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Diffusive Particle Acceleration inShocked, Viscous Accretion Disks
Peter A. Becker (GMU)Santabrata Das (IIT)Truong Le (STScI)
Diffusive Particle Acceleration inShocked, Viscous Disks
Talk Outline
o Why do we frequently see outflows from radio-loud AGNs and galactic black-hole candidates?
o How are these outflows produced, powered, and collimated?o Can shock acceleration in the accretion disk play a role?
Diffusive Particle Acceleration inShocked, Viscous Disks
Talk Outline
o How would the presence of a shock affect the dynamical structure and stability of the disk?
o How is particle acceleration in the disk related to supernova-driven cosmic ray acceleration?
o Conclusions and plans for future work
o Advection-dominated inflows are frequently used to model underfed black-hole accretion
o The stability of these models is called into question by large Bernoulli parameters
o The Bernoulli parameter is always positive in ADAF models without outflows – hence these are not self-consistent
o Can shock-powered outflows carry away excess binding energy, and stabilize the disk?
Diffusive Particle Acceleration inShocked, Viscous Disks
Theory: Background
Narayan, et al., ApJ, 476, 49, 1997
o Shocks in luminous X-ray disks will heat the gas, but not accelerate particles – the gas is too dense
o Previous work obtained shock solutions in inviscid ADAF disks
o Hot, tenuous ADAF disks are ideal sites for particle accelerationo When dynamically possible, shocks should be the preferred
solutions according to the Second Law of Thermodynamics
Diffusive Particle Acceleration inShocked, Viscous Disks
Theory: Background
Le & Becker, ApJ, 632, 476, 2005
o Narayan et al. (1997) modeled viscous ADAF disks without shocks
Diffusive Particle Acceleration inShocked, Viscous Disks
Theory: Background
sonic pointsub-Keplerian
Narayan, et al., ApJ, 476, 49, 1997
Diffusive Particle Acceleration inShocked, Viscous Disks
Theory: Background
o Using the Narayan equations, we showed that shocks can occur in viscous disks
Das, Becker, & Le, ApJ, 702, 649, 2009
o Using the Narayan equations, we showed that shocks can occur in viscous disks
Diffusive Particle Acceleration inShocked, Viscous Disks
Theory: Background
Das, Becker, & Le, ApJ, 702, 649, 2009
o But are shocks really present in disks??o General relativistic hydrodynamical models confirm the presence
of shocks – but these models do not consider the consequences for nonthermal particle acceleration
Diffusive Particle Acceleration inShocked, Viscous Disks
Theory: Background
De Villiers & Hawley, ApJ, 599, 1238, 2003Shock forms at funnel wall – centrifugal barrier
Diffusive Particle Acceleration inShocked, Viscous Disks
o ADAF disks contain hot, tenuous gas
o Collisionless plasma allows Fermi acceleration of relativistic particles
o Small fraction of particles get boosted via multiple scatterings with MHD waves
Theory: Background
Diffusive Particle Acceleration inShocked, Viscous Disks
o Most efficient acceleration mechanism is first-order Fermi at a discontinuous shock
o Shock-driven acceleration is augmented by additional acceleration due to bulk compression of the background gas
Theory: Background
Don Ellison, NCSU
Diffusive Particle Acceleration inShocked, Viscous Disks
Theory: Background
Diffusive Particle Acceleration inShocked, Viscous Disks
Mass and Angular Momentum TransportMass and Angular Momentum Transport
Radial momentum conservationRadial momentum conservation
Torque and viscosityTorque and viscosity
Theory: Conservation Equations
Diffusive Particle Acceleration inShocked, Viscous Disks
Keplerian angular velocity in Pseudo-Newtonian potentialKeplerian angular velocity in Pseudo-Newtonian potential
Disk half-thickness and sound speedDisk half-thickness and sound speed
Angular momentum gradientAngular momentum gradient
Theory: Conservation Equations
Diffusive Particle Acceleration inShocked, Viscous Disks
Total energy transportTotal energy transport
Thermal energy densityThermal energy density
Entropy variationEntropy variation
Theory: Conservation Equations
Jumpsat shock
Diffusive Particle Acceleration inShocked, Viscous Disks
Combining energy and angular momentum transport Combining energy and angular momentum transport equations yieldsequations yields
Combining torque and angular momentum transport Combining torque and angular momentum transport equations yieldsequations yields
These are supplemented by the “wind equation”These are supplemented by the “wind equation”
Theory: Differential Equations
Jumpsat shock
Theory: Critical Behavior
Diffusive Particle Acceleration inShocked, Viscous Disks
The Wind Equation can be written in the formThe Wind Equation can be written in the form
The Numerator and Denominator functions areThe Numerator and Denominator functions are
Diffusive Particle Acceleration inShocked, Viscous Disks
Theory: Critical Conditions
Setting Setting NN=0 and =0 and DD=0 yields the critical conditions=0 yields the critical conditions
These must be satisfied simultaneously so that the flows These must be satisfied simultaneously so that the flows passes smoothly through the critical point (or points)passes smoothly through the critical point (or points)
Diffusive Particle Acceleration inShocked, Viscous Disks
Theory: Transport Equation
Transport equationTransport equation
Specific FluxSpecific Flux
Convection-Diffusion EquationConvection-Diffusion Equation
Fermi acceleration Spatial diffusion Source termComoving time derivative Escape term
Localized to shock
High-energy tail ofMaxwellian...
or pre-accel due toreconnection at shock?
