Theory of Diffusive Shock Acceleration

Transcription

1 Theory of Diffusive Shock Acceleration Tony Bell University of Oxford Rutherford Appleton Laboratory SN1006: A supernova remnant 7,000 light years from Earth X-ray (blue): NASA/CXC/Rutgers/G.Cassam-Chenai, J.Hughes et al; Radio (red): NRAO/AUI/GBT/VLA/Dyer, Maddalena & Cornwell; Optical (yellow/orange): Middlebury College/F.Winkler. NOAO/AURA/NSF/CTIO Schmidt & DSS

2 Cosmic ray spectrum arriving at earth Mainly protons

3 CR populations (I) Galactic Supernova remnants (II) Probably galactic (III) Extragalactic? knee ankle ~E -3 ~E -2.6

4 Why shocks?

5 Historical shell supernova remnants Tycho 1572AD Kepler 1604AD SN1006 Chandra observa?ons Cas A 1680AD NASA/CXC/Rutgers/ J.Hughes et al. NASA/CXC/Rutgers/ J.Warren & J.Hughes et al. NASA/CXC/NCSU/ S.Reynolds et al. NASA/CXC/MIT/UMass Amherst/ M.D.Stage et al.

6 Cassiopeia A Radio (VLA) Infrared (Spitzer) Optical (Hubble) X-ray (Chandra) NASA/JPL-Caltech/O. chandra.harvard.edu/photo/0237/0237_radio.jpg NASA/JPL Krause (Steward Observatory) chandra.harvard.edu/photo/ 0237/0237_radio.jpg NASA/JPL-Caltech/ O Krause(Steward Obs) NASA/ESA/ Hubble Heritage (STScI/AURA)) NASA/CXC/MIT/UMass Amherst/ M.D.Stage et al. Mixture of line radiation & synchrotron continuum Synchrotron in magnetic field ~ 0.1-1mG Radio (hν~10-5 ev): electron energy ~1 GeV X-ray (hν~10 3 ev): electron energy ~ 10 TeV

7 HESS: γ-rays directly produced by TeV particles SNR RX J Aharonian et al Nature (2004)

8 Cygnus A Active galaxies Centaurus A is the closest powerful radio galaxy (5Mpc) optical Radio jets X-ray (Chandra)

9 Strong shock: high Mach number In shock rest frame density pressure velocity Upstream Downstream Conserved across shock (Rankine Hugoniot relations) Mass flux Momentum flux Energy flux Shock turns kinetic streaming energy Into random thermal energy Divert part of thermal energy Into high energy particles

10 Cosmic ray acceleration by shocks

11 Cosmic ray acceleration CR track B 1 B 2 High velocity plasma Low velocity plasma Due to scattering, CR recrosses shock many times Gains energy at each crossing

12 Shock acceleration energy spectrum: energy gain CR θ Change in fluid velocity across shock shock Change in momentum from upstream to downstream Mean increase in momentum Similar increase in momentum on recrossing into upstream Average fractional energy gained at each crossing is

13 Shock acceleration energy spectrum: loss rate Upstream ISM Downstream shocked plasma B 1 B 2 B 2 >B 1 Shock velocity: v s High velocity plasma Low velocity plasma CR density at shock: n CR cross from upstream to downstream at rate nc/4 CR carried away downstream at rate nv downstream = nv s /4 Mean number of shock crossings = (nc/4)/(nv s /4)=c/v s Fraction lost at each shock crossing is v s /c

14 Shock acceleration energy spectrum Upstream ISM Downstream shocked plasma B 1 B 2 B 2 >B 1 Shock velocity: v s High velocity plasma Low velocity plasma CR density at shock: n Fractional CR loss per shock crossing Fractional energy gain per shock crossing Turn into differential equation integrated spectrum Differential energy spectrum

