2004/3/1 3 N 1 Antonov Problem & Quasi-equilibrium State in N-body N Systems A. Taruya (RESCEU, Univ.Tokyo) M. Sakagami (Kyoto Univ.)
2 Antonov problem N-body study of quasi-attractivity M15
T Antonov Problem 1/ 2 free ~ ( Gρ) Vest.Leningrad Gros.. Univ. 7 (1962) 135 (English trans.) IAU sympo.. 113 (1985) 525 Padmanabhan, ApJS 71 (1989) 651 virialized T relax ( N / 8ln N) ~ T (fully relaxed) free 3 ( )
Idealized Setup 4 re N Self-gravitating N-body system re re rc re E, MM Small re Small D=rc/re Large D Large re
Statistical Mechanical Analysis 5 = M,E S BG = d 3 xd 3 v f(x, v) ln f(x, v) βε f ( ε ) e ; ε = v + Φ ( ) f (x,v): 1 2 x 2 λ= ree/gm 2 stable unstable Critical density:709 D= ρc/ ρe >709
6 Historical Remarks 1962 1968 Antonov Lynden-Bell & Wood 1980 Lynden-Bell & Eggleton Cohn Self-similar core-collapse found by Fokker-Planck simulation 1983 Sugimoto & Bettwieser 1996 Makino Confirmed by N-body simulation
Main Focus 7 Self-similar collapse Gravothermal oscillation equal-mass component self-similar similar collapse? Fokker-Planck simulation by Cohn (1980): Self-similar collapse 1-
8 Tsallis entropy S q 1 3 3 = d q 1 xd v ( f q f ) f ε n 3/ 2 1 3 ) [ Φ0 ε ] ; n = + q 1 2 ( 1- A.T & Sakagami, Physica A 307 (2002) 185; 318 (2003) 387; 322 (2003) 285 ( t >> Trelax ) Reality of stellar polytropes & A.T & Sakagami, Phys.Rev.Lett. 90 (2003) 181101
9 Equilibrium Sequence of Stellar Polytropes n=6 n > 5 D > Dcrit stable unstable 2 δ S >0
10 Summary of N-body N Study n (run C1)
11 Result from run A 4 (n=3,d=10 ) Density profile Phase-space space distribution function Time evolution of Lagrangian radii ρ(r) f(ε) core-collapse takes place at T~450 r ε Fitting to the stellar polytrope is quite good until T~300. scaled time 1 v 2 + Φ ( x ) 2
N-body Study of Quasi-attractivity attractivity 12 AT & Sakagami (2004) in prep. Simulation setup Force calculation: GRAPE-6 # of particles : N=8,192 Softening length: ε = 1/Ν 4/N Perfectly reflecting Adiabatic wall G=M=re=1 =1 re
Initial Condition 13 A family of stellar models with cusped density profile: 1 ( r + ρ ( r) 3 η 1 + η r a) Tremaine et al. AJ 107 (1994) 634 f( ) =1.5 =2 =3 2 σ v(r) =1 =1 f ( ε) ( ε (3+ η)/2/( η 1) * ε) =1.5 =2 =3 a/re=0.5 = v 2 / 2 + Φ( r) a/re=0.5 r
Survey Results 14 Fitting failed Fitting to polytrope is good Final state is isothermal Previous work Quasi-attractive behaviors appear when 1< η & 0.3 < λ < 0.8
Cases with a/re=0.5 (1) =1 singular isothermal ( ) 15 =1.5 =3 ( )
Cases with a/re=0.5 (2) 16 =1 ( ) =1.5 =3
Physical Reason 17 Gravothermal expansion Local relaxation time Heat flows inward and flat core is formed. (Negative specific heat) Timescale becomes shorter for denser region. t r 3 σ v = 0.065 2 G mρ ln Λ Power-law feature of f(ε) Polytropic behavior of ρ(r), σ v(r) are rapidly attained. =1.5 Degree of this behavior depends on the amount of heat-flow (a, η)
18 S BG = d 3 xd 3 v f(x, v) ln f(x, v) SBG (SBG, D)
19 Condition of quasi-attractive behavior in N-body system Power-law type distribution naturally arises when the sufficient amount of the inward heat-flow is supplied. For more rigorous argument, long-range attractivity (negative specific heat) A large N-body N simulation (N=16k~32k) with a more sophisticated N-body N code Analytic treatment based on the Fokker-Planck model (now in progress)