The Plasma Physics of Cosmic Rays Ellen Zweibel [email protected]

Departments of Astronomy & Physics University of Wisconsin, Madison and Center for Magnetic Self-Organization in Laboratory and Astrophysical Plasmas

The Plasma Physics of Cosmic Rays – p.1/23

Overarching Question How do cosmic rays interact (collisionlessly) with their environment? How are cosmic rays accelerated? What determines their spectrum? How do cosmic rays propagate away from their sources? What controls the confinement time & energy density of cosmic rays in galaxies? Can cosmic rays transfer momentum & energy to the thermal background and to magnetic fields? We won’t answer all of these questions. We just do linear theory. The Plasma Physics of Cosmic Rays – p.2/23

A Good Time for This

Fermi image of the LMC in γ -rays. Abdo et al. 2010. The Plasma Physics of Cosmic Rays – p.3/23

Historical Roots Several lines of evidence suggest cosmic rays are strongly scattered Nearly isotropic at Earth Abundance of spallation products suggests confinement length ∼ 103 X size of Galaxy If unconfined, energy source would be comparable to luminosity of the Galaxy There are several ways cosmic rays can be scattered. Probably all contribute.

The Plasma Physics of Cosmic Rays – p.4/23

Fieldline Wandering

Assuming the cosmic rays follow the fieldlines, a tangled field will confine them. But we know Br /Bu ∼ 1 ≪ 103 .

The Plasma Physics of Cosmic Rays – p.5/23

Scattering By Galactic Turbulence

If the spectrum of fluctuation wavenumbers is present, cosmic rays follow the long wavelength ones, ignore the short wavelength ones, and interact most strongly with the gyroresonant ones, with kv = ±ωc .

The Plasma Physics of Cosmic Rays – p.6/23

Alfven Wave Cascade !

Alfven waves in the inertial range of an MHD cascade have k⊥ /kk ≫ 1; reduces scattering efficiency. Could scatter from

other modes. The Plasma Physics of Cosmic Rays – p.7/23

Self-Generated Turbulence Generated by CR anisotropy. min ) Growth rate Γcr = ωci ncr (>E ni



vD vA

 −1 ,

Emin ≡ mcωci /k . Driven by super-Alfvenic streaming. Waves are excited by gyroresonant particles: kvk = ±ωc

Waves propagate in direction of streaming. Only appears to work up to ∼ 100 GeV because growth rate decreases as energy increases & is overwhelmed by damping.

Assumes Γcr /kvA ≪ 1. The Plasma Physics of Cosmic Rays – p.8/23

Diffusion Coefficient Gyroresonant turbulence of relative amplitude δB/B → scattering rate  2 δB ν ∼ ωc B and corresponding spatial diffusivity c2 . κ∼ 3ν δB/B ∼ 10−3 suffices to explain propagation.

The Plasma Physics of Cosmic Rays – p.9/23

Implications of Self-Confinement In a steady state, energy & momentum pass from cosmic rays to waves to thermal plasma Force k to B is -∇k Pcr Heating rate is -vA · ∇Pcr

Calculate κ by balancing wave driving against damping Ion - neutral friction Nonlinear Landau damping Turbulent damping

The Plasma Physics of Cosmic Rays – p.10/23

Example

Left: a simple model cloud with n increasing. T , ne /n decreasing. Right: Wave damping by ion-neutral friction sets up a large ∇Pcr which drives strong scattering in a thin skin beyond which there are no waves. From Everett & EZ 2011.

The Plasma Physics of Cosmic Rays – p.11/23

Strong Self-Confinement On scales ≫ mean free path, cosmic rays can be described as a fluid, coupled to the thermal fluid, which Modifies shock structure Drives galactic winds Can destabilize acoustic waves Affects Parker & buoyancy instabilities etc...

The Plasma Physics of Cosmic Rays – p.12/23

Strong Growth What if Γcr /kvA ≥ 1? This is equivalent to UB vD . ≤ Ucr c

This suggests that for large cosmic ray fluxes/weak magnetic fields, the instability is modified.

