Self-consistent models of quasi-relaxed rotating stellar systems

0.87 MB·English

by  A. L. Varri

Astronomy&Astrophysicsmanuscriptno.VB˙2011f c ESO2012
(cid:13)
January10,2012
Self-consistent models of quasi-relaxed rotating stellar systems
A.L.VarriandG.Bertin
DipartimentodiFisica,Universita`degliStudidiMilano,ViaCeloria16,I-20133Milano,Italy
e-mail:[email protected]; [email protected]
Received19October2011;toappear
ABSTRACT
2 Aims.Twonewfamiliesofself-consistentaxisymmetrictruncatedequilibriummodelsforthedescriptionofquasi-relaxedrotating
1 stellarsystemsarepresented.Thefirstextendsthewell-knownsphericalKingmodelstothecaseofsolid-bodyrotation.Thesecond
0 ischaracterizedbydifferentialrotation,designedtoberigidinthecentralregionsandtovanishintheouterparts,wheretheimposed
2 energytruncationbecomeseffective.
Methods.ThemodelsareconstructedbysolvingtherelevantnonlinearPoissonequationfortheself-consistentmean-fieldpotential.
n
Forrigidlyrotatingconfigurations,thesolutionsareobtainedbyanasymptoticexpansionbasedontherotationstrengthparameter,
a
followingaproceduredevelopedearlierbyusforthecaseoftidallygeneratedtriaxialmodels.Thedifferentiallyrotatingmodelsare
J
constructedbymeansofaspectraliterativeapproach,withanumericalschemebasedonaLegendreseriesexpansionofthedensity

9 andthepotential.
Results.Thetwoclassesof modelsexhibit complementary properties. Therigidlyrotatingconfigurations areflattenedtowardthe
] equatorialplane,withdeviationsfromsphericalsymmetrythatincreasewiththedistancefromthecenter.Formodelsofthesecond
A
family,thedeviationsfromsphericalsymmetryarestrongestinthecentralregion,whereastheouterpartstendtobequasi-spherical.
G The relevant parameter spaces are thoroughly explored and the corresponding intrinsic and projected structural properties are de-
scribed.Specialattentionisgiventotheeffectofdifferentoptionsforthetruncationofthedistributionfunctioninphasespace.
.
h Conclusions.Modelsinthemoderaterotationregimearebestsuitedtoapplicationstoglobularclusters.Forgeneralinterestinstellar
p dynamics,athighvaluesoftherotationstrengththedifferentiallyrotatingmodelstendtoexhibitatoroidalcoreembeddedinanoth-
- erwisequasi-sphericalconfiguration.Physicallysimpleanalyticalmodelsofthekindpresentedhereprovideinsightsintodynamical
o
mechanismsandmaybeausefulbasisformorerealisticinvestigationscarriedoutwiththehelpofN-bodysimulations.
r
t
s Keywords.Methods:analytical–globularclusters:general
a
[

1 1. Introduction the spherical isotropic King (1966) models, which incorporate
v inaheuristicwaytheconceptoftidaltruncation,provideastan-
9 A large class of stellar systems is expected to be in a quasi- dard reference framework (see Zocchi et al. 2012 for a photo-
9 equilibriumstatecharacterizedbyadistributionfunctionnotfar metric and kinematic study of a sample of Galactic globular
8 froma Maxwellian. Mechanismsthatare thoughtto operate in clusters under different relaxation conditions). Recently, King
1
this direction range from standard two-body gravitationalscat- models have been extended to the full triaxial case, to include
.
1 tering (Chandrasekhar1943) to violentrelaxation, instabilities, explicitlythethree-dimensionalroleofanexternaltidalfieldin
0 andphasemixingforcollisionlesssystems(Lynden-Bell1967). thesimplecase ofa circularorbitofthe clusterwithinthehost
2 In particular, for relatively small stellar systems, such as glob- galaxy (Bertin & Varri 2008, Varri & Bertin 2009, in the fol-
1 ularclusters, two-starcollisionalrelaxationis thoughtto bere- lowingdenotedasPaperI andII,respectively;see also Heggie
:
v sponsibleforsuchquasi-relaxedcondition,whileforlargestel- &Ramamani1995);thesetriaxialmodelscanthusbeseenasa
i lar systems, such as elliptical galaxies, incomplete violent re- collisionlessanaloguesofthepurelytidalRocheellipsoids.
X
laxation has likely acted during the formation process so as to
For astronomical applications, one of the main drivers (or
r bringthesystemtoastateinwhichsignificant(radiallybiased)
a empiricalclues) in the abovemodelinginvestigationsis the is-
anisotropy is present only for weakly bound stars. To be sure,
sue of the origin of the observed geometry of stellar systems,
full relaxation would lead to a singular state, characterized by
beingitknownthatpressureanisotropy,tides,orrotationcanbe
infinitemass;theso-calledisothermalspheresolutionmaypro-
responsible,separately,fordeviationsfromsphericalsymmetry.
videanapproximaterepresentationonlyoftheinnerpartsofreal
In the above-mentioned studies, very little attention was actu-
stellarsystems.
ally placed on the role of rotation. For ellipticals, most of the
Inlinewiththeaboveconsiderations,severalphysicallymo- attentionthatled tothedevelopmentofstellar dynamicalmod-
tivatedself-consistentfinite-massmodels(thuscharacterizedby els, after the first kinematical measurements became available
significant,butnotarbitrary,deviationsfromapureMaxwellian) in the mid-70s, was taken by the study of the curious behav-
have been constructed and studied, with interesting astronom- iorofpressure-supportedsystemsinthepresenceofanisotropic
ical applications. For the description of partially relaxed sys- orbits(seealsoSchwarzschild1979,1982;deZeeuw1985).In
tems, self-consistent anisotropic models have been introduced contrast, very little effort has been made in modeling rotation-
to describe the products of incomplete violent relaxation (see dominatedellipticals,eventhoughtheentirelow-massendofthe
Bertin & Stiavelli 1993 and Trenti et al. 2005; and references distributionofellipticalgalaxiesmightbeconsistentwithapic-
therein).Forthedescriptionofquasi-relaxedglobularclusters, ture of rotation-inducedflattening (e.g.,see Davieset al. 1983,
1
A.L.VarriandG.Bertin:Self-consistentmodelsofquasi-relaxedrotatingstellarsystems
andmorerecentlyEmsellemetal.2011);similarcommentsap- kinematical measurements, in which rotation, when present, is
plytobulges. clearlyidentified).
