On Stationary, Self-Similar Distributions of a Collisionless, Self-Gravitating, Gas

We study systematically stationary solutions to the coupled Vlasov and Poisson equations which have `self-similar' or scaling symmetry in phase space. In particular, we find analytically {\it all} spherically symmetric distribution functions where the mass

OnStationary,Self-SimilarDistributions

ofaCollisionless,Self-Gravitating,Gas

R.N.HenriksenandLawrenceM.Widrow

arXiv:astro-ph/9412047v1 14 Dec 1994DepartmentofPhysicsQueen’sUniversity,Kingston,CanadaDecember14,1994AbstractWestudysystematicallystationarysolutionstothecoupledVlasovandPoissonequa-tionswhichhave‘self-similar’orscalingsymmetryinphasespace.Inparticular,we ndanalyticallyallsphericallysymmetricdistributionfunctionswherethemassdensityandgravitationalpotentialarestrictpowerlawsinr,thedistancefromthesymmetrypoint.Wetreatasspecialcases,systemsbuiltfrompurelyradialorbitsandsystemsthatareisotropicinvelocityspace.Wethendiscusssystemswitharbitraryvelocityspaceanisotropy ndinganewandverygeneralclassofdistributionfunctions.Thesedistributionsmayproveusefulinmodellinggalaxies.Distributionfunctionsincylin-dricalandplanargeometriesarealsodiscussed.Finally,westudyspatiallyspheroidalsystemsthatagainexhibitstrictpower-lawbehaviourforthedensityandpotentialand

ndresultsinagreementwithresultspublishedrecently.

1

We study systematically stationary solutions to the coupled Vlasov and Poisson equations which have `self-similar' or scaling symmetry in phase space. In particular, we find analytically {\it all} spherically symmetric distribution functions where the mass

1Introduction

Starclusters,darkmattergalactichalos,andclustersofgalaxiesareessentiallycol-lisionlessself-gravitatingsystemsobeyingthecoupledPoisson-Vlasovequationsandthereforeequilibriumsolutionstotheseequationsareofgreatimportance.Whilesub-stantialprogresshasbeenmadethroughnumericalsimulation,therehasalsobeenapersistentschoolofanalysisthatseekstoobtainexact,analyticresults,particularlyinasymptoticor‘stationary’limits.

Onebranchofthislatterschoolhasstudiedtheevolutionofcollisionlessmatterinanexpandinguniverse(Fillmore&Goldreich1984;Bertschinger1985;Gurevich&Zybin1988,1990;Ryden1993).Thesolutionsaretimedependentofnecessitybutshowsteadyoratleastadiabaticbehaviouratlatetimes.Thetreatmentsmakeuseofanintuitiveself-similarsymmetrywhichcanseemratheradhoc(evenifexceedinglyclever)andso,di culttoaccessandgeneralize.Moreover,therearesomeremaininginconsistenciesinthevariouspublishedresults.Ourintentionhereistostudysuchcollisionlessself-similarityinasimpleandsystematicway,beginningwithstrictlystationaryexamples.Self-similarsymmetryhasalsobeenfoundinthestudyofcollisionalsystemssuchasthecoresofglobularclusters(e.g.Lynden-Bell1967;Lynden-BellandEggleton1980;InagakiandLynden-Bell1983,1990).Thesesystemsallowthestudyoftheevolutiontowardscore-halocon gurationsbutneveryieldtruethermodynamicequilibriaintheformsayofstationarypowerlawbehaviour.Neverthelessinvariousintermediate(spa-tiallyandtemporally)stagessimplepowerlawdoappearandmayinfactcorrespondtothesolutionswe ndforstationarycollisionlesssystems.Itiswellknowninthehydro-dynamicliterature(e.g.BarenblattandZel’dovitch1972,hereafterBZ)thatself-similarsolutionsariseasintermediateasymptoticsbetweenboundaries,anditisprobablyforthisreasonthattheyarefoundinthepreandpostcollapsephasesofthecollisionalsystems.Onemightspeculatethattheyareascloseto‘equilibrium’assuchsystemsget.Inadditiontoglobularclusters,thesesystemsmayariseintheintermediatestagesofcollapseofastarclustertoablackholeandintheultimatestateofcollisionlessdarkmatterhalos.

