Pasquale Blasi INAF/Arcetri Astrophysical Observatory 4th School on Cosmic Rays and Astrophysics UFABC - Santo André - São Paulo – Brazil

Acceleration of charged particles The presence of non-thermal particles is deduced in a myriad of situations in Nature (from the solar wind to the AGNs, from SNRs to GRBs, from pulsars to m QSO) PARTICLE ACCELERATION BUT usually (through not always) in the same regions there is evidence for Thermalized plasmas, therefore the questions arises WHICH PROCESSES DETERMINE WHETHER A PARTICLE IS GOING TO BE ACCELERATED OR RATHER BE THERMALIZED ?

Acceleration of charged particles All acceleration processes we are aware of are of electro-magnetic nature – but magnetic fields DO NOT MAKE WORK on charged particles WHAT IS THE ORIGIN OF THE ELECTRIC FIELDS THAT PRODUCE ACCELERATION? ACCELERATION MECHANISMS ARE CLASSIFIED ACCORDING WITH THE ORIGIN OF THE ELECTRIC FIELDS REGULAR ACCELERATION STOCHASTIC ACCELERATION

REGULAR ACCELERATION Large mean scale electric fields are produced on some spatial scale L reg DIFFICULT TO CREATE NET ELECTRIC FIELD IN ASTROPHYSICS BECAUSE OF HIGH CONDUCTIVITY, BUT SOME EXCEPTIONS: Unipolar Inductor Magnetic reconnection

STOCHASTIC ACCELERATION The stochastic electric field may result from random fluctuations on a typical scale L st but with random orientations so that on average the field vanishes. If both regular and stochastic acceleration occur: Most astrophysical acceleration processes belong to this class

2 nd order Fermi Acceleration (Fermi, 1949) PROBABILITY OF ENCOUNTER LOSSES AND GAINS ARE PRESENT BUT DO NOT COMPENSATE EXACTLY

WHY WOULD MAGNETIC CLOUDS ACCELERATE PARTICLES? WHERE ARE THE ELECTRIC FIELDS? In the Fermi example the electric fields are induced by the motion of the magnetized moving clouds In reality we need to go back to our example of motion of a charged particles in a group of Alfven waves…what if we do not sit in the reference frame of the waves? As usual:

Where you should recall that: Therefore: The time for diffusion in momentum space is then: DIFFUSION IN SPACE IMPLIES THAT A (2 nd ORDER) DIFFUSION IN MOMENTUM TAKES PLACE (ACCELERATION)

A PRIMER ON SHOCK WAVES For σ~ cm 2 and density n~1 cm -3 the typical interaction length is ~3 Mpc >> than the typical size of astrophysical objects and even Larger than the Galaxy!!! COLLISIONLESS SHOCKS UPSTREAMDOWNSTREAM U 1 U 2 -∞ 0 +∞

STATIONARY SHOCKS 4 M→∞ SHOCK WAVES ARE MAINLY HEATING MACHINES!

BOUNCING BETWEEN APPROACHING MIRRORS UPSTREAMDOWNSTREAM U 1 U 2 -∞ 0 +∞ V=U 1 -U 2 >0 Relative velocity INITIAL ENERGY DOWNS: E -1< μ <0 0< μ’ <1 TOTAL FLUX

FIRST ORDER A FEW IMPORTANT POINTS: I.There are no configurations that lead to losses II. The mean energy gain is now first order in V III. The energy gain is basically independent of any detail on how particles scatter back and forth!

RETURN PROBABILITIES AND SPECTRUM OF ACCELERATED PARTICLES UPSTREAMDOWNSTREAM U 1 U 2 -∞ 0 +∞ Return Probability from Downstream HIGH PROBABILITY OF RETURN FROM DOWNSTREAM BUT TENDS TO ZERO FOR HIGH U 2

ENERGY GAIN: E 0 → E 1 → E 2 → --- → E K =[1+(4/3)V] K E0 N 0 → N 1 =N 0 *P ret → --- → N K =N 0 *P ret K

Putting these two expressions together we get: Therefore: THE SLOPE OF THE DIFFERENTIAL SPECTRUM WILL BE γ+1=(r+2)/(r-1) → 2 FOR r → 4 (STRONG SHOCK)

THE TRANSPORT EQUATION APPROACH UP DOWN U 1 U 2 -∞ ∞ Integrating around the shock: Integrating from upstr. infinity to 0-: and requiring homogeneity downstream:

