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introduction self-gravitating systems King model statistical mechanics summary & outlook
caloric curve of King modelswith a short-distance cutoff on the interactions
Lapo Casetti
Dipartimento di Fisica e Astronomia & CSDC, Universita di Firenze, ItalyINFN, 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, 061105 (2012)
introduction self-gravitating systems King model statistical mechanics summary & outlook
introduction & motivation
self-gravitating systems: natural examples of long-range systems
truly long-range interactions, unscreenedalmost ideal samples: globular clusters, elliptical galaxies...seemingly obvious testing ground for theoretical predictions
standard equilibrium statistical mechanics does not work!
short-distance singularityescape of particles
idealized systems & toy models
very interesting theoretical featuresclustering, phase transitions, ensemble inequivalence, Cv < 0...
encoded in the caloric curve T (E)relevant for real systems?
first step
caloric curve analysis of observationally probed models (King)introduction of a short-distance cutoff
introduction self-gravitating systems King model statistical mechanics summary & outlook
equilibrium statistical mechanics of self-gravitating systems
H (r1, . . . , rN , v1, . . . , vN ) =m
2
N∑i=1
v2i − Gm2
N∑i=1
N∑j>i
1
|ri − rj |
short-distance singularity =⇒ no true equilibrium statemetastable states may still exist (local entropy maxima)
“easy” solution: regularization via short-distance cutoff (more soon)
unbounded space =⇒ escape of particlesfinite escape velocity incompatible with maxwellian velocity distribution
stationary maxwellian distribution in unbounded space =⇒ infinite mass
solution (not so easy...): put the system in a boxor consider an expanding background, but that’s another story
regularized & confined =⇒ equilibrium exists
[Kiessling, Chavanis]
introduction self-gravitating systems King model statistical mechanics summary & outlook
equilibrium statistical mechanics of self-gravitating systems
introduction self-gravitating systems King model statistical mechanics summary & outlook
isothermal sphere
forget about regularizationit 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
f (r , v) = C e−βv2/2e−βϕ(r)
%(r) =
∫dv f (r , v)
∇2ϕ(r) = 4πG%(r)
that is, for given β and %c = %(0),
d2ϕ(r)
dr2+
2
r
dϕ(r)
dr= 4πG%c e−β[ϕ(r)−ϕ(0)]
[Antonov, Lynden-Bell & Wood, Padmanabhan, Chavanis]
introduction self-gravitating systems King model statistical mechanics summary & outlook
isothermal sphere: caloric curve
energy & temperature (kB = 1)
K =1
2
∫dr dv v2f (r , v) =
3
2β=
3T
2
U = −G
2
∫dr dv dr′ dv′
f (r , v)f (r ′, v ′)
|r − r′| =1
2
∫dr %(r)ϕ(r)
E = K + U
energy unit GM2/R
M =
∫dr %(r)
dimensionless energy & temperature
ε =RE
GM2
ϑ =RT
GM2
introduction self-gravitating systems King model statistical mechanics summary & outlook
isothermal sphere: caloric curve
!
"!0.2 0.0 0.2 0.4
0.4
0.5
0.6
0.7
minimal energy & temperature
εmin ' −0.335 ϑmin ' 0.4
ε < εmin =⇒ “gravothermal catastrophe”
[Antonov, Lynden-Bell & Wood, Padmanabhan, Chavanis]
introduction self-gravitating systems King model statistical mechanics summary & outlook
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 principleself-gravitating fermions [Chavanis & Ispolatov]
classical particles: new interactions at small scalesstars & planets have a finite size!
many possible implementationshard-core/soft-core particles, truncated/softened potential...
