The efficiency of electron acceleration by ICME-driven shocks

Electron acceleration is a vital topic in space and astrophysical plasmas. The well-known efficient accelerators are collisionless shocks which are able to accelerate electrons to high energies. It is widely accepted that diffusive shock acceleration (DSA; Axford et al., 1977; Krymsky, 1977; Bell, 1978; Blandford & Ostriker, 1978), i.e., the combination of first-order Fermi acceleration (FFA) and shock drift acceleration (SDA), remains the dominant particle acceleration mechanism at collisionless shocks. Recently many authors have studied the acceleration of electrons. Tsuneta & Naito (1998) proposed that in solar flares the formation of an oblique fast shock below the reconnection region provides the acceleration site for nonthermal electrons of 20–100 keV by first-order Fermi acceleration (FFA) mechanism. Guo & Giacalone (2012) studied electron acceleration at the flare termination shock in large-scale magnetic fluctuations and found that electrons can be accelerated to a few MeV which can qualitatively explain the observed hard X-ray emissions. Mann et al. (2001) suggested that in the solar corona, high energetic electrons of 1 MeV can be produced by shock waves if strong magnetic field fluctuations appear near the shock transition so that electrons are allowed to cross the shock front many times by field-line meandering. Klassen et al. (2002) investigated the origin of 0.25–0.7 MeV electrons in solar energetic particle events through observations from COSTEP/SOHO and Wind 3DP instruments and concluded that the electrons measured are accelerated by coronal shock waves. Mann et al. (2006) proposed that hard X- and $\gamma$-rays can be generated by electrons that are accelerated by shock drift acceleration (SDA) mechanism at the termination shock during solar flares. Warmuth et al. (2009) developed a quantitative model based on radio and hard X-ray observations to study the acceleration of electrons in solar flares, and showed the possibility of shock drift acceleration at the reconnection outflow termination shock. In addition, Miteva & Mann (2007) proposed that in the solar corona energetic electrons at quasi-perpendicular shocks can be accelerated by resonant whistler wave-electron interaction. Saito & Umeda (2011) performed particle-in-cell simulations for electron acceleration at a quasi-perpendicular shock, and showed that the parallel scattering by the kinetic Alfv$\acute{\text e}$n turbulence, which is generated in the shock transition, suppresses the reflection of electrons during the shock drift acceleration process.

Dresing et al. (2016) determined the particle acceleration efficiency at interplanetary shock crossings observed by STEREO spacecraft, and found that only $1\%$ (five events) of all analyzed shocks shows a shock-associated electron intensity increase for 65–75 keV electrons. Among the five events, they found that four were associated with ICME-driven quasi-perpendicular shocks and the rest one was associated with an SIR ¹¹1stream interaction region(with an ICME embedded in) forward quasi-parallel shock. It is possible that the observed shock-associated electron intensity increase events have more occurrence frequency in quasi-perpendicular shocks than in quasi-parallel ones, although there is very poor statistics in their studies. Furthermore, Yang et al. (2018) studied the acceleration of suprathermal electrons in the energy range of 0.3–40 keV at an ICME-driven quasi-perpendicular shock with the spacecraft observations, and obtained the $90^{\circ}$ pitch angle enhancements and a larger downstream electron spectral index. In the statistical work of Yang et al. (2019), for those selected quasi-perpendicular and quasi-parallel shock events they found a positive correlation between the average downstream suprathermal electron flux and magnetosonic Mach number, and the downstream flux enhancement and shock compression ratio. These results implys the importance of shock drift acceleration in the acceleration of electrons at quasi-perpendicular shocks. In addition, Yang et al. (2019) concluded that there are more SEP events associated with quasi-perpendicular shocks compared to quasi-parallel shocks.

