caloric curve of King models

Transcription

1 caloric curve of King models with a short-distance cutoff on the interactions Lapo Casetti Dipartimento di Fisica e Astronomia & CSDC, Università di Firenze, Italy INFN, sezione di Firenze, Italy Dynamics & Kinetic Theory of Self-Gravitating Systems IHP, Paris, France, November 6, 2013 joint work with Cesare Nardini (ENS Lyon) Physical Review E 85, (2012)

2 introduction & motivation self-gravitating systems: natural examples of long-range systems truly long-range interactions, unscreened almost ideal samples: globular clusters, elliptical galaxies... seemingly obvious testing ground for theoretical predictions standard equilibrium statistical mechanics does not work! short-distance singularity escape of particles idealized systems & toy models first step very interesting theoretical features clustering, phase transitions, ensemble inequivalence, C v < 0... encoded in the caloric curve T (E) relevant for real systems? caloric curve analysis of observationally probed models (King) introduction of a short-distance cutoff

3 equilibrium statistical mechanics of self-gravitating systems H (r 1,..., r N, v 1,..., v N ) = m 2 N v 2 i Gm 2 i=1 N N i=1 j>i 1 r i r j short-distance singularity = no true equilibrium state metastable states may still exist (local entropy maxima) easy solution: regularization via short-distance cutoff (more soon) unbounded space = escape of particles finite escape velocity incompatible with maxwellian velocity distribution stationary maxwellian distribution in unbounded space = infinite mass solution (not so easy...): put the system in a box or consider an expanding background, but that s another story regularized & confined = equilibrium exists [Kiessling, Chavanis]

4 equilibrium statistical mechanics of self-gravitating systems

5 isothermal sphere forget about regularization it is implied, and will come back shortly... continuum (mean-field) limit & spherical box of radius R (m = 1) S[f ] = dr dv f (r, v) log f (r, v) local extrema of S spherically symmetric that is, for given β and ϱ c = ϱ(0), d 2 ϕ(r) dr 2 f (r, v) = C e βv 2 /2 e βϕ(r) + 2 r ϱ(r) = dv f (r, v) 2 ϕ(r) = 4πGϱ(r) dϕ(r) dr = 4πGϱ ce β[ϕ(r) ϕ(0)] [Antonov, Lynden-Bell & Wood, Padmanabhan, Chavanis]

6 isothermal sphere: caloric curve energy & temperature (k B = 1) U = G 2 K = 1 2 dr dv v 2 f (r, v) = 3 2β = 3T 2 dr dv dr dv f (r, v)f (r, v ) r r E = K + U = 1 2 dr ϱ(r)ϕ(r) energy unit GM 2 /R M = dr ϱ(r) dimensionless energy & temperature ε = ϑ = RE GM 2 RT GM 2

7 isothermal sphere: caloric curve ϑ minimal energy & temperature ε ε min ϑ min 0.4 ε < ε min = gravothermal catastrophe [Antonov, Lynden-Bell & Wood, Padmanabhan, Chavanis]

8 short-distance cutoff short-distance regularization regularization + confinement = equilibrium states exist necessary to justify the mean-field procedure required by physics quantum particles: effective cutoff due to Pauli exclusion principle self-gravitating fermions [Chavanis & Ispolatov] classical particles: new interactions at small scales stars & planets have a finite size! many possible implementations hard-core/soft-core particles, truncated/softened potential... Gm 2 V (r i, r j ) = ri r j 2 + a [a] = l 2

9 models with cutoff mean-field in a spherical box (isothermal sphere + cutoff) [Aronson & Hansen, Chavanis, Ispolatov & Cohen, Alastuey & coworkers] shell model [Youngkins & Miller] self-gravitating ring [Sota et al., Tatekawa et al.] self-gravitating particles on S 2 [Kiessling] minimalistic models [Thirring, Lynden-Bell, Chavanis, LC & Nardini] N stars in a box (MC simulations of N self-gravitating particles in a 3D box) [De Vega & Sanchez] common features short-distance cutoff + confinement in a box or in a compact configuration space

10 caloric curve ϑ cutoff-dominated C < 0 gaslike ε common features (small cutoff) the cutoff stabilizes a low-energy phase (clustered phase) no gravothermal catastrophe, minimal energy related to real lower bound on potential energy negative specific heat in a region of the clustered phase phase transition to high-energy phase (quasi-uniform, perfect-gas-like) the order of the phase transition depends on the cutoff, as do the details of the phases

11 caloric curve ϑ cutoff-dominated C < 0 gaslike ε question what about real self-gravitating systems? no box, no thermal velocity distribution

12 globular clusters ω Cen the largest Milky Way globular cluster

13 King model phenomenological & stationary mean-field-like model spherically symmetric cluster of equal stars globulars & open clusters & elliptical galaxies... assumptions 1 single particle distribution function f (r, v) 2 ϱ(r) 0 if r r t 3 relaxed system = f (v) as close to thermal equilibrium as it can be 4 constraint: v v e(r) = escape velocity [King 1966]