For globular clusters, given the fact that they only exhibit On the theoretical side, two general questions provide fur-
modestamountsofflatteningandgiventhesuccessofthespher- ther motivation to study quasi-relaxed rotating stellar systems.
icalKingmodels,littleworkhasbeencarriedoutinthedirection Ontheonehand,manypapershavestudiedtheroleofrotation
ofstationaryself-consistentrotatingmodels(withsomenotable in the general context of the dynamical evolution of globular
exceptions,thatis Wolley & Dickens 1962, Lynden-Bell1962, clusters, but a solid interpretation is still missing. Early inves-
Kormendy& Anand1971, Lupton& Gunn1987, andLagoute tigations(Agekian1958, Shapiro& Marchant1976) suggested
&Longaretti1996).Therefore,asfarasrotation-dominatedsys- thatinitiallyrotatingsystemsshouldexperiencealossofangular
tems are concerned, much of the currently available modeling momentuminducedbyevaporation,thatis,angularmomentum
tools go back to the pioneering work of Prendergast & Tomer would be removed by stars escaping from the cluster. Because
(1970),Wilson(1975),andToomre(1982),intendedtodescribe of the small numberof particles, N-bodysimulationswere ini-
ellipticals,andofJarvis&Freeman(1985)andRowley(1988), tially (Aarseth 1969, Wielen 1974, and Akiyama & Sugimoto
devoted to bulges. In general, we may say that only very few 1989)unabletoclearlydescribethecomplexinterplaybetween
rotatingmodelswithexplicitdistributionfunctionarepresently relaxationandrotation.Laterinvestigations,primarilybasedon
known(fora recentexample,see Monarietal. in preparation). a Fokker-Planckapproach(Goodman1983, Einsel & Spurzem
In this context, one should also mention the interesting work 1999,Kimetal.2002,andFiestasetal.2006)haveclarifiedthis
by Vandervoort (1980) on the collisionless analogues of the point, not only by testing the proposed mechanism of angular
MaclaurinandJacobiellipsoids. momentumremovalbyescapingstars,butalsobyshowingthat
On the empirical side, a deeper study of quasi-relaxed ro- rotation accelerates the entire dynamical evolution of the sys-
tating stellar systems is actuallyencouragedbythe widespread tem. More recent N-body simulations (Boily 2000, Ernst et al.
convictionthatthegeometryoftheinnerpartsofglobularclus- 2007,andKimetal.2008)confirmtheseconclusionsandshow
ters,wheresomeflatteningisnoted,shouldnotbetheresultof that, when a three-dimensionaltidal field is included, such ac-
tides, but rather of rotation (King 1961). In other words, it is celerationis enhancedevenfurther.The mechanismof angular
frequentlybelievedthattidesandpressureanisotropy(anddust momentumremovalisgenerallyconsideredtobethereasonwhy
obscuration),eventhoughplayingsomeroleinindividualcases, Galactic globularclustersare muchrounderthanthe (younger)
shouldnotbe consideredas the primaryexplanationof the ob- clustersintheMagellanicClouds,forwhichanage-ellipticityre-
servedflatteningofGalactic globularclusters.Suchconclusion lationhasbeennoted(Frenk&Fall1982),butothermechanisms
issuggestedbytheWhite&Shawl(1987)databaseofelliptici- might operate to producethe observed correlations(Meylan &
ties.However,forthisdatabasewenotethat:(i)theclusterflat- Heggie1997,vandenBergh2008).
teningvaluesdonotallrefertoastandardisophote,suchasthe Ontheotherhand,theroleofangularmomentumduringthe
clusterhalf-lightradius(asalsonotedbyvandenBergh2008), initial stages of cluster formation should be better clarified. In
(ii) the data mostly refer to the inner regions. Such limitations the contextof dissipationlesscollapse, relativelyfew investiga-
are crucial because there is observational evidence that the el- tionshaveconsideredtheroleofangularmomentuminnumeri-
lipticity of a cluster dependson radius(see Geyer et al. 1983). calexperimentsofviolentrelaxation(e.g.,thepioneeringstudies
In turn, recentstudies by Chen & Chen (2010), in which these byGott1973;seealsoAguilar&Merritt1990).Interestingly,the
restrictions are addressed more carefully, suggest that Galactic finalequilibriumconfigurationsresultingfromsuchcollisionless
globularclusters are less roundthan previouslyreported,espe- collapseshowacentralregionwithsolidbodyrotation,whilethe
ciallythoseintheregionoftheGalacticBulge;atvariancewith externalpartsarecharacterizedbydifferentialrotation.
previous analyses, it seems that their major axes preferentially Inconclusion,masteringtheinternalstructureofspheroidal
point toward the Galactic Center, as naturally expected if their andtriaxialstellarsystemsthroughafullspectrumofmodels,in-
shapeisoftidalorigin. cludingrotation,isaprerequisiteforstudiesofmanyempirical
Theconnectionbetweenflatteningandinternalrotationhas and theoretical issues. In addition, it is required for the inter-
been discussed in detail by means of nonspherical dynamical pretationoftherelevantscalinglaws(suchasthe Fundamental
models in just a handful of cases, in particular for ω Cen (for Plane,whichappearstoextendfromthebrightest,pressuresup-
an oblate rotator nonparametric model, see Merritt 1997; for portedellipticalsdowntothelow-luminosityendofthedistribu-
an orbit-based analysis, see van de Ven 2006; for an applica- tionofearly-typegalaxies,andpossiblyfurtherdowntothedo-
tion of the Wilson 1975 models, see Sollima et al. 2009), 47 mainofglobularclusters) andforinvestigationsaimedatiden-
Tuc(Meylan&Mayor1986),M15(vandenBoschetal.2006), tifying the presence of invisible matter (in the form of central
andM13(Luptonetal.1987).Inaddition,specificallydesigned massive black holes or diffuse dark matter halos) from stellar
2D Fokker-Planck models (Fiestas et al. 2006) have been ap- dynamicalmeasurements.Itwouldthusbedesirabletoconstruct
plied to the study of M5, NGC 2808, and NGC 5286. We re- rotatingmodelstobetestedonlowluminosityellipticals,bulges,
call that the detection of internal rotation in globular clusters andglobularclusters,especiallynowthatimportantprogresshas
is a challenging task, because the typical value of the ratio of been made in the collection and analysis of kinematical data.
meanvelocitytovelocitydispersionisonlyofafewtenths,for Presumably,manyofthesestellarsystemsarequasi-relaxed.In
example V/σ 0.26,0.32for 47 Tuc and ω Cen, respectively whichdirectionsshouldweexploredeviationsfromthestrictly
≈
(MeylanandMayor1986;forasummaryoftheresultsforsev- relaxedcase?
eral Galactic objects, see also Table 7.2 in Meylan & Heggie In the present paper, we consider two families of axisym-
1997).However,greatprogressmadeintheacquisitionofpho- metricrotatingmodels:thefirstoneischaracterizedbythepres-
tometric and kinematical information, and in particular of the ence of solid-body rotation and isotropy in velocity space (see
propermotionofthousandsofstars(forωCen,seevanLeeuwen AppendixBinPaperIforabriefintroduction).Indeed,fullre-
etal.2000,Anderson&vanderMarel2010;for47Tucanae,see laxation in the presence of nonvanishingtotal angular momen-
Anderson&King2003andMcLaughlinetal.2006),makesthis tum suggests the establishment of solid-body rotation through
goalwithinreach(seeLaneetal.2009,Laneetal.2010fornew the dependence of the relevant distribution function f = f(H)
2
A.L.VarriandG.Bertin:Self-consistentmodelsofquasi-relaxedrotatingstellarsystems
on the Jacobi integral H = E ωJ (see Landau & Lifchitz
z
−
1967, p. 125). But, for applicationsto real stellar systems, one
maytakeadvantageofthefactthatthecollisionalrelaxationtime
may be large in the outer regions,so thatin the outerparts the
constraintofsolid-bodyrotationmightbereleased.Inparticular,
forglobularclusterswe may arguethattheouterpartsfallinto
a tide-dominated regime, for which evaporation tends to erase
systematicrotationevenifinitiallypresent,asconfirmedbythe
above-mentionedstudybasedontheFokker-Planckmethod.For
thetruncation,wemaythenconsideraheuristicprescriptionto
simulate the effects of tides, much like for the spherical King
models.