In the previous work (Kong & Qin, 2020), using test-particle simulations, we investigated the acceleration of suprathermal electrons at a quasi-perpendicular shock studied by Yang et al. (2018), suggesting the importance of shock drift acceleration in the acceleration of electrons. In this paper, using spacecraft data analysis and test particle simulations, we expand to study statistically the effect of shock parameters, such as shock angle (the angle between the shock normal and upstream magnetic field), upstream Alfv$\acute{\text e}$n Mach number, and shock compression ratio, on the ICME-driven shock acceleration efficiency. In addition, the theoretical models for shock acceleration efficiency are derived to compare with the observations and simulations. In Section 2, we show the observational data. We describe the numerical model in Section 3, and the models for efficient shock acceleration in Section 4. The results of observational analysis and simulations are presented in Section 5. Summary and discussion are given in Section 6.

In the survey of Yang et al. (2018), 74 ICME-driven shocks are observed by Wind over the period from 1995 to 2004. The electron electrostatic analyzer (EESA) in the 3DP instrument (Lin et al., 1995) onboard Wind provides three-dimensional (3D) data for eight pitch angle channels in the energy range of $\sim$3 eV to $\sim$30 keV. For the shock events, the flux data of energetic electrons in the upstream and downstream of the shock (i.e., before and after the shock arrival) can be obtained from the Wind/3DP/EESA instrument.

We choose the shock event on 2000 Feb 11 observed by Wind as an example. The upstream energy spectra in all pitch angle directions averaged in a period of 10 minutes (23:14 UT–23:24 UT) prior to the shock are plotted in Figure 1, multiplied by $2^{0}$, $2^{1}$, $2^{2}$, $2^{3}$, $2^{4}$, $2^{5}$, $2^{6}$, and $2^{8}$ for the pitch angles of $14^{\circ}$, $34^{\circ}$, $56^{\circ}$, $78^{\circ}$, $102^{\circ}$, $124^{\circ}$, $145^{\circ}$, and $165^{\circ}$, respectively. These spectra are assumed to be used as the initial upstream electron distribution before the shock acceleration in the simulations. In this work, we focus on the observations of the electron energy channels at $\sim$0.428, 0.634, 0.920, 1.339, 1.952, 2.849, and 4.161 keV indicated by $E_{k}$ with $k=1$, 2, 3, …, 7, as listed in Table 1, from the Wind/3DP/EESA measurement. Using the data observed by Magnetic Field Investigation (MFI) (Farrell et al., 1995; Kepko et al., 1996) and Solar Wind Experiment (SWE) (Ogilvie et al., 1995) instruments onboard Wind we obtain the upstream average magnetic field $B_{01}=7.0$ nT and proton number density $n_{\text{p}}=5.19~{}\text{cm}^{-3}$. Thus, the upstream Alfv$\acute{\text e}$n speed is $V_{\text A1}=67~{}\text{km}~{}\text s^{-1}$ with the formula $V_{\text A1}=B_{01}/\sqrt{\mu_{0}n_{\text p}m_{\text p}}$, where $\mu_{0}$ is space permeability and $m_{\text p}$ is proton mass.

We study the acceleration of electrons at a plane shock by using spacecraft data analysis and test-particle simulations similar to our previous work (Kong et al., 2017, 2019). Under the given shock parameters, the trajectories of test particles are traced by calculating the motion of equation of particles in the electromagnetic field

where $\bm{p}$ is the particle momentum, $q$ is the electron charge, $\bm{v}$ is the particle velocity, and t is time. The electric field $\bm{E}$ is the convection electric field $\bm{E}=-\bm{U}\times\bm{B}$. The shock is located at $z=0$, and the plasma flows from the upstream region with a speed $\bm{U}_{1}$ to downstream region with a speed $\bm{U}_{2}$ in the shock reference frame, as shown in Figure 2 (similar to Figure 1 in Kong & Qin (2020)). Here, the upstream speed $U_{1}$ and downstream speed $U_{2}$ can be determined by

