Diffusive Particle Acceleration in Shocked, Viscous Accretion Disks Peter A. Becker (GMU) Santabrata...

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