NON-EXTENSIVE THEORY OF DARK MATTER AND GAS DENSITY DISTRIBUTIONS IN GALAXIES AND CLUSTERS

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NON-EXTENSIVE THEORY OFDARK MATTER AND GAS
DENSITY DISTRIBUTIONS IN GALAXIES AND
CLUSTERS
COSMO-05, BONN 2005

• M. P. LEUBNER
• Institute for Astrophysics
• University of Innsbruck, Austria

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c o r e h a l o ?
? 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
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Empirical fitting relations - DM
Burkert, 95 / Salucci, 00 non-singular
Navarro, Frenk White, 96, 97 NFW, singular
Fukushige 97, Moore 98, Moore 99
Zhao, 1996 singular
Ricotti, 2003 good fits on all scales dwarf
galaxies ? clusters
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Empirical fitting relations - GAS
Cavaliere, 1976 single ß-model
Generalization convolution of two ß-models ?
double ß-model Aim resolving ß-discrepancy
Bahcall Lubin, 1994 good representation of hot
plasma density distribution galaxies /
clusters Xu Wu, 2000, Ota Mitsuda, 2004 ß
2/3 ...kinetic DM energy / thermal gas energy
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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
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FROM EXPONENTIAL DEPENDENCETO POWER - LAW
DISTRIBUTIONS
Standard Boltzmann-Gibbs statistics based on
extensive entropy measure piprobability of
the ith microstate, S extremized for
equiprobability
Assumtion particles independent from e.o.
? no correlations Hypothesis isotropy of
velocity directions ? extensivity Consequence
entropy of subsystems additive ? Maxwell
PDF 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

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NON - EXTENSIVE STATISTICS

• Subsystems A, B EXTENSIVE
• ? ?
• non-extensive statistics
• Renyi, 1955 Tsallis,85
• 
  ? ? ?
• 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/? ? long range
  interactions / mixing
• ? quantifies degree of non-extensivity
  /couplings
• ? accounts for non-locality / correlations

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FROM ENTROPY GENERALIZATION TO PDFs
S? extremizing entropy under conservation of
mass and energy
power-law distributions, bifurcation ? ? 0
HALO
CORE
? gt 0
? lt 0
normalization
different generalized 2nd moments
Leubner, ApJ 2004 Leubner Vörös, ApJ 2005
restriction
thermal cutoff
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EQUILIBRIUM OF N-BODY SYSTEM NO CORRELATIONS

• spherical symmetric, self-gravitating,
  collisionless
• Equilibrium via Poissons equation
• f(r,v) f(E) mass distribution

(1) relative potential ? - F F0 , vanishes
at systems boundary Er -v2/2 ? and
?? - 4p G ? (2) exponential mass
distribution extensive,
independent 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
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EQUILIBRIUM OF N-BODY SYSTEM CORRELATIONS

• long-range interactions ? non-extensive systems
• extremize non-extensive entropy,
• conservation of mass and energy
• ? corresponding distribution

negative ? again energy cutoff v2/2 ? s2
?, integration limit
bifurcation
integration over v
limit ? 8
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DUALITY OF EQUILIBRIA AND HEAT CAPACITY IN
NON-EXTENSIVE STATISTICS

• (A) two families (?,?) of STATIONARY STATES
  (Karlin et al., 2002)
• non-extensive thermodynamic equilibria, ? gt 0
• non-extensive kinetic equilibria, ? lt 0
• related by ? - ?
• limiting BGS state for ? 8 ? self-duality
  ? extensivity
• (B) two families of HEAT CAPACITY (Almeida, 2001)
• ? gt 0 finite positive thermodynamic systems
• ? lt 0 finite negative self-gravitating
  systems

non-extensive bifurcation of the BGS ? 8,
self-dual state requires to identify ? gt
0 thermodynamic state of gas ? lt 0
self-gravitating state of DM
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NON-EXTENSIVE SPATIAL DENSITY VARIATION
combine
Leubner, ApJ, 2005
?(r) radial density distribution of spherically
symmetric hot plasma and dark
matter ? 8 BGS selfduality, conventional
isothermal sphere
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Non-extensive family of density profiles
Non-extensive family of density profiles ? (r) ,
? 3 10 Convergence to the selfdual BGS
solution ? 8
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Non-extensive DM and GAS density profiles

• Non-extensive GAS and DM density
• profiles, ? 7 as compared to
• Burkert and NFW DM models
• and single/double ß-models

• Integrated mass of non-extensive
• GAS and DM components, ? 7
• as compared to
• Burkert and NFW DM models
• and single/double ß-models

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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 ß-models
• Solid simulation (?1, ?2 ... relaxation times),
  dashed non-extensive

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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