- 1Southwest Jiaotong University, Chengdu, China
- 2Key Laboratory of Dark Matter and Space Astronomy, Purple Mountain Observatory, Chinese Academy of Sciences, Nanjing, China
- 3Department of Planetary Sciences, University of Arizona, Tucson, AZ, United States
The origin of high-energy particles in the Universe is one of the key issues of high-energy solar physics, space science, astrophysics, and particle astrophysics. Charged particles in astrophysical plasmas can be accelerated to very high energies by electric fields. Based on the characteristics of interactions between charged particles and electric fields carried by the background plasma, the mechanisms of charged particle acceleration can be divided into several groups: resonant interactions between plasma waves and particles, acceleration by electric fields parallel to magnetic fields, and acceleration caused by drift of the guiding center of particle gyro-motion around magnetic fields in magnetic field in-homogeneity-related curvature and gradient, etc. According to macroscopic energy conversion mechanisms leading to acceleration of particles, several theories of particle acceleration have been developed: stochastic particle acceleration by turbulent electromagnetic fields, diffusive shock acceleration of particles, and particle acceleration during magnetic re-connections. These theories have their own assumptions and characteristics and find applications in different astrophysical contexts. With advances in high-energy astrophysical observations and in combination with analyses of characteristics of high-energy particle acceleration and radiation, we can better understand the underlying physical processes in dramatically evolving astrophysical environments.
1 Introduction
The origin of high-energy astrophysics can be traced back to the early 20th century when cosmic rays, high energy particles that can ionize gases in the atmosphere, were discovered by Victor Francis Hess (1936 Nobel Laureate in Physics)1. Later with the birth and advances of radio astronomy, X-ray astronomy, γ-ray astronomy, as an important branch of astrophysics, high-energy astrophysics has been playing an increasingly important role in astrophysics research. High-energy particles are the main emitter of varieties of high-energy astrophysical sources. Given the high energy of these particles, they can emit efficiently in all electromagnetic wave bands ranging from radio to gamma rays (Blumenthal and Gould, 1970; Kelner et al., 2006). High-energy astrophysical sources therefore can be readily detected and constitute an important kind of astrophysical objects ubiquitous in the Universe.
With the development of space astronomy, people started to realize that plasma disturbances in solar flares, Earth’s magnetosphere, and interplanetary space can also produce a large amount of high-energy particles. The acceleration of energetic particles then becomes a key issue in high-energy space science, high-energy solar physics, high-energy astrophysics, and particle astrophysics. The radiative characteristics of high-energy astrophysical sources depend on properties of high-energy particles they produced. High-energy particles from the Sun or outer space can also be detected directly by satellites in space. Cosmic rays with energies exceeding ∼1013 eV can be detected indirectly via observation of the cascading shower they produce in the atmosphere. Observation and analysis of properties of high-energy particles in high-energy astrophysics sources play an essential role in exploring the underlying physical processes.
It is generally accepted that most of the observed high-energy particles originate from acceleration of charged particles in the background plasma ubiquitous in the Universe. These high-energy charged particles may collide with nuclei in the background producing energetic secondaries. They can also produce energetic neutrals via charge exchange process2. Since Lorentz force of a magnetic field does not change the energy of charged particles, acceleration of charged particles needs to be realized through an electric field. In most astrophysical environment, conductivity of the background plasma is high and the corresponding electric field parallel to magnetic field is too weak to accelerate particles to observed high energies. Considering the fact that particle acceleration always occurs in intense energy dissipation processes of magnetized plasmas, motion of the background plasma with respect to magnetic field can produce inductive electric fields, and some electromagnetic fluctuations can effectively suppress the conductivity giving rise to electric field parallel to the magnetic field. In a strong magnetic field with very low density of charged particles, a strong electric field parallel to the magnetic field may also exist due to lack of current carriers to drive evolution of the magnetic field. All in all, under appropriate physical conditions, electric fields generated in a free energy dissipation process can effectively accelerate charged particles.
