Gravitational Instability of Isothermal and Polytropic Spheres Gral Is Dominated by the Distribution Φ0 Which Maximizes 2

Home , Vlasov equation

arXiv:astro-ph/0207080v4 2 May 2003 prxmto skont be to mean-field known the is interactions, approximation long-range me- with tested. systems statistical and developed For of be can ideas thermodynamics and which chanics for interaction long-range that in contributions other analogies and applica- book). 2002e of the Chavanis domains by (see numerous and tion the forces among differences long-range and with repre- treatment systems thermodynamic is plasma a of of challenge construction physics, the main by nuclear sented The as two- turbulence... differ- clusters, such dimensional atomic in condensates, physics rediscovered Bose-Einstein of physics, recently fields been 1968), ent have (Padmanabhan Wood galaxies They and & 1990). 1974) (Eddington (Hawking (Lynden-Bell stars holes clusters black for the globular thermodynamics during 1926), a literature been of astrophysical have the elaboration behaviours in curious discussed and first These negative ensembles transitions. as statistical such of phase inequivalence features heats, peculiar specific long- exhibits via forces interacting systems range of mechanics statistical The Introduction 1. nnt ass httesse utb ofie within confined be must structure system the stellar that so of mass have context & spheres infinite isothermal (Binney the well-known, is structure As in 1987). galactic Tremaine time and 1942) long stud- (Chandrasekhar a been have for ther- which ied a configurations of isothermal approach maximization to leads the thermodynamical This of equilibrium potential. terms statistical modynamical in stabil- analyzed at and be systems can structure self-gravitating the of Therefore, ity limit. modynamic DI ilb netdb adlater) hand by inserted be will (DOI: Astrophysics & Astronomy rvt rvdsafnaetleapeo unshielded of example fundamental a provides Gravity rvttoa ntblt fiohra n oyrpcsp polytropic and isothermal of instability Gravitational u td rvdsa etei n ipeapoc oti o this words. to Key approach simple and thermodynamica aesthetic an and provides dynamical study a between Our methods relation analytical the by p discuss instability approach gravitational thermodynamical of generalized onset a ensemb with isobaric connexion and in microcanonical grand canonical, grand Abstract. later included be To Sabat Universitée-mail: Paul Quantique, Physique de Laboratoire [email protected] ecmlt rvosivsiain ntethermodynamics the on investigations previous complete We tla yaishdoyais instabilities dynamics-hydrodynamics, Stellar aucitno. manuscript exact nasial ther- suitable a in ..Chavanis P.H. dgahclcntutosi h in ln.W also We plane. Milne the in constructions graphical nd n aetetemdnmclapoc ioos tis corre- to It only spond configurations rigorous. isothermal approach that well-known thermodynamical evaporation also prevent the to make order in and problem) (Antonov box a C,C n C epcieya ahitga o a for integral path a as field respectively states, in formal GCE of function and partition density CE grand the the MCE, write and and function theory partition Katz the field & a Horwitz technics. use different (1978) by studied be can tems 1990). (Padmanabhan transition at phase systems a (extensive) of normal verge phase to the a similar virial near in are systems except equilibrium gravitating that limit, suggests This thermodynamic transition. the range equiv- are at short ensembles alent with statistical (GCE). the systems, which canonical for ordinary de- interactions, grand with instability or contrasts of (CE) This onset canonical the microcanon- (MCE), considered: that ical ensemble statistical is the on realize pends cru- to The points. point thermodynami- saddle cial unstable the the become of equilibria, and maxima of potential local cal series be the to con- in cease solutions large point sufficiently some at at sta- centrations: low occurs thermodynamically sufficiently instability is For However, system system. ble. the the of contrasts, cen- boundary density the the between and contrast density ter the by parametrized be metastable such in probably are states. described models clusters Michie-King globular contemplated by particular, timescales In the astrophysics. for in expected however, relevant, are be states equilibrium to equilibrium true metastable not These potential), states. thermodynamical the of e,18rued abne302Tuos,France Toulouse, 31062 Narbonne de route 118 ier, e.W lodsustesaiiyo oyrpcspheres polytropic of stability the discuss also We les. h hroyaia tblt fsl-rvttn sys- self-gravitating of stability thermodynamical The can spheres isothermal finite of equilibria of series The ooe yTals edtriei ahcs the case each in determine We Tsallis. by roposed tblt fselrssesadgsosspheres. gaseous and systems stellar of stability l hriecmlctdsubject. complicated therwise metastable fsl-rvttn ytm ysuyn the studying by systems self-gravitating of φ ntema-edapoiain h inte- the approximation, mean-field the In . qiiru tts(.. oa maxima local (i.e., states equilibrium coe 4 2018 24, October heres 2 P.H. Chavanis: Gravitational instability of isothermal and polytropic spheres gral is dominated by the distribution φ0 which maximizes 2. Thermodynamics of self-gravitating systems a specific action A[φ]. In the grand canonical ensemble, 2.1. Thermodynamical ensembles AGCE[φ] is the Liouville action. Horwitz & Katz (1978) solve the problem numerically and find that the series of Consider a system of N particles, each of mass m, in- equilibria become unstable for a density contrast 1.58, 32.1 teracting via Newtonian gravity. Let f(r, v,t) denote the and 709 in GCE, CE and MCE respectively. It is normal distribution function of the system and ρ(r,t) = fd3v that the critical density contrast (hence the stability of the spatial density. The total mass and total energy can the system) increases when more and more constraints are be expressed as R added on the system (conservation of mass in CE, conservation of mass and energy in MCE). This field theory has M = ρd3r, (1) been rediscussed recently by de Vega & Sanchez (2002) Z who confirmed previous results and proposed interesting 1 1 E = fv2d3rd3v + ρΦd3r = K + W, (2) developements. 2 2 The thermodynamical stability of self-gravitating sys- Z Z tems can also be settled by studying whether an isother- where K is the kinetic energy and W the potential energy. mal sphere is a maximum or a saddle point of an appropri- The gravitational potential Φ is related to the spatial den- ate thermodynamical potential: the entropy in MCE, the sity by the Poisson equation free energy in CE and the grand potential in GCE. The change of stability can be determined very easily from the ∆Φ=4πGρ. (3) topology of the equilibrium phase diagram by using the The Boltzmann entropy is given by the standard formula turning point criterion of Katz (1978) who has extended (within an additional constant term) Poincaré’s theory on linear series of equilibria (see also f f f Lynden-Bell & Wood 1968). This method is very powerful S = k ln d3rd3v, (4) but it does not provide the form of the perturbation pro- − m m − m Z file that triggers the instability. This perturbation profile which can be obtained by a combinatorial analysis can be obtained by computing explicitly the second order (Ogorodnikov 1965). For the moment, we shall assume variations of the thermodynamical potential and reducing that the system is confined within a spherical box of ra- the problem of stability to the study of an eigenvalue equa- dius R so that its volume is fixed. tion. This study was first performed by Antonov (1962) in In the microcanonical ensemble, the system is isolated MCE and revisited by Padmanabhan (1989) with a sim- and conserves energy E and particle number N (or mass pler mathematical treatment. This analysis was extended M = Nm). Statistical equilibrium corresponds to the in CE by Chavanis (2002a) who showed in addition the state that maximizes the entropy S at fixed E and N. equivalence between thermodynamical stability and dy- Introducing Lagrange multipliers 1/T and µ/T for each namical stability with respect to Navier-Stokes equations constraint, the first order variations of entropy− satisfy the (Jeans problem). Remarkably, this stability analysis can relation be performed analytically by using simple graphical con- 1 µ structions in the Milne plane. δS δE + δN =0, (5) In the present paper, we propose to extend these an- − T T alytical methods to more general situations in order to where T is the temperature and µ the chemical potential. provide a complete description of the thermodynamics of Eq. (5) can be regarded as the first principle of thermo- spherical self-gravitating systems. In Sec. 2, we review the dynamics. stability limits of isothermal spheres in different ensembles In the canonical ensemble, the temperature and the by using the turning point criterion. In Sec. 3, we consider particle number are fixed allowing the energy to fluctu- specifically the grand canonical and grand microcanon- ate. In that case, the relevant thermodynamical parame- ical ensembles and evaluate the second order variations ter is the free energy (more precisely the Massieu func- 1 of the associated thermodynamical potential. In Sec. 4, tion) J = S T E which is related to S by a Legendre we briefly discuss the connexion between thermodynam- transformation.− According to Eq. (5), the first variations 1 µ ics and statistical mechanics (and field theory) for self- of J satisfy δJ = Eδ( T ) T δN. Therefore, at statistical gravitating systems. In Sec. 5, we consider the case of gen- equilibrium, the system− is− in the state that maximizes the eralized thermodynamics proposed by Tsallis (1988) and free energy J at fixed particle number N and temperature leading to stellar polytropes (Plastino & Plastino 1997, T . Taruya & Sakagami 2002a,b, Chavanis 2002b). In Sec. In the grand canonical ensemble, both temperature 6, we discuss the stability of isothermal and polytropic and chemical potential are fixed, allowing the energy and spheres under an external pressure (Bonnor problem). We the particle number to fluctuate. The relevant thermody- 1 provide a new and entirely analytical solution of this old namical potential is now the grand potential G = S T E+ µ 1 − µ problem. Finally, in Sec. 7 we discuss the relation between T N. Its first variations satisfy δG = Eδ( T )+ Nδ( T ). dynamical and thermodynamical stability of stellar sys- At statistical equilibrium, the system is− in the state that tems and gaseous spheres. maximizes G at fixed T and µ/T . P.H. Chavanis: Gravitational instability of isothermal and polytropic spheres 3

The microcanonical, canonical and grand canonical The density field associated with the distribution func- ensembles are the most popular. However, we are free tion (6) is given by to define other thermodynamical ensembles. For example, Lecar & Katz (1981) have introduced a grand mi- 2π 3/2 ρ = ze−βΦ, (7) crocanonical ensemble (GMCE) in which the fugacity and β the energy are fixed. This corresponds to a thermodynam- µ where we have defined the fugacity z and the inverse tem- ical potential = S + T N whose first variations satisfy δ = 1 δE+NδK( µ ). At statistical equilibrium, the system perature β by the relations K T T is in the state that maximizes at fixed E and µ/T . m K z = meµ/kT , β = . (8) kT 2.2. The isothermal distribution The distribution function (6) can then be written in terms of ρ as For extensive systems, it is well-known that the statisti- 3/2 cal ensembles described previously are all equivalent at 2 m − mv the thermodynamic limit. Therefore, the choice of a par- f = ρ(r)e 2kT . (9) 2πkT ticular ensemble (in general the grand canonical one) is only dictated by reasons of convenience. This is not the This distribution function is a global maximum of the ther- case for systems with long-range interactions such as grav- modynamical potential under the usual constraints and itational systems. Indeed, the statistical ensembles are not for a given density field ρ(r) (see Padmanabhan 1989, interchangeable and the stability limits differ from one en- Chavanis 2002a). Substituting this optimal distribution semble to the other. Therefore, the choice of the ensemble in Eqs. (2) and (4), we get is imposed by the physical properties of the system under consideration. If we want to describe in terms of statis- 3 1 E = NkT + ρΦd3r, (10) tical mechanics globular clusters (Lynden-Bell & Wood 2 2 Z 1968) and elliptical galaxies (Lynden-Bell 1967, Hjorth & Madsen 1993, Chavanis & Sommeria 1998), which can S 5N 3N 2πkT ρ ρ = + ln ln d3r. (11) be treated as isolated systems in a first approximation, k 2 2 m − m m the good choice is the microcanonical ensemble. If now Z the system is in contact with a radiation background It will be more convenient in the sequel to study the sta- imposing its temperature, the relevant ensemble is the bility problem directly from these equations, without loss canonical one. This ensemble may be appropriate to star of generality. or galaxy formation (Penston 1969, Chavanis 2002a,b), white dwarf and neutron stars (Hertel & Thirring 1971, 2.3. The Emden equation and the Milne variables Chavanis 2002c,d) and dark matter made of massive neu- trinos (Bilic & Viollier 1997, Chavanis 2002g). This canon- Inserting the relation (7) in Eq. (3), we find that the gravi- ical description is also exact in a model of self-gravitating tational potential Φ is a solution of the Boltzmann-Poisson Brownian particles (Chavanis et al. 2002, Sire & Chavanis equation 2002). Finally, the grand canonical ensemble may be ap- −βΦ propriate to the interstellar medium (or possibly the large- ∆Φ=4πGAe . (12) scale structures of the universe), assuming that a statisti- If we introduce the function ψ = β(Φ Φ ) where Φ is cal description is relevant (de Vega et al. 1998). Possible − 0 0 applications of the grand microcanonical ensemble are dis- the gravitational potential at r = 0, the density field can cussed by Lecar & Katz (1981). be written If we just cancel the first order variations of the ther- −ψ ρ = ρ0e , (13) modynamical potential (under constraints appropriate to the ensemble considered), it is straightforward to check where ρ0 is the central density. Introducing the normal- 1/2 that each ensemble yields the Maxwell-Boltzmann distri- ized distance ξ = (4πGβρ0) r, the Boltzmann-Poisson bution equation can be written in the standard Emden form (Chandrasekhar 1942) 2 µ/kT − m ( v +Φ) f = m e e kT 2 . (6) 1 d dψ ξ2 = e−ψ, (14) ξ2 dξ dξ Therefore, the statistical equilibrium state of a self- gravitating system is isothermal whatever the ensemble with ψ = ψ′ =0 at ξ = 0. This equation has to be solved considered. The differences will come from the second between ξ = 0 and ξ = α corresponding to the normalized order variations of the thermodynamical potential. The box radius choice of the statistical ensemble will affect the stability of 1/2 the system. α = (4πGβρ0) R. (15) 4 P.H. Chavanis: Gravitational instability of isothermal and polytropic spheres

We also recall the deﬁnition of the Milne variables that 3 will be used in the sequel (CE) ξe−ψ u = , v = ξψ′. (16) ψ′ These variables satisfy the identities 2 (GCE) 1 du 1

