials. It also corresponds to an observed belt,the3:2,4:3,and1:1resonanceswithJupiterarepopulated
characteristic of spiral arms, since Elmegreen&Elmegreen byclumpsofasteroids.Similarly,thestarscapturedwithinres-
(1995) observed that most spiral galaxies have two prominent onances could produce a local increase of density and turn it
symmetricarmsintheirinnerregions,insideapproximately0.5 intoakindofarm.Ifthismodeliscorrect,whenappliedtoour
R ,whereR wasdefinedbythemastheradiusforwhichthe Galaxy,itwouldpredictthatsomeofthearmsseeninthesolar
25 25
surface brightness is 25 mag arcsec−2. These authors suggest neighborhoodhavetheformofresonantorbits.
that the termination of these arms could coincide with corota- Directevidence of an arm with the shape of a 4:1 resonant
tion.Thisrobuststructure,with a similaraspectin manyspiral stellarorbitwasfoundbyLe´pineetal.(2011a)usingastracers
galaxies,ispossiblytheonethatisabletodeterminethepattern molecular CS sources, Cepheids with a period longer than six
speed. That the pattern speeds are found to have a distribution days, and open clusters with ages less than 30 Myr. The map
8
T.C.Junqueiraetal.:ThespiralstructureoftheGalaxy
obtained in that paper is reproduced in Fig. 9. Here, the same
symbolhas been used forall the kindsof tracers, since we are
onlyinterestedinthe generalstructure.Itmustberemembered
thatnowadaysonlyintheSolarneighborhoodwecanhaveade-
taileddescriptionofthespiralstructure,sinceinterstellarextinc-
tiondoesnotallowopenclustersandCepheidstobeobservedat
morethanafewkpc.
Figure 9 can beseen as a zoomofthe upperpartof Fig. 8,
in the region aroundthe Sun. One can observe the presence of
a structure with an angle of about 90◦ (labeled “a” and brown
inthefigure),whichwasidentifiedbyLe´pineetal.(2011a)with
the 4:1resonance.Theargumentwasnotonlythe fitpresented
in this figure, but also much more the points observed along a
straightportionofabout7kpcin length,inadditiontothe fact
thattheresonanceisexactlywhereitispredictedtobe,basedon
the knownpatternspeed.Now,we also presenta fit ofan orbit
atthe6:1resonancetomanyoftheobservedpoints(curvewith
label“b”,inblue).Thisgreatlyresemblesa predictedstructure
inFig.8,whichpassesclosetothecornerofthearmatthe4:1 Fig.11.Poincare´ diagramsandtheirrespectivefamilies(x1,x2)
resonanceandclosetotheSun,inadditiontopresentinganacute ofperturbedperiodicorbits(seeContopoulos1975,fordetails).
cornerinthesecondquadrantoftheGalaxy. InbothcaseswesettheenergyatRc = 2.2kpc,andthepertur-
bationparametersare the same asTable 1. The onlydifference
Stableperiodicorbitsattractaroundthemquasi-periodicor- is the bulgefraction f ;at thetop we have f = 3.2and below
bits,asindicatedbythecontoursoftheislandsinthePoincare´’s b b
wehave f =0.
surface of section. It seems that in the region of the disk dis- b
cussed here, quasi-periodic orbits are trapped around the sta-
ble periodic orbits of the 4:1 and 6:1 families, with very small region of the galactic plane but worse in another. So, we also
deviations from the periodic orbits themselves. As a result the leavethistaskforfuturework.
