Acceleration of relativistic protons in a CME-perturbed solar wind

Ahmed Houeibib; Filippo Pantellini; Lea Griton

REVIEW 3 major objections 6 minor 22 references

Acceleration of relativistic protons in a CME-perturbed solar wind

T0 review · 3 major / 6 minor · reviewed 2026-08-02 · deepseek-v4-flash

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Pith's one-line read CME-driven shocks can re-energize 5 GeV protons via gradient drift in the compressed downstream field, with scattering enabling repeated crossings and a λ∥^{-3/2} scaling of the accelerated fraction.

desk verdict Plausible mechanism, but the λ∥^-3/2 scaling headline is not derived correctly and the numerical support is a two-point ratio. read the letter →

arxiv 2602.23723 v2 pith:OK67V6PO submitted 2026-02-27 astro-ph.SR astro-ph.HEphysics.space-ph

Ahmed Houeibib , Filippo Pantellini , Lea Griton This is my paper

classification astro-ph.SRastro-ph.HEphysics.space-ph

keywords solarenergeticparticlescoronalmassejectionsgradientdriftaccelerationpitch-anglescatteringrelativisticprotonsMHDsimulationinterplanetarymagneticfieldparticle

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper tries to establish that a coronal mass ejection can do more than block high-energy protons; it can measurably re-energize them. Tracking 5 GeV protons through a 3D MHD simulation of a CME-driven shock, the authors argue that the energy gain comes from gradient drift while the proton streams along the compressed field downstream of the quasi-perpendicular part of the shock, not from the shock crossing itself. The gain is proportional to the time spent in that region, which peaks when the shock is near 0.3 AU. When pitch-angle scattering lets protons cross the accelerating region many times, a small tail—about 0.1% of injected particles at a mean free path of 0.1 AU—gains a factor of six in energy in four days, and the number accelerated to a given energy grows as the inverse 3/2 power of the scattering mean free path. If true, this is a concrete route by which interplanetary CMEs contribute to the multi-GeV proton population observed near Earth.

What carries the argument

The central object is the gradient-drift term in the guiding-center energy equation, ΔE ≈ (μB/γ) Δt_c v_E·∇B, where μB is the magnetic moment, γ the Lorentz factor, Δt_c the time spent in the region with v_E·∇B > 0, and v_E = E×B/B^2 is the flow-induced drift. This term converts the CME's inductive electric field into proton energy as magnetic field lines are advected into stronger field regions. The second load-bearing device is pitch-angle scattering, modeled as isotropic hard-sphere collisions (each collision randomizes the particle's direction in the solar-wind frame) with a parallel mean free path λ∥; it allows particles to pass through the acceleration region repeatedly, between mirror

What would settle it

Compute the pitch-angle diffusion coefficient at 5 GeV from the measured magnetic turbulence in a well-observed CME sheath; if the resulting mean free path exceeds 0.5 AU, the claimed 0.1% sixfold-energy tail cannot appear within 4 days, and the uniform-scattering premise is falsified.

Watch

Extended reading notes

Core claim

Using a 3D MHD simulation of a solar wind containing a spheromak-driven CME, the authors integrate the trajectories of 5 GeV protons with guiding-center equations (tracking the gyration-averaged guiding center) and optional pitch-angle scattering. Their central claim is that the dominant acceleration is drift acceleration in the compressed plasma downstream of the shock: as the field line guiding a proton is advected by the flow into regions of increasing field strength, the gradient drift produces an energy gain ΔE ≈ (μB/γ) Δt_c v_E·∇B, so the gain is proportional to the time the proton spends where v_E·∇B > 0. The gain per crossing peaks when the shock is near 0.3 AU, reaching up to about

Load-bearing premise

The whole quantitative story rests on the premise that 5 GeV protons are pitch-angle scattered with a uniform, energy-independent mean free path of 0.1–0.5 AU, the same range inferred from lower-energy particles; if the real GeV mean free path is much longer or spatially patchy, the predicted factor-six tail and its λ∥^{-3/2} scaling shrink accordingly.

