COSMO-05, BONN 2005
NON-EXTENSIVE THEORY OF
DARK MATTER AND GAS
DENSITY DISTRIBUTIONS IN
GALAXIES AND CLUSTERS
M. P. LEUBNER
Institute for Astrophysics
University of Innsbruck, Austria
core

–
halo

leptokurtic long-tailed
NON-GAUSSIAN
DISTRIBUTIONS
PERSISTENT FEATURE OF DIFFERENT
ASTROPHYSICAL ENVIRONMENTS
standard Boltzmann-Gibbs statistics not applicable
 thermo-statistical properties of interplanetary medium
 PDFs of turbulent fluctuations of astrophysical plasmas
 self – organized criticality ( SOC ) - Per Bak, 1985
 stellar gravitational
equilibrium
Empirical fitting relations - DM
Burkert, 95 / Salucci, 00
non-singular
 DM ~
Navarro, Frenk & White, 96, 97
NFW, singular
 DM ~
1
(1  r / rs )(1  r 2 / rs2 )
1
( r / rs )(1  r / rs ) 2
Fukushige 97, Moore 98, Moore 99…
Zhao, 1996
singular
 DM ~
1
(r / rs ) (1  r / rs )  (3 )
Ricotti, 2003: good fits on all scales: dwarf galaxies  clusters
Empirical fitting relations - GAS
Cavaliere, 1976:
single β-model
GAS ~ (1  r / rc ) 3/ 2 
Generalization
convolution of two β-models  double β-model
Aim: resolving β-discrepancy: Bahcall & Lubin, 1994
good representation of hot plasma density distribution
galaxies / clusters
Xu & Wu, 2000, Ota & Mitsuda, 2004
β ~ 2/3 ...kinetic DM energy / thermal gas energy
Dark Matter - Plasma
DM halo  self gravitating system of weakly interacting
particles in dynamical equilibrium
hot gas  electromagnetic interacting high temperature
plasma in thermodynamical equilibrium
any astrophysical system

long-range gravitational / electromagnetic interactions
FROM EXPONENTIAL DEPENDENCE
TO POWER - LAW DISTRIBUTIONS
Standard Boltzmann-Gibbs statistics
based on extensive entropy measure
S B  k B  pi ln pi
pi…probability of the ith microstate, S extremized for equiprobability
 no correlations
Hypothesis: isotropy of velocity directions
 extensivity
Consequence: entropy of subsystems additive  Maxwell PDF
Assumtion:
particles independent from e.o.
microscopic interactions short ranged, Euclidean space time
not applicable accounting for long-range interactions
THUS
 introduce correlations via non-extensive statistics
 derive corresponding power-law distribution
NON - EXTENSIVE STATISTICS
Subsystems A, B:
EXTENSIVE

non-extensive statistics
Renyi, 1955; Tsallis,85
  1 /(1  q )

Sq ( A  B)  Sq ( A)  Sq ( B) 


1
Sq ( A) S q ( B)


PSEUDOADDITIVE NON-EXTENSIVE ENTROPY BIFURKATION
Dual nature
+ tendency to less organized state, entropy increase
- tendency to higher organized state, entropy decrease
generalized entropy (kB = 1, -     )
1/
S   ( pi11/   1)
 long – range interactions / mixing
 quantifies degree of non-extensivity /couplings
 accounts for non-locality / correlations
FROM ENTROPY GENERALIZATION TO PDFs
S … extremizing entropy under conservation of mass and energy
power-law distributions, bifurcation   0

v 
f ch  Bch 1 
2



2
HALO  > 0
N
( )
Bh  1 / 2
 vth 1 / 2 (  1 / 2)
h  vth

 3/ 2
  3/ 2
restriction

normalization
CORE  < 0
N
(  3 / 2)
Bc  1 / 2
 vth  1 / 2 (  1)
different
generalized 2nd moments
Leubner, ApJ 2004
Leubner & Vörös, ApJ 2005
c  vth