V (ri , rj ) = − Gm2√∣∣ri − rj
∣∣2 + a
[a] = `2
introduction self-gravitating systems King model statistical mechanics summary & outlook
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 S2
[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 + confinementin a box or in a compact configuration space
introduction self-gravitating systems King model statistical mechanics summary & outlook
caloric curve
ε
ϑ
gaslikeC < 0cutoff-dominated
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
introduction self-gravitating systems King model statistical mechanics summary & outlook
caloric curve
ε
ϑ
gaslikeC < 0cutoff-dominated
question
what about real self-gravitating systems?no box, no thermal velocity distribution
introduction self-gravitating systems King model statistical mechanics summary & outlook
globular clusters
ω Cen – the largest Milky Way globular cluster
introduction self-gravitating systems King model statistical mechanics summary & outlook
King model
phenomenological & stationary mean-field-like model
spherically symmetric cluster of equal starsglobulars & open clusters & elliptical galaxies...
assumptions
1 single particle distribution function f (r , v)
2 %(r) ≡ 0 if r ≥ rt
3 relaxed system =⇒ f (v) as close to thermal equilibrium as it can be
4 constraint: |v | ≤ v e(r) = escape velocity
[King 1966]
introduction self-gravitating systems King model statistical mechanics summary & outlook
King model
f (r , v) =
{C e−2βϕ(r)
[e−βv2 − e−βv2
e (r)]
if v2 < v2e(r)
0 otherwise
%(r) =
∫dv f (r , v)
C ←→ M
β 6= T−1
v2e(r) = −2ϕ(r)
ϕ(rt ) = 0
for the moment no short-distance cutoff on the gravitational interaction...
∇2ϕ(r) = 4πG%(r)
...and go on as in the isothermal spherert →∞ (M →∞) =⇒ King→ isothermal sphere
[King 1966]
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
introduction self-gravitating systems King model statistical mechanics summary & outlook
“statistical mechanics” of King models
energy & temperature (kB = 1)
K =1
2
∫dr dv v2f (r , v) =
3T
2
U = −G
2
∫dr dv dr′ dv′
f (r , v)f (r ′, v ′)
|r − r′| =1
2
∫dr %(r)ϕ(r)
E = K + U
energy unit GM2/rt
dimensionless energy & temperature
ε =rt E
GM2
ϑ =rt T
GM2
introduction self-gravitating systems King model statistical mechanics summary & outlook
caloric curve of King model without cutoff
!
"!2.0 !1.5 !1.0 !0.5
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
virial theorem for purely gravitational interactions
K = −E =⇒ ϑ = −2
3ε
energy & temperature are bounded
ε ∈ [−2.13,−0.60] ϑ ∈ [0.40, 1.42]
introduction self-gravitating systems King model statistical mechanics summary & outlook
caloric curve of King model without cutoff
!0
"!2.0 !1.5 !1.0 !0.5
0.5
1.0
1.5
2.0
increasing ϕ(r = 0) data points reach εmin then go back and forth in a“collapsed spiral” pattern
plotting e.g. ϑ0 = ϑ(r = 0) the spiral pattern opens up
introduction self-gravitating systems King model statistical mechanics summary & outlook
switching on the cutoff
short-range cutoff
1
|r − r′| −→1√
|r − r′|2 + a
all definitions of f , %, ϕ, U, K , E , T formally as before
same adimensionalization: dimensionless cutoff α
α =a
rt2
no analogue of Poisson equation =⇒ no differential formulation
self-consistent iterative procedureconceptually straightforward, numerically less efficient
“reasonable” cutoff values
star size < cutoff length < average interstellar separation
10−9 .√α . 5× 10−2
introduction self-gravitating systems King model statistical mechanics summary & outlook
caloric curve with cutoff
!
"!15 !5!10 0
0.0
0.5
1.0
1.5
α = 10−3
effect of the short-distance cutoff
stabilization of a low-energy phaseenergy range much larger than without cutoff
high-energy region ' model without cutoffalready for moderate cutoff α . 10−5
close analogy to confined models with cutoffno gas-like phase at high energy (no container!)
introduction self-gravitating systems King model statistical mechanics summary & outlook
caloric curve with cutoff
!
