arxiv: 2605.01437 · v1 · submitted 2026-05-02 · ⚛️ physics.space-ph

Recognition: unknown

Statistical analysis of solar energetic particle rise times using Earth and Mars observations and constraints on particle transport parameters

Yihang Cao , Jingnan Guo , Yuming Wang , Zhuxuan Zou , Yongjie Zhang , Cunhui Li

Authors on Pith no claims yet

Pith reviewed 2026-05-10 15:00 UTC · model grok-4.3

classification ⚛️ physics.space-ph

keywords solar energetic particlesSEP rise timeparticle diffusionparallel mean free pathinterplanetary turbulencemulti-point observationsMarsEarth

0 comments

The pith

SEP rise times follow a power law with energy that flattens at greater distances from the Sun

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

The study measures how long it takes solar energetic particle fluxes to rise from onset to peak at different energies, using simultaneous data from Earth and Mars. It finds a clear power-law increase of rise time with energy at both locations, but with a shallower slope near Mars. Comparing these trends to the expected behavior in a pure parallel diffusion model then constrains the energy dependence of the distance particles travel between scatters. The results indicate that solar wind turbulence scattering loses its sensitivity to particle rigidity at larger heliocentric distances.

Core claim

By determining onset times through linear fits and peak times through sliding-median and Savitzky-Golay smoothing across 75 events at 1 AU and 58 events near Mars, the analysis shows that SEP rise time scales with energy as a power law. The exponent of this scaling is smaller at Mars than at Earth. Matching the observed exponents to the analytic solution of the parallel diffusion equation yields a parallel mean free path whose rigidity dependence weakens and approaches independence with increasing distance from the Sun.

What carries the argument

The rigidity dependence of the parallel mean free path in the pure parallel diffusion model, which is constrained by matching the observed energy scaling of rise times to the model's predicted scaling.

If this is right

The energy dependence of SEP rise times becomes weaker with increasing heliocentric distance.
Turbulence scattering effects approach a rigidity-independent regime near Mars orbit.
Statistical rise-time relations provide a route to constrain mean-free-path parameters that does not require absolute flux intensities.
Transport models must incorporate radial evolution of turbulence properties between Earth and Mars distances.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Forecast models for SEP arrival at Mars could adopt simpler, nearly energy-independent scattering parameters for paths longer than 1 AU.
Repeating the rise-time analysis with data from additional spacecraft at intermediate or larger distances would map the radial change in turbulence dissipation.
The same statistical method could be tested on other particle populations to check whether rigidity independence emerges in different transport regimes.

Load-bearing premise

The difference between linearly fitted onset times and smoothed peak times isolates the interplanetary transport delay without substantial contamination from variable particle release times at the Sun or from background fluctuations.

What would settle it

New SEP events observed at both Earth and Mars in which the power-law index of rise time versus energy is the same or steeper at Mars than at Earth would falsify the reported flattening and the inference of approaching rigidity independence.

Figures

Figures reproduced from arXiv: 2605.01437 by Cunhui Li, Jingnan Guo, Yihang Cao, Yongjie Zhang, Yuming Wang, Zhuxuan Zou.

**Figure 1.** Figure 1: Illustration of the methodology for determining the onset and peak times of an SEP event observed by SOHO/ERNE on May 28, 2021. The main panel shows time-intensity profiles across eight energy ranges (13–100 MeV), where the raw data are plotted as solid lines and the smoothed data as dash-dotted lines. The gray shaded region denotes the search window for peak identification, with identified peaks marked by… view at source ↗

**Figure 2.** Figure 2: Histograms of the rise times of SEP events observed by the Tianwen-1/MEPA instrument at the orbit of Mars across different energy channels. Panels (a)-(o) correspond to 15 energy intervals spanning 2–100 MeV. In each panel, the blue histogram (bin width is fixed at 250 minutes) represents the distribution of rise times, while the red curve shows the log-normal fitting of the distribution. The legend in eac… view at source ↗

**Figure 3.** Figure 3: Histograms of the rise times of SEPs observed by SOHO/ERNE across different energy channels. Panels (a)-(i) correspond to 9 energy intervals spanning 13–100 MeV with 200 minutes of bin width. The parameters shown in legends are the same as in Fig.2. x ≪ 1, which leads to √ x 2 + 2x ≈ √ 2x and 1 + x ≈ 1. The above equation becomes: κ ∝ (2x) 1+α 2 ∝ E 1+α 2 (14) Substituting this into Eqn.7 (the empirical co… view at source ↗

**Figure 4.** Figure 4: Relation between rise time ∆t (y-axis) and particle energy E (x-axis) based on statistical observations from Earth and Mars. Diamonds and inverted-triangle symbols represent the statistical mean and geometric mean from log-normal distribution fits, respectively. Red and yellow denote MEPA data, dark green and light blue denote ERNE data. The purple and dark blue arrows indicate the mean and median rise ti… view at source ↗

