Cosmic ray transport and acceleration with magnetic mirroring

ABSTRACT

We analyse the transport of cosmic rays (CR) in magnetic fields that are structured on scales greater than the CR Larmor radius. We solve the Vlasov–Fokker–Planck equation for various mixes of mirroring and small-angle scattering and show that relatively small deviations from a uniform magnetic field can induce mirroring and inhibit CR transport to levels that mimic Bohm diffusion in which the CR mean-free path is comparable with the CR Larmor radius. Our calculations suggest that shocks may accelerate CR to the Hillas energy without the need for magnetic field amplification on the Larmor scale. This re-opens the possibility, subject to more comprehensive simulations, that young supernova remnants may be accelerating CR to PeV energies, and maybe even to higher energies beyond the knee in the energy spectrum. We limit our discussion of CR acceleration to shocks that are non-relativistic.

1 INTRODUCTION

There has been considerable progress in understanding the origins of cosmic rays (CR) during the nearly half century since the development of the theory of diffusive shock acceleration (DSA) by Krymsky (1977), Axford, Leer & Skadron (1977), Bell (1978), and Blandford & Ostriker (1978). Observations across the whole range of energies have improved immensely and we now have a credible theoretical understanding of how and where CR are accelerated in the range of energies up to 100 TeV. However, there are many questions to be answered about the origins of Galactic CR at energies of PeV and above. At the very highest energies, 1–200 EeV, CR must originate outside the Milky Way Galaxy since these have a Larmor radius exceeding the size of the Galaxy. Observations of ultra-high-energy CR (UHECR) are limited by their rarity and the lack of secondary radiation, although the Pierre Auger Observatory and the smaller Telescope Array are beginning to provide composition and anisotropy data that constrain their origin (Tsunesada et al. 2022; Plotko et al. 2023). The most challenging remaining mystery is the origin of Galactic CR at and above the knee in the energy spectrum at a few PeV.

From both observation and theory, it appears that CR acceleration by the historical supernova remnants (SNR) tails off at a few hundred TeV (Zirakashvilii & Ptuskin 2008; Bell et al. 2013; Cristofari 2021). One possibility is that PeV CR are accelerated by SNR in their first decades when a high velocity shock expands into a dense circumstellar medium. Other possibilities are that PeV accelerated near the Galactic centre are transported to the Earth without being lost from the Galaxy (Abramowski et al. 2016; Adams et al. 2021; Muena et al. 2024), or that star-forming regions with strong stellar winds or repeated SNR expanding into a strong magnetic field provide enhanced conditions for first- or second-order Fermi acceleration beyond 1 PeV (Vieu, Reville & Aharonian 2022; Vieu & Reville 2023; Vink 2024; Aharonian et al. 2024).

We use SI units throughout this paper, except that eV is used for CR energy and values of the magnetic field are quoted in Gauss. The Hillas energy |$E=ZuBR$| (Lagage & Cesarsky 1983a, b; Hillas 1984) imposes a basic limit on the maximum energy to which CR can be accelerated. In this expression, |$Ze$| is the particle charge, B is the magnetic field, R is the size of the accelerating region, and E is the CR energy. A more incontrovertible but less stringent limit that |$E< ZcBR$| arises from the need for the Larmor radius to be less than R. The tighter limit |$E< ZuBR$| arises from the need for the distance over which acceleration occurs to be smaller than R or for the time required for acceleration to be less than the available time |$R/u$|⁠. One form of the spatial constraint is that a CR passing a distance R through the magnetohydrodynamic (MHD) electric field |$-{\boldsymbol u}\times {\boldsymbol B}$| traverses a maximum potential difference |$uBR$|⁠. This limit most clearly applies to acceleration at perpendicular shocks where the magnetic field is perpendicular to the shock normal. It also applies to neutron star magnetospheres and nebulae where |$uBR$| is the potential difference between pole and equator. In principle, the path of a CR through turbulence may sample the MHD electric field preferentially to gain an energy exceeding |$ZuBR$|⁠, but this is difficult to arrange. To reach the higher Larmor limit |$E=ZcBR$|⁠, a CR trajectory would need to be microscopically aligned with the |$-{\boldsymbol u}\times {\boldsymbol B}$| MHD electric field throughout a time |$R/u$| during which the CR travels a distance |$(c/u)R$|⁠. The Hillas limit, |$E< ZuBR$|⁠, is most easily understood as a competition between the acceleration rate and the available time-scale |$t_{\rm limit}=R/u$| as discussed below. A recent discussion of the general applicability of the Hillas condition is given by Oka et al. (2025).

The maximum energy to which DSA can accelerate CR is determined by CR transport, especially in the upstream plasma. Fluctuations in the magnetic field scatter the CR and stop them escaping far upstream of the shock. Scattering is strongest when the magnetic field is structured on the scale of the Larmor radius |$r_g$| and scatters CR with a mean-free path on the scale of the Larmor radius. Scattering on the Larmor scale leads to Bohm diffusion with a diffusion coefficient |$D_B\sim r_gc$|⁠. In DSA at non-perpendicular shocks, a condition for reaching the Hillas energy is that CR diffuse with the Bohm diffusion coefficient.

As shown by Lagage & Cesarsky (1983a, b), even with Bohm diffusion, the characteristic ambient magnetic field in the interstellar medium (ISM) of 1–30 |$\mu$|G is insufficient for acceleration to 1 PeV. Magnetic field amplification by a non-resonant instability (Bell 2004) can provide the hundred-fold increase needed for the Hillas limit to reach 1 PeV. Given sufficient time to evolve, non-resonant amplification saturates on the spatial scale of the Larmor radius of the CR driving the instability as required. However, the growth rate is too slow for the instability to grow to the Larmor scale of CR with PeV energies (Zirakashvilii & Ptuskin 2008; Bell et al. 2013).

Standard DSA allied with non-resonant magnetic field amplification may possibly account for PeV acceleration in special environments such as SNR in their first few decades of existence, the Galactic centre, or young stellar clusters, but it does not provide a basis for CR acceleration to 1 PeV in any currently observed SNR in the Galaxy.

Here, we examine the possibility that CR transport might be inhibited by magnetic mirrors which confine CR more closely to the shock in the upstream plasma and thereby facilitate more rapid acceleration. This mirror-dominated shock acceleration (MDSA) may ease the challenge of explaining Galactic CR acceleration to PeV, and may even open up the possibility of CR acceleration beyond the Hillas limit.

