Revisiting the Thermal Stability of Radiation ... - Users' Pages

Revisiting the Thermal Stability ofRadiation-dominated Thin DisksWei-Min GuDepartment of Physics, Xiamen University, ChinaCollaborators: Sheng-Ming Zheng (XMU)Feng Yuan (SHAO)Ju-Fu Lu (XMU)

Outline• Background and Motivation• Standard accretion disk theory confronts the reality• Accretion disk model with toroidal magnetic fields• Thermal instability analysis of the newaccretion disk model• Equations• Stability properties• Summary & Discussion1

I.1 Instability of the SSD• Shakura-Sunyaev Disk(SSD)• Secular Instability•∂ M< 0∂Σ• Thermal Instability+ −∂ ( Q − Q )> 0∂TΣ• The Critical Pointβ = P gasP tot= 0.4LLEdd=. .log M/M Edd•M•M EddR = 5 Rgα M~ 0.05( ⋅0.1 10MBHsun)1−8β = 2 5(5RRg)21162

I.1 “Black hole accretion discs: reality confronts theory”High Luminosity X-ray BinariesL = 0.01~0.5 L EddGierliński & Done,MNRAS, 20043

I.2 Accretion Disks with Toroidal Magnetic Fields• Inclusion of the effect of magnetic fields(e.g., Oda et al., ApJ, 2009)• The ViscosityThe α viscosity law is qualitatively right (Hirose etal. 2009, Pessah et al. 2007), when considering the MRIdriven turbulence.Trϕ= −2αH( Pgas+ Prad+ Pmag)• The vertical hydrostatic equilibrium12Ω 2KΣH = P + P +gasradPmagP mag=2Bϕ / 8π4

I.2 Magnetically Supported Accretion DiskStable?Or not?Abramowiczet al. (1995)Standard model withno magnetic fieldOdaet al. (2009)toroidal magnetic fields incorporated5

II.1 Equations of Local Disk SolutionsPtot=12Ω2KΣH····vertical hydrostatic equilibriumPtotTQrϕ=+visPgas+Prad+Pmag=kmΣTH1•∧= −2αPtotH= − ΩKM (1 − 3/ R)2π− −= Qrad+ QadvBH+13aT24< Bϕ>•2 23Q+ − 16cPradvis= − ΩKTrϕ, Q =2κΣ, − ξ M ΩKH( radQadv=2 )2πR+8π····equation of state····viscosity prescription & angular momentum conservation····energy equation6

Constraint on the relation between B φ and HBHa perturbation in the vertical directionPresumption: duringathermalperturbationR• δ Bϕ δ H , γ isafree parameter which is positive.Bϕ= −γH•γThe integrated formof this relation Φγ≡ BϕH = const holds . true withinathermal timescale.• Alargerγ denotesstronger reactionof B φ onthe perturbationofH(orT).BHthe scale height increases and the meantoroidal magnetic field becomes weakerPartially supported by the simulations ofmagnetic thick disks (Machida et al. 2006)RHB φ7

II.2 Perturbation Analysis• Relations between the first order perturbationδof each physical quantityP1H1Σ= , 1Trϕ,1 P= 0,1H P1 mag,1H= + , = −2γP0H0 Σ0 Trϕ,0P0H0Pmag,0H(t vis >> t th )PP1T1Σ1H1T1mag ,1= βgas⋅(+ − ) + 4(1 − βgas− βmag) ⋅ + βmag⋅P0T0Σ0H0T0Pmag, 010BBϕϕδ= −γβ gas= P gas/Pβ mag= P mag/PHHQQ+vis,1−vis,0T=Trϕ,1rϕ,0P1=P0H+H10,QQ−rad ,1−rad ,0T= 4T10Σ−ΣWe express the perturbation of eachphysical quantity in terms of T 1 /T 0 , i.e.10,QQ−adv,1−adv,0T=Trϕ,1rϕ,00H+ 2 HX1 =X10TF(T10)8

II.2 Thermal Instability Criterion• ⎡⎢⎣∂+( Q − Q∂T−) ⎤⎥⎦Σ>0⎡Q⎢⎣Q+vis,1+vis,0−fadvQ⋅Q−adv,1−adv,0− (1 −fadvQ) ⋅Q−rad ,1−rad ,0⎤⎥⎦TT10>0f /− +adv= QadvQvisThermal instability criterion:∆ =2 − 5β − 4(1 + γ ) β − 6 f + 8 f β + (8 + 4γ)gasmagWhen Δ>0, disk solutions are thermally unstable.advadvgasfadvβmag9

II.3 Thermal Instability PropertiesΔ < 0Δ > 0Critical mass accretion rates of the magnetized thin disk with different γ.The curves correspond to the states where Δ=0. The disk is thermallystable when the ratio of magnetic to total pressure is over 0.25.10

II.3 S-curves of Local Disk Solutions~ L Eddstandard solutionswithout magnetic fieldS-curves with different values ofΦ 1= B ϕHOur model may help raise the critical accretion rate (or luminosity)for the thermal instability, which may be consistent with theobserved X-ray sources with high luminosities.11

III. Summary• By assuming that the field responds negatively to a positivetemperature perturbation, we find that the critical massaccretion rate increases significantly. This accounts for thestability of the observed sources with high luminosities.• Our model also presents a possible explanation as to why onlyGRS 1915+105 seems to show thermally unstable behavior.This peculiar source holds the highest accretion rate (orluminosity) among the known high state sources.12

III. Discussion• Compared with simulation: two problems• Requiring stronger magnetic fields (0.1 < β mag < 0.25)Simulations: β mag < 0.1 (e.g., Hirose et al. 2009)• Magnetic field reversal: beyond explanation by our modelOur paper: ApJ, 732, 52 (05/2011)13