Gravothermal Catastrophe: the dynamical stability of a fluid model

2.2 MB·English

by  M. C. Sormani

Astronomy&Astrophysicsmanuscriptno.SormaniBertinRevised (cid:13)c ESO2013
January28,2013
Gravothermal Catastrophe: the dynamical stability of a fluid model
M.C.Sormani1 andG.Bertin2
1 ScuolaNormaleSuperiore,PiazzadeiCavalieri7,56126Pisa,Italy(cid:63)
e-mail:[email protected]
2 Universita`degliStudidiMilano,DipartimentodiFisica,viaCeloria16,I-20133Milano,Italy
e-mail:[email protected]
ReceivedXXX;acceptedYYY
ABSTRACT
3
Are-investigationofthegravothermalcatastropheispresented.Bymeansofalinearperturbationanalysis,westudythedynamical
1
stabilityofasphericalself-gravitatingisothermalfluidoffinitevolumeandfindthattheconditionsfortheonsetofthegravothermal
0 catastrophe,underdifferentexternalconditions,coincidewiththoseobtainedfromthermodynamicalarguments.Thissuggeststhat
2
thegravothermalcatastrophemayreducetoJeansinstability,rediscoveredinaninhomogeneousframework.Wefindnormalmodes

n andfrequenciesforthefluidsystemandshowthatinstabilitydevelopsonthedynamicaltimescale.Wethendiscussseveralrelated
a issues.Inparticular:(1)Forperturbationsatconstanttotalenergyandconstantvolume,weintroduceasimpleheuristicterminthe
J energybudgettomimictheroleofbinaries.(2)Weoutlinetheanalysisofthetwo-componentcaseandshowhowlinearperturbation
analysiscanbecarriedoutalsointhismorecomplexcontextinarelativelystraightforwardway.(3)Wecomparethebehaviorofthe
5
fluidmodelwiththatofthecollisionlesssphere.Inthecollisionlesscasetheinstabilityseemstodisappear,whichisatvariancewith
2
thelinearJeansstabilityanalysisinthehomogeneouscase;wearguethatakeyingredienttounderstandthedifference(aspherical

stellarsystemisexpectedtoundergothegravothermalcatastropheonlyinthepresenceofsomecollisionality,whichsuggeststhatthe
]
A instabilityisdissipativeandnotdynamical)liesintheroleofthedetailedangularmomentuminacollisionlesssystem.
Finally, we briefly comment on the meaning of the Boltzmann entropy and its applicability to the study of the dynamics of self-
G
gravitatinginhomogeneousgaseoussystems.
.
h Keywords.hydrodynamics–instabilities–gravitation–galaxies:clusters:general
p
-
o
r 1. Introduction models, collisionless models, and fluid models) admit equilib-
t
s rium configurations that are spatially truncated self-gravitating
a Corecollapseinan N-bodysystemisaproblemrelevanttothe isothermalspheres,butitisnotcleartowhatextenttheycanbe
[ dynamics of globular clusters, because these are considered to
consideredasrepresentativeoftheequilibriumstatesofthepure
betheonlystellarsystemsthatpossessthenecessarydegreeof
1 N-body problem. Several studies have addressed the stability
collisionalitytorelaxthermally.Thediscoveryofsomeclusters
v problemofisothermalspheresusingthermodynamicalmethods,
withcuspycoreswasinterpretedasasignthattheyindeedexpe-
8 startingwithAntonov(1962)andLynden-Bell&Wood(1968)
3 riencedthegravothermalcatastrophe.Inthecontextofglobular andcontinuingwithalonglistofpapers(forexample,Hachisu
0 clusters,thethreemainphenomenacausedbycollisionalityare: & Sugimoto 1978; Nakada 1978; Katz 1978; Inagaki 1980;
6 core collapse, evaporation, and mass segregation. The focus of
Padmanabhan 1989; Chavanis 2002, 2003, see also Thirring
. thispaperisoncorecollapse.
1 1970).Inthethermodynamicalapproachthestudyofisothermal
Consider the gravitational N-body problem, where N (cid:29) 1.
0 spheresisbasedonaformofentropyknownastheBoltzmann
3 Thatis,considerasetofN classicalpointmasses,eachofmass entropy:
1 m,mutuallyinteractingthroughNewtoniangravity.Theparticles
: mayormaynotbeconfinedwithinasphericalvolumeofradius (cid:90)
v R.Weareinterestedinthefollowingquestions: Sb[f]≡−k f(r,v)ln f(r,v)d3rd3v, (1)
i
X
– Can the system reach some sort of equilibrium? Can this
r where f is the one-particle distribution function,1 r and v the
a equilibriumbecalledthermal?
position and velocity vector respectively, and k the Boltzmann
– Can a model (for example, collisionless or fluid) of the N-
constant.
bodyproblemreachequilibrium?Howdoestheequilibrium
Unfortunately, the thermodynamics of self-gravitating sys-
ofamodelrelatetotheequilibriumofthepureN-bodyprob-
temsstilldependsonanumberofunresolvedissues,partlybe-
lemandofarealstellarsystem?
cause of the long-range nature of the force and partly because
– Aretheseequilibriastable?
ofthedivergentbehavioroftheforceatshortdistance(e.g.,see
Thistopichasbeenstudiedbymanyauthors(forrecentreviews Padmanabhan1990;Katz2003;Chavanis2006;Mukamel2008;
seeforexampleHeggie&Hut2003;Binney&Tremaine2008). Campaetal.2009;Bouchetetal.2010).Acriticalanalysisofthe
Different models of the N-body problem (in particular gaseous useoftheBoltzmannentropyhasbeenmadebyMiller(1973).
WewillbrieflycommentonthispointinSect.2.
(cid:63) Present address: Rudolf Peierls Centre for Theoretical Physics, 1
(cid:82)
KebleRoad,OxfordOX13NP 1 Normalizedinsuchawaythat fd3rd3v=N
1
M.C.SormaniandG.Bertin:GravothermalCatastrophe:thedynamicalstabilityofafluidmodel
Therefore,itwouldbeinterestingtostudythegravothermal Thepaperisorganizedasfollows.InSect.2wecommenton
catastrophe by limiting as much as possible the use of thermo- the meaning of the Boltzmann entropy. In Sect. 3 we write the
dynamicalarguments.Inthispaperwereconsiderthegravother- basic hydrodynamic equations in the Eulerian and Lagrangian
mal catastrophe and analyze the dynamical stability of self- representation.InSect.4weshowtheresultsofthelinearanaly-
gravitatingfluidsgovernedbytheEulerequationbyfindingthe sisofthehydrodynamicequationsbystudyingthepropertiesof
normalmodesandfrequenciesofsphericalsystemsundervari- spherically-symmetric perturbations. We find the relevant nor-
ousexternalconditions,inparticular:constanttotalenergyEand malmodesandfrequenciesunderdifferentboundaryconditions.
volumeV(thiscasecorrespondstothegravothermalcatastrophe In the analysis of the gravothermal catastrophe, we propose a
ofLynden-Bell&Wood1968),constanttemperatureT andvol- modifiedexpressionfortheenergy,whichcanbeconsideredas
umeV(isothermalcollapse),constanttemperatureT andbound- a simple way to incorporate the energy generation from bina-
ary pressure P (isobaric collapse, Bonnor 1956; Ebert 1955). ries in the linear regime; we show that the catastrophe can in-
Note that, in contrast to thermodynamical arguments, the lin- deedbehaltedifweconsidersuchamodifiedexpression.Inthe
earmodalanalysisautomaticallygivesthetimescaleforthede- lastsubsectionandinAppendixE,webrieflyconsiderthetwo-
velopment of the instability. The constant {T,V} case has been componentcase.InSect.5.1wediscusstherelevanttimescales.
addressedbySemelinetal.(2001),whostudiedthestabilityof In Sect. 5.2 we discuss the difference between the collisionless
thesystemnumerically,andbyChavanis(2002),whofoundan model and the fluid model of the N-body problem. In Sect. 6
analytical solution for the marginally stable perturbations us- wedrawourconclusionsandidentifysomeopenquestionsand
ing methods developed by Padmanabhan (1989). The constant issues.
