Calculation of diffusive particle acceleration by shock waves using asymmetric skew stochastic processes

Transcription

1 Calculation of diffusive particle acceleration by ock waves using asymmetric skew stochastic processes Astronum June 9, 9 Ming Zhang Florida Institute of Technology

2 Cosmic ray or energetic particle transport in interstellar magnetic and interplanetary magnetic field is most likely a diffusion process in phase space. The distribution function of particles can be described by a Fokker-Planck equation. The applicable problems include:. Solar modulation of cosmic rays. Cosmic ray propagation through interstellar medium with nuclear interaction network 3. Diffusive ock acceleration 4. Solar energetic particles 5. Second order Fermi

3 Einstein Theory of Brownian Motion HO Pollen F for steady state kv F ma F dt ( White k Noise) dt Langevin Equation: d(t) dt V(,t) (,t)*"noise" Stochastic differential equation: d( t) V (, t) dt (, t) dw ( t) (Ito type) where dw ( t) " noise"* dt with (dw(t)) dt, a Wiener (Gaussian) processs

4 Fokker-Planck Equation For a stochastic process: (Ito) d(t) V(,t)dt (,t)dw(t) The probability density to find the process at a given time and location follows a Fokker-Planck equation: P(,t ;,t) t [ diffusion coefficient is: (,t) V(,t)]P(,t ;,t)] (Time forward) P(,t ;,t) t [ (,t ) V(,t ) ]P(,t ;,t)] (Time backward)

5 Solar Modulation of Cosmic Ray Clima Neutron Monitor (Cosmic Ray Intensity) Sunspot Number Cosmic Ray Intensity, Percent of 954 Minimum Modulation Level CS Tilt S+ N- S- N+ N-S+ N+S- <<< Solar Polar Reversals 5 5 Smoothed Sunspot Number Current Sheet Tilt (Degree)

6 Modulated cosmic ray spectra Hydrogen 5 Flu (particles/(m sr s Mev/n) Helium Carbon + Oygen E.75 Energy (GeV/n)

7 Cosmic ray transport mechanisms in the heliosphere Drift V d Solar wind Convection V sw Adiabatic deceleration Inward diffusion ISM f (, p, t ) t with and f f f ism (f ) Diffusion V sw f Convection V d f Drift ( V sw ) p f Adiabatic energy change 3 p f ( D ) Second Fermi pp p p p ( p ) at outer at inner boundary boundary

8 Stochastic method to solve cosmic ray modulation spectra Interstellar Spectrum f ism (p) (p e ) (p e ) Backward trajectory of particles 3 d(s) i dw i (s) i ( V V d )ds dp(s) (V) pds 3 all starting at (,p,t) Modulated spectrum f (, p, t) f b f ism ( p eit )

9 Particle drift in inhomogeneous magnetic field Gradient drift Drift path of protons Drift in the current eet V d B pcv B 3q B A r ( rˆ r sin ˆ) H ( ) cs Vsw V dcs pcv ( cs ) 3qB (singularity)

10 Distribution of particle ( GeV/c) intensity on a sphere at AU in an asymmetric heliospheric field Latitude (deg) Longitude (deg) Ulysses/IMP-8 >9 MeV p E>9 MeV GCR North pole South pole Ulysses Latitude

11 3-d MHD model heliosphere Termination ock MHD model heliosphere provided by Linde et al. (998)

12 Diffusive Shock Acceleration Nearly isotropic distribution f t f V f 3 Vp f p Stochastic simulation (time forward): d dw ( V) dt p dp Vdt 3 At ock V ( V n V n ) ( ( ) nˆ ( ) ) (3- d) (singularity) upstream : downstream : V n V n V n V ( n ( ), ( ), ( ) )

13 How to integrate ( ) dt In normal calculus: d ( ) dt ( ) dt v dt v fied time In stochastic calculus: d ( ) dt ( ) dt? dw( t) v dt random time Remove the singularity in stochastic differential equation s( )( ), s( ),, for for for d d d! s( )[ dw( t) Vdt] s ) d d s ( )( ) ( ) dt (

