Mario A. Riquelme, Anatoly Spitkovsky Department of Astrophysical Sciences, Princeton University Generation of magnetic field upstream of shocks: the cosmic ray current-driven (CRCD) instability

Motivation Observation of X-ray synchrotron emission from rims in SNRs suggest that: Observation of X-ray synchrotron emission from rims in SNRs suggest that: -Electrons are accelerated to ultrarelativistic energies in these environments. -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. -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” (~3x10 15 eV). Such amplification would ease the acceleration of galactic CRs in SNRs until the “knee” (~3x10 15 eV).

Possible mechanisms Idea: field is amplified by the CRs themselves. Idea: field is amplified by the CRs themselves. Resonant instability: Resonant instability: amplification of Alfven waves due to their resonant interaction with CRs ( R L,CR ~  amplification of Alfven waves due to their resonant interaction with CRs ( R L,CR ~  In 2004 A. Bell predicted that plasma waves can be amplified non- resonantly (R L,CR >>  due to the presence of  the cosmic ray current (J CR ). In 2004 A. Bell predicted that plasma waves can be amplified non- resonantly (R L,CR >>  due to the presence of  the cosmic ray current (J CR ). This cosmic ray current-driven (CRCD) instability would have a growth rate much faster than the resonant instability. This cosmic ray current-driven (CRCD) instability would have a growth rate much faster than the resonant instability.

The CRCD instability JeJe v v J e x B tr y x z J cr B0B0 Right handed, circularly polarized when B 0 || J cr

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 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. 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.

The CRCD waves properties One-dimensional + constant CR current One-dimensional + constant CR current We calulate analytically a non-linear, kinetic dispersion relation and obtain that We calulate analytically a non-linear, kinetic dispersion relation and obtain that the waves will grow exponentially with: the waves will grow exponentially with:  max = cB 0 / J cr and  max  =  V a,0 / max until V a ~ V d,cr. Our model allows for evolution of the phase velocity of the wave, V . Our model allows for evolution of the phase velocity of the wave, V . V   V a 2 /V d,cr V   V a 2 /V d,cr This would explain the saturation at V a ~ V d,cr, since at that point the plasma This would explain the saturation at V a ~ V d,cr, since at that point the plasma moves at a velocity of about V d,cr so, from the point of view of the plasma, the moves at a velocity of about V d,cr so, from the point of view of the plasma, the CR current has stopped. CR current has stopped. Our model also predicts a transverse velocity of the plasma V tr  f V a,0, where Our model also predicts a transverse velocity of the plasma V tr  f V a,0, where f=B tr /B 0 (important when multidimensional effects are considered). f=B tr /B 0 (important when multidimensional effects are considered). (confirmed by one-dimensional PIC simulations)

Multidimensional effects The possible initial filamentation: The possible initial filamentation: (V d,cr / V a,0 ) (n cr / n i ) = 4 J cr JeJe B0B0 z x y V a,0 /V d,,cr = 1/100 n cr /n i = 0.04 m i /m e = 10 V d,cr = c (See Niemiec et al. 2008)

Multidimensional effects The possible initial filamentation: The possible initial filamentation: (V d,cr / V a,0 )(n cr / n i ) = 0.4 (V d,cr /V a,0 )(n cr / n i ) Requirement (V d,cr /V a,0 )(n cr / n i ) << 1 J cr JeJe B0B0 z x y V a,0 /V d,,cr = 1/100 n cr /n i = m i /m e = 10 V d,cr = c

The 3D structure of the instability

x y z electrons CRs BoBo Remember: V tr ~ f V a,0, where f=B tr /B 0 V a,0 /V d,,cr = 1/40 n cr /n i = m i /m e = 10 V d,cr = c Growth rate,  decreases but V a ~V d,cr at saturation. Dominant wavelength, d, grows. Plasma accelerates. (CR current still constant)

The 3D structure of the instability z y electrons CRs BoBo x V a,0 /V d,,cr = 1/40 n cr /n i = m i /m e = 10 V d,cr = c Remember: V tr ~ f V a,0, where f=B tr /B 0 Growth rate,  decreases but V a ~V d,cr at saturation. Dominant wavelength, d, grows. Plasma accelerates. (CR current still constant)

