Plasma Physics Course

Total: 8 parts covering plasma fundamentals through advanced topics

Part VII, Chapter 6

Cosmic Rays

Fermi acceleration, diffusive shock acceleration, and cosmic ray transport

6.1 The Cosmic Ray Energy Spectrum

Cosmic rays are relativistic charged particles (mostly protons) observed with energies spanning from ∼109 eV to beyond 1020 eV — over 11 orders of magnitude. The differential energy spectrum follows a remarkable power law:

$$\frac{dN}{dE} \propto E^{-\gamma}$$

with spectral index γ ≈ 2.7 below the "knee" (∼3 × 1015 eV) and γ ≈ 3.1 above it. This non-thermal power-law spectrum demands a non-thermal acceleration mechanism.

Key Features of the Spectrum

  • • Knee (∼3 × 1015 eV): γ steepens from 2.7 to 3.1 — galactic accelerators reach their limit
  • • Second knee (∼5 × 1017 eV): heavy nuclei reach their maximum energy
  • • Ankle (∼5 × 1018 eV): γ flattens — transition to extragalactic origin
  • • GZK cutoff (∼5 × 1019 eV): interaction with CMB photons (pγ → Δ⁺ → pπ⁰)

The total energy density of cosmic rays in the Galaxy is:

$$u_{CR} \approx 1\text{ eV/cm}^3 \approx \frac{B_{ISM}^2}{2\mu_0} \approx u_{photon} \approx u_{turb}$$

This remarkable equipartition suggests cosmic rays are dynamically important in the ISM.

6.2 Second-Order Fermi Acceleration

Enrico Fermi (1949) proposed that cosmic rays gain energy by repeatedly bouncing off moving magnetic clouds (mirrors) in the interstellar medium.

6.2.1 Derivation of Energy Gain

Derivation

Step 1. A relativistic particle (energy E, momentum p) collides with a magnetic cloud moving at velocity V. In the cloud frame, the particle energy is (Lorentz transformation):

$$E' = \Gamma(E + Vp\cos\theta) = \Gamma E(1 + \beta\cos\theta)$$

where β = V/c, Γ = (1 − β²)−1/2, and θ is the angle between the particle velocity and the cloud velocity. For ultrarelativistic particles, p ≈ E/c.

Step 2. The particle scatters isotropically in the cloud frame (elastic), then transforms back to the lab frame. The exit angle θ' is uniformly distributed. Transforming back:

$$E_{out} = \Gamma E'(1 + \beta\cos\theta') \approx \Gamma^2 E(1+\beta\cos\theta)(1+\beta\cos\theta')$$

Step 3. Average over isotropic exit angles (⟨cos θ'⟩ = 0) and over the collision rate (head-on collisions are more frequent than overtaking ones). The collision rate ∝ |v − V cos θ|, so ⟨cos θ⟩rate = −β/3:

$$\left\langle\frac{\Delta E}{E}\right\rangle = \Gamma^2(1 + \langle\beta\cos\theta\rangle)(1 + 0) - 1 \approx \frac{4}{3}\beta^2$$
$$\boxed{\left\langle\frac{\Delta E}{E}\right\rangle = \frac{4}{3}\left(\frac{V}{c}\right)^2 \quad \text{(2nd order Fermi)}}$$

The energy gain is proportional to β² — hence "second order." Since V/c ∼ 10−4for ISM clouds, each collision gives ΔE/E ∼ 10−8. This process is very slow and produces spectra steeper than observed. However, it establishes the key concept: repeated stochastic acceleration produces a power law.

6.2.2 Power-Law Spectrum from Stochastic Acceleration

If the average fractional energy gain per collision is α = ⟨ΔE/E⟩ and the probability of remaining in the acceleration region per collision is P, then after k collisions:

$$E = E_0(1+\alpha)^k, \qquad N(>E) \propto P^k$$

Eliminating k:

$$N(>E) \propto \left(\frac{E}{E_0}\right)^{-s} \quad \text{where} \quad s = -\frac{\ln P}{\ln(1+\alpha)} \approx \frac{1-P}{\alpha}$$

The differential spectrum is dN/dE ∝ E−(s+1). A power law emerges naturally frommultiplicative energy gains with a constant loss probability.