= (3 v u), (17) v u dξ ξ − − 1 dv 1 = (u 1), (18) 1 (MCE) v dξ ξ − which can be directly derived from Eq. (14). The phase portrait of classical isothermal spheres is represented in Fig. 1 and forms a spiral. The spiral starts 0 at (u, v) = (3, 0) for ξ = 0 and tends to the limit point 0 1 2 3 (u , v )=(1, 2) as ξ + . This limit point corresponds s s → ∞ u to the singular solution e−ψs = 2/ξ2. For finite isothermal spheres, we must consider only the part of the curve Fig. 1. This graphical construction, first introduced by between ξ = 0 and ξ = α. Padmanabhan (1989) in MCE, shows very simply the existence of an upper bound for Λ, η and χ above which there 2.4. The equilibrium phase diagram is no hydrostatic equilibrium for an isothermal gas (dotted line: Λ > Λc; solid line: Λ = Λc; dashed line: Λ < Λc with The thermodynamical parameters can be expressed in similar convention for η and χ). terms of the values of the Milne variables at the normalized box radius α. Writing u0 = u(α) and v0 = v(α), the energy and the temperature can be written as (which corresponds to sufficiently negative energies), we (Padmanabhan 1989, Chavanis 2002a): have the well-know gravothermal catastrophe (Antonov ER 1 3 1962, Lynden-Bell & Wood 1968). Similarly, for η ηc = Λ 2 = u0 , (19) ≥ ≡−GM v0 2 − 2.52 (which corresponds to small temperatures or large masses), we have an isothermal collapse (Chavanis 2002a, βGM Chavanis et al. 2002). We can show the existence of such η = v0. (20) ≡ R bounds by a simple graphical construction, without nu- We need also to express α in terms of the fugacity z. merical work. Indeed, the equilibrium structure of the system (characterized by its degree of central concentration Using the expression of the gravitational potential at the box radius Φ(R)= GM/R and comparing Eqs. (7) and α) is determined by the intersection between the spiral in (13), we get − the (u, v) plane and the curves defined by Eqs. (19), (20) and (22). If there is no intersection, the system cannot be 2π 3/2 in statistical equilibrium. This graphical construction is zeη = ρ e−ψ(α), (21) β 0 made explicitly in Fig. 1 in MCE, CE and GCE. In the microcanonical ensemble, the control parame- Expressing ρ0 in terms of α by Eq. (15) and introducing ters are the energy E and the particle number N. The the Milne variables (16), we obtain parameter conjugate to the energy (with respect to the 4(2π)5G2R4z2 entropy S) is the inverse temperature β. The curve giving χ = u2v2e−2v0 . (22) the inverse temperature as a function of energy for a fixed ≡ β 0 0 particle number N is drawn in Fig. 2. The parameter con- We can also normalize the fugacity by the energy instead jugate to the particle number N is the ratio µ/T , related of the temperature. Combining Eqs. (22), (20) and (19), to the fugacity z. In order to represent the− fugacity as a we get function of N, for a fixed energy, we use Eqs. (19) and (23). The µ/T vs N curve can then be deduced from Figs. 3 3 − ν 16(2π)10G5R7z4E = u4v5e−4v0 u . (23) and 4. According to standard turning point arguments ≡− 0 0 2 − 0 (Katz 1978), the series of equilibria becomes unstable at It is easy to show that statistical equilibrium states the point of minimum energy (for a given mass) or at the only exist for sufficiently small values of the control pa- point of minimum mass (for a given negative energy). The rameters Λ, η, χ and ν defined previously. Above a critical condition dΛ/dα = 0 implies (Padmanabhan 1989) value, there is no possible equilibrium state and the system is expected to collapse. For example, for Λ Λ = 0.335 4u2 +2u v 11u +3=0. (24) ≥ c 0 0 0 − 0 P.H. Chavanis: Gravitational instability of isothermal and polytropic spheres 5

0.008 2.6 CE α=8.99 R=32.1 η=2.52 Fixed E Fixed M 0.006 2.4 E 4 z

7 0.004 2.2 GMCE2 R α

5 =34.4 GM/R G R=709 MCE 10 Λ

η=β =0.335 MCE π) α =5.95 2 0.002 2

16(2 R2=11.6 α=34.4 ν 2=0.0032 R=709 ν=− GMCE3 Λ=0.335 1.8 α 0 3=13.8 R3=92.7 ν 3=0.0017 1.6 −0.4 −0.2 0 0.2 0.4 −0.002 Λ=−ER/GM2 −0.05 0.05 0.15 0.25 0.35 Λ=−ER/GM2 Fig. 2. Temperature vs energy plot at ﬁxed particle number. The series of equilibria becomes unstable at MCE in Fig. 4. Enlargement of Fig. 3 near the spiral. In the grand the microcanonical ensemble and at CE in the canonical microcanonical ensemble, a change of stability occurs at ensemble. GMCE2 and GMCE3. Between these points the series of equilibria is stable again. After GMCE3, the series becomes and remains unstable. −0.0336 Fixed E 3

(GCE) α=8.99 E 4 z

7 −0.2336 R

5 α=34.4 G

10 2 π) 16(2 v

ν=− −0.4336

α 1=1.89 1 R1=1.66 α=1.78 ν (CE) GMCE1 1=−0.542 −0.6336 −3.775 −2.775 −1.775 −0.775 0.225 Λ=−ER/GM2 (MCE)

Fig. 3. Fugacity vs particle number at fixed energy. The 0 series of equilibria becomes unstable at GMCE1 in the 0 1 2 3 u grand microcanonical ensemble. Fig. 5. Graphical construction determining the turning points of the control parameter at which a change of sta- The values of α at which the parameter Λ is extremum are bility occurs in MCE, CE and GCE. determined by the intersections between the spiral in the (u, v) plane and the parabole defined by Eq. (24), see Fig. 5. The change of stability occurs for the smallest value can represent the fugacity as a function of mass, for a of α. Its numerical value is α = 34.4 corresponding to a given temperature, by using the relations (22) and (20). density contrast of 709. Gravitational instability occurs in The corresponding curve can be deduced from the η vs χ plot represented in Fig. 6. In the canonical ensemble, MCE when the specific heats CV = (∂E/∂T )N,V becomes zero, passing from negative to positive values. the series of equilibria becomes unstable at the point of In the canonical ensemble, the control parameters are minimum temperature (for a given mass) or at the point of the inverse temperature β and the particle number N. maximum mass (for a given temperature). The condition The parameter conjugate to the inverse temperature (with dη/dα = 0 is equivalent to (Chavanis 2002a) respect to the free energy J) is E. The curve giving u0 =1. (25) E as a function of β for a fixed− particle number can be− deduced from Fig. 2. The parameter conjugate to the The values of α at which the parameter η is extremum are particle number is µ/T , related to the fugacity z. We determined by the intersections between the spiral in the − 6 P.H. Chavanis: Gravitational instability of isothermal and polytropic spheres

1 0.008

Fixed β Fixed z GCE α=1.78 0.8 R=1.58 χ=0.812 0.006 E 4 β z / 7 2 0.6 z R 4 5 α 2=5.95 R G 2

10 0.004 R2=11.6 G 5

π) GMCE2 ν 2=0.0032 π)

0.4 16(2 4(2 χ= ν=− 0.002 α =8.99 α =13.8 0.2 GMCE3 3 R=32.1 R =92.7 η=2.52 ν 3 3=0.0017 CE 0 0 0.02 0.07 0.12 0 1 2 3 χ=4(2π)5G2R4z2/β η=βGM/R Fig. 8. Enlargement of Fig. 7 near the spiral. Fig. 6. Fugacity vs mass at ﬁxed temperature. The series of equilibria becomes unstable at GCE in the grand 1 canonical ensemble.

0.0115 0.8 Fixed z

0.6 E

4 −0.1885 z χ 7 R 5 G

10 0.4 π) 16(2 −0.3885 ν=− α=1.78 0.2 α =1.89 R=1.58 1 χ R =1.66 =0.812 ν 1 1=−0.542 GCE GMCE1 0 −0.5885 −5 0 5 10 0 0.2 0.4 0.6 0.8 1 ln(α) χ=4(2π)5G2R4z2/β Fig. 9. Evolution of χ as a function of the central concen- Fig. 7. Energy vs temperature at fixed fugacity. tration α. In dashed line we have represented the asymp- −4 totic value χs =4e corresponding to the singular solu- (u, v) plane and the straight line defined by Eq. (25), see tion. The Λ(α) and η(α) curves have a similar behaviour. Fig. 5. The change of stability occurs for the smallest value of α. Its numerical value is α = 8.99 corresponding to a of maximum temperature (for a given fugacity) or at the density contrast of 32.1. Gravitational instability in CE point of maximum fugacity (for a given temperature). The occurs when the specific heats becomes negative passing condition dχ/dα = 0 implies by C = . V ∞ In the grand canonical ensemble, the control param- u0v0 =2. (26) eters are the inverse temperature β and the ratio µ/T related to the fugacity z. The parameter conjugate to the The values of α at which the parameter χ is extremum inverse temperature (with respect to the grand potential (see Fig. 9) are determined by the intersections between G) is E. The curve giving E as a function of β for a the spiral in the (u, v) plane and the hyperbole defined fixed fugacity− z is determined− by Eqs. (22) and (23) and by Eq. (26), see Fig. 5. The change of stability occurs for can be deduced from Figs. 7 and 8. The parameter conju- the smallest value of α. Its numerical value is α = 1.78 gate to the ratio µ/T is the particle number N. The curve corresponding to a density contrast of 1.58. giving N as a function of z for a given temperature can In the grand microcanonical ensemble, the control pa- be deduced from Fig. 6. In the grand canonical ensem- rameters are the energy E and the ratio µ/T related to the ble, the series of equilibria becomes unstable at the point fugacity z. The parameter conjugate to the energy (with P.H. Chavanis: Gravitational instability of isothermal and polytropic spheres 7

0.01 3

α Towards maximum at 1 α α =5.95 3=13.8 2 0.005

ν 0 −

α v 3

α 2 1 (GMCE) α =1.89 −0.005 1

α 4 −0.01 −5 0 5 10 ln(α) 0 0 1 2 3 u Fig. 10. Evolution of ν as a function of the central concentration α. In dashed line we have represented the asymp- 4 −8 Fig. 11. Graphical construction determining the turning totic value νs =2 e corresponding to the singular solu- points of the control parameter ν in GMCE. The series of tion. Contrary to Fig. 9, the second extremum α2 is above equilibria is stable for α α1, unstable for α1 α α2, the asymptote ν = νs. This appears to be the reason for ≤ ≤ ≤ stable again for α2 α α3 and unstable for α α3. the regain of stability between α2 and α3. ≤ ≤ ≥ 3. Thermodynamical stability of self-gravitating systems We now determine the condition of thermodynamical sta- respect to the potential ) is β. The curve giving β as a bility by computing the second order variations of the function of E for a fixedK fugacity z can be deduced from thermodynamical potential explicitly. This study has al- Figs. 7 and 8. The parameter conjugate to the ratio µ/T ready been done in MCE (Padmanabhan 1989) and CE is the particle number N. The N vs µ/T plot for a given (Chavanis 2002a). In the following, we perform the anal- energy can be deduced from Figs. 3 and 4. In the grand ysis in GCE and GMCE. microcanonical ensemble, the series of equilibria becomes unstable at the point of maximum energy (for a given fugacity) or at the point of maximum fugacity (for a given 3.1. The grand canonical ensemble positive energy). The condition dν/dα = 0 implies The second order variations of the grand potential G are k (δρ)2 1 2 δ2G = d3r δρδΦd3r. (28) 8u0v0 10u0v0 17u0 +21=0. (27) −m 2ρ − 2T − − Z Z An isothermal sphere is stable in the grand canonical en- The values of α at which the parameter ν is extremum semble if δ2G 0 for all variations δρ of the density pro- (see Fig. 10) are determined by the intersections between file. We shall≤ restrict ourselves to spherically symmetric the spiral in the (u, v) plane and the curve defined by perturbations since only such perturbations can trigger Eq. (27), see Fig. 11. The series of equilibria becomes un- gravitational instability for non rotating systems (Horwitz stable for α1 = 1.89 (density contrast 1.66) which corre- & Katz 1978). In the grand canonical ensemble, the mass sponds to the first intersection. At that point, the ν vs of the system is not conserved. Therefore, the potential at χ curve rotates clockwise (see Fig. 7). However, at the the wall changes according to second turning point α = 5.95 (density contrast 11.6) 2 GδM the ν vs χ curve rotates anticlockwise (see Fig. 8) so δΦ(R)= . (29) that stability is regained (this is proved in Katz 1978). − R Stability is lost again at α3 = 13.8 (density contrast 92.7) On the other hand, according to the Gauss theorem, we and never recovered afterwards. This curious behaviour in have GMCE was first noted by Lecar & Katz (1981). A sim- dδΦ GδM (R)= . (30) ilar lost-gain-lost stability behaviour also occurs for self- dr R2 gravitating fermions and hard sphere models in MCE and These boundary conditions suggest the introduction of the CE (Chavanis & Sommeria 1998, Chavanis 2002c). In that new function case, it is related to the occurence of phase transitions between “gaseous” and “condensed” states. X(r)= rδΦ(r), (31) 8 P.H. Chavanis: Gravitational instability of isothermal and polytropic spheres which satisfies the differential operator which occurs in Eq. (40). Using the Emden equation (14), we readily check that X(0) = 0, X′′(0) = 0, X′(R)=0, (32) (ξ)= ξ, (ξ2ψ′)=2ξ, (ξψ)= ξ + ξψ. (43) where we have used Eqs. (29) and (30) to obtain the third L L L equality (the second equality is obvious for a spherically Therefore, the general solution of Eq. (40) which satisfies symmetric system). the condition X = 0 at the origin is The variations δΦ and δρ are related to each other by X(ξ)= c ξ(ξψ′ 2), (44) the Poisson equation 1 − 1 d dδΦ where c1 is an arbitrary constant. The boundary condition r2 =4πGδρ. (33) X′(α) = 0 implies r2 dr dr α2e−ψ(α) 2=0. (45) In terms of the new variable (31), Eq. (33) becomes − X′′ This relation determines the value of α at which the se- =4πGδρ. (34) ries of equilibria becomes unstable in the grand canoni- r cal ensemble. In terms of the Milne variables, Eq. (45) is Expressing the second order variations of G in terms of equivalent to X, we get u0v0 =2, (46) k 1 R (X′′)2 1 R δ2G = dr X′′Xdr. (35) −m 8πG2 ρ − 2T G which is precisely the condition (26) obtained with the Z0 Z0 turning point criterion. We are now in a position to deter- Integrating by parts, we can put Eq. (35) in the form mine the structure of the perturbation profile that triggers R the instability in GCE. Using Eq. (34), we have 2 1 ′ G k d 1 d ′ δ G = drX + X . (36) ′′ 2G2 T m dr 4πρ dr δρ X 0 = , (47) Z ρ 4πξ The boundary terms arising from the integration by parts 0 are seen to vanish by virtue of Eq. (32). We can therefore with Eq. (44) for X. This yields reduce the stability analysis of isothermal spheres in the δρ c1 grand canonical ensemble to the eigenvalue problem = e−ψ(2 ξψ′), (48) ρ0 4π − k d 1 d G + X′ (r)= λX′ (r). (37) or, introducing the Milne variables, m dr 4πρ dr T λ λ δρ c = 1 (2 v). (49) If all the eigenvalues λ are negative, then the isothermal ρ 4π − sphere is a maximum of the thermodynamical potential G (and is therefore stable in GCE). If at least one eigenvalue We note that Eq. (49) coincides with the expression found is positive, it is an unstable saddle point. The condition in the canonical ensemble (Chavanis 2002a). The number of marginal stability λ = 0 determines the transition be- of nodes in the profile δρ is determined by the number tween stability and instability. We thus have to solve the of intersections between the spiral in the (u, v) plane and equation the line v = 2. We see in Fig. 5 that there is no intersection for ξ α = 1.78, where 1.78 is the stability limit in k d 1 dX′ GX′ ≤ + =0. (38) GCE. Therefore, collecting the results obtained by differ- m dr 4πρ dr T ent authors, the perturbation that triggers gravitational Integrating once and using the boundary conditions (32) instability has two nodes in MCE (“core-halo” structure), at r = 0, the constant of integration is seen to vanish and one node in CE and no node in GCE (see Fig. 12). we get k 1 GX 3.2. The grand microcanonical ensemble X′′ + =0. (39) m 4πρ T The grand microcanonical ensemble is interesting in its Introducing the dimensionless variables defined in Sec. 2.3, own right but also because it exhibits a lost-gain-lost sta- we are led to solve the problem bility behaviour similar to that found for self-gravitating fermions and hard sphere models (Chavanis 2002c). Now, eψX′′ + X =0, (40) the stability analysis can be conducted analytically in GMCE for classical isothermal spheres while only numer- ′ X(0) = X (α)=0. (41) ical results are available for fermions and hard sphere systems. On a technical point of view, GMCE is the most Let us denote by complicated and richest of all thermodynamical ensem- d2 bles. We can argue, however, that GMCE may not be re- eψ +1, (42) L≡ dξ2 alized in practice. It is not clear, indeed, what mechanism P.H. Chavanis: Gravitational instability of isothermal and polytropic spheres 9

