Non-Linear Theory of Particle Acceleration at Astrophysical Shocks

Transcription

1 Non-Linear Theory of Particle Acceleration at Astrohysical Shocks Pasquale Blasi INAF/Arcetri Astrohysical Observatory, Firenze,, Italy TeV Worksho, Madison, August 6

2 First Order Fermi Acceleration: a Primer x x Test Particle dµ Pu, The article is always advected back to the shock ( µ µ ) dµ Return Probability from UP The article may either diffuse back to the shock or be advected downstream d µ Pd, ( µ µ ) dµ < Return Probability from DOWN< N ( E ) N γ γ logp logg r 3 P Total Return Probability from DOWN G Fractional Energy Gain er cycle r Comression factor at the shock

3 The Return Probability and Energy Gain for Non-Relativistic Shocks U Down At zero order the distribution of (relativistic) articles downstream is isotroic: f(µ)f Return Probability Escaing Flux/Entering Flux Φ out f ( u +µ ) dµ ( u )² Φ i f ( u +µ ) dµ ( +u )² P return ( u ) ² 4u ( + u ) ² Newtonian Limit Close to unity for u <<! The extraolation of this equation to the relativistic case would give a return robability tending to zero! The roblem is that in the relativistic case the assumtion of isotroy of the function f loses its validity. G E E 4 3 ( ) u u SPECTRAL SLOPE log P γ log G return log 4 3 ( 4u ) 3 ( ) r u u

4 The Diffusion-Convection Equation: A more formal aroach ADIABATIC COMPRESSION OR SHOCK COMPRESSION f du f f f + Q( x,,t ) D u + x 3 dx t x x INJECTION TERM ADVECTION WITH THE FLUID SPATIAL DIFFUSION The solution is a ower law in momentum 3u N inj f ( ) u u 4πinj 3u u u inj The sloe deends ONLY on the comression ratio (not on the diffusion coef.) Injection momentum and efficiency are free aram.

5 The Need for a Non-Linear Theory The relatively Large Efficiency may break the Test Particle Aroximation What haens then? Cosmic Ray Modified Shock Waves Non Linear effects must be invoked to enhance the acceleration efficiency (roblem with Emax) Self-Generation of Magnetic Field and Magnetic Scattering

6 Going Non Linear: Part I Particle Acceleration in the Non Linear Regime: Shock Modification

7 Why Did We think About This? Divergent Energy Sectrum CR ( E) ln( E E ) E deen / max min At Fixed energy crossing the shock front ρu tand at fixed efficiency of acceleration there are values of Pmax for which the integral exceeds ρu (absurd!)

8 If the few highest energy articles that escae from ustream carry enough energy, the shock becomes dissiative, therefore more comressive If Enough Energy is channelled to CRs then the adiabatic index changes from 5/3 to 4/3. Again this enhances the Shock Comressibility and thereby the Modification

9 The Basic Physics of Modified Shocks Shock Front Undisturbed Medium v subshock Precursor ( x ) u ( x ) ρ u ρ u ρ ( ) ( ) ( ) ( ) Conservation of Mass f t + P x g, D ρ f x x u x + P f u + x g 3 x + P CR du f dx x + Q Conservation of Momentum ( x,, t ) Equation of Diffusion Convection for the Accelerated Particles

10 Main Predictions of Particle Acceleration at Cosmic Ray Modified Shocks Formation of a Precursor in the Ustream lasma The Total Comression Factor may well exceed 4. The Comression factor at the subshock is <4 Energy Conservation imlies that the Shock is less efficient in heating the gas downstream The Precursor, together with Diffusion Coefficient increasing with -> NON POWER LAW SPECTRA!!! Softer at low energy and harder at high energy

11 Sectra at Modified Shocks Very Flat Sectra at high energy Amato and PB (5)

12 Efficiency of Acceleration (PB, Gabici & Vannoni (5)) This escaes out To UPSTREAM Note that only this Flux ends u DOWNSTREAM!!!

13 Suression of Gas Heating max 3 m c Rankine-Hugoniot Increasing Pmax max 5 mc M, 5, PB, Gabici & Vannoni (5) The suressed heating might have already been detected (Hughes, Rakowski & Decourchelle ())

14 Going Non Linear: Part II Coing with the Self-Generation of Magnetic field by the Accelerated Particles

15 The Classical Bell (978) - Lagage-Cesarsky (983) Aroach Basic Assumtions:. The Sectrum is a ower law. The ressure contributed by CR's is relatively small 3. All Accelerated articles are rotons The basic hysics is in the so-called streaming instability: of articles that roagates in a lasma is forced to move at seed smaller or equal to the Alfven seed, due to the excition of Alfven waves in the medium. Excitation Of the C V instability A

16 Pitch angle scattering and Satial Diffusion The Alfven waves can be imagined as small erturbations on to of a background B-field The equation of motion of a article in this field is B B B r d dt Ze c r v r r ( B + ) B In the reference frame of the waves, the momentum of the article remains unchanged in module but changes in direction due to the erturbation: dµ dt Zev ( ) / µ B cos ( Ω kvµ ) c [ t + ψ] Ω ZeB / mcγ D ( ) 3 vλ 3 cr F ( ) ( ) L F( ( k)) k( δb / B) The Diffusion coeff reduces To the Bohm Diffusion for Strong Turbulence F()~