{T,P} case has been addressed previously with synthetic argu-
mentsbyBonnor(1956)andEbert(1955),byYabushita(1968),
2. Thethermodynamicalapproachandthe
whostudiedthestabilityofthesystemnumerically,byLombardi
& Bertin (2001), who extended the analysis by Bonnor (1956) Boltzmannentropy
to the nonspherically symmetric case, and by Chavanis (2003),
Antonov (1962) and Lynden-Bell & Wood (1968) based their
who gave an analytical solution for the case of marginally sta-
stability analysis of self-gravitating spatially-truncated isother-
bleperturbationsusingthesamemethodashehadusedforthe
mal spheres on the use of the Boltzmann entropy (1). Starting
constant {T,V} case. The constant {E,V} case has been studied
fromakineticdescription,theylookedforstationarystatesofthe
usingamodelbasedontheSmoluchowski-Poissonsystem(dif-
Boltzmannentropywithrespecttothedistributionfunction f,at
ferentfromtheEuler-Poissonsystemconsideredinthispaper), fixed2 total mass M, total energy E, and volume V = 4πR3/3,
byChavanisetal.(2002).
whereRistheradiusofasphericalbox.Thesestationarystates
Inthispaper,weextendtheseanalysesoftheconstant{T,V} arespatiallytruncatedisothermalspheres.Itisfoundthat,fora
and {T,P} cases to the general calculation of eigenfrequencies givensingle-particlemassm,eachspatiallytruncatedisothermal
andeigenfunctionsforconditionsoutsidethoseofmarginalsta- sphere is identified by two dimensional scales (e.g., the central
bility, and study in detail the constant {E,V} case. We use a densityρ (0)andthetemperatureT)andonedimensionlesspa-
0
EulerianorLagrangianrepresentationofhydrodynamicsassug- rameter{forexampleΞ=R/λ,whereλ=[kT/m4πGρ (0)]1/2is
0
gestedbytheboundaryconditionstobeimposedonthesystem reminiscentoftheJeanslength(Jeans1902);anotherequivalent
and we provide a unified treatment for all cases. Surprisingly, choice for the dimensionless parameter is the density contrast
wefindthatthesystembecomesdynamicallyunstableinallthe ρ (0)/ρ (R),thatis,theratioofthecentraldensitytothedensity
0 0
cases considered exactly at the same points found by Lynden- atthetruncationradiusR}.Conversely,eachchoiceofthethree
Bell & Wood (1968) by means of a thermodynamical analysis, quantities{T,ρ(0),Ξ}identifiesoneparticularisothermalsphere.
thatisforvaluesofthedensitycontrast(theratioofthecentral Thisone-to-onecorrespondencemaybelostforotherchoicesof
densitytotheboundarydensity)≈14,32.1,and709respectively thequantitiescharacterizingthesystem.
fortheconstant{T,P},{T,V},and{E,V}case.Theseresultssug- In this thermodynamical approach the necessary condition
gest that the gravothermal catastrophe may reduce to the Jeans forstabilityisthatthestationarystatecorrespondstoa(localat
(1902) instability, rediscovered in the inhomogeneous context. least)maximumoftheBoltzmannentropyfunctional.Antonov
Thenwebrieflyoutlinetheproblemofthelineardynamicalsta- (1962) found that if Ξ < 34.4 [i.e., if ρ (0)/ρ (R) < 709] the
0 0
bilityofasphericalfluidgeneralizedtothetwo-componentcase, equilibrium configurations are local maxima, whereas for Ξ >
byperturbingthetwo-componentspatiallytruncatedisothermal 34.4 they are saddle points. Therefore, he concluded that self-
sphereconfigurationspreviouslyconsideredbyTaffetal.(1975), gravitatingisothermalspheresareunstableifΞ>34.4.
Lightman (1977), Yoshizawa et al. (1978), de Vega & Siebert TheworkofAntonov(1962)wasextendedbyLynden-Bell
(2002), who studied the stability of these systems with a ther- &Wood(1968).Theseauthorsstudiedthethermodynamicalsta-
modynamicalapproach,andSopiketal.(2005),whoaddressed bility of the system under various conditions by applying the
alsothedynamicalstabilitywithamodeldifferentfromtheone relevant thermodynamical potentials (based on the Boltzmann
used in this paper. We show that the onset of thermodynamical entropy) and gave a physical interpretation of the instability
anddynamicalstabilityoccursatthesamepointinthesimplest in terms of negative specific heats, as a characteristic feature
case (constant {T,V}) and confirm that the component made of of self-gravitating systems, thus creating the paradigm of the
heavierparticlesistheprimarydriveroftheinstability.Thisre- gravothermal catastrophe. They found that the critical value of
sultislikelytoberelatedtoasimilarfindingbyBreen&Heggie Ξ that corresponds to the onset of instability depends on the
(2012a,b)fortwo-componentgravothermaloscillations.Finally
(cid:82)
we focus on the following puzzling phenomenon: in the colli- 2 Intermsofthedistributionfunction f,wehaveM =m fd3xd3v
sionlessmodeltheinstabilitydisappears.Thephenomenonisat and
variancewith whathappens inhomogeneous systems(e.g., see (cid:90) mv2 1 (cid:90) f(r,v)f(r(cid:48),v(cid:48))
Bertin2000).Inthelastpartofthepaperwearguethatthecause E= f(r,v)d3rd3v− Gm2 d3rd3vd3r(cid:48)d3v(cid:48),
ofthedifferenceliesintheroleoftheangularmomentumofthe 2 2 |r−r(cid:48)|
individualparticles. wheremisthesingle-particlemass.
2
M.C.SormaniandG.Bertin:GravothermalCatastrophe:thedynamicalstabilityofafluidmodel
adoptedconditions.Forexample,Ξ=34.4holdsforasystemat temwhent→∞(tistime)hasaneffectiveequationofstate3not
constanttotalenergyE andvolumeV,Ξ = 8.99forasystemat affected by the attractive nature of the gravitational force: thus
constanttemperatureT andvolumeV,andΞ=6.45forasystem the Boltzmann entropy may not be applicable to find the true
atconstanttemperatureT andexternalpressureP.Inthispaper, thermalequilibriumofaself-gravitatingN-bodysystem,sinceit
wewillshowthatthethreepointscanbeidentifiedbymeansof assumestheequationofstateofanidealgas.Moreover,because
adynamicalstabilityanalysisofthefluidmodel. gravitational forces have long range, each particle feels the in-
fluenceofallotherdistantparticles.Thus,itmaybethat,strictly
Now we briefly comment on the use of the Boltzmann ex-
speaking, in the case of self-gravitating systems an equation of
pression for the entropy (1). If we refer to systems for which
statecannotbedefinedintermsoflocalquantities.
the traditional thermodynamic limit (N → ∞ and V → ∞ at
AdifferentwaytojustifytheuseoftheBoltzmannentropy
N/V fixed)iswelldefinedandleadstohomogeneousequilibria,
is to show that it can be obtained from the Gibbs microcanoni-
itiswellknown(Jaynes1965)thattheBoltzmannentropycor-
calentropybymeansoftheso-calledmeanfieldapproximation
respondstotheentropydefinedinphenomenologicalthermody-
(Katz 2003; Padmanabhan 1990). However, the range of appli-
namicsonlywhentheinterparticleforcesdonotaffectthether-
cability of this approximation is still not clear, also because a
modynamic properties; that is, the Boltzmann entropy neglects
small-scale cut-off is necessary to avoid divergences in the mi-
the interparticle potential energy and the effect of the interpar-
crocanonicalentropy(Padmanabhan1990).TheBoltzmannen-
ticle forces on the pressure. If the equation of state is different
tropyisalsothe H quantityoftheBoltzmann H-theorem;from
from that of an ideal gas, the Boltzmann entropy is in error by
thispointofview,acriticalanalysisoftheBoltzmannentropyin
anon-negligibleamount.Thecorrectexpressionfortheentropy
thecontextofthegravothermalcatastrophehasbeenperformed
(corresponding to phenomenological thermodynamics) of sys-
byMiller(1973).
tems with a well-defined thermodynamic limit is the one given
The discussion in this section supports the hypothesis that
byGibbs[forthedefinitionofGibbsentropy,seeJaynes(1965)].
self-gravitating isothermal spheres are not true thermal equi-
Thegeneralizationoftheaboveresulttosystemswithinho- librium states of the pure N-body problem (i.e., the states that
mogeneous equilibria, such as self-gravitating systems, is that would be found in a spherical box containing N gravitating
the Boltzmann entropy is valid (i.e., it corresponds to entropy particles eventually, after an infinite amount of time), but are
asdefinedinphenomenologicalthermodynamics)ifandonlyif only metastable states of which the significance is still not
(1) the equation of state of the ideal gas applies locally to the completely understood (for example, see Padmanabhan 1990,
equilibriumstates;(2)hydrostaticequilibriumholds.Indeed,the Chavanis2006andreferencestherein).
mostelementarywaytounderstandtheBoltzmannentropyinthe
case of self-gravitating systems is to think of a fluid in hydro-
staticequilibrium,withtheequationofstateofanidealgas p= 3. Basicequationsofthedynamicalapproach
ρkT/m, subject to reversible transformations between equilib-
riumstates(i.e.,betweendifferenttruncatedisothermalspheres Inthissectionwederivethelinearizedhydrodynamicequations
that govern the evolution of a fluid system for small deviations
inthesphericallysymmetriccase).AsshownbyLynden-Bell&
fromthetruncatedisothermalsphereequilibriumconfigurations.
