2.2.1 Second-order Fermi acceleration
In 1949, Fermi proposed a stochastic mechanism in which charged particles are accel-erated through collisions with magnetic clouds in the interstellar medium [39]. These magnetic clouds move randomly with speedV and reflect charged particles. The energy of these particles is increased with each reflection in a head-on collision, see Fig. 2.2.1 for a graphical description of the process. If the particles remain in the acceleration zone for some time τesc before the escape, the energy spectrum turns out to be a power-law.
The energy gain and the spectrum can be derived using relativistic equations, see e.g.
the approach of Ref. [40]. A simple formulation of the process is presented next. The average energy of the charged particle, with initial energy E0, after one collision is E =βE0. The probability that it remains in the accelerating region isP. For an initial population of particlesN0 and afterkcollisions, the number of particles that remain in the accelerating region is N = N0Pk, while their energies are E = βkE0. We can get a relation between the number of particles and the energies by computing the ratio of logarithms,
This equation was derived for k collisions, but some of the particles can still be accel-erated after that number, making N = N(≥E), and thus the energy distribution is a power-law,
N(E)dE = constant×E−1+(lnP /lnβ)dE. (2.2.2) Comparing this equation with the original work from Fermi, the parameterP is related to τesc, while β is proportional to (V /c)2. This is why this acceleration mechanism is known as second-order Fermi acceleration.
2.2.2 Diffusive shock acceleration
The acceleration mechanism could be more efficient if the fractional energy increase would be ∼V /c instead of ∼(V /c)2. This is called first-order Fermi acceleration. By the end of the 1970s, diffusive shock acceleration, a first-order Fermi acceleration pro-cess, had gained a lot of attention in astrophysics. It was discovered independently by different people [41, 42, 43, 44]. Strong shock waves propagating through the interstel-lar medium, discussed in Ref. [45], are common in astrophysical environments. In this section, we follow the physical derivation from Ref. [42]. Ultra-relativistic particles are expected to be in both sides of the shock, which moves much slower compared to the particles. The thickness of the shock is usually much smaller than the gyro-radius of the HE particles, hence the particles barely notice its effects. When a particle crosses the shock in any direction, it is scattered due to streaming instabilities. The velocity distribution of particles becomes isotropic in the reference frame of the moving gas on both sides of the shock.
The dynamics of the problem is shown in Figure 2.2.1. Let us consider a strong shock
wave propagating at speedU through stationary interstellar gas. In the reference frame in which the shock front is at rest, the upstream (front of the shock) gas flows into the shock at velocityu1 =U. After crossing the shock, the downstream (back of the shock) gas has velocity u2. For a mono-atomic gas in the strong shock limit, the equation of mass continuity yields u2 = (1/4)U. When a particle passes through the shock from the upstream side to the downstream side, the gas of the downstream side has a velocity V = (3/4)U, relative to the upstream side. The energy increase of the particle in the upstream-downstream crossing can be computed with relativistic expressions, and it is given by,
*∆E E
+
= 2 3
V c = 1
2 U
c, (2.2.3)
after averaging over all crossing angles of the particles with respect to the shock wave. In the opposite process, when a particle crosses the shock downstream-upstream, the gas moving towards the shock has the same velocityV = (3/4)Utowards the shock, relative to the downstream gas. The particle gains the same amount of energy in both cases.
If the same particle travels back and forth through the shock, the fractional energy increase is doubled. The energy of the particle is always increased when crossing the shock, no matter the side. In contrast to the original Fermi acceleration process, there are never crossings that result in energy loss for the particle that is being accelerated in this scenario.
Figure 2.2.1: Left: representation of the 2nd-order Fermi acceleration mechanism. A particle is scattered many times in different magnetic mirrors labeled as "B". The particle gains energy in the green and red regions, and it loses energy in the blue cloud.