Many authors have recognized that simple diffusion theory, as assumed in derivations of the Hillas limit, does not tell the whole story since CR trajectories follow magnetic field lines if their Larmor radius is smaller than distances over which the magnetic field changes in magnitude or direction. This can lead to subdiffusion (Duffy et al. 1995), superdiffusion (Lazarian & Yan 2014), cross-field diffusion in perpendicular shocks (Jokipii 1982, 1987), and trapping in magnetic loops (Decker 1993). The contribution of magnetic mirrors to CR transport has recently received growing attention (Lazarian & Xu 2021; Barreto-Mota et al. 2024; Zhang & Xu 2024; Reichherzer et al. 2025).

The possibility of CR acceleration when trapped between an upstream mirror and a shock, or between an upstream mirror and a downstream mirror, was considered by Jokipii (1966) in the context of the solar wind. Jokipii showed that rapid acceleration is possible leading to CR energy increases that are essentially adiabatic as the distance contracts between an upstream mirror and the shock or a mirror downstream of the shock.

The problem with mirrors for general acceleration is that CR are too rapidly swept away downstream by the mirror, and that mirroring only applies to CR with particular pitch angles, thus limiting the overall acceleration efficiency and failing to produce the extended |$p^{-2.3-2.6}$| spectrum required to explain Galactic CR. Jokipii’s theory was developed before the advent of modern DSA theory which showed that CR return to the shock typically |$c/u$| times, and some CR return to the shock many more times as determined by diffusion theory or random walk theory. If mirror reflection confines some CR more closely to the shock while still allowing CR to the traverse the shock many times in an overall diffusive manner, then energies of a few CR may be boosted to, and possibly beyond, the Hillas limit by rapid reflection. CR acceleration by MDSA may be less efficient, and this would be consistent with the observed steepening at the knee.

Jokipii (1966) noted that non-conservation of the magnetic moment at a shock, might lead to de-trapping of CR, allow CR to stay with the shock for longer and be accelerated to higher CR energies. It is a variant on this line of thinking that we pursue in this paper. Angular scattering by sub-Larmor-scale fluctuations in the magnetic field might allow CR to pass semirandomly through mirrors. CR might benefit from multiple acceleration episodes as a population of mirror-trapped CR is overtaken by a shock advancing into a plasma containing mirrors. In Section 5–8 below, we examine transport when there is both mirroring and small-scale scattering.

2 COSMIC RAY TRANSPORT THROUGH A SINGLE MIRROR: THE EQUATIONS

We begin with two idealized analyses (Sections 2–4) to set the scene for the more comprehensive analysis that follows, and to offer physical explanations to underpin our computational model. First, we analyse CR interaction with a mirror when two conditions both apply: (i) no small-angle scattering and (ii) the mirror scale length exceeds the Larmor radius, |$L\gg r_g$|⁠, such that cross-field drifts are small and the magnetic moment is conserved. We consider only highly relativistic protons (⁠|$Z=1$|⁠). The theory can easily be extended to nuclei with |$Z>1$|⁠. CR travel along magnetic field lines at a speed |$\mu v$|⁠, where |$|{\boldsymbol v}|=c$|⁠, and circulate around the field line at speed |$v_\perp = (1- \mu ^2)^{1/2}v$| where |$\mu =\cos \theta$| and |$\theta$| is the angle between the CR velocity and the local magnetic field. The gyration radius is |$(v_\perp /c) r_g$| where |$r_g = p/eB$|⁠. In this section, we use x as the local direction parallel to |$\boldsymbol B$|⁠. The instantaneous separation (in y and z) of the CR from its gyrocentre on its magnetic field line we call |${\boldsymbol r}$|⁠.

|${\boldsymbol B}_\parallel$| (⁠|$B_\parallel =|{\boldsymbol B}|$|⁠) is the magnetic field on the field line around which the CR circulates. We assume in this section that the magnitude of |${\boldsymbol B}_\parallel (x)$| varies along the magnetic field line but its direction does not change. Variation in direction can be added to the theory, but it adds only small terms if |$\boldsymbol B$| varies over distances much greater than |$r_g$|⁠.

In order to maintain |$\nabla . {\boldsymbol B}=0$|⁠, we must include the variation of |$\boldsymbol B$| (with components |$B_y$| and |$B_z$|⁠) in the perpendicular y- and z-directions. We follow a conventional analysis of CR trajectories along field lines and through a mirror. The instantaneous magnetic field experienced by a CR on sub-Larmor time-scales is

The CR experiences a force |${\boldsymbol F}_\parallel =e{\boldsymbol v}_\perp \times {\boldsymbol B}_\perp ^{\prime }$| in the x-direction where |${\boldsymbol B}_\perp ^{\prime }= {\boldsymbol r}.\nabla {\boldsymbol B} _\perp$| is the instantaneous vector component perpendicular to the field line

When averaged over the gyration around the magnetic field line, this reduces to

Applying |$\nabla . {\boldsymbol B}=0$| gives

In the absence of electric field, the Vlasov equation for the CR distribution function takes the form

With minor algebraic manipulation, this becomes

Equation (6) is a well-known gyrokinetic equation neglecting cross-field drifts and using the notation that |$\mu$| is the cosine of the angle between the CR velocity and the x-direction. |$\mu$| is not the magnetic moment (a notation used in other analyses of mirroring). We represent the normalized magnetic moment for a monoenergetic CR distribution by the symbol M:

It can easily be demonstrated by substitution in equation (6) that |$f=f(M)$| is a steady-state solution as expected for consistency with conservation of the magnetic moment.

Also, it can easily be shown by integrating equation (6) that the CR current density is proportional to B in steady state. This dependence on |$B(x)$| is correct because the cross-sectional area associated with a magnetic field line is proportional to |$1/B(x)$|⁠. The current density in the mirror has to increase in proportion to |$B(x)$| to carry the same total current.

3 TRANSPORT THROUGH A SINGLE MAGNETIC MIRROR

We now apply magnetic moment conservation (⁠|$f=f(M)$| is a solution) to a simple idealized case. We solve for f along a magnetic field line between two points, |$x=-L$| and |$x=L$|⁠, at which |$B=B_{0}$|⁠. We refer to these points as ‘isotropic plates’ for reasons now to be explained. The magnetic field is larger between the plates (⁠|$-L< x< L$|⁠) with a maximum |$B_{\max }$| at the midway point |$x=0$|⁠. We assume that there is no CR angular scattering in the space |$-L< x< L$|⁠. We assume that CR injected into the space |$-L< x< L$| from either of the plates are part of a locally isotropic monoenergetic distribution. The difference between the plates is that the CR density is larger at the left-hand plate than at the right-hand plate. The magnetic field is spatially symmetric about |$x=0$|⁠.

