Shock geometry and particle acceleration timescales in gradual SEP

events e.g., 1/20/05 event

G.P. Zank, Gang Li, and Olga Verkhoglyadova

Institute of Geophysics and Planetary Physics (IGPP)

University of California, Riverside

Also V. Florinski, Q. Hu, D. Lario, and CW. SmithCritical to understanding the particle acceleration timescale in diffusive shock acceleration is the nature of the turbulence upstream of the shock.

This can take essentially two forms: the one is turbulence excited self-consistently by the particles experiencing energization themselves, and the second is pre-existing in situ turbulence.

perp

parallel

Mewaldt et al. 2005)

Saiz et al. inferred an interplanetary scattering mean free path of 0.9 + 0.1 AU at the start of the event.  Cane et al. (Fall 2005 AGU Abstract #SH21A-05) have suggested that a low level of scattering might reflect an extremely quiet interplanetary medium within an interplanetary CME that originated less than 2 days earlier from the same active region.

The problems: 1) High injection threshold necessary 2) No self-excited waves

2

00 0 00 02 2

00 0

; ;3

4 4; ,

3 3/g

p

f f p u f f A Au u A A

t r r r r r t r

p pB p cp r c

A k B eBm c p p p

Quasi-parallel shocks:

Understanding particle acceleration at perpendicular shocks has been complicated by the absence of a good understanding of field-line “wandering” or turbulence, which amounts to understanding the perpendicular spatial diffusion coefficient.

Quasi-perpendicular shocks:

INTERPLANETARY MAGNETIC FIELD

TERMINATION SHOCKU_down

U_up

Turbulence in the Solar Wind

• Matthaeus et al, 1990: 2-point correlation matrix in co-ord system parallel to mean magnetic field (see also Carbone et al. 2000).• Parallel mean free path for cosmic rays in the inner heliosphere (e.g. Bieber et al., 1994)

• Conclusion: turbulence in the SW comprised of admixture of 2D and SLAB

0.01

Left: Plot of the parallel (solid curve) and perpendicular mfp (dashed curve) and the particle gyroradius (dotted) as a function of energy for 100 AU (the termination shock) and 1 AU (an interplanetary shock). Right: Different format - plots of the mean free paths at 1 AU as a function of particle gyroradius and now normalized to the correlation length. The graphs are equivalent to the ratio of the diffusive acceleration time to the Bohm acceleration time, and each is normalized to gyroradius. Solid line corresponds to normalized (to the Bohm acceleration time scale) perpendicular diffusive acceleration time scale, the dashed-dotted to parallel acceleration time scale, and the dashed to Bohm acceleration time scale (obviously 1).

Maximum and injection energies

Remarks: 1) Parallel shock calculation assumes wave excitation implies maximum energies comparable 3) Injection energy at Q-perp shock much higher than at Q-par therefore expect signature difference in composition

Parallel and perpendicular diffusion coefficients

Can utilize interplanetary shock acceleration models of Zank et al., 2000 and Li et al., 2003, 2005 for perpendicular shock acceleration to derive spectra, intensity profiles, etc.

K_perp = blackK_par = red Remarks: 1) K_par includes

waves 2) The diffusion coefficients as a function of kinetic energy at various heliocentric distances. The inclusion of wave self-excitation makes K_parallel significantly smaller than K_perp at low energies, and comparable at high energies.

Intensity profiles emphasize important role of time dependent maximum energy to which protons are accelerated at a shock and the subsequent efficiency of trapping these particles in the vicinity of the shock. Compared to parallel shock case, particle intensity reaches plateau phase faster for a quasi-perpendicular shock – because K_perp at a highly perpendicular shock is larger than the stimulated K_par at a parallel shock, so particles (especially at low energies) find it easier to escape from the quasi-perpendicular shock than the parallel shock.

The time interval spectra for a perpendicular (solid line) and a parallel shock (dotted line). From left to right and top to bottom, the panels correspond to the time intervals t = (1 - {8/9})T, t = (1 - {7/9})T, … t = (1 -{1/9}) T, where T is the time taken for the shock to reach 1 AU. Note particularly the hardening of the spectrum with increasing time for the perpendicular shock example.