Right: representation of charged particles in a shock. a): the shock moves at speed u in the observer’s frame. b): reference frame of the shock. c): frame in which the medium is unperturbed in the downstream region and the charged particles velocities are isotropic. d): frame in which the medium is unperturbed in the upstream region and the charged particles velocities are isotropic. Adaptated from Ref. [46].
The average number of particles crossing the shock in any direction is (1/4)N c, where N is the density of particles. In upstream-downstream crossings, particles can be lost in the flow of gas behind the shock. This is due to the isotropy of the velocity distributions of the particles with respect to the gas in that zone. The flux of particles removed from the system is (1/4)N U. The probability of losing particles is then the loss flux divided by total flux that crosses the shock, (1/4)N U/(1/4)N c = U/c. The probability of particles remaining in the accelerating region is P = 1−U/c. Comparing these results to the second-order acceleration parameters, the energy increase parameter in a round trip is β = E/E0 = 1 +U/c. With these values of P and β, the differential energy spectrum is:
N(E)dE= constant×E−2dE. (2.2.4) The predicted spectrum in first-order Fermi acceleration is a power-law with index−2.
These results are for the simplest diffusive shock acceleration model.
A full treatment of the problem requires the use of the Fokker-Planck equation, taking into account more elements in the problem, such as the effects of magnetic fields in the plasma or the adiabatic and radiative cooling [47]. The effects of the charged particles on the shock itself and the stability of the flows have also to be considered, making the
process of acceleration non-linear. A review of the non-linearity problem is given in Ref. [48], while the efficiency of non-linear models is widely studied with simulations in Ref. [49]. Another problem with this model that the particles need to exceed a threshold energy in order to be further accelerated. For shock acceleration to work, charged particles have to reach an initial energy high enough so that Larmor radius becomes much larger than the size of the shock. This is known as the injection problem [50].
2.2.3 Magnetic reconnection
Magnetic reconnection is a physical process that can take place in highly-conducting plasmas in which the magnetic topology of a system is reorganized. The result of this rearrangement is a conversion of magnetic energy to kinetic and thermal energy. A review on the topic can be found in, e.g. Ref. [51]. The mechanism was first suggested in 1964 [52] in order to explain solar flares particle acceleration.
The first quantitative two-dimensional model, the Parker-Sweet model [53, 54], was developed a decade after the initial model. The schematics of the model are depicted in Figure 2.2.2. Over a region with plasma of density ρ, the magnetic field lines of opposite magnetic fields ±B0 are steadily brought together in a boundary layer. The size of the region is 2Land the thickness of the reconnection layer is 2δ, with 2δ2L.
Plasma, to which the magnetic field is frozen according to Alfven’s theorem [55], flows into the boundary layer from both sides at a speed vin. This is the same speed at which the lines are steadily merging. A large electric current that heats the plasma is induced in the boundary, leading to locally small Reynolds numbers, thus the field lines velocity deviates from the plasma velocity in the reconnection region. The tension force due to the bend in the reconnected field lines accelerates the plasma flow. Equating the pressure of the heated gas to the magnetic tensions in a steady-state model, the resulting speed of the outflow is the Alfven speed vA = B0/(4πρ) [23]. Assuming that the plasma is incompressible, a relation between the initial flow speed of the plasma and the ejection speed can be foun from the conservation of mass:
ρvinL=ρvAδ → vin
vA
= δ
L (2.2.5)
Figure 2.2.2: Geometry of the Sweet-Parker reconnection model. Magnetic field lines (blue) are brought together and merge in the boundary layer (orange). The plasma moves in the direction of the vertical lines and charged particles are accelerated across the horizontal lines. Adapted from Ref. [51]
These values are typically small, thus more complex models of magnetic reconnection are required to explain cosmic ray acceleration. Such models predict instabilities in the flow between the two magnetic fields that generate magnetic islands [56, 57]. In this scenario, particles are first accelerated by the electric fields in the reconnection region and then proceed to further acceleration in the boundaries of the magnetic islands through the Fermi mechanism [58].