Migration to longer wavelengths Since V turb ~ V tr ~ f V a,0, then for a wavelength Since V turb ~ V tr ~ f V a,0, then for a wavelength Time scale of suppression ~ fV a,0. Time scale of suppression ~ fV a,0. Time scale of growth ~    from the dispersion relation)  Time scale of growth ~    from the dispersion relation)     d  ≈  d  fV a,0 =>  2    d  ≈  d  fV a,0 =>  2  d  max ((f/3) 2 + 1)/2 d  max ((f/3) 2 + 1)/2 (solid) where f = B tr /B 0 This migration is faster than suggested by previous MHD studies (Bell 2004). V a,0 /V d,cr = 1/40, V d,cr =c (dash-dotted) V a,0 /V d,cr = 1/20, V d,cr =c/2 (dashed) V a,0 /V d,cr = 1/10, V d,cr =c (dotted)

The back-reaction on the CRs Red and green lines represent B tr 2 for one-dimensional runs with CR’s Lorentz factors,  of 20 and 40. Here V a,0 /V d,cr =1/40 Orange lines show a three-dimensional simulation with  =30 (solid is B x 2 and dotted is B tr 2 ). Here V a,0 /V d,cr =1/20. ≈ V a ~ V d,cr In all three simulations saturation occurs when R L,cr ≈ d. Thus, in general, the CRCD instability will saturate either when V a ~ V d,cr or when R L,cr ~ d, whichever happens first. Also, many CRs are scattered back in the –x direction => efficient scattering mechanism x V cr (semi-isotropic velocity distribution, => V d,cr =c/2)

Application An estimate for the magnetic amplification in SNRs (only considering the most energetic CRs that escape from the remnant): f ((f/3) 2+ 1) = 130 (V sh /10 4 km/sec) 3 (10km/sec/V a,0 ) 2 (  esc /0.05), f ((f/3) 2+ 1) = 130 (V sh /10 4 km/sec) 3 (10km/sec/V a,0 ) 2 (  esc /0.05), where  esc = F E,cr /(n i m i V sh 3 /2). This would imply a typical amplification factor due only to the most energetic “escaping” particles of f ≈ 10.

Conclusions Using PIC simulations, we confirm the existence of the CRCD instability predicted by Bell (2004). Using PIC simulations, we confirm the existence of the CRCD instability predicted by Bell (2004). One-dimensional geometry + constant CR: One-dimensional geometry + constant CR: CRCD waves grow exponentially until V a ~ V d,cr (intrinsic saturation is due to plasma moving at the drift velocity of CRs) CRCD waves grow exponentially until V a ~ V d,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 (B tr ~ B 0 ). Including multidimensional effects we see the formation of significant turbulence in the plasma when the instability becomes non-linear (B tr ~ B 0 ).

Conclusions Turbulence makes the instability evolve rapidly into longer wavelengths ( d ≈ max ((f/3) 2 + 1)/2), where f = B tr /B 0. Turbulence makes the instability evolve rapidly into longer wavelengths ( d ≈ max ((f/3) 2 + 1)/2), where f = B tr /B 0. Turbulence also reduces the growth rate of the field, but intrinsic saturation still happens due to plasma acceleration at V a ~V d,cr. Turbulence also reduces the growth rate of the field, but intrinsic saturation still happens due to plasma acceleration at V a ~V d,cr. However, the back-reaction on the CRs can stop the CR current and cause saturation when However, the back-reaction on the CRs can stop the CR current and cause saturation when R L,cr ≈ d  The magnetic amplification in SNRs (only considering the most energetic, or “escaping” CRs) could reach a factor of ~10. The magnetic amplification in SNRs (only considering the most energetic, or “escaping” CRs) could reach a factor of ~10.

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

Motivation Example: Cassiopeia A (Cas A) Red: infrared (Spitzer). Yellow: optical (Hubble). Blue and green: X-ray (Chandra).

The CRCD waves properties J cr B0B0 JeJe One-dimensional + constant CR current One-dimensional + constant CR current

Numerical confirmation (one-dimensional simulations): Numerical confirmation (one-dimensional simulations): V a,0 /c = 1/10 V a,0 /c = 1/10 Our model also predicts a transverse velocity of the plasma V tr ~ f V a,0, where Our model also predicts a transverse velocity of the plasma V tr ~ f V a,0, where f=B tr /B 0 (important for turbulence generation). f=B tr /B 0 (important for turbulence generation). The CRCD waves properties V d,cr = Solid-yellow: V d,cr =  1c V d,cr  Solid-green: V d,cr  0.9c V d,cr  Solid-red: V d,cr  0.8c V d,cr  Dotted-yellow: V d,cr  0.6c V d,cr  Dotted-green: V d,cr  0.4c V d,cr  Dotted-red: V d,cr  0.2c