Observations and theory – 20 Jan ‘05

perp

parallel

Intensity profiles emphasize important role of time dependent maximum energy to which protons are accelerated at a shock and the subsequent efficiency of trapping these particles in the vicinity of the shock. Compared to parallel shock case, particle intensity reaches plateau phase faster for a quasi-perpendicular shock – because K_perp at a highly perpendicular shock is larger than the stimulated K_par at a parallel shock, so particles (especially at low energies) find it easier to escape from the quasi-perpendicular shock than the parallel shock.

11 hours 24 hours

31 hours 36 hours (at 1 AU)

Observations:“Standard” quasi-parallel example

From top to bottom (1 - 5), plot of the magnetic field (top, 1) upstream and downstream of a shock showing three intervals upstream for which the magnetic power spectral density was computed (2), (3) the energetic ion fluxes at various energy intervals using 1-minute spin-averaged intensities as measured by the LEMS120 telescope of the EPAM instrument on board ACE, (4) energy spectra measured at times indicated by the vertical lines in the intensity panel, and (5) the evolution of the spectral index gamma obtained by fitting a power-law E^{-\gamma} to the intensities measured in the four lowest energy channels (47-321 keV). The time of the shock passage is indicated by the gray vertical line. The shock parameters are given in the panel (4).

Observations

Thanks to David Lario for particle data

Perpendicular shock Quasi-perp shock

CONCLUDING REMARKS

• The NLGC model shows that the perpendicular mfp is related simpy to the parallel mfp.

• The perpendicular acceleration timescale at high energies is significantly smaller than the Bohm acceleration timescale, which implies rapid acceleration at perpendicular shocks.

• The perpendicular mfp at 100 AU is not significantly different from that at 1 AU, especially at high energies. One implication is that the perpendicular acceleration timescale for perpendicular shocks is similar throughout the heliosphere. By contrast, three orders of magnitude separate the parallel mfp at 100 AU and 1 AU.

• Highly perpendicular interplanetary shocks are more likely to accelerate a pre-existing energetic particle population (possibly flare accelerated particles) than in situ solar wind particles.

• With the inclusion of wave excitation at parallel shocks, the acceleration timescale for parallel shocks can be faster than at a perpendicular shock -- a subtlety that has not been appreciated before. Consequently, we find that parallel interplanetary shocks can accelerate protons to higher energies than comparable perpendicular interplanetary shocks. The maximum energy to which a parallel shock can accelerate protons falls fairly rapidly with increasing heliocentric distance whereas it remains almost constant for highly perpendicular shocks.

CONCLUDING REMARKS

• Find that the evolving spectra and intensity profiles observed at 1 AU can be quite different for parallel and perp cases. Parallel shock spectra extend to higher energies but highly perpendicular shock accelerated spectra are more power law-like than the more exponential parallel shock spectra. Easier injection at the parallel shock ensures that corresponding spectra have a greater relative number density than those corresponding to highly perpendicular shocks. The absence of a self-excited wave field at highly perpendicular shocks implies more efficient particle trapping at the parallel shock. Consequently, rise times for intensity profiles are much more abrupt for particles accelerated at a highly perpendicular shock than for quasi-parallel shocks. Thus, a plateau is established much more rapidly for highly perpendicular shocks, and, for similar reasons, intensity profiles can decay more rapidly for high energy particles, in marked contrast to the parallel shock case.

• Observations support notion of particle acceleration at shocks in absence of stimulated wave activity.

Some basics:

For diffusion theory to be applicable at shocks, probability of escape downstream of shock must be small:

Vinj

n

n c

V

rV

r

RS||T||

40

4 190

12

12

2

Parallel shock

Perpendicular shock

Evidently, a critical dependence on the strength of scattering.

INTERPLANETARY MAGNETIC FIELD

TERMINATION SHOCKU_down

U_up

ACCELERATION TIME SCALE

acc xx

p

dp

dt

r

u r FHG

IKJ

1 3

1

1

2

b g

|| Bohmgvr

3

||

1 2cHard sphere scattering: c gr 3 v

Particle scattering strength

Weak scattering: 1gr 2 1gr

Strong scattering: 1gr 2 2B B

Acceleration time at quasi-par shock much greater than at quasi-perp shock.