14 King model ϱ(r) = f (r, v) = C M β T 1 { dv f (r, v) v 2 e(r) = 2ϕ(r) ϕ(r t) = 0 C e 2βϕ(r) [e βv 2 e βv 2 e (r)] if v 2 < v 2 e (r) 0 otherwise for the moment no short-distance cutoff on the gravitational interaction... 2 ϕ(r) = 4πGϱ(r)...and go on as in the isothermal sphere r t (M ) = King isothermal sphere [King 1966]

15 introduction self-gravitating systems King model statistical mechanics summary & outlook King model vs. observations good fit of density profiles for roughly 80% of Milky Way globulars bad fit or no fit for the remaining 20% of globulars post-core-collapsed globulars [Djorgovic & King 1986] M 13 in Hercules King cluster M 15 in Pegasus post-core-collapsed cluster

16 statistical mechanics of King models energy & temperature (k B = 1) K = 1 2 dr dv v 2 f (r, v) = 3T 2 U = G 2 dr dv dr dv f (r, v)f (r, v ) r r E = K + U = 1 2 dr ϱ(r)ϕ(r) energy unit GM 2 /r t dimensionless energy & temperature ε = ϑ = rte GM 2 rtt GM 2

17 caloric curve of King model without cutoff ϑ ε virial theorem for purely gravitational interactions energy & temperature are bounded K = E = ϑ = 2 3 ε ε [ 2.13, 0.60] ϑ [0.40, 1.42]

18 caloric curve of King model without cutoff ϑ ε increasing ϕ(r = 0) data points reach ε min then go back and forth in a collapsed spiral pattern plotting e.g. ϑ 0 = ϑ(r = 0) the spiral pattern opens up

19 switching on the cutoff short-range cutoff 1 r r 1 r r 2 + a all definitions of f, ϱ, ϕ, U, K, E, T formally as before same adimensionalization: dimensionless cutoff α α = a r t 2 no analogue of Poisson equation = no differential formulation self-consistent iterative procedure conceptually straightforward, numerically less efficient reasonable cutoff values star size < cutoff length < average interstellar separation 10 9 α

20 caloric curve with cutoff ϑ ε α = 10 3

21 caloric curve with cutoff ϑ ε α = 10 5

22 caloric curve with cutoff 1.5 ϑ ε α = 10 3 α = 10 5 no cutoff

23 caloric curve with cutoff effect of the short-distance cutoff stabilization of a low-energy phase energy range much larger than without cutoff high-energy region model without cutoff already for moderate cutoff α 10 5 close analogy to confined models with cutoff no gas-like phase at high energy (no container!) ϑ cutoff-dominated C < 0 gaslike ε

24 caloric curve with cutoff effect of the short-distance cutoff stabilization of a low-energy phase energy range much larger than without cutoff high-energy region model without cutoff already for moderate cutoff α 10 5 close analogy to confined models with cutoff no gas-like phase at high energy (no container!) ϑ cutoff-dominated C < 0 gaslike ε

25 density profiles α = ψ(x) x C < 0, high energy C < 0, intermediate energy C > 0, low energy

26 density profiles α = ψ(x) x C < 0, high energy C < 0, intermediate energy C > 0, low energy

27 phase transition? 4 3 ϑ ε α = no cutoff

28 phase transition? 3.5 α = ϑ ε

29 phase transition? α = ψ(x) x C < 0, high energy C < 0, intermediate energy C > 0, low energy

30 summary & outlook summary statistical-mechanical approach to King phenomenological model of star clusters study of the caloric curve short-range cutoff stabilizes a low-energy phase caloric curve analogous to confined self-gravitating systems, without high-energy gas phase low-energy density profile with core-halo structure qualitatively similar to post-core-collapsed clusters and many elliptical galaxies phase transition between King and core-halo structure for small cutoff? preliminary result precise understanding still lacking

31 summary & outlook summary statistical-mechanical approach to King phenomenological model of star clusters study of the caloric curve short-range cutoff stabilizes a low-energy phase caloric curve analogous to confined self-gravitating systems, without high-energy gas phase low-energy density profile with core-halo structure qualitatively similar to post-core-collapsed clusters and many elliptical galaxies phase transition between King and core-halo structure for small cutoff? preliminary result precise understanding still lacking outlook differential formulation using soft-core particles regularization (Yukawa-like)? improvement of numerics, test of robustness against different regularizations and better understanding of the phase transition (work in progress) possible physical origin of effective cutoff? e.g. formation of hard binaries (work in progress) quantitative comparison with observations of collapsed globulars and ellipticals? density profiles does not seem to work for globulars but might work for ellipticals improved models? (starting collaboration with A. Marconi)

32

33 globular clusters platonic self-gravitating systems clusters of stars, almost spherical orbiting (all?) galaxies # of Milky Way globulars in Andromeda galaxy, > 10 4 in giant elliptical M87 finite size r t 50 pc tidal effect of the host galaxy no gas, no dust no dark matter too... very old (age > 10 Gyr) may have undergone collisional relaxation