quasi-periodic orbits could be considered as building a “thick Like the previous works on self-consistent spiral structure,
periodic orbit” structure. These arms, which are composed of we did not extend the calculation of stable orbits beyond the
similarquasi-periodicorbits,wouldhaveverysmalldispersions corotationradius.We expectto see resonantorbitsas well, but
ofvelocities. the comparisonswith real arms will be even more difficult be-
Finally,onecanseeinFig.8,justbelowthecornerofthe4:1 causethedistancesandshapesoftheexternalarmsofourGalaxy
resonance,anorbitthatcrossesitintwopoints.Thisresembles arepoorlyknown.Whatisknown,fromanobservationalpoint
the arm-like structure with label C in Fig. 9. We consider that ofview,isthatthespiralarmsshowstrongdeparturesfromloga-
thisobservedstructureisprobablyreal,duetothelargenumber rithmicspiralsintheexternalregionsofgalaxies.Inmostcases,
of tracers that fall on it. This suggests an interesting concept. the arms look more like a sequence of straight segments that
Sinceresonantorbitscancrosseachother,itisalsopossibleto roughlyfollowaspiralpath(e.g.,Chernin1999).
havearm-likestructureswhichcorrespondtothemcrossingeach
other. This contrasts with the classical view of spiral arms, in
6.4.Thebulgeandthebar
whichnosuchintersectionsexist.
Ofcourse,theinterpretationwegiveofsomelocalstructures About50%ofspiralgalaxieshaveabar,eitherstrongorweak,
may not seem convincing. One possible argument against it is and there are both kinematic and photometric evidences that
that the resonantorbitsdo notseem to be uniformlypopulated the MilkyWay hasa bar(Binney&Tremaine2008, Sect. 6.5).
by stars all around them. However, the mechanism of the cap- Shenetal. (2010) arguethat the Galactic bulge is a part of the
ture of stars by resonancesdeservesdeeper study to determine diskandsuggestthatthebulge(inreality,aboxypseudobulge)
ifsomeregionsoftheorbitsshouldpresentalargerpopulation. oftheMilkyWay isanedge-on,buckledbarthatevolvedfrom
The solarneighborhoodis rich in structuresthathave notbeen acold,massivedisk.
explainedby any model.One couldarguethat they are not ex- Inthissection,weanalyzetheconnectionbetweenthespher-
plainablebecausetheyareduetostochasticevents.Theideathat ical central bulge and the inner stellar orbits. Figure 10 shows
weproposehere,thatsomeofthesestructurescorrespondtores- the rotation curves from Eq. 18. The scale lengths for the disk
onantorbits,canbeverified,sinceateachpointofsuchanorbit and the bulge are fixed at ǫ−1 = 2 kpc−1 and ǫ−1 = 0.4 kpc−1,
d b
we have a particular velocity of the stars that are trapped into V = 220 km s−1. The parameter f , defined in Eq. 18 as
max b
it. However, this is not the scope of the present paper. We did “the bulge fraction”, is equivalent to a measure of the “bulge
notmakeanyadjustmentoftheparametersofthemodelinorder strength”, since the disk component is kept constant (only the
tofit observedstructures.Parameterslike V (whichaffectsthe bulgeisvaried),oreventoameasureofthebulgedensity,since
0
rotation curve), the exact depth and location of the local mini- its scale size is kept constant. We will refer to it as the bulge
mumin the rotationcurve, Ω , andthe width, depth,andpitch strength.Wevaried f from3.2downto0.Beloweachrotation
p b
angleoftheimposedperturbationcouldallhavebeensubmitted curveinFig.10,weshowthefamilyofperturbedperiodicorbits
tominorvariations.Ourexperienceisthatthisisalengthytask, trappedaroundunperturbedperiodicorbitsfrom R = 2upto4
c
with minor variations of the structures produced and some re- kpc;seeSect.5.1formoredetails.Foragivenenergy h,which
dundancy(differentcombinationscangivesimilarresults).The correspondtoaradiusR ,wemayhavemorethenoneperiodic
c
comparisonwith the observeddata isdifficultin partdue to its orbit.Thefamilyofperiodicorbitsliesatthecenteroftheisland
poorquality.Besides,somefittedmodelsmaylookbetterinone inthePoincare´diagram.Wecallitfamily x ifthecenterofthe
1
9
T.C.Junqueiraetal.:ThespiralstructureoftheGalaxy
islandliesatR≤R ,andfamily x ifR≥R ,(seeContopoulos We must add to the list of similarities between our model
c 2 c
1975). andtherealGalaxytheagreement,inFig.8,betweentheorien-
tationoftheelongatedinnerorbits,wherethespiralarmsseem
The parameters for the spiral perturbation are the same as
to begin, with respect to the Sun. This is very similar with the
giveninTable1.Theorbitsintheinnerregion,almostperpen-
observed angle between the Sun and the extremities of the bar
diculartothemoreexternalorbits,belonginFig.10,tothefam-
(see,e.g.,Le´pineetal.2011a).