THE TRANSPORT EQUATION APPROACH INTEGRATION OF THIS SIMPLE EQUATION GIVES: 1.THE SPECTRUM OF ACCELERATED PARTICLES IS A POWER LAW EXTENDING TO INFINITE MOMENTA 2.THE SLOPE DEPENDS UNIQUELY ON THE COMPRESSION FACTOR AND IS INDEPENDENT OF THE DIFFUSION PROPERTIES 3.INJECTION IS TREATED AS A FREE PARAMETER WHICH DETERMINES THE NORMALIZATION NOTE THAT THIS IS IN P SPACE NAMELY N(p)dp=4π p 2 f(p)dp Therefore the slope is 3r/(r-1)

TEST PARTICLE SPECTRUM

SOME IMPORTANT COMMENTS  THE STATIONARY PROBLEM DOES NOT ALLOW TO HAVE A MAX MOMENTUM!  THE NORMALIZATION IS ARBITRARY THEREFORE THERE IS NO CONTROL ON THE AMOUNT OF ENERGY IN CR  AND YET IT HAS BEEN OBTAINED IN THE TEST PARTICLE APPROXIMATION  THE SOLUTION DOES NOT DEPEND ON WHAT IS THE MECHANISM THAT CAUSES PARTICLES TO BOUNCE BACK AND FORTH  FOR STRONG SHOCKS THE SPECTRUM IS UNIVERSAL AND CLOSE TO E -2  IT HAS BEEN IMPLICITELY ASSUMED THAT WHATEVER SCATTERS THE PARTICLES IS AT REST (OR SLOW) IN THE FLUID FRAME

A FREE ESCAPE BOUNDARY CONDITION UP DOWN x0x0 THE ESCAPE OF PARTICLES AT X=X0 CAN BE SIMULATED BY TAKING THIS REFLECTS IN AN EXP CUTOFF AT SOME MAX MOMENTUM ESCAPE FLUX TOWARDS UPSTREAM INFINITY!!!

ESCAPE FLUX IN TEST PARTICLE THEORY FOR D(E) PROPORTIONAL TO E (BOHM DIFFUSION):

SOME FOOD FOR THOUGHT  WHAT DETERMINES THE MAX MOMENTUM IN REALITY?  IF THE RETURN PROBABILITY FROM UPSTREAM IS UNITY, WHAT ARE COSMIC RAYS MADE OF?  ARE WE SURE THAT THE 10-20% EFFICIENCY WE NEED FOR SNR TO BE THE SOURCES OF GALACTIC CR ARE STILL COMPATIBLE WITH THE TEST PARTICLE REGIME?

MAXIMUM MOMENTUM OF ACCELERATED PARTICLES THE ACCELERATION TIME IS GIVEN BY: AND SHOULD BE COMPARED WITH THE AGE OF THE ACCELERATOR, FOR INSTANCE A SUPERNOVA REMNANT AS AN ESTIMATE: E max

IF THE SHOCK IS PROPAGATING IN THE ISM ONE WOULD BE TEMPTED TO ASSUME D(E)=D gal (E) WHERE TYPICALLY: A=(1-10) cm 2 /s α= FOR ALL CHOICES OF PARAMETERS THE MAX FRACTIONS ENERGY OBTAINED IN THIS WAY IS FRACTIONS OF GeV OF GeV, THEREFORE IRRELEVANT !!! …BUT IT WOULD BE HIGHER IF D(E) WERE MUCH SMALLER…CAN IT HAPPEN?

DIFFERENT PHASES OF A SNR MASS OF THE EJECTA: M ej FREE EXPANSION VELOCITY: TOTAL KINETIC ENERGY: E 51 THERE IS AN INITIAL PERIOD DURING WHICH THE SHELL OF THE SN EXPANDS FREELY (FREE EXPANSION PHASE -BALLISTIC MOTION):

SEDOV PHASE: The sound speed in the ISM is about 10 6 cm/s STRONGSHOCK BUT THE SHOCK SWEEPS THE MATERIAL IN FRONT OF IT AND AT SOME POINT IT ACCUMULATES ENOUGH MATERIAL TO SLOW DOWN THE EXPANDING SHELL:

Simple implications During free expansion the shock fron moves with constant speed Therefore its position scales with t The diffusion front moves proportional to t 1/2 During the free expansion phase the particles are not allowed to Leave the acceleration box, which is the reason why the maximum Energy increases

During the Sedov-Taylor expansion the radius of the blast waves Grows as t 2/5, slower than the diffusion front THE MAXIMUM ENERGY OF ACCELERATED PARTICLES DECREASES WITH TIME THIS IS THE PHASE DURING WHICH THE PARTICLES CAN POSSIBLY BECOME COSMIC RAYS

OVERLAP OF ESCAPE FLUXES: A SIMPLE ESTIMATE BE VERY CAREFUL…THIS IS JUST A WAY TO SHOW HOW YOU GET ROUGHLY A POWER LAW BUT SUMMING NON-POWER LAWS. MORE DETAILED CALC’S SHOW DEPARTURES FROM THIS SIMPLE TREND