# Recent Progress in the Theory of Electron Injection in Collisionless Shocks

## Abstract

The injection problem in diffusive shock acceleration theory is discussed with particular focus on electrons. The following issues are addressed: Why it has been considered to be so difficult, what is the required condition, and how it can be resolved. It is argued that there exists a critical Mach number above which the electron injection is achieved. Above the threshold, back-streaming electrons reflected back upstream by the shock front can self-generate high-frequency whistler waves, which can scatter themselves as required for subsequent acceleration. The theoretical estimate is found to be well consistent with in-situ measurements, indicating this could provide a possible solution to the long standing problem in the diffusive shock acceleration theory.

## Keywords

Shock Front Electron Injection Whistler Wave Loss Cone Collisionless Shock## 1 Introduction

The acceleration of charged particles at collisionless shocks has been a subject of great interest in connection with the early conjecture that cosmic rays are produced in association with Galactic supernovae. A more concrete theoretical background was given by the theory of diffusive shock acceleration (DSA) proposed in the late 1970s by several independent research groups (see [1] for review). It is actually the most successful theory in that it naturally predicts a power-law type energy spectrum, which is close to the source spectrum inferred from cosmic-ray measurements on the earth. Indeed, there is direct observational evidence for the presence of relativistic electrons in young supernova remnants (SNRs) as observed by radio and X-ray synchrotron emission. These cosmic-ray electrons can therefore be used as an important probe to investigate the properties of the particle acceleration sites. It is, on the other hand, also known that the shock acceleration of electrons is rarely observed in the heliosphere [2, 3]. These shocks are generally weaker than astrophysical shocks which are radiating strong synchrotron emission. Protons or other heavier nuclei seem to be accelerated rather efficiently even by such relatively weak shocks, although there are many events showing no evidence for the ion acceleration as well. Identification of the astrophysical proton acceleration sites is of significant importance, because it is this component that constitutes a large fraction of cosmic rays. It is, however, still difficult even with present-day observation facilities due to their intrinsic low radiation efficiency.

The difficulty for the detection of the proton component is to some extent related to a theoretical flaw. The DSA theory can predict the spectral index of accelerated particles, but does not tell anything about the fraction of accelerated particles (or the normalization factor),—the issue known as the injection problem. The injection fraction seems to depend on particle species and parameters of the shock as implied by observations. This actually makes it difficult to identify cosmic-ray protons, because the expected radiative signature of protons in the γ-ray spectrum is contaminated by electron contributions with unknown amount. It is important to give a theoretical constraint for the number of accelerated electrons and protons independently, to reduce this ambiguity as much as possible. To do so, we must address the injection problem, in particular, for electrons. Because in contrast to relatively well understood ion injection, the electron injection is known to be much more difficult [4, 5]. In this review, we outline the basic concept of the injection problem, and describe the difficulty of the electron injection. We then introduce our recent idea in which the electron injection is achieved by exploiting high-frequency whistler waves. A future perspective on the theory of electron injection is also discussed.

## 2 The Injection Problem

The DSA theory, as its name suggests, is based on the assumption that the transport of energetic particles around the shock is described by the diffusion process. Since the medium is a highly ionized collisionless plasma, the particles have to be scattered by waves so that they diffuse in space otherwise their transport would be ballistic. Assuming the isotropy of the distribution function as a result of strong scatterings, energetic particles will diffuse in the local fluid rest frame. The diffusive transport makes it possible for them to propagate against the downstream flow and cross the shock front. This means that the energetic particles are essentially confined in the vicinity of the shock before being advected far away by the downstream flow (escape toward upstream can be ignored under this assumption.) Across the shock, there exists a compression of the flow, indicating converging motion of scattering centers. This converging motion brings the particle energy gain during their confinement around the shock, just like a fluid adiabatically heated by the compression. As is evident from the above consideration, the scattering by waves is crucial for the confinement and acceleration of particles. The required conditions for the injection are therefore summarized that the energetic particles (1) should have velocities large enough to travel for a long distance to repeatedly cross the shock front, and (2) need to be scattered by waves so that the transport becomes diffusive. Below, we discuss these conditions in more detail.

### 2.1 Shock Internal Structure

Since one must consider the dynamics of thermal and/or suprathermal particles to understand the injection, the internal structure of the shock determined primarily by low-energy particles needs to be taken into account. Nevertheless, we will not go into details of rich variety of collective phenomena associated with collisionless shocks, and look for the simplest possible explanation essential for understanding the injection.