In view of possible applications to globular clusters, we
thus consider a second class of axisymmetric rotating models
based on a distribution function dependent only on the energy
and on the z-component of the angular momentum f = f(I)
where I = I(E,J ), with the propertythat I E for stars with
z
∼
relatively high z-component of the angular momentum, while
I H = E ωJ forrelativelylow valuesof J . Suchmodels Fig.1. The boundary surface, defined implicitly by ψ(rˆ) = 0,
z z
are∼indeedde−finedinordertohavedifferentialrotation,designed of a critical second-order rigidly rotating model with Ψ = 2.
Theconfigurationisaxisymmetric;the pointsontheequatorial
to be rigid in the center andto vanishin the outer parts, where
the energy truncation becomes effective. As far as the velocity planearesaddlepoints(seeSect.2.2fordetails).Rotationtakes
placearoundthezˆ-axis.Thespatialcoordinatesareexpressedin
dispersionisconcerned,thisfamilymayshowavarietyofpro-
appropriatedimensionlessunits(seeEq.(4)).
files (dependingon the values of the relevantfree parameters),
allofthemcharacterizedbythepresenceofisotropyinthecen-
tral region. We thus add two classes of self-consistent models
to the relatively short list of rotating stellar dynamical models 2. Rigidlyrotatingmodels
currentlyavailable.
2.1.Thedistributionfunction
Theconstructionofrigidlyrotatingconfigurationscharacterized
One aspect that plays an importantrole in defining a phys- by nonuniform density is a classical problem in the theory of
ically motivated distribution function, which often goes unno- rotating stars, starting with Milne (1923) and Chandrasekhar
ticed(butseeHunter1977,Davoust1977,andRowley1988),is (1933), but it basically remainedlimited to the study of a fluid
thechoiceofthetruncationprescriptioninphasespace.Thead- with polytropic equation of state, for which the solution of
vantagesandthe limitationsof alternativeoptionsavailable for the relevant Poisson equation can be obtained by means of a
the secondfamily ofmodelswill be discussed in detail. Inthis semi-analyticalapproach(for a comprehensivedescription,see
context,wewillalsoaddresstheissueofwhetherthesedifferen- Chaps.5 and10in Tassoul1978; foran enlighteningpresenta-
tiallyrotatingmodelsfallwithintheclassofsystemsforwhich tionofthegeneralproblemofrotatingcompressiblemasses,see
rotationisconstantoncylinders. Chap. 9 in Jeans 1929). The reader is referred to Paper I for a
discussionoftheapplicationofsomeofthemathematicalmeth-
odsdevelopedinthatcontexttotheconstructionofnonspherical
Thepaperisstructuredasfollows.Thepropertiesofthefam- truncated self-consistent stellar dynamical models. The devia-
ilyofrigidlyrotatingmodels,constructedonthebasisofgeneral tions from spherical symmetry studied in Paper I are induced
statistical mechanics considerations, are illustrated in Sect. 2. by the presence of a stationary perturbationcharacterized by a
The family of differentially rotating models, designed for the quadrupolarstructure,thatis,eitheranexternaltidalfieldorin-
applicationtorotatingglobularclusters,isintroducedinSect.3, ternalsolid-bodyrotation.Inparticular,inAppendixBofPaper
where we briefly describe the method used for the solution of IwebrieflyoutlinedtheextensionoftheKing(1966)modelsto
the self-consistent problem and discuss the relevant parameter the case of internal solid-bodyrotation, of which, after a short
space.Section4isdevotedtoastudyoftheintrinsicproperties summaryof the relevantdefinitions, we now providea fullde-
and Sect. 5 to the projected observablesderivedfrom differen- scriptionintermsoftherelevantintrinsicandprojectedproper-
tially rotating models. After illustrating in Sect. 6 the effect of ties.
differenttruncationprescriptionsinphasespace,wesummarize TheextensionofthefamilyofKingmodelsisperformedby
the results and present our conclusions in Sect. 7. The appen- consideringthedistributionfunction:
dices are devoted,respectively,to a discussion of the nonrotat-
itnivgeltirmunitcoatfioonurofpatmioinlifeosrotfhemsoedceolns,dtfoamthielyd,eatnadilstoofatdheescarlitpertinoan- fKr(H)= Ae−aH0 e−a(H−H0)−1 (1)
h i
of the codeused for the constructionof the differentiallyrotat- forH H and fr(H)=0otherwise,where
ingconfigurations.Astudyofthedynamicalstabilityandofthe ≤ 0 K
long-term evolution of the families of models introduced here H = E ωJ (2)
z
willbeaddressedbymeansofanextensivesurveyofspecifically −
designedN-bodysimulationsandwillbepresentedinfollowing denotes the Jacobi integral, with ω the angular velocity of the
papers(Varrietal.2011;Varrietal.,inpreparation). rigid rotation (hence the superscript r), assumed to take place
3
A.L.VarriandG.Bertin:Self-consistentmodelsofquasi-relaxedrotatingstellarsystems
aroundthez axis.ThequantitiesE and J arethespecificone-
z
−
starenergyandz-componentoftheangularmomentum,H rep-
0
resents a cut-off constant of the Jacobi integral, while A and a
are positive constants. In the corotating frame of reference the
Jacobiintegralcan be written as the sum of the kinetic energy,
the centrifugalpotential Φ (x,y) = (x2 +y2)ω2/2 (with the
cen
−
equatorialplane(x,y)perpendiculartotherotationaxisgivenby
the z axis), and the cluster mean-field potential Φ , to be de-
C
−
terminedself-consistently.Therefore,the dimensionlessescape
energycanbeexpressedas:
ψ(r)=a H [Φ (r)+Φ (x,y)] , (3)
0 C cen
{ − }
andtheboundaryofthecluster,implicitlydefinedbyψ(r) = 0,
isanequipotentialsurfaceforthetotalpotentialΦ +Φ .Note
C cen
that,byconstruction,inthelimitofvanishinginternalrotation,
this family of models reduces to the family of spherical King
(1966)models(seeAppendixAforasummaryofthemainprop-
ertiesofthefamilyinthenonrotatinglimit).