Editorial extensions

If this is right

At 1 AU, a CME encounter should leave a measurable high-energy tail in the GeV proton spectrum even if the shock itself does not accelerate particles, because pre-existing protons are re-energized downstream.
The most efficient acceleration occurs not at the shock front but in the compressed downstream region, and it peaks at a specific shock distance (~0.3 AU in this setup), so the timing of CME observations matters.
Shorter scattering mean free paths harden the energy spectrum and increase the number of accelerated particles, following a quantifiable λ∥^{-3/2} scaling.
The yield depends strongly on field-line geometry: among the three equatorial lines studied, the one with the longest downstream compressed section accelerates particles most, with westward lines favored.
On hour-to-day time scales, the mechanism produces multi-GeV gains in a small fraction of the population, with the largest gains requiring repeated crossings enabled by the mirror force and pitch-angle scattering.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Extension: if real turbulence in the CME sheath differs from the uniform hard-sphere assumption, the λ∥^{-3/2} scaling implies that the accelerated fraction could vary by more than an order of magnitude from one CME to the next, while the drift mechanism itself would survive.
Extension: the same gradient-drift route should apply to lower-energy protons and heavier ions whenever the guiding-center approximation holds, suggesting a scattering-regulated acceleration channel that connects solar energetic particles and galactic cosmic rays near CMEs.
Extension: because the simulation uses a monopole magnetic field without a current sheet, real field-line disconnection and reconnection would change which field lines reach a detector; multi-spacecraft observations of a single CME could test how strongly this selection effect modifies the predicted factor-six tail.
Extension: a direct comparison with diffusive shock acceleration at the same quasi-perpendicular shock would clarify whether the downstream drift term is additive to or competitive with the shock-front mechanism in global SEP models.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

You'll want to know two things about this manuscript. First, the core mechanism — 5 GeV protons gaining energy via gradient drift in the compressed region downstream of a quasi-perpendicular CME shock, with scattering enabling repeated crossings — is clearly argued and worth taking seriously. Second, the abstract's headline scaling, efficiency ∝ λ∥^-3/2, does not follow from the paper's own diffusion argument, and the numerical 'confirmation' is a single ratio between two runs. That quantitative claim needs to be revised or dropped.

The paper does several things well. The energy-gain estimate in Eqs. (6)-(8) is transparent and parameter-free; the authors split curvature vs. gradient drift, show the gradient term dominates, and identify the vE·∇B > 0 downstream region as the acceleration site. They state their assumptions (E∥≪E⊥, static fields during a crossing, vE≪c) and even correct a typo in their earlier GC equation. The multi-crossing statistics, with ~0.1% of particles gaining a factor of six in four days, are a concrete prediction that could be tested against observations. The work extends their 2025 electron study to time-dependent CME fields and GeV protons; the quantitative results are new.

The soft spots are real, and one is load-bearing. The λ∥^-3/2 scaling does not come out of the 1D diffusion model as written. The density decays as (πλ∥ct)^-1/2; with Nc = ct/λ∥ collisions and ΔE = Nc δE, reaching a fixed energy requires t ∝ λ∥, which gives n ∝ λ∥^-1, not λ∥^-3/2. The printed intermediate (ct/λ∥^3)^1/2 is dimensionally wrong and looks like a mix-up between particle counts and collision counts. The numerical check in Fig. 10 is two runs, one ratio, with the quoted factor-6 tail coming from roughly ten particles; that cannot distinguish λ∥^-3/2 from λ∥^-1 or Poisson noise. So the abstract and Conclusion item 5 overstate the scaling. The scattering parameters themselves are also extrapolated from lower-energy fits (Palmer 1982; Bieber et al. 1994) and applied uniformly at 5 GeV, which is a reasonable first cut but not justified in detail. The idealized monopole-field wind and the hand-picked best of three field lines mean the absolute efficiencies are likely upper bounds.

None of that breaks the mechanism claim. The acceleration site and the multi-crossing picture are supported by the simulation and the parameter-free estimate. This is a decent mechanism study that deserves a careful referee — not a desk reject — but the authors should fix or soften the scaling law, and ideally add error bars or more runs before the numbers are cited.

Referee Report

3 major / 6 minor

Summary. The paper presents a coupled 3D MHD simulation of a spheromak-driven CME in a Parker-type solar wind and uses guiding-center test-particle integration to study the transport and acceleration of 5 GeV protons. The central claim is that protons gain energy when they stream along field lines in the compressed region downstream of the quasi-perpendicular part of the CME-driven shock, with the energy gain dominated by the gradient-drift term in Eq. (6)-(8). The authors identify the condition vE·∇B > 0 as defining the acceleration site, show a peak per-passage gain when the shock reaches ~0.3 AU, and argue that pitch-angle scattering combined with the mirror force allows multiple crossings and substantially larger total gains. They further claim that the efficiency scales as λ∥^{-3/2} and that the energy spectra harden for smaller parallel mean free path.