From the assumption of isotropy, f is the same for all CR emerging from the left-hand plate. This allows us to set |$f=f_L$| for |$\mu >0$| at |$x=-L$|⁠. Similarly at the right-hand plate, |$f=f_R$| for |$\mu <0$|⁠. Some of the CR injected from the left-hand plate, with |$\mu$| less than some value |$\mu _L$| (see Fig. 1), are reflected by the magnetic mirror and return to the left-hand plate with |$\mu$| reversed in sign, but with the same |$\mu ^2$| to conserve the magnetic moment M, and therefore having the same f. Hence, at the left-hand plate, |$f=f_L$| for all |$-\mu _L< \mu < 1$|⁠.

CR arriving at the left-hand plate with |$\mu < - \mu _L$| have been transmitted through the mirror from the right-hand plate, and therefore have |$f=f_R$| where |$f_R$| (smaller than |$f_L$|⁠) is the value of f for CR emitted from the right-hand plate. Consequently, at the left-hand plate,

Similarly at the right-hand plate

By symmetry, |$\mu _L=\mu _R$|⁠.

In the region between the plates, there is a boundary in |$\mu$| at |$\mu _b(x)$| such that |$\mu _b(-L)=-\mu _L$| and |$\mu _b(L)=\mu _R$|⁠. CR with |$\mu >\mu _b$| are emitted by the left-hand plate, and CR with |$\mu <\mu _b$| are emitted from the right-hand plate. The function |$\mu _b(x)$| is determined by whether or not CR are able to pass through the mirror at |$x=0$| where |$B(0)=B_{\max }$|⁠. Hence, |$\mu _b(0)=0$|⁠, and conservation of magnetic moment determines |$\mu _b$| at all other x such that

where |$\mp \rightarrow -$| for |$x<0$| and |$\mp \rightarrow +$| for |$x>0$|⁠. The overall solution for the distribution function for all |$\mu$| between the plates, |$-L< x< L$|⁠, is

From this, it follows that

We choose a magnetic field with x-dependence |$B(x)=B_{0}+{ (B_{\max } -B_{0})}/{\cosh (x/h)}$| with |$B_{\max }=4B_{0}$| and |$h=L/4$| as plotted in Fig. 1. This actually makes |$B(L)=1.03B_0$|⁠, but this small departure from |$B_0$| is inconsequential.

Equations (11) could be generalized to a CR distribution that is not monoenergetic by allowing |$f_L$| and |$f_R$| to be functions of momentum p.

We take moments of the distribution function (equations 11) to derive the density |$n_L(x)$| and current density |$j_L(x)$| profiles. The subscript L (in |$n_L$|⁠, |$j_L$|⁠) denotes CR entering from the left. To avoid confusion on the meaning of f for a monoenergetic distribution, we make f a function of momentum p but restrict the momentum range to |$p_{\min } < p < p_{\max }$| where |$\Delta p= p_{\max }-p_{\min }$|⁠, and |$\Delta p$| is vanishingly small. First, we consider only CR originating at the left-hand plate by setting |$f_R=0$|⁠. The x-dependent number and current densities of the left-originating CR are

where |$\pm \rightarrow +$| for |$x<0$|⁠, and |$\pm \rightarrow -$| for |$x> 0$|⁠, and

where the |$\pm$| sign disappears from the derivation of |$j_L(x)$|⁠. The form of the expression for the current density is the same for both |$x<0$| and |$x>0$|⁠. As anticipated in Section 2, |$j_L(x)$| is proportional to |$B(x)$|⁠.

Similar expressions can be derived for the density |$n_R(x)$| and current density |$j_R(x)$| of right-originating CR. The total current density of CR entering and leaving the system at each of the end-plates, at |$x=-L$| and |$x=L$|⁠, is the same:

The CR density at the plates is

where |$\pm \rightarrow +$| at the left-hand plate and |$\pm \rightarrow -$| at the right-hand plate.

The difference between the number densities at the left- and right-hand plates is

The relation between the current density and the difference between the plates in the number density is

This relationship can be made to look like a diffusion process if we write, with gross approximation, that the density gradient is equal to the density difference |$\Delta n_{\rm plate}$| between the plates divided by the distance |$2L$| between the plates:

and define a mirror-induced diffusion coefficient |$D_m$| as the constant in the diffusion equation

With these definitions,

Expressed in this way, |$D_m \rightarrow \infty$| in a uniform magnetic field when propagation becomes ballistic, and the diffusion model breaks down.

|$D_m$| can be compared to a representative Bohm diffusion coefficient |$D_B=cr_{g,\max }$| where |$r_{g,\max }=p/eB_{\max }$| is the CR Larmor radius in the magnetic field at the centre of the mirror:

The mirror only acts as a mirror if L is larger than the CR Larmor radius, |$L> r_{g,\max }$|⁠. Equation (22) suggests the possibility of sub-Bohm transport, |$D_m< D_B$|⁠, if the magnetic field |$B_0$| between mirrors (at the end-plates in our calculation) is much smaller than the magnetic field |$B_{\max }$| in the mirror. The ideal condition, |$B_0\ll B_{\max }$|⁠, for strong mirror-inhibited transport may be more easily satisfied by having small |$B_0$| than by having large |$B_{\max }$|⁠.

Another interesting feature of mirror-dominated transport is that the CR pressures are variable and anisotropic on the spatial scale of the mirrors. This contrasts with diffusive transport where the CR pressure is smooth on the scale of diffusion length |$D/u$|⁠. This raises the possibility that mirror-induced CR pressure gradients may drive turbulence on the mirror scale which can be much smaller than the diffusion length |$D/u$|⁠. CR undergoing shock acceleration have large energy densities and pressures that can be as much as 10 per cent or more of the hydrodynamic energy density |$\rho u^2$|⁠. Hence, there can be large pressure differences across a mirror that might feed back onto the hydrodynamical structures of mirrors, possibly enhancing the mirrors and further inhibiting CR transport. This can be seen in Fig. 2 which plots the spatial profiles of density, current density, and the parallel and perpendicular pressures when CR originate only at the left-hand mirror (⁠|$f_R=0$|⁠).

where

4 AN ILLUSTRATIVE MODEL FOR CR SHOCK ACCELERATION BETWEEN CONVERGING MIRRORS IN THE ABSENCE OF SMALL-ANGLE SCATTERING

Consider a configuration in which two ideal mirrors upstream and downstream of a parallel shock (magnetic field aligned with the shock normal) are represented by perfectly reflecting plates. In this model, the magnetic field is uniform between the reflecting plates acting as mirrors. This model is far from any realistic astrophysical situation, but it clarifies salient aspects of shock acceleration when CR are reflected between the upstream and downstream plasmas by mirrors each side of the shock. In particular, it shows that significant acceleration is only possible if the upstream plasma contains multiple semipermeable mirrors.

