Mario A. Riquelme, Anatoly Spitkovsky

1
Generation of magnetic field upstream of
shocks the cosmic ray current-driven (CRCD)
instability

• Mario A. Riquelme, Anatoly Spitkovsky
• Department of Astrophysical Sciences, Princeton
  University

2
Motivation

• Observation of X-ray synchrotron emission from
  rims in SNRs suggest that
• -Electrons are accelerated to
  ultrarelativistic energies in these environments.
• -Magnetic field can be a factor of 100
  bigger than the typical ISM field in the
  downstream medium.
• Such amplification would ease the acceleration of
  galactic CRs in SNRs until the knee (3x1015
  eV).

3
Possible mechanisms

• Idea field is amplified by the CRs themselves.
• Resonant instability
• amplification of Alfven waves due to their
  resonant interaction with CRs ( RL,CR l ).
• In 2004 A. Bell predicted that plasma waves can
  be amplified non-resonantly (RL,CR gtgt l) due to
  the presence of the cosmic ray current (JCR) .
• This cosmic ray current-driven (CRCD) instability
  would have a growth rate much faster than the
  resonant instability.

4
The CRCD instability
Right handed, circularly polarized when B0 Jcr
z
Je x Btr
Dv
B0
Jcr
y
Je
x
5
In this study...

• We combine an analytical, kinetic model of the
  CRCD waves valid in the non-linear regime, with
  particle-in-cell (PIC) simulations.
• We study the non-linear properties of the
  instability, mainly focused on its possible
  saturation mechanisms and its applications to the
  case of SNRs shocks.

6
The CRCD waves properties

• One-dimensional constant CR current
• We calulate analytically a non-linear, kinetic
  dispersion relation and obtain that
• the waves will grow exponentially with
• lmax cB0 / Jcr and gmax 2p Va,0 /
  lmax until Va Vd,cr .
• Our model allows for evolution of the phase
  velocity of the wave, Vf.
• Vf ? Va2/Vd,cr
• This would explain the saturation at Va
  Vd,cr, since at that point the plasma
• moves at a velocity of about Vd,cr so, from
  the point of view of the plasma, the
• CR current has stopped.
• Our model also predicts a transverse velocity of
  the plasma Vtr ? f Va,0, where
• fBtr/B0 (important when multidimensional
  effects are considered).
• (confirmed by one-dimensional PIC simulations)

7
Multidimensional effects
Va,0/Vd,,cr 1/100 ncr/ni 0.04 mi/me 10 Vd,cr
c
Jcr
z
Je

• The possible initial filamentation
• (Vd,cr/ Va,0) (ncr / ni) 4

B0
y
x
(See Niemiec et al. 2008)
8
Multidimensional effects
Va,0/Vd,,cr 1/100 ncr/ni 0.004 mi/me
10 Vd,cr c
Jcr
z
Je

• The possible initial filamentation
• (Vd,cr / Va,0)(ncr / ni) 0.4

B0
y
x
Requirement (Vd,cr /Va,0 )(ncr / ni) ltlt 1
9
The 3D structure of the instability
10
The 3D structure of the instability
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The 3D structure of the instability
y
(CR current still constant)
electrons
x
CRs
z
Bo
Va,0/Vd,,cr 1/40 ncr/ni 0.0125 mi/me
10 Vd,cr c
Remember Vtr f Va,0, where fBtr/B0
Growth rate, g, decreases but Va Vd,cr at
saturation. Dominant wavelength, ld ,
grows. Plasma accelerates.
12
The 3D structure of the instability
y
x
CRs
(CR current still constant)
electrons
Bo
z
Va,0/Vd,,cr 1/40 ncr/ni 0.0125 mi/me
10 Vd,cr c
Remember Vtr f Va,0, where fBtr/B0
Growth rate, g, decreases but Va Vd,cr at
saturation. Dominant wavelength, ld ,
grows. Plasma accelerates.
13
Migration to longer wavelengths