15 Derivation from Boltzmann equation Krimskii 1977 Axford, Leer & Skadron 1977 Blandford & Ostriker 1978

16 The Vlasov-Fokker-Planck (VFP) equation Vlasov equation (collisioness) Collisions Fokker-Planck = number of CR in phase space volume VFP equation: 1) Advection: at velocity v in r-space at velocity e(e+vxb) in p-space 2) Collisions: small angle scattering B on scale > CR Larmor radius Due to B on scale < CR Larmor radius

17 Cosmic ray acceleration Large scale B CR track B 1 B 2 Small scale B scatters CR

18 Parallel shock CR track Only diffusion along B matters Large scale field irrelevant Same is if no large scale field

19 Redefine f in local fluid rest frame CR track f fluid rest frame Fluid moves at velocity u Scattering in angle advection with fluid Frame transformation

20 Sub-relativistic shocks: small u/c u 1 =u shock u 2 =u shock /4 To first approximation in u/c VFP equation reduces to isotropic drift Scattering frequency Advection Diffusion adiabatic compression

21 Steady state solution u 1 =u shock u 2 =u shock /4 No escape upstream: Downstream: no drift relative to background Boundary condition at shock

22 Acceleration efficiency

23 Efficiency Has to be efficient (10-50%) to explain galactic CR energy density Solar wind shocks can be >10% efficient Shock processes produce many suprathermal protons

24 At high efficiency: non-linear feedback onto shock Drury & Voelk (1981) CR pressure decelerates flow into shock

25 High efficiency: concave spectrum Low energy, small mfp CR see smooth shock: weak acceleration, steep spectrum Steep spectrum High energy spectrum flatter than p -4 Shock compression > 4 Due to mildly relativistic equation of state Flat spectrum

26 Maximum CR energy

27 CR upstream of shock u shock n cr upstream L=D/u Balance between: flow into shock diffusion away from shock Exponential density Scaleheight

28 u CR acceleration time shock n cr upstream L=D/u Number of CR upstream: Flow rate into shock: Average time spent upstream: Average time spent upstream+downstream: Time needed for acceleration (Lagage & Cesarsky)

29 Maximum CR energy Acceleration time where Diffusion coefficient SNR radius Smallest possible mfp: Limit on CR momentum: Typically for young SNR ISM mag field: 3µG u shock =c/30 R ~ m Hillas parameter Max CR energy ~ ev under favourable assumptions

30 Hillas diagram (condition on RuB) (Hillas, 1984) Get original version

31 Perpendicular shocks (Jokipii 1982, 1987)

32 CR trajectory at perpendicular shock (no scattering) B into screen CR gain energy by drifting in E field CR drift velocity shock

33 CR acceleration at perpendicular shock CR trajectory divides into Motion of gyrocentre Gyration about gyrocentre Without diffusion: Every CR gets small adiabatic gain due to compression at shock shock

34 CR acceleration at perpendicular shock: with scattering Diffusive shock theory applies Provided gyrocentre diffuses over distances greater than Larmor radius during shock transit Same power law (see later) No scattering Weak scattering Strong scattering Not to scale

35 CR acceleration at perpendicular shock B Transit between pole & equator: energy gain ~ Hillas parameter as with parallel shock: similar max CR energy

36 The case of SN1006 Polar x-ray synchrotron emission? SN1006 B? At perpendicular shocks Acceleration is faster potentially higher CR energy CR energy limited to eubr (Hillas) by space rather than time Injection is more difficult at a perpendicular shock CR scattering frequency has to be in right range Room for discussion!