Initially at time |$t=0$|⁠, the upstream plate is at a distance |$L_0$| from the shock and moves towards the shock at velocity u in the shock rest frame. The downstream plate is initially coincident with the shock and moves away from the shock at a velocity |$u/4$|⁠. After time t, the distance between the two mirrors is |$L=L_0-3ut/4$|⁠.

A CR has momentum |${\boldsymbol p}$| with a component |$p_\parallel$| parallel to the shock normal and consequently also parallel to the magnetic field. |$p_\parallel$| increases by |$\Delta p_\parallel = (3 u/2c)p$| on each cycle between upstream and downstream taking a time |$\Delta t = (2L/c) (p/p_\parallel)$| for each cycle. In the limit of |$u\ll c$| and small |$\Delta p_\parallel$| and |$\Delta t$|⁠,

|$p^2=p_\perp ^2+p_\parallel ^2$|⁠. The magnitude of the perpendicular part of the momentum, |$p_\perp =p_0 \sqrt{1-\mu _0^2}$|⁠, is constant for perfect mirrors and no small-angle scattering. |$p_0$| and |$\mu _0$| are the initial values of p and |$\mu$|⁠. Accordingly, |$\mu$| and p after time t are

giving

Ignoring all other effects, acceleration terminates when the upstream mirror passes through the shock at time |$t=L_0/u$|⁠. On termination,

However, the mirrors are transparent for CR with |$\mu > (1-B_{\rm trap}/B_{\rm mirror})^{1/2}$| in which event the CR escapes the mirror when its momentum reaches

where |$B_{\rm mirror}$| and |$B_{\rm trap}$| are the magnetic fields at and between the mirrors, respectively. The momentum gain from the converging mirrors is the lower value of that imposed by (i) the upstream mirror passing through the shock, and (ii) the CR escaping through the mirror:

The energy gain is plotted in Fig. 3 which shows that an energy increase by a typical factor of 2 is reasonable. Multiple trapping episodes are needed if the CR energy is to increase by an order of magnitude or more.

A more realistic calculation will be needed to allow for changes in the magnetic moment due to small-angle scattering by turbulence between the mirrors and at shocks which are oblique rather than parallel. Scattering in angle would diffuse the CR in |$\mu$| and counteract the otherwise monotonic increase in |$\mu$| during acceleration. This would allow some CR to remain in the accelerating region for a longer period of time without escaping through the mirror.

It should be noted that the shock compresses the component of the magnetic field that is perpendicular to the shock normal. This will increase the magnetic field at downstream mirrors with the consequences that the downstream mirrors are stronger than the upstream mirrors and that CR are more likely to escape upstream than downstream.

5 A VLASOV–FOKKER–PLANCK MODEL FOR TRANSPORT IN STATIC MIRROR FIELDS WITH SMALL-ANGLE SCATTERING

An ideal next step would be to solve the full momentum-dependent transport equation for CR as they cross and recross a shock embedded in an evolving turbulent magnetic field which is structured both on a sub-Larmor scale causing small-angle scattering and on a super-Larmor scale causing mirroring. Such a calculation is beyond the scope of the present paper. Here, we limit ourselves to a calculation of CR transport in a 2D (⁠|$x,y$|⁠) stationary plasma in which the magnetic field is unchanging. The magnetic field consists of a uniform field |$B_0$| in the x-direction to which components |$B_x(x,y)$| and |$B_y(x,y)$| are added to locally vary the direction and magnitude of the field lines. (Note: the x direction is fixed in this section, unlike in Section 2 in which x followed the field lines.) The aim of the calculation is to assess whether mirroring and trapping can inhibit CR transport to levels comparable with small-angle Bohm diffusion and thereby provide a route for CR acceleration to, and conceivably beyond, the Hillas limit.

The crucial point of comparison is that the DSA acceleration time-scale is |$D/u^2$| which is the time taken in diffusion theory for the shock to overtake a CR precursor of height |$D/u$| (Lagage & Cesarsky 1983a, b). We test whether mirroring can, by inhibiting CR transport, reduce the precursor scale height and thereby increase the acceleration rate and the energy to which CR can be accelerated.

In this and subsequent sections, we present solutions to the Vlasov–Fokker–Planck (VFP) equation for the transport of monoenergetic CR in a pre-determined static magnetic field. We include small-angle scattering as an additional term in the VFP equation so that we can compare confinement by mirrors with confinement by mixtures of scattering and mirroring.

An inherent assumption in our model is that the magnetic field can be separated into super-Larmor structures modelled on a spatial grid and sub-Larmor structures modelled as sub-grid damping of CR anisotropy. Maybe the most interesting case would be the intermediate case in which structures on the Larmor scale are included, but that would require a more complicated calculation, and we find that much can be learned from the present model.

The non-zero components of the magnetic field, |$B_x$| and |$B_y$|⁠, can be represented as the curl of a vector potential |$A_z$| in the z-direction. We impose |$B_z=0$|⁠, and |$A_x=A_y=0$|⁠. |$B_y$| is set to zero at each boundary in x.

CR are initially placed in a small region close to the left-hand reflective boundary at |$x=0$|⁠. No further CR are injected during the calculation. CR reaching the right-hand boundary in x are allowed to escape freely without reflection.

Because the magnetic field is constant in time, there is no electric field and the energy of each CR is constant. The imposed small-angle scattering is constant in space and time. We perform calculations with a range of different scattering rates.

In our approximation that the magnetic field varies on scales much larger than the Larmor radius, CR trajectories are confined to field lines. Trajectories are the same for CR of any energy. Gyration about field lines removes anisotropies across the magnetic field. In contrast to cross-field transport, which is negligible and neglected, CR are free to move along field lines with a degree of anisotropy that can reach beam-like levels.

We adopt a formalism, often used in solution of the VFP equation, in which the CR distribution function is expanded in spherical harmonics

The equations for the evolution of the coefficients |$f_n^m$| in 2D |$x,y$| space are given in Bell et al. (2006), see also Reville & Bell (2013). Our case is simpler than the general case considered by Bell et al. (2006) because the electric field is zero, the distribution is monoenergetic, and the only collision term we consider is small-angle scattering. However, we complicate the evolution equations by aligning the |$\theta =0, \ \mu =1$| axis with the local magnetic field. The benefit of this move is that all |$m>0$| terms can be ignored in the limit in which CR follow the field lines. The |$m=1$| terms describe drifts across the magnetic field, but these are small when the Larmor radius is small compared with other scale lengths. We keep only the |$m=0$| terms, in which case |$f(x,y,\mu)$| is an expansion in Legendre polynomials alone with |$\mu ={\boldsymbol b}.{\boldsymbol v}/|{\boldsymbol v}|$| where |${\boldsymbol b}={\boldsymbol B}/|{\boldsymbol B}|$|⁠.