1 The stability analysis reduces therefore to the study of the eigenvalue equation R ′ ′ ′ ′ ′ (GCE) dr K(r, r )Xλ(r )= λXλ(r). (55) 0 0.5 Z The condition of marginal stability (λ = 0) reads GX′ k d 1 dX′ 2 3kT ′ δρ/ρ + = Φ+ r T m dr 4πρ dr 3NkT 2 2m (MCE) R ′ 0 3kT Φ+ r X′dr. (56) × 2m Z0 Integrating once, we obtain (CE) GX k X′′ 2 3kT + = Φ+ r −0.5 T m 4πρ 3NkT 2 2m 0 0.2 0.4 0.6 0.8 1 r/R R 3kT ′ Φ+ r X′dr. (57) × 2m Fig. 12. First mode of instability in MCE, CE and GCE. Z0 The gravitational potential Φ is related to the Emden function ψ(ξ) by the relation ψ = β(Φ Φ0). Using can be devised so that two systems can share particles Φ(R)= GM/R and Eq. (20), we get − without also sharing energy. One example might be the − βΦ= ψ ψ(α) v . (58) case of relativistic particles created and destroyed at equi- − − 0 librium in a cosmological settling. Inserting this relation in Eq. (57) and introducing the di- In GMCE, we have to maximize the thermodynamical mensionless variables defined in Sec. 2.3, we find that potential ψ ′′ 1 3 X + e X = ψ ψ(α) v0 + ξ 5 3 2πkT µ ρ αv0 − − 2 K = + ln + d3r k 2 2 m kT m 2 α Z Xξe−ψdξ + X(α) , (59) ρ ρ × −3 ln d3r, (50) Z0 − m m Z where we have integrated by parts and used the identity ′′ −ψ at fixed energy (ξψ) = ξe equivalent to the Emden equation (14). Writing

3 ρ 3 1 3 α E = kT d r + ρΦd r. (51) 2 3 −ψ 2 m 2 V = X(α) Xξe dξ , (60) 3v α 2 − Z Z 0 Z0 We vary the density profile and use Eq. (51) to determine 3 the corresponding variation of temperature. After some W = ψ(α)+ v0 V, (61) − − 2 algebra, we find that the second order variations of the thermodynamical potential can be written we are led to solve the problem ψ ′′ 1 k (δρ)2 e X + X = V ψξ + W ξ, (62) δ2 = δρδΦd3r d3r K −2T − m 2ρ X(0) = X′(α)=0. (63) Z Z 2 1 3kT 3 Φ+ δρ d r . (52) Seeking a solution of Eq. (62) in the form −3NkT 2 2m Z X = a ξ + a ξ2ψ′ + a ξψ, (64) Introducing the variable X defined previously and inte- 1 2 3 grating by parts, we can put the problem in the form and using the identities (43), we find that 2 ′ 1 R R X = (W V )ξ + b(ξ ψ 2ξ)+ V ξψ. (65) δ2 = dr dr′X′(r)K(r, r′)X′(r′), (53) − − K 2G2 The constant b is determined by the boundary condition Z0 Z0 X′(α) = 0. We thus find that the general solution of Eqs. with (62) and (63) is G k d 1 d K(r, r′)= + δ(r r′) 1 T m dr 4πρ dr − X(ξ)= ψ(α)+ v0 V ξ + V ξψ − − 2 ′ ′ 2 3kT 3kT V 1 Φ+ r (r) Φ+ r (r′). (54) (ξ2ψ′ 2ξ), (66) −3NkT 2 2m 2m − 2 u v 2 − 0 0 − 10 P.H. Chavanis: Gravitational instability of isothermal and polytropic spheres where we have used Eq. (61). The values of α at which a change of stability occurs (λ = 0) are obtained by inserting the solution (66) for X in Eq. (60). We need the identities α α 3 2 α v 2 −ψ 4 ξ e dξ = αv0, (67) 2 0 Z v α 3 4 α 3 ′ −ψ v1 ξ ψ e dξ = αv (3 u ), (68) v 0 − 0 Z0 α 2 −ψ ξ ψe dξ = αv0(ψ(α)+ v0 6+2u0). (69) α − 1 Z0 which are derived in Appendix A. After simplification, we obtain the condition

2 8u v0 10u0v0 17u0 +21=0, (70) 0 0 − − 0 1 2 3 which coincides with the condition (27) deduced from the u turning point criterion. The perturbation profile at the critical points at which λ = 0 is given by Eq. (47) with Fig. 13. Graphical construction determining the number Eq. (66) for X. Using Eqs. (13), (62) and (66), we get of nodes for the different modes of instability in GMCE. The horizontal lines correspond to the condition (72) for δρ c = 1 (2u v 6+ v), (71) the four first critical points. ρ 4π 0 0 − α where c1 is a constant. The nodes of the perturbation pro- 2 file are determined by the condition 1.5 v(ξ )=6 2u v , (72) i − 0 0 with ξi α. The number of nodes can be found by a graphical≤ construction. In Fig. 13, we plot the lines de-

fined by Eq. (72) in the (u, v) plane for the four first crit- δρ/ρ α 1 ical points in the series of equilibria. The corresponding 0.5 perturbation density profiles are plotted in Fig. 14. The α first mode of instability α1 has no node. The profile of the 4 second mode α2 increases (in absolute value) with distance. The absence of node at α2 is a signature of the fact α 3 that stability is regained at that point. The mode α3 at which stability is lost again has one node, the mode α4 −0.5 0 0.5 1 two nodes etc... r/R

4. Connexion with statistical mechanics Fig. 14. Density perturbation profiles for the four first critical points in the series of equilibria in GMCE. We In this section, we briefly discuss the relation between the have fixed the constant c1 such that δρ/ρ =1 at r = 0. thermodynamics and the statistical mechanics (and field theory) of self-gravitating systems. We shall just discuss the physical ideas without entering into technical details. (specified by the position and the velocity of all N parti- In the microcanonical ensemble, the object of funda- cles) with energy between E and E + dE. The entropy is mental interest is the density of states defined by

N S(E) = ln g(E). (75) g(E)= δ(E H) d3r d3v , (73) − i i i=1 Z Y As is customary in statistical mechanics, we decompose where phase space into macrocells and divide these macrocells into a large number of microcells. A macrostate is char- N 2 1 2 Gm acterized by the number of particles in each macrocell H = mvi , (74) 2 − ri rj (irrespective of their precise position and velocity in the i=1 i≤j X X | − | cell), or equivalently by the smooth distribution function is the Hamiltonian of the self-gravitating gas. By defini- f(r, v). If we call Wi the number of microstates corre- tion, g(E)dE is proportional to the number of microstates sponding to the macrostate i, we can rewrite Eq. (73) P.H. Chavanis: Gravitational instability of isothermal and polytropic spheres 11 formally in the form influence of a small-scale cut-off on the form of the equilibrium phase diagram is discussed by Chavanis (2002c) Si Nsi g(E)= Wi = e = e , (76) and Chavanis & Ispolatov (2002). When the small-scale Ei=E Ei=E Ei=E cut-off a 0, which is the limit relevant for globular clus- X X X → where the sum runs over all macrostates with energy E ters since the size of the stars is in general much smaller (and mass M). In addition, Si = ln Wi is the Boltzmann than the interstellar distance, the entropy S[f] possesses entropy (4) of the macrostate i (i.e., S S[f]) and two maxima (for sufficiently high energies). The local en- i ≡ si = Si/N the entropy per particle. The thermodynamic tropy maximum describes a “gaseous” phase insensitive limit for self-gravitating systems corresponds to N + to the small scale cut-off. It corresponds to the isother- in such a way that all the dimensionless control parame-→ ∞ mal spheres studied previously. The global entropy max- ters defined in Sec. 2.4 remain finite. This can be written imum is dominated by the small scale cut-off and corre- for example N, R + with N/R finite (and E/N 1, sponds to a “condensed” phase made of a few stars close β 1). This is a→ very∞ unusual thermodynamic limit∼ due together (or simply a single binary) surrounded by a dif- to∼ the long-range nature of the interactions. Now, at the fuse halo. This “core-halo” structure is the most probable thermodynamic limit, the sum in Eq. (76) is dominated configuration of a stellar system so it is expected to be by the macrostate f∗ which maximizes the Boltzmann en- reached at equilibrium. Since the observed globular clus- tropy (4) at fixed mass and energy: ters, described by the Michie-King model, do not possess this core-halo structure, we conclude that they are not in g(E) eS[f∗], S(E)= S[f ]. (77) ≃ ∗ true equilibrium but that they are slowly evolving in that It has to be noted that the sum in Eq. (76) is sharply direction. However, the formation of binaries can take extremely long times, much larger than the age of the uni- peaked around f∗, which corresponds to what is called a large deviation property in mathematics. Therefore, the verse. Indeed, equilibrium statistical mechanics tells noth- mean-field approximation is exact for self-gravitating sys- ing about the relaxation time. For the timescales contem- tems at the thermodynamic limit. plated in astrophysics, globular clusters can be considered metastable In fact, if we look at the problem more carefully, the as equilibrium states corresponding to local en- argument leading to Eq. (77) is not so obvious because tropy maxima. Indeed, the probability of transition from the entropy S[f] has no global maximum, except if we in- a gaseous state to a condensed state (corresponding to troduce small-scale and large-scale cut-offs. The introduc- the formation of a binary) is extremely small (Katz & tion of these cut-offs is necessary to define correctly the Okamoto 2000, Chavanis & Ispolatov 2002). Therefore, in density of state (73), otherwise the integral would diverge the calculation of the density of states (73), we must dis- (Padmanabhan 1990). The divergence at large scales is as- card by hands the states which correspond to collapsed sociated to the fact that a stellar system has the tendency configurations (binaries) since they cannot be reached in to evaporate under the effect of encounters. In reality, stel- the timescales of interest. Accordingly, Eq. (77) is correct lar systems are not allowed to extend to infinity because with f∗ being the local maximum of S[f]. This is precisely they interact with the surrounding. For example, globular the situation studied in Secs. 2 and 3. In fact, this picture clusters are subject to the tides of a nearby galaxy so that is correct only for globular clusters with high energy E, the Boltzmann distribution has to be truncated at high well above the Antonov limit Ec. Because of evaporation, energies. The Michie-King model, the energy of a globular cluster slowly decreases until it reaches the point at which the system cannot be in equilib-