Significance. If correct, the proposed mechanism is a genuinely useful addition to the understanding of how CMEs re-energize pre-existing GeV protons in the heliosphere. The derivation in Section 3 is transparent and parameter-free: Eqs. (6)-(8) clearly separate curvature and gradient drift, identify the dominant gradient-drift contribution, and tie the energy gain to measurable field properties without fitting. The forward-model approach, in which the MHD fields are not adjusted to produce acceleration, and the explicit statement of assumptions (E∥ ≪ E⊥, static fields during a crossing, vE ≪ c) are strengths. The paper also gives a physical explanation for why scattering increases the yield. However, one of the headline quantitative claims, the λ∥^{-3/2} scaling, is based on an algebraic error and is not supported by the presented numerical evidence, so the paper needs revision before the quantitative conclusions can be accepted.

major comments (3)

[Section 4, last paragraph; Conclusion 5; Abstract] The derivation of the λ∥^{-3/2} scaling is algebraically incorrect. The paper starts from the 1D diffusion density n(t) ∝ (πλ∥ct)^{-1/2} and Nc = ct/λ∥, and then states that the number of particles gaining a given energy ΔE scales as (ct/λ∥^3)^{1/2} ∝ λ∥^{-3/2}. But for a fixed target energy ΔE0, the required number of collisions N0 = ΔE0/δE is fixed, so the required time is t0 = N0λ∥/c. Substitution gives n(t0) ∝ (λ∥ct0)^{-1/2} = (N0λ∥^2)^{-1/2} ∝ λ∥^{-1}, not λ∥^{-3/2}. The printed expression (ct/λ∥^3)^{1/2} is, up to constants, n(t)×Nc — a weighted count of collision events, not the number of particles reaching a given energy; it also has dimensions of inverse length. The numerical check in Fig. 10 is a comparison of only two values of λ∥, and the quoted factor-6 tail at 0.1% of 10^4 particles is about 10 counts, so it cannot distinguish λ∥^{-1} from λ∥^{-3/2}. Since this scaling appe
[Section 4, first paragraph] The quantitative results depend sensitively on the scattering model, but the chosen mean free paths are not well justified for the particles studied. The values λ∥ = 0.1-0.5 AU are cited from Palmer (1982) and Bieber et al. (1994), which are determinations at much lower energies, and no rigidity correction is applied for 5 GeV protons. The mean free path is also taken to be spatially uniform and energy-independent, although the CME sheath is expected to have different turbulence properties from the ambient wind. The multiple-crossing statistics, the factor-six tail, and the λ∥ scaling all follow from this assumption. Please provide a sensitivity analysis with energy-dependent or spatially varying λ∥, or at least a quantitative discussion of how such effects would alter the spectra and yields.
[Section 5, items 3-4; Fig. 9] The headline quantitative claims (e.g., 0.1% of particles reaching a factor-six energy gain) are based on a single field line, line 0, which the authors acknowledge is the most favorably oriented among the three selected lines. The simulation uses a monopole wind with no current sheet and a hand-selected equatorial geometry. The generalizability of the numbers is therefore limited. The conclusions should either present these as line-0-specific results or include robustness tests over more field lines and topologies before making broader statements about the efficiency of the mechanism.

minor comments (6)

[Abstract] 'increment the protons energy' is awkward; consider 'increase the proton energy'. Also 'scales asλ−3/2 ∥' should be typeset consistently as λ∥^{-3/2}.
[Introduction] 'Lara et al. (2024) end references therein' contains a typo; should be 'and references therein'.
[Section 2.1] 'the wind is starts supersonic' should be 'the wind starts supersonic'.
[Section 3, Fig. 8 caption] The sentence 'which is the for the particle in Fig. 6' is incomplete; please revise.
[Section 4, Figs. 9-10] The cumulative spectra are shown without statistical uncertainties. Given the small counts in the high-energy tail, Poisson error bars or confidence bands should be included.
[Conclusion 4] 'particle particle' is duplicated; remove one occurrence.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: central mechanism and scaling law are derived and checked, not fitted.

full rationale

The paper's central claim is a forward-model result. No parameter is fitted to produce the acceleration: the parallel mean free path values are taken from external literature (Palmer 1982; Bieber et al. 1994), and the guiding-center equations are stated in the paper. Equation (7) is an analytic approximation derived from the energy equation and checked against the full GC trajectory integration (Fig. 6) rather than tuned to reproduce it. The claimed lambda_parallel^-3/2 scaling is presented as a semi-quantitative estimate from a standard 1D diffusion model and then compared with the two simulated cases; even if the algebraic derivation is questionable, this is a correctness/verification issue, not circularity, because the simulation output is not constructed to equal the analytic formula. Self-citations to Houeibib et al. (2025) are used for numerical setup and scattering implementation details, but the relevant equations and assumptions are restated here, and the acceleration mechanism does not rest on an unverified result from that paper. No uniqueness argument, imported ansatz, or definitional equivalence forces the stated conclusions.

