Diffusive Shock Acceleration and magnetic fields in SNR

Transcription

1 Diusive Shock Acceleration and magnetic ields in SNR Tony Bell University o Oxord Rutherord Aleton Laboratory SN16: A suernova remnant 7, light years rom Earth X-ray (blue): NASA/CXC/Rutgers/G.Cassam-Chenai, J.Hughes et al; Radio (red): NRAO/AUI/GBT/VLA/Dyer, Maddalena & Cornwell; Otical (yellow/orange): Middlebury College/F.Winkler. NOAO/AURA/NSF/CTIO Schmidt & DSS

2 Cosmic ray sectrum arriving at earth Mainly rotons

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

4 Why shocks?

5 Historical shell suernova remnants Tycho 1572AD Keler 164AD SN16 Chandra observations Cas A 168AD 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 Cassioeia A Radio (VLA) Inrared (Sitzer) Otical (Hubble) X-ray (Chandra) NASA/JPL-Caltech/O. chandra.harvard.edu/hoto/237/237_radio.jg NASA/JPL Krause (Steward Observatory) chandra.harvard.edu/hoto/ 237/237_radio.jg NASA/JPL-Caltech/ O Krause(Steward Obs) NASA/ESA/ Hubble Heritage (STScI/AURA)) NASA/CXC/MIT/UMass Amherst/ M.D.Stage et al. Mixture o line radiation & synchrotron continuum Synchrotron in magnetic ield ~.1-1mG Radio (hν~1-5 ev): electron energy ~1 GeV X-ray (hν~1 3 ev): electron energy ~ 1 TeV

7 HESS: γ-rays directly roduced by TeV articles SNR RX J Aharonian et al Nature (24)

8 Cygnus A Active galaxies Centaurus A is the closest owerul radio galaxy (5Mc) otical Radio jets X-ray (Chandra)

9 Strong shock: high Mach number In shock rest rame density ressure velocity Ustream Downstream 1 u = u shock ρ = ρ u = u ρ = 4ρ shock 4 kt = P = 3 A 2 kt = 3 2 P = ρu 16 1 m u shock Z shock + 4 Conserved across shock (Rankine Hugoniot relations) Mass lux Momentum lux Energy lux ρu 2 P + ρu 5 1 Pu + ρu Shock turns kinetic streaming energy Into random thermal energy Divert art o thermal energy Into high energy articles

10 Cosmic ray acceleration by shocks

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

12 Shock acceleration energy sectrum: energy gain CR θ shock Change in luid velocity across shock v = v v s s = 4 3v 4 s Change in momentum rom ustream to downstream 1 = = v cosϑ c Mean increase in momentum π / 2 1 cosϑ sinϑdϑ < 1 > = = ( vs / 2c) π / 2 cosϑ sinϑdϑ Similar increase in momentum on recrossing into ustream < > ( v s / 2c) 2 = Average ractional energy gained at each crossing is ε = ε < > + < > 1 2 = vs c

13 Shock acceleration energy sectrum: loss rate Ustream ISM Downstream shocked lasma B 1 B 2 B 2 >B 1 Shock velocity: v s High velocity lasma Low velocity lasma CR density at shock: n CR cross rom ustream to downstream at rate nc/4 CR carried away downstream at rate nv downstream = nv s /4 Mean number o shock crossings = (nc/4)/(nv s /4)=c/v s Fraction lost at each shock crossing is v s /c n n = vs c

14 Shock acceleration energy sectrum Ustream ISM Downstream shocked lasma B 1 B 2 B 2 >B 1 Shock velocity: v s High velocity lasma Low velocity lasma CR density at shock: n Fractional CR loss er shock crossing Fractional energy gain er shock crossing n = n ε = ε vs c v s c Turn into dierential equation dn dε n n = ε ε 1 n ε integrated sectrum Dierential energy sectrum N( ε ) dε ε 2 dε

15 Derivation rom Boltzmann equation Krimskii 1977 Axord, Leer & Skadron 1977 Blandord & Ostriker 1978

16 The Vlasov-Fokker-Planck (VFP) equation + t v. e r ( E + v B). = C( ) Vlasov equation (collisioness) Collisions Fokker-Planck ( x, y, z,,,, t) x y z dx dy dz d x d y d = number o CR in hase sace volume x y z z dx dy dz d d d VFP equation: 1) Advection: at velocity v in r-sace at velocity e(e+vxb) in -sace 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 t + v. e r ( E + v B). = C( )

18 Parallel shock CR track ) (. C t = + r v Only diusion along B matters Large scale ield irrelevant Same is i no large scale ield ( ) ) (.. C e t = + + B v E r v