−βǫ −βǫm A(e e ) ǫ<ǫm, rium anymore (e.g., Katz 1980). At that point, an isother- f = − (78) mal sphere becomes unstable and the system undergoes ( 0 ǫ ǫm, ≥ a gravitational collapse (gravothermal catastrophe). The which is a truncated isothermal, can take into account evolution proceeds self-similarly with the formation of a the evaporation of high energy stars and provides a good high density core with a shrinking radius and a small mass description of about 80% of globular clusters (Binney (Hénon 1961, Cohn 1980, Lynden-Bell & Eggleton 1980). & Tremaine 1987). It can be derived from the Fokker- This core collapse concerns typically 20% of globular clus- Planck equation (taking into account the encounters be- ters. In theory, the central density becomes infinite in a tween stars) by imposing that the distribution function finite time. In practice, the formation of binaries can re- vanishes at the escape energy ǫ = ǫm. In that case, the lease sufficient energy to stop the collapse (Hénon 1961) system is not truly static since it gradually loses stars and even drive a reexpansion of the system (Inagaki & but we can consider that a globular cluster passes by a Lynden-Bell 1983). Then, a series of gravothermal oscilla- succession of quasi-equilibrium configurations. Instead of tions should follow (Bettwieser & Sugimoto 1984). working with truncated models, it is more convenient to In the canonical ensemble, the object of interest is the confine the system within a box of radius R, the box radius partition function playing the role of a tidal radius. The divergence at small scales of the density of states reflects the natural tendency N of a stellar system to form binaries. In this extreme case, −βH 3 3 Z(β)= e d rid vi. (79) the size a of the stars provides a small-scale cut-off. The i=1 Z Y 12 P.H. Chavanis: Gravitational instability of isothermal and polytropic spheres

2 The free energy is defined by Gm 2 2 7/2 Teff =4π , µ = zGm √T. (86) 1 T rπ F (β)= ln Z. (80) −β for the Liouville action 1 1 The partition function is related to the density of states A[φ]= d3r ( φ)2 µ2eφ . (87) −kT {2 ∇ − } by the Laplace transform eff Z +∞ It should be emphasized that φ is a formal field which is −βE Z(β)= dE g(E)e . (81) not the gravitational field. In the mean-field approxima- Z−∞ tion, the path integral is dominated by the contribution Using Eq. (76), it can be rewritten of the field φ0 which maximizes the Liouville action (87). The cancellation of the first order variations gives Z(β)= eSi−βEi = eJi , (82) 2 φ0 i i ∆φ0 = µ e , (88) X X − where J = S βE is the Massieu function and the which is similar to the Boltzmann-Poisson equation (12). i i − i sum runs over all macrostates with mass M. In the ther- This allows one to identify φ0 with the equilibrium gravi- modynamic limit, the partition function is dominated by tational potential Φ (up to a negative proportionality fac- the contribution of the macrostate which maximizes the tor). The second order variations of A at the critical point Massieu function Ji J[f] at fixed mass. In this limit, must be negative for A[φ0] to be a maximum. This yields the mean field approximation≡ is exact and we have the condition

Z(β) eJ[f∗], F = E T S. (83) d3r δφ ∆+ µ2eφ0 δφ 0, (89) ≃ − { } ≤ Z Again, the Massieu function J[f] has no global maximum which is equivalent to the maximization of the grand po- in the absence of cut-off. If we introduce a small scale cut- tential G (see Sec. 3.1). Therefore, the thermodynamical off a and a large scale cut-off R and consider the limit approach is equivalent to the statistical mechanics or field a/R 1, it is found that the Massieu function J[f] has ≪ theory approach. We note, however, that the Liouville ac- two maxima (for sufficiently high temperatures). The local tion A[φ] is not the same functional as the grand potential maximum corresponds to isothermal “gaseous” configura- G[ρ] although they give the same critical points and the tions and the global maximum to a “condensed” structure same conditions of stability. in which all the particles are packed together (Aronson & Hansen 1972, Chavanis 2002c, Chavanis & Ispolatov 2002). As a 0, the density profile of the condensed 5. Generalized thermodynamics and Tsallis state tends to→ a Dirac peak (Kiessling 1989). As in the entropy microcanonical ensemble, the “gaseous” configurations are metastable but they can be long lived and relevant for as- 5.1. General considerations trophysical purposes. This justifies why we only select the It has been recently argued that the classical Boltzmann contribution of the local maximum of J[f] in Eq. (83). entropy may not be relevant for non-extensive systems This treatment is, however, valid only for sufficiently high and that Tsallis entropies, also called q-entropies, should temperatures. Below Tc, a phase transition similar to the be used instead. These entropies can be written gravothermal catastrophe occurs in the canonical ensem- 1 ble and leads to a condensed structure containing almost S = (f q f)d3rd3v, (90) q −q 1 − all the mass. The inequivalence of statistical ensembles re- − Z garding the formation of binaries (in MCE) or Dirac peak where q is a real number. For q 1, this expression re- (in CE) is further discussed in Appendices A and B of Sire turns the classical Boltzmann entropy→ (4) as a particular & Chavanis (2002). case. Finally, the grand canonical partition function is de- The first application of Tsallis generalized thermody- fined by namics in stellar systems was made by Plastino & Plastino +∞ N (1997) who showed that the extremization of Tsallis en- zN Z = d3r d3v e−βHN , (84) tropy leads to stellar polytropes with an index GC N! i i N=0 Z i=1 3 1 X Y n = + . (91) where z is the fugacity. Applying the standard Hubbard- 2 q 1 − Stratanovich transformation, it can be rewritten in For q 1, or n + , we recover isothermal spheres. the form of a path integral (Horwitz & Katz 1978, For q >→ 9/7 (i.e. →n < ∞5), the density drops to zero at a Padmanabhan 1991, de Vega & Sanchez 2002) finite distance so that the mass is finite. Then, Taruya & − 1 d3r{ 1 (∇φ)2−µ2eφ} Sakagami (2002a) considered the stability problem in the Z = φ e kTeff 2 , (85) GC D microcanonical ensemble by extending Padmanabhan’s Z R P.H. Chavanis: Gravitational instability of isothermal and polytropic spheres 13 1 classical analysis of the Antonov instability to the case of p =4√2πA(α Φ)n+1 B(3/2,n 1/2), (95) polytropic distribution functions. A configuration is sta- − n +1 − ble in the sense of generalized thermodynamics if it cor- with B(a,b) being the β function and use has been made responds to a maximum of Tsallis entropy at fixed mass of Eq. (91). Furthermore, in obtaining Eq. (95), we have and energy. They considered polytropic spheres confined used the identity B(m +1,n)= mB(m,n)/(m + n). The within a box and showed that polytropes with index n 5 integrability condition fd3v < requires that n> 1/2. become unstable above a certain density contrast while≥ Eliminating the gravitational potential∞ between the rela- polytropes with n 5 are always stable. On the other tions (94) and (95), weR recover the well-known fact that hand, if we consider≤ the canonical ensemble, we have to stellar polytropes satisfy the equation of state select maxima of Tsallis free energy Jq = Sq βE at fixed mass and β (the generalized inverse temperature).− It is 1 p = Kργ, γ =1+ , (96) found (Taruya & Sakagami 2002b; see also Sec. 5.5) that n polytropes with index n 3 become unstable above a like gaseous polytropes (note, however, that these systems certain density contrast while≥ polytropes with n 3 are ≤ do not have the same phase space distribution function, always stable. In our sense, the main interest of the q- unlike isothermal spheres). In the present context, the entropies (and their attractive nature) is to offer a simple polytropic constant is given by generalization of classical thermodynamics which leads to models that are still analytically tractable since exponen- 1 −1/n K = 4√2πAB(3/2,n 1/2) . (97) tials are replaced by power laws. This approach is interest- (n + 1) − ing to develop, at least formally, as it offers a nice connexion between polytropic and isothermal spheres, which are Using Eqs. (92) and (94), the distribution function can the most popular mathematical models of self-gravitating be written as a function of the density as systems. We shall therefore devote a section to this gen- 1 v2/2 n−3/2 eralization. However, in Sec. 7, we shall argue that Tsallis f = ρ1/n , (98) Z − (n + 1)K entropies are just a particular case of H-functions S[f] whose maximization at fixed mass and energy determines with nonlinearly stable stationary solutions of the Vlasov equation (Tremaine et al 1986). We shall also argue that the Z =4√2πB(3/2,n 1/2)[K(n + 1)]3/2. (99) maximization of the J-function J[f] = S βE at fixed − mass determines nonlinearly stable stationary− solutions Using the foregoing relations, it is possible to express of the Euler-Jeans equations. This discussion should de- the total energy and the entropy in terms of ρ and p ac- mystify the concept of generalized thermodynamics intro- cording to (Taruya & Sakagami 2002b; see also Sec. 7.8) duced by Tsallis (1988). 3 1 E = pd3r + ρΦd3r, (100) 2 2 Z Z 5.2. Stellar polytropes 3 1 S = n pd3r M . (101) The extremization of Tsallis entropy at fixed mass and en- − − 2 T − ergy yields (Plastino & Plastino 1997,Taruya & Sakagami Z 2002a) 5.3. The Lane-Emden equation 1 v2 q−1 The equilibrium structure of a polytrope is determined by f = A α Φ , (92) − − 2 substituting the relation (94) between ρ and the gravitational potential Φ inside the Poisson equation (3). This 1 (q 1)β q−1 1 + (q 1)(µ/T ) is equivalent to using the condition of hydrostatic equilib- A = − , α = − , (93) q (q 1)β rium p = ρ Φ with the equation of state (96). Letting − ∇ − ∇ 4πGρ1−1/n 1/2 where β = 1/T and µ/T (related to α) are Lagrange ρ = ρ θn, ξ = 0 r, (102) 0 K(n + 1) multipliers which can be called inverse temperature and chemical potential in the generalized sense. The distribu- where ρ0 is the central density, we can reduce the condition tion function (92) corresponds to stellar polytropes that of hydrostatic equilibrium to the Lane-Emden equation were first introduced by Plummer (1911). Their derivation (Chandrasekhar 1942) from a variational principle was discussed by Ipser (1974) among others. Note that Eq. (92) is valid for v2 2(α Φ). 1 d dθ ≤ − ξ2 = θn, (103) For v2 2(α Φ), we set f = 0. The spatial density ξ2 dξ dξ − ρ = fd≥3v and− the pressure p = 1 fv2d3v can be ex- 3 ′ pressed as with boundary conditions θ(0) = 1, θ (0) = 0. R R For n < 5, the density vanishes at some radius iden- ρ =4√2πA(α Φ)nB(3/2,n 1/2), (94) tified with the radius of the polytrope (in terms of the − − 14 P.H. Chavanis: Gravitational instability of isothermal and polytropic spheres

1 normalized distance, this corresponds to θ =0 at ξ = ξ1). Such polytropic configurations (with infinite density con- n=10 trasts) will be refered to as complete polytropes. We shall n=2 also consider the case of incomplete polytropes, with arbi- 0.5 trary index, enclosed within a box of radius R. For these configurations, the Lane-Emden equation must be solved between ξ = 0 and ξ = α with n=6

χ 0 4πGρ1−1/n 1/2 n=5 α = 0 R. (104) K(n + 1) We also recall the expression of the Milne variables −0.5 n=4 that will be used in the sequel, ξθn ξθ′ u = , v = . (105) − θ′ − θ −1 0 0.2 0.4 0.6 0.8 1 These variables satisfy the identities η 1 du 1 = (3 nv u), (106) Fig. 15. Fugacity vs mass (for a given temperature) in u dξ ξ − − the framework of generalized thermodynamics for diﬀerent 1 dv 1 polytropic index. For 1 5, χ and≤η have an inﬁnity of turning points. which can be derived from Eq. (103). The phase portrait For n< 5, complete polytropes correspond to the terminal of polytropes in the (u, v) plane and the properties that point of the η χ curve. For n> 5, the curve has a spiral − we shall need in the following are summarized in Chavanis behaviour towards the limit point (ηs,χs) corresponding (2002b). n−3 2 to the singular sphere (us, vs) = ( n−1 , n−1 ).

5.4. Turning point criterion Eliminating A between Eqs. (93) and (97), we find that entropy Sq is the inverse temperature β (microcanoni- 2 cal description) and the parameter conjugate to the in- 4πQn−1 2n−3 β = , (108) verse temperature with respect to Tsallis free energy Kn Jq = Sq βE is E (canonical description). The E T − − − where we have defined (Taruya & Sakagami 2002b) curve has been plotted and discussed in Chavanis (2002b) for different values of the polytropic index. According to n− 3 1 n 1 2 n−1 the turning point criterion, the series of equilibria becomes Q = − 2 . (109) 16√2π2B(3/2,n 1/2)(n + 1)n unstable in the canonical ensemble for dη/dα = 0. We have − shown that this condition is equivalent to In a preceding paper (Chavanis 2002b), we have estab- lished that the mass of the configuration is related to the n 3 central density ρ (through the parameter α) by the rela- u = − = u . (112) 0 0 n 1 s tion − n n−1 M 4πG 1 1 For n 3, the polytropes are always stable since the η(α) n n−1 η n−3 = (u0v0 ) . (110) ≤ ≡ 4π K(1 + n) R n−1 curve is monotonic. The polytropes start to be unstable (for sufficiently high density contrasts) when n> 3. This We had conjectured that η plays the role of a tempera- generalized thermodynamical stability criterion coincides ture in the context of Tsallis generalized thermodynam- with the Jeans dynamical stability criterion based on the ics. This is indeed the case since, using Eq. (108), η can Navier-Stokes equations for a polytropic gas (Chavanis be expressed in terms of β as 2002b). Since K is considered as a constant when we an- 1 n n−1 n−1 alyze the stability of polytropic spheres with respect to 1 G M n− 3 1 η = β 2 . (111) the Navier-Stokes equations, and since K is related to the Q Rn−3 (n + 1)n temperature via Eq. (108), this corresponds to a sort of For n + , this parameter becomes equivalent to the “canonical situation”. This explains why the condition of parameter→ η∞= βGM/R introduced in the study of isother- dynamical stability for gaseous polytropes is equivalent mal spheres (Chavanis 2002a). to the condition of thermodynamical stability for stellar In the framework of generalized thermodynamics, the polytropes in the canonical ensemble. This equivalence is parameter conjugate to the energy with respect to Tsallis further discussed in Sec. 7. P.H. Chavanis: Gravitational instability of isothermal and polytropic spheres 15