Assumptions & free parameters 4 free parameters · 7 assumptions · 2 invented entities

The central result rests on three pillars: (a) the scattering model (uniform hard-sphere λ∥=0.1/0.5 AU, values extrapolated from lower-energy fits), which controls the multiple-crossing statistics, the factor-6 tail, and the λ∥^{-3/2} law; (b) the MHD background (monopole Parker field with no current sheet; spheromak driver with B0=144 nT, Rs=10.5 R⊙, Vs=1028 km/s), which produces the quasi-perpendicular shock and the vE·∇B>0 downstream compression; (c) the GC approximation with stated validity conditions (gyroradius small vs. field scales, vE≪c, E∥≪E⊥, static fields during a crossing). None is fitted to the target result — they are forward-model choices — but they set the quantitative claims, and the paper provides no observational validation of the wind or CME parameters.

free parameters (4)

parallel mean free path λ∥ = 0.1 AU and 0.5 AU
Central control parameter of the scattering model; the efficiency scaling λ∥^{-3/2} and the factor-6 multi-crossing gains are statements about this parameter. Values are taken from fits (Palmer 1982; Bieber et al. 1994) at much lower energies and applied to 5 GeV protons without rigidity correction.
Spheromak CME driver (B0, Rs, Vs) = B0=144 nT, Rs=10.5 R⊙, Vs=1028 km/s, injected at 0.139 AU
Chosen by hand following Verbeke et al. 2019 and others; the peak-gain phase (shock at ~0.3 AU) and the energy-gain magnitudes depend on these.
Ambient wind base state = T=2 MK, ρ=1e-19 kg/m^3, monopole B=2e-4 T at surface, Ω=2π/30d
Chosen; yields a wind that is 'somewhat underdense' (Table 1) and a monopole field with no current sheet; the shock compression ratios and field-line geometry derive from these.
Particle injection conditions = E=5 GeV, α=180° (170° in §3), injected at r=3 AU on 3 selected field lines
Initial conditions chosen by the authors; the measured spectra and the 0.1%-factor-6 claim are conditional on them. Boundary re-injection at the same state keeps particle number constant but artificially resets accelerated particles that exit the domain.

assumptions (7)

domain assumption Guiding-center approximation valid for 5 GeV protons in the simulated fields
Invoked in §1 and §2.2; requires gyroradius small vs. field gradient/CME scales. Reasonable for 5 GeV protons in nT-scale fields but not proven for the thin quasi-perpendicular shock region.
domain assumption |vE| = |E×b/B| ≪ c and E∥ ≪ E⊥
Stated in §2.2 and §3; needed for the GC equations (3)-(4) and for dropping the parallel electric field term in Eq. (6).
domain assumption Static-field approximation during a single crossing (∂/∂t = 0)
Stated in §3: a 5 GeV proton crosses the CME in much less than its expansion time; used to derive the energy-gain equation (6).
domain assumption Hard-sphere, isotropic pitch-angle scattering, uniform λ∥, scattering centers at rest in the solar wind frame
§4 and §2.3 of Houeibib et al. (2025); the multiple-crossing statistics, the residence time, and the λ∥^{-3/2} scaling all follow from this model.
domain assumption Ideal MHD, polytropic index 5/3, fully ionized proton-electron plasma with Te = Tp
§2.1; standard but an idealization (no heat conduction, no turbulence, no pickup ions in the wind).
domain assumption Monopolar Parker field with no heliospheric current sheet is a representative CME propagation background
§2.1; the authors state this simplification explicitly; the favorable quasi-perpendicular shock geometry and the vE·∇B > 0 downstream region depend on it.
domain assumption Force-free spheromak is an adequate CME driver
§2.1, following Verbeke et al. (2019), Kataoka et al. (2009), Shiota & Kataoka (2016), Singh et al. (2020), Koehn et al. (2022); an inherited modeling convention, not observationally validated in this paper.

invented entities (2)

Spheromak CME structure (Eqs. 1-2)
purpose: Drive a CME-like shock and magnetic compression through the simulated solar wind
A standard simulation construct in the cited literature, but its parameters (B0=144 nT, Rs=10.5 R⊙, Vs=1028 km/s) are chosen, not constrained by observations; the quantitative acceleration results are conditional on it.
Hard-sphere scattering centers at rest in the solar wind frame
purpose: Represent sub-grid pitch-angle scattering; the device that makes multiple CME crossings (and hence large gains) possible
A toy model for turbulence; the mean free path values come from lower-energy fits, and the model's energy- and space-independence is an assumption, so the multi-crossing statistics have no external handle.