The construction of the models requires the integration of
the associated nonlinear Poisson equation, which, after scaling
Fig.2. Parameter space for second-order rigidly rotating mod-
thespatialcoordinateswithrespecttothescalelength
els. The solid line represents the critical values of the rotation
strengthparameterχ andthe greyregionidentifiesthe values
9 1/2 cr
r = , (4) (Ψ,χ) for which the resulting models are boundedby a closed
0 4πGρ0a! constant-ψsurface(subcriticalmodels).
canbewrittenas
ρˆ (ψ(int)) valid solution over the entire space, an asymptotic matching is
ˆ2ψ(int) = 9 K 2χ , (5) performedbetweentheinternalandtheexternalsolution,using
∇ − " ρˆ (Ψ) − #
K the Van Dyke principle (see Van Dyke 1975). This method of
solution is basically the same as proposedby Smith (1975) for
where
theconstructionofrotatingconfigurationswithpolytropicindex
ω2 n=3/2.Wehavecalculatedthecompletesolutionuptosecond-
χ (6)
≡ 4πGρ orderin the rotationstrengthparameterχ. The final solutionis
0
expressedinsphericalcoordinatesrˆ = (rˆ,θ,φ)andtheresulting
is the dimensionless parameter that characterizes the rotation configurationsare characterized by axisymmetry (i.e., the den-
strengthand sitydistributionandthepotentialdonotdependontheazimuthal
angleφ).Forthedetailsofthemethodfortheconstructionofthe
Ψ ψ(int)(0) (7)
≡ solutionthereaderisreferredtoPaperIandtoVarri(2012),in
isthedepthofthepotentialwellatthecenter.Thedimensionless whichthecompletecalculationisprovided.
densityprofileisgivenby
2.2.Theparameterspace
ρ (ψ) 5
ρˆ (ψ)= K =eψγ ,ψ , (8)
K Aˆ 2 ! The resulting models are characterized by two dimensional
scales (e.g., the total mass and the core radius) and two di-
where
mensionless parameters. As in the spherical King models, the
8π21/2A firstparametermeasurestheconcentrationoftheconfiguration;
Aˆ = e−aH0 (9) we thus consider the quantity1 Ψ (see Eq. (7)) or equivalently
3a3/2
c log(r /r ),wherer isthetruncationradiusofthespherical
tr 0 tr
≡
and γ denotes the incomplete gamma function. Therefore, the KingmodelassociatedwithagivenvalueofΨ.Theseconddi-
centraldensityisgivenbyρ = Aˆρˆ (Ψ).Outsidethecluster,for mensionlessparameterχ(seeEq.(6))characterizestherotation
0 K
negativevaluesofψ,weshouldrefertotheLaplaceequation strengthmeasuredintermsofthefrequencyassociatedwiththe
centraldensityofthecluster.
ˆ2ψ(ext) =18χ. (10) For every value of the dimensionless central concentration
∇
Ψthereexistsamaximumvalueoftherotationstrengthparam-
The relevantboundaryconditionsare given by the require-
eter, correspondingto a criticalmodel,for which the boundary
mentof regularityof the solution at the originand by the con-
isgivenbythecriticalconstant-ψsurface.Theboundarysurface
dition that ψ(ext) + aΦ aH at large radii. The Poisson
cen → 0 of a representative critical model (with Ψ = 2) is depicted in
(internal)andLaplace(external)domainsarethusseparatedby
Fig. 1; the surface is such that all the points on the equatorial
theboundarysurface,whichisunknownapriori.Therefore,we
planearesaddlepoints,wherethecentrifugalforcebalancesthe
have to solve an elliptic partial differential equation in a free
self-gravity.Werefertotheirdistancefromtheoriginasrˆ ,the
boundary problem. In particular, here we illustrate the proper- B
ties of the solutions of the Poisson-Laplace equation obtained 1 Intheliterature,theparameterΨisoftendenotedbyW ;herewe
0
by using a perturbationmethod,which also requires an expan- prefertokeepthesamenotationusedinPaperI,PaperII,andinother
sion of the solution in Legendre series. To obtain a uniformly articles.
4
A.L.VarriandG.Bertin:Self-consistentmodelsofquasi-relaxedrotatingstellarsystems
break-offradius.Intheconstant-ψfamilyofsurfacesassociated 0.012
with a given value of Ψ, the critical surface thus separates the
open from the closed surfaces. Consistent with the assumption
0.010
of stationarity, only configurationsboundedby closed surfaces
are considered here. This unique geometrical characterization
suggests that the effect of the rotation may be expressed also 0.008
intermsofanextensionparameter
δ= rˆtr , (11) t 0.006
rˆ
B
which provides an indirect measure of the deviations from 0.004
sphericity of a configuration, by considering the ratio between
thetruncationradiusofthecorrespondingsphericalKingmodel
0.002
and the break-off radius of the associated critical surface.
Therefore,a givenmodelmay be labelled bythe pair (Ψ,χ) or
equivalentlybythepair(Ψ,δ).ForgivenΨ,thereisthusamaxi-
0.000
mumvalueoftheallowedrotation,whichwemayexpressasχ
cr 0.0 0.2 0.4 0.6 0.8 1.0
orδ .Amodelwithδ<δ maybecalledsubcritical.
cr cr Χ(cid:144)Χ
ForeachvalueofΨ,thecriticalvalueoftherotationparam- cr
etercanbefoundbynumericallysolvingthesystem2 Fig.3. Values of the ratio between ordered kinetic energy and
gravitational energy t = K /W for selected rigidly rotating
∂rˆψ(rˆ=rˆB,θ=π/2;χcr)=0 (12) modelscharacterizedby Ψo=rd1|,3|,5,7(solid lines, from top to
(ψ(rˆ=rˆB,θ=π/2;χcr)=0, bottom)andχintherange[0,χcr].Forcomparison,thedashed
lineindicatesthevaluesoftforthesequenceofMaclaurinoblate
where the unknowns are rˆ and χ . In terms of the extension
B cr spheroidsinthelimitofsmalleccentricity.
parameter,foragivenΨ,thecriticalconditionoccurswhenδ =
δ =rˆ /rˆ 2/3.ThisvalueisobtainedbyinsertinginEq.(12)
cr tr B
≈
thezeroth-orderexpressionfortheclusterpotential,asdiscussed
selectedvaluesofΨandincreasingvaluesofχ,upto thecriti-
indetailinSect.2ofPaperIIforthetidalproblem(seep.251-
calconfigurationcharacterizedbyχ .Theparametertincreases
255 in Jeans 1929 for an equivalent discussion referred to the cr
linearly for increasing values of χ (with a slope dependent on
purelyrotatingRoche model,thatis a rotatingconfigurationin
the concentrationparameterΨ). A similar linear relationis ob-
which a small region with infinite density is surrounded by an
servedforsmallvaluesofeccentricity(e << 1)inthesequence
“atmosphere”ofnegligiblemass).
of Maclaurin oblate spheroids t(e) χ(e) 2e2/15 (for the
The parameter space for the second-order models is pre- ∼ ∼
definitions of the two parameters in the context of Maclaurin
sentedinFig.2(whichcorrespondstoFig.1ofPaperIIdescrib-
spheroids,see Eqs.(10.20)and(10.24)inBertin2000);inFig.
ing the tidal models). Two rotation regimes exist, namely the
3 the relevant linear relation is normalized with respect to the
regimeoflow-deformation(δ δ ,bottomleftcorner),where
≪ cr maximumvalueoftherotationstrengthparameterattainedinthe
internalrotationdoesnotaffectsignificantlythemorphologyof
sequenceofMaclaurinspheroidsχ = 0.11233(seeEq.(10),
theconfiguration,whichremainsveryclosetosphericalsymme- max
p.80inChandrasekhar1969).