Consider upstream particles penetrating into a thin shock transition layer whose scale length comparable to the ion inertial length. Because of the inertial difference between ions and electrons, electrons are easily decelerated at the leading edge of the shock front, while ions are not. As a result, an electrostatic potential is produced, which has to be large enough to decelerate upstream ions otherwise the shock would not form. Although the actual value of the potential is difficult to determine, it is estimated to be 10−20% of the upstream bulk ion energy by in-situ measurements of the bow shock [6], which is more or less consistent with kinetic numerical simulations. The structure of the shock depends strongly on upstream parameters, in particular, Alfvén Mach number \({M}_{A} = {V }_{1}/{v}_{A}\) (*V* _{1} and *v* _{ A } are the shock and Alfvén speeds), and the shock angle θ_{ Bn } defined as the angle between the shock normal and the upstream magnetic field. It is known that, whenever the Alfvén Mach number exceeds a few, a fraction of ions are reflected by the shock front and then dominate the shock structure. Conventionally, shocks with θ_{ Bn } ≲ 45 and θ_{ Bn } ≳ 45 are respectively called quasi-parallel and quasi-perpendicular shocks, and have different characteristics.

### 2.2 Escape Condition: Thermal Leakage

Since particles have to cross the shock front many times to be accelerated significantly, the lower limit for the threshold energy is obviously characterized by the condition that particles can traverse the shock almost freely. Here, we consider the condition that downstream thermal particles can escape toward upstream.

A particle in the downstream region whose velocity parallel to the magnetic field line denoted by *v* _{ ∥ } can travel toward the shock when \({v}_{\parallel }\geq \frac{{V }_{1}} {r} \sqrt{1 + {r{}^{2 } \tan }^{2 } {\theta }_{Bn}}\), where *r* is the shock compression ratio. Assuming Maxwellian distributions and strong shocks, the downstream particle velocity is characterized by the thermal velocity determined by Rankine-Hugoniot relations (or roughly ∼ *V* _{1}). Then, it is easy to confirm that the downstream thermal particles can escape toward upstream when θ_{ Bn } ≲ 30, while the threshold energy becomes higher as increasing the shock angle. Although this simple argument gives a rather good estimate for the escape condition of ions, one must take into account the effect of the electrostatic potential for electrons. Since the potential develops so as to decelerate the upstream ions, it actually prevents the downstream electrons from escaping upstream. If the downstream temperature is determined solely by the conversion of the bulk energy into the thermal energy and there is no energy transfer between ions and electrons, the thermal velocities of electrons and ions would be the same. In this case, the downstream electron kinetic energy ∼ *m* _{ e } *V* _{1} ^{2} ∕ 2 is not enough to overcome the potential of the order of ∼ η*m* _{ i } *V* _{1} ^{2} ∕ 2, where \(\eta = 0.1\mathrm{-}0.2\) is the normalized electrostatic potential, and *m* _{ j } is the mass of particle species *j*. In reality, the energy conversion from ions to electrons is known to occur, although the efficiency, especially the dependence on upstream parameters, is not known very well. Nevertheless, in-situ measurements have shown that the temperatures of downstream electrons are mostly less than several tens of percent of the ion temperature, i.e., of the order of the electrostatic potential energy. This indicates that the escape condition of electrons is more or less similar to that of ions. One thus finds that the thermal leakage is possible for both electrons and ions at quasi-parallel shocks, while it is not in quasi-perpendicular shocks unless the downstream distribution develops into one which possesses a high energy tail. Note that, as mentioned earlier, there exist ions that are directly reflected by the shock in the upstream, which can also be considered as a seed population. Qualitatively, however, the condition for ion injection is not so much different, while the processes become much more complicated.

### 2.3 Resonance Condition

Suppose there exist back-streaming particles in the upstream region. In the absence of upstream waves, these particles would just stream away freely and never get back to the shock. For a self-consistent description of the injection process, one has to consider waves generated by these back-streaming particles themselves through beam instabilities. Since they are streaming along the ambient field line, the condition for instability may qualitatively be analyzed by considering the cyclotron resonance \(\omega = k{V }_{b} - {\Omega }_{j}\) (*V* _{ b } is the beam velocity and *Ω* _{ j } is the cyclotron frequency of particle species *j*) with a normal mode (ω, *k*) of a magnetized plasma.

*k*< 0 in Fig. 1). Consequently, the ion (electron) beam will be unstable against the excitation of a wave on the R-mode Alfvén/whistler (L-mode Alfvén/ion cyclotron) branch.