6.3 Diffusive Shock Acceleration (1st Order Fermi)

The key improvement came from recognizing that shock waves provide a much more efficient accelerator. At a shock, the upstream and downstream flows always converge, ensuring head-on collisions every time — the energy gain is first order in V/c.

6.3.1 Energy Gain Per Shock Crossing

Derivation

Step 1. A particle crosses the shock from upstream to downstream. In the downstream frame, the upstream gas approaches at velocity ΔV = v1 − v2(in the shock frame, v1 and v2 are upstream and downstream velocities).

Step 2. Each crossing is like a collision with a "cloud" approaching at ΔV. For an upstream → downstream crossing, the particle always meets a head-on converging flow. The fractional energy gain for one complete cycle (upstream → downstream → upstream):

$$\left\langle\frac{\Delta E}{E}\right\rangle_{\text{cycle}} = \frac{4}{3}\frac{\Delta V}{c} = \frac{4}{3}\frac{v_1 - v_2}{c}$$

Step 3. Using the Rankine-Hugoniot compression ratio r = v1/v2 = ρ2/ρ1:

$$\Delta V = v_1 - v_2 = v_1\left(1-\frac{1}{r}\right) = v_s\left(\frac{r-1}{r}\right)$$

where vs = v1 is the shock speed.

$$\boxed{\left\langle\frac{\Delta E}{E}\right\rangle = \frac{4}{3}\frac{v_s}{c}\frac{r-1}{r} \quad \text{(1st order Fermi)}}$$

For a strong shock (Mach ≫ 1), the compression ratio r → (γ+1)/(γ−1) = 4 (for γ = 5/3), giving:

$$\frac{\Delta E}{E} = \frac{4}{3}\frac{v_s}{c}\cdot\frac{3}{4} = \frac{v_s}{c}$$

6.3.2 The Universal Power-Law Index

Derivation of the Spectral Index

Step 1. The probability of escape per cycle is the probability that the particle is advected downstream. The flux of particles crossing the shock is isotropic with rate ∝ c/4. The rate of advection downstream is n·v2. So:

$$P_{esc} = \frac{4v_2}{c} = \frac{4v_1}{rc}$$

Step 2. Using the power-law formula s = (1 − Premain)/α with Premain = 1 − Pesc and α = (4/3)(v1/c)(r−1)/r:

$$s = \frac{P_{esc}}{\alpha} = \frac{4v_1/(rc)}{(4/3)(v_1/c)(r-1)/r} = \frac{3}{r-1}$$

The differential spectral index is γ = s + 1 = (r + 2)/(r − 1).

$$\boxed{\frac{dN}{dE} \propto E^{-\gamma}, \quad \gamma = \frac{r+2}{r-1}}$$

For a strong shock (r = 4): γ = (4+2)/(4−1) = 2.

The Remarkable Result

The spectral index γ = 2 is universal — it depends only on the compression ratio, not on the shock speed, magnetic field, or diffusion coefficient! The observed γ ≈ 2.7 is steeper because of energy-dependent escape from the Galaxy (higher-energy CRs escape faster), which steepens the source spectrum by ∼0.5−0.7.

6.4 Cosmic Ray Transport

6.4.1 Diffusion-Convection Equation

Cosmic rays scatter off magnetic irregularities, undergoing a random walk. Their transport is described by the Parker transport equation:

$$\boxed{\frac{\partial f}{\partial t} + \mathbf{v}\cdot\nabla f = \nabla\cdot(\kappa\nabla f) + \frac{1}{3}(\nabla\cdot\mathbf{v})\frac{\partial f}{\partial \ln p} + Q}$$

where f(x, p, t) is the isotropic distribution function, κ is the spatial diffusion tensor, and Q is the source term. The terms represent: advection, spatial diffusion, adiabatic energy changes (compression/expansion), and injection.