try,andthatofhigh-deformation(δ δ ,closetothesolidline),
cr
≈
wherethemodelishighlyaffectedbythenearlycriticalrotation
velocity,especiallyintheouterparts.Notethattheactualregime 2.3.Intrinsicproperties
dependsonthecombinedeffectofrotationstrengthandofcon-
centration.Inotherwords,themodelsdescribedherebelongto Thegeometryofthemodels,reflectingthepropertiesofthecen-
the class of rotating configurations characterized by equatorial trifugal potential, is characterized by symmetry around the zˆ-
break-off(“regionofequatorialbreak-off”,seeFig.44,p.267in axisandreflectionsymmetrywithrespecttotheequatorialplane
Jeans 1929), for which the limiting case is givenby the purely (xˆ,yˆ). As expected, compared to the corresponding spherical
rotatingRochemodel. King models, the rotating models stretch out on the equatorial
Aglobalkinematicalcharacterization,complementarytothe planeandareslightlyflattenedalongthedirectionoftherotation
information provided by the rotation strength parameter, is of- axis(seeFig.4forthedensityprofilesofselectedcriticalsecond
feredbytheparametert= K /W,definedastheratiobetween ordermodelsevaluatedontheequatorialplaneandalongthezˆ-
ord
orderedkinetic energyand grav|ita|tionalenergy.Figure 3 illus- axis). In general,configurationsin the low-deformationregime
trates the relation between the two parametersfor models with (δ δcr),regardlessofthevalueofconcentrationΨ,arealmost
≪
indistinguishable from the corresponding spherical King mod-
2 Forbrevity,hereweomittheexplicitstructureofthesystem.The els.
readerisreferredtoEqs.(8)-(14)inPaperII,withrespecttowhichsev- Models in the intermediate and high-deformation regime
eraldifferencesoccur,becauseintheaxisymmetricrotationproblemthe (δ δ ) show modest deviations from spherical symmetry
cr
coefficientsoftheasymptoticseriesarebestexpandedin(normalized) in th≈e central regions, while they are significantly flattened in
Legendrepolynomials(ratherthansphericalharmonics).Inparticular,
the outer parts. The intrinsic eccentricity profile, defined as
withrespecttoPaperII:(i)thecoefficientswithm,0mustbedropped; e = [1 (bˆ/aˆ)2]1/2, where aˆ and bˆ are semi-major and semi-
(ii)thecoefficientswithl,0mustbemultipliedbythefactor(π/4)1/2, −
minoraxesoftheisodensitysurfaces,isamonotonicallyincreas-
because of the different normalization used in the two systems of or-
thonormalfunctions;(iii)theexpressionsareevaluatedatrˆ insteadof ingfunctionofthesemi-majoraxis(seeFig.5forthecomputed
B
rˆ . As in Paper I, for the definition of the Legendre polynomials we eccentricity profiles of selected critical second-order models).
T
refertoEqs.(22.3.8)and(22.2.10)ofAbramowitz&Stegun(1965). We recall that the geometry of the boundary surface of a crit-
5
A.L.VarriandG.Bertin:Self-consistentmodelsofquasi-relaxedrotatingstellarsystems
Fig.4. Intrinsicdensityprofiles(normalizedtothecentralvalue) Fig.5. Eccentricity profiles of the isodensity surfaces for se-
evaluated on the equatorialplane (solid lines) and along the zˆ- lected critical second-order rigidly rotating models, with Ψ =
axis(dashedlines)forcriticalsecond-orderrigidlyrotatingmod- 1,3,5,7(fromtop to bottom);dottedhorizontallines show the
elswithΨ=1,2,...,10(fromlefttoright). centraleccentricitiesvaluesestimatedanalytically(seeEq.(14)).
ical model depends only slightly on the value of the concen-
Therefore, the central value of the eccentricity is finite, of or-
tration parameter. In particular, in the critical case, the break-
der (χ1/2), andstrictly vanishesonlyin the limitof vanishing
offradiusrˆ representsthe distancefromthe centerofthe out-
B O
rotation strength. This result is nontrivialbecause the centrifu-
ermost points of the boundary surface on the equatorial plane
galpotential(whichinducesthedeviationsfromsphericity)isa
and the truncation radius rˆ is approximately the distance of
tr
homogeneousfunctionofthespatialcoordinates.Therefore,we
thelastpointonthepolaraxis(i.e.,thezˆ-axis).Therefore,since
mightnaivelyexpectthat,intheircentralregions,themodelsre-
δ =rˆ /rˆ 2/3,thevalueoftheterminationpointsoftheec-
cr tr B ≈ ducetoaperfectlysphericalshape(i.e.,e = 0),evenforfinite
centricity profiles of critical models is approximatelythe same 0
valuesoftherotationstrength.Thispropertyhasbeennotedalso
(e 0.75;seetheterminationpointsofthesolidlinesinFig.5).
≈ inthefamilyoftriaxialtidalmodels,inwhichthetidalpotential
Inaddition,byusingthemultipolarstructureofthesolution
playstheroleofthecentrifugalpotential(seeSect.3.1ofPaper
ofthe Poisson-Laplaceequationobtainedwith the perturbation
II).