Inthispaperwestudysystematicallythefamilyofdistributionsthatareexactlystation-aryandthatpossessbothprecisegeometric(usuallyspherical)andscalingsymmetries.ThesedistributionsaresolutionstothecoupledVlasovandPoissonequations:

3 H

i=1 xi H vi

=0(1.1)

2Φ=4πGfd3v.(1.2)

Herefistheusualmassdistributionfunction,Φisthegravitationalpotential,HistheHamiltonianandvi,xiarecanonicalpairs.Thesolutionswe ndincludetheintermediateasymptoticlimitsoftherelevantprecedingstudies.Theyoverlapdirectly

2

We study systematically stationary solutions to the coupled Vlasov and Poisson equations which have `self-similar' or scaling symmetry in phase space. In particular, we find analytically {\it all} spherically symmetric distribution functions where the mass

withtherecentworkofEvans(1994,hereafterE94)on“powerlawgalaxies”andtoalesserextentwiththe‘η’modelsofTremaineetal.(1994).Theselattermodelsaresimilarinspirittoourownbutarenotstrictlyscalefreeandsoprovetoberathermorecomplicatedtoexpressanduse.Neverthelesstheysharetheobservationallyimportantpropertyofacuspedcentraldensityandpossessthesubstantialadvantageofhaving nitemass.

Thepower-lawgalaxiesofE94(seealsoEvansanddeZeeuw1994,hereafterEdZ)arescaleinvariantandthereforesimpletouse.Theywereconstructedtoprovideafamilyofversatilemodelsofgalaxiesthatcouldbeusedtoanalyseobservablepropertiesofgalaxiessuchaslinepro lesfromstellarabsorption-linespectra.Ourapproachismoresystematicandinfact,forthecaseofsphericalsymmetrywithvelocityspaceanisotropy,yieldsamoregeneralclassofsolutions.WemostlyleaveapplicationssuchasthosefoundinvanderMarel&Franx(1993),E94andEdZforfuturepublications.

Wealsousethispapertointroducetotheastronomicalcommunityasystematicmethodofclassifyingscalingorself-similarsymmetry rstgivenbyCarterandHenriksen(1991,hereafterCH)inaratherformalstyle.Thetechniqueiseasytouseandyieldsthereducedequations(withmanifestscalingsymmetry)inastandardform.Inaddition,themethodincludesthevariousscalingsymmetriesthatproduceirrationalpowerlawsformerlyreferredtoasscalingsymmetriesofthe‘second’kind(e.g.BZ).

2RadialOrbitsinSphericalSymmetry;

AnIntroductiontotheCHAnalysisoftheVlasov-PoissonEquationsWeintroducethecanonicaldistributionfunctionFwhere

F(r,vr)f(r,v)≡

We study systematically stationary solutions to the coupled Vlasov and Poisson equations which have `self-similar' or scaling symmetry in phase space. In particular, we find analytically {\it all} spherically symmetric distribution functions where the mass

where

Lk≡kj j≡δr r+νvr vr(2.5)

istheLiederivativewithrespecttothe(phasespace)vectoroperatork.Inotherwords,kisthescalingdirectioninphasespace.Itisconvenientintheseformulaetoimaginethatrandvrarescaledintermsof ducialvalues(nottobeconfusedwithrealconstantlengths).

Equation(2.4)holdssolongasF=F(ζ)whereζ≡r(ν/δ)/vr.Thedimensionlessrealnumberδ/νgivesthesimilarity‘class’ofthesymmetryinthesenseofCHanditisgenerally xedonlybyboundaryconditionsorbythedimensionsofconservedquantities.

Our rststepistochoosenewphasespacecoordinatestoreplacerandvr.FollowingCHwede neanew‘radial’coordinateR(r)suchthat

Lk=kj j≡ R.

kj jR=1andweobtainthetransformationlaws

r|δ|=eδR,

and

dR(2.7)(2.6)

We study systematically stationary solutions to the coupled Vlasov and Poisson equations which have `self-similar' or scaling symmetry in phase space. In particular, we find analytically {\it all} spherically symmetric distribution functions where the mass

andmass.ThereisthenathreeparametermultiplicativerescalinggroupwithelementsA≡(eδ,eν,eµ)thatdescribesthescalingsofdimensionalquantities.Alternatively,ifweconsiderchangesinthelogarithmsofthesequantities,wehaveanadditiverescalinggroupwithelementsa=(δ,ν,µ).ThevectorcomponentsofAoracorrespondtothescalinginlength,velocityandmassrespectively.