"!150 !100 !50 0
0
5
10
15
α = 10−5
effect of the short-distance cutoff
stabilization of a low-energy phaseenergy range much larger than without cutoff
high-energy region ' model without cutoffalready for moderate cutoff α . 10−5
close analogy to confined models with cutoffno gas-like phase at high energy (no container!)
introduction self-gravitating systems King model statistical mechanics summary & outlook
caloric curve with cutoff
!
"!2.5 !2.0 !1.5 !1.0
0.5
1.0
1.5
−−− α = 10−3 − · − α = 10−5 ——– no cutoff
effect of the short-distance cutoff
stabilization of a low-energy phaseenergy 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 cutoffno gas-like phase at high energy (no container!)
introduction self-gravitating systems King model statistical mechanics summary & outlook
caloric curve with cutoff
effect of the short-distance cutoff
stabilization of a low-energy phaseenergy range much larger than without cutoff
high-energy region ' model without cutoffalready for moderate cutoff α . 10−5
close analogy to confined models with cutoffno gas-like phase at high energy (no container!)
ε
ϑ
gaslikeC < 0cutoff-dominated
introduction self-gravitating systems King model statistical mechanics summary & outlook
caloric curve with cutoff
effect of the short-distance cutoff
stabilization of a low-energy phaseenergy range much larger than without cutoff
high-energy region ' model without cutoffalready for moderate cutoff α . 10−5
close analogy to confined models with cutoffno gas-like phase at high energy (no container!)
ε
ϑ
gaslikeC < 0cutoff-dominated
introduction self-gravitating systems King model statistical mechanics summary & outlook
density profiles
α = 10−3
x
!(x
)
10!6
10!4
10!2
100
102
104
0.01 0.1 1
——– C < 0, high energy −−− C < 0, intermediate energy − · − C > 0, low energy
introduction self-gravitating systems King model statistical mechanics summary & outlook
density profiles
α = 10−5
x
!(x
)
10!6
10!4
10!2
100
102
104
106
108
0.001 0.01 0.1 1
——– C < 0, high energy −−− C < 0, intermediate energy − · − C > 0, low energy
introduction self-gravitating systems King model statistical mechanics summary & outlook
phase transition?
!
"!8 !6 !4 !2
1
2
3
4
− · − α = 7.5× 10−6 ——– no cutoff
introduction self-gravitating systems King model statistical mechanics summary & outlook
phase transition?
α = 5× 10−6
!
"!6 !5 !4 !3 !2 !1
1.0
1.5
2.0
2.5
3.0
3.5
introduction self-gravitating systems King model statistical mechanics summary & outlook
phase transition?
α = 5× 10−6
x
!(x
)
10!6
10!4
10!2
100
102
104
106
108
0.001 0.01 0.1 1
——– C < 0, high energy −−− C < 0, intermediate energy − · − C > 0, low energy
introduction self-gravitating systems King model statistical mechanics summary & outlook
summary & outlook
summary
statistical-mechanical approach to King phenomenological model of star clustersstudy of the caloric curve
short-range cutoff stabilizes a low-energy phasecaloric curve analogous to confined self-gravitating systems, without high-energy gas phase
low-energy density profile with core-halo structurequalitatively 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)
introduction self-gravitating systems King model statistical mechanics summary & outlook
summary & outlook
summary
statistical-mechanical approach to King phenomenological model of star clustersstudy of the caloric curve
short-range cutoff stabilizes a low-energy phasecaloric curve analogous to confined self-gravitating systems, without high-energy gas phase
low-energy density profile with core-halo structurequalitatively 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)
introduction self-gravitating systems King model statistical mechanics summary & outlook
globular clusters
“platonic” self-gravitating systems
clusters of 105 ÷ 106 stars, almost sphericalorbiting (all?) galaxies
# of Milky Way globulars & 150500 in Andromeda galaxy, > 104 in giant elliptical M87
finite size rt . 50 pctidal effect of the host galaxy
no gas, no dustno dark matter too...
very old (age > 10 Gyr)may have undergone “collisional” relaxation