Acceleration mechanisms

1Cosmic-ray accelerators¶

1.1Maximum energy: the Hillas criterion¶

A necessary condition for accelerating particles up to a given maximum energy is that their astrophysical source is large enough, or magnetised enough, to confine them for at least one radius of gyration. This relatively simple criterion, attributed to Michael Hillas, is used to classify astrophysical accelerators in a so-called Hillas diagram, as shown in figure 1.

The gyro-radius, $r_\mathrm{g}$, of a particle of four-velocity $(\gamma, \gamma\vec{\beta})$ derives from the equation of motion $\frac{\dd}{\dd t}\big(\gamma m \vec{v} \big) = Ze \vec{v}\times \vec{B}$, where $m$ and $Z$ are the mass and charge of the particle. As the magnetic field does no work, $\gamma m$ is constant so that the particle moves along an helicoidal trajectory characterized by $\gamma m \frac{v_\perp^2}{r_\mathrm{g}} = Z e B v_\perp$, where $p_\perp = \gamma m v_\perp$ is the momentum in the plane perpendicular to the $B$ field. Thus

where $R = \frac{p_\perp c}{Z e} \approx \frac{E}{Z e}$ is the magnetic rigidity in volts.

The Hillas criterion states that accelerating a particle up to a given rigidity $R$ within a region of size $L$ is only possible if $r_\mathrm{g}<r$, which imposes $R < r B c$. Accounting for relativistic bulk motion of the emitting region $(\Gamma, \Gamma\vec{\beta})$, the relativistic Hillas criterion on the observed rigidity reads

The classes of Galactic sources satisfying this condition up to the magnetic rigidity corresponding to the knee and second knee of the cosmic-ray spectrum, i.e. a proton energy of $\approx 4 \times 10^{15}\,\mathrm{eV}$ and an energy of $\approx 10^{17}\,\mathrm{eV}$ for fully ionized iron are shown in figure 1. Extragalactic sources are also shown and can be compared to the Hillas criterion up to the ankle and cut-off of the cosmic-ray spectrum, i.e. for maximum proton energy of $\approx 4 \times 10^{18}\,\mathrm{eV}$ and iron energy of $\approx 10^{20}\,\mathrm{eV}$.

1.2Cosmic-ray production rate¶

Cosmic-ray accelerators must not only be able to achieve a given maximum rigidity, but must also be sufficiently luminous that their cumulative emission reproduces the observed intensity, $I_\mathrm{CR} = \frac{c}{4\pi} \varepsilon_\mathrm{CR}$.

A particularly useful quantity for studying the origin of Galactic cosmic rays is the energy production rate: $w_\mathrm{GCR} = \varepsilon_\mathrm{GCR}({>}\,1\,\mathrm{GeV})/\tau_\mathrm{esc}$, where $\varepsilon_\mathrm{GCR}({>}\,1\,\mathrm{GeV})\approx 1.2 \times 10^6\,\mathrm{eV\,m}^{-3} \approx 6 \times 10^{45}\,\mathrm{J\,kpc}^{-3}$ was determined in exercise 1 and where $\tau_\mathrm{esc} \gtrsim 15\,$Myr is the residence time of cosmic rays in the Galaxy. This residence time is estimated from so called cosmic-ray clocks (see e.g. Lipari (2014)), e.g. through the ratio between ¹⁰Be ($t_{1/2} = 1.4\,$Myr) and its stable isotope ⁹Be, both formed by the fragmentation of carbon and oxygen nuclei confined in the Milky Way.

If we model the Milky Way as a disk with diameter $2r_\mathrm{MW} = 25\,$kpc and height $h_\mathrm{MW} = 0.3\,$kpc, then the energy production rate of Galactic cosmic rays integrated over the volume of the Milky Way yields the cumulative luminosity of the cosmic-ray sources:

The kinetic energy of a core-collapse supernova can be estimated to

and their explosion rate in the Milky-Way is estimated as $\lambda_\mathrm{CC} = (1.6 \pm 0.5)\times 10^{-2}\,\mathrm{yr}^{-1}$ (Rozwadowska et al. (2021)).

If core-collapse supernovae are responsible for accelerating the majority of cosmic rays to energies greater than 1 GeV, then the efficiency of the conversion of kinetic energy into cosmic rays, $\eta_\mathrm{GCR}$, should satisfy the relation

a reasonable constraint suggesting that core-collapse supernovae may be responsible for accelerating the bulk of Galactic cosmic rays.