Since the discovery of cosmic rays at the beginning of last century, several mechanisms have been proposed for quantitative modelling of acceleration processes. Generally speaking, intense release of free energy of a magnetized plasma will produce a variety of plasma disturbances. Stochastic interaction of charged particles with electric field fluctuations on average will lead to an increase in the particle energy, which constitutes the base of the theory of stochastic particle acceleration. Some of the electric field fluctuations may be described as waves that can exchange energy with charged particles via resonant interactions. The energy distribution of charged particles accelerated by electric field fluctuations depends on statistical properties of these electric fields. The theory of stochastic particle acceleration is developed to address this dependence. The stochastic particle acceleration process is generic to all free energy dissipation in magnetized plasmas. However as a second order Fermi acceleration process (Fermi, 1949), which corresponds to diffusion in the momentum space, the corresponding acceleration efficiency may not be very high.
Over the past few decades, the so-called diffusive shock theory of particle acceleration has been introduced and studied extensively. Based on theories of magneto-hydrodynamics (MHD) and transport of charged particles in a magnetized plasma, the diffusive shock acceleration theory shows that particles can be accelerated in shock dissipation process via repeated crossing of the shock front. In particular, if the particle distribution is isotropic in the co-moving frame of the background plasma, it can be shown that the energy distribution of accelerated particles follows a power law with the spectral index determined by the compression ratio of the shock. Since the theory naturally links the energy distribution of accelerated particles to macroscopic properties of the fluid with rather general assumptions, this theoretical result is considered generic to energy dissipation via collionless shocks. In fact, it can be shown that as far as the particle distribution is isotropic, the energy of particles can only be changed by compressible modes of a fluid for the first order processes, which correspond to convection in the momentum space. Moreover high-energy astrophysical observations are generally consistent with this theory. This theory was developed independently by several teams in the 1970s (Blandford and Ostriker, 1978; Bell, 1978a, Bell, 1978b; Axford, 1981) and has drawn widespread attentions in high-energy astrophysics community (Drury, 1983).
When there is a large-scale magnetic field, charged particles in the background plasma can be accelerated by parallel electric fields associated with the reconnection current sheet and by inductive electric fields via coupling with curvature and/or gradient of the magnetic field. Compression of a large scale magnetic field can also lead to betatron acceleration (Bogachev and Somov, 2005), the underlining process for betratron accelerators that, as the magnetic gradient draft acceleration, leads to an increase of the perpendicular momentum of accelerated particles. Detailed processes of particle acceleration depends on the structure of the large scale magnetic field, which may lead to a variety of energy distribution of accelerated particles. However, given the importance of magnetic field in the process of particle acceleration, particle acceleration during magnetic reconnection is considered generic to magnetic energy dissipation process.
It should be noted that not all processes pertaining to energy gain of charged particles in the background plasma are considered as particle acceleration process. If an energization process is not selective resulting in comparable energy gain of all charged particles in the background plasma, it is considered as a plasma heating process. The corresponding distribution function of charged particles can be approximated as a Gaussian in velocity (Maxwellian in energy). The theory of particle acceleration mainly studies momentum distribution and transport of particles with their energy much higher than the mean energy of the background particles. The selective acceleration of some low-energy charged particles in the background plasma to high energies so that these particles may participate in the acceleration processes at larger scales is called the injection problem. The above-mentioned resonant wave-particle interaction, resonant interaction of waves and charged dust, and some particular configuration of the magnetic field can cause charged particles in the background plasma to be selectively accelerated to high energies (Ellison et al., 1997; Meyer et al., 1997). Neutral atoms moving with a high speed relative to the background plasma will appear as high-energy charged particles when being ionized, which can also be an effective mechanism for injection under appropriate astrophysical conditions. In high-energy solar physics and space science, where the energy spectrum can be measured from the thermal background to very high energies, injection of particles into the acceleration process is an important issue.
When charged particles with increasing energy decouple from background particles, they can interact with electromagnetic fields on larger scales due to their relatively long mean free paths. An important aspect of high-energy astrophysics research is to analyze acceleration and radiation mechanisms of these energetic particles via interactions with the background plasma. The stochastic particle acceleration mechanism originally proposed by Fermi (i.e., charged particles are accelerated gradually to higher energies via stochastic interactions with electromagnetic field fluctuations carried by the background plasma) is still the starting point of most macroscopic theories of particle acceleration (Fermi, 1949; Parker, 1958; Toptygin, 1980; Petrosian and Liu, 2004). Current researches in this area focus on using the associated radiation mechanisms to infer distribution function of energetic particles from observations, then in combination with the relevant energy release processes (such as magnetic reconnection, shocks, MHD instabilities, etc.), analyzing the related particle acceleration and transport, plasma heating, and magnetic field amplification processes.