In reality, one must also consider the effect of cyclotron damping by thermal particles. Waves with which thermal particles can satisfy the cyclotron resonance condition are actually strongly damped, and thus cannot be recognized as the normal modes of the plasma. Since high frequency waves are more strongly damped, the velocity of back-streaming particles needs to be large enough so that the interaction occurs at sufficiently low frequencies. More specifically, the cyclotron resonance condition defined by using the thermal velocity \(\omega = \pm k{v}_{j} - {\Omega }_{j}\) (where *v* _{ j } denotes the thermal velocity) may be used to estimate the strongly damped regions (shaded regions in Fig. 1).

First, let us consider the case of back-streaming ions that are leaked from the downstream. Since they have upstream directed velocities in the shock frame, their streaming velocities in the upstream rest frame is larger than the Alfvén speed at least by a factor of *M* _{ A }. For *M* _{ A } greater than a few, the interaction will occur well within the magnetohydrodynamics (MHD) regime, so that the ion cyclotron damping is virtually negligible. The back-streaming ions can thus easily excite Alfvén waves. The excitation of low-frequency Alfvén waves in the upstream region (the foreshock) and associated pitch-angle scattering of ions have been confirmed both by numerical simulations and in-situ observations.

*Ω*

_{ i }) where the ion cyclotron damping is significant unless the electron velocity is extremely large. More specifically, the required condition that the electrons interact with low-frequency MHD waves (

*kv*

_{ A }∕

*Ω*

_{ i }≲ 1) may be written as

*V*

_{ b }∼

*c*to satisfy the above condition. This is the reason why the electron injection has been believed to be so difficult. For instance, the electron temperatures downstream of young SNR shocks are estimated to be ∼ 1 keV, which are far too small for the injection. These shocks, on the other hand, are radiating radio and X-ray photons by synchrotron emission from ultra-relativistic electrons. This apparent contradiction has not yet been resolved so far.

In order to achieve the electron injection, one must have a pre-acceleration mechanism that energizes thermal electrons to mildly relativistic energies. Several possible mechanisms have been proposed so far that could accelerate to the required energies involving microinstabilities in the shock transition region [7, 8, 9, 10, 11, 12, 13]. Or, alternatively, there must be a mechanism that produces high-frequency whistler waves that can scatter low-energy electrons [14]. In any case, there has been no general consensus about what would be the most probable mechanism for the electron injection. In the following section, we discuss our recent work on the electron injection problem, which is found to be consistent with existing observations.

## 3 Electron Injection Mechanism: Recent Progress

### 3.1 Shock Drift Acceleration

We have seen that the thermal leakage does not provide sufficient energies for the electron injection. It is known that electrons can also be directly reflected by the shock front, as in the case of ions, and thus be energized more efficiently. Shock drift acceleration (SDA), or a fast Fermi process, is known as an adiabatic magnetic mirror reflection process in the Hoffmann-Teller frame (HTF) [15, 16]. The HTF can be defined as the frame where the motional electric field vanishes. Because of the absence of electric fields, the particle energy measured in this frame is a conserved quantity. An upstream electron traveling toward the shock will be reflected by the shock acting as a magnetic mirror if the particle pitch-angle defined in the HTF is large enough (Fig. 2). Notice that, however, a finite electrostatic potential at the shock enlarges the loss cone angle at low energies, while it has little effects on sufficiently high-energy particles.

*V*

_{ b }≃ 2

*V*

_{1}∕ cosθ

_{ Bn }(see [15, 16] for detail), which is larger than that of a thermal leakage population by a factor of ∼ 2 ∕ cosθ

_{ Bn }(strictly speaking, this holds only for low-energy electrons having gyroradii much smaller than the shock thickness). Therefore, this effect alone can to some extent help the situation in quasi-perpendicular shocks. The condition that the reflected electron beam resonates with low-frequency Alfvén waves at

*kv*

_{ A }∕

*Ω*

_{ i }≲ 1 may be written as

_{ Bn }≪ 1.