19 Redeine in local luid rest rame CR track luid rest rame Fluid moves at velocity u t + u ( v x + u) x = C( x x x ) advection with luid Frame transormation

20 u 2 =u shock /4 u 1 =u shock x ) ( ) ( 1 + = To irst aroximation in u/c isotroic drit VFP equation reduces to v 1 = + + x u x x u t 1 v x = ν Scattering requency v = + x u x x x u t ν Advection Diusion adiabatic comression Sub-relativistic shocks: small u/c

21 Steady state solution x = ( ) + 1( ) u 1 =u shock u 2 =u shock /4 No escae ustream: u c + = Downstream: no drit relative to background 1 = Boundary condition at shock 1 u c ustream = u2 c 1 1 downstream ( u1 u2 ) = 3u1 4

22 Acceleration eiciency

23 Eiciency Has to be eicient (1-5%) to exlain galactic CR energy density Solar wind shocks can be >1% eicient Shock rocesses roduce many surathermal rotons

24 At high eiciency: non-linear eedback onto shock Drury & Voelk (1981) CR ressure decelerates low into shock

25 High eiciency: concave sectrum Stee sectrum SN16 (Allen et al 28) Flat sectrum

26 Maximum CR energy

27 CR ustream o shock u shock n cr ustream L=D/u Balance between: low into shock diusion away rom shock n t cr = u n x cr D 2 n x cr 2 = Exonential density n cr = n e ux / D Scaleheight L = D u

28 u CR acceleration time shock n cr ustream Number o CR ustream: Average time sent ustream: n cr D ustream u shock t = 4D cu L=D/u ustream shock Rate CR cross shock: 1 n cr c 4 (neglect time sent downstream) Average energy gain er shock crossing: Acceleration rate: ε ε u = t 4 D 2 shock ustream ε = ε u shock c Time needed or acceleration (Lagage & Cesarsky) τ = 4D u ustream 2 shock 4Ddownstream + 2 ( ushock / 4) downstream time

29 Maximum CR energy Acceleration time Diusion coeicient τ = 8D 2 u shock D = λc 3 Smallest ossible m: λ = cr eb 3 u λ shock R where R = u shock τ 8 c SNR radius Limit on CR momentum: Limit on CR energy in ev: Tyically or young SNR ISM mag ield: ew µg u shock =c/3 R ~ 1 17 m cr = 3 8 c e cr u shock = c 3 8 ebr Bu shock R Max CR energy ~ 1 14 ev under avourable assumtions

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

31 Perendicular shocks (Jokiii 1982, 1987)

32 CR trajectory at erendicular shock (no scattering) B into screen E = ushock B v drit = E B 2 B CR gain energy by driting in E ield CR drit velocity shock

33 CR acceleration at erendicular shock CR trajectory divides into Motion o gyrocentre Gyration about gyrocentre Without diusion: Every CR gets small adiabatic gain due to comression at shock shock

34 CR acceleration at erendicular shock: with scattering Diusive shock theory alies Provided gyrocentre diuses over distances greater than Larmor radius during shock transit Same ower law (see later) No scattering Weak scattering Strong scattering Not to scale

35 CR acceleration at erendicular shock B E = u shock B Transit between ole & equator: energy gain ~ eer = eu shock BR Hillas arameter as with arallel shock: similar max CR energy

36 The case o SN16 SN16 B? Polar x-ray synchrotron emission? (Rothenlug et al 24) At erendicular shocks Acceleration is aster otentially higher CR energy CR energy limited to eubr (Hillas) by sace rather than time Injection is more diicult at a erendicular shock CR scattering requency has to be in right range Room or discussion!

37 CR scattering what is the mean ree ath?

38 CR drive a resonant instability CR trajectory Alven wave B Satial resonance between wavelength and CR Larmor radius wave delects CR CR current drives wave Skilling (1975) Wave growth (energy density I) I t I + u = x v A c U M n x cr I δb = B 2 2 CR scattering n t cr + u n x cr = x D n x cr D = 4 cr 3π I g

39 Turbulence ustream o shock n cr Skilling (1975) L=D/u I t Wave growth (amlitude I) I + u = x v A c U M n x cr n t cr CR scattering n + u x cr = x n D x cr D = 4 3π cr I g Solution I shock = M A U ρu cr 2 shock 4 r g m = π I L = 4 3 u c shock r g I Alven Mach number ~1 CR eiciency ~.1 Question: What does I > 1 tell us??imlies?: m < Larmor radius Waves non-linear: I >> 1

40 Evidence that magnetic ield exceeds tyical interstellar value 1) x-ray observations Tycho 1572AD Keler 164AD 2) Needed ti accelerate galactic CR to a ew PeV 3) Theory breaks down because o large wave growth I shock = U M A SN16 ρu cr 2 shock Cas A 168AD Alven Mach number ~1 CR eiciency ~.1