This review is organized as the following: General Characteristics of HIGH-ENERGY Particle Acceleration in a Fluid discusses microscopic interactions leading to changes in particle energy; Specific Acceleration Mechanisms summarize particle acceleration in specific astrophysical conditions; Conclusions are drawn in Conclusion.
2 General Characteristics of HIGH-ENERGY Particle Acceleration in a Fluid
In what follows, we take the energetic particles to sufficiently rare that they do not affect the background thermal plasma or electromagnetic field, so the electric and magnetic fields are determined by the ambient thermal plasma. This test-particle limit is usually a good approximation for the energetic particles emphasized in this review. In reality, there are situations (notably supernova blast waves) where the effects of cosmic rays on the background plasma and magnetic field must be considered (Lucek and Bell, 2000; Blasi, 2004; Bell et al., 2013).
2.1 Energy Change and Instantaneous Acceleration Rate
Let us begin with some general comments regarding charged particle acceleration in astrophysical plasmas. Since particle-particle collisions are extremely rare in the plasmas which we are considering, and neglecting the gravitational force, motion of a particle with charge q velocity w and moment p is determined by the electric and magnetic fields E and B. We have
where c is the speed of light and the upper dot indicates a derivative with respective to time t. Then, the rate change of the energy T of a charged particle in such a plasma is via the electric field
The electric field may be ambient or associated with scattering by turbulent fluctuations. Clearly, in order to evaluate the energy change, we must know the particle trajectory in the electromagnetic field. This leads to the general requirements that acceleration and spatial transport be intimately coupled and both
It turns out that the spatial scales of the plasma and magnetic-field variations which are most important in determining cosmic-ray motions are those comparable to or larger than the particle gyro-radii, which are much larger than the thermal particle gyro-radii. Therefore, the relevant plasma scales are such that the MHD approximation is valid and the ambient electric field in a nonrelativistic plasma may be determined from the fluid velocity
where
In this case, the rate of energy gain of the particle is given by
which, upon using a basic vector identity to rearrange the terms, can be rewritten as
Then the instantaneous energy change rate of a particle can be written as
where the first term on the right-hand side gives the acceleration rate by the inductive electric field:
Note that
for
In the presence of electric fields parallel to the magnetic field or electromagnetic fields varying on the same scales of the particle gyro-motion, the acceleration process will depend on the charge and/or mass of the particles, and therefore become selectively. These processes are essential at low energies when the particle velocity is comparable to the flow velocity and the particle distribution may not be isotropic. The details of these accelerations are complicated, depending on properties of the plasma waves and parallel electric fields. The former usually leads to stochastic particle acceleration, a second order process, while the latter leads to direct acceleration, a first order process.