### 3.2 Generation of Whistler Wave

*V*

_{ b }≳

*v*

_{ e }, which leads

_{ e }is the electron thermal pressure to magnetic pressure ratio. Notice for the different dependence on the mass ratio between Eqs. (2) and (3) (see also Fig. 3). One can see that the required condition is now much less stringent. In particular, the bow shock can be super-critical in the quasi-perpendicular regime. This actually gives a firm theoretical background for a statistical analysis of bow shock measurements done by Oka et al. [17], who found an almost the same dependence. According to them, the spectral index of energetic electrons in the shock transition region becomes systematically harder when an approximately (within a factor of ∼ 2) the same condition to Eq. (3) is satisfied. (More specifically, they claimed the dependence on the so-called whistler critical Mach number [18], which happens to have a similar form. However, since it has a different theoretical background, the authors of [17] could not find any reasonable explanations.) Based on the observational fact, we believe that it is actually this condition that determines the electron acceleration efficiency in the bow shock. This also explains naturally the reason why strong SNR shocks are efficient electron accelerators as opposed to weaker heliospheric shocks. A more detailed numerical investigation of the instability given in [19] basically confirmed this argument.

### 3.3 Non-adiabatic Effects

We have discussed the process by which the electron injection is achieved. However, we still do not know the fraction of electrons being injected into the acceleration cycle. Answering this question probably needs understanding of nonlinear collective plasma phenomena occurring around the collisionless shocks. This is because the adiabatic theory is not necessarily a good description for the dynamics of suprathermal electrons being considered. This does not mean that the adiabatic description is totally wrong; rather, we think that non-adiabatic corrections to SDA need to be taken into account for quantitative discussion. Furthermore, the situation will increasingly become complicated with increasing the Mach number of the shock.

As seen in Fig. 2, the SDA operates only for electrons having sufficiently large initial pitch angles because otherwise they would be just transmitted to the downstream. The required threshold particle velocity (defined in the upstream rest frame) is roughly proportional to *V* _{1} ∕ cosθ_{ Bn }. The condition for the injection Eq. (3), on the other hand, can be rewritten as *V* _{1} ∕ cosθ_{ Bn } ≳ *v* _{ e } (apart from a factor of order unity). Taking this literally, the reflection of thermal particles will hardly occur for strong shocks well above the critical Mach number such as SNR shocks. This implies the need for taking into account non-adiabatic effects that could lead to a non-negligible electron injection rate.

Indeed there have been a lot of discussion on plasma instabilities in the vicinity of the shock. In the regime *V* _{1} ∕ cosθ_{ Bn } ≫ *v* _{ e }, one can actually expect stronger instabilities in the shock transition region driven by the reflected ion beam [7, 20]. These plasma waves can play a role for the acceleration of thermal electrons above the threshold energy so that the electron injection rate is dramatically increased even in a highly super-critical regime. We have actually demonstrated by using particle-in-cell (PIC) simulations that such can actually occur in high Mach number quasi-perpendicular shocks [12], although there is still much work to be done to understand quantitatively the complicated electron injection process dominated by nonlinear plasma phenomena, particularly dependence on the Mach number and multidimensional effects. Large scale kinetic numerical simulations of high Mach number shocks will further improve our understanding in the future.

## 4 Discussion

We have presented a possible mechanism for the electron injection, which is proven to be consistent with in-situ measurements of the bow shock. We believe this resolves, at least qualitatively, the observational discrepancy found between shocks associated with high-energy astrophysical objects and those directly measured in the heliosphere. One might have concern about the possibility of back-streaming electron scattering toward the shock because the instability is fed by the scattering of electrons away from the shock. In the steady state, however, we anticipate that the pitch-angle distribution of electrons will become closer to isotropic. This indicates there will also be back scattering of electrons as well, so that the diffusion approximation is justified. This needs to be clarified in relation to issues discussed below.

Our discussion has been restricted to the linear stability analysis, which indicates that the growth rate of the instability is so large that the acceleration of electrons can occur within the thin shock layer. This idea is consistent with the fact that the observed energetic electrons associated with quasi-perpendicular shocks are typically confined within the shock [17, 21]. Nevertheless, the acceleration process will probably be modified by the inhomogeneity, affecting the propagation of whistler waves (through refraction and mode conversion), and the transport of electrons by the lowest order mirror force.

It is particularly important to mention that the electron injection will be influenced by the ion injection when the latter is so efficient that the upstream medium is strongly disturbed. Although the local electron injection rate is likely to be determined only by local shock parameters, the total number of injected population will be affected by the global modification of the shock. Comprehensive understanding of the whole process is needed for elucidation of cosmic-ray acceleration in astrophysical shocks.

## Notes

### Acknowledgements

T. A. is supported by the Global COE program of Nagoya University (QFPU) and KAKENHI 22740118 from JSPS and MEXT of Japan. This manuscript was written while one of the authors (T. A.) was visiting Max-Planck-Institut für Kernphysik at Heidelberg, Germany.

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