14 Skew stochastic process ] ) ( [ ) ( time Local d d s dt L t ) )( ( s ) )( ( s d d dt ( ) ) ( skew ] ) ( )[ ( Vdt dw t s d (no singularity) ( ) s )] ( ) ( [ ) ( scheme Euler n n n n sign sign t

15 Full code for time-dependent diffusive ock acceleration implicit real*8 (a-h,o-z) implicit integer (i-n) dimension p(),pb() character ttime*4,cmon()*3 integer iyr,imon(),idom,idoy,ihr,imn,isc data imon/,3,59,9,,5,8,,43,73,34,334/ data cmon/"jan","feb","mar","apr","may","jun", + "Jul","Aug","Sep","Oct","Nov","Dec"/ common /param/v,v,rb,gm,gp,gc,b,iseed,nz,nm data ep/e-/ iseed=-88 c start the injection ntp=5 do i=,ntp p()=-ep p()=. pb()=rb te=. call walk(p,pb,te,ns) write(6,96) pb,te,ns 96 format(3e.4,i) enddo stop end ccccccccccccccccccccccccccccccccccccccccccccccccccccc cccc subroutine walk(p,pb,te,ns) c random walk in the ock frame ( in Cartisian) implicit real*8 (a-h,o-z) implicit integer (i-n) c p = (, p) dimension p(),pb() common /param/v,v,rb,gm,gp,gc,b,iseed,nz,nm data ts/e-/,ts/.e-4/ dt=*gc/v/v*ts dt=*gc/v/v*ts srdt=sqrt(dt) srdt=sqrt(dt) =p() p=p() t=. n= c find the convection speed and sign of, scaling factor, and y if(.lt.) then v=v nsgn=- sf=gp/(gp+gm) y=sf* else if(.eq.) then v=(v+v)/. nsgn= sf=.5 y=sf* else v=v nsgn= sf=gm/(gp+gm) y=sf* endif endif c find the diffusion coeficients call getdc(,p,gc,dgc) g=gc+.5*(gp-gm)*nsgn dg=dgc c c find time step side c to increase accrucy near the ock dma=3*sqrt(*g)*srdt dmin=3*sqrt(*g)*srdt if(nsgn*.gt.dma) then dt=dt srdt=srdt else if(nsgn*.lt.dmin) then dt=dt srdt=srdt else dt=dt+(dt-dt)/(dma*dmadmin*dmin)*(*-dmin*dmin) srdt=sqrt(dt) endif endif t=t+dt n=n+ c step forward dw=srdt*gasdev(iseed) dy=sf*((v+dg)*dt+sqrt(*g)*dw) y=y+dy c find the convection sign of and convert y to c and speed, scaling factor for net step, if(y.lt.) then v=v nsgn=- sf=gp/(gp+gm) =y/sf else if(y.eq.) then v=(v+v)/. nsgn= sf=.5 =y/sf else v=v nsgn= sf=gm/(gp+gm) =y/sf endif endif c local time dlt=(.5/gm+.5/gp)*(nsgn-nsgn)*y dp=./3.*(v-v)*p*dlt p=p+dp c write the trajectory to file 8 c write(8,98),p,t 98 format(3e.4) c if(nsgn*.lt.pb()) then if(t.lt.te) then nsgn=nsgn goto endif c eit from the boundary pb()= pb()=p ns=n te=t return end ccccccccccccccccccccccccccccccccccccccccccccccccccccc ccccccccccccc subroutine getdc(,p,gc,dgc) c subroutine to calculate diffusion coeffient at (,p) and its gradient implicit real*8 (a-h,o-z) implicit integer (i-n) common /param/v,v,rb,gm,gp,gc,b,iseed,nz,nm gc=gc*p**b*f() dgc=gc*p**b*df() return end function f() c position dependence of the diffusion coefficient implicit real*8 (a-h,o-z) f=. return end function df() c derivative of f() implicit real*8 (a-h,o-z) df=. return end