We now turn to the grand canonical situation. high density contrasts are unstable in MCE for n> 5, in Combining Eqs. (94), (93) and (109), we can write the CE for n> 3 and in GCE for all n. Complete polytropes density field as are always stable in MCE, stable in CE only for n < 3 and never stable in GCE. Q1−n 3 µ n ρ = n + βΦ . (113) In the following, we analyze the stability of polytropic 4π(n + 1)nβ3/2 − 2 T − spheres by explicitly calculating the second order varia- This expression suggests to define a generalized fugacity tions of the thermodynamical potential. This study has by the relation nz1/n n 3/2+ µ/T so that z is equiv- been done by Taruya & Sakagami (2002a,b) in MCE and ≡ − µ alent to the usual fugacity e T when n + . Taking CE. We shall consider CE in relation with our previous r = R in Eq. (113) and comparing with Eq.→ (102),∞ we get study (Chavanis 2002b), GCE and GMCE. Q1−n βGM n ρ θ(α)n = n z1/n + , (114) 0 4π(n + 1)nβ3/2 R 5.5. Generalized thermodynamical stability in the canonical ensemble where we have used Φ(R)= GM/R. Expressing the central density in terms of α, using− Eqs. (104) and (108), we In the canonical ensemble, the thermodynamical parameter is the free energy (Massieu function) J = S 1 E find − T 1 which is explicitly given by Q (n + 1)nα2n n−1 ρ0 = 3 . (115) 4π n 2n n− 2 n 1 G R β J = pd3r ρΦd3r, (120) −T − 2T From Eqs. (114) and (115), we get Z Z

1 where we have used Eqs. (100) and (101). Using the equa- n 1/2 n−1 1/n (n + 1) β 2 βGM n z = Q α n−1 θ(α) . (116) tion of state (96), the second order variations of free energy GR2 − R can be put in the form Eliminating M thanks to Eq. (111) and introducing the 1 n +1 p Milne variables (105), we finally obtain δ2J = (δρ)2d3r + δρδΦd3r . (121) −2T n ρ2 1 Z Z 2 n−1 1 n 1/n GR − χ z = (u0v0) n 1 (1 v0).(117) As in previous papers, we introduce the function q(r) de- ≡ Q (n + 1)nβ1/2 − fined by For n + , this parameter becomes equivalent to the → ∞ 1 dq one introduced in Sec. 2.4 for isothermal spheres (note δρ = , (122) that for sufficiently small index n, the parameter χ can 4πr2 dr become negative when v > 1; therefore, the definition of 0 and satisfying the perturbed Gauss theorem z given previously is not valid for small n). According to the turning point criterion, the series of equilibria becomes dδΦ Gq unstable in the grand canonical ensemble when dχ/dα = = . (123) dr r2 0. This gives the condition The conservation of mass imposes q(0) = q(R) = 0. After 2(1 v ) (n 1)u v =0. (118) − 0 − − 0 0 an integration by parts, we can rewrite Eq. (121) in the The values of α at which χ is extremum can be determined form by a graphical construction like in Chavanis (2002b). They 1 R d ργ−2 d G correspond to the intersections between the hyperbole de- δ2J = drq Kγ + q. (124) fined by Eq. (118) and the solution curve in the (u, v) 2T dr 4πr2 dr r2 Z0 plane (which depends on the index n). From these graphical constructions, we deduce that the function χ(α) has The point of marginal stability is therefore determined by one maximum for n 5. For n = 5, the Lane-Emden the condition equation can be solved≤ analytically and we get d ργ−2 dq Gq Kγ + =0. (125) α4 dr 4πr2 dr r2 χ = , (119) (1 + 1 α2)12 3 This is the same equation as that determining the dy- which is maximum for α = (3/5)1/2. For n> 5, the func- namical stability of gaseous polytropes (Chavanis 2002b). tion χ(α) has an infinite number of extrema. The onset We have solved this equation and showed that the bound- of gravitational instability corresponds to the first max- ary condition q(R) = 0 returns the condition of Eq. (112) imum. The fugacity vs mass plot (for a given tempera- brought by the turning point criterion. The structure of ture) can be deduced from Fig. 15. Collecting the results the density perturbation profile that triggers instability is obtained by different authors, incomplete polytropes with also discussed in our previous paper. 16 P.H. Chavanis: Gravitational instability of isothermal and polytropic spheres

5.6. Generalized thermodynamical stability in the with vs = 2/(n 1). Equation (135) coincides with the grand canonical ensemble expression found− in CE. The number of nodes of the perturbation profile can be determined by a graphical con- The thermodynamical potential in the grand canonical en- struction like in Chavanis (2002b). First, consider the case semble is G = S 1 E + µ M, i.e. − T T n = 5 which can be solved analytically. The solution of the 1/2 n 1 equation v(ξ)= vs is ξ(1) = √3. Since ξ(1) > α = (3/5) , G = nz1/n ρd3r pd3r ρΦd3r. (126) − T − 2T we conclude that the perturbation profile in GCE has no Z Z Z node contrary to the equivalent situation in CE. This con- Its second order variations are given by Eq. (121) like for clusion remains true for all index n 5. Indeed, the crit- ≤ the free energy. Only the boundary conditions are differ- ical case ξ(1) = α would occur for an index n such that ent since the mass is not conserved anymore. Introducing (us, vs) belongs to the solution curve (substitute v0 = vs the function X(r) defined by Eq. (31), we obtain after an in Eq. (118)). This is never the case for n 5 (see integration by parts Chandrasekhar 1942). Therefore, the structure of≤ the density profile for n< 5 is the same as for n =5 so it has no 1 R d ργ−2 d δ2G = drX′ G + Kγ X′. (127) node. For n> 5, the first mode of instability has no node, 2T G2 dr 4π dr Z0 the second mode of instability one node etc... The point of marginal stability is determined by the equation 5.7. Generalized thermodynamical stability in the d ργ−2 grand microcanonical ensemble Kγ X′′ + GX′ =0, (128) dr 4π In the grand microcanonical ensemble, the fixed parame- with the boundary conditions (32). Integrating once, we ters are the fugacity and the energy. The normalized en- get ergy is given by (Taruya & Sakagami 2002a) ργ−2 ER 1 3 1 n 2 u Kγ X′′ + GX =0. (129) Λ = 1 + − 0 . (136) 4π ≡−GM 2 −n 5 2 − v n +1 v − 0 0 In terms of the dimensionless variables introduced in Sec. Eliminating β in Eq. (117) in profit of E, using Eqs. (111) 5.3, we have to solve the problem and (136), we find that the relevant control parameter 1 θ1−nX′′ + X =0, (130) (n z1/n)2(2n−3) 1 n ν G5R7E, (137) ≡− (n + 1)4n Q4(n−1) ′ X(0) = X (α)=0. (131) is related to α by

Let us introduce the differential operator ν = u4v6(1 v )2(2n−3) − 0 0 − 0 1 d2 1 3 1 n 2 u0 = θ1−n +1. (132) 1 + − . (138) L n dξ2 ×n 5 2 − v0 n +1 v0 − Using the Lane-Emden equation (103), it is easy to check In the grand microcanonical ensemble, we have to max- that imize the thermodynamical potential = S+ µ M at fixed K T 2 n 1 E and µ/T . The entropy and the energy are expressed in (ξ)= ξ, (ξ2θ′)= ξθ, (ξθ)= − ξθ. (133) terms of ρ and K by Eqs. (100), (101) and (96). We vary L L −n L n the density and use the energy constraint to determine Therefore, the general solution of Eq. (130) satisfying the corresponding variation of K (which is related to the X(0) = 0 is inverse temperature β). After some algebra, we find that 2 the second order variations of are given by X = ξ ξθ′ + θ . (134) K n 1 1 1 p 2 3r 2 3r − δ = δρδΦd γ 2 (δρ) d The critical value of α at which the series of equilibria K −2 − 2 ρ Z Z 2 becomes unstable in GCE is determined by the boundary 2n 1 3γp Φ+ δρ d3r . (139) condition X′(α) = 0. Using Eq. (134) and introducing the −3(2n 3) p d3r 2ρ Milne variables (105), we find that this condition is equiv- − Z alent to Eq. (118) in agreement with the turning point (To simplifyR the formulae, we have not written the positive proportionality factor in front of δ2 ). Introducing the criterion. The density perturbation profile triggering in- K stability can be deduced from Eqs. (47) and (134). We variable X defined previously and integrating by parts, get we can put the problem in the form δρ 3 R R = c (v v), (135) δ2 = dr dr′X′(r)K(r, r′)X′(r′), (140) ρ 4π 1 s − K Z0 Z0 P.H. Chavanis: Gravitational instability of isothermal and polytropic spheres 17 with 6. The isobaric ensemble d p d 4n K(r, r′)= G + γ δ(r r′) We now consider the stability of isothermal spheres in dr 4πρ2 dr − − 3(2n 3) contact with a medium exerting a constant pressure P − ′ ′ on their boundary. In this isobaric ensemble, the volume 1 3γp 3γp ′ Φ+ r (r) Φ+ r (r ). (141) of the system can change unlike in the Antonov prob- × p d3r 2ρ 2ρ lem. This situation was first considered by Ebert (1955), TheR condition of marginal stability reads Bonnor (1956) and McCrea (1957) who found that an isothermal gas becomes gravitationally unstable above a γp ′′ 4n 1 3γp GX + X = Φ+ r critical pressure Pmax or above a maximum compression. 4πρ2 3(2n 3) p d3r 2ρ We shall complete these results by supplying an entirely − ′ R 3γp analytical derivation of the stability criterion and by show- Φ+R r X′dr. (142) × 0 2ρ ing the equivalence between thermodynamical and dy- Z namical stability (Jeans problem). We shall also consider We now need to relate the gravitational potential Φ to the case of polytropic gas spheres under external pressure. the Lane-Emden function θ. Integrating the equation of hydrostatic equilibrium for a polytropic equation of state, we readily find that 6.1. Isothermal gas spheres under external pressure p (n + 1) = Φ + Const. (143) Let us consider an isothermal gas sphere with mass M at ρ − temperature T and under pressure P . The first principle of thermodynamics can be written δS 1 δE p δV = 0. The constant can be determined from the boundary con- − T − T dition Φ(R)= GM/R. Using the relation In the isothermal-isobaric ensemble (TBE), the relevant − thermodynamical potential is the Gibbs energy = S 1 P V 1 G P− GM 1/n ′ E . Its first variations satisfy δ = Eδ( ) Vδ( ) = (n + 1)Kρ αθ (α), (144) T T T T R − 0 so that,− at statistical equilibrium, theG system− is in the− state which results from Eqs. (110) and (104), we find that Eq. that maximizes at fixed temperature T and pressure P . G k (143) can be written For an isothermal gas, p = m ρT . Using Eq. (13), the pressure on the boundary of the sphere is 1/n ′ Φ = (n + 1)Kρ0 (θ(α) θ + αθ (α)). (145) kT ρ − P = ρ(R)= 0 e−ψ(α). (150) Inserting this relation in Eq. (142) and introducing the m β dimensionless variables defined in Sec. 5.3, we find that The central density ρ0 can be eliminated via the relation 1 (15). Eliminating the radius R via the relation (20) and X + θ1−nX′′ = n introducing the Milne variables (16), we obtain after simplification 2(n + 1)ξ 3 2n ′ α θ(α)+ − θ + αθ (α) θ1+nξ2dξ 2n 3 4 2 3 0 4πG β M P = u0v0 . (151) 1 α αθ′(α)+ θ(α) P ≡ R ξθnXdξ + X(α) , (146) Similarly, the normalized volume of the sphere is given by × −3 2n 3 Z0 − 3V 1 where we have integrated by parts and used the iden- 3 3 3 = 3 . (152) tity (ξθ)′′ = ξθn equivalent to the Lane-Emden equation V ≡ 4πβ G M v0 − (103). We are led therefore to solve a problem of the form Using the expansion of the Milne variables for α 0, → 1 1 1 θ1−nX′′ + X = V θξ + W ξ, (147) u =3 α2 + ..., v = α2 ..., (153) n 0 − 5 0 3 − X(0) = X′(α)=0. (148) we find that the first order correction to Boyle’s law due to self-gravity is Using the identities (133), we find that the general solution GM 2 of Eq. (147) satisfying X(0) = 0 is P V = NkT λ , (154) − V 1/3 1 2 ′ n 2 ′ X = W ξ + b ξθ + (n 1)ξ θ V ξ θ . (149) with λ = 1 ( 4 π)1/3. This formula can also be derived from 2 − − 2 5 3 the Virial theorem The constant b is determined by the boundary condition 1 X′(α) = 0. Substituting the resulting expression for X(ξ) P V = p d3r + W, (155) 3 in Eq. (146), and using the identities of Appendix B, we Z obtain after tedious algebra a condition for α which is by using the isothermal equation of state and by assuming equivalent to the one obtained from the turning point ar- that the density ρ is uniform (in which case the potential gument, i.e. by setting dν/dα = 0. energy W = 3GM 2/5R). As reported by Bonnor (1956), − 18 P.H. Chavanis: Gravitational instability of isothermal and polytropic spheres

20 and TBE β d ln 1 α=6.45 Fixed M, P = (2u0 v0). (158) CE R=14.0 dα α − P=17.6 15 Therefore, the “compressibility” κ = (∂ ln P/∂ ln V )N,T can be expressed as − P 2 1 2u0 v0 M 3 10 κ = − . (159) G 4 3 u0 1