Wood (1968) in their Appendix I, the classical thermodynamic
entropy defined from the relation dS = dQ/T and calculated We assume spherical symmetry, that is, we consider only ra-
c
dial perturbations. The configurations are known to be stable
from such transformations coincides with S . For the general
b
against nonradial perturbations (Semelin et al. 2001; Chavanis
case of a system of particles interacting through an arbitrary
2002;Binney&Tremaine2008).
two-bodypotential(notnecessarilygravitational),itispossible
Consider a self-gravitating fluid governed by the Navier-
toshowthatthestationarystatesoftheBoltzmannentropyhave
Stokes and continuity equations, together with the equation of
thesamedensitydistributionasthatofafluidwiththeequation
stateofanidealgas(forexample,seeLandau&Lifshitz1987):
ofstateofanidealgasinhydrostaticequilibrium.
The above considerations suggest that the validity of the ∂ρ
+∇·(ρu)=0,
Boltzmannentropyisstrictlyrelatedtothevalidityoftheequa-
∂t
tionofstateofanidealgas.Inakineticdescription,theequation ∂u ∇p η ζ+ η
ofstateofanidealgasisobtainedbyconsideringthelocalone- +(u·∇)u=− −∇Φ+ ∇2u+ 3∇(∇·u) , (2)
particle distribution function to be a Maxwellian (i. e., of the ∂t ρ ρ ρ
form f(r,v) = Aexp[−mv2/2kT(r)],whereAisanormalization kT
(cid:82) p=ρ ,
constant)andbydefiningthepressureasp≡ mf(v2/3)d3rd3v. m
Thisdefinitionofpressure,whichisobtainedbyconsideringpar-
ticlesthatwouldreversetheirmomentumwhenhittinganimag- whereρisthedensity,uisthefluidvelocity, pisthelocalpres-
inary wall, ignores the effects of interparticle forces. If, for ex- sure,Φisthegravitationalpotential,ηandζareviscositycoeffi-
ample, particles repelling one another were confined in a box, cients,mistheone-particlemass,T isthetemperature,tistime,
theywouldexertsomepressureonthewallsoftheboxevenifat andkistheBoltzmannconstant.Wewillkeeptrackofviscosity
rest:thiscontributioniscompletelyneglectedbytheequationof termuntiltheendofSubsection3.1;then(fromSubsection3.2
stateoftheidealgas(theneglectedpressureissimilartothatof on), for simplicity, we will set η = ζ = 0. In this paper, we do
rigid spheres when packed too closely; thus the Boltzmann en- not perform an analysisof the effects of viscosity. Inparticular
tropycannotgiveacorrectresultforagasofalmostrigidspheres we do not discuss whether it has a stabilizing or destabilizing
whentheirdensityistoohigh). effect:hopefully,theroleofviscositywillbestudiedindetailin
asubsequentpaper.
In the case of particles interacting through gravity, the ne-
glected contribution is attractive and should therefore decrease 3 Byeffectiveequationofstatewemeananequationofstatethat,by
thepressurecomparedtothatofanidealgas.Therefore,itisun- imposing hydrostatic equilibrium, would reproduce the density distri-
likelythatthetruethermalequilibriumstateofan N-bodysys- butionoftheequilibriumstate.
3
M.C.SormaniandG.Bertin:GravothermalCatastrophe:thedynamicalstabilityofafluidmodel
Under the assumption of spherical symmetry, the gravita- Inthefollowing,weshalldropthesymbol˜tokeepthenotation
tionalpotentialobeysthefollowingequations: simpler. After some manipulations (see Appendix B.1) and us-
ing the unperturbed density profile (4) we obtain the following
GM(r)
∇Φ= rˆ, linearizedequation:
r2
(cid:90) r (3)
M(r)= ρ(s)4πs2ds, iω kT
0 Lf = Lf + 1ψ(cid:48)e−ψ (8)
4πGλ m
whicThhearehyedqruoivstaalteinctetoquthileibProiaissoofnseuqcuhatfliouni.d system are spa- + iω m (cid:40)η 1 [ξ2(feψ)(cid:48)](cid:48)+(ζ+ η)(cid:34) 1 (cid:16)ξ2feψ(cid:17)(cid:48)(cid:35)(cid:48)(cid:41),
ρ (0)kT ξ2 3 ξ2
tially truncated isothermal spheres (Chandrasekhar 1967) and 0 0
are briefly described in Appendix A. As mentioned in Sect. 2,
where f(ξ)=ρ (ξ)u (ξ)istheunknownfunction,
eachself-gravitatingtruncatedisothermalsphereisidentifiedby 0 1
two dimensional scales (for example the central density ρ (0)
0 ω2
and the temperature T) and one dimensionless parameter Ξ = L= (9)
4πGρ (0)
R/λ, which is the value of the dimensionless radius ξ ≡ r/λ 0
at the truncation radius (the scale λ was defined at the begin-
representsthedimensionless(squared)eigenfrequencyandLis
ningofSect.2).Conversely,eachchoiceofthethreequantities thefollowingdifferentialoperator:
{ρ (0),T,Ξ}identifiesaparticularisothermalsphere.
0
We shall denote unperturbed quantities by subscript 0 and d2 (cid:32)2 (cid:33) d (cid:32) 2 2ψ(cid:48) (cid:33)
the perturbations by subscript 1. Thus the unperturbed density L≡− − +ψ(cid:48) + − −e−ψ . (10)
dξ2 ξ dξ ξ2 ξ
profile (the density profile of a truncated isothermal sphere) is,
asafunctionofthedimensionlessradiusξ(seeAppendixA):
Equation (8) is the equation to be solved to find the nor-
(cid:40)ρ (ξ)=ρ (0)e−ψ(ξ) ifξ ≤Ξ mal modes and frequencies of the system, under the appropri-
0 0 (4) ate boundary conditions and constraints. The boundary condi-
0 ifξ >Ξ,
tions for Eq. (8) are defined in the following way. The absence
of sinks and sources of mass, together with the assumption of
where ρ (0) is the unperturbed central density and ψ(ξ) is the
0 sphericalsymmetry,implies f(0)=0.SinceintheEulerianrep-
regular solution to the Emden equation (the symbol (cid:48) denotes
resentationweshallconsideronlysystemsinasphericalboxof
derivativewithrespecttotheargumentξ):
constant volume, the radial velocity at the edge must be zero,
d whichimplies f(Ξ)=0.Thustheboundaryconditionsare:
(ξ2ψ(cid:48))=ξ2e−ψ, ψ(0)=ψ(cid:48)(0)=0. (5)
dξ f(0)= f(Ξ)=0. (11)
When we perturb the hydrodynamic equations it is convenient
Tosolvetheequations,westillneedtospecifythefunction
to work in the Eulerian or Lagrangian representation of hydro-
T (t).Thisspecificationdiscriminatesbetweentheconstanten-
dynamics,dependingonwhichboundaryconditionsweimpose. 1
ergyandtheconstanttemperaturecase.Oncethisspecificationis
IntheEulerianrepresentationtheindependentvariablesarethe
made,Eq.(8)isaneigenvalueproblem:forfixedω,theequation
timetandthepositionvectorr.IntheLagrangianrepresentation
admitsasolutiononlyfordiscretevaluesof L.Thesesolutions
theindependentvariablesarethetimetandtheoriginalposition
aretheradialnormalmodesofthesystem.
r that the fluid element under consideration had at the initial
0 The operator L has the following properties, which can be
timet .