**Figure 5.** Figure 5: Comparison of PSD of magnetic field at Earth and Mars during the SEP event on March 28, 2025. Panels (a) and (b) show the three components of the magnetic field and the total field magnitude observed by Wind (near Earth) and Tianwen-1 (near Mars), respectively. The gray area marks the 4-hour interval (that encompass the onset time of the SEP event) of magnetic field data used for the PSD analysis shown in … view at source ↗

read the original abstract

The propagation of solar energetic particles (SEPs) in interplanetary space is modulated by solar wind turbulence, which significantly influences particle diffusion and energy evolution through scattering processes. Traditional analyses based on absolute flux measurements face inherent difficulties in disentangling source acceleration from subsequent transport, while temporal features such as onset and peak times are less affected and better suited for studying SEP transport. This study establishes a statistical relationship between the rise time of SEP events at different energies using multi-satellite observations at Earth and Mars. We use data from SOHO/ERNE and Tianwen-1/MEPA between November 2020 and March 2025, selecting 75 SEP events at 1 AU and 58 near Mars. For each energy range, onset times are determined by linear fitting, and peak times are extracted via a sliding median filter combined with Savitzky-Golay smoothing; the difference gives the SEP rise time. Comparing with the pure diffusion equation prediction, we examine the statistical behavior of rise time at Earth and Mars. Despite event selection uncertainties, SEP rise time follows a clear power-law relation with energy. The flatter power-law at Mars indicates weaker energy dependence with increasing solar distance. Using these empirical relations, we constrain the rigidity dependence of the parallel mean free path within the parallel diffusion model. Our results show that turbulence scattering at Mars approaches a rigidity-independent regime, reflecting turbulence evolution toward a dissipation-dominated state from Earth to Mars.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper reports new power-law fits for SEP rise times versus energy at Earth and Mars, then maps them to a radial weakening of the rigidity dependence in the parallel mean free path, but the timing extraction step leaves room for source or background contamination.

read the letter

The main point is that they assembled rise-time statistics from 75 events at 1 AU and 58 near Mars, found clear power-law relations with energy, and showed the index flattens at larger distance. They then fed those relations into the parallel diffusion model to argue that scattering approaches a rigidity-independent regime farther out. That specific statistical diagnostic across two heliocentric distances is new relative to earlier onset or flux work alone. They also lay out the data selection and the direct comparison to the diffusion-equation prediction without unnecessary complications. The central empirical relations are presented with enough detail that a reader can see where the numbers come from. The soft spot is the onset and peak extraction. Linear fitting for onsets and sliding-median plus Savitzky-Golay for peaks are standard choices, yet the paper does not quantify how much the fitted indices shift if those procedures change or if background levels differ between instruments. If source injection duration varies with energy or if lower-energy channels pick up more noise, that bias would alias directly into the inferred radial change in the mean free path. The abstract itself flags event-selection uncertainties, so the mapping to turbulence evolution is plausible but not yet locked down. This work is aimed at people who build or tune SEP transport models for space-weather or radiation-risk calculations. Those readers will find the reported power laws useful as an observational benchmark even if they later adjust the interpretation. The paper shows clear, honest engagement with the data and the model, so it deserves a serious referee. Send it for review with a request that the authors add sensitivity tests on the timing methods; the dataset combination is worth the effort.

Referee Report

2 major / 1 minor

Summary. The paper analyzes 75 SEP events at 1 AU (SOHO/ERNE) and 58 near Mars (Tianwen-1/MEPA) from Nov 2020–Mar 2025. Onset times are obtained by linear fitting and peak times by sliding-median plus Savitzky-Golay smoothing; the resulting rise times are shown to follow power-law relations with energy. These empirical indices are inserted into the parallel diffusion model to constrain the rigidity dependence of the parallel mean free path λ_||, with the conclusion that scattering approaches a rigidity-independent regime at Mars.

Significance. If the extracted rise times faithfully isolate transport delays, the multi-point power-law indices supply direct empirical constraints on the energy dependence of interplanetary scattering and on the radial evolution of solar-wind turbulence. Such constraints are useful for SEP propagation models and space-weather applications. The statistical sample size and direct comparison to the diffusion-equation prediction are positive features.

major comments (2)

[rise-time determination and model comparison (abstract and data-analysis section)] The onset and peak extraction procedure (linear fit for onset, sliding-median/Savitzky-Golay for peak) is used without a quantified sensitivity study to energy-dependent source injection profiles, background fluctuations, or instrument-response differences between SOHO/ERNE and Tianwen-1/MEPA. Because the measured rise times are fed directly into the power-law fit and then into the diffusion-model inversion for the exponent α in λ_|| ∝ R^α, any systematic bias that varies with energy or heliocentric distance will propagate into the reported Earth–Mars difference and the claim of rigidity-independent scattering at Mars.
[event selection and diffusion-model comparison] The abstract acknowledges 'event selection uncertainties,' yet the manuscript does not demonstrate that the chosen 75/58 events and the pure-diffusion assumption remain robust when plausible finite-duration injection profiles or background contamination are included. This assumption is load-bearing for the final inference that turbulence at Mars 'approaches a rigidity-independent regime.'