EachdimensionalquantityΨintheproblemhasitsdimensionsrepresentedbyadi-mensionality(co)vectordΨinthedimensionspace,andthechangeinthelogarithmofthequantityisgivenby(CH)

LkΨ= RΨ=(dΨ·a)Ψ.(2.10)

Strictlyspeaking,thevectorkshouldcorrespondtotherescalingvectora.Inotherwords,weshouldreplaceequation(2.5)with

kj j=δr r+νvr vr+µm m.(2.11)

Howeveraswewillsoonsee,theinvarianceofGunderrescalingimpliesthatmassrescalingisnotindependentoflengthandvelocityrescaling.Thisallowsustoreduceourscalingalgebraelementtoa=(δ,ν).

ThedimensionalquantitiesinthecurrentproblemF,ΦandGhavethefollowingdimensionalitycovectorsinthechosendimensionspace(length,velocity,mass);

dF=( 1, 1,1),

dΦ=(0,2,0),

dG=(1,2, 1).(2.12)

TherequirementthatGbeinvariantundertherescalinggroupactionimpliesa·dG=0,oronperformingthescalarproduct(directmultiplicationsincedisacovector)

µ=2ν+δ.(2.13)

Consequentlythedimensionspacemaybereducedtothesub-spaceof(length,velocity)whereintherescalinggroupelementbecomesa=(δ,ν)and

dF=(0,1),

dΦ=(0,2).

5(2.14)

We study systematically stationary solutions to the coupled Vlasov and Poisson equations which have `self-similar' or scaling symmetry in phase space. In particular, we find analytically {\it all} spherically symmetric distribution functions where the mass

Thesub-spacealgebraelementanowcorrespondstoourchoiceofthescalingvectork.ThescalingsymmetrycanbeimposedonFandΦbyequation(2.10)which,withequation(2.14)requires

F(X,R)=

Φ(X)e2νR.(2.15)

Theseequationscanbesimpli edfurtherbynotingthatat xedrthepotentialisindependentofvrorequivalently

F

X2+2

ν+2 ν 2

F=C|X2+2

Φe2R=

a

whereΦahasunitsofvelocity2andreplaces 2/δ(2.20)

We study systematically stationary solutions to the coupled Vlasov and Poisson equations which have `self-similar' or scaling symmetry in phase space. In particular, we find analytically {\it all} spherically symmetric distribution functions where the mass

C=2|δ+2|

Φ(orequivalentlyΦa)mustbenegativeandthereforeδ< 2.

Formallyδ>0and

2> 3.Inaddition,thevelocitymomentsofthedistributionfunctionalsofollowsimplepowerlaws:δ

We study systematically stationary solutions to the coupled Vlasov and Poisson equations which have `self-similar' or scaling symmetry in phase space. In particular, we find analytically {\it all} spherically symmetric distribution functions where the mass

masslessparticlesthatneverthelessaredistributedinenergyspaceaccordingtoequation(2.19).

Havingillustratedourmethodindetailinthissimpleexamplewithradialorbits,weproceedtogiveourresultsbrie yforothercasesofinterest.

3IsotropicOrbitsinSphericalSymmetry

Here,thefull3Ddistributionfunctionhasthefunctionalformf=f(r,v),where222v2=vr+vθ+vφ.TheVlasovandPoissonequationsarenowrespectively(vr=0)

v rf ( rΦ) vf=0,

1(3.1)

f(X)e (2δ+ν)R,

Φ=

Φ=constant.Substitutingintoequations(3.1)and(3.2)yields

dln

dX

and

8(2δ+1)XΦ= (3.5)

We study systematically stationary solutions to the coupled Vlasov and Poisson equations which have `self-similar' or scaling symmetry in phase space. In particular, we find analytically {\it all} spherically symmetric distribution functions where the mass

2

2(δ+2)

Equation(3.5)easilyintegratesandwe ndf(X)dX.(3.6)

Φ| (δ+1/2).

Itbecomesclearfromthislastequationthatforaboundsystem((3.7)

Φ(|δ|r)2/δ≡Φa r

Φ.ThenormalizationconstantCisdeterminedfromthePoissonequation:

(1+δ/2)(1 δ)Γ(1 δ)C=Ga2.(3.10)

Itisinterestingtonotethatthecorrespondingdensitylawhasthesamefunctionalformasinthepureradialcasediscussedabove.Moreover,

We study systematically stationary solutions to the coupled Vlasov and Poisson equations which have `self-similar' or scaling symmetry in phase space. In particular, we find analytically {\it all} spherically symmetric distribution functions where the mass

Wenotethatonceagainthelimitδ= 2correspondstoaninvariantcentralmasssurroundedbymasslessparticles.ThisistheKeplerian,andpossiblythecentralblackhole,limit.