πβ − 4 1/3 1/3 Using u0 =3PV/NkT and v0 = (4π/3) GMm/kTV according to Eqs. (151) and (152), we can check that Eq. 5 (159) is equivalent to Eq. (2.16) of Bonnor (1956), but it has been obtained here in a much more direct manner. Gravitational instability in TBE occurs when the com- 0 pressibility becomes negative passing by κ = 0. 0 0.1 0.2 0.3 0.4 0.5 We now consider the dynamical stability of isothermal πβ3 3 3 3V/4 G M gas spheres under external pressure and show the equivalence with the thermodynamical criterion. This problem Fig. 16. The P V diagram for isothermal spheres. was studied numerically by Yabushita (1968) but we pro- The series of equilibria− becomes unstable at TBE in the vide here an entirely analytical solution. In a preceding isothermal-isobaric ensemble. paper, we have found that the equation of radial pulsations for a gas with an isothermal equation of state could be written (Chavanis 2002a) Eq. (154) was first suggested by Terletsky (1952) as a simple attempt to take into account gravitational effects in k d 1 dq Gq λ2 + = q, (160) the perfect gas law. It appears here as a systematic ex- m dr 4πρr2 dr T r2 4πρTr2 pansion in terms of the concentration factor α. For large concentrations, Eq. (154) is not valid anymore and Eqs. where λ is the growth rate of the perturbation (such that (151)-(152) must be used instead. The P V curve is δρ eλt) and q is defined by Eq. (122). In the isobaric ∼ plotted in Fig. 16. It presents a striking spiral− behaviour ensemble, the boundary conditions are given by Eq. (C.4) towards the limit point ( s, s) = (1/8, 8) corresponding of Appendix C. Introducing dimensionless variables and to the singular sphere. ThisV isP similar to the E T curve considering the case of marginal stability λ = 0, we are and other curves studied in Sec. 2.4. This is interesting− to led to solve the problem note, for historical reasons, because this diagram was dis- d eψ dF F (ξ) coved before the fundamental papers of Antonov (1962) + =0, (161) dξ ξ2 dξ ξ2 and Lynden-Bell & Wood (1968) on the gravothermal catastrophe. Like the minimum energy of Antonov, there 4 4 3 2 4 with F (0) = 0 and exists a maximum pressure Pmax = 1.40 k T /G M m above which no hydrostatic equilibrium for an isothermal dF dψ + F =0, at ξ = α. (162) gas is possible. Alternatively, this corresponds to a min- dξ dξ 3/4 1/2 1/4 imum temperature Tmin = 0.919 G mM P /k for a given mass and pressure. From the turning point crite- Now, the function F (ξ) can be expressed as (Chavanis rion, the series of equilibria becomes unstable in TBE at 2002a) the point of maximum pressure. The condition d /dα =0 3 −ψ 2 ′ F (ξ)= c1(ξ e ξ ψ ). (163) leads to P − Substituting this relation in the boundary condition (162), 2u0 v0 =0. (156) using the Emden equation (14) and introducing the Milne − variables (16), we obtain the condition The critical value α is determined by the intersection between the spiral in the (u, v) plane and the straight 2u0 v0 =0, (164) − line (156). Its numerical value is α = 6.45 correspond- which coincides with the criterion (156) of thermodynam- ing to a density contrast of 14.0. More compressed ical stability. The expressions of the density perturbation isothermal spheres, ie. those for which V V = min profile and of the velocity profile at the point of marginal 0.29 G3M 3m3/k3T 3 are thermodynamically unstable.≤ stability are given in Chavanis (2002a). Only the value of Using Eqs. (151)-(152) and the identities (17)-(18), we the critical point α changes. Due to the different bound- readily obtain ary conditions, we now have δv = 0 at the surface of the 6 d ln 3 gaseous sphere since the sphere can oscillate. The per- V = (u0 1), (157) turbation profile δρ/ρ has one node for the first mode of dα −α − P.H. Chavanis: Gravitational instability of isothermal and polytropic spheres 19

2.2 For the ideal gas without gravity (u0, v0) = (3, 0) we re- 5 TBE cover the well-known result CP = 2 Nk. Gravitational in- Fixed M,P stability occurs in the isenthalpic-isobaric ensemble when 2 CP = 0 passing from negative to positive values and in the isothermal-isobaric ensemble when CP < 0. β

1/4 1.8 P

1/2 6.2. Polytropic gas spheres under external pressure M

3/4 HBE

G We now consider a polytropic gas sphere of index n (or 1/4 1.6 ) π α=25.8 γ =1+1/n) and mass M under pressure P . Using Eqs. R=390 ’=(4 Λ (96) and (102), the pressure on the boundary of the sphere η ’=0.262 1.4 is given by

1 γ 1+ n n+1 P = Kρ(R) = Kρ0 θ(α) . (169) 1.2 −1.2 −0.7 −0.2 0.3 0.8 The central density ρ0 can be eliminated via the relation Λ’=−H/(4π)1/4G3/4M3/2P1/4 (104). Eliminating the radius R via the relation (110) and introducing the Milne variables (105), we obtain after sim- Fig. 17. Temperature vs enthalpy plot at fixed pressure plification and mass. The series of equilibria becomes unstable at n+1 3 2 n−3 HBE in the isenthalpic-isobaric ensemble. 1 4πG M 3 n+1 P = (u v ) n−3 . (170) P ≡ K K3(n + 1)3 0 0 instability, two nodes for the second mode etc... The ve- Similarly, the normalized volume of the sphere is given by locity profile δv has no node for the first mode (except at n 3 3 n−3 the origin), one node for the second mode etc... K (1 + n) 1 3V 3(n−1) = 3 . (171) V ≡ 3 n n−3 We could also consider the situation in which the pres- 4πG M n (u0v0 ) sure P and the enthalpy H = E + P V are fixed (instead Using the expansion of the Milne variables for α 0, of the temperature). This corresponds to the isenthalpic- → 1 V n 1 isobaric ensemble (HBE). Since δS = T δH T δP , the 2 2 − u0 =3 α + ..., v0 = α ..., (172) entropy must be a maximum at fixed H and P at statisti- − 5 3 − cal equilibrium. Eliminating the radius R in Eqs. (19) and we find that the first order correction to the classical adi- (20) in profit of the pressure P , we get after some algebra abatic law due to self-gravity is H 1 3 4 2 Λ′ − = u , γ γ GM 1/4 3/4 3/2 1/4 3 1/4 0 P V = KM λ − , (173) ≡ (4π) G M P (u0v ) 2 − 3 n 3 0 − V 3n (165) Equation (173) can also be obtained from the Virial theorem (155) as before. The P V curve is plotted in Figs. 18 ′ 1/4 3/4 1/2 1/4 3 1/4 − η (4π) G M P β = (u0v ) . (166) and 19 for different values of the polytropic index n. The ≡ 0 volume is extremum at points α such that d /dα = 0. The T H curve in HBE is plotted in Fig. 17. It spirals This leads to the condition u = u . The pressureV is ex- around− the fixed point (Λ′ , η′ ) = (21/4/12, 23/4) corre- 0 s s s tremum at points α such that d /dα = 0. This leads to sponding to the singular solution. There is no equilibrium the condition P state for Λ′ > 0.262. This implies a minimum enthalpy for a fixed pressure or a maximum pressure for a fixed 2u0 + (3 n)v0 =0. (174) enthalpy. This is similar to the Antonov (1962) criterion, − the pressure playing the same role as the box radius. The The number of extrema can be obtained by a graphical series of equilibria becomes unstable in HBE at the point construction in the (u, v) plane like in Chavanis (2002b). of minimum enthalpy. The condition dΛ′/dα = 0 gives For n< 3, there is no extremum for and . The pressure is a decreasing function of the volumeP andV vanishes for 2 16u +8u v 38u +3v =0. (167) a maximum volume Vmax corresponding to the complete 0 0 0 − 0 0 polytrope (such that α = ξ1). For n = 1, the solution of The critical value of α is 25.8 corresponding to a density the Lane-Emden equation is θ = sin ξ/ξ for ξ ξ1 = π contrast of 390. and we explicitly obtain ≤ The specific heat at fixed pressure CP = (∂H/∂T )N,P sin2 α can be deduced from Eqs. (165) and (166). After simplifi- = , = α3, (175) cation, we find P α2(sin α α cos α)2 V − 1 16u2 +8u v 38u +3v for α π. For 3

0.5 α=0 Using Eqs. (170), (171) and the identities (106) and n=1 (107), we readily obtain 0.4 d ln 3 n 1 V = − u 1 , (177) dα −α n 3 0 − − 0.3 and

P d ln n +1 P = (3 n)v +2u . (178) dα (n 3)α − 0 0 0.2 − Therefore, the “compressibility” κ = (∂ ln P/∂ ln V )N,K n=4 can be expressed as 0.1 α=0

1 2u0 + (3 n)v0 α=ξ1 α=ξ1 κ = (n + 1) − . (179) 0 −3 (n 1)u0 n +3 0 2 4 6 8 10 − − V We now consider the dynamical stability of polytropic gas spheres under external pressure. The equation of radial Fig. 18. The P V diagram for polytropes with 1 n< 3 pulsations for a gas with a polytropic equation of state can (speciﬁcally n =1)− and 3

P 0.002 d θ1−n dF nF (ξ) + =0, (181) dξ ξ2 dξ ξ2 0.001 with F (0) = 0 and

dF dθ θ n F =0, at ξ = α. (182) 0 dξ − dξ 0 500 1000 1500 2000 2500 3000 V Now, the function F (ξ) can be expressed as (Chavanis 2002b) Fig. 19. The P V diagram for polytropes with n > 5 − (specifically n = 10). n 3 F (ξ)= c ξ3θn + − ξ2θ′ . (183) 1 n 1 − Substituting this relation in the boundary condition (182), and a minimum volume Vmin. In particular, for n = 5, the using the Lane-Emden equation (103) and introducing the Lane-Emden equation can be solved analytically and we Milne variables (105), we obtain the condition get 2u + (3 n)v =0, (184) 0 − 0 α18 36 1 9 = , = 1+ α2 . (176) which coincides with the criterion (174) of thermodynam- P 36(1 + 1 α2)12 V α15 3 3 ical stability. The expressions of the density perturbation profile and of the velocity profile at the point of marginal The pressure is maximum for α = 3 and the volume is stability are given in Chavanis (2002b). Only the value of minimum for α = √15. For n > 5 there is an infinity of the critical point α changes. For 3 < n 5, the density extrema for and . The P V curve spirals towards the perturbation profile has one node and the≤ velocity profile P V − limit point ( s, s) corresponding to the singular sphere. no node (except at the origin). For n> 5, the perturbation From turningV pointP arguments, the series of equilibria be- profile δρ/ρ has one node for the first mode of instability, comes thermodynamically unstable at the point of max- two nodes for the second mode etc... The velocity profile imum pressure (for n > 3). Polytropes with n < 3 are δv has no node for the first mode (except at the origin), always stable in the isobaric ensemble. one node for the second mode etc... P.H. Chavanis: Gravitational instability of isothermal and polytropic spheres 21

7. Stability of galaxies and stars Vlasov-Poisson system. Elliptical galaxies form therefore a continuous Hamiltonian system. Now, due to a compli- In this section, we discuss in detail the relation between cated mixing process in phase space, the Vlasov-Poisson the dynamical and the thermodynamical stability of stel- system can achieve a metaequilibrium state (on a coarse- lar systems and gaseous stars following our previous inves- grained scale) on a very short timescale. A statistical the- tigations (Chavanis 2002a,b). This discussion should clar- ory appropriate to this violent relaxation was developed ify the notion of generalized thermodynamics introduced by Lynden-Bell in 1967 (see a short review in Chavanis by Tsallis (1988). 2002f). For collisionless stellar systems, the infinite mass problem is solved by realizing that violent relaxation is in- 7.1. Collisional vs collisionless relaxation complete so that the galaxies are more confined that one would expect on the basis of statistical considerations. Basically, a stellar system is a collection of N stars in Collisionless and collisional relaxations are therefore gravitational interaction. Due to the development of en- two distinct processes in the life of a self-gravitating sys- counters between stars, this system is expected to relax to- tem and they must be studied by different types of statis- wards a statistical equilibrium state. The relaxation time tical mechanics. Note that a globular cluster first experi- can be estimated by t (N/ ln N)t where t is relax ∼ D D ences a collisionless relaxation leading to a virialized state the dynamical time, i.e. the time it takes a star to cross and then evolves more slowly towards another equilibrium the system (Binney & Tremaine 1987). For globular clus- state due to stellar encounters. We can discuss these two 5 5 ters, N 10 , tD 10 years so that the relaxation successive equilibrium states in a more formal manner. Let time is of∼ the same∼ order as their age ( 1010 years). ∼ us consider a fixed interval of time t and let the number of Therefore, globular clusters form a discret Hamiltonian particles N + . In that case, the system is rigorously system which can be studied by ordinary methods of sta- described by→ the∞ Vlasov equation, and a metaequilibrium tistical mechanics. In particular, it can be shown rigor- state is achieved on a timescale independant on N as a ously that the Boltzmann entropy SB[f] is the correct result of violent relaxation. Alternatively, if we fix N and form of entropy for these systems (in a suitable ther- let t + , the system is expected to relax to an equi- modynamic limit) although they are non-extensive and librium→ state∞ resulting from a collisional evolution. These non-additive. The absence of entropy maximum in an un- two equilibrium states are of course physically distinct. bounded domain simply reflects the natural tendency of a This implies that the order of the limits N + and stellar system to evaporate under the effect of encounters. t + is not interchangeable for self-gravitating→ ∞ sys- However, evaporation is a slow process so that a glob- tems→ (Chavanis∞ 2002e). ular cluster passes by a succession of quasi-equilibrium states corresponding to truncated isothermals (see Sec. 4). It should be emphasized that statistical mechanics is 7.2. The Vlasov-Poisson system based on the postulate that the evolution is ergodic. Now, In Secs. 2-4, we have implicitly considered the case of col- the kinetic theory of self-gravitating systems is not well- lisional stellar systems, such as globular clusters, relax- understood (Kandrup 1981). Indeed, it is possible to prove ing via two-body encounters. We now turn to the case a H-theorem for the Boltzmann entropy only if simplify- of collisionless stellar systems, such as elliptical galaxies, ing assumptions are implemented (Markovian approxima- described by the Vlasov equation tion, local approximation, regularization of logarithmic di- vergences,...). It would be of interest to determine more ∂f ∂f ∂f precisely the influence of non-ideal effects on the relax- + v + F =0, (185) r v ation of stars. However, these effects are not expected to ∂t ∂ ∂ induce a strong deviation to Boltzmann’s law. In particu- coupled with the Poisson equation (3). The Vlasov equa- lar, the diffusion of stars is normal or slightly anomalous, tion (185) simply states that, in the absence of encoun- i.e. polluted by logarithmic corrections (Lee 1968). Hence, ters, the distribution function f is conserved by the flow Tsallis entropies, which often arise in the case of anoma- in phase space. This equation conserves the usual invari- lous diffusion, do not seem to be justified in the context of ants (1) and (2) but also an additional class of invariants “collisional” stellar systems. However, there are many ef- called the Casimirs fects that can induce a lack of ergodicity and a deviation to Boltzmann’s law. To a large extent, the dynamics of 3 3 self-gravitating systems remains misunderstood and addi- Ih = h(f)d rd v, (186) tional work (using numerical simulations) is necessary to Z clarify the process of collisional relaxation. where h is any continuous function of f. This infinite set For elliptical galaxies, on the other hand, the relax- of additional constraints is due to the collisionless nature ation time due to close encounters is larger than their age of the evolution and results from the conservation of f 11 8 by about ten orders of magnitude (N 10 , tD 10 and the incompressibility of the flow in phase space. As years, age 1010 years). Therefore, their∼ dynamics∼ is es- we shall see, they play a crucial role in the statistical me- sentially encounterless∼ and described by the self-consistent chanics of violent relaxation. We note that the conserva- 22 P.H. Chavanis: Gravitational instability of isothermal and polytropic spheres