In these coordinates, and neglecting cross-field drifts and higher order cross-field anisotropies, the VFP equation takes the form

This equation simplifies when gyro-averaged since any component of |${\boldsymbol v}$| perpendicular to |$\boldsymbol B$| averages to zero, and its square averages to |$v_\perp ^2/2$|⁠. Using |${\boldsymbol b}.[ ({\boldsymbol b}.\nabla) {\boldsymbol b} ]=0$|⁠, gyro-averaging yields

From |$\nabla . {\boldsymbol B}=0$|⁠, |$\ \nabla _\perp . {\boldsymbol B}_\perp = - \nabla _\parallel B_\parallel$| where the subscripts |$_ \perp$| and |$_ \parallel$| refer to components perpendicular to and parallel to the magnetic field at the point of interest. Consequently, the VFP equation takes the form

This is the equivalent of equation (6) in Section 2. The corresponding equations for the evolution of the coefficients |$f_n$| are

6 A CANDIDATE SITE FOR MIRROR-DOMINATED TRANSPORT: CASSIOPEIA A

The above VFP equation might be solved for many different problems in and beyond astrophysics. With appropriate scaling, solution of the same VFP equation might be relevant to UHECR transport in the intergalactic medium or CR transport in the ISM, or to energetic particles in the Solar system. We could solve a representative problem in dimensionless units, but we prefer to adopt a particular application and solve the VFP equation in dimensional units. The particular application we choose is the transport of PeV protons, which may or may not exist, ahead of the outer shock of the iconic SNR Cassiopeia A (Cas A).

The approximately 340-yr-old SNR, Cas A, is the brightest extra-solar radio source in the sky, comparable with Cygnus A. The radio emission is generated by synchrotron-emitting energetic electrons, so there is good prima facie reason to suppose that Cas A might be a good accelerator of high-energy CR. However, it has been found that its gamma-ray spectrum steepens at around 10 TeV (Ahnen et al. 2017; Abeysekara et al. 2020), indicating a turnover in the CR spectrum at around 100 TeV. Consistent with results from MAGIC and VERITAS, recent results from LHAASO impose upper limits on CR densities in Cas A at energies up to 1 PeV (Cao et al. 2024). Gamma-ray emission depends on the presence of a dense target for the CR-background interaction. As Cao et al. point out, any substantial population of PeV CR produced now or earlier in the evolution of Cas A would need to be outside the dense shell at present.

A turnover in the CR spectrum at 100 TeV accords well with theories of magnetic field amplification and DSA. Zirakashvilii & Ptuskin (2008) and Bell et al. (2013) showed that CR above 100–200 TeV are unable to amplify the Larmor-scale magnetic field needed for acceleration beyond this energy. Our aim here is to explore MDSA as a process by which CR might be accelerated to, or possibly beyond, 1 PeV.

In MDSA, a combination of mirroring and angular scattering may confine CR relatively close to the shock and allow acceleration to continue beyond 100 TeV. Because MDSA depends on the presence of mirror-inducing magnetic field structures, it can be expected to be spasmodic and less effective than standard DSA, leading to a steepening of the gamma-ray spectrum which would be in accordance with observations. As discussed above, magnetic mirrors are strengthened as they pass through the shock and CR preferentially escape upstream instead of being carried downstream into the heart of the SNR. Gamma-ray emission by PeV CR will therefore be weakened by the lack of target material with which to interact. This would further contribute to a reduction in gamma-ray emission that would otherwise indicate their presence.

Cas A is a good candidate for MDSA since the plasma upstream of the shock is strongly disturbed by high velocity, high density, knots penetrating beyond the shock and, maybe more importantly, by the presence of dense disordered circumstellar material thrown off by the pre-supernova (Arias et al. 2018; Vink, Patnaude & Casro 2022; Milisavljevic et al. 2024). The strong activity seen in Cas A is probably caused by the shock passing through a dense shell of circumstellar material. Consequently, there is good reason to suppose that the magnetic field ahead of the shock is far from uniform with the likely result that mirroring may restrict CR transport during acceleration by the outer shock. The question is whether mirroring can confine CR close to the shock, but not confine them so well that CR get lodged in magnetic traps which are overtaken by the shock. Or equivalently, viewed in the shock rest frame, CR acceleration would be terminated at low CR energy if strong mirroring prematurely carries CR through the shock and away downstream.

7 SOLUTION OF THE VFP EQUATION WITH PARAMETERS BASED ON CAS A

We now solve the VFP equation for CR transport in a 2D stationary plasma with no hydrodynamic motion and a non-evolving magnetic field as described in Section 5. The calculation can be thought of as loosely applying to a region immediately ahead of the outer shock in Cas A. The left-hand boundary in our figures at |$x=0$| is reflective to represent strong mirroring at or immediately behind a shock. The shock velocity in Cas A is of the order of 60 times smaller than the velocity of the CR, so it does not concern us unduly that the plasma is stationary with respect to the boundary. An improved model, work for the future, would have the CR and the upstream magnetic field flowing into and passing through a shock. The improved model would include CR angular scattering at the shock and in the immediately downstream plasma where the magnetic field is compressed.

In our calculation, the right-hand boundary in x at a distance of |$L_x=5\times 10^{16}\, {\rm m}$| (= 1.7 pc, approximately the radius of the shell of Cas A) is a free escape boundary to represent CR being lost to the surrounding ISM. The boundaries in y are periodic and similarly separated by |$L_y=5\times 10^{16} \,{\rm m}$| for calculations with a non-uniform field. For calculations with a uniform magnetic field there is no dependency on y and we set |$L_y=0.05\times 10^{16}\,{\rm m}$| for more rapid computation.

The magnetic field is |${\boldsymbol B}= \nabla \times \left(A_z \hat{\boldsymbol z} \right)$|⁠. The magnetic field consists of a uniform component (⁠|$B_0$|⁠) in the x-direction and additional non-uniform components in the x- and y-directions. We choose |$B_0=50 \,\mu {\rm G}$| which is intermediate between a field of |$3\ - \ 30 \,\mu {\rm G}$| expected in an undisturbed ISM (Beck 2015) and the field of 100–|$500 \,\mu {\rm G}$| observed at the shock (Vink & Laming 2003; Völk, Berezhko & Ksenofontov 2005).