17 Maximum Energy a la Lagage-Cesarsky In the LC aroach the lowest diffusion coefficient, namely the highest energy, can be achieved when F()~ and the diffusion coefficient is Bohm-like. D ( ) 3 vλ 3 cr F ( ) ( ) L ( E) D 6 τ acc 3.3 EGeV B µ u shock u sec For a life-time of the source of the order of yr, we easily get E max ~ 4-5 GeV We recall that the knee in the CR sectrum is at 6 GeV and the ankle at ~3 9 GeV. The roblem of accelerating CR's to useful energies remains... BUT what generates the necessary turbulence anyway? Wave growth HERE IS THE CRUCIAL PART! F t F + u x σf ΓF Bell 978 Wave daming

18 Standard calculation of the Streaming Instability (Achterberg 983) R,L χ + ω k c ( ) ( ) µ f µ ω kv + f k v ω ± Ω' v µ d µ d ω e π χ L R, 4 µ There is a mode with an imaginary art of the frequency: CR s excite Alfven Waves resonantly and the growth rate is found to be: z P CR v A σ

19 Maximum Level of Turbulent Self- Generated Field F F + u σf t x Stationarity u F z v A P CR z Integrating δb B M P A CR ρu >> Breaking of Linear Theory For tyical arameters of a SNR one has δb/b~.

20 Non Linear DSA with Self-Generated Alfvenic turbulence (Amato & PB 6) We Generalized the revious formalism to include the Precursor! We Solved the Equations for a CR Modified Shock together with the eq. for the self-generated Waves We have for the first time a Diffusion Coefficient as an outut of the calculation

21 Sectra of Accelerated Particles and Sloes as functions of momentum

22 Magnetic and CR Energy as functions of the Distance from the Shock Front Amato & PB 6

23 Suer-Bohm Diffusion Amato & PB 6 Sectra Sloes Diffusion Coefficient

24 How do we look for NL Effects in DSA? Curvature in the radiation sectra (electrons in the field of rotons) (indications of this in the IR-radio sectra of SNRs by Reynolds) Amlification in the magnetic field at the shock (seen in Chandra observations of the rims of SNRs shocks) Heat Suression downstream (detection claimed by Rakowski & Decourchelle ) (detection claimed by Hughes, All these elements are suggestive of very efficient CR acceleration in SNRs shocks. BUT similar effects may be exected in all accelerators with shock fronts

25 A few notes on NLDSA The sectrum observed at the source through non- thermal radiation may not be the sectrum of CR s The sectrum at the source is most likely concave though a convolution with max (t) ) has never been carried out The sectrum seen outside (ustream infinity) is eculiar of NL-DSA: F(E) F(E) E max E

26 CONCLUSIONS Particle Acceleration at shocks occurs in the non-linear regime: these are NOT just corrections, but rather the reason why the mechanism works in the first lace The efficiency of acceleration is quite high Concave sectra, heating suression and amlified magnetic fields are the main symtoms of NL-DSA More work is needed to relate more strictly these comlex calculations to the henomenology of CR accelerators The future develoments will have to deal with the crucial issue of magnetic field amlification in the fully non-linear regime, and the generalization to relativistic shocks

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28 Exact Solution for Particle Acceleration at Modified Shocks Basic Equations f for Arbitrary Diffusion Coefficients ρ ( x ) u ( x ) ρ u ρ( x) u( x) + Pg ( x) + Pc ( x) ρu + Pg, (,x) f ( ) (,x) f ( ) (,x) du f (,x) D,x u x + t x 4π d 3 x ( x) v( ) f (,x) P c 3 x 3 dx Amato & Blasi (5) + Q ( x,,t) f ( x, ) f ( ) ( ) u( x' ) ( ) dx' D x', q ex 3 q( ) dlnf dln f ( ) R tot 3R U tot ( ) U ( ' ) ( ' ) ηn d' 3Rtot ex 3 4π ' RtotU inj DISTRIBUTION FUNCTION AT THE SHOCK

29 F() η 3π 4 3 ξ ( R ) / sb ξ e Recie for Injection INJECTED PARTICLES It is useful to introduce the equations in dimensionless form: ξ c γ ( x) + U ( x) U ( x) ξ ( x) γm γm c P ρ c ( x) u ξ c 3 ( x) d v( ) f ( ) 3ρ 4π u ex dx' U x ( x' ) ( x' ) x ( x) 3D q (, x) ( ) u One should not forget that the solution still deends on f which in turn deends on the function U ( ) dxu ( x) x ( x) ex dx' Differentiating with resect to x we get... U x ( x' ) ( x' )

30 dξ c dx λ( x) ξ ( x) U( x) c λ ( x) d d 3 x 3 ( ) ( ) ( ) U( x' ) v f ex dx' x x ( x' ) v ( ) f ( ) ex dx' U x ( x' ) ( x' ), The function ξ c (x) has always the right boundary conditions at the shock and at infinity but ONLY for the right solution the ressure at the shock is that obtained with the f that is obtained iteratively (NON LINEARITY)