3 1 2 3 tion of the Casimirs is equivalent to the conservation of fd v = ρ(Φ), p = 3 fv d v = p(Φ), hence p = p(ρ). the moments Writing explicitly the pressure in the form R R 1 +∞ M = f nd3rd3v. (187) p = F (βǫ + α)4π[2(ǫ Φ)]3/2dǫ, (193) n 3 − Z ZΦ It is well-known that any distribution function of the and taking its gradient, we obtain the condition of hydro- v2 static equilibrium form f = f(ǫ), where ǫ = 2 + Φ is the individual energy of a star, is a stationary solution of the Vlasov equation. p = ρ Φ. (194) This is a particular case of the Jeans theorem (Binney ∇ − ∇ & Tremaine 1987). We shall assume furthermore that Therefore, the Jeans equations of stellar dynamics reduce f ′(ǫ) < 0 so that high energy stars are less probable than in these cases of spherical systems with isotropic pressure low energy stars. Any such distribution function can be to the equation of hydrostatics for a barotropic gas. obtained by extremizing a functional of the form 7.3. Violent relaxation of collisionless stellar systems S[f]= C(f)d3r d3v, (188) − The Vlasov equation does not select a universal form of Z entropy S[f], unlike the Boltzmann equation for example. at fixed mass M and energy E, where C(f) is a convex Therefore, the notion of equilibrium is relatively subtle. function, i.e. C′′(f) 0. Furthermore, maxima of S[f] at ≥ When coupled to the Poisson equation, the Vlasov equa- fixed mass and energy are dynamically stable with respect tion develops very complex filaments as a result of a mix- to the Vlasov-Poisson system (Ipser 1974, Ipser & Horwitz ing process in phase space. In this sense, the fine-grained 1979). This is a sufficient condition of nonlinear stabil- distribution function f(r, v,t) will never converge towards ity. We are tempted to believe that it is also a necessary a stationary solution f(r, v). This is consistent with the condition of nonlinear stability although this seems to go fact that the functionals S[f] are rigorously conserved by against the theorem of Doremus et al (1971) which states the flow (they are particular Casimirs). However, if we ′ that all distribution functions with f (ǫ) < 0 are dynam- introduce a coarse-graining procedure, the coarse-grained ically stable (even if they are minima or saddle points of distribution function f(r, v,t) will relax towards a metae- S at fixed E and M). In fact, this criterion is only a con- quilibrium state f(r, v). This collisionless relaxation is dition of linear stability (Binney & Tremaine 1987) and called violent relaxation (Lynden-Bell 1967) or chaotic it may not be valid for box confined models (its general mixing. It can be shown that S[f] calculated with the applicability has also been criticized). In the following, we coarse-grained distribution function at time t> 0 is larger shall consider that f = f(ǫ) is dynamically stable “in a than S[f] calculated with the fine-grained distribution strong sense” if and only if it is a maximum of S[f] at fixed function at t = 0 (a monotonic increase of H is not im- E and M. This is equivalent to f = f(ǫ) being a minimum plied). This is similar to the H-theorem in kinetic theory of E at fixed S and M (Ipser & Horwitz 1979). It can be except that S[f] is not necessarily the Boltzmann entropy. shown that stellar systems satisfying this requirement are For that reason, S[f] is sometimes called a H-function necessarily spherically symmetric. (Tremaine et al 1986). Boltzmann and Tsallis functionals Introducing appropriate Lagrange multipliers and SB[f] and Sq[f] are particular H-functions corresponding writing the variational principle in the form to isothermal stellar systems and stellar polytropes. It is important to note, furthermore, that among the δS βδE αδM =0, (189) − − infinite set of integrals conserved by the Vlasov equation, one finds that some are more robust than others. For example, the mo- c.g. n 3 3 ments Mn [f]= f d rd v calculated with the coarse- C′(f)= βǫ α. (190) grained distribution function vary with time for n > 1 n − − since f n = f . ByR contrast, the mass and the energy Since C′ is a monotonically increasing function of f, we are approximately6 conserved on the coarse-grained scale. can inverse this relation to obtain We shall therefore classify the collisionless invariants in two groups: the energy E and the mass M will be called f = F (βǫ + α), (191) robust integrals since they are conserved by the coarse- where F (x) = (C′)−1( x). From the identity grained dynamics while the moments Mn with n > 1 − will be called fragile integrals since they vary under the f ′(ǫ)= β/C′′(f), (192) operation of coarse-graining. Note that, in practice, the − moments Mn>1 are also altered at the fine-grained scale resulting from Eq. (190), f(ǫ) is a monotonically decreas- because, intrinsically, the system is not continuous but ing function of energy if β > 0. made of stars. Therefore, the graininess of the medium We note also that for each stellar systems with f = can break the conservation of some moments, in general f(ǫ), there exist a corresponding barotropic gas with those of high order, even if we are in a regime where the the same equilibrium density distribution. Indeed ρ = evolution is essentially encounterless. P.H. Chavanis: Gravitational instability of isothermal and polytropic spheres 23

7.4. Lynden-Bell’s theory of violent relaxation H-function S[f] whose extremization at fixed mass and energy returns Eq. (199). Furthermore, it can be shown that Now, the question of fundamental interest is the following: the condition (198) of thermodynamical stability (max- given some initial condition, can we predict the equilib- imum of S[ρ] at fixed E and M ) implies that f is a rium distribution function that the system will eventually n nonlinearly stable solution of the Vlasov-Poisson system. achieve? Lynden-Bell (1967) tried to answer this question Therefore, it is a maximum of S[f] at fixed mass and by using statistical mechanics arguments. He introduced energy. This functional depends on the initial conditions the density probability ρ(r, v, η) of finding the value of the through the Lagrange multipliers α , or equivalently distribution function η in (r, v) at equilibrium. Then, us- n>1 through the function χ(η), which are determined by the ing a combinatorial analysis he showed that the relevant fragile moments M . Therefore, S[f] is non-universal measure is the Boltzmann entropy n>1 and can take a wide variety of forms. In general, S[f] dif- fers from the ordinary Boltzmann entropy S [f] derived S[ρ]= ρ ln ρd3rd3vdη, (195) B − in the context of collisional stellar systems. This is due Z to the additional constraints (Casimir invariants) brought which is a functional of ρ. This mixing entropy has been by the Vlasov equation. For a given initial condition, the justified rigorously by Michel & Robert (1994), using the H-function S[f] maximized by the system at statistical concept of Young measures. Therefore, although the sys- equilibrium is uniquely determined by the statistical the- tem is non-extensive, the Boltzmann entropy is the correct ory according to Eqs. (190) and (199). measure. Assuming ergodicity (which may not be realized in practice, see Sec. 7.5) the most probable equilibrium state, i.e. most mixed state, is obtained by maximizing S[ρ] taking into account all the constraints of the dynam- In general, S[f] is not an entropy, unlike S[ρ], as it ics. The optimal equilibrium state can be written cannot be directly obtained from a combinatorial analy- 1 sis. There is a case, however, where S[f] is an entropy. ρ(r, v, η)= χ(η)e−(βǫ+α)η, (196) If the initial condition is made of patches with value Z f0 = η0 and f0 = 0 (two-levels approximation), then n where χ(η) exp( n>1 αnη ) accounts for the conser- the entropy S[ρ] is equivalent to the Fermi-Dirac en- ≡ − n 3 3 vation of the fragile moments Mn>1 = ρη dηd rd v and tropy S [f] = [f/η ln f/η + (1 f/η ) ln(1 P F.D. 0 0 0 α, β are the usual Lagrange multipliers for M and E (ro- f/η )]d3rd3v and− Lynden-Bell’s distribution− (199) co-− R 0 R bust integrals). The distinction between these two types of incides with the Fermi-Dirac distribution (Lynden-Bell constraints is important (see below) and was not explicitly 1967, Chavanis & Sommeria 1998). Moreover, all the mo- made by Lynden-Bell (1967). The “partition function” ments can be expressed in terms of the mass alone as n +∞ Mn = η0 M. In the non-degenerate (or dilute) limit f −(βǫ+α)η ≪ Z = χ(η) e dη, (197) η0, the Fermi-Dirac entropy reduces to the Boltzmann en- 0 tropy S [f]= f ln fd3rd3v. Quite generally, in the di- Z B − is determined by the local normalization condition lute limit, the equilibrium distribution is a sum of isother- R ρdη = 1. The condition of thermodynamical stability mal distributions. reads R 1 (δρ)2 1 δ2J d3rd3vdη β δΦδfd3rd3v 0, ≡−2 ρ − 2 ≤ We can note some general predictions of the statistical Z Z (198) theory, which are valid for any initial condition f0(r, v). (i) The predicted distribution function is a function f = f(ǫ) for all variation δρ(r, v, η) which does not change the con- of the stellar energy alone in the non-rotating case and a straints to first order. The equilibrium coarse-grained dis- function f = f(ǫJ ) of the Jacobi energy ǫJ = ǫ Ω (r v) tribution function f ρηdη can be expressed as alone in the rotating case. (ii) f ′(ǫ) < 0 so that− high· × en- ≡ ergy stars are less probable than low energy stars. (iii) 1 ∂ ln Z R f = = F (βǫ + α)= f(ǫ). (199) f(ǫ) is strictly positive for all energies (it never vanishes). −β ∂ǫ max (iv) f(ǫ) is always smaller than the maximum value f0 Taking the derivative of Eq. (199), it is easy to show that of the initial condition. (v) According to Eqs. (200) and ′′ (192), we have the general relation f2 =1/C (f). (vi) For ′ f (ǫ)= βf , f ρ(η f)2dη > 0, (200) a box-confined system, there always exist a global maxi- − 2 2 ≡ − Z mum of entropy. Therefore, contrary to collisional stellar where f2 is the centered local variance of the distribu- systems, it is not necessary to introduce a small-scale cut- tion ρ(r, v, η). The integrability condition fd3v < off to make the thermodynamical approach rigorous. This requires that the temperature is positive (β > 0). Then,∞ is due to the Liouville theorem which puts an upper bound R from Eq. (200), we deduce that f is a monotonically de- on the value of the distribution function like the Pauli ex- creasing function of ǫ. Therefore, from Sec. 7.2, we know clusion principle in quantum mechanics (see Chavanis et that for each function f(ǫ) of the form (199) there exists a al 1996, Chavanis & Sommeria 1998). 24 P.H. Chavanis: Gravitational instability of isothermal and polytropic spheres