For the benefit of easy comparison we use the same structure for the non-uniform part of the magnetic field in every calculation.

where k is even but j takes both odd and even values. |$A^c_{jk}$| and |$A^s_{jk}$| are chosen by a random number generator and scaled by |$(j^2+k^2)^{-1/2}$|⁠. Additionally, the non-uniform part of |$A_z$| is artificially reduced by an ad hoc multiplier close to the boundaries in x. Without this, the amplitude of the turbulent field is greater at the x boundaries due to the in-phase (or antiphase) addition of all its |$\cos (j\pi x/L_x)$| harmonic components; components of the form |$\sin (j\pi x/L_x)$| are disallowed by the reflective boundary conditions. The field lines and the magnitude of the magnetic field in the reference calculation (⁠|$\alpha _B=0.1$| in equation 37) are plotted in Fig. 4.

CR are initialized as a monoenergetic isotropic population within 0.05 pc of the left-hand x boundary. No further CR are injected during the course of the calculation. The calculation is run for 50 yr, during which time a shock with velocity |$5000\,{\rm km \ s}^{-1}$| would advance 0.25 pc. In the reference calculation, the angular scattering rate is set to |$0.01 \nu _{\rm B}$| where we define |$\nu _{\rm B} = \omega _g/2\pi$| and |$\omega _g$| is the Larmor frequency of a PeV proton in the magnetic field |$B_0$|⁠. Defined in this way, |$\nu _{\rm B}$| represents Bohm-like exponential decay of the |$f_1$| current anisotropy in one Larmor gyration. The expansion in Legendre polynomials is extended to 40th order for calculations with the lowest small-angle scattering rate (⁠|$\nu =0.001 \nu _{\rm B}$|⁠). For calculations with larger scattering rates an expansion to 20th Legendre polynomial is sufficient. The number of spatial gridpoints is 500 in each of the x- and y-dimensions, corresponding to a cell size of |$10^{14}\,{\rm m}$|⁠.

Fig. 5 plots the first three coefficients, |$f_0,\ f_1,\ -f_2$|⁠, of the Legendre polynomial expansion after 50 yr. The normalization of each coefficient is the same in each case. Noticeably, |$|f_2|$| can be larger than |$f_0$|⁠, although not reaching the maximum allowed value of |$5f_0$| for a perfectly beamed distribution. We perform calculations with the same magnetic field but with four different small-angle scattering rates, |$\nu =\nu _{\rm B}$|⁠, |$0.1 \nu _{\rm B}$|⁠, and |$0.01 \nu _{\rm B}$| (the reference calculation), and |$\nu =0.001 \nu _{\rm B}$|⁠. Fig. 6 plots |$f_0$| for each of these cases. The numerical normalization of f is the same throughout in Figs 5–7. The |$\nu = 0.01 \nu _B$| data are the same in both Figs 5 and 6. We also repeat calculations with our reference scattering rate, |$\nu =0.01 \nu _B$|⁠, but with different magnetic field non-uniformities, including the case of a uniform magnetic field. These are plotted in Fig. 7 in which the magnetic fields are set by multiplying the non-uniform part of the magnetic field (⁠|$\alpha _B=0.1$|⁠) by a further factor |$\beta _B$| where |$\beta _ B =0,\ 0.5, \ {\rm and}\ 1.5$|⁠. In the case of a uniform magnetic field (⁠|$\beta _B=0$|⁠) the problem is 1D and the CR profiles are smooth. Even a relatively small magnetic non-uniformity (⁠|$\beta _B=0.5$|⁠) produces a significant non-uniformity in the CR number density which translates into non-uniform CR pressure profiles. As noted above in Section 3, since CR may be accelerated with efficiencies of 10 per cent or more, the consequently large CR pressure gradients may be sufficient to drive or amplify turbulence in the plasma upstream of an SNR outer shock. This may suggest a fruitful future inquiry into the evolution of the turbulence.

8 MIRROR-DOMINATED TRANSPORT INHIBITION AND SHOCK ACCELERATION BY SNR

The scale height of the CR precursor ahead of an SNR shock increases with CR energy because the CR Larmor radius increases with energy. CR at low energy are accelerated rapidly because their precursor scale height is much smaller than the SNR shock radius. At higher energies, acceleration is slower because the scale height is larger. Acceleration terminates when the scale height approaches the radius of the SNR.

Effective acceleration by a steadily propagating shock depends on the comparison of two lengths: (i) the scale height h of the CR precursor (ii) the distance |$L_{\rm shock}=ut$| propagated by the shock. Acceleration is effective if |$h< L_{\rm shock}$|⁠.

Our calculation does not include a shock, and the left-hand boundary does not advance into the plasma, so we cannot directly assess the competition between the advance of a shock and the advance of the CR precursor scale height. However, we can compare the advance of the CR precursor with the distance a shock would advance if it were included in the model.

In a diffusive system |$h\propto t^{1/2}$|⁠, while a shock would advance linearly with time, |$L_{\rm shock }\propto t$|⁠. Initially h increases more rapidly than |$L_{\rm shock}$|⁠, but a time |$t_{\rm overtake}$| is reached at which |$L_{\rm shock}$| overtakes h. When applied to SNR, CR are accelerated if |$t_{\rm overtake}$| is less than the age of the SNR.

In a mirror-dominated system, the time dependence of h does not have a simple analytic expression, but we can still compare the distance travelled by CR with the distance a shock would travel. Our procedure is to calculate the fraction of the initially injected CR that remain within a distance |$L_{\rm shock}$| (⁠|$\propto t$|⁠) of their point of origin close to |$x=0$|⁠.

We define the function |$F_{\rm shock}(t)$| (as plotted in Fig. 8) to represent the number of CR within the distance |$L_{\rm shock}=ut$| that a shock would propagate:

|$h_0$| is set to the distance (0.05 pc) in which CR are contained at |$t=0$|⁠. The ratio |$F_{\rm shock}(t)/F_{\rm shock}(0)$| is the fraction of CR remaining at time t within the shock-propagation distance from the left-hand boundary. While acknowledging that the real situation is much more complicated, we use this to compare CR confinement in different combinations of mirroring and small-angle scattering.

The plots of |$F_{\rm shock}(t)/F_{\rm shock}(0)$|⁠, in Fig. 8, with and without magnetic non-uniformity, show the effect of mirroring for different rates of small-angle scattering. The curves in Fig. 8 have a characteristic structure of an initial decrease in |$F_{\rm shock}(t)/F_{\rm shock}(0)$|⁠, while the CR precursor is advancing more quickly than the shock. In diffusion theory, this is the equivalent of h increasing more rapidly than |$L_{\rm shock}$|⁠. At later times, |$F_{\rm shock}(t)/F_{\rm shock}(0)$| increases, or decreases less rapidly, when the opposite condition pertains as the shock overtakes the CR precursor.