and

with the upstream Alfv$\acute{\text e}$n Mach number $M_{\text{A1}}$ and shock compression ratio $s$. It will be later shown that the downstream fluxes from observations and simulations are measured in the range of $z$ from 0 to $z_{1}=2.7\times 10^{-3}$ au. We assume the plasma speed in the shock transition is in the form (e.g., Qin et al., 2018)

where $L_{\text{th}}$ is the shock thickness and assumed to be $L_{\text{th}}=2\times 10^{-6}$ au in this work. It is noted that $L_{\text{th}}$ is too small to be shown in Figure 2. The magnetic field, $\bm{B}$, is given by

where $\bm{B_{0}}$ is a background magnetic field assumed to lie in $x$–$z$ plane, and $\bm{b}$ is a turbulent field perpendicular to the background field. Note that the magnetic field $\bm{B}$ is taken to be a static magnetic field without the flow-advected effect for simplicity considering the high speed of particles (e.g., Fraschetti & Giacalone, 2015). The turbulent field, composed of a slab and two-dimensional (2D) components (Matthaeus et al., 1990; Mace et al., 2000; Qin et al., 2002a, b), is given by

where $z^{\prime}$-axis in the Cartesian coordinate ($x^{\prime},y^{\prime},z^{\prime}$) system, which is used to generate the slab and 2D turbulence, is parallel to the background magnetic field $\bm{B}_{0}$.

In all simulations, the slab turbulence has a correlation length of $\lambda=0.02$ au. The ratio of slab to 2D correlation length is 2.6 according to previous studies (Osman & Horbury, 2007; Weygand et al., 2009, 2011; Dosch et al., 2013). We define the turbulence in a periodic box of size [10$\lambda$,10$\lambda$] for the 2D component turbulence and size 25$\lambda$ for the slab component turbulence. A dissipation range in which low-energy electrons resonate for the slab turbulence is required, and the break wavenumber is fixed at $k_{\text b}=10^{-6}~{}\text m^{-1}$ from the slab inertial to dissipation ranges. The spectral indices of the inertial and dissipation ranges are $\beta_{\text i}=5/3$ and $\beta_{\text d}=2.7$, respectively. Note that the pitch angle scattering corresponds to the parallel diffusion, and the resonance between particles and turbulence is not affected by the 2D turbulence assuming weak nonlinear effects. Hence, the dissipation range of 2D turbulence is ignored. We take the turbulence level, $(b/B_{0})^{2}$, of 0.25 in the upstream and 0.36 in the downstream of the shock. Note that the turbulent magnetic fluctuations $\bm{b}$ is supposed to be perpendicular to the background magnetic field $\bm{B}_{0}$ and have a zero-average in both upstream and downstream. We assume that the enhancement of the turbulence level, $(b/B_{0})^{2}$, from upstream to downstream, is caused by the strong disturbance of the shock wave. The detail of the varying turbulence level throughout the shock structure is very complicated. If the upstream turbulence level is set, the value of downstream turbulence level has to be obtained with theoretical modeling depending on values of different parameters, e.g., the density compression. However, for the simplicity, we use a fixed turbulence level in this work to study the acceleration efficiency of electrons by the interplanetary shock. In addition, the energy density ratio of the slab to 2D components is assumed to $E_{\text{slab}}:E_{\text{2D}}=20:80$.

In the simulations of shock acceleration of energetic electrons, we use a method of backward tracing electron trajectories (i.e., backward-in-time method, Kong et al., 2017; Kong & Qin, 2020). For each simulation case, we want to obtain the downstream distribution of accelerated electrons under a given initial distribution (e.g., upstream flux taken as the initial distribution in this work). Therefore, we put a large number of test particles in the downstream and calculate the trajectories of those particles backward. After a period of time, the particles move back to the upstream at the initial time $t_{0}$. The downstream distribution would be obtained with Equation 5 in Kong & Qin (2020) by calculating all the test particles. In this work each of the simulation cases does not correspond to any of the 74 shock events from the observations. It is complicated to obtain all the physical parameters for simulations in each of the observational shock events. In addition, the initial pitch angle distributions of all the observed events are difficult to be obtained; in particular, several events are even lack of observational data in some pitch angle directions. Therefore, we do not simulate the 74 observed shock events. Instead, we construct a parameter space with the shock angle $\theta_{\text{Bn}}$, upstream Alfv$\acute{\text e}$n Mach number $M_{\text A1}$, and shock compression ratio $s$ (listed in Table 2). This allows us to obtain a total of $9\times 7\times 5=315$ virtual shock event cases, for each of which we perform shock acceleration simulations of electrons. Then, the acceleration efficiency of electrons at the shock with different parameters is statistically analyzed, and compared with the result from the observations.