7.5. Incomplete relaxation a model of incomplete violent relaxation which provides a depletion of stars in the high energy tail of the distribu- The prediction of Lynden-Bell’s statistical theory crucially tion function. If the system is isolated, as most elliptical depends on the initial conditions. This is because we need galaxies are, Hjorth & Madsen (1993) have proposed a to know the value of the moments Mn>1 which can only dynamical scenario attempting to take into account in- be deduced from the fine-grained distribution function at complete relaxation in the halo. They introduce a two- t = 0 (say). Indeed, these moments are fragile, or mi- step process: (i) in a first step, they assume that violent croscopic, constraints which cannot be mesured from the relaxation proceeds to completion in a finite spatial re- coarse-grained flow at t> 0 since they are altered by the gion, of radius rmax, which represents roughly the core of coarse-graining procedure. This is a specificity of contin- the galaxy (where the fluctuations are important). At this uous Hamiltonian systems which contrasts with ordinary stage, the escape energy represents no special threshold (discrete) Hamiltonian systems for which the constraints so that negative as well as positive energy states are pop- are robust, or macroscopic, and can be evaluated at any ulated in that region. (ii) After the relaxation process is time (as the mass M and the energy E). Now, Lynden- over, positive energy particles leave the system and par- Bell (1967) gives arguments according to which elliptical ticles with Φ(rmax) <ǫ< 0 move in orbits beyond rmax, galaxies should be non-degenerate (except possibly in the thereby changing the distribution function to something nucleus). In that case, the dilute limit of his theory applies significantly ‘thinner’ than a Boltzmann distribution. The in the main body of the galaxy and leads to an isothermal crucial point to realize is that the differential energy dis- distribution. The theoretical justification of an isothermal tribution N(ǫ), where N(ǫ)dǫ is the number of stars with distribution for collisionless stellar systems was considered energy between ǫ and ǫ + dǫ, will be discontinuous at as a triumph in the 1960’s. Indeed, Lynden-Bell’s statisti- the escape energy ǫ = 0 since there is a finite number cal theory of violent relaxation could explain the observed of particles within rmax after the relaxation process. It isothermal core of elliptical galaxies without recourse to can be shown that this discontinuity implies necessarily collisions that operate on a much longer timescale. that f(ǫ) ( ǫ)5/2 for ǫ 0− (Jaffe 1987). Therefore, However, it is easy to see that the statistical theory the distribution∼ − function consistent→ with this scenario is of violent relaxation cannot describe the halo of elliptical (Hjorth & Madsen 1993) galaxies. Indeed, at large distances, the density decreases Ae−βη0ǫ Φ ǫ<ǫ′, as r−2 resulting in the infinite mass problem. Lynden- 0 ≤ f = B ( ǫ)5/2 ǫ′ ǫ< 0, (201) Bell (1967) has related this mathematical difficulty to the ( −0 ≤ǫ 0. physical problem of incomplete relaxation and argued that ≥ his distribution function has to be modified at high en- This model corresponds formally to a composite configu- ergies. Indeed, the relaxation is effective only in a finite ration with an isothermal core in which violent relaxation region of space and persists only for a finite period of is efficient and a polytropic halo (with index n = 4) which time. Therefore, there is no reason to maximize entropy is only partially relaxed (in reality, there is a small devia- in the whole available space. The ergodic hypothesis which tion to the ideal value n = 4 because the potential created sustains the statistical theory applies only in a restricted by the core is not exactly Keplerian; see Hjorth & Madsen domain of space, in a sort of “maximum entropy bubble”, 1993). For this model, the density decreases as r−4 like in surrounded by an unmixed region which is only poorly elliptical galaxies. Furthermore, this model can reproduce sampled by the system. Accordingly, the physical picture de Vaucouleur’s R1/4 law for the surface brightness of el- that emerges is the following: during violent relaxation, lipticals (Binney & Tremaine 1987). Alternatively, if the the system has the tendency to reach the most mixed system is subject to tidal forces (this may be the case for state described by the distribution (199). However, as it dark matter halos), one can propose an extended Michie- approaches equilibrium, the mixing becomes less and less King model of the form (Chavanis 1998) − − efficient and the system settles on a state which is not the e βη0ǫ−e βη0ǫm η − ǫ<ǫ , most mixed state. f = 0 λ+e βη0ǫ m (202) 0 ǫ ǫ , How can we take into account incomplete relaxation? ( ≥ m A first possibility is to confine the system artificially in- which takes into account the specificities of Lynden-Bell’s side a box, where the box delimits the typical region in distribution function (in particular the exclusion principle which the statistical theory applies (Chavanis & Sommeria associated with Liouville’s theorem) and leads to a con- 1998). Another possibility is to use a parametrization of finement of the density at large distances (i.e. the density the coarse-grained dynamics of collisionless stellar systems drops to zero at a finite distance identified with the tidal in the form of relaxation equations involving a space and radius). time dependant diffusion coefficient related to the fine- grained fluctuations of the distribution function (Chavanis 7.6. Tsallis entropies et al 1996, Chavanis 1998). This space and time dependant diffusion coefficient can freeze the system in a “maximum The statistical theory of Lynden-Bell encounters some entropy bubble” and lead to the formation of self-confined problems because the distribution function f(ǫ) never van- stellar systems. Finally, a third possibility is to construct ishes. Therefore, we have to invoke incomplete relaxation P.H. Chavanis: Gravitational instability of isothermal and polytropic spheres 25 and introduce dynamical constraints in order to explain at fixed mass and energy. Therefore, their model is consis- the observation of confined galaxies and solve the infinite tent with a composite Tsallis model with q = 1 (complete mass problem. This is probably the correct approach to mixing) in the core and q (2n 1)/(2n 3) = 7/5 the problem. Another line of thought, defended by Tsallis (incomplete mixing) in the halo.≡ More− generally,− we could and co-workers, is to change the form of entropy in order argue that q varies in space in order to take into account a to obtain a distribution function that vanishes at some variable mixing efficiency (very similarly, stars are some- maximum energy. Therefore, Tsallis generalized thermo- times approximated by composite polytropic models with dynamics is essentially an attempt to take into account a space dependent index in order to distinguish between incomplete mixing and lack of ergodicity in complicated convective and radiative regions). However, this gives to systems. If we assume that the system is classically de- Tsallis distribution a practical, not a fundamental justi- scribed by the Boltzmann entropy SB[f] (which is true fication. It can just fit locally the distribution function for collisionless stellar systems only in a coarse-grained of collisionless stellar systems resulting from incomplete sense and in a dilute limit), then Tsallis proposes to re- violent relaxation. place SB[f] by a q-entropy Sq[f]. The parameter q mea- In addition, not all stable stationary solutions of the sures the efficiency of mixing (q = 1 if the system mixes Vlasov equation can be obtained by maximizing a H- well) but is not determined by the theory. function at fixed mass and energy. This maximization This approach is attractive but the justification given problem leads to f = f(ǫ) with f ′(ǫ) < 0. For spheri- by Tsallis and co-workers is misleading. Fundamentally, cal stellar systems, the distribution function can depend ordinary thermodynamics (in the sense of Lynden-Bell ) on the specific angular momentum j = r v in addi- × is the correct approach to the problem. The Boltzmann tion to the stellar energy ǫ and this is what happens entropy S[ρ] has a clear physical meaning as it is propor- in certain models of incomplete violent relaxation with tional to the logarithm of the disorder, where the disorder anisotropic velocity distribution (Lynden-Bell 1967). On is equal to the number of microstates consistent with a the other hand, for rotating stellar systems, the maximiza- given macrostate. Of course, if the system does not mix tion of S[f] at fixed mass, energy and angular momentum ′ well, statistical mechanics loses its power of prediction. predicts that f = f(ǫJ ) with f < 0, where ǫJ is the However, the state resulting from incomplete violent re- Jacobi energy. However, it is well-known that such distri- laxation is always a nonlinearly stable stationary solution bution function cannot account for the triaxial structure of the Vlasov equation on a coarse-grained scale (i.e., for of elliptical galaxies. More general distribution functions f). Therefore, if f = f(ǫ), it maximizes a H-function must be constructed in agreement with the Jeans theorem S[f] = C(f)d3rd3v at fixed mass M and energy E. (Binney & Tremaine 1987). However, it is possible that rel- The function− C(f) is influenced by thermodynamics (mix- evant distribution functions maximize a H-function with R ing) but it cannot be determined by pure thermodynami- additional constraints. These constraints may correspond cal arguments because of incomplete relaxation. to adiabatic invariants that are approximately conserved during violent relaxation. For example, anisotropic veloc- The H-function S[f] selected by the system depends ity distributions can be obtained by maximizing a H- both on the initial condition and on the type of mixing. function while conserving the individual distributions of Tsallis entropy S [f] is just a particular H-function lead- q angular momentum L = f(r v)2d3rd3v (or other dis- ing to confined structures (stellar polytropes). Now, it 2 tribution) in addition to mass× and energy. This leads to is well-known that stellar polytropes give a poor fit of 2 j R ′ f = f(ǫa) with ǫa ǫ+ and f (ǫa) < 0 (ra is called the elliptical galaxies. In particular, they cannot reproduce ≡ 2ra de Vaucouleur’s law, neither the r−4 density decrease of anisotropy radius). It is expected that this maximization these objects. Indeed, for stellar polytropes, the density problem is a condition of nonlinear stability. For rotating decreases as r−5 for n = 5, the density drops to zero at systems, it is not known whether we can construct distri- a finite distance for n < 5 and the mass is infinite for bution functions yielding triaxial bodies by maximizing a n > 5. We conclude, therefore, that observations of ellip- H-function at fixed M, E, L and additional (adiabatic) tical galaxies do not support the prediction of Tsallis. The invariants. These ideas may be a route to explore. model of Hjorth & Madsen (1993) based on the statistical theory of Lynden-Bell improved by a physically motivated 7.7. Dynamical stability of collisionless stellar systems scenario of incomplete relaxation provides a much better agreement with observations. This equilibrium state can- On a formal point of view, the maximization problem de- not be described by Tsallis distribution which fails to re- termining the nonlinear stability of a collisionless stellar produce simultaneously an isothermal core and an exter- system is similar to the one which determines the thermo- nal confinement. In fact, the model of Hjorth & Madsen dynamical stability of a collisional stellar system. In ther- (1993) can be obtained by maximizing a H-function with modynamics, we maximize the Boltzmann entropy SB[f] at fixed mass M and energy E in order to determine the most probable equilibrium state. In stellar dynamics, the f ln f, f>f ; C(f)= 0 (203) maximization of a H-function S[f] at fixed mass and en- 1 (f q f), f 3). We note, in particular, that gaseous polytropes Vlasov equation. with 3 5. In that case, they the disorder is equal to the number of microstates consis- are maxima of Sq[f] at fixed M and E. According to Sec. tent with a given macrostate. However, SB[f] is not maxi- 7.7, this implies that stellar polytropes are dynamically mized by the system because of incomplete relaxation (this stable solutions of the Vlasov equation for n < 5 (and if is independant on the fact that SB[f] has no maximum!). the density contrast is sufficiently low for n> 5). On the In any case, the state resulting from incomplete violent other hand, self-gravitating systems described by Tsallis relaxation is a nonlinearly stable solution of the Vlasov entropy are thermodynamically stable in the canonical en- equation (on a coarse-grained scale). If f = f(ǫ), it max- semble if n< 3 or if the density contrast is sufficiently low imizes a H-function S[f] = C(f)d3rd3v, where C(f) − for n> 3. In that case, they are maxima of Jq[f] at fixed is a convex function, at fixed mass and energy. Therefore, M and β. According to Sec. 7.8, this implies that poly- we can use a thermodynamicalR analogy to study the dy- tropic gaseous spheres are dynamically stable solutions namical stability of collisionless stellar systems. Tsallis en- of the Jeans-Euler equations if, and only, if n < 3 (or tropies Sq[f] are a particular class of H-functions leading if, and only, if the density contrast is sufficiently low for to stellar polytropes. Stellar polytropes do not give a good P.H. Chavanis: Gravitational instability of isothermal and polytropic spheres 29

fit of elliptical galaxies. A better model of incomplete vio- Appendix B: Some useful identities for polytropic lent relaxation, motivated by precise physical arguments, spheres consists of an isothermal core and a polytropic halo with index n 4 (Hjorth & Madsen 1993). This can be fitted Using integrations by parts similar to those performed in by a composite≃ Tsallis (polytropic) model with q = 1 in Appendix A, we can easily establish the identities the core (complete mixing) and q 7/5 in the envelope α 3 ′ 2 ′ 2 2 α θ (α) u0 6 (incomplete mixing). More generally,≃ the state resulting (θ ) ξ dξ = n +1+2 , (B.1) 5 n v0 − v0 from incomplete violent relaxation does not necessarily Z0 − maximize a H-function at fixed mass and energy. Due to α α3θ′(α)2 u n +1 the Jeans theorem, the distribution function can depend θn+1ξ2dξ = n +1+2 0 , (B.2) on other integrals than the stellar energy or the Jacobi 5 n v0 − v0 Z0 − energy. It is possible, nevertheless, that relevant distribution functions arising from incomplete violent relaxation α 3 ′ 2 α θ (α) u0 3 maximize a H-function at fixed mass, energy, angular mo- θnθ′ξ3dξ = 3 + , (B.3) 5 n − − v v mentum and additional (adiabatic) invariants. Z0 − 0 0

Acknowledgements. I am grateful to J. Katz and T. α Padmanabhan for stimulating discussions and encourage- θnξ2dξ = α2θ′(α). (B.4) − ment. Z0 Appendix C: Equation of pulsations and boundary Appendix A: Some useful identities for isothermal conditions spheres In this paper and previous ones, we have studied the dy- Using the Emden equation (14), we have namical stability of barotropic gaseous spheres with the α α 2 −ψ 2 ′ ′ 2 ′ equation of pulsation ξ e dξ = (ξ ψ ) dξ = α ψ (α)= αv0, (A.1) Z0 Z0 d p′(ρ) d G λ2 + q (r)= q (r), (C.1) which establishes Eq. (67). On the other hand, using an dr 4πρr2 dr r2 λ 4πρr2 λ integration by parts, we obtain where λ is the growth rate of the perturbation (such α α d ψ′ξ3e−ψdξ = ξ3 (e−ψ)dξ that δρ eλt, δv eλt etc...) and q(r) δM(r) = 0 − 0 dξ r δρ4πr2∼dr represent∼ the mass perturbation≡ within the Z Z α 0 = α3e−ψ(α) + 3ξ2e−ψdξ. (A.2) sphere of radius r which is related to the velocity by − R(Chavanis 2002a) Z0 Combining with Eq. (A.1) and introducing the Milne vari- λ δv = q. (C.2) ables, we establish Eq. (68). To establish Eq. (69), we start −4πρr2 from the identity 3 α ξ2ψ′ d Obviously q(0) = 0 and more precisely q r for r 0. (ξ2ψ′)dξ = α3ψ′(α)2 If the system is confined within a sphere∼ of fixed radius→ ξ dξ Z0 R, the conservation of mass imposes q(R) = 0. This can α ξ2ψ′ d α (ξ2ψ′)dξ + ξ2ψ′2dξ, (A.3) also be obtained from Eq. (C.2) if we set δv(R) = 0. If the − ξ dξ system is free and the density vanishes at its surface, i.e. Z0 Z0 which results from a simple integration by parts. Thus, ρ(R) = 0, like for a complete polytrope, then Eq. (C.2) α α again implies that q(R) = 0 (Chavanis 2002b). If, finally, ξ2ψ′2dξ = α3ψ′(α)2 +2 ψ′ξ3e−ψdξ, (A.4) the gaseous sphere supports a fixed external pressure (iso- − Z0 Z0 baric ensemble) the boundary condition requires that the where we have used the Emden equation (14). Combining Lagrangian derivative of the pressure vanishes at the sur- this relation with Eq. 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