41 Evidence that magnetic ield exceeds tyical interstellar value 1) x-ray observations Tycho 1572AD Keler 164AD 2) Needed ti accelerate galactic CR to a ew PeV 3) Theory breaks down because o large wave growth Need to look more closely at CR interaction with magnetic ield I shock = U M A SN16 ρu cr 2 shock Cas A 168AD Alven Mach number ~1 CR eiciency ~.1

42 Diusive Shock Acceleration and magnetic ields in SNR Tony Bell University o Oxord Rutherord Aleton Laboratory SN16: A suernova remnant 7, light years rom Earth X-ray (blue): NASA/CXC/Rutgers/G.Cassam-Chenai, J.Hughes et al; Radio (red): NRAO/AUI/GBT/VLA/Dyer, Maddalena & Cornwell; Otical (yellow/orange): Middlebury College/F.Winkler. NOAO/AURA/NSF/CTIO Schmidt & DSS

43 magnetic ield amliication and cosmic ray scattering

44 Streaming CR excite instabilities SNR CR re-cursor CR streaming ahead o shock Excite instabilities Amliy magnetic ield Shock ustream downstream

45 Streaming instabilities amliy magnetic ield Lucek & Bell (2) CR CR treated as articles Thermal lasma as MHD

46 Electric currents carried by CR and thermal lasma R L j cr CR re-cursor Density o 1 15 ev CR: ~1-12 cm -3 Current density: j cr ~ 1-18 Am m -2 CR current must be balanced by current carried by thermal lasma j thermal = - j cr j thermal xb orce acts on lasma to balance j cr xb orce on CR

47 Three equations control the instability 1) 2) 3) Equation or j cr in terms o erturbed B B CR j cr j cr jxb driving orce slits into two arts: du ρ dt 1 = B ( B) jcr B jcr B µ

48 Resonant Alven instability u ρ t = 1 µ B ( B) j B j cr cr B B CR j cr j cr j cr B drives Alven waves Perturbed cosmic ray current

49 Non-resonant instability u ρ t = 1 µ B ( B) j B j cr cr B B j cr CR j cr j cr j cr B dominates or shock acceleration in SNR Perturbed magnetic ield

50 Disersion relation ω log1(omega) log1(k) -2 Red line is growth rate Im(ω) Re(ω) k k in units o r g -1 ω in units o v S2 /cr g Wavelength longer than Larmor radius CR ollow ield lines. jxb drives weak instability -4 du ρ = jcr B dt B = t ( u B) Magnetic tension inhibits instability γ = kb ρ jcr 1/ 2

51 The essence o the non-resonant instability

52 Siral exands leaving central cavity j x B j x B Same without vertical ield j x B j x B

53 Simlest orm: exanding loos o B B j x B j x B CR current jxb exands loos stretches ield lines more B more jxb

54 Simlest orm: exanding loos o B B j x B j x B CR current jxb exands loos stretches ield lines more B more jxb

55 Non-linear growth exanding loos Slices through B - time sequence (ixed CR current) Field lines: wandering sirals Cavities and walls in B & ρ

56 How large does the magnetic ield grow?

57 Historical SNR Tycho 1572AD Keler 164AD SN16 Chandra observations Cas A 168AD 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.

58 Instability growth a) b) c) No reason or non-linear saturation o a single mode d)

59 Saturation (back o enveloe) Magnetic ield grows until 1) 1 µ ( B) j B B CR Magnetic tension CR driving orce 2) eb CR Larmor radius L scalelength = 1 L Set and eliminate between 1) & 2) 2 B µ L j e CR eiciency 3 ρu c shock

60 Inerred downstream magnetic ield (Vink 28) B 2 /(8πρ) (cgs) Data or RCW86, SN16, Tycho, Keler, Cas A, SN1993J velocity Fit to obs (Vink): B 7 1 3/ 2 1/ 2 n µ 4 1 e 3 u kms cm G Theory: B 4 1 3/ 2 1/ 2 1/ 2 n µ 4 1 e 3 u kms cm η.1 G

61 The cosmic ray sectrum revision rom -4

62 Shock acceleration energy sectrum: loss rate Ustream ISM Downstream shocked lasma B 1 B 2 B 2 >B 1 High velocity lasma Low velocity lasma CR cross rom ustream 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 n n = v c s n downstream n shock In diusive limit n = n downstream shock