At the end of the simulation time $t_{\text{acc}}=10$ min, the downstream flux in the range $[z_{0},z_{1}]$ in the $90^{\circ}$ pitch angle for a target energy channel is obtained based on the initial upstream distributions as shown in Figure 1, where $z_{0}=L_{\text{th}}/2$ and $z_{1}=2.7\times 10^{-3}$ au. Note that the value of $z_{1}$ is set to a distance that a shock with a speed of 682 km s^-1 travels within 10 min. It is also noted that the speed of 682 km s^-1 happens to be equal to that of the shock on 2000 Feb 11. Actually, the exact values of $z_{0}$ and $z_{1}$ do not change the general findings in this work. More details about the backward-in-time method are given in Section 3 in Kong & Qin (2020). In order to calculate the trajectory of each test particle, we use a fourth order Runge-Kutta scheme with adaptive time stepping. Our calculation is regulated by a fifth order error estimate step with the error parameter $10^{-9}$. If we have a higher number of test particles and higher accuracy of the trajectories of test particles, we can get a more accurate downstream flux from the simulations. In our previous work (Kong & Qin, 2020), the downstream flux from the simulations are compared with the observed downstream flux. While in this work the simulated shock cases with the artificial shock parameters do not correspond to any of the observational events. Therefore, we do not directly perform comparisons of downstream flux between the simulations and observations. The input parameters for the shock and turbulence in the simulations are summarized in Table 3. The turbulence level of observations usually becomes greater near the shock surface (e.g., Kong et al., 2017), and it is not the same in different shock events. Note that the turbulence level $(b/B_{0})^{2}$ in Table 3 is set to a fixed value upstream and downstream of the shock for a qualitative analysis. Besides, the ratio of shock thickness $L_{\text{th}}$ to the gyroradius of the electrons of minimal and maximal energy in the simulations are 30 and 10, respectively, which implies a small gyroradius of electrons compared with the shock thickness, so that the gyro-cycles of energetic electrons we study could stay in the shock transition region for a period of time.

Next, we show some qualitative models for the efficient shock acceleration. Suppose that particles are accelerated by shock drift acceleration, the drift time $T_{\text{drift}}$ can be written as (Kong & Qin, 2020)

here we assume electrons are in a completely scatter-free regime. Note that Equation (7) is strictly valid only in the shock with a perpendicular geometry. Furthermore, in order to achieve efficient acceleration, particles should not move away from the shock transition within the drift time, i.e.,

where the factor $1/2$ in the left-hand side is used to consider the average over the pitch angle of particle velocity. Note that in the left hand side of Equation (8) a change of the angle between the magnetic field and the shock normal across the shock within $L_{\text{th}}$ should be taken into account. However, we ignore the angle change for simplicity, because it is very difficult to consider this effect. Thus, for the critical value $\cos\theta_{\text{Bn,c}}$ with efficient acceleration, we have

On the other hand, Drury (1983) showed that for each cycle of the particle crossing of the shock front the average momentum change of particles is

The work done by electric field force is equal to the energy change, i.e.,

Substituting Equation (10) into Equation (11), and considering Equations (2) and (3), we get