method, the asymptotic behavior of the eccentricity profiles in
Deviationsfromsphericalsymmetrycan also be described,
thecentralregionscanalsobeevaluatedanalytically.Sincethe
inaglobalway,bythequadrupolemomenttensor,definedas
distribution function depends only on the Jacobi integral (i.e.,
the isolating energy integral in the corotating frame of refer-
ence),thedensityand thevelocitydispersionprofilesarefunc- Qij = (3xixj r2δij)ρ(r)d3r=
Z −
tions of only the escape energy (see Eqs. (8) and (17), respec- V
tively).Therefore,thereisaone-to-onecorrespondencebetween Aˆr5 (3xˆ xˆ rˆ2δ )ρˆ(rˆ)d3rˆ = Aˆr5Qˆ , (15)
equipotential,isodensity,andisobaricsurfacesandtheeccentric- 0Z i j− ij 0 ij
V
ityprofilescanbecalculatedwithreferencetojustoneofthese
wheretheintegrationisperformedinthevolumeV oftheentire
familiesofsurfaces. Infact, if expandedto secondorderin the
configuration.Itcanbeeasilyshownthatinourcoordinatesys-
dimensionless radius, the escape energy in the internal region
temthetensorisdiagonalandthatQ = Q .Thecomponents
reducesto: xx yy
ofthetensorcanbecalculatedexplicitly
1
ψ(int)(rˆ,θ) = Ψ(B++21(cid:2)−)U3+(θ6)χχ2+rˆ22A+2U2(r(ˆθ4)),χ+ (13) Qˆ(x2x) = Qˆ(y2y) =−Qˆ2(z2z) =2π√910ρˆK(Ψ) a2χ+b2χ22! , (16)
2 2
O
i
whereU (θ)denotesthenormalizedLegendrepolynomialwith The quantities a2 and b2 are appropriate (positive) coeffi-
2
l= 2andA ,B areappropriate(negative)coefficients,depend- cients, depending on Ψ, resulting from the asymptotic match-
2 2
ing on Ψ, which are determined by asymptotically matching ingoftheinternalandexternalsolutionofthePoisson-Laplace
equation. The sign of the components are consistent with the
the internal and external solution, in order to have continuity
abovementionedcompressionandstretchingofthedensitydis-
on the entire domain (see Eqs. (62) and (68) in Paper I, to be
tribution. The expression in Eq. (16) is calculated from the
interpreted as indicated in Appendix B of Paper I). By setting
ψ(int)(aˆ,π/2)=ψ(int)(bˆ,0),wethusfindthat,intheinnermostre- second-order external solution; the first-order expression is re-
coveredby droppingthe quadratic term in the parameter χ. At
gion,theeccentricitytendstothefollowingnonvanishingcentral
variancewiththetidalcase,theratioQˆ /Qˆ isindependentof
value zz xx
the rotation parameter χ. This analytical result has been com-
e = [6A2χ+3(B2+1)χ2]1/2 . (14) pared to the ratio of the components of the quadrupole tensor
0 [6√2/5(2χ 1)+4A χ+2(B +1)χ2]1/2 determinedbydirectnumericalintegration(performedbymeans
2 2
−
6
A.L.VarriandG.Bertin:Self-consistentmodelsofquasi-relaxedrotatingstellarsystems
Fig.6. Intrinsic velocity dispersion profiles (normalized to the Fig.7. Mean rotation velocity profiles on the equatorial plane
centralvalue)evaluatedontheequatorialplane(solidlines)and fortheselectedcriticalsecond-orderrigidlyrotatingmodelswith
alongthezˆ-axis(dashedlines)forselectedcriticalsecond-order Ψ = 1,2,...,10(fromlefttoright).Themeanvelocityincreases
rigidlyrotatingmodelswithΨ=1,2,...,10(fromlefttoright). linearly with radius because the rotation is rigid (see Eq. (18);
notethelogarithmicscaleofthehorizontalaxis).Thevaluesof
the terminationpointsof the curvesdependonthe valueofχ
cr
(which decreases as Ψ increases, see Fig. 2) and on the exten-
ofthealgorithmVEGAS,seePressetal.1992)andgoodagree-
sionofthemodelsontheequatorialplane(whichincreasesasΨ
menthasbeenfound.3Forthetidalcase,thedetailedcalculation
increases,seeFig.4).
canbefoundinSect.3.3andAppendixBofPaperII;thecalcu-
lationiseasilyadaptedtotherotatingcase.
By construction,the modelsareisotropicin velocityspace,
2.4.Projectedproperties
withthedimensionlessscalarvelocitydispersiongivenby
For a comparison of the models with the observations (under
2γ(7/2,ψ) the assumptionof a constantmass-to-lightratio),we havethen
σˆ2(ψ)= . (17)
K 5γ(5/2,ψ) computedsurface(projected)densityprofilesandisophotes.The
projectionhasbeenperformedalongselecteddirections,identi-
Asnotedfortheintrinsicdensityprofiles,a compressionalong fied bythe viewingangle(θ,φ)correspondingto thezˆP axisof
theverticalaxisandastretchingalongtheequatorialplaneoccur a new coordinate system related to the intrinsic system by the
alsoforthevelocitydispersionprofiles(theprofilesofselected transformationxˆP = Rxˆ; the rotation matrixR = R1(θ)R3(φ) is
criticalsecond-ordermodelsareshowninFig.6). expressedintermsoftheviewingangles,bytakingthe xˆP axis
Themeanrotationvelocity(whichissubtractedawaywhen as the line of nodes (i.e., the same projection rule we adopted
the corotating frame of reference is considered) characterizing forthetriaxialtidalmodels,seeSect.3.4inPaperII).Sincethe
themodelsisdefinedas v =ωeˆ reˆ = v eˆ ;intheadopted rigidly rotating models are characterized by axisymmetry with
z r φ φ
dimensionlessunits,thehaziimuthal×componhenticanbewrittenas respect to the zˆ-axis and reflection symmetry with respect to
the equatorial plane, it is sufficient to choose the viewing an-
gles from the (xˆ,zˆ)-plane of the intrinsic coordinatesystem. In
vˆ = v a1/2 =3χ1/2rˆsinθ. (18)
h φi h φi particular,weusedthelineofsightsdefinedbyθi = i(π/8)and
φ = 0 with i = 0,..,4, and we calculated (by numerical inte-
Asexpected,the meanvelocityisconstantoncylinders(inour
gration,using the Romberg’srule) the dimensionlessprojected
coordinate system, the cylindrical radius is defined by Rˆ =
density
rˆsinθ).Therelevantdimensionlessangularvelocityislinkedto
the rotation strength parameter by the following relation ωˆ =
zˆsp
3χ1/2 (thenumericalfactor3isduetotheadoptedscalelength, Σˆ(xˆ ,yˆ )= ρˆ(rˆ )dzˆ , (19)
P P P P
seeEq.(4)).Therotationprofilesofselectedsecond-ordercrit- Z−zˆsp
icalmodelsarerepresentedin Fig.7. Forallthemodels,aswe
aqpupicrkolaychdivtheergbeosusinndcaeryatothfethbeoucnodnafirgyutrhaetiroonta,ttihoenvraetlioocihtvˆyφtie/nσˆdKs walhoenrgethzˆsepxˆ=ax(ixˆs2eo−f txhˆ2Pe−intyˆr2Pin)1s/i2ccwoiothrdxiˆneattheesyesdtgeemo.fTthheepmroojdece-l
toa finitevaluewhilethevelocitydispersionvanishes;thisbe- tionplane(xˆP,yˆP)hasbeensampledonanequally-spacedsquare
havior is observed in every direction (except for the zˆ-axis, on cartesiangridcenteredattheorigin.
whichtherotationisabsentbydefinition). Thefirstfive panelsofFig. 8 showthe projectedimagesof
acriticalsecond-ordermodelwithΨ = 2;thefirstandthefifth
3 Strictlyspeaking,theanalyticalexpressionsforthecomponentsof panel correspond, respectively, to the least and to the most fa-
thequadrupole moment tensor referonly totheinner region, because vorableline of sightfor the detectionof the intrinsic flattening
thecontributionfromtheboundarylayer,whereψisoforder (χ),is ofthemodel(thezˆandxˆ-axisoftheintrinsiccoordinatesystem,
neglected. O thatis“face-on”and“edge-on”view).
7
A.L.VarriandG.Bertin:Self-consistentmodelsofquasi-relaxedrotatingstellarsystems
Fig.8. Projectionsalongdirectionsidentifiedbyφ=0andθ=i(π/8)withi=0,..,4(fromlefttoright,toptobottom)ofacritical
second-orderrigidlyrotatingmodelwithΨ = 2;thefirstandthefifthpanelrepresenttheprojectionsalongthezˆ-axis(“faceon”)
andthexˆ-axis(“edge-on”)oftheintrinsiccoordinatesystem,respectively.Solidlinesmarktheisophotes,correspondingtoselected
values of Σ/Σ in the range [0.9,10 7]. The last panel shows the ellipticity profiles, as functions of the semi-major axis of the
0 −
projectedimageaˆ ,referredtothelinesofsightconsideredinthepreviouspanels(frombottomtotop).Dotsrepresentthelocations
P
oftheisophotesandthearrowmarksthepositionofthehalf-lightisophote(practicallythesameforeveryprojectionconsideredin
thefigure).