ilyx ofperiodicorbit.Theyaremoreprominentwherewehave
2
astrongcentralbulge(f =3.2);asthebulgeweakens,thefam-
b
iliesoforbits x disappear,aswecanseeinFig.11.Inthecase 7. Conclusions
2
where f , 0, we havetwo familiesofstable orbits(x ,x ).In
b 1 2 We have presented a new description of the spiral structure of
Fig. 10, we plotthe orbits correspondingto families forwhich
galaxies,basedontheinterpretationofthearmsasregionswhere
wehaveonlysmalldeviationsfromR andthosewitharealistic
c thestellarorbitsofsuccessiveradiicomeclosetogether,produc-
distribution of velocities. As we can see in Fig. 11 on top, the
ing large stellar densities. In other words, the arms are seen as
familyx hasalargedeviationfromR withavelocityintheor-
derof p1≈−150kms−1generatingavceryeccentricorbit,while groovesinthepotentialenergydistribution.Suchanapproachis
thefamirlyx isclosetoR with p ≈20kms−1;thus,weexpect not new (e.g., Contopoulos&Grosbøl 1986; Amaral&Lepine
2 c r 1997). The innovation is that the potential energy perturbation
thatitwouldbethefavoritefamilytopopulatethisregion.Itis
thatwe haveadoptedisitself likea groovethatfollowsaloga-
clearthatstrongbulgesfavorthe x familyin theinnerregions
2 rithmicspiral,withaGaussianprofileinthedirectionperpendic-
oftheGalaxy,whilewithoutabulgewehaveonlythefamilyx .
1 ulartothearms.Thisrepresentsanimportantstepinthedirec-
Avisualinspectionoftheorbitsassociatedwithdifferentro- tion ofself-consistency,since this potentialperturbationgener-
tationcurvesshowselongatedstellarorbitsintheinnerregions ates,bymeansofthestellarorbits,spiralarmswithasimilarpro-
ofthedisk forrotationcurveswiththe presenceofa peaknear file.Inpreviousclassicalmodels,thepotentialperturbationwas
thecenter,suggestingtheformationofabarwithahalf-size∼3 representedby a sine function(ora sum of two sine functions,
kpc.Thismodelshowstheexistenceofacorrelationbetweenthe if four arms are presented) in the azimuthal direction, but the
formationof a barand the magnitudeof the centralbulge.The responsepotentialresembledthegroove-likeoneofthepresent
peakintherotationcurvebecomesvisiblewhen f isrelatively work, so that the self-consistency was poorer. One of the clas-
b
large(3.2or2.2).Astrongbulgegeneratesaprominentvelocity sicalmodels(Pichardoetal.2003)doesnotuseasinefunction
peak in the rotation curve and modifies the inner stellar orbits. forthe imposedpotential,butits complexitydoesnotallow an
Inotherwords,thestrengthofthebulgehasaninfluenceonthe easycheckofself-consistency.
existenceofabar.Thisbarappearsasaconsequenceofthesame Anew parameterappearsin ourdescription,allowingusto
potentialperturbationthatweintroducedforthespiralarms,so controlthewidthof thearms.We founda relationbetweenthe
thatthebarandthespiralarmshavethesamerotationvelocity. density contrastbetweenthe arm andinter-arm region,and the
Thetypeofbarthatwearediscussingherecorrespondstowhat perturbationamplitude,undertheassumptionofanexponential
isunderstoodasaweakbar.Inprinciple,thiswouldseemtobe disk.