0
proveddirectlyfromtheEmdenequation(5):
3.1. Eulerianrepresentation L(ψ(cid:48))=−e−ψψ(cid:48)
. (12)
L(ξe−ψ)=−e−ψψ(cid:48)
In this subsection we derive the linearized perturbation equa-
tions in the Eulerian representation. So we write each quantity
Thesepropertiesallowustoobtainanalyticalsolutionsinsome
as the sum of an unperturbed part (characterizing the truncated
cases. They are equivalent to those found by Padmanabhan
isothermalsphere)andaperturbation:
(1989) in his review of the thermodynamical analysis of
ρ(r,t)=ρ (r)+ρ (r,t) Antonov (1962) and later also used by Chavanis (2002, 2003)
u(r,t)=u0(r,t) 1 (6) toobtainanalyticalsolutionsofthepresenthydrodynamicprob-
T(t)=T 1+T (t), lemfortheconstant{T,V}(Sect.4.1.1)andconstant{T,P}(Sect.
0 1 4.2.1)cases.
whereuistheradialcomponentofthefluidvelocity(recallthat Notethatviscositydisappearswhenω = 0,whichisthesit-
weconsideronlyradialperturbations).WesubstituteEq. (6)in uationofmarginalstability.Thusviscositydoesnotmodifythe
Eq.(2)andexpandtofirstorderintheperturbedquantities.We pointsoftheonsetofinstability(Semelinetal.2001;Chavanis
allow the temperature to vary in time while remaining uniform 2002).
inspace:thisconstraintwillbeusedinordertoimposethecon-
ditionofconstanttotalenergy(seeSubsection4.1.2).Tofindthe
3.2. Lagrangianrepresentation
normalmodesofthesystem,welookforsolutionsofthefollow-
ingform: Inthissectionwepresentthelinearizedperturbationequationsin
ρ (r,t)=ρ˜ (r)e−iωt theLagrangianrepresentationintheinviscidcase.Inthisrepre-
1 1
u1(r,t)=u˜1(r)e−iωt (7) sentation,theindependentvariableisthepositionr0 ofthefluid
T1(t)=T˜1e−iωt . elementunderconsiderationattheinitialtimet0.ThusinEq.(2)
4
M.C.SormaniandG.Bertin:GravothermalCatastrophe:thedynamicalstabilityofafluidmodel
we have to perform the following change of independent vari- 4.1.1. Constant{T,V}case(isothermalcollapse)
ables:
(cid:40) HereweconsiderafluidatconstanttemperatureT containedina
r→r (r,t)
0 (13) sphereoffixedradiusR.Theconditionofconstanttemperature
t→t. is satisfied by imposing T = 0 in Eq. (8). The condition of
1
constant volume has been discussed in Sect. 3.1 and leads to
IntheLagrangianrepresentation,eachquantityisafunctionof
theboundarycondition f(Ξ) = 0.ThusEq.(8)fortheconstant
thenewindependentvariablesr andt.
0 {T,V}casebecomes:
ThecalculationsaresummarizedinAppendixB.2.1.Forlin-
Lf = Lf (18)
earperturbations,theresultingcontinuityandEulerequationsin
theLagrangianrepresentationare(neglectingthetermquadratic withtheboundaryconditions(11).
inthevelocity): Equation(18)isaneigenvalueequationthat,atgivenΞ,ad-
mits solutions only for discrete values of L. If the lowest value
of LatfixedΞispositive,thenallmodesarestable(becauseω
 
∂∂t − rr22ρρ u∂∂r ρ+ r12ρρ ∂∂r (cid:16)r2ρu(cid:17)=0 iastaalgwivayensrveaalul)eaonfdΞthisenseygsatetimvei,sthsteanbulen.sItfabthleemloowdeesstavraelupereosefnLt
 0 0 0 0 0 0 (14) andthesystemisunstable.
∂∂t − rr22ρρ u∂∂r u=−rr22∂∂rρ ρ1 kmT − GMr2(r0). Bymeansofthestandardtransformation
0 0 0 0 0 0 (cid:34) (cid:90) ξ(cid:32)1 ψ(cid:48)(ξ(cid:48))(cid:33) (cid:35)
f(ξ)= f˜(ξ)exp − + dξ(cid:48) , (19)
AsfortheEuleriancaseweseparateeachquantityinanun- ξ¯ ξ(cid:48) 2
perturbedpartandaperturbation:
whereξ¯isanarbitrarypointinthedomainof f,Eq.(18)canbe
ρ(r ,t)=ρ (r )+ρ (r ,t) recastintheformofaSchro¨dingerequation:
0 0 0 1 0
u(r ,t)=u (r ,t)
r(r0,t)=r1+0r (r ,t) (15) − f˜(cid:48)(cid:48)(ξ)+ f˜(ξ)U(ξ)= Lf˜(ξ), (20)
0 0 1 0
T(t)=T0+T1(t). whereU(ξ)istheeffectivepotentialgivenby:
We substitute Eq. (15) in Eqs. (14) and expand to first order in U(ξ)≡ 2 + 1ψ(cid:48)(ξ)2− 2ψ(cid:48)(ξ)− 1e−ψ(ξ). (21)
quantities with subscript 1. Then to find the normal modes we ξ2 4 ξ 2
take:
ρ (r ,t)=ρ˜ (r )e−iωt Theboundaryconditionsbecome:
1 0 1 0
u1(r0,t)=u˜1(r0)e−iωt (16) f˜(0)= f˜(Ξ)=0. (22)
r (r ,t)=r˜ (r )e−iωt .
1 0 1 0
The effective potential is shown in Fig. 1. From the boundary
In the following we shall drop the symbol ˜ for simplicity of conditions(22)weseethatchoosingaspecificΞrequiresusto
notation.Afterintroducingthedimensionlessradiusξ0 ≡ r0/λ, consider a potential that is infinite for ξ ≥ Ξ. The existence of
using the density profile (4) of the unperturbed state and after negative eigenvalues implies that the system is unstable. From
somemanipulationsweobtainthefollowingequationforρ1(ξ0) theformofthepotentialitisclearthatforsmallvaluesofΞneg-
(seeAppendixB.2.2foranoutlineofthecalculations): ative eigenvalues (i.e., unstable modes) do not exist. Negative
− ρ0ρ(10) + eξ−02ψddξ0 (cid:32)ddξξξdd40003(e+−ρ0ψρ1((0ξξ)002))ψ+L(cid:48)(ξ0TT)01(cid:33)=0, (17) teisnhnaiagefimmenpeniicrtvpeeaasollneiuannuepttmsspprbwaroepoahrpbceeolhreaefm(rnsnoeeetenghwlaiKystuiavonftoecsztcrae1ubsi9rguls7eeffi8nam)vtc.aioΞeldune≥etslsy8aap.lp9app9rege.aeaWrr,vihnapelrtunheeceΞsisthoe→elfyrΞm∞a.ot,Fdtahyone-r
LetusnowconsiderthesolutionstoEq.(18)ingreaterdetail.
whereL=ω2/4πGρ (0)asfortheEuleriancase.Boundarycon- Figure2showstheminimumvalueofLatgivendimensionless
0 radius Ξ as a function of Ξ. We see that L is negative, that is,
ditions are discussed in Subsection 4.2.1. Solving Eq. (17) al-
the system is unstable, for Ξ > 8.99, which is the same point
lowsustofindthenormalmodesintheLagrangianrepresenta-
found by Lynden-Bell & Wood (1968) in the thermodynamical
tion.WehaveallowedthetemperaturetovaryintimeinEq.(17),
approach.Highermodes,thatis,highervaluesof L atgivenΞ,
but in the following we shall analyze only the isothermal case
withT =0. would be represented by lines above the plotted curve. These
1 lineswouldintersecttheΞaxisatsomepoints,whicharetheze-
rosoftheanalyticalsolutionG describedbelow[seeEq.(23)].