For the qualitative analysis, we suppose that the particle guiding center displacement in the shock normal direction during the shock drift acceleration is $L_{\text{th}}/2$ only considering upstream of the shock. Then, the number of particle crossings of the shock front is

where $T_{\text{gyro}}$ is the gyro-period of particles. In order for energetic particles to be efficiently accelerated, the total energy increase is assumed to be large enough compared to the particle kinetic energy $E_{\text{k}}$, i.e.,

Substituting Equation (12) into Equation (14), we obtain

or

Assuming the compression ratio $s$ is given by a representative value $s_{\text r}$, from Equation (15) we have the critical value of the upstream Alfv$\acute{\text e}$n Mach number with efficient acceleration as

Similarly, assuming the upstream Alfv$\acute{\text e}$n Mach number $M_{\text{A1}}$ is given by a representative value $M_{\text r}$, from Equation (16) we have the critical value of the compression ratio with efficient acceleration as

Here, we obtain some qualitative models for efficient shock acceleration for critical values by assuming shock drift acceleration and quai-perpendicular shocks. It is suggested that particles are efficiently accelerated in quasi-perpendicular shocks, so these models for critical values can be used for the qualitative analysis purpose.

We study the shock acceleration efficiency of suprathermal electrons, using spacecraft observational data for some ICME-driven shock events during 1995–2014 and test-particle simulations for cases of different shock parameters. It is suggested by Yang et al. (2018) (see also, Kong & Qin, 2020) that there exists a stream of energetic particles anti-sunward downstream of the shock in some SEP events. We check the 74 ICME-driven shock events listed in Yang et al. (2019), and find that for 65 events with complete data there is an anti-sunward beam of electrons downstream of the shock. This downstream anti-sunward beam may originate from the background solar wind rather than the shock acceleration. We need to use the sunward particles to study energetic particles accelerated by the shock along the magnetic field direction. It is shown that for most observational cases the acceleration of electrons in the sunward direction is much less efficient than in the perpendicular direction. Consequently, we suggest that the strongest acceleration occurs in the perpendicular pitch-angle direction. Therefore, in this paper the shock acceleration efficiency of electrons at $90^{\circ}$ pitch angle is investigated by comparing the simulations with observations.

For the 74 shock events listed in Yang et al. (2019), using data analysis on the spacecraft observations in seven energy channels ranging from 0.428 keV to 4.161 keV, we study how the efficiency of shock acceleration depends on different shock parameters, including the shock angle $\theta_{\text{Bn}}$, upstream Alfv$\acute{\text e}$n Mach number $M_{\text A1}$, and compression ratio $s$. In addition, we perform a similar research using data from 315 test-particle simulation cases with different shock parameters. With data from both the observations and simulations, we obtain the ratio of the average downstream to upstream flux $R_{a}$ for each shock acceleration case. Next, we get the ratio $N_{1}/N_{0}$ of the events with a large $R_{a}$, i.e., $R_{a}>R_{t}$ (the threshold value $R_{t}$ set to $4$ here). We find that for both the observations and simulations, the ratio $N_{1}/N_{0}$ decreases with the increase of shock angle cosine and increases with upstream Mach number and compression ratio. This indicates that a large shock angle, Mach number, and compression ratio can contribute to enhance the shock acceleration efficiency. Furthermore, the variation of ratio $N_{1}/N_{0}$ with respect to the shock angle cosine $\cos\theta_{\text{Bn}}$, upstream Alfv$\acute{\text e}$n Mach number $M_{\text A1}$, and compression ratio $s$ is fitted by a straight line of slope $\xi$ through the point $(1,0)$ for the observations and simulations in different energy channels. The value of slope $\xi$ by fitting $N_{1}/N_{0}$ –$\cos\theta_{\text{Bn}}$, –$M_{\text A1}$, and –$s$, respectively, shows that the shock acceleration is more efficient for lower energy particles compared to higher energy particles from the observations and simulations, which is obtained over a relatively small energy range. The reason for the lower acceleration efficiency in higher energy may be the following. With stronger scattering of particles by turbulence, the period of each cycle of the particle crossing of the shock front is shorter and the acceleration is more efficient (Drury, 1983; Kong et al., 2019). Furthermore, the turbulence scatters lower energy particles more easily compared to higher energy particles, so low-energy particles are accelerated easily to high energies by the shock. In addition, lower energy particles are more likely to stay in the shock transition within drift time. On the other hand, particles accelerated to higher energies may be firstly accelerated to lower energies. It is noted that if $R_{t}$ is set to some other values, the ratio $N_{1}/N_{0}$ probably vary, but the general findings of this paper will be the same.