The Plasma Physics of Cosmic Rays – p.13/23

Bell Instability When the strong growth condition UB vD ≤ Ucr c

is satisfied, a nonresonant instability grows faster than the resonant instability. The fastest growing mode has kf gm rcr > 1 and is driven by the electron current that is assumed

to cancel the cosmic ray current in the unperturbed state. Simulations show strong magnetic amplification, filamentation, secondary instabilities.(A. Bell, J. Kirk, B. Reville, A. Spitkovsky; EZ& J. Everett 2010) The Plasma Physics of Cosmic Rays – p.14/23

Schematic Domains of Instability Magnetic Field Strength Too Large for Non-Resonant Instability

kwicercr > 1

nt na o s y Re bilit a n t No l Ins l Be

Thermally Modified Bell Instability

Thermal Ions Unmagnetized

Log10@ Magnetic Field StrengthD

No Non-Resonant Instability

kwiceri < 1

Log10@Cosmic Ray FluxD

Top wedge: resonant instability. Bottom wedge: thermally modified nonresonant instability. Middle wedge: “classical" nonresonant instability. Left & right boundaries are the magnetization conditions for cosmic rays & thermal ions.

The Plasma Physics of Cosmic Rays – p.15/23

Dispersion Relation c2 k 2 ω2

c2

2 ωpi



2 ωpe



ncr (ω − kvD ) ζr Z − ωci = 2 2 ni ω ωkvi vA

(1)

ωkve

Z

ωci + ω kvi



+



,

ωce + ω kve

where we have dropped the displacement current, Z is the plasma dispersion function, representing the thermal plasma response (2)

1 Z(z) ≡ √ π

Z

∞ −∞

2 −s e

s−z

ds,

and ζr measures the cosmic ray response (Z2003). The Plasma Physics of Cosmic Rays – p.16/23

Solutions 10-3 10-4 Ωi -5 -1 10 @rad s D 10-6 10-7 -2 10

10-1

1

101 krcr

102

103

104

Growth rate vs scaled wavenumber for B = 3µG, ni = 1 cm−3 , ncr vD = 104 cm−2 s−1 , and T = 104 K (solid), T = 106 K (long dashed), T = 107 K (short dashed). The Plasma Physics of Cosmic Rays – p.17/23

Standard Bell Instability When 2 3 vA vA ni < ncr vD < ni 2 c vi

the fastest growing mode has scaled wavenumber krcr

c 1 ncr cvD −2 = kBell ∝ B ≡ 2 ωci 2 ni vA

and frequency ω = ωBell = kBell vA ∝ B 0 .

The Plasma Physics of Cosmic Rays – p.18/23

Thermally Modified Instability When 3 vA ncr vD > ni 2 , vi

the fastest growing mode is at scaled wavenumber c = vi



ωwice ∼ ωci



kwice rcr

1/3

∝ B0,

2/3

∝ B.

ncr vD ni vi

and has growth rate ncr vD ni vi

The Plasma Physics of Cosmic Rays – p.19/23

Growth Rates – ISM -4

-13.5 -16.2

-8.1 -10.8

-5

-6

-3.6

Log@BGaussD

-4.5 -5.4 -11.8 -14.4

-7

-6.3

-9

-8

-9

-10 -7.2

-9.9 -12.6

-15.3

-8

-6

-4 -2 0 -2 -1 Log@ncrvDcm s D

2

4

Contour plot of maximum growth rate for ICM parameters, ni = 1 cm−3 , T = 104 K . The Plasma Physics of Cosmic Rays – p.20/23

Application to Shock Acceleration For maximally efficient acceleration the time to cross c2 −1 the cosmic ray layer is v2 ωci . D

The Bell instability growth time is less than this if 2 vD ncr vD ni vA > c2 . The thermally modified Bell instability growth time is 3 vD ncr vD less if ni vi > c3 .

The Plasma Physics of Cosmic Rays – p.21/23

Examples – ISM An estimated young SNR cosmic ray flux of 104 cm−2 s−1 excites the Bell instability for 1.1µG < B < 87µG if ni = 1, T = 104 K . The Galactic flux (ncr vD ∼ 10−2 ) excites the standard Bell instability for 1.1 × 10−8 G < B < 8.7 × 10−8 G and the modified Bell instability for B < 1.1 × 10−8 G.

In low density ISM (ni = 3 × 10−3 ), if ncr /ni ∼ 10−5 , ωBell > ωci vS2 /c2 requires B < 1.3µG.

The Plasma Physics of Cosmic Rays – p.22/23

Summary The classical gyroresonant cosmic ray streaming instability couples cosmic rays & thermal gas in galactic environments. High cosmic ray fluxes and low magnetic fields allow faster growing, nonresonant instability which has been studied in shock acceleration context, where it amplifies B. At lower fieldstrengths, e.g. in young galaxies, this could be the dominant instability in the ISM, ICM, or IGM at large. The thermally modified instability is probably most relevant, & its growthrate scales with ωci . Could nonresonant instabilities amplify weak magnetic fields in global environments? The Plasma Physics of Cosmic Rays – p.23/23