If the variation scale of
where
It can be shown that drift of the guiding center in magnetic field gradients (3rd term on the right-hand side of Eq. 8) and curvatures (2nd term on the right-hand side of Eq. 8) will lead to changes of energies associated with motions in perpendicular and parallel to B, respectively (Zhou et al., 2015):
where the last terms on the right-hand side of Eqs 9, 10 can lead to changes in the particle speed and may be rewritten as
In principle, in combination with particle trajectory, Eqs. 5 or 9 and 10 can be used to study acceleration of individual particles (Turkmani et al., 2006; Zhou et al., 2015). However, the results heavily depend on the structure of
The problem can be simplified dramatically for test particles in MHD. Jones (1990) first pointed out that the microscopic energy exchange between charge particles and electromagnetic fields discussed above can be expressed in terms of compression of the fluid
For isotropic particle distributions, the last two terms cancel if averaged over the direction of motion. The mean energy gain is then proportional to the divergence of
Using Eqs 1, 3 for
Again by simply rearranging terms this may then be rewritten as
Noting that
where the particle velocity in the fluid frame
where we have used the notation that for an arbitrary vector
One may also obtain Eq. 17 by considering the Lorentz force in the fluid frame (Zhang and Lee, 2013):
where the first term on the right hand side corresponds to acceleration of the fluid. Then from
For energetic particles,
This is precisely the rate of energy change appearing in the Parker equation (Parker, 1965) and corresponds to the adiabatic thermodynamic process. The averaged acceleration rate is then determined by the compression rate and does not depend on the particle charge and mass as far as the particle distribution is isotropic. Although for an isotropic particle distribution, Eq. 18 shows that the momentum change rate averaged over angle
Eq. 19 for energy change clearly also applies at discontinuities in the flow and/or magnetic field, such as shocks or current sheets, if scattering is sufficient to keep the anisotropies small. In what follows, we will suppress the angular brackets in Eq. 19. Also, in most applications,
In most cases, the anisotropies will, like the electric field, be proportional to the flow speed or its derivatives. Hence in these cases, the acceleration rate will be of the second order in the flow velocity
However, the amount of energy gain via the adiabatic process is limited by the extent of compression of the background plasma (Schatzman, 1963; Drury, 1983). The acceleration also does not depend on the charge and mass of accelerated particles. Without consideration of other physical processes, the compression process alone clearly cannot account for the observed high energies of cosmic ray particles and spectral variations among different particle species. As we will see below, the acceleration rate will be further reduced from
when considering other processes.
2.2 Transport Equations of Cosmic Rays and Energetics
As mentioned above, the acceleration process is intimately related to the spatial transport of particles. A robust and widely applicable transport theory, applicable to particles with
FIGURE 1. Computed orbits of two particles in a specified magnetic field that is axisymmetric along the z axis. The two particles are started with nearly the same initial velocity and position, but behave quite differently. This sensitivity to initial conditions results in chaotic or random trajectories (Buechner and Zelenyi, 1989; Xie and Liu, 2020).
The resulting spatial diffusion coefficient can be obtained in terms of the statistical properties of the turbulent magnetic field by using a variety of approximations: quasi-linear theory (Ptuskin, 1988; Bykov and Toptygin, 1993) and integration of particle orbits in a synthesized magnetic field are the most often used. The trajectories become effectively randomized and are made isotropic in the local frame (Figure 1). Under a wide range of conditions this drives them to near-isotropy in the fluid frame (or, more generally, the frame of the scatter) with a time scale τ, which usually depends on the charge to mass ratio of particles. Unfortunately, a general theory for lower-energy particles whose speeds are comparable to U or smaller is not yet available3 and observations of narrow band radio bursts from solar flares suggest that these lower-energy particles can be highly an-isotropic (Kliem et al., 2000).
We work in terms of the energetic-particle distribution function
where
Proceeding to the next order in
where we have used Eq. 19 for
The associated streaming flux (in the observer’s frame) of the particles may be written as:
where the second term on the right hand side is due to the Doppler effect. The associated anisotropy is given by:
The drift velocity of the guiding center
In this equation we have the spatial diffusion contained in the zeroth order Eq. 21, with new terms containing the advection and acceleration/deceleration caused by the fluid flow. Note that the electric field does not appear explicitly in Eq. 22. However, as shown above on the energy change, it is nonetheless contained in the terms involving the flow velocity
Upon carrying out the expansion to second order in
where the source and guiding-center drift terms have been ignored and the coefficient of viscous momentum transfer
In Eq. 30, the first term on the right-hand side describes diffusion, the second: convection, the third: adiabatic energy change and the last four terms describe energy changes due to velocity shear and fluid acceleration. The order of magnitude of the new terms are of the second order of