We develop a model for the value of critical shock angle $\theta_{\text{Bn,c}}$ with high efficiency of shock acceleration, assuming shock drift acceleration of particles. We also obtain models for the critical values of upstream Mach number $M_{\text{A1,c}}$ and compression ratio $s_{\text{c}}$ with efficient acceleration, based on the average momentum change of particles proposed by Drury (1983) for each cycle of the particle crossing of the shock front. Although there are undetermined parameters, the models may be used to qualitatively show the trend of the shock acceleration of particles. It is assumed that the shock acceleration efficiency is prominent when the shock angle, upstream Alfv$\acute{\text e}$n Mach number, and compression ratio are larger than their respective critical values. Consequently, if the critical values of shock angle, upstream Alfv$\acute{\text e}$n Mach number, and compression ratio are smaller, the electron shock acceleration is more efficient. The modeling results show that the critical values of shock angle, upstream Alfv$\acute{\text e}$n Mach number, and compression ratio increase with the increase of particle energy, implying stronger shock acceleration for lower energy particles. Therefore, the data of observations and simulations, as well as the theoretical modeling results, suggest that shock drift acceleration is efficient in the electron acceleration by ICME-driven shocks (Yang et al., 2018, 2019; Kong & Qin, 2020).

It should be noted that, the upper limit of slab turbulence wavenumbers in the simulations is set to $k_{M}\sim 1.75\times 10^{-4}$ m^-1. It is, however, not significantly larger than the minimum resonant wavenumber ($k_{\text{rm}}=eB_{0}/p_{\text e}$, where $p_{\text e}$ is the electron momentum) of low energy electrons. For example, for electrons with an energy of 0.428 and 0.634 keV, the minimum resonant wavenumber is $k_{\text{rm}}\sim 1\times 10^{-4}$ m^-1 and $8\times 10^{-5}$ m^-1, respectively. The break wavenumber $k_{\text b}$ for the slab turbulence between the inertial and dissipation ranges is $10^{-6}~{}\text{m}^{-1}$. Therefore for electrons in the energy range of $0.428$ keV to $4.161$ keV the resonant wavenumber locates in the dissipation range of the slab turbulence. In addition, the upper limit of the wavenumber of 2D magnetic turbulence component is $6.07\times 10^{-7}~{}\text m^{-1}$, and the dissipation range of 2D turbulence is ignored. It is considered that energetic particles are in resonance with slab turbulence in the resonant wavenumber $k_{r}$, and parallel diffusion depends on the slab component with the assumption of quasi-linear theory, so the upper limit of the wavenumber of slab component of the magnetic turbulence is very important. We also consider that perpendicular diffusion depends on both the 2D and slab turbulence by the non-linear effects. It is more difficult to construct a 2D component turbulence with a very large wavenumber range. In addition, lower wavenumber range of 2D component can influence the perpendicular diffusion because of the non-linear effects. Therefore, we set a 2D component with a low upper limit of wavenumber.

As a result, in our simulations lower energy electrons have a narrow resonant range in the power spectrum of the turbulence compared to the observations. The simulations of the acceleration of high energy electrons, which have a large velocity, require more computational resources in order to ensure numerical accuracy. Therefore, we expect to use more powerful computational resources to increase the upper limit of the slab turbulence and the computing accuracy in the simulations in future works.