4SystemswithCylindricalandPlanarSymmetry

Itisstraightfowardtoextendthepreviousanalysistosystemswithcylindricalandplanarsymmetry.Suchsolutionsmightprovideinsightintothe lamentsandsheetsseenbothinN-bodysimulationsandinlarge-scalegalaxysurveys.OursolutionsshouldcorrespondtothosefoundinFillmoreandGoldreich(1984)thoughoursolutionsarestationaryanddonotincludecosmologicalexpansion.

Forsystemswithcylindricalsymmetry,wechoosethecanonicaldistributionfunction

f(r,v)≡F( ,v )

We study systematically stationary solutions to the coupled Vlasov and Poisson equations which have `self-similar' or scaling symmetry in phase space. In particular, we find analytically {\it all} spherically symmetric distribution functions where the mass

2 zΦ=G Fdvz.(4.7)

Inthiscase,thefamilyofsolutionsis

f∝|E|(1 2δ)/2Φ∝z2/δ.(4.8)

1<δ<2isenoughtoinsurethatthedensitywilldecreasewithincreasingzandthemassperunityareawillvanishforz→0.

5AnisotropicOrbitsinSphericalSymmetry

Thisexampleismorechallengingthentheotherstreatedinthispaperandthere-sultsaresomewhatsurprising.Thedistributionfunctiondependsonthreephasespace22coordinates(r,vr,j2)wherej2≡r2(vθ+vφ)isthesquareofthetransverseangularmomentum(Fujiwara1983).TheVlasovandPoissonequationsbecomerespectively

vr rf+j2

X(ζ)×vre R,

11i(5.4)

We study systematically stationary solutions to the coupled Vlasov and Poisson equations which have `self-similar' or scaling symmetry in phase space. In particular, we find analytically {\it all} spherically symmetric distribution functions where the mass

where

andX(1)=1

f(X,Y)e(1 λ)R,

Φ=

Φ=constant.

TheVlasovandPoissonequationsnowbecome

(2δ+1)

2f 2(1+δ)Y Yδ+2X2(δ3Y 2f=0,f(X,Y)=0.(5.9)(5.10)

where

λ=2(1+δ)(5.11)

followsfromtherequirementthattheseequationsbeindependentofR.Weseefromequation(5.10)thatδ< 2forthesystemtobebound.

Equation(5.9)isaquasi-linear, rst-orderpartialdi erentialequationthatcanbesolvedexactly.Thecharacteristicequationsare

d

(2δ+1)=XdX

Φ),(5.12) 2(1+δ)Y

the rstofwhichisclearlyintegrableandgives

F(ξ)Y 2δ+1

We study systematically stationary solutions to the coupled Vlasov and Poisson equations which have `self-similar' or scaling symmetry in phase space. In particular, we find analytically {\it all} spherically symmetric distribution functions where the mass

whereξisafunctionconstantontheintegralcurvesofthesecondequationof(5.12).Thissecondequationmaybeintegratedbyintroducingthevariablesu,ssuchthat

s≡X2+2

ds+(2+δ)u+(1+δ)s=0,(5.16)

whichisofatypealreadyknowntoLeibnitzin1691.Thesolutionisimmediatefors=0bychangingthedependentvariabletosayy≡u/s.We ndtherebythatthequantityξ(X,Y)maybetakentobe

ξ=Y 1|δ2Y+X2+2

F(ξ).The

constant

Φ

Inthisformula 1δ2 =4πG2 2 2f. 2(5.18)

δ+1)

F(ξ),equations(2.20)and(2.22)continuetogivethe

F(ξ)potentialanddensitypowerlaws.Theamplitude

throughequation(5.18)andonemustinsurethattheintegralsinthisequationexist.Equation(5.20)representsanewclassofdistributionfunctionswithvelocityspaceanisotropy.Modelswith