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Cite this review

Pith. "Pith review of Acceleration of relativistic protons in a CME-perturbed solar wind." pith.science (2026). https://pith.science/paper/OK67V6PO

@misc{pith2026260223723,
  author       = {Pith},
  title        = {Pith review of: Acceleration of relativistic protons in a CME-perturbed solar wind},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OK67V6PO}},
  note         = {Machine review of arXiv:2602.23723}
}

abstract

We investigate the impact of a Coronal Mass Ejection (CME) on the transport and acceleration of relativistic protons in the solar wind using a coupled 3D Magnetohydrodynamics (MHD) simulation and a test-particle approach. The CME is driven by a spheromak injected into a Parker solar wind at a heliocentric distance of 0.139 AU. The trajectories of 5 GeV protons, injected toward the CME from 3 AU, are integrated in the guiding-center approximation and scattered in velocity space with a mean free path $\lambda_{\|}$. Our results show that the CME can increase the protons' energy by several GeV. The acceleration occurs during the time particles stream along the portion of a magnetic field line downstream of the quasi-perpendicular portion of the CME-driven shock. In our configuration, the maximum energy gain, which is of the order of a few percent per passage through the acceleration region, occurs when the shock approaches 0.3 AU. Large energy gains require multiple passes through the acceleration region, which is made possible by the combined action of the mirror force and pitch angle scattering. The efficiency of the acceleration on time scales of the order of hours scales as $\lambda_{\|}^{-3/2}$. Energy spectra harden for decreasing parallel mean free path $\lambda_{\|}$.

Figures

Figures reproduced from arXiv: 2602.23723 by the authors.

**Figure 1.** Top and side views of the magnetic field lines in a spheromak. Note that despite appearances the magnetic field strength of the spheromak increases monotonically towards its center where it peaks at 2B0/3 (see equations (1) and (2)). At the boundary of the spheromak, at r ′ = Rs , there is only a non-zero poloidal component B ′ p = 0.2172 B0 sin(θ ′ ). perature Te = Tp = T and same number density ne = np = n/2 imply… view at source ↗

**Figure 2.** Injection of the spheromak into the numerical domain. Top panel : the spheromak is injected in the equatorial plane at a constant radial speed Vs in a pre-existing stationary wind. – Shock arrival: The shocks reaches the observation point at t ≈ 28 hours after the spheromak’s injection when the density rises by a factor 3 and the magnetic field intensity from ∼ 5 nT to ∼ 11 nT. – Sheath region : Between t ≈ 28 h an… view at source ↗

**Figure 3.** Equatorial cuts of the simulation domain as a function of time, where the time t=0 corresponds to the injection time of the spheromak. The color maps are for the radial fluid velocity (top panel) and temperature (bottom panel) [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

Figures from the paper (5 more)

**Figure 4.** Figure 4: CME induced variations of the plasma parameters at 1 AU. The vertical lines approximately delimit the time interval during which the local plasma can be considered to be perturbed by the CME. magnetic field along which the particle is moving. The gain of energy is prop…

**Figure 5.** Figure 5: Time evolution of three selected magnetic field lines, denoted 0,1 and 2 from est to west [PITH_FULL_IMAGE:figures/full_fig_p005_5.png]

**Figure 6.** Figure 6: Contributions to the energy variations of a 5GeV proton moving throughout the CME perturbed plasma along line 0. In this figure, t = 0, corresponds to the time of injection of the particle into the system. The first increment at t ≈ 0.6h occurs as the particle moves to…

**Figure 7.** Figure 7: Sketch representing the configuration of the perturbed field line. The gray section of length ℓc = c∆tc is the portion of trajectory where vE · ∇B > 0 and particles can gain energy according to equation (7). most two times; the first time as they stream towards the Sun…

**Figure 9.** Figure 9: Cumulative energy spectra measured at 1 AU during a time interval of ∼ 4 days for 5GeV protons injected on the three different field lines subject to scattering with a mean free path λ∥ = 0.5. CME, with line 0 being the one with the longest portion in the region of st…

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Pith tools

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