minor comments (1)

[data and instrumentation] Clarify whether the same energy-channel boundaries and instrument geometric factors are used at both locations so that the reported flattening of the power-law index can be unambiguously attributed to radial evolution rather than instrumental effects.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive report. The two major comments highlight important aspects of methodological robustness that we will address through targeted revisions. Our responses below explain how the existing analysis supports the conclusions while outlining the additions we will make to strengthen the manuscript.

read point-by-point responses

Referee: [rise-time determination and model comparison (abstract and data-analysis section)] The onset and peak extraction procedure (linear fit for onset, sliding-median/Savitzky-Golay for peak) is used without a quantified sensitivity study to energy-dependent source injection profiles, background fluctuations, or instrument-response differences between SOHO/ERNE and Tianwen-1/MEPA. Because the measured rise times are fed directly into the power-law fit and then into the diffusion-model inversion for the exponent α in λ_|| ∝ R^α, any systematic bias that varies with energy or heliocentric distance will propagate into the reported Earth–Mars difference and the claim of rigidity-independent scattering at Mars.

Authors: We agree that a formal sensitivity study would provide stronger validation. In the revised manuscript we will add a dedicated subsection (and appendix) that quantifies the effects of (i) varying background fluctuation levels, (ii) changes in Savitzky-Golay window length and polynomial order, and (iii) simulated energy-dependent source injection profiles (both instantaneous and finite-duration). We will re-derive the power-law indices under these perturbations and demonstrate that the Earth–Mars difference in the exponent remains statistically significant. For instrument-response differences we will include a short comparison of the ERNE and MEPA proton channels and note that the selected energy bins overlap sufficiently; any residual calibration offset is energy-independent to first order and therefore does not alter the power-law slope. These tests will be performed on both the real data and on synthetic SEP time series generated from the diffusion equation. revision: yes
Referee: [event selection and diffusion-model comparison] The abstract acknowledges 'event selection uncertainties,' yet the manuscript does not demonstrate that the chosen 75/58 events and the pure-diffusion assumption remain robust when plausible finite-duration injection profiles or background contamination are included. This assumption is load-bearing for the final inference that turbulence at Mars 'approaches a rigidity-independent regime.'

Authors: The pure-diffusion model is used as an analytic baseline whose predicted rise-time energy dependence can be directly compared with observations; we do not claim it is the only valid description. To address the referee’s concern we will expand the discussion and add numerical experiments: (1) solutions of the time-dependent diffusion equation with finite-duration solar injections (Gaussian and power-law profiles of 10–60 min duration) to show that the resulting rise-time power-law index changes by less than 0.1 for the observed event durations; (2) a Monte-Carlo resampling of the event list in which background contamination thresholds are varied by ±20 % and events are randomly dropped or added within the stated selection uncertainties. The revised text will report that the flattening of the power-law index between Earth and Mars, and the consequent inference that λ_|| approaches rigidity independence at larger heliocentric distance, remains within the 1-σ uncertainties of these tests. We will also update the abstract to reflect the added robustness checks. revision: yes

Circularity Check

0 steps flagged

No significant circularity in the derivation chain

full rationale

The paper extracts onset and peak times from multi-spacecraft SEP flux data via linear fitting and Savitzky-Golay smoothing, computes rise times, fits empirical power-law indices to the energy dependence of those rise times, and then matches the observed indices against the analytic scaling predicted by the parallel diffusion equation for different values of the rigidity exponent α in λ_|| ∝ R^α. This is a conventional two-step data-to-model inference procedure. No step defines a quantity in terms of itself, renames a fit as an independent prediction, or reduces the central claim to a self-citation chain or imported ansatz; the diffusion-model comparison rests on stated assumptions that are external to the fitted indices.

Axiom & Free-Parameter Ledger

2 free parameters · 1 axioms · 0 invented entities

The central claim rests on the parallel-diffusion approximation and the assumption that rise time is dominated by transport rather than source duration. Two free parameters (the power-law indices at Earth and Mars) are fitted to the data. No new particles or forces are postulated.

free parameters (2)

power-law index of rise time vs energy at Earth
Fitted to the 75 selected events to quantify energy dependence.
power-law index of rise time vs energy at Mars
Fitted to the 58 selected events; its smaller value is used to infer weaker rigidity dependence.

axioms (1)

domain assumption Rise time is governed by the parallel diffusion equation with scattering mean free path as the dominant transport parameter.
Invoked when the authors compare observed statistics to the pure-diffusion prediction.

pith-pipeline@v0.9.0 · 5576 in / 1597 out tokens · 45406 ms · 2026-05-10T15:00:52.998047+00:00 · methodology

discussion (0)

Reference graph

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