morecompatiblewiththemodelofspiralarmspresentedinthis We confirm the conclusion of previous works that the 4:1
work,inwhichnoeffectofabaristakenintoaccountinthelarge inner resonanceis a fundamentalstructure of the disk and that
“spiral” regionthat we described.A baris weak if it generates similar strong arms do not appear beyond this resonance. This
onlyasmallfractionofthetotalgravitationalfieldaroundit.In resultissupportedbyobservations,since itis knownthatmost
suchsystems,thestellarorbitscanbedescribedbytheepicyclic spiralgalaxieshavetwosymmetricprominentarmsintheirinner
theoryplus a weak drivingforce due to the periodic motionof regions(Elmegreen&Elmegreen1995).
the bar. In our Galaxy, the mass associated with the spherical ThemodelisappliedtoourGalaxy,usingadescriptionofthe
bulgeismuchlargerthanthatassociatedtothebar,asdeduced axisymmetricpotentialsimilartothatderivedfromtheobserved
frominfraredluminosity(e.g.,Le´pine&Leroy(2000)).Onecan rotationcurve,alongwiththeknownpatternrotationspeed and
evenexpectthatatadistanceoftheorderof1kpcfromthecen- pitchangle.The rangeof valuesforthe perturbationamplitude
ter,outsidethebulgebutunderitsstronginfluence,thepotential compatiblewithobservationalevidencesofthecontrastdensity
is Keplerian,with a dominantmass atthe center.In such a po- is 400−800 km2 s−2 kpc−1. Using a density contrast of about
tential, the orbits are elliptic, with the center of the Galaxy at 20%,anacceptedaveragevalue,the perturbationamplitude ζ ,
0
one focus of the ellipse, like the orbits of comets in the solar is equalto 600 km2 s−2 kpc−1. It is foundthatthe orbitsin the
system.Inepicycletheory,theseare1:1orbits.Twofamiliesof regionbetweenthe4:1resonanceandcorotationdonotreinforce
suchorbitsalignedalongastraightline,oneoneachsideofthe spiralarmslikeintheinnerregions,butonlyfeaturessimilarin
bulge,wouldbeinpartresponsibleforthebar.Thealignmentof shapewiththeperiodicorbitsexistinginthisregion.Ifthemodel
suchelongated,one-sidedorbitswiththeinnerextremityofthe can validly explain the spiral arms in the solar neighborhood,
spiralarmswouldoccurnaturally,sincestarstravelslowlywhen thenthedynamicsinthatregionisdeterminedbythemotionof
theyareattheapogalacticpartsoftheirorbitsandthisfacilitates starsinquasi-periodicorbits.
synchronization.Of course, we are not presenting here a com- Interestingly,anumberofobservedstructuresareverysim-
pletemodelofaweakbar,butonlyarguingthattheexistenceof ilar to the predicted resonant orbits. The similarities of the
aweakbarinourGalaxyshouldnotberejected.Ifweconsider model with the Galaxy include the orientation of the bar, the
thattheoriginofthespiralstructureisrelatedtotheinteraction size and orientation of the 4:1 resonance orbits as revealed by
withanexternalgalaxy,itismorelogicaltothinkthatthebaris Le´pineetal.(2011a),andpossiblyaresonantorbitwithoutward
inducedbythe spiralstructure,andnotvice versa.Aninterest- peaksor“corners”and invertedcurvaturebetweenthe corners,
ingpossibilityisthatsincetheonlyimposedperturbationhereis situatedclosetotheSun.
thatofthe spiralarms,ourresults supportthe ideathata pseu- The idea that the spiral structure can be self-consistent (a
dobulge could have evolved from a perturbed disk (Shenetal. certain perturbing potential gives rise to stellar orbits that re-
2010). producethisperturbation)pointstowardsalong-livedstructure.
10