TV
4. Modalstabilityofaself-gravitatinginviscidfluid In Appendix D.1 a few density and velocity profiles of numer-
ically calculated eigenfunctions are shown. Most of them are
sphereunderdifferentboundaryconditions
formodesofminimum LatgivenΞ.Densityprofilesofhigher
Hereweanalyzetheequationsobtainedintheprevioussection modesexhibitoscillationsnotpresentinthelowestmode.
byimposingdifferentboundaryconditions. For the case of marginal stability (L = 0), with the help of
properties(12),therelevanteigenfunctioncanbeexpressedan-
alytically(Chavanis2002).Thefunction
4.1. Eulerianrepresentation
G (ξ)≡ψ(cid:48)(ξ)−ξe−ψ(ξ) (23)
TV
InthissubsectionweanalyzeEq.(8)byimposingtwokindsof
boundary conditions: constant temperature T and volume V or is indeed a solution to Eq. (18) with L = 0, which satisfies
constanttotalenergyEandvolumeV. G (0)=0.ThevaluesofΞforwhichG (Ξ)=0arethosefor
TV TV
5
M.C.SormaniandG.Bertin:GravothermalCatastrophe:thedynamicalstabilityofafluidmodel
1.0 It has two terms, which represent the thermal and gravitational
contributions. The condition of constant energy is imposed in
0.8
thefollowingway.Whenthefluidisperturbed,itsgravitational
0.6 energychangesasaconsequenceoftheredistributionofmatter.
We suppose that the temperature varies in time, while remain-
0.4
ing uniform in space, so as to keep the total energy (thermal
U
0.2 plusgravitational)constant(foradifferentmodel,basedonthe
Smoluchowski-Poissonsystemofequations,thesamenonstan-
0.0
dard assumption has been made by Chavanis et al. 2002). The
(cid:45)0.2 thermalenergyexpressionattimetisthusgivenby3NkT(t)/2.
Indoingso,weareassuminginfinitethermalconductivity(see
(cid:45)0.4
also Sect. 5.1 for the relation of this fact to the relevant time
0 10 20 30 40 50
scales).
Ξ
ToreduceEq.(8)totheconstant{E,V}caseweneedtofind
Fig.1.TheeffectivepotentialU [seeEqs.(20)and(21)]forthe the expression for the temperature as a function of the density
constant {T,V} case. To find the frequencies of normal modes, distribution at fixed total energy, in the linear regime of small
the Schro¨dinger equation (20) has to be solved with boundary perturbations. Starting from Eq. (24) for the total energy and
conditions(22),whichmeansthattheeffectivepotentialisU(ξ) recallingthatΦ(r)=−G(cid:82)d3r(cid:48)ρ(r(cid:48))/|r−r(cid:48)|,wehave:
for ξ ≤ Ξ and taken to be infinite for ξ > Ξ. States with nega-
tive energies, which exist for Ξ ≥ 8.99, correspond to unstable 3 G (cid:90) ρ(r)ρ(r(cid:48))
E = NkT − d3rd3r(cid:48). (25)
modes. 2 2 |r−r(cid:48)|
By substituting T = T +T , ρ = ρ +ρ in Eq. (25), keeping
0 1 0 1
0.20 onlyfirst-orderquantitiesandimposingthattheenergyremains
constant,weobtainthefollowingexpressionforT :
1
0.15 (cid:82)
ρ (r)Φ (r)d3r
T =− 1 0 . (26)
0.10 1 3Nk
2
HereΦ isthegravitationalpotentialoftheunperturbeddensity
0.05 0
distribution ρ . After some manipulations (see Appendix B.3)
0
weobtain:
0.00
(cid:82)Ξ
-0-.000-5-- -0.042 T1 =−i1ω 0{d3[ξρ2(f0(ξ)λ)]Ξ/d2ξψ}(cid:48)ψ(Ξ(ξ))dξT0. (27)
2 0
0 20 40 60 80
By substituting Eq. (27) in Eq. (8), recalling the definition λ =
Fig.2. The minimum value of the eigenvalue L at given Ξ (the (cid:2)kT0/4πGρ0(0)m(cid:3)1/2,andneglectingviscosity,weobtain:
dΞi,mfeonrstihoenlceossnsratadniuts{Tch,Var}acctaesreiz.inIfgtthheemsyisntiemmu)masLafisunncetgioantivoef Lf =Lf −ψ(cid:48)e−ψ 1 (cid:90) Ξψ(ξ) d (cid:104)ξ2f(ξ)(cid:105) dξ. (28)
then the system is unstable. The system becomes unstable at 3Ξ2ψ(cid:48)(Ξ) dξ
Ξ = 8.99, the same value obtained from the thermodynamical 2 0
approach.Fromthefigurewecanreadthetypicaltimescaleof AsdiscussedinSect. 3.1theboundaryconditionsaregivenby
theinstability.AsΞ→∞,theasymptoticvalueL =−0.042is Eq.(11).ByintegratingthelasttermofEq.(28)bypartsunder
∞
thesameasfortheothercases(seeFigs.3and4). theboundaryconditions(11),weobtain:
(cid:90) Ξ
1
Lf =Lf +ψ(cid:48)e−ψ ξ2ψ(cid:48)(ξ)f(ξ)dξ. (29)
which the boundary conditions (11) are satisfied; thus they are 3Ξ2ψ(cid:48)(Ξ)
thevaluesofΞatwhicheachnewunstablemodeappears.The 2 0
first zero ofG occurs where the first unstable mode appears, Note that Eq. (29) [or (28)] contains an integral global con-
TV
atΞ=8.99.Fromtheasymptoticbehaviorofψitispossibileto straint. Equation (29) is to be solved to find the normal modes
obtainanasymptoticapproximationofthezeros:theyapproxi- andcorrespondstoEq.(18).Thenumericalprocedurefollowed
√
mately follow a geometric progression of ratio e2π/ 7 (see also tosolveEq.(29)isrelativelystraightforwardandthusisnotre-
Semelinetal.2001;Chavanis2002). portedhere.
Figure3showstheminimumvalueof LatgivenΞ.Wesee
that the system is unstable for Ξ > 34.36, which corresponds
4.1.2. Constant{E,V}case(gravothermalcatastrophe) to a density contrast ρ (0)/ρ (Ξ) > 709 (Antonov 1962). As Ξ
0 0
increases,newunstablemodesappear,similarlytothebehavior
Here we consider a self-gravitating fluid sphere at constant to-
observedintheconstant{T,V}case.AsΞ→∞,theasymptotic
tal energy E and volume V. In this case the instability has
valueoftheminimumLisfoundnumericallytobethesameas
beennamedgravothermalcatastrophebyLynden-Bell&Wood
intheconstant{T,V}case,L (cid:39)−0.042.Suchasymptoticvalue
(1968).ThetotalenergyEofthefluidisdefinedas: ∞
isthesamealsointheconstant{T,P}case(seeSect.4.2.1).
3 1(cid:90) In the case of marginal stability (L = 0), Eq. (29) is equiv-
E = NkT + ρ(r)Φ(r)d3r. (24)
alent to that found and solved analytically by Padmanabhan
2 2
6
M.C.SormaniandG.Bertin:GravothermalCatastrophe:thedynamicalstabilityofafluidmodel
0.5 instability: by numerically solving the algebraic equation (36),
this minimum value is found to be Ξ = 34.36 (as shown by
0.4 Antonov 1962). As for the constant {T,V} case, the number of
unstablemodesforeachΞcoincideswiththeresultsofthether-
0.3 modynamical analysis [see Katz (1978)]. Once a value of Ξ is
(cid:76)0 obtained,thevalueofa/bandthenananalyticalsolutionG is
min(cid:72)Ρ0 0.2 determined. EV
2ΩΠG Density profiles for modes of minimum L at given Ξ are
4
0.1 showninAppendixD.2.Sincetheequationsinvolvedareequiv-
alent, the density profile for the marginally stable perturbation
0.0 (cid:88)=34.363 (L = 0) is the same as that found by Padmanabhan (1989). He
pointed out that it has a “core-halo” structure, that is, an oscil-
-0.042
lationinρ :thedensityperturbationispositiveintheinnerpart
(cid:45)0.1 1
0 20 40 60 80 100
(nucleus),negativeinthemiddle(emptyingarea),andthenposi-
(cid:88)
tiveagain(halo).Thecore-halostructurehasbeenphysicallyin-
Fig.3. Theminimumvalueoftheeigenvalue LatgivenΞ,the terpretedintheframeworkofthegravothermalcatastrophegiven
dimensionlessradiuscharacterizingthesystem,asafunctionof by Lynden-Bell & Wood (1968), using the concept of negative
Ξ,fortheconstant{E,V}case.WhentheminimumLisnegative specificheat.However,thisinterpretationisnotapplicabletothe
the system is unstable. The system becomes unstable at Ξ = presentcontext.