37 CR scattering what is the mean free path?

38 CR drive a resonant instability CR trajectory Alfven wave B Spatial resonance between wavelength and CR Larmor radius wave deflects CR CR current drives wave Skilling (1975) Wave growth (energy density I) CR scattering

39 Turbulence upstream of shock n cr Skilling (1975) Wave growth (amplitude I) L=D/u CR scattering Solution Alfven Mach number ~1000 CR efficiency ~0.1 Question: What does I > 1 tell us??implies?: mfp < Larmor radius Waves non-linear: I >> 1

40 Re-examine CR scattering

41 Streaming CR excite instabilities SNR CR pre-cursor CR streaming ahead of shock Excite instabilities Amplify magnetic field Shock upstream downstream

42 Streaming instabilities amplify magnetic field Lucek & Bell (2000) CR CR treated as particles Thermal plasma as MHD

43 Electric currents carried by CR and thermal plasma R L j cr CR pre-cursor Density of ev CR: ~10-12 cm -3 Current density: j cr ~ Amp m -2 CR current must be balanced by current carried by thermal plasma j thermal = - j cr j thermal xb force acts on plasma to balance j cr xb force on CR

44 Three equations control the instability 1) 2) 3) Equation for j cr in terms of perturbed B B CR jxb driving force splits into two parts:

45 Resonant Alfven instability B CR drives Alfven waves Perturbed cosmic ray current

46 Non-resonant instability B j cr CR dominates for shock acceleration in SNR Perturbed magnetic field

47 Dispersion relation Red line is growth rate ω Im(ω) Re(ω) k k in units of r g -1 ω in units of v S2 /cr g Wavelength longer than Larmor radius CR follow field lines. jxb drives weak instability Magnetic tension inhibits instability

48 The essence of the non-resonant instability

49 Spiral expands leaving central cavity j x B j x B Same without vertical field j x B j x B

50 Simplest form: expanding loops of B B j x B j x B CR current jxb expands loops stretches field lines more B more jxb

51 Simplest form: expanding loops of B B j x B j x B CR current jxb expands loops stretches field lines more B more jxb

52 Non-linear growth expanding loops Slices through B - time sequence (fixed CR current) Field lines: wandering spirals Cavities and walls in B & ρ

53 How large does the magnetic field grow?

54 Historical SNR Tycho 1572AD Kepler 1604AD SN1006 Chandra observa?ons Cas A 1680AD NASA/CXC/Rutgers/ J.Hughes et al. NASA/CXC/Rutgers/ J.Warren & J.Hughes et al. NASA/CXC/NCSU/ S.Reynolds et al. NASA/CXC/MIT/UMass Amherst/ M.D.Stage et al.

55 Instability growth a) b) c) No reason for non-linear saturation of a single mode d)

56 Saturation (back of envelope) Magnetic field grows until 1) Magnetic tension CR driving force 2) CR Larmor radius scalelength Set and eliminate between 1) & 2)

57 Inferred downstream magnetic field (Vink 2008) B 2 /(8πρ) (cgs) Data for RCW86, SN1006, Tycho, Kepler, Cas A, SN1993J velocity Fit to obs (Vink): Theory:

58 The cosmic ray spectrum revision from p -4

59 Shock acceleration energy spectrum: loss rate Upstream ISM Downstream shocked plasma B 1 B 2 B 2 >B 1 High velocity plasma Low velocity plasma CR cross from upstream to downstream at rate n shock c/4 CR carried away downstream at rate = n downstream v s /4 Fraction lost at each shock crossing In diffusive limit

60 Parallel shock u/c=0.1 ν/ω g =0.1 f 0 f 2 f x/r g 140

61 Vlasov-Fokker-Planck (VFP) analysis for oblique magnetic field with Klara Schure & Brian Reville

62 Equivalent form of solution for small u/c Extra term Upstream solution Downstream solution Cannot match & across the shock

63 Equivalent form of solution for small u/c Extra term NOT VALID SOLUTION Upstream solution Downstream solution Cannot match & across the shock