Using Eq. 33, Earl et al. (1988) (ELM) derived the average rate of momentum change:
and the average rate of momentum dispersion:
Note that this momentum or energy change is caused by the fluctuating electric fields associated with the moving magnetic scattering centers, and results in both deceleration and acceleration. The associated viscosity introduced by high-energy particle acceleration is given by:
where the integration is for p satisfying
This mechanism is distinguished from previous discussions in which the energy change was associated with compression. The transport equation of EJM has been solved using stochastic integration for an idealized, divergence-free flow similar to that in 2-dimensional reconnection with the flow vector in the x-z plane given by the expression
It is readily shown that the values of
Although the derivation of Eq. 30 assumes that
Finally, the acceleration of particles is directly related to dissipation of energy stored in the magnetic field and/or damping of the flow velocity (Melia et al., 2001). From MHD equations, we have
where ρ and P are the mass density and pressure of the background plasma, respectively, the second and the last two terms on the right-hand side of Eqs 40, 41 are associated with dissipation via resistive and inductive electric fields, respectively,
and the coefficient of viscous momentum transfer
Both the acceleration via drift in magnetic gradient and/curvature (Eqs. 9, 10) and the acceleration due to compression (Eq. 19) are associated with kinetic energy dissipation via the inductive electric field. The Fermi mechnism and shock drift acceleration are associated with the inductive electric field as well (Fermi, 1949; Jokipii, 1982). In the MHD, the dissipation of magnetic energy via the inductive electric field also leads to acceleration of the flow and vice versa via the third terms on the right-hand side of Eqs. 40, 41. The particle acceleration by the shear and by deceleration of the flow is associated with scattering of particles by the background plasma leading to viscosity and cosmic ray pressure which can cause damping of the shear and a force acting to decelerate the background flow, respectively. These accelerations are caused by inductive electric fields at small scales directly, reminiscence of acceleration by compression.
Resistive dissipation in MHD usually leads to heating of the background plasma:
where
Therefore in the context of MHD, the acceleration of high-energy particles can be classified into two kinds: direct acceleration by resistive electric fields, which can be very efficient and only operates in particular astrophysical environments where the resistivity may be high, and acceleration by inductive electric fields via effective scattering with the background plasma, which appears to be ubiquitous and is relatively less efficient than acceleration by resistive electric fields. Statistical properties of the latter can be described with the diffusion coefficient κ, which also determines the spatial transport of high-energy particles. The issue of high-energy particle acceleration in MHD then can be fully addressed with κ and the source term Q, which describes the particle acceleration at relatively low energies and therefore smaller scales where MHD may not be appropriate to describe the characteristics of particle energizing electric fields.
3 Specific Acceleration Mechanisms
The basic transport equation discussed above has been used in a variety of contexts to discuss the acceleration of energetic particles. The most successful and widely applicable is that of standard diffusive shock acceleration, which results from the application of the first-order Eqs. 22 to a planar shock wave. As we will see, this has the singular property of producing a power law momentum spectrum which is extremely insensitive to model parameters and is close to that observed.
3.1 Diffusive Shock Acceleration
Consider a steady, plane shock propagating in a uniform medium. Define the x-direction as the direction of propagation and let particles be introduced uniformly and steadily at the shock with a rate of A at an injection momentum
where
where H is the Heaviside step function. Note that in the limit of a strong non-relativistic shock,
One should note that since the diffusion coefficient κ usually depends on the momentum, the particle distribution in the upstream of the shock is usually not a power law. The distribution function integrated in the upstream is given by
where we have assumed that κ is independent of x in the upstream. Therefore in the shock precursor, one may have a distribution harder than that in the downstream of the shock. If one has an escape boundary in the upstream, the particle distribution will be softer (Zhang and Liu, 2020).
Solving the time-dependent version of Eq. 22 with particles injected at a low momentum
which is higher than the rate caused by shearing by a factor of
Although the Parker equation and the discussion leading to Eq. 22 apply for arbitrary direction of the local average magnetic field with appropriate diffusion tensor, the acceleration at a shock is physically quite different for quasi-parallel shocks (upstream plasma flow velocity nearly parallel to the magnetic field) than for perpendicular shocks, where the upstream flow is nearly perpendicular to the magnetic field (Jokipii, 1982; Drury, 1983; Jokipii, 1987). For quasi-parallel shocks the particle acceleration is best thought of as resulting from particles being scattered by upstream and downstream scattering centers which in the shock frame are converging and accelerating the particles which cross the shock more than once. Statistically, some particles will cross the shock many times and others not so many. This process can be shown to produce the power-law spectrum (Drury, 1983).