TRANSVERSE COMPLEXITYQin et al. [2002a,b] - perpendicular diffusion can occur only in the presence of a transverse complex magnetic field. Flux surfaces with high transverse complexity are characterized by the rapid separation of nearby magnetic field lines.

Slab turbulence only – no development of transversely complex magnetic field.

Superposition of 80% 2D and 20% slab turbulence, with the consequent development of a transversely complex magnetic field.

Pressure Balanced Structures RevisitedEquilibrium solutions to ideal MHD

2

2

10

4

18

4

0 8 .

and

0

Pc

c

P B

P B const

P P

U U J B

B J

B B

B B

B J

0 0

0

0

2

ˆ

, ,

0

,D slab

B

x y z

x y z

B z

B B b

b B

b b b

Hence B and J lie along surfaces of constant pressure.

Consider axisymmetric magnetic fluctuations b transverse to a uniform mean magnetic field:

Consider simple 2-component model “slab” + “2D”

0

0P

Pz

b

B Passive scalar equation for pressure

INTEGRAL FORM OF THE NONLINEAR GUIDING CENTER THEORY

Matthaeus, Qin, Bieber, Zank [2003] derived a nonlinear theory for the perpendicular diffusion coefficient, which corresponds to a solution of the integral equation

32 2

2 2 20 03

xxxx

xx z zz

S da v

B v k k

k k

2D slabxx xx z xxS S k k S k k

Superposition model: 2D plus slab

Solve the integral equation approximately (Zank, Li, Florinski, et al, 2004):

2/3

2/3 1/322 22/3 22 2/3 1/3

2 2/3 1/32/32 2/3 1/32 20 2 0

min , 33 1 4.33 3 3.091 3

3

slabD slab

xx D slab slabslabD

a Cb ba C H H

B b B

modeled according to QLT.

ANISOTROPY AND THE INJECTION THRESHOLD

Diffusion tensor: 2 2sin cosxx bn bn

Since , the anisotropy is defined by

1/ 22 2 2 2 2

2 2 2

( cos )sin31

3 ( sin cos )d bn bn

bn bn

u q

v

For a nearly perpendicular shock sin 1bn

1/ 22 2 2

2 2 2

cos3 1

( cos )1

g bn

bn

ru

v r

DIFFUSION APPROXIMATION VALID IF 1

ANISOTROPY AND THE INJECTION THRESHOLD

Anisotropy as a function of energy(r = 3)

Injection threshold as a function of angle for 1

Remarks: 1)Anisotropy very sensitive to2) Injection more efficient for quasi-parallel and strictly perpendicular shocks

90bn

PARTICLE ACCELERATION AT PERPENDICULAR SHOCKS

• STEP 1: Evaluate K_perp at shock using NLGC theory instead of wave growth expression.

2/32

2/3 22 2/3 1/322

0

2/31/32 2

2/3 1/32/3 2/3 1/32 22 0

33

min , 31 4.33 3 3.091 3

3

D

xx D

slabslab

slab slabslabD

bva C

B

a C bH H

b B

2 2 22

2 3

20% :80%slab Db b b

b R

Parallel mfp evaluated on basis of QLT (Zank et al. 1998.

PARTICLE ACCELERATION AT PERPENDICULAR SHOCKS

• STEP 2: Evaluate injection momentum p_min by requiring the particle anisotropy to be small.

1/ 22 2 2

2 2 2

cos13

( cos )1

g bninj

bn

rv u

r

PARTICLE ACCELERATION AT PERPENDICULAR SHOCKS

• STEP 3: Determine maximum energy by equating dynamical timescale and acceleration timescale.

R t

R t

q t

up d p

p

pafaf

af a f a fb g lnmax

z12

inj

3/ 4

2 1/32 1/3

max min2

14

3 2.243slabsh slabbmV r eB R

p pr B c R

Remarks: 2

12

slabbR

B

22 3shV Rt

R

Like quasi-parallel case, p_max decreases with increasing heliocentric distance.