  3/ 2
vmax  vth 
thermal cutoff
EQUILIBRIUM OF N-BODY SYSTEM
NO CORRELATIONS
spherical symmetric, self-gravitating, collisionless
Equilibrium via Poisson’s equation
f(r,v) = f(E) … mass distribution
1
  4 G  f ( v 2   )d 3v
2
(1) relative potential Ψ = - Φ + Φ0 , vanishes at systems boundary
Er = -v2/2 + Ψ
and
(2) exponential mass distribution
extensive, independent
ΔΨ = - 4π G ρ
0
v2 / 2  
f ( Er ) 
exp(
)
2 3/ 2
2
(2 )

f(Er)… extremizing BGS entropy, conservation of mass and energy
isothermal, self-gravitating sphere of gas ==
phase-space density distribution of collisionless system of particles
EQUILIBRIUM OF N-BODY SYSTEM
CORRELATIONS
long-range interactions  non-extensive systems
extremize non-extensive entropy,
conservation of mass and energy
 corresponding distribution
0 
v
(2) / 2   
B 

 2 3/ 2 3/ 1
2
( E ) (2B ) 1  (  3 / 2)
f


r
negative κ again energy cutoff v2/2 ≤ κ σ2 – Ψ,
B 
0
( )
(2 2 )3/ 2  3/ 2 (  3/ 2)
integration over v


bifurcation
B 
 1 
   0 1 
2 
  
limit κ =

2


integration limit
0
(  5 / 2)
(2 2 )3/ 2  3/ 2 (  1)
3/ 2 
∞
  0 exp( /  2 )
DUALITY OF EQUILIBRIA AND HEAT CAPACITY
IN NON-EXTENSIVE STATISTICS
(A) two families (κ’,κ) of STATIONARY STATES (Karlin et al., 2002)
non-extensive thermodynamic equilibria, Κ > 0
non-extensive kinetic equilibria, Κ’ < 0
related by κ’ = - κ
limiting BGS state for κ = ∞  self-duality

extensivity
(B) two families of HEAT CAPACITY (Almeida, 2001)
Κ > 0 … finite positive … thermodynamic systems
Κ < 0 … finite negative … self-gravitating systems
non-extensive bifurcation of the BGS κ = ∞, self-dual state
requires to identify
Κ > 0 … thermodynamic state of gas
Κ < 0 … self-gravitating state of DM
NON-EXTENSIVE
SPATIAL DENSITY VARIATION
  4 G 
combine


   0 1 
1/(3/ 2  )


 


1 d  2 d

4 G 


r
1







r 2 dr  dr  0 
 2





d  2 d 
1
 1  d 


1


 
 
2
dr
r dr  3 / 2      dr 
2
2
1 
  2 
3/ 2 
Leubner, ApJ, 2005
4 G  3 / 2   

2
  
2  
 0 
1/  3/ 2  
0
ρ(r) … radial density distribution of spherically symmetric
hot plasma and dark matter
κ = ∞ … BGS selfduality, conventional isothermal sphere
Non-extensive family of density profiles
Non-extensive family of density profiles ρ± (r) , κ = 3 … 10
Convergence to the selfdual BGS solution κ = ∞
Non-extensive DM and GAS density profiles
Non-extensive GAS and DM density
profiles, κ = ± 7 as compared to
Burkert and NFW DM models
and single/double β-models
Integrated mass of non-extensive
GAS and DM components, κ = ± 7
as compared to
Burkert and NFW DM models
and single/double β-models
Comparison with simulations
dark matter (N – body)
gas (hydro)
Kronberger, T. & van Kampen, E.
Mair, M. & Domainko, W.
DM
popular phenomenological: Burkert, NFW
GAS
popular phenomenological: single / double β-models
Solid: simulation (1, 2 ... relaxation times), dashed: non-extensive
SUMMARY
Non-extensive entropy generalization generates a bifurcation
of the isothermal sphere solution into two power-law profiles
The self-gravitating DM component as lower entropy state resides
beside the thermodynamic gas component of higher entropy
The bifurcation into the kinetic DM and thermodynamic gas branch is
controlled by a single parameter accounting for nonlocal correlations
It is proposed to favor the family of non-extensive distributions,
derived from the fundamental context of entropy generalization,
over empirical approaches when fitting observed density profiles
of astrophysical structures