The acceleration process is physically quite different for quasi-perpendicular shocks. In this case the average magnetic field magnitude jumps suddenly at the shock by the shock ratio r, which causes a gradient drift along the shock face in the direction of the electric field
FIGURE 2. Cartoon illustrating the difference in the acceleration of charged particles at parallel (left) and perpendicular (right) shocks. In the former, the particles can move back and forth across the shock moving along the magnetic field, whereas in the latter, the particles cannot move as easily normal to the shock front and the energy gain at the shock comes in large part from drift. The electric field is perpendicular to the magnetic field and the flow velocity.
A discontinuous change in the flow speed, such as at a shock, is not necessary for this acceleration to occur. As demonstrated by Giacalone et al. (2002), when a finite-scale compression (with characteristic scale
Because large-scale astrophysical fluids are generally turbulent, with large-amplitude fluctuations in fluid parameters, the theory of diffusive shock acceleration has been applied to shocks propagating through a turbulent medium. This gives one possible explanation of the Voyager 1 observations as it crossed the termination shock (Figure 3). Recently Arthur and le Roux (2013) have considered a similar model with similar results.
FIGURE 3. Left: Shock (red dashed line) and magnetic field (blue lines) geometry for an upstream average magnetic field perpendicular to the shock normal. U1 and U2 give the upstream and downstream flow velocities, respectively. Right: Profiles of density of the accelerated particles, for low (black lines), medium (blue dashed lines), and high (red dot-dashed lines) energy ranges at different locations z = 0.0 (top) and π (bottom), respectively (Guo et al., 2010).
The Parker transport equation has acceleration of particles occurring only if the term
Consider the case that the diffusive skin depth
The physical basis of the acceleration is the interplay between 1) the energy change caused by the compression or expansion of the fluid and 2) the diffusion into or away from the region of compression or expansion. Rapid diffusion leads to the particle being able to diffuse away from a region of compression or expansion before the compensating expansion or compression can occur. Hence, statistically, some few particles will be fortunate enough to gain energy in several compression regions. In this process, for large κ, the accelerations dominate the particle energy change, even in those cases where the compression and expansions are equally present in the fluid flow. This is because, statistically, some particles can reach very high energies, but they cannot be decelerated to energies lower than zero.
Note also that this acceleration can take place for any orientation of the magnetic field. Gradient and curvature drifts can in general significantly affect the particle trajectories as they are accelerated. To illustrate this process, consider the simple, periodic one-dimensional velocity profile
Using the quasi-linear approximation to the Parker Eq. 22, for small-amplitude compression, Jokipii and Lee (2010) find an equation which is diffusive in the momentum space. For no spatial variations except for the fluctuating density, they find
where
3.2 Stochastic Acceleration
Acceleration by fluctuating electric and magnetic fields, such as those found in a turbulent medium has been discussed ever since the seminal paper by Fermi (1949), who considered acceleration by randomly moving magnetic clouds. This has been subsequently generalized to apply to turbulent fluctuations (Bykov and Toptygin, 1993), and is also often termed the second-order Fermi acceleration. This form of acceleration, contained as the diffusion in momentum term in Eq. 30, has an extensive history. It is quite often the mechanism of choice for possible diffuse re-acceleration of cosmic rays in the interstellar medium and in solar flares (Petrosian and Liu, 2004). Nonetheless, it has at least one considerable disadvantage. Being of the second order in the wave or fluid flow velocity, this term, which is generally much slower than shock acceleration at high energies, may simply be added to the right side of Eq. 22 and is often invoked where shocks are believed not to be present or the diffusion coefficient in the shock upstream is too large to accelerate particles efficiently.
This acceleration can arise in two physically distinct processes. One can consider the effect of randomly moving waves which scatter the particle resonantly (Tverskoǐ, 1968) or one can consider larger-scale turbulent fluid motions where the energy change results from the stochastic shear or compression as discussed in the previous section (Ptuskin, 1988; Jokipii and Lee, 2010; Ohira, 2013). The former does not produce a power law spectrum with the desired value in a robust way since both acceleration time and scattering time, which is inversely proportional to the diffusive escape time, is inversely proportional to the wave intensity. The shape of the spectrum depends sensitively on the transport parameters in the acceleration region. Hence it is probably not the source of most of the observed cosmic-ray spectra which are almost-always power laws. Nonetheless, it may produce power-law with variable indexes and remains popular in some applications. The latter can lead to a power law distribution since both the acceleration and diffusive escape times are inversely proportional to the scattering time (see discussion below).