We study systematically stationary solutions to the coupled Vlasov and Poisson equations which have `self-similar' or scaling symmetry in phase space. In particular, we find analytically {\it all} spherically symmetric distribution functions where the mass

f=Cja|E|b(5.21)

whereaandbarerelatedtoαandδinasimpleway.(Actually,distributionswithenergyandangularmomentumdependencegivenbyequation(5.21)were rstdiscussedbyCamm(1952).There,thespatialdependenceofthemassdensityandpotentialaregivenbysolutionstotheEmden-Fowlerequationwithappropriateboundaryconditions.)AsdiscussedbyE94,itisstraightfowardtocalculatevelocitymomentsforthesedistributionfunctions.Forexample,onecancalculatethevelocityspaceanisotropyparameter

2:vrβ≡1

β=2δ+1

F(ξ)aretheystable?FridmanandPolyachenko(1984)

showthattheCammmodelsarelinearlyunstableforsu cientlylargeanisotropy,butarestabletowardstheisotropiclimit.Butlinearstabilityisnotthesameasnon-linearstabilitywhichhasreallytodowiththeexistenceorotherwiseofasymptoticdistribu-tionstowardswhichftends.Bystudyingthestabilityofourstationarysolutionswemightdiscoverahintastohowtode neauseful‘entropy’functionthatcharacterizes

14

We study systematically stationary solutions to the coupled Vlasov and Poisson equations which have `self-similar' or scaling symmetry in phase space. In particular, we find analytically {\it all} spherically symmetric distribution functions where the mass

suchasymptoticequilibria.ThislatterquestionhasrecentlybeengivennewlifebythestudiesofTremaineetal.(1986),Wiechenetal.(1988)andAly(1989,1993).Thislastauthorhasshownforexamplethatthe‘softened’Plummermodelwithδ= 4inournotationactuallyattainstheminimumenergysubjecttoa xedmassanda xedvalueofthe‘entropy’.Ifthisvalueisusednaivelyinourmodels,oneobtainsρ∝r 5/2andΦ∝r 1/2.FinallyonemightaskhowcloselycanastablemodelapproachMaxwelliantypedistributions.

6AxiallySymmetricSolutionswithEllipsoidal

andHyperboloidalSymmetries

InordertomakecontactwiththerecentstudiesofE94andofEdZweshowherehowourmethodyieldsthestrictlyscalingsubsetoftheiraxiallysymmetricsolutionsinaverydirectway.Thescalingsymmetryinphasespaceforthiscaseisactuallysimplerthanthatfortheanisotropicsphericalgeometrydealtwithabove.Weusecylindricalcoordinates ,φ,zandwritetheVlasovequationinthesymmetricform

2vφ vφf zΦ vzf=0,(6.1)v f+vz zf+

whilethePoissonequationbecomes

1

q2.(6.3)

Wedi erfromE94inthatwedonotintroducea‘core’radiusintotheproblem,sincestrictlyspeakingthiswouldprohibittheexistenceoftruegeometricallyscalingsolutions.Itwillbeclearhoweverfromourmethodthatonecanaddacoreradiussquaredtou2withoutchangingthesolution,providedthatitisignoredinthescalingsymmetry.ThusoursolutionsaswellasthoseofE94donot‘really’knowofitsexistenceandsoweprefertosuppressit.Itisamusingtonotealsothatthesubstitutionsq2→ q2andu→vwherez222v≡

We study systematically stationary solutions to the coupled Vlasov and Poisson equations which have `self-similar' or scaling symmetry in phase space. In particular, we find analytically {\it all} spherically symmetric distribution functions where the mass

WeproceedwithadirectsubstitutionofΦ=Φ(u)intotherighthandsideof(6.2)followedbyacollectionoftermsinequalpowersof todiscoverthenecessityforadistributionfunctionintheansatzform

f=f1(u,v ,vφ,vz)+f2(u,v ,vφ,vz) 2.

HencethePoissonequationsplitsintothetwoequations

2Φ′,4πGf1dv dvφdvz=q2 14πGf2dv dvφdvz=(6.5)(6.6a)u3 Φ′′

f2=f22 2vz+Φ,v ,vφ ,

2+Φ ≡f1(E).(6.8)

Thevanishingofthecoe cientof yieldsthendirectlyforf2

2f2=vφF2(E),(6.9)

afterwhichthecoe cientofthelasttermin 3vanishesidentically.ConsequentlyweconcurwithE94thatthenecessaryansatz(6.5)isinfactoftheform

22f=f1(E)+ 2vφF2(E)≡f1(E)+jzF2(E),(6.10)

althoughatpresentf1andF2arearbitraryfunctionsoftheenergyE.TheVlasovequationisnowidenticallysatis ed,inaccordancewithJeans’theoremforstationarysolutions.