34.36,whichcorrespondstoadensitycontrast(cid:39) 709,thesame WefoundthatformodesofminimumLatgivenΞthecore-
value obtained from the thermodynamical approach. From the halo structure is present if L < 0.021, disappears between L =
figure we can read the typical time scale of the instability. As 0.021andL = 0.022,andisabsentforL > 0.022,asillustrated
Ξ→∞,theasymptoticvalueofL,L∞ =−0.042,isthesameas inthefigurespresentedinAppendixD.2.Highermodesalways
fortheothercases(seeFigs.2and4). exhibitoneoremoreoscillations,asintheconstant{T,V}case.
Wehavethusshownthatforthepresentcasethebehaviorof
theeigenvaluesLasafunctionofΞissimilartothatofthecon-
(1989) in the thermodynamical approach. Here, for complete-
stant{T,V}case,withthedifferencethattheinstabilitythreshold
ness, we record how the solution is derived, by adapting the
is higher in the constant {E,V} case. The interpretation of this
method of Padmanabhan (1989) to our choice of variables and
factinthecontextofthefluidmodelisasfollows.Whenthefluid
unknowns.
is compressed, the gravitational potential energy decreases and
TosolveanalyticallyEq.(29)for L = 0werewriteitinthe
the temperature increases in order to maintain the total energy
followingform:
Lf =−Cψ(cid:48)e−ψ, (30) constant. Therefore, the tendency toward collapse is weakened
and instability can take place only at higher values of Ξ (with
where respecttothecaseinwhichthetemperatureremainsconstant).
(cid:90) Ξ
1 Infact,ifinsteadofexpression(24)forthetotalenergywe
C = ξ2ψ(cid:48)(ξ)f(ξ)dξ (31)
3Ξ2ψ(cid:48)(Ξ) consider a modified expression in such a way that the temper-
2 0 ature increase is greater, the collapse can be halted completely.
isaconstant(withrespecttoξ)thatdependsgloballyon f.Using Considerthefollowingheuristicexpressionforthetotalenergy:
theproperties(12),welookforasolutionoftheform (cid:90)
3 1
E = NkT + ρ(r)Φ(r)d3r−υσ2R3ρ(0), (37)
GEV(ξ)=aψ(cid:48)(ξ)+bξe−ψ(ξ), (32) 2 2 0
where a and b are real numbers. Since Eq. (30) is linear in f, whichcontainsanadditionaltermproportionaltothedimension-
onlytheratioa/bisrelevant.Susbtituting(32)in(30)weobtain less parameter υ and to the central density ρ(0). The quantity
thefirstconditiononaandb: σ20 = kT0/mistheunperturbedthermalspeed.Werepeatedthe
analysis of this subsection with such a modified expression for
a+b=C. (33) theenergyandfoundthenewvalueofΞforthethresholdofthe
instability. In this analysis the expression for T (27) obtained
1
UsingtheexpressionofC(31)anddividingbyb,weobtain: previously,tobesubstitutedinEq.(8),isreplacedwiththeex-
a +1= 1 (cid:90) Ξξ2ψ(cid:48)(ξ)(cid:18)aψ(cid:48)(ξ)+ξe−ψ(ξ)(cid:19) dξ. (34) pTre=ssTio0n+Tfo1r,ρT1=tρh0a+t ρis1oanbdtabinyeidmfproosmingEqth.a(t3t7h)ebvyarisautbiosntitouftitnhge
b 32Ξ2ψ(cid:48)(Ξ) 0 b totalenergyE iszero.Wefoundthatwhenυ>0(i.e.,whenthe
fluid is compressed the new term contributes to make the tem-
Asecondconditionona/bfollowsfromtheboundarycondition perature increase) the(linear) instability is postponed to higher
f(Ξ)=0: valuesofΞforsmallυandiscompletelyhaltedforυ>5×10−4.
a
ψ(cid:48)(Ξ)+Ξe−ψ(Ξ) =0. (35) Wealsoverifiedthatifυ<0(i.e.,whenthefluidiscompressed
b
the new term contributes in the opposite direction), instability
Substitutinga/bfrom(35)in(34)weobtain: occursatlowervaluesofΞ.
In the case of the pure N-body problem, it is believed that
3Ξ2(cid:16)ψ(cid:48)(Ξ)−Ξe−ψ(Ξ)(cid:17)=(cid:90) Ξξ2ψ(cid:48)(ξ)(cid:32)ξe−ψ(ξ)−ψ(cid:48)(ξ)Ξe−ψ(Ξ)(cid:33) dξ collapse can be halted by energy “generation” through bina-
2 ψ(cid:48)(Ξ) ries, giving rise to a phenomenon called gravothermal oscil-
0
(36) lations (for a review, see Heggie & Hut 2003). Gravothermal
The values of Ξ satisfying Eq. (36) are those for which oscillations were discovered by Sugimoto & Bettwieser (1983)
Eq. (29) admits a solution for L = 0. In particular, the lowest using a gaseous model. These authors introduced in the model
value of Ξ which satisfies Eq. (36) determines the threshold of of Lynden-Bell & Eggleton (1980) a phenomenological energy
7
M.C.SormaniandG.Bertin:GravothermalCatastrophe:thedynamicalstabilityofafluidmodel
generation term (to represent the role of binaries), proportional 0.5
toapowerofthedensityandfoundthatcollapsecanbehalted
and reversed. Similarly, our υ term shows in a simple manner 0.4
thattheinstabilitycanbehaltedinthelinearregimewithasuit-
ableterminthetotalenergybudgetthatmimicseffectspresumed 0.3
tobeassociatedwiththepresenceofbinaries. (cid:76)0
Gravothermal oscillations have been confirmed by N-body 2Ωmin(cid:72)ΡG0 0.2
simulations (Makino 1996). We will comment further on this Π4
topicinSect.5.2. 0.1
(cid:88)=6.45
0.0
4.1.3. Constant{T,V}:two-componentcase
-0.042
(cid:45)0.1
Webrieflystudiedtheproblemofdynamicalstabilitybymeans 0 5 10 15 20 25 30
ofalinearmodalanalysisofatwo-componentidealfluid.Each (cid:88)
component is assumed to interact with the other only through Fig.4. Theminimumvalueoftheeigenvalue LatgivenΞ,the
thecommongravitationalpotential.Thehydrostaticequilibrium dimensionlessradiuscharacterizingthesystem,asafunctionof
configurationsarespatiallytruncatedtwo-componentisothermal Ξ, for the constant {E,V} case. When the minimum L is neg-
spheres, as considered by Taff et al. (1975), Lightman (1977), ative the system is unstable. The system becomes unstable at
andYoshizawaetal.(1978);seealsodeVega&Siebert(2002) Ξ = 6.45,thesamevalueasfoundinthethermodynamicalap-
andSopiketal.(2005). proach.Fromthefigurewecanreadthetypicaltimescaleofthe
Wecallthesingle-particlemassesm andm ,withm >m . instability.AsΞ→∞,theasymptoticvalueL =−0.042isthe
A B B A ∞
Two-component isothermal spheres are characterized by two sameasfortheothercases(seeFigs.2and3).
additional dimensionless parameters (with respect to the one-
componentcase):theratioofthesingle-particlemassesm /m
B A
andtheratioM /M ofthetotalmassesassociatedwiththetwo
B A Boundary conditions are as follows. The condition u (0) = 0
components.ThereaderisreferredtoAppendixEforadescrip- 1
translates into the condition ρ(cid:48)(0) = 0 [using Euler equation
tionoftheequationsused. 1
(14), under the assumption that ∂u/∂r does not diverge and
We considered the constant {T,V} case, which is the sim- 0
∂u/∂t =−iωu].Theconditionofconstantpressurerequiresthat
plest from the mathematical point of view, and found that the
afixedLagrangianfluidshell(whichfollowsthefluidduringthe
equivalenceofthethermodynamicalanddynamicalapproaches
motion and thus does not have a fixed position in space) feels
stillholds.Itispossibletoshowanalyticallythattheonsetofdy-
constant pressure. Constant pressure requires constant density
namicalinstabilitytakesplaceatexactlythesamepointsasthose
because of the ideal gas equation of state. Therefore, the rele-
found with the thermodynamical approach. The analysis in the
vantboundaryconditionsare:
thermodynamicalapproachwasperformedbygeneralizingina
straightforwardmannertheanalysisofChavanis(2002).