In three of the plots of |$F_{\rm shock}(t)/F_{\rm shock}(0)$| in Fig. 8 (⁠|$\nu =1,0.1,0.001 \nu _B$|⁠), we compare the standard magnetic field (⁠|$\beta _B=1$|⁠) with a uniform magnetic field (⁠|$\beta _B =0$|⁠). For the standard reference scattering rate (⁠|$\nu = 0.01 \nu _B$|⁠), we also plot results for |$\beta _B=0.5$| and |$1.5$|⁠.

Non-uniformities in the magnetic field make little difference to |$F_{\rm shock}(t)/F_{\rm shock}(0)$| when |$\nu =\nu _{\rm B}$| since small-angle diffusion dominates trapping by mirrors. Bohm scattering rapidly damps any anisotropy, and the pitch angle is randomized in the time taken to travel the scale length of magnetic field variation.

The magnetic uniformities have greatest impact when scattering is small (⁠|$\nu = 0.001 \nu _{\rm B}$|⁠) and mirroring is allowed to dominate. When |$\nu = 0.001 \nu _{\rm B}$|⁠, CR have a mean-free path |$80 \times$| larger than the size of the computational box. As expected for a uniform magnetic field and nearly unscattered propagation of an initially isotropic distribution, the CR density evolves to become spatially uniform with a density that is proportional to |$1/t$|⁠. The shock propagation distance |$L_{\rm shock}$| increases in proportion to t. This accounts for the time-independent flat horizontal line for |$F_{\rm shock}(t)/F_{\rm shock}(0)$| in Fig. 8 for |$\nu = 0.001 \nu _{\rm B}$| and |$\beta _B=0$|⁠.

Trapping appears particularly effective when |$\nu = 0.001 \nu _{\rm B}$| and |$\beta _B=1$|⁠. However, the spatial plot for these parameters in Fig. 6 shows that the trapped CR stay close to the left-hand boundary where they would rapidly be overtaken by a shock and be carried away downstream with limited acceleration as considered in Section 4.

According to Fig. 8, increasing the scattering from |$\nu =0.001 \nu _{\rm B }$| to |$0.01 \nu _{\rm B }$| reduces the confinement, but it may be more conducive to acceleration because CR are not as strongly trapped and are less likely to be carried away into the downstream plasma. A small amount of scattering allows CR to pass through mirrors. If the escape rate is matched to the advance of the shock, this may allow some CR to stay with the shock for longer and gain more energy.

Increasing the scattering rate to |$\nu =0.1 \nu _{\rm B}$| further reduces the trapping, thus reducing the confinement, but also allowing the CR to travel further towards the right-hand boundary. A more sophisticated calculation, including a travelling shock, is needed to discover the optimal level of scattering for CR acceleration.

Overall, our calculations suggest that a suitable combination of magnetic non-uniformity and small-angle scattering can produce Bohm-level confinement without the need for Larmor-scale structure in the magnetic field. Of all the curves plotted in Fig. 8, the case of |$\nu =0.01 \nu _{\rm B }, \ \beta _B=1.5$| in the top-right plot in Fig. 8 appears most suitable for CR acceleration. The form of the curve resembles that in the top-left plot for Bohm diffusion (⁠|$\nu =\nu _B$|⁠) in a uniform field (⁠|$\beta _B=0$|⁠). |$F_{\rm shock}(t)/F_{\rm shock}(0)$| is smaller in other plots in Fig. 8, but not always by a large margin.

We have chosen configurations in which the magnetic field varies on scales characteristically |$10\times$| larger than the CR Larmor radius (⁠|$6.7 \times 10^{14}\,{\rm m}$|⁠). This relatively large non-uniformity scale length, compared with the Larmor radius, has been needed to justify the assumption that CR are tied to magnetic field lines. A reduction in the magnetic scale length would be expected to increase confinement, further inhibit transport, and accelerate CR more rapidly and to higher energies. The important regime of magnetic non-uniformity on a scale slightly greater than the Larmor radius will need an improved computational model that includes cross-field drifts and |$m>0$| anisotropies. Lemoine (2022, 2023) has previously demonstrated the importance of this theoretically and computationally challenging regime for CR transport and second-order Fermi acceleration.

9 A GENERALIZED HILLAS LIMITING ENERGY

The Hillas energy, |$E=uBR$|⁠, for CR acceleration by DSA can be derived by assuming that the minimum CR diffusion coefficient is |$D_{\rm Bohm} =r_gc$| and that the acceleration time is |$h/u$| where the CR precursor scale height is |$h=D_{\rm Bohm}/u$|⁠. A different derivation of the maximum CR energy is needed if mirroring is important.

Suppose that CR are confined in traps of characteristic length |$L_{\rm trap}$| which is the distance between the mirrors that control entry and exit from a trap. Suppose that a CR passes from one trap to an adjacent trap with a characteristic escape probability |$q_{\rm trap}$|⁠. Omitting numerical factors of order 1, the CR performs a random walk of step length |$L_{\rm trap}$| and time-step |$L_{\rm trap}/c q_{\rm trap}$|⁠. The resulting order-of-magnitude diffusion coefficient (⁠|${\rm steplength}^2/{\rm timestep}$|⁠) is

The corresponding precursor scale height ahead of the shock is |$h_{\rm trap}= (c/u)L q_{\rm trap}$|⁠, and the CR acceleration time is |$\tau _{\rm trap}=q_{\rm trap} cL_{\rm trap}/u^2$|⁠. If R is the spatial extent of the system, the Hillas condition is determined by |$\tau _{\rm trap}< R/u$|⁠. This generalized Hillas condition can be written in the form

where B is a characteristic field in mirrors and |$r_g$| is the Larmor radius in the same field. Note that B cancels from equation (40), since |$r_g$| is inversely proportional to B. Comparison of equation (39) with equation (21) in Section 3, suggests that a characteristic escape probability from the trap is

Fig. 9 is a plot of the ratio of the maximum CR energy to the Hillas energy |$E_{\rm Hillas}=uBR$| as defined by combining equations (40) and (41) when |$L_{\rm trap}/r_g=10$|⁠. For the parameters adopted for the VFP calculation in Section 7 (⁠|$u=5000\, {\rm km\,s}^{-1}$|⁠, |$B=50 \,\mu {\rm G}$|⁠, and |$R=1.7\,{\rm pc}$|⁠), the Hillas energy |$E_{\rm Hillas}$| is 1.3 PeV.

The analysis in Section 3, and hence this estimate of |$q_{\rm trap}$|⁠, starts from the assumption that CR emerging from the left- and right-hand plates are isotropic. The left- and right-hand plates in Section 3 are the equivalent of the mid-points between mirrors. This assumption of isotropy between mirrors may be incorrect since CR with large |$| \mu |$| at the mid-points escape through the mirrors, while CR with |$\mu$| close to zero are confined in the trap. The transfer of CR between trapped and untrapped trajectories requires a more sophisticated calculation.