For scattering by Alfvén waves propagating both along the magnetic field and opposite to the field with a speed
The corresponding acceleration rate is given by
which is lower than the acceleration rate by shocks by a factor of
Here, τ must be significantly less than
Because of the term
Stochastic particle acceleration focuses on addressing the gradual and statistical energy gain of charged particles via interactions with a spectrum of turbulent electromagnetic fields. It is essentially a second order Fermi acceleration process, where the acceleration of a population of high energy particles is described with a diffusive term in the momentum space Webb (1989). For the case where the turbulence is homogeneous and isotropic with the spectrum given by
where
T and S correspond to the transverse (sharing) and the longitudinal (compressible) component, respectively. Then we have the ensemble averaged turbulence energy density per unit mass:
and
As long as the velocity change of the background plasma over a particle scattering mean-free-path λ is much smaller than the particle speed w, the particle distribution in the comoving frame of the fluid is approximately isotropic with an anisotropy on the order of
where
For acceleration by acoustic wave with a wave speed of
If the waves have a length of L so that
Diffusion of particles in a shear flow can also introduce particle diffusion in the momentum space (Earl et al., 1988; Ohira, 2013):
The last integration diverges as
where we have assumed
Although stochastic acceleration of high energy particles is slower than that by shocks by a factor of
Observations of cosmic rays abundance show that heavier elements and those in refractories are strongly enhanced. Figure 4 shows the corresponding results (Meyer et al., 1997). This can be attributed to the small charge to mass ratio of these elements and/or corresponding dust (Ellison et al., 1997) so that they may resonantly interact with larger scale waves than those with higher charge to mass ratio. For a given magnetic field, the gyro-radius is inversely proportional to the change to mass ratio and waves at larger scales usually carry more energies.
FIGURE 4. Dependence of the cosmic ray abundance enhancement on the atomic number A (Meyer et al., 1997).
The best example to illustrate resonance wave-particle interactions is perhaps the prominent enhancement of MeV 3He abundance relative to 4He in small impulsive solar energetic particle events (SEPs) (Liu et al., 2006). Figure 5 shows the relative abundance of energetic helium isotopes for some SEPs (Ho et al., 2005). It can be seen that the fluence of 3He is distributed in a relative narrow range, which the fluence of 4He can spead out over 4 order of magnitude. The enhancement of 3He is most prominent for small events with a lower 4He fluence.
FIGURE 5. Relative abundance of Helium isotopes accelerated in solar flares (Ho et al., 2005; Petrosian et al., 2009).
Liu et al. (2006) attributed this phenomenon to resonant interaction of 3He with proton cyclotron waves. Figure 6 shows dispersion relations with different Alfvén speed for waves propagating along the magnetic field lines and the corresponding resonance conditions for 3He and 4He. It can be seen different helium isotopes interact with different modes of waves and this interaction also depends on the particle energy.
FIGURE 6. Resonant interactions of Helium 3 and 4 with plasma waves in different background plasmas indicated with α, the ratio of the electron plasma frequency to gyro-frequency (Liu et al., 2006). The straight lines stand for resonance conditions for helium isotopes with different energies. The proton gyro-frequency is given by
If one approximates the spatial diffusion as an escape term from the acceleration region, the theory of stochastic particle acceleration can be simplified as a diffusion equation in the momentum space with both source and escape terms (Petrosian and Liu, 2004). This model has been applied to observation of SEPs with prominent 3He enhancement. Figure 7 shows such an example. It can be seen that due to selective acceleration of 3He, all 3He in the background are accelerated to the MeV energy range. For 4He ions, as a major component of the background plasma, they affect the dispersion relations (Figure 6) and cause strong damping of waves in resonance with the background 4He. As a result, only a small fraction of 4He is accelerated to MeV, explaining the observed enhancement of high energy 3He. This also can also explain the fluence distribution of 3He and 4He (Petrosian et al., 2009).