Themorphologyoftheisophotesofagivenprojectedimage 3. Differentiallyrotatingmodels
can be described in terms of the ellipticity profile, defined as
3.1.Choiceofthedistributionfunction
ε = 1 bˆ /aˆ where aˆ and bˆ are the principal semi-axes,
P P P P
−
as a function of the semi-major axis aˆ . As already noted for As indicated in the Introduction, theoretical and observational
P
the(intrinsic)eccentricityprofile,thedeviationfromcircularity motivations have brought us to look for more realistic config-
increaseswiththedistancefromtheorigin.Intheinnerregion, urations, characterized by differential rotation. Thus we focus
the central value of the ellipticity is consistent with the central our attentionon axisymmetric systems, within the class of dis-
eccentricity e calculated in the previous subsection. The last tribution functions that depend only on the energy E and the
0
panelofFig.8illustratestheellipticityprofilescorrespondingto z-componentoftheangularmomentumJ ,andweconsiderthe
z
theprojectionsdisplayedinthepreviouspanels.Inaddition,the integral
isophotesofmodelsinthehighdeformationregime(δ δ ),if
cr ωJ
projectedalongappropriatelineofsights,showclearde≈partures I(E,J )= E z , (20)
z − 1+bJ2c
fromapureellipse,thatcanbecharacterizedasa“disky”overall z
trend (e.g., see Jedrzejewski 1987), particularly evident in the where ω, b, and c > 1/2 are positive constants. The quantity
outerparts(seefourthandfifthpanelsinFig.8). I(E,J )reducestotheJacobiintegralforsmallvaluesofthez-
z
component of the angular momentum and tends to the single-
star energy in the limit of high values of J . Therefore, if we
z
refer to a distribution function of the form f = f(I), we may
arguethatω is related to the angularvelocityin the centralre-
gion of the system, characterized by approximatelysolid-body
Because the family of rigidly rotating models is character- rotation, whereas the positive constants b,c will determine the
ized by simple kinematical properties (pressure isotropy and shapeoftheradialprofileoftherotationprofile.Inviewofthe
solid-bodyrotation), that have been already presented in detail argumentsthathaveledtothetruncationprescriptionthatchar-
with referenceto three-dimensionalconfigurations,for brevity, acterizes King model, we decided to introduce a truncation in
thederivationoftheprojectedkinematicalpropertiesisomitted phase-spacebasedexclusivelyonthesingle-starenergywithre-
here;afulldescriptioncanbefoundinVarri(2012). specttoacut-offconstantE .
0
8
A.L.VarriandG.Bertin:Self-consistentmodelsofquasi-relaxedrotatingstellarsystems
Table1.Summaryofthepropertiesofthefamiliesofmodelsstudiedinthepresentpaper.
Familyof Distribution Dimensionless Internal Anisotropy Isophote Paper
models function parameters rotation profile shape sections
fKr(H) Ae−aH0[e−a(H−H0)−1] Ψ,χ solid-body (0,0,0) disky 2
fWdT(I) Ae−aE0[e−a(I−E0)−1+a(I−E0)] Ψ,χ,b¯,c differential (0,>0,−2) boxy 3,4,5,6
[ fPdT(I) Ae−aE0e−a(I−E0) Ψ,χ,b¯,c differential (0,>0,0) boxy 3,6,App.B]
Notes.TherelevantintegralsaredefinedasH = E ωJ andI = E ωJ/(1+bJ2c),withthecorrespondingcut-offconstantsgivenbyH and
E .Thedimensionlessparametersaredefinedasfol−lows:zΨ=ψ(0)re−spreszentsamezasureoftheconcentration,χ=ω2/(4πGρ )ameasureo0fthe
0 0
(central)rotationstrength,and,forthefamilyofdifferentiallyrotatingmodels,b¯ =br2ca candc>1/2determinetheshapeoftherotationprofile.
0 −
Thepressureanisotropyprofilesarecharacterizedintermsofthevaluesoftheanisotropyparameterα=1 σ2 /σ2 inthecentral,intermediate,
− φφ rr
andouterregionsofamodel;valuesofαgreaterthan,lowerthan,andequaltozeroindicateradially-biased,tangentially-biasedanisotropy,and
isotropyinvelocityspace,respectively.Foreachfamilyofmodels,thefirstandthirdvaluesofαarecalculatedanalyticallyasthelimitingvalues
forsmallandlargeradii.ForphysicalreasonsdiscussedinSect.3.1,thefocusofthepaperisonthefirsttwofamilies.
Forsimplicity,weconsidertwofamiliesofdistributionfunc- a summary of the properties of the families of models studied
tions.Thefirstfamilyisdefinedas inthepresentpaper,seeTable1).Theintrinsicpropertiesofthe
familyofmodelsdefinedby fd (I)aresummarizedinAppendix
fWdT(I)= Ae−aE0 e−a(I−E0)−1+a(I−E0) (21) B. In this investigation, we dPeTcided to briefly mention and to
h i keepalsothesecondfamilynotonlybecauseitextendsawell-
if E E and fd (I) = 0 otherwise, so that both fd (I) and knownfamilyofmodels,butalsobecauseitallowsustocheck
≤ 0 WT WT
its derivativewith respectto E are continuous.We referto this directlyanimportantaspectofmodelconstructionthathadbeen
truncationprescriptionasWilsontruncation(hencethesubscript notedbyHunter(1977).Thisisthatthetruncationprescription
WT)because,inthelimitofvanishinginternalrotation(ω 0), affects the density distribution in the outer parts of the models
→
thisfamilyreducestothesphericallimitofthedistributionfunc- significantly.Thisgeneralpointisevenmoreimportantifweare
tion proposed by Wilson (1975). The superscript d in Eq. (21) interesting in modelingthe outermostregionsof globular clus-
indicatesthepresenceofdifferentialrotation. ters (and the issues that are often discussed under the name of
Thesecondfamilyisdefinedbythedistributionfunction studiesof“extra-tidallight”,e.g.,seeJordi&Grebel2010).
fPdT(I)= Ae−aE0e−a(I−E0) (22)
3.2.Theconstructionofthemodels
if E E and fd (I) = 0 otherwise; therefore the function,
≤ 0 PT Theconstructionofthemodelsrequirestheintegrationoftherel-
characterized by plain truncation (hence the subscript PT), is
evantPoissonequation,supplementedbyasetofboundarycon-
discontinuous with respect to E. In the limit of vanishing in-
ditionsequivalenttotheonedescribedinSect.2forrigidlyro-
ternal rotation, it reduces to the spherical limit of the function
tatingmodels.Inthiscase,weobtainthesolutionbymeansofan
proposed by Prendergast & Tomer (1970), which leads to the
iterativeapproach,basedonthemethodproposedbyPrendergast
truncatedisothermalsphere(seealsoWoolley&Dickens1962).
