Mario A. Riquelme, Anatoly SpitkovskyMario A. Riquelme, Anatoly Spitkovsky

Department of Astrophysical Sciences, Princeton Department of Astrophysical Sciences, Princeton UniversityUniversity

Generation of magnetic field upstream of shocks:

the cosmic ray current-driven (CRCD) instability

MotivationMotivation Observation of X-ray synchrotron emission from Observation of X-ray synchrotron emission from

rims in SNRs suggest that:rims in SNRs suggest that:

--Electrons are accelerated to ultrarelativistic Electrons are accelerated to ultrarelativistic energies in these environments.energies in these environments.

-M-Magnetic field agnetic field can be a can be a factor of ~100factor of ~100 bigger bigger than the typical ISM field in the downstream than the typical ISM field in the downstream medium.medium.

Such amplification would ease the acceleration of Such amplification would ease the acceleration of galactic CRs in SNRs until the “knee” (~3x10galactic CRs in SNRs until the “knee” (~3x1015 15 eV).eV).

Possible mechanismsPossible 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 amplification of Alfven waves due to their resonant interaction with CRs ( Rresonant interaction with CRs ( RL,CRL,CR ~ ~

In 2004 A. Bell predicted that plasma waves In 2004 A. Bell predicted that plasma waves can can be amplified be amplified non-resonantlynon-resonantly (R (RL,CRL,CR >> >> due to due to the presence ofthe presence ofthe the cosmic ray currentcosmic ray current (J (JCRCR) ) ..

This This cosmic ray current-driven (CRCD)cosmic ray current-driven (CRCD) instability instability would have a growth rate much faster than the would have a growth rate much faster than the resonant instability.resonant instability.

The CRCD instabilityThe CRCD instability

Je

v

Je x Btr

y

x

z

Jcr

B0

Right handed, circularly polarized when B0 || Jcr

In this study...In this study...

We combine an analytical, kinetic model of the We combine an analytical, kinetic model of the CRCD waves valid in the non-linear regime, with CRCD waves valid in the non-linear regime, with particle-in-cell (PIC) simulations.particle-in-cell (PIC) simulations.

We study the non-linear properties of the We study the non-linear properties of the instability, mainly focused on its possible instability, mainly focused on its possible saturation mechanisms and its applications to the saturation mechanisms and its applications to the case of SNRs’ shocks.case of SNRs’ shocks.

The CRCD waves propertiesThe CRCD waves properties One-dimensional + constant CR currentOne-dimensional + constant CR current We We calulate analytically a non-linear, kinetic dispersion relation and calulate analytically a non-linear, kinetic dispersion relation and

obtain thatobtain that ththee waves will grow exponentially with waves will grow exponentially with::

maxmax = cB = cB0 0 // JJcr cr and and maxmax= = VVa,0 a,0 / / maxmax untiluntil VVa a ~ V~ Vd,crd,cr ..

Our model allows for evolution of the phase velocity of the wave, VOur model allows for evolution of the phase velocity of the wave, V. .

VVVVaa22/V/Vd,crd,cr

This would explain the saturation at This would explain the saturation at VVa a ~ V~ Vd,crd,cr, since at that point the , since at that point the plasma plasma

moves at a velocity of about moves at a velocity of about VVd,crd,cr so, from the point of view of the plasma, so, from the point of view of the plasma, the the

CR current has stopped.CR current has stopped. Our model also predicts a transverse velocity of the plasmaOur model also predicts a transverse velocity of the plasma V Vtr tr f Vf Va,0a,0, ,

where where f=Bf=Btrtr/B/B0 0 (important (important when multidimensional effects are consideredwhen multidimensional effects are considered)).. (confirmed by one-dimensional PIC simulations)(confirmed by one-dimensional PIC simulations)

Multidimensional effectsMultidimensional effects The possible initial The possible initial

filamentation:filamentation:

((VVd,crd,cr/ / VVa,0a,0)) (n(ncr cr / n/ nii)) = 4 = 4

JcrJe

B0

z

xy

Va,0/Vd,,cr = 1/100

ncr/ni = 0.04

mi/me= 10

Vd,cr = c

(See Niemiec et al. 2008)

Multidimensional effectsMultidimensional effects The possible initial The possible initial

filamentation:filamentation:

((VVd,crd,cr // VVa,0a,0))(n(ncr cr / n/ nii)) = 0.4 = 0.4Requirement ((VVd,crd,cr //VVa,0 a,0 ))(n(ncrcr / / nnii)) << 1

JcrJe

B0

z

xy

Va,0/Vd,,cr = 1/100

ncr/ni = 0.004

mi/me= 10

Vd,cr = c

The 3D structure of the The 3D structure of the instabilityinstability

The 3D structure of the The 3D structure of the instabilityinstability

The 3D structure of the The 3D structure of the instabilityinstability

x

y

z

electrons

CRs

Bo

Remember:Remember:VVtr tr ~ f V~ f Va,0a,0, , where f=Bwhere f=Btrtr/B/B00

Va,0/Vd,,cr = 1/40

ncr/ni = 0.0125

mi/me= 10

Vd,cr = c

Growth rate, decreases but Va

~Vd,cr at saturation.

Dominant wavelength, d , grows.

Plasma accelerates.

QuickTime™ and aYUV420 codec decompressor

are needed to see this picture.

(CR current still constant)

The 3D structure of the The 3D structure of the instabilityinstability

z

yelectrons

CRs

Bo

x

Va,0/Vd,,cr = 1/40

ncr/ni = 0.0125

mi/me= 10

Vd,cr = c

Remember:Remember:VVtr tr ~ f V~ f Va,0a,0, , where f=Bwhere f=Btrtr/B/B00

Growth rate, decreases but Va

~Vd,cr at saturation.

Dominant wavelength, d , grows.

Plasma accelerates.

QuickTime™ and aYUV420 codec decompressor

are needed to see this picture.

(CR current still constant)

Migration to longer Migration to longer wavelengthswavelengths

Since VSince Vturbturb ~ V~ Vtrtr ~ f V~ f Va,0a,0, then for a wavelength , then for a wavelength Time scale of suppression Time scale of suppression ~ ~ fVfVa,0a,0 .. Time scale of growTime scale of growth th ~ ~ from the dispersion from the dispersion

relation)relation)

dd ≈ ≈ ddfVfVa,0a,0 => => 2 2

dd maxmax((f/3)((f/3)22 + + 1)/21)/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,cr=c (dash-dotted)

Va,0/Vd,cr = 1/20, Vd,cr=c/2 (dashed)

Va,0/Vd,cr = 1/10, Vd,cr=c (dotted)

The back-reaction on the CRsThe back-reaction on the CRs

Red and green lines represent Btr2 for

one-dimensional runs with CR’s Lorentz factors, of 20 and 40. Here Va,0/Vd,cr=1/40

Orange lines show a three-dimensional simulation with =30 (solid is Bx

2 and dotted is Btr

2). Here Va,0/Vd,cr=1/20.

In all three simulations saturation occurs when RL,cr ≈≈ d. Thus, in general, the CRCD instability will saturate either when VVaa ~ ~ VVd,crd,cr or when RL,cr ~ d, whichever happens first.

Also, many CRs are scattered back in the –x direction => efficient scattering mechanism

x

Vcr(semi-isotropic velocity distribution, => Vd,cr =c/2)

ApplicationApplication

An estimate for the magnetic amplification in SNRs An estimate for the magnetic amplification in SNRs (only(only

considering the considering the most energeticmost energetic CRs that CRs that escapeescape from thefrom the

remnant):remnant):

ff ((f/3)((f/3)2+2+1) = 130 (V1) = 130 (Vshsh/10/1044km/sec)km/sec)33(10km/sec/V(10km/sec/Va,0a,0))22((escesc/0.05),/0.05),

where where esc esc = F= FE,crE,cr/(n/(niimmiiVVshsh3 3 /2). This would imply a/2). This would imply a

typical amplification factor due only to the most typical amplification factor due only to the most energeticenergetic

““escaping” particles of escaping” particles of f ≈ 10f ≈ 10..