2Acceleration processes¶

2.1Astrophysical plasmas¶

Consider a fully ionized plasma, composed e.g. of protons and electrons. Astrophysical plasmas are usually neutral on large scales, i.e. $n_p = n_e = n$ where $n_p$ and $n_e$ are the number density of protons and electrons respectively.

Neutrality on a large spatial scale, $L$, means $L \gg \lambda_\mathrm{D}$, where $\lambda_\mathrm{D}$ is the Debye length, beyond which charge screening occurs (see Chap. 11.1 in Longair (2011)). The Debye length depends on the thermal velocity of the electrons, $v_e$, and the plasma frequency, $\omega_e$:

This microphysics-relevant scale can be compared to the much larger astrophysical source sizes in Figure 1.

In a rarefied plasma, it is improbable for a test particle to encounter another particle. The system is said to be colisionless, i.e. the motion of a charged particle is determined by its overall interaction with the plasma and the magnetic fields attached to it.

A few properties of astrophysical plasmas are worth noting here (Spitzer (1962)). Firstly, astrophysical plasmas have a high conductivity (low resistivity), which is comparable to that of metals. This high conductivity leads to the short-circuiting of any large-scale electric field and to the phenomenon of magnetic-flux freezing, i.e. magnetic field lines behave as if anchored into the plasma.

A second notable feature of a plasma is the speed at which disturbances propagate along the field lines. The equivalent of the speed of sound, called the Alfvén speed, is provided by the ratio of the magnetic energy density, $u_B$, to the mass density of the medium $\rho = n m_p$, i.e. $v_\mathrm{A} \propto \sqrt{\frac{u_B}{\rho}}$. More specifically, $v_\mathrm{A} = \frac{B}{\sqrt{\rho\mu_0}}$ that is

The diffusion of charged particles in an astrophysical plasma is determined by their resonance with magnetic-field perturbations such as Alfvén waves and other hydromagnetic waves.

2.2Fermi mechanisms¶

Electromagnetic interaction is responsible for the acceleration of charged particles in astrophysical environments, via the Lorentz force $\vec{F} = q(\vec{E} + \vec{v}\times\vec{B})$. Since the magnetic term does not work, an effective electric field is required for acceleration. Note, however, that an electromagnetic field such that $\vec{E} = \vec{0}$ and $\vec{B} \neq \vec{0}$ can always be transformed to $\vec{E'} \neq \vec{0}$ and $\vec{B'} \neq \vec{0}$ through a change of frame. The relevant quantities for discussing the different types of acceleration mechanisms are the Lorentz invariants of the electromagnetic tensor $F_{\mu\nu}$, namely $F_{\mu\nu}F^{\mu\nu} = 2\left[ (Bc)^2-E^2 \right]$ and $\mathrm{det}(F) = \vec{E} \cdot \vec{B}$ (Lemoine (2019)).

The acceleration mechanism most commonly used by humans on Earth is linear acceleration, i.e. $\vec{E} \cdot \vec{B} \neq \vec{0}$ or $E^2 > (Bc)^2$. These cases, for which ideal magnetohydrodynamics does not apply (ideal Ohm’s law with no resistive term: $\vec{E} + \vec{v}\times\vec{B} = \vec{0}$), are only found in a limited number of astrophysical environments, such as the gaps in the magnetospheres of pulsars.

The case more generally found in astrophysical environments, for which $E^2 < (Bc)^2$ and $\vec{E} \cdot \vec{B} = \vec{0}$, allows us to describe acceleration by shock waves, magnetic turbulence or in sheared flows. Two seminal publications by Enrico Fermi described the concepts in the 1950s (Fermi (1949)Fermi (1954)). In both cases, acceleration results from particle scattering on magnetic-field perturbations such as Alfvén waves, which move with a characteristic velocity $v_\mathrm{A}$. A head-on / receding collision with a pertuburation leads to a gain / loss of momentum $\frac{\Delta p}{p} \approx \pm \frac{v_\mathrm{A}}{c}$. In the case of an isotropic distribution of the velocities of the scattering centers, found for example in a turbulent magnetic field, head-on collisions are more frequent by a factor $\frac{v_\mathrm{A}}{c}$, so that the rate of energy gain is favored over the rate of loss (see chapter 7.3 in Longair (2011)). This mechanism is known as second-order Fermi acceleration. In the case where only head-on collisions take place, for the geometry of a contracting magnetic bottle found for example on either side of a shock front, the mechanism is said to be first-order.