Taken at face value, equation (40) suggests that the highest energy is reached if |$q_{\rm trap}$| is indefinitely small. However, |$q_{\rm trap}$| must not be so small that CR are locked into traps and carried away downstream through the shock. The condition for CR escaping the trap before being overtaken by the shock is that |$q_{\rm trap}> {u}/{c}$| since the CR passes between the mirror |$\sim c/u$| times in the time |$L_{\rm trap}/u$| taken for the shock to overtake the trap. Equation (41) suggests that for realistic SNR-relevant magnetic field configurations and shock velocities, |$q_{\rm trap}$| safely exceeds this lower limit. Once again, a more sophisticated calculation is needed to resolve these issues.

10 MIRROR-DOMINATED TRANSPORT IN OTHER CONTEXTS

This paper has been framed around the question of the maximum energy to which CR can be accelerated by shocks. Our arguments may have a wider application to CR transport in the Galaxy and to UHECR transport between galaxies.

Reichherzer et al. (2022; their fig. 1) identify four different regimes of charged particle transport. The mirror-dominated transport considered here corresponds most closely to their ‘mirror regime’. Previous treatments of mirroring use a variety of approaches, variously starting from turbulence with a defined k-spectrum, or ‘micro-mirrors’, or calculating CR trajectories with a particle-in-cell code, or using a mixture of approaches (Cesarsky & Kulsrud 1973; Lazarian & Xu 2021; Barreto-Mota et al. 2024; Zhang & Xu 2024, Reichherzer et al. 2025). How these compare with our use of VFP calculations and approximate configuration-space models may be a matter for future investigation.

Following from equations (39) and (41), a diffusion coefficient for CR transport in the ISM might take the form

where |$B_{\max }$| and |$B_{\min }$| are the characteristic maximum and minimum magnetic field in any length of magnetic field line equal to the CR mean-free path for angular scattering. The justification for this dependence is that CR lose their memory of the magnetic field structure over distances larger than the CR mean-free path. Transport is controlled by the ratio |$B_{\rm min}/B_{\rm max}$| in this distance.

Depending upon the k-spectrum of magnetic fluctuations in the ISM, there may be some scale length |$L_{\rm trap}$| which dominates transport at each CR energy. In this model, |$B_{\rm max}/B_{\rm min}$| and |$L_{\rm trap}$|⁠, both of which will in general be energy-dependent, determine the CR diffusion coefficient and propagation time, which in turn affects the composition and energy spectrum of CR arriving at the Earth.

Magnetic fields as large as 30–60 nG have been discovered in filaments connecting galaxies (Vernstrom et al. 2021). A proton with an energy of 50 EeV has a Larmor radius of |$\sim 1$| Mpc in such a field, and a proton at the ankle in the UHECR spectrum at 5 EeV has a Larmor radius of |$\sim 100$| kpc. Such a spatially structured intergalactic field is sufficient to affect UHECR transport, playing a role in forming the energy spectrum and generating anisotropy (Condorelli, Biteau & Adam 2023; Abdul Halim et al. 2024; Marafico et al. 2024).

UHECR anisotropy is structured on both large angular scales, such as the observed dipole, and on smaller angular scales (hotspots) as in the case of arrivals from the direction of Centaurus A or from echoes by magnetized galactic haloes (Bell & Matthews 2022; Taylor, Matthews & Bell 2023). Transport dominated by localized mirrors may explain the co-existence of isotropy and anisotropy on many scales. It can produce large-angle scattering for UHECR encountering a mirror while at the same time allowing other UHECR to follow essentially straight line trajectories if they do not encounter a mirror. Mirror-dominated transport allows high-order multipole anisotropies to co-exist with a strong isotropic component and with low-order dipole and quadrupole anisotropies. A conventional diffusion model with a uniform small-angle scattering rate is much more constraining in the anisotropy-mode spectrum that it allows. Future observations of UHECR anisotropy may be able to distinguish between diffusion and mirror models of transport.

11 CONCLUSIONS

The theory of DSA is usually developed on the assumption that CR trajectories are determined by small-angle scattering by fluctuations in the magnetic field on scales smaller than the CR Larmor radius. With this assumption it can be shown that the maximum energy to which CR can be accelerated is the Hillas energy, |$E=uBR$|⁠. Except in the case of perpendicular shocks, the Hillas energy is only achieved if the CR mean-free path is of the order of the Larmor radius resulting in Bohm diffusion with a coefficient |$D_{\rm Bohm}\sim r_g c$|⁠. A non-resonant instability, driven by CR currents, amplifies the magnetic field with a wavelength that grows during amplification, and saturates when the wavelength reaches the Larmor radius. The combination of non-resonant field amplification and DSA satisfactorily accounts for CR acceleration by SNR to energies of the order of a few hundred TeV, but it cannot account for acceleration to the knee at a few PeV.

Special sites such as very young SNR (none are known in the Galaxy at present), star-forming regions where the magnetic field is large and there are repeated supernovae and strong stellar winds, or the Galactic centre where similar conditions apply, may account for PeV acceleration. However, we instead question the assumption that small-angle scattering dominates CR transport. We show that mirroring by magnetic fluctuations on scales larger than the Larmor radius can be equally as good as Bohm diffusion at confining CR close to a shock. We show that the Hillas energy may be achievable without the generation of magnetic fluctuations on the Larmor scale. Mirror-inducing magnetic fluctuations may be expected in the environment of SNR such as Cas A where the medium upstream of the shock is disturbed by clumpy pre-supernova ejections and by fast knots.

MDSA is not as reliably efficient as standard DSA because the required mirrors may not always be present. Spasmodic acceleration may be expected, leading to a steeping of the spectrum beyond energies at which standard DSA operates. The spectrum of gamma-rays produced by high-energy CR will further steepen because CR preferentially escape upstream of the MDSA process, thereby not entering the dense interior of the SNR where gamma-rays are efficiently produced. Hence, acceleration to PeV, and possibly higher energies, is not necessarily precluded by the observed steepening of the gamma-ray and CR spectrum of young SNR.

We also briefly consider the role of mirror-dominated transport in the ISM and the intergalactic medium (IGM), and suggest the possibility that the strong CR pressure gradients generated by mirrors may act to amplify the mirrors.

ACKNOWLEDGEMENTS

JHM acknowledges a Royal Society University Research Fellowship. AT acknowledges support from DESY (Zeuthen, Germany), a member of the Helmholtz Association HGF. Computing resources were provided by STFC Scientific Computing Department’s SCARF clusters.

DATA AVAILABILITY

The data underlying this article are available from the authors on reasonable request.

REFERENCES