• Since Vturb Vtr f Va,0, then for a
  wavelength l
• Time scale of suppression l /fVa,0 .
• Time scale of growth g-1(l) (from the
  dispersion relation).
• 
  g-1(ld) b ld/fVa,0 gt b ? 2
• 
  
• 
  

ld ? lmax((f/3)2 1)/2
(solid)
where f Btr/B0 This migration is faster than
suggested by previous MHD studies (Bell
2004). Va,0/Vd,cr 1/40, Vd,crc
(dash-dotted) Va,0/Vd,cr 1/20, Vd,crc/2
(dashed) Va,0/Vd,cr 1/10, Vd,crc (dotted)
14
The back-reaction on the CRs
Vcr
(semi-isotropic velocity distribution, gt Vd,cr
c/2)
x
Red and green lines represent Btr2 for
one-dimensional runs with CRs Lorentz factors,
G, of 20 and 40. Here Va,0/Vd,cr1/40 Orange
lines show a three-dimensional simulation with
G30 (solid is Bx2 and dotted is Btr2). Here
Va,0/Vd,cr1/20.
In all three simulations saturation occurs when
RL,cr ld. Thus, in general, the CRCD
instability will saturate either when Va Vd,cr
or when RL,cr ld, whichever happens
first. Also, many CRs are scattered back in the
x direction gt efficient scattering mechanism
15
Application

• An estimate for the magnetic amplification in
  SNRs (only
• considering the most energetic CRs that escape
  from the
• remnant)
• f ((f/3)21) 130 (Vsh/104km/sec)3(10km/sec/V
  a,0)2(hesc/0.05),
• where hesc FE,cr/(nimiVsh3 /2). This would
  imply a
• typical amplification factor due only to the most
  energetic
• escaping particles of f 10.

16
Conclusions

• Using PIC simulations, we confirm the existence
  of the CRCD instability predicted by Bell (2004).
• One-dimensional geometry constant CR
• CRCD waves grow exponentially until Va
  Vd,cr (intrinsic saturation is due to plasma
  moving at the drift velocity of CRs)
• Including multidimensional effects we see the
  formation of significant turbulence in the plasma
  when the instability becomes non-linear (Btr
  B0).

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Conclusions

• Turbulence makes the instability evolve rapidly
  into longer wavelengths (ld lmax((f/3)2
  1)/2), where f Btr/B0.
• Turbulence also reduces the growth rate of the
  field, but intrinsic saturation still happens due
  to plasma acceleration at Va Vd,cr.
• However, the back-reaction on the CRs can stop
  the CR current and cause saturation when RL,cr
  ld.
• The magnetic amplification in SNRs (only
  considering the most energetic, or escaping
  CRs) could reach a factor of 10.

18
Conclusions

• Open questions -Does the CR current really
  exist? i..e. are CR
• only positively
  charged particles? (injection
• problem).
• -What happens in
  the region close to the
• shock? Can we
  expect further magnetic
• amplification due
  to the diffusing CRs current?
• -Why in almost all
  the cases (except SN1006)
• the amplification
  seems to happen
• symmetrically all
  around the remnant?

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20
Motivation
Example Cassiopeia A (Cas A)
Red infrared (Spitzer). Yellow optical
(Hubble). Blue and green X-ray (Chandra).
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The CRCD waves properties

• One-dimensional constant CR current

B0
Jcr
Je
22
The CRCD waves properties

• Numerical confirmation (one-dimensional
  simulations)
• 
  Va,0 /c
  1/10
• Our model also predicts a transverse velocity of
  the plasma Vtr f Va,0, where
• fBtr/B0 (important for turbulence
  generation).

Solid-yellow Vd,cr 1c Solid-green
Vd,cr 0.9c Solid-red Vd,cr
0.8c Dotted-yellow Vd,cr 0.6c Dotted-green
Vd,cr 0.4c Dotted-red Vd,cr 0.2c