6.4.2 Diffusion Coefficient

Particles with gyroradius rL = pc/(ZeB) scatter resonantly off magnetic fluctuations with wavelength k−1 ∼ rL. The parallel diffusion coefficient is:

$$\kappa_\parallel = \frac{1}{3}\frac{v\, r_L}{F(k_{res})} = \frac{1}{3}v\lambda_{mfp}$$

where F(kres) is the power in magnetic fluctuations at the resonant scale. For a Kolmogorov turbulence spectrum F(k) ∝ k−5/3:

$$\kappa_\parallel \propto E^{1/3} \quad \text{(Kolmogorov)}, \qquad \kappa_\parallel \propto E^{1/2} \quad \text{(Kraichnan)}$$

6.4.3 Confinement in the Galaxy

Cosmic rays are confined within the galactic disk (thickness L ∼ few kpc) by scattering. The escape time for a CR of energy E is:

$$\tau_{esc} \sim \frac{L^2}{\kappa(E)} \propto E^{-\delta}$$

where δ ≈ 1/3−1/2 depending on the turbulence model. Higher-energy CRs escape faster, steepening the observed spectrum from the source spectrum E−2 to:

$$\frac{dN}{dE}\bigg|_{obs} \propto E^{-(2+\delta)} \approx E^{-2.7}$$

6.5 Maximum Energy & the Hillas Criterion

A particle can only be accelerated if its gyroradius fits within the acceleration region. This gives the Hillas criterion for the maximum energy:

Derivation

Step 1. The Larmor radius of a particle with energy E and charge Ze in field B:

$$r_L = \frac{E}{ZeB c} \quad \text{(ultrarelativistic)}$$

Step 2. Confinement requires rL < R (source size). Additionally, the acceleration time must be shorter than the source lifetime or escape time. For DSA:

$$t_{acc} = \frac{E}{\dot{E}} = \frac{E}{(v_s/c)E/t_{cycle}} \sim \frac{\kappa}{v_s^2} \sim \frac{r_L c}{3v_s^2}$$

Step 3. Setting rL = R for the maximum energy:

$$E_{max} = ZeBRv_s/c \approx ZeBR \quad \text{(for } v_s \sim c \text{)}$$
$$\boxed{E_{max} \approx Z e B R \beta_s \approx 10^{18}\, Z\left(\frac{B}{1\,\mu\text{G}}\right)\left(\frac{R}{1\,\text{kpc}}\right)\beta_s \text{ eV}}$$

Hillas Diagram: Candidate Sources

  • • Supernova remnants: B ∼ 10 μG, R ∼ 10 pc → Emax ∼ Z × 1014 eV (knee!)
  • • Galactic center: B ∼ 100 μG, R ∼ 100 pc → Emax ∼ Z × 1017 eV
  • • Radio galaxy lobes: B ∼ 10 μG, R ∼ 100 kpc → Emax ∼ Z × 1019 eV
  • • GRB jets: B ∼ 105 G, R ∼ 1010 m → Emax ∼ Z × 1020 eV
  • • AGN jets: B ∼ 1 G, R ∼ 0.1 pc → Emax ∼ Z × 1020 eV

Supernova remnants can explain cosmic rays up to the knee. Ultra-high-energy cosmic rays (UHECRs) above 1018 eV require extragalactic sources.

6.5.1 The GZK Cutoff

Protons with E > 5 × 1019 eV interact with CMB photons via pion production:

$$p + \gamma_{CMB} \to \Delta^+ \to \begin{cases} p + \pi^0 \\ n + \pi^+ \end{cases}$$

The threshold energy in the proton rest frame requires Eγ' > mπc² ≈ 140 MeV. For a head-on collision with a CMB photon (ε ≈ 6 × 10−4 eV):

$$E_{GZK} \approx \frac{m_\pi m_p c^4}{4\epsilon_{CMB}} \approx \frac{140\times938\times10^6}{4\times6\times10^{-4}} \approx 5\times10^{19} \text{ eV}$$

The mean free path for this interaction is ∼50 Mpc, so UHECRs above the GZK energy must originate within ∼100 Mpc — limiting the "GZK horizon."

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