64 Expand in spherical harmonics audio.com/images/sphericalharm.jpg f 0 0 f 1 m f 2 m f 3 m

65 Equation for evolution of each spherical harmonic Solve numerically

66 Parallel shock θ = 0 o u/c=0.1 ν/ω g =0.1 f 0 0 f 2 0 f x/r g 140

67 Oblique shock (nearly parallel) θ = 30 o u/c=0.1 ν/ω g =0.1 f 0 0 Im(f 11 ) f 2 0 f x/r 100 g Density spike at shock as seen by Ostrowski MNRAS (1991) Ruffalo, ApJ (1999) Gieseler, Kirk, Heavens & Achterberg A&A (1999)

68 More perpendicular, less parallel u/c=0.1 θ = 60 o ν/ω g =0.1 f 0 0 Im(f 11 ) f 1 0 f 2 0 Re(f 11 ) -60 x/r 35 g Re(f 11 ) represents cross-field drift Im(f 11 ) represents drift along oblique field lines

69 Peperpendicular shock θ = 90 o u/c=0.1 ν/ω g =0.1 f 0 0 CR density reduced at shock Im(f 11 ) f 2 0 f 1 0 Re(f 11 ) -2.8 x/r 1.7 g

70 CR density profiles near shock u/c=0.1 θ = 90 o ν/ω g =0.1 CR density θ = 72 o θ = 60 o shock upstream downstream distance from shock (CR Larmor radius)

71 Spectral index plotted against shock obliquity Shock compression x4 in all cases Perpendicular shock Parallel shock γ (α) 6 (1.5) 5 (1.0) u=c/10 ν/ω g =0.03 ν/ω g =0.1 ν/ω g =0.3 ν/ω g =1 4 (0.5) cosθ 0 1 Radio spectral index In brackets ν = collision frequency ω g = Larmor frequency

72 Spectral index plotted against shock obliquity Shock compression x4 in all cases Perpendicular shock Parallel shock 6 (1.5) γ (α) 5 (1.0) u=c/30 ν/ω g =0.03 ν/ω g =0.1 ν/ω g =0.3 ν/ω g =1 u=c/10 ν/ω g =0.03 ν/ω g =0.1 ν/ω g =0.3 ν/ω g =1 ν/ω g =0.03 ν/ω g =0.1 ν/ω g =0.3 u=c/5 ν/ω g =1 4 (0.5) cosθ cosθ cosθ Radio spectral index In brackets ν = collision frequency ω g = Larmor frequency

73 Observations

74 Cosmic Ray spectrum arriving at earth (Nagano & Watson 2000) E -2.7 E -2 Leakage from galaxy accounts for some of difference (Hillas 2005)

75 Observed radio spectral index v. mean expansion velocity (Klara Schure following Glushak 1985)

76 Spectral steepening suggests quasi-perpendicular shocks For random field orientation, steepening & flattening nearly cancel out Steepening at high velocity might be due to γ (α) 6 (1.5) u=c/10 ν/ω g =0.03 ν/ω g =0.1 1) expansion into Parker spiral 2) magnetic field amplification jxb stretches field perpendicular to shock normal 5 (1.0) ν/ω g =0.3 ν/ω g =1 4 (0.5) cosθ 0 1

77 SNR morphology: spectral steepening/flattening SN1006 (Chandra) CR electrons (10-100TeV) Quasi-perpendicular shocks accelerate fewer CR to high energy

78 What we know, and what we know we don t know What we know (but not totally proved observationally) Galactic CR are accelerated to ev by diffusive shock acceleration by SNR Streaming CR amplify the magnetic field which confines CR near shock CR spectra are often steeper than p -4 (E -2 ): non-linear effects, quasi-perpendicular shocks What we don t know How CR reach ev When CR are accelerated to what energy at different stages of SNR evolution How CR escape SNR without losing energy adiabatically When & where non-linear effects are important Why typically the spectrum is flatter than p -4 in older SNR Why is the galactic CR spectrum so straight? Whether second order Fermi acceleration contributes substantially Whether perpendicular shocks are good injectors of low energy CR How the above applies extra-galactically - the origin of ev CR