ρ(cid:48)(0)=0
Aninterestingresultofthisanalysisisthattheinstabilityap- ρ1(Ξ)=0. (39)
pears to be driven by the heavier component, in the following 1
sense.Considertheratiosρ /ρ andρ /ρ ofthedensityper-
turbationofeachcomponen1tAto0thetota1lBloc0alunperturbedden- Frommasscontinuityandbyimposingu(cid:48)(0) (cid:44) 0(whichistrue
1
sity.Theratioreferringtotheheaviercomponentcanbehigher in the Eulerian representation and thus we expect to be true in
even if the total mass of the heavier component is very small. thepresentcase),wealsohavetheconditionρ1(0)(cid:44)0.
For example, for m /m = 3, we found that the ratios ρ /ρ , The numerical procedure to solve Eq. (38) is relatively
B A 1A 0
ρ /ρ were comparable for M /M (cid:39) 0.066 (see Fig. E.1). straightforward and is thus not reported here. Figure 4 repre-
1B 0 B A
Forhighervaluesof M /M theheaviercomponentdominates. sentstheminimumvalueof LatgivenΞ.Thesystembecomes
B A
Independent indications that the heavier component dominates unstableforΞ > 6.45,whichisthesameconditionasfoundby
thecollapsehavebeenfoundbySopiketal.(2005)foradiffer- Bonnor(1956),Ebert(1955),andLynden-Bell&Wood(1968).
entmodelbasedontheSmoluchowski-Poissonsystemofequa- As Ξ increases, other unstable modes appear, similarly to the
tions. Breen & Heggie (2012a,b) found that the heavier com- casesdescribedpreviously.AsΞ → ∞,theasymptoticvalueof
ponent dominates gravothermal oscillations in two-component the minimum L is found numerically to be the same as for the
clusters.Thisislikelytoberelatedtothepresentanalysis. constant {E,V} and {T,V} cases, L∞ = −0.042, but, in contrast
totheconstant{E,V}and{T,V}cases,itisreachedfrombelow.
ThereisaminimumatL≈−0.043forΞ≈15.
4.2. Lagrangianrepresentation
In Appendix D.3 density profiles of normal modes are
shown. Most of them are for modes of minimum L at given
4.2.1. Constant{T,P}case(isobariccollapse)
Ξ. Higher modes present oscillations. It would be interest-
Inthissectionweanalyzeperturbationsatconstanttemperature ing to show whether the analysis of this subsection might be
T and constant boundary pressure P. Therefore, T = 0 and generalized to the nonspherically-symmetric case by following
1
Eq.(17)becomes: Lombardi&Bertin(2001).
Asforpreviouscases,wecanobtainanalyticalsolutionsfor
 d ( ρ1 )  themarginallystableperturbations.ForL=0,Eq.(38)reads:
− ρ0ρ(10) + e−ξψ02(ξ0)ddξ0 ξ403dd+ξξd00eξ−ρ020ψψ((0ξL(cid:48))0()ξ0)=0. (38) L(cid:32)ρ0ρ(10)(cid:33)=0, (40)
8
M.C.SormaniandG.Bertin:GravothermalCatastrophe:thedynamicalstabilityofafluidmodel
whereLisdefinedas: 5.2. Thedynamicsofacollisionlessself-gravitating
 d  isothermalsphere
L≡−1+ eξ−02ψddξ0  dξd0dξ4ξ0e30−ψ=0. (41) IigdsroeatavhliietzaretmidnagmliossdpoehthlesereormsftaahlreesppehuqereureiNlsib-cbraionudmbyepcsrtoounbdfilieegmdur.aaIstnisoptnaatsritoifncouarrlaysre,svtsaeetrleafs-l
ofthecollisionlessBoltzmannequation:
FromEmdenEq. (5),theoperatorLhasthefollowingproper-
ties: ∂f(r,v,t) ∂f(r,v,t) ∂Φ(r,t) ∂f(r,v,t)
L(ψ(cid:48)2)=−e−ψ +v· − · =0. (47)
2 (42) ∂t ∂r ∂r ∂v
L(e−ψ)=−e−ψ .
4
Therefore,itisnaturaltoaskwhethersuchcollisionlessisother-
Hence, an analytical solution of Eq. (40) is the following (for
mal spheres are or can be unstable. It can be shown (Binney
a derivation in the Eulerian representation, see also Chavanis
& Tremaine 2008) that the unbounded collisionless isothermal
2003):
G (ξ )=2e−ψ−ψ(cid:48)2. (43) sphere is linearly stable, in contrast to the results of the fluid
TP 0 counterpart (in the fluid model we recover the unbounded case
This solution satisfies the boundary conditions ρ1(0) = 2 and bytakingΞ → ∞).Itisgenerallybelievedthatthecollisionless
ρ(cid:48)1(0) = 0, as required by Eq. (39). The zeros ofGTP(ξ0) allow isothermalsphereisalsostableinthenonlinearregime,although
ustoidentifythemarginallystablenormalmodesthatsatisfythe toourknowledgearigorousproofofthisstatementisstilllack-
correctboundaryconditions(39).ThefirstzeroofGTP(ξ0)isat ing. Moreover, if only spherically symmetric perturbations are
Ξ = 6.45. Other zeros correspond to the values of Ξ at which considered,itispossibletoshow,throughanargumentbasedon
new unstable modes appear and are found to be the same as in theconservationofthedetailedangularmomentum4 (Appendix
thestandardthermodynamicalapproach. C), that a collisionless sphere (bounded or unbounded) cannot
collapse,becauseeachparticlehasaminimumradiusitcanat-
tain.Incontrast,thefluidsystemanalyzedinSects.3and4isun-
5. Thedifferentbehaviorofacollisionless
stableonlywithrespecttosphericallysymmetricperturbations,
self-gravitatingsphere
with the subsequent nonlinear evolution presumably leading to
5.1. Timescales acollapse.Theaboveconsiderationssupportthehypothesisthat
self-gravitatingcollisionlessisothermalspheres(boundedorun-
Inthissubsectionwebrieflydiscussthetypicaltimescalesthat bounded)arestablewithrespecttoallkindsofperturbationsand
characterize gaseous and fluid models and compare them to cannot collapse (Kandrup & Sygnet 1985; Kandrup 1990; Batt
thoseofglobularclusters.Toalargeextent,ourdiscussionfol- etal.1995).
lowsthatofInagaki(1980). The different behavior, between the collisionless and the
Letτ bethelocalrelaxationtime,τ theglobalrelax-
LTE GTE fluidcase,emergedinthispaperisactuallyquitesurprising,be-
ationtime,andτ thedynamicaltimescale.Foragaseoussys-
d causeforthehomogeneouscasethecollisionlessandfluidmod-
tem,weidentifyτ withthetime(oftheorderoftheinverse
LTE elsbehaveinthesamewaywithrespecttoJeansinstability(e.g.,
mean collision frequency) needed to reach a local Maxwellian
see Bertin 2000), in the sense that both the fluid and the colli-
distribution, τGTE with the time in which thermal conduction sionless systems are linearly unstable under the same criterion
balances the temperatures of different parts of the system, and
forinstability.