The denomination of first- and second-order Fermi acceleration is more explicit when we consider the acceleration time, $t_\mathrm{acc}$, which depends on the mean free path between scattering centers, $l_\mathrm{mfp}$:

3Spectrum of accelerated particles¶

3.1The Fokker-Planck equation¶

The equation that governs the evolution of the momentum and position of a collection of particles is a partial differential equation known as the diffusion-loss equation. Its formal derivation uses the diagram provided in Figure 2 (see also chapter 7.5 in Longair (2011)). The reasoning is limited here to one spatial and one momentum dimension, but can easily be extended to 6D if one wishes to consider anisotropic phenomena such as the different diffusion coefficients along and perpendicular to a magnetic-field line.

Consider an initial Maxwellian momentum distribution for a number of particles $n(x, p, t)\dd x \dd p$. The 2D phase-space density, $n$, is analoguous to the 6D phase-space density $\dd N/\dd^3 x\dd^3 p$ defined in chapter Jetted and non-jetted astroparticle sources in the universe.

The rate at which the number of particles in the box changes is given by:

i.e.

where $Q$ is the source term characterizing the injection of new particles and τ is the time scale of catastrophic energy losses, relevant e.g. to radioactive decay, spallation, or simply the escape of the particle from the region of interest.

$\phi_p$ is the flux of particles of momentum between $p$ and $p + \dd p$ through $\dd x$. Assuming, for simplicty, no momentum diffusion, it reads:

where $\dot p = \frac{p}{t_\mathrm{acc}} - \frac{p}{t_\mathrm{loss}}$ for Fermi acceleration at a rate $t_\mathrm{acc}^{-1}$ and continuous energy losses, such as synchrotron losses, at a rate $t_\mathrm{loss}^{-1}$.

$\phi_x$ is the flux of particles of position between $x$ and $x + \dd x$ through $\dd p$:

i.e. an advection term, with $v_\mathrm{adv}$ the advection velocity, and a diffusion term from Fick’s law with $D$ the diffusion coefficient. For isotropic diffusion, $D = \frac{1}{3} c l_\mathrm{mfp}$. Writing $\eta = l_\mathrm{mfp}/r_\mathrm{g}$ the number of gyroradii in a mean free path:

where $\eta \geq 1$ as the mean free path is at least a gyroradius. The mimimum diffusion coefficient, obtained for $\eta=1$, is referred to as the Bohm limit.

The terms of Equation (10) being defined, we get the Fokker-Planck equation describing the diffusion-loss of the particles:

3.2Where power laws come from¶

The simplest application of the Fokker-Planck equation is highly relevant to understanding acceleration (see also chapter 7.4 in Longair (2011) for an introduction to diffusive shock acceleration). Consider a stationary system, with no diffusion or advection, no energy loss, for which catastrophic losses are limited to escape ($\tau = t_\mathrm{esc}$) and with particle injection in the past, then:

where $t_\mathrm{acc}$ does not depend on $p$ for Fermi acceleration. This gives

That is a power-law spectrum of index $s=1 + \frac{t_\mathrm{acc}}{t_\mathrm{esc}}$.

The value of the index can be determined with multiple approaches, depending on the considered astrophysical environment. In the case of shock acceleration or magnetic reconnection, one can use box models such as developped by Drury et al. (1999)Drury (2012) and show that $\frac{t_\mathrm{acc}}{t_\mathrm{esc}} = \frac{3}{|r-1|}$, where $r=\rho_2/\rho_1$ is the compression ratio between the shocked medium and the undisturbed medium.

For a shock acceleration at high Mach number and assuming a plasma described as a perfect gaz of adiabatic index $\gamma = \frac{c_P}{c_V}$, the compression ratio is equal to $r = \frac{\gamma +1}{\gamma -1}$. As $\gamma = 5/3$ for a monoatomic gaz of protons or electrons, we get $r = \frac{\frac{5}{3} +1}{\frac{5}{3} -1} = 4$. In such a system, the power-law spectrum has an index $s = 1 + \frac{3}{|4-1|} = 2$, that is:

The universality of this power law of index 2, whose presence is inferred in many non-thermal astrophysical systems, is one of the great successes of diffusive shock acceleration theory.