16 Comparison with analytical solution -d ock, monoenergetic particle injection at ock at t=

17 Evolution of particle spectrum at the ock

18 Transport equation for particles with large anisotropy f ( t, r, p, ) t f cross - field diffusion ( Vsw Vd vbˆ) f convection, drift and streaming f D pitch angle diffusion ( ) v ( ) ( 3 ˆ ˆ f Vsw bb : Vsw) focusing LB - ( ˆ ˆ : ) ˆ ˆ : f qe p V energy gain sw bb Vsw bb Vsw p Stochastic differential equation solver d dp d κ D dw( s) ( κ ( V sw bb ˆ ˆ : V D dw( s) V sw sw V B d vbˆ) ds ) bb ˆ ˆ : V sw pds ( ) v ( ) ( V L sw 3ˆ bbˆ : V sw ) ds

19 Shock layer -3 Particles injected at 5 kev, Bn =6 o -3 Particles injected at 5 kev, Bn =87 o Intensity kev p 64 kev p 79 kev p Intensity kev p 64 kev p 79 kev p - - Distance (AU) Distance (AU) Particles injected at 5 kev, Bn =6 o -3 Particles injected at 5 kev, Bn =6 o Intensity kev p 64 kev p 79 kev p Intensity kev p 64 kev p 79 kev p - - Distance (AU) Distance (AU) 3 4 5

20 Particles injected at 5 kev, Bn =6 o Particles injected at 5 kev, Bn =87 o - downstream upstream.4 AU upstream - downstream upstream.4 AU upstream f=j/p^ - -3 f=j/p^ Energy (MeV) Energy (MeV) Bn =6 o downstream upstream.4 AU upstream..8.6 downstream upstream.4 AU upstream Bn =87 o J front /J back J front /J back Energy (MeV) Energy (MeV) 3 4 5

21 Particle acceleration by the termination ock and helioeath Voyager observations ow that anomalous cosmic rays are accelerated beyond the termination ock Acceleration mechanisms of particle acceleration in the helioeath adiabatic heating by plasma compression in the helioeath Second order Fermi acceleration with D pp p VA v D p V A A

22 Particle acceleration by the termination ock and helioeath A. Plain ock acceleration Speed (km/s) 4 3 Solar wind Alfven dlog(p)/dt X (AU) X (AU) B. Shock acceleration with second-order Fermi acceleration Speed (km/s) 4 3 Solar wind Alfven dlog(p)/dt X (AU) X (AU)

23 C. Shock acceleration with adiabatic post-ock compression Speed (km/s) 4 3 Solar wind Alfven dlog(p)/dt X (AU) X (AU) D. Shock acceleration with adiabatic post-ock compression and stochastic acceleration Speed (km/s) 4 3 Solar wind Alfven dlog(p)/dt X (AU) X (AU)

24 A B 9 9 Radial distance (AU) Radial distance (AU) 8. Energy (MeV) 8. Energy (MeV) Energy (MeV). Energy (MeV) Radial distance (AU) Radial distance (AU) C - - log(j arbitrary unit) D -3-4

25 A 3 He spectra at the ock B 3 He spectra at AU downstream Log(J) - Log(J) Model A Model B Model C Model D - -3 Model A Model B Model C Model D Energy (MeV). Energy (MeV)

26 Stochastic differential equation vs. Fokker-Planck equation Monte-Carlo simulation with SDE can be used to solve Fokker- Planck equation. Track stochastic trajectories to look into the details of particle transport. Problems in high dimensions do not necessarily increase computation demand. Programming is etremely simple and less numerical instability. Singularity in stochastic differential equations can be solved by skew Brownian motion Application to ock acceleration (both isotropic and focus transport) Application to particle drift in heliospheric current eet