16

We study systematically stationary solutions to the coupled Vlasov and Poisson equations which have `self-similar' or scaling symmetry in phase space. In particular, we find analytically {\it all} spherically symmetric distribution functions where the mass

Asintheprevioussectionsweturnnowtotheimpositionofascalingsymmetryinphasespace.TheLiederivativewillbealongthevector

kj j≡uδ u+v v +vφ vφ+vz vz≡ R,

whereR=R(u)andsoasusual

u=

dueRδ(6.13)(6.11)=e Rδsgn(δ).

ProceedingexactlyasbeforetosolvefortheinvariantsX(i)fromkj jX(i)=0givestheconvenientchoices

X(1)≡e Rv ,

X(2)≡e Rvφ,

X(3)≡e Rvz.(6.14)

Moreoverstartingonceagainwiththedimensionspacecovectora≡(δ,ν,µ),preservingG(butnotanycharacteristiclength)and ndingthedimensionvectorsoff1,F2andΦinthereduceddimensionspacegives

f1=

F2(X(i))e (4δ+3ν)R,

Φ=(6.15)

2+E(X(j))e2R.(6.16)

Thenthecompatibilityofequations(6.8)and(6.15)(togetherwiththeisotropicchar-acteroff1invelocityspace)requires

f1(E)=

f1(

f1(kx)=kα

We study systematically stationary solutions to the coupled Vlasov and Poisson equations which have `self-similar' or scaling symmetry in phase space. In particular, we find analytically {\it all} spherically symmetric distribution functions where the mass

α= (δ+1/2).(6.18)

Inonedimensionsuchahomogeneousfunctionisapowerlawofpowerαsothat

4πGB f1(X(j))e (2δ+1)R+jz2EαdX(1)dX(2)dX(3)=,Φ q21

Φ<0,werequire

δ<sgn(A)1(6.23)

We study systematically stationary solutions to the coupled Vlasov and Poisson equations which have `self-similar' or scaling symmetry in phase space. In particular, we find analytically {\it all} spherically symmetric distribution functions where the mass

andbyequation(6.24) 1

Φbothpositive,wenotethatδshouldbepositiveifthegravitationalaccelerationistobedirectedinwards.

Theevaluationoftheintegrals(6.24)and(6.25)hasbeenextensivelydiscussedbyE94andEdZandareeasilydonebychangingvariablesto

We study systematically stationary solutions to the coupled Vlasov and Poisson equations which have `self-similar' or scaling symmetry in phase space. In particular, we find analytically {\it all} spherically symmetric distribution functions where the mass

Insection5wehavefoundthepower-lawsolutionsthataresphericallysymmetricthoughanisotropicinvelocityspace.Asubsetofthesemodelsarethe‘power-lawgalaxies’ofE94.However,ourmodelsaremoregeneralasillustratedinFigures2(a)-2(d).Themodelsthereforeallowformorefreedominmodellinggalaxiesand ttingobservablessuchasabsorption-linespectratotheoreticalpredictions.

Finallysection6hasenabledustodemonstratetheeasewithwhichthespheroidalso-lutionsmaybefoundwhenthespatialsymmetryandthescalingsymmetryareimposedseparately.OurconclusionsarethesameasthoseofE94andofEdZ94,althoughwedopointoutthatthecoreradiusisnotactuallyplayingaroleinthesesolutionsanditisapparentthatthesesolutionsformpartofthewiderfamilyofsolutionsdiscussedinthispaper.Wealsoremarkthathyperboloidalsolutionsofthistypeobviouslyalsoexist.

FutureworkwillfocusonthetimedependentsolutionswithandwithoutanexpandingbackgroundpartlyinhopesofobtainingtheresultsofFillmoreandGoldreich(ibid)withoutthesingularitiesinthedensity,butalsoinordertomakeasurveysimilartothepresentsurveyofthestationarysolutions.Moredetailedinvestigationofthestabilityandotherpropertiesofthesolutioninsection5isalsorequired.

Acknowledgements

ItisapleasuretoacknowledgehelpfulconversationswithDrsW-YChau,MartinDun-can,andScottTremaine.

20

We study systematically stationary solutions to the coupled Vlasov and Poisson equations which have `self-similar' or scaling symmetry in phase space. In particular, we find analytically {\it all} spherically symmetric distribution functions where the mass

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