& Tomer (1970), in which an improved solution ψ(n) of the
A summary of the main properties and definitions of the rele-
dimensionless Poisson equation is obtained by evaluating the
vantnonrotatinglimitofthetwofamiliesofmodelsispresented
sourcetermontheright-handsidewiththesolutionfromtheim-
inAppendixA.
mediatelypreviousstep(foranapplicationofthesamemethod
In both casesthe distributionfunctionsare positivedefinite
totheconstructionofconfigurationsshapedbyanexternaltidal
fd (I), fd (I) 0,byconstruction.Curiously,anaiveextension
WT PT ≥ field,seeSect.5.2inPaperI)
of King models f = fd(I) with a similar truncation in energy
K
alone would define a distribution function that is not positive 9
ˆ2ψ(n) = ρˆ rˆ,θ,ψ(n 1) ; (23)
definiteinthewholedomainofdefinition. ∇ −ρˆ −
0 (cid:16) (cid:17)
Sharpgradientsordiscontinuitiesinphasespace(suchasthe
ones associated with the truncation prescription of fd (I)) are herethedimensionlessescapeenergyis
PT
expected to be associated with evolutionary processes dictated
ψ(r)=a[E Φ (r)], (24)
0 C
eitherbycollectivemodesorbyanysmallamountofcollisional- −
ity.Therefore,thefirsttruncationprescription,correspondingto the dimensionlessradius is defined as rˆ = r/r , with the same
0
asmootherdistributioninphasespace,istobepreferredfroma scale length introducedin Eq. (4), and ρˆ indicates the dimen-
0
physicalpointofviewasthebasisforarealisticequilibriumcon- sionless central density. The relevant density profile ρ = Aˆρˆ,
figuration(inprinciple,wemighthavereferredtoevensmoother with Aˆ definedasin Eq. (9), resultsfromthe integrationin ve-
functions;seeDavoust1977).Inaddition,afullanalysisofthe locity space of the distributions function defined by fd (I). It
WT
configurationsdefinedby fd (I)showsthatthisfamilyofmod- isclearthatthegeneralstrategyfortheconstructionoftheself-
WT
elsexhibitsanumberofinterestingintrinsicandprojectedprop- consistentsolutionisapplicablealsotothedensityderivedfrom
erties,moreappropriateforapplicationtoglobularclusters,with fd (I). The iteration is seeded by the corresponding spherical
PT
respecttothemodelsdefinedby fd (I). models, that is, the Wilson and the Prendergast-Tomerspheres
PT
Therefore,thefollowingSects.4and5aredevotedtothefull respectively, and is stopped when numerical convergence is
characterizationofthe familyofmodelsdefinedby fd (I)(for reached(seeAppendixCfordetails).Ateachiterationstep,the
WT
9
A.L.VarriandG.Bertin:Self-consistentmodelsofquasi-relaxedrotatingstellarsystems
0.20
0.15
t
0.10
0.05
0.00
0.0 0.2 0.4 0.6 0.8 1.0
Χ(cid:144)Χ
max
Fig.9. Two-dimensionalparameterspace, givenby central ro- Fig.10. Valuesoftheratiobetweenorderedkineticenergyand
tation strength χ vs. concentration Ψ, of differentially rotating gravitationalenergyt = K /W forselectedsequencesofdif-
ord
models defined by fd (I); the remaining parameters are fixed ferentiallyrotatingmodelscha|rac|terizedbyΨ = 1,2,..,7(from
WT
at b¯ = c = 1. The upper solid line marks the maximum ad- bottom to top) and χ in the range [0,χmax]; the remaning pa-
mitted values of χ, for given valuesof concentrationΨ, that is rametersarefixedatb¯ = c = 1.Thearrowsmarkthethreshold
theunderlyingareaindicatespairs(Ψ,χ)forwhichmodelscan values of χ for the moderate and rapid rotation regimes, illus-
beconstructed.Theintermediateandthelowersolidlinesmark trated in Fig. 9. For comparison,the dashedline representsthe
the values of ωˆ/ωˆ = 0.4,0.2, respectively. The gray, wide- valuesoftforthesequenceofMaclaurinoblatespheroids(with
max
striped,andthin-stripedareasrepresenttheextreme,rapid,and e < 0.92995; this eccentricity value corresponds to spheroids
moderaterotationregimes,respectively. withmaximumvalueoftherotationparameterχmax =0.11233).
sacnhdetmheepreoqteunirteiasltheexpansioninLegendreseriesofthedensity ψ(ln)(rˆ) = (2l+91)ρˆ0 "rˆlZrˆ∞rˆ′1−lρˆ(ln−1)(rˆ′)drˆ′
1 rˆ
ψ(n)(rˆ)= ∞ ψ(ln)(rˆ)Ul(cosθ), (25) +rˆl+1 Z0 rˆ′l+2ρˆ(ln−1)(rˆ′)drˆ′# . (32)
Xl=0
Thefactor √2appearinginEqs.(28)and(31)isduetothenor-
ρˆ(n)(rˆ)= ∞ ρˆ(n)(rˆ)U(cosθ). (26) malizationassumedfortheLegendrepolynomials.
l l
Xl=0
3.3.Theparameterspace
The associated Cauchy problems for the radial functions
ψ(n)(rˆ)aretherefore Much like in the case of rigidly rotating models, in both fami-
l lies fd (I)and fd (I),theresultingmodelsarecharacterizedby
WT PT
d2 2 d l(l+1) 9 two scales, associated with the positive constants A and a, and
"drˆ2 + rˆdrˆ − rˆ2 #ψl(n) =−ρˆ ρˆ(ln−1), (27) two dimensionless parameters (Ψ,χ), measuring concentration
0
andcentralrotationstrength,respectively.Inaddition,twonew
supplementedbythefollowingboundaryconditions dimensionlessparameters,namelycand
ψ(0n)(0)=Ψ√2, (28) b¯ =br02ca−c, (33)
ψ(n)(0)=0,forl,0 (29) determinetheshapeoftherotationprofile.Variationsinthepa-
l rameterb¯ andcarefoundtobelessimportant.Minorchangesin
themodelpropertiesarefounduptob¯, c 4,abovewhichthe
ψ(n)′(0)=ψ(n)′(0)=0, (30) ≈
0 l precisevalueofchasonlylittleimpactonthepropertiesofthe
configurations.
whereΨ is the depth ofthe dimensionlesspotentialwellat the
For each family of differentially rotating models, for given
center. By usingthe methodof variationof arbitraryconstants,
values(Ψ,b¯,c)thereexistsa maximumvalueofthe centralro-
theradialfunctionscanbeexpressedinintegralformasfollows
tationstrengthparameterχ ,correspondingtothelastconfig-
max
9 rˆ urationforwhichtheiterationdescribedinSect.3.2converges.
ψ(0n)(rˆ) = Ψ√2− ρˆ0 "Z0 rˆ′ρˆ(0n−1)(rˆ′)drˆ′ Smuocrhphmoaloxgimieasl,lcyhraortaactitnergizceodnfibgyutrhaetipornesseexnhceiboitfhaigsihzlaybdleefcoernmtreadl
1 rˆ toroidal structure, which will be described in detail in the fol-
−rˆ Z rˆ′2ρˆ(0n−1)(rˆ′)drˆ′# , (31) lowingsections.
0
10