FIGURE 7. Selective acceleration of 3He in impulsive solar flares (Liu et al., 2006).
4 Conclusion
The acceleration of charged particles from low energies in the background to high energies covers a broad energy range. The associated spatial and temporal scales are also wide. These multi-scale couplings make the study of particle acceleration a very challenging task. To have quantitative modeling, dramatically simplifications are necessary.
If one concerns very high-energy particles with a very low number density, the test particle and fluid approaches can be taken. Then the leading order acceleration is associated with compression. In the context of acceleration by diffusive shocks, a simple power-law distribution can be obtained for the lack of scales in this model. In the case of strong non-relativistic shocks in supernova remnants (SNRs), one may consider the feedback of the cosmic ray acceleration to the shock structure (Bell et al., 2013). Multi-wavelength observations of a sample of SNRs appears to favor such a model (Zeng et al., 2019; Zhang and Liu, 2019).
Stochastic particle acceleration is generic to magnetic energy dissipation processes. Although it is not as efficient as diffusive shocks, it may dominate the acceleration of low energy particles (Petrosian and Liu, 2004). In cases where the shock acceleration is suppressed due to lack of scattering in the upstream, the stochastic acceleration may dominate in the shock downstream (Fan et al., 2010). The most prominent feature of this mechanism is selective acceleration of background particles with different change to mass ratio. It can not only explain the abundance enhancement of heavy elements in cosmic rays, but also account for enhancement of MeV 3He relative to 4He by up to 4 orders of magnitude in some impulsive SEPs. Such kind of resonant interactions have been realized in plasma devices (Zhang et al., 2017). Since both the acceleration and scatter rates of particles by waves are proportional to the wave intensity, the resulting particle distribution is very sensitive to the wave intensity, which can lead to a broad range of spectral index and may explain the impulsive nature of high energy burst.
Study of particle acceleration in magnetic reconnection and in general astrophysical context rely heavily on numerical simulations due to complexity of the related electromagnetic field configuration (Kowal et al., 2011, 2012; Guo et al., 2014; Dahlin et al., 2015; Zhou et al., 2015; Beresnyak and Li, 2016; Comisso and Sironi, 2018). On the other hand, power-law distribution may result from an emergence phenomenon due to the lack of characteristic scales in the system (Mewaldt et al., 2007).
Author Contributions
All authors listed have made a substantial, direct, and intellectual contribution to the work and approved it for publication.
Funding
This work was partially supported by the National Key R&D Program of China (Grant No. 2018YFA0404203), NSFC grants (Nos. U1738122, U1931204, and 11761131007), and by the International Partnership Program of the Chinese Academy of Sciences (Grant No. 114332KYSB20170008). SL thanks the support of a startup grant at Southwest Jiaotong University.
Conflict of Interest
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
Acknowledgments
This study benefited from discussions within the International Space Science Institute (ISSI) team on “Origins of 3He-Rich Solar Energetic Particles.”
Footnotes
1Although supernovae as high-energy astrophysical phenomena had been observed before, due to lack of observations of high-energy radiation at that time, it is not considered as the birth of high-energy astrophysics here.
2Although decay or annihilation of some heavy particles beyond the standard model of elementary particles may also produce energetic particles, conclusive evidence for these processes has not been discovered yet.
3See the section on stochastic particle acceleration below.
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Keywords: cosmic rays, particle acceleration, the Lorentz force, turbulence, high-energy astrophysics, high-energy particle radiation, electromagnetic fields
Citation: Liu S and Jokipii JR (2021) Acceleration of Charged Particles in Astrophysical Plasmas. Front. Astron. Space Sci. 8:651830. doi: 10.3389/fspas.2021.651830
Received: 11 January 2021; Accepted: 01 June 2021;
Published: 23 June 2021.
Edited by:
Elisabete M. De Gouveia Dal Pino, University of São Paulo, BrazilReviewed by:
Yasuhito Narita, Austrian Academy of Sciences (OeAW), AustriaGrzegorz Kowal, University of São Paulo, Brazil
Copyright © 2021 Liu and Jokipii. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.
*Correspondence: Siming Liu, liusm@swjtu.edu.cn; J. Randy Jokipii, jokipii@lpl.arizona.edu