ConclusionsConclusions

Using PIC simulations, we confirm the existence of Using PIC simulations, we confirm the existence of the CRCD instability predicted by Bell (2004).the CRCD instability predicted by Bell (2004).

OOne-dimensional geometry ne-dimensional geometry ++ constant CR constant CR:: CRCD waves grow eCRCD waves grow exponentialxponentiallyly until until VVa a ~ V~ Vd,cr d,cr

((intrinsicintrinsic saturationsaturation is due to plasma moving at the is due to plasma moving at the drift velocity of CRs)drift velocity of CRs)

IncludIncludinging multidimensional effects we see the multidimensional effects we see the formation of significant formation of significant turbulenceturbulence in the plasma in the plasma whenwhen the instability becomes non-linear the instability becomes non-linear (B (Btr tr ~ B~ B00))..

ConclusionsConclusions

TTurbulenceurbulence makes the instability evolve rapidly into makes the instability evolve rapidly into longer wavelengthslonger wavelengths ( (dd ≈ ≈ maxmax((f/3)((f/3)22 + 1)/2 + 1)/2), where f = ), where f = BBtrtr/B/B00..

Turbulence also reduces the growth rate of the field, but Turbulence also reduces the growth rate of the field, but intrinsic saturationintrinsic saturation still happens due to plasma still happens due to plasma acceleration at Vacceleration at Va a ~V~Vd,crd,cr..

However, the back-reaction on the CRs can However, the back-reaction on the CRs can stop the CR stop the CR current and current and cause saturation cause saturation whenwhen RL,cr ≈ d

The magnetic amplification in SNRs (only considering The magnetic amplification in SNRs (only considering the the most energeticmost energetic, or , or “escaping”“escaping” CRs) could reach a CRs) could reach a factor of factor of ~~1010..

ConclusionsConclusions

Open questions: -Does the CR current really exist? i..e. are Open questions: -Does the CR current really exist? i..e. are CR CR

only only positively charged particles?positively charged particles? (injection (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 amplification due to the diffusing CRs current?current?

-Why in almost all the cases (except -Why in almost all the cases (except SN1006) SN1006)

the amplification seems to happen the amplification seems to happen

symmetrically all around the remnant?symmetrically all around the remnant?

QuickTime™ and aYUV420 codec decompressor

are needed to see this picture.

MotivationMotivationExample: Cassiopeia A (Cas A) Red: infrared

(Spitzer).

Yellow: optical (Hubble).

Blue and green: X-ray (Chandra).

The CRCD waves propertiesThe CRCD waves properties

Jcr

B0

Je

One-dimensional + constant CR One-dimensional + constant CR currentcurrent

Numerical confirmation (one-dimensional simulations):Numerical confirmation (one-dimensional simulations):

VVa,0 a,0 /c = 1/10/c = 1/10

Our model also predicts a transverse velocity of the plasmaOur model also predicts a transverse velocity of the plasma V Vtr tr ~ f ~ f

VVa,0a,0, where , where f=Bf=Btrtr/B/B0 0 (important for turbulence generation)(important for turbulence generation)..

The CRCD waves propertiesThe CRCD waves properties

Solid-yellow: VVd,crd,cr = =1c

Solid-green: VVd,crd,cr0.9c

Solid-red: VVd,crd,cr0.8c

Dotted-yellow: VVd,crd,cr0.6c

Dotted-green: VVd,crd,cr0.4c

Dotted-red: VVd,crd,cr0.2c