τ withthesoundtraveltime.Gaseousmodelsgenerallyassume
d If the self-gravitating collisionless isothermal sphere were
thefollowingordering:
unstable, its instability would develop on the dynamical time
τ (cid:28)τ (cid:28)τ . (44) scale; in turn, it is commonly believed that the gravothermal
LTE d GTE
catastrophemayoccuronlyifthesystemisatleastweaklycol-
For a globular cluster, because the mean free path is very large lisional and thus it is thought to develop on the collision time
andstarscancrossthesystemmanytimesbeforeactuallycollid- scale.
ing,theorderingoftimescalesisdifferent.Starsdonotcollide In the spherically symmetric case a difference between the
significantly with neighboring stars, following the mechanisms twomodelsisthefollowing:whileinthefluidmodeleachfluid
thatusuallycharacterizethermalconduction;instead,theytend element is sustained against gravity by pressure of the inner
toreleasetheirenergythroughtheentirecluster(alsobecauseof parts,inthecollisionlessmodelstarsaresustainedbytheirindi-
thelongrangenatureoftheforce).Thusforglobularclusterswe vidualangularmomentumrelativetothecenter,thatis,bytheir
have: velocity dispersion.5 In moving from a kinetic description to a
τd (cid:28)τLTE (cid:39)τGTE. (45) fluid description, all the information about microscopic veloci-
In the fluid model analyzed in this paper we assumed infinite ties and detailed angular momentum is lost: in particular, each
thermalconductivity,becausethetemperaturewasalwaystaken fluid element has zero angular momentum (see also Appendix
to be and to remain uniform. Moreover, we implicitly assumed C).Inthisrespect,therealsituationofaglobularclusterresem-
that the distribution function is locally Maxwellian. Therefore, bles more the collisionless case: strictly speaking, stars are not
ourfluidmodelfollowstheordering: sustainedbypressure,butratherbytheirvelocitydispersion,be-
cause the mean free paths are long and stars cross the cluster
τLTE (cid:28)τd manytimesbeforefeelingtheeffectsofcollisions.
(46)
τ (cid:28)τ .
GTE d
4 Bydetailedangularmomentumwemeantheangularmomentumof
Note that assumptions (44) and (46) are not applicable to the theindividualparticles.
situation of a globular cluster (45). This difference and the re- 5 Evenifthetotalangularmomentumvanishes,thesumofthemag-
latedlimitationsmustbekeptinmindifwewishtoapplythese nitudesoftheangularmomentaoftheindividualparticlesisdifferent
modelstounderstandtheevolutionofglobularclusters. fromzero.
9
M.C.SormaniandG.Bertin:GravothermalCatastrophe:thedynamicalstabilityofafluidmodel
If we come back to the original pure N-body problem, it is stablemodes.Indeed,asnotedintheIntroduction,someresults
thereforenaturaltoask:whatisinthiscasetheroleofdetailed alongtheselineshavebeenobtainedpreviouslybyotherauthors.
angularmomentum?Wemayarguethataquantityrelatedtothe Themainnewresultsobtainedinthispaperarethefollowing:
detailed angular momentum should characterize the process of
core collapse. If the collapse can happen in an N-body system, – UsingthefluidmodelbasedontheEulerequation,wehave
it may be accompanied by a slow decrease of the sum of the extended previous studies of the constant {T,V} and {T,P}
magnitudes of the angular momenta of the individual particles casestotheconstant{E,V}case(gravothermalcatastrophe),
slowlysinkingtowardthecenter. proving that the onset of Jeans instability occurs exactly at
For a particle with angular momentum J that moves in a the same point identified in the thermodynamical approach
gravitational potential generated by a mass M a radius related also in this case. For this constant {E,V} case, we have in-
toangularmomentum(whichistheradiusofthecircularorbitif troduced a heuristic term to incorporate effects akin to the
Misapointmass)canbedefinedas: stabilizingroleofbinaries.
– For all the three cases described above, we have calculated
J2 numericallyeigenfrequenciesandeigenfunctionsoftherele-
d ≡ . (48)
vantmodesalsooutsidetheconditionsofmarginalstability.
Gm2M
The time scale for the instability that we have found is the
Therefore, for a stellar system of N stars and total mass M we dynamical time scale. The excitation of higher modes has
candefinearadiusrelatedtodetailedangularmomentuminthe beenillustratedinasimpleway,byreferringtoaneffective
followingway: potentialthatgovernsthestructureofthelinearmodalanal-
A
ysis.
R ≡ , (49)
am NGm2M – We have found that for all the cases treated in our investi-
whereAisthesumofthesquaresoftheangularmomentaofthe gation,asthedimensionlessradiusoftheisothermalsphere
individualstars,thatis,thefollowingquantity: Ξbecomeslargerandlarger,thevalueofthedimensionless
growth rate of the most unstable mode tends to a univer-
(cid:88)N sal asymptotic constant value, independent of the adopted
A≡ J2. (50) boundaryconditions.
i
i=1 – We have briefly shown that the correspondence between
the stability in the dynamical and in the thermodynamical
Thus we may argue that the typical time scale for core col-
approach also holds for the two-component case and have
lapseshouldcorrelatewith:
foundindicationsthattheheaviercomponentisthemoreim-
A portantdriveroftheinstability.
tcc ≡ dA/dt . (51) – As a general discussion, we have commented on the mean-
ing and applicability of the Boltzmann entropy for self-
The main mechanism through which A varies with time is ex- gravitating systems and argued that the main difference be-
pected to be that of two-body collisions. Thus t should be of tween the dynamical behavior of a fluid and a collisionless
cc
theorderofthetwo-bodyrelaxationtime. sphere,inrelationtotheirapplicationasmodelsofthepure
By monitoring the quantity A(t) in N-body simulations, it N-body problem or a real weakly collisional stellar system
would be interesting to test the role of detailed angular mo- suchasglobularclusters,istobeascribedtotheroleofthe
mentuminthemechanismofgravothermaloscillations(Makino detailedangularmomentumbehaviorinthecollisionlessand
1996) [A(t) may reach an equilibrium value, around which the weaklycollisionalcases.
systemgravothermallyoscillates;thiswouldbeconsistentwith
The role of the viscosity and its consequences outside the con-
the fact that the typical time scale of gravothermal oscillations
dition of marginal stability have not been examined; hopefully,
is the two-body relaxation time], its relevance to the studies of
thisissuewillbeaddressedinafuturepaper.
core-collapse in the gaseous model (Lynden-Bell & Eggleton
Another interesting question is how the instability depends
1980; Sugimoto & Bettwieser 1983), and its connection with
on the particle-particle interaction, for non-Newtonian cases
thephenomenonofthegyro-gravothermalcatastrophe(Hachisu
(Padmanabhan1989).Potentialsthatexhibitasofteningatsmall
1979).
radii,suchas1/(r2+r2)1/2,wherer isaconstant(Chavanis&
0 0
Ispolatov2002,seealsoCasetti&Nardini2012),orthatdecline
6. Discussionandconclusions withadifferentpowerlawatlargeradii,suchas1/rα,withα(cid:44)1
(Ispolatov&Cohen2001),haveindeedbeenconsidered.Based
Inthispaperwehavestudiedsystematicallythedynamicalsta-
on the present article, we may argue that the results obtained
bilityofaself-gravitatingisothermalfluidspherebymeansofa
fromthethermodynamicalapproachwouldbereinterpretedand
linearmodalanalysiswithrespecttosphericallysymmetricper-
clarified as the analogue of the Jeans instability, by studying a
turbations. In this sense, we have studied the Jeans instability
fluid model for the potential considered, with the equation of
intheinhomogeneouscontextofasphereoffinitesize.Within
stateofaperfectgas.
a unified framework, by imposing the boundary conditions of
In general, this paper strengthens the view that the applica-
constant{T,V}(isothermalcollapse),constant{E,V}(gravother- bility of different idealized models to describe the process of
mal catastrophe), and constant {T,P} (isobaric collapse), we
corecollapseinsystemsmadeofafinitenumberofstarsismore
have proved that the onset of dynamical instability occurs ex-
subtlethancommonlyreportedandthat,ingeneral,thestudyof
actly at the points identified in the thermodynamical approach
Jeansinstabilityofinhomogeneousstellarsystemsstillleavesa
(see Antonov 1962; Lynden-Bell & Wood 1968; Bonnor 1956;
numberofquestionsopen.
Ebert 1955) and, by adapting derivations from other authors
(Padmanabhan 1990; Chavanis 2002, 2003), we have provided Acknowledgements. We would like to thank Marco Lombardi, Francesco
an analytic expression for the eigenfunctions of the marginally PegoraroandStevenN.Shoreformanyinterestingcommentsanddiscussions.
10