caloric curve of King models - uma.ensta-paris.fr · introduction self-gravitating systems King model statistical mechanicssummary & outlook caloric curve of King models with a short-distance

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 & 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


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]


equilibrium statistical mechanics of self-gravitating systems


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]


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


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]


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


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


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


caloric curve

ε

ϑ


question

what about real self-gravitating systems?no box, no thermal velocity distribution


globular clusters

ω Cen – the largest Milky Way globular cluster


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]


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]


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


“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


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]


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


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


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!)



!

"!150 !100 !50 0

0

5

10

15

α = 10−5







!

"!2.5 !2.0 !1.5 !1.0

0.5

1.0

1.5

−−− α = 10−3 − · − α = 10−5 ——– no cutoff



high-energy region ' model without cutoff

already for moderate cutoff α . 10−5








ε

ϑ








ε

ϑ



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


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



phase transition?

!

"!8 !6 !4 !2

1

2

3

4

− · − α = 7.5× 10−6 ——– no cutoff


phase transition?

α = 5× 10−6

!

"!6 !5 !4 !3 !2 !1

1.0

1.5

2.0

2.5

3.0

3.5


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



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)


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)



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

Documents

caloric curve of King models - uma.ensta-paris.fr · introduction self-gravitating systems King model statistical mechanicssummary & outlook caloric curve of King models with a short-distance