63 Parallel shock 1.5 u/c=.1 1 ν/ω g = x/r g

64 Vlasov-Fokker-Planck (VFP) analysis or oblique magnetic ield

65 ) (. ) v ( C e x u x u t x x x = B v Extra term z z y y x x ) ( ) ( ) ( ) ( = Equivalent orm o solution or small u/c Cannot match & across the shock y z Ustream solution 1 3 c u x = c u x z y ω ν νω + = c u x z z ω ν ω + = Downstream solution = = = z y x m e γ B ω =

66 Extra term z z y y x x ) ( ) ( ) ( ) ( = Equivalent orm o solution or small u/c Cannot match & across the shock y z Ustream solution 1 3 c u x = c u x z y ω ν νω + = c u x z z ω ν ω + = Downstream solution = = = z y x m e γ B ω = NOT VALID SOLUTION ) (. ) v ( C e x u x u t x x x = B v

67 Exand in sherical harmonics 1 m 2 m 3 m ( ) m x,, t P (cosϑ)ex( im ) = m ( x,, t) l l φ l, m

68 Equation or evolution o each sherical harmonic Solve numerically

69 Parallel shock 1.5 θ = o 1.5 u/c=.1 ν/ω g = x/r g

70 Oblique shock (nearly arallel) u/c=.1 ν/ω g =.1 θ = 3 o 1 Im( 11 ) x/r 1 g Density sike at shock as seen by Ostrowski MNRAS (1991) Rualo, AJ (1999) Gieseler, Kirk, Heavens & Achterberg A&A (1999)

71 More erendicular, less arallel 1.5 θ = 6 o 1 u/c=.1 ν/ω g =.1.5 Im( 11 ) 1 2 Re( 11 ) x/r g Re( 11 ) reresents cross-ield drit Im( 11 ) reresents drit along oblique ield lines

72 Peerendicular shock u/c=.1 ν/ω g =.1 θ = 9 o Im( 11 ) 2 CR density reduced at shock Re( 11 ) x/r 1.7 g

73 CR density roiles near shock u/c=.1 ν/ω g =.1 θ = 9 o CR density 1 θ = 6 o θ = 72 o.5 shock 1 ustream downstream -1 1 distance rom shock (CR Larmor radius) 11

74 Sectral index lotted against shock obliquity Shock comression x4 in all cases Perendicular shock Parallel shock γ (α) (1.5) (1.) u=c/1 ν/ω g =.3 ν/ω g =.1 ν/ω g =.3 ν/ω g =1 4 (.5) cosθ 1 Radio sectral index In brackets ν = collision requency ω g = Larmor requency

75 Sectral index lotted against shock obliquity Shock comression x4 in all cases Perendicular shock Parallel shock (1.5) γ (α) 5 (1.) 4 (.5) u=c/3 6 ν/ω g = ν/ω g =.1 ν/ω g =.3 5 ν/ω g = u=c/1 6 ν/ω g = ν/ω g =.1 ν/ω g =.3 5 ν/ω g = cosθ cosθ u=c/5 ν/ω g =.3 ν/ω g =.1 ν/ω g =.3 ν/ω g =1 cosθ Radio sectral index In brackets ν = collision requency ω g = Larmor requency

76 Observations

77 Cosmic Ray sectrum arriving at earth (Nagano & Watson 2) E -2.7 E -2 Leakage rom galaxy accounts or some o dierence (Hillas 25)

78 Observed radio sectral index v. mean exansion velocity (ollowing Glushak 1985)

79 Sectral steeening suggests quasi-erendicular shocks For random ield orientation, steeening & lattening nearly cancel out Steeening at high velocity might be due to γ (α) (1.5) u=c/1 ν/ω g =.3 ν/ω g =.1 1) exansion into Parker siral 2) magnetic ield amliication jxb stretches ield erendicular to shock normal 5 (1.) ν/ω g =.3 ν/ω g =1 4 (.5) cosθ 1

80 SNR morhology: sectral steeening/lattening SN16 (Chandra) CR electrons (1-1TeV) bi-olar emission (Rothenlug et al 24) Could be: 1) Quasi-erendicular shocks accelerate ewer CR to high energy 2) Poor injection at quasi-erendicular shocks

81 What we think we know, and what we know we don t know What we think we know Galactic CR are accelerated to 1 15 ev by diusive shock acceleration by SNR Streaming CR amliy the magnetic ield which conines CR near shock CR sectra are oten steeer than -4 (E -2 ) What we don t know How CR reach ev When CR are accelerated to what energy at dierent stages o SNR evolution How CR escae SNR without losing energy adiabatically When & where non-linear eects are imortant Why tyically the sectrum is latter than -4 in older SNR Why is the sectrum so straight? Whether second order Fermi acceleration contributes substantially Whether erendicular shocks are good injectors o low energy CR How the above alies extra-galactically - the origin o 1 2 ev CR