Diffusive Particle Acceleration inShocked, Viscous Disks
Theory: Transport Equation
Eigenfunction expansionEigenfunction expansion
ODE for Separation FunctionsODE for Separation Functions
Eigenfunction OrthogonalityEigenfunction Orthogonality
Expansion CoefficientsExpansion Coefficients
gives high-energy slope
Diffusive Particle Acceleration inShocked, Viscous Disks
Theory: Results
Energy jump conditionEnergy jump condition
Velocity jump conditionVelocity jump condition
Injected energy Injected energy EE00 = 0.002 ergs = 0.002 ergs Injection from Maxwellian tail? Or via pre-acceleration Injection from Maxwellian tail? Or via pre-acceleration
due to reconnection at shock?due to reconnection at shock? Injected proton Lorentz factor Injected proton Lorentz factor 1.3 1.3
Diffusive Particle Acceleration inShocked, Viscous Disks
Theory: Resultso M87 parameters:
o Sgr A * parameters:
xsun
sun/yearLjet=xergs/sec
xsun
xsun/yearLjet=xergs/sec
Diffusive Particle Acceleration inShocked, Viscous Disks
Theory: Results
gives high-energy slope
shock
(smooth)
(shock),
(smooth)
o Eigenvalues
Diffusive Particle Acceleration inShocked, Viscous Disks
Theory: Results
o Green’s function inside the disk
Diffusive Particle Acceleration inShocked, Viscous Disks
Theory: Results
o We can compute the number and energy densities by integrating the Green’s function, or by solving independent equations
o Solution accuracy is confirmed
Diffusive Particle Acceleration inShocked, Viscous Disks
Theory: Results
o Outflowing particle spectrum from shock location
L=5.5 x 1043
ergs/secL=5.0 x 1038
ergs/sec
∞=6.3 (proton) =104 (electron)
∞=5.9 (proton) =104 (electron)
o Interesting to compare with equivalent cosmic-ray case
o Supernova-driven plane-parallel shock with compression ratio R has spectral index
o Mean energy of SN-driven shock is
Diffusive Particle Acceleration inShocked, Viscous Disks
Theory: Results
R=2.04,
CR
=5.90
R=2.04,CR
o Disk Stability: Bernoulli parameter
o Shock-driven outflow carries away energy, allowing the remaining gas to accrete
Diffusive Particle Acceleration inShocked, Viscous Disks
Theory: Results
Diffusive Particle Acceleration inShocked, Viscous Disks
o Compare particle pressure with background pressure:
Particle pressure is significant near shock
Theory: Results
Theory: Results
Diffusive Particle Acceleration inShocked, Viscous Disks
o Examine vertical momentum flux:
Diffusive vertical momentum flux drives escape
Diffusive Particle Acceleration inShocked, Viscous Disks
Conclusions
oDiffusive shock acceleration in the disk can power the energetic outflows in M87 and Sgr A*
oGreen's function for the accelerated particles is obtained using eigenfunction expansion – and verified
oAccretion-driven shock acceleration is similar to the standard model of supernova-driven cosmic-ray acceleration
oBoth processes efficiently channel energy into a small population of relativistic particles
oThe spectrum is a relatively flat power-law, much harder than would be expected for a SN-driven shock with the same compression ratio
oThe disk environment plays a key role in enhancing the efficiency of the acceleration process
Diffusive Particle Acceleration inShocked, Viscous Disks
o In two-fluid model, multiple dynamical modes may occur
oPossible occurrence of sharp sub-shocks, with state transitions?
oMay be relevant for Sgr A* X-ray flares – 10 fold increase in
X-rays for a few hours...possible state transition?
Conclusions
Diffusive Particle Acceleration inShocked, Viscous Disks
o In two-fluid model, multiple dynamical modes may occur
oPossible occurrence of sharp sub-shocks, with state transitions?
Conclusions
One shocksolution
3 shocksolutions 2 shock, 1 smooth
solutions
One smoothsolution
Becker & Kazanas, ApJ, 546, 429, 2001
Diffusive Particle Acceleration inShocked, Viscous Disks
o In two-fluid model, multiple dynamical modes may occur
oPossible occurrence of sharp sub-shocks, with state transitions?
Conclusions
1 smooth solution
Becker & Kazanas, ApJ, 546, 429, 2001
Diffusive Particle Acceleration inShocked, Viscous Disks
o In two-fluid model, multiple dynamical modes may occur
oPossible occurrence of sharp sub-shocks, with state transitions?
Conclusions
1 shock solution
Becker & Kazanas, ApJ, 546, 429, 2001
Diffusive Particle Acceleration inShocked, Viscous Disks
o In two-fluid model, multiple dynamical modes may occur
oPossible occurrence of sharp sub-shocks, with state transitions?
Conclusions
2 shock solutions,plus 1 smooth
solution
Becker & Kazanas, ApJ, 546, 429, 2001
Diffusive Particle Acceleration inShocked, Viscous Disks
o In two-fluid model, multiple dynamical modes may occur
oPossible occurrence of sharp sub-shocks, with state transitions?
Conclusions
3 shock solutions
Becker & Kazanas, ApJ, 546, 429, 2001
Diffusive Particle Acceleration inShocked, Viscous Disks
oThe shock stabilizes the disk by reducing the Bernoulli parameter (are flows convectively stable??)
oExcess binding energy is channeled into outflows
oParticle pressure exceeds background pressure near the shock
oParticle source at shock (local reconnection? Pickup from Maxwellian? We need a trans-relativistic model.)
oRelax test-particle approximation and include dynamical effect of particles (“two-fluid” model)
oAllow the outflow to carry away angular momentum
oCompute primary and secondary radiation
o Include the effect of stochastic wave scattering Could be important in CR shocks too, especially for trans-relativistic model
Future Work
Conclusions