Longitudinal SEP Intensity Dependence

• Longitudinal dependence of SEP intensities is defined by interactions between CME-driven shock acceleration, Parker spiral magnetic fields, and turbulent cross-field transport.
• Modeling shows that key parameters like the parallel scattering mean free path and perpendicular diffusion coefficient determine the Gaussian width and westward shift of SEP profiles at 1 AU.
• Combined analyses of early-time transport, east-west asymmetries, and corotation effects yield practical insights for forecasting SEP event profiles and mitigating space weather risks.

The longitudinal dependence of Solar Energetic Particle (SEP) intensities refers to the pronounced variation in SEP fluxes as a function of solar longitude at a given heliocentric distance (typically 1 AU). This spatial structure emerges from the interactions between particle acceleration at coronal mass ejection (CME)-driven shocks, the 3D heliospheric magnetic field (especially the Parker spiral geometry), interplanetary turbulence, and various cross-field transport mechanisms. Modern observations and modeling reveal that SEP profiles are shaped not only by injection physics and shock strength but decisively by stochastic field-line wandering, perpendicular diffusion, corotation, and the dynamical evolution of the observer's connection to the acceleration region.

1. Early-Time Cross-Field Transport: Meandering Field Lines and Turbulence

At the onset of SEP events, particles are rapidly distributed across longitudes that far exceed expectations from pure parallel transport along Parker-spiral field lines. Early full-orbit simulations (the FP+FLRW paradigm) demonstrated that, in a turbulent solar wind, particles initially propagate not diffusively but are tightly tied to individual, stochastically meandering field lines. The cross-field displacement of these field lines, described by a field-line diffusion coefficient $D_{FL}$, is set by turbulence strength through $D_{FL} \propto 1/\sqrt{\lambda_{\parallel}^*}$, where $\lambda_{\parallel}^*$ is the parallel scattering mean free path at 1 AU.

The resulting longitudinal width of the SEP intensity profile at 1 AU, typically fit with a Gaussian form $$I(\phi) = I_{\max} \exp[-(\phi-\phi_0)^2/(2\sigma_\phi^2)]$$, scales as

$$\sigma_\phi \propto (1/\lambda_{\parallel}^*)^{1/4}.$$

Strong turbulence (small $\lambda_{\parallel}^*$) thus produces both slower parallel transport and a wider SEP footprint in longitude. Typical observed widths for 10 MeV protons are $\sigma_\phi \sim 30^\circ$–$50^\circ$, requiring values as low as $\lambda_{\parallel}^* \sim 0.1$ AU, with westward centroid shifts ($\phi_0$) of up to $10^\circ$–$15^\circ$ for the strongest scattering regimes (Laitinen et al., 2018).

2. The Role of Perpendicular Diffusion

Perpendicular diffusion, arising from the stochastic wandering of magnetic field lines and particle scattering, is the dominant process governing the effective longitudinal spread of SEPs. Both first-principles transport modeling and recent empirical fits show that with realistic perpendicular diffusion coefficients ($\kappa_\perp/\kappa_\parallel \sim 0.01$–$0.1$), the Gaussian width of the longitudinal SEP intensity distribution at 1 AU matches in-situ multipoint measurements (Strauss et al., 2018, Strauss et al., 2018, Young et al., 19 Sep 2025).

Key parameterizations include:

• The FLRW scaling for the perpendicular diffusion coefficient,
• A dimensionless field-line wandering parameter $a \approx 0.2$,
• Energy-dependent cross-field mean free paths $\lambda_\perp \sim 0.005$–$0.03$ AU for $\lambda_\parallel = 0.15$–$1.0$ AU,
• A close link between the steepness of initial particle gradients and the early-time effectiveness of cross-field transport, which rapidly decreases as the SEP event fills in.

Perpendicular diffusion also produces characteristic east-west asymmetries due to Parker spiral geometry, as detailed below.

3. Longitudinal Asymmetries: East–West Effects and Parker Spiral Geometry

SEP intensity maxima at 1 AU are systematically shifted west of the nominal Parker-spiral magnetic connection to the source. Two robust, statistically and numerically confirmed asymmetries emerge (He et al., 2015, He et al., 2016, Laitinen et al., 2023):

• For equal angular separation from the observer's footpoint, SEP events originating east of the nominal connection yield higher peak intensities and earlier onsets than those from equivalent western sources.
• The intensity distribution is not symmetric; westward locations (relative to the footpoint of the acceleration region) generally exhibit both earlier SEP arrivals and higher peaks.

This asymmetry results from the Parker spiral's azimuthal tilt, which causes field lines from eastern sources to traverse a shorter effective cross-field distance at 1 AU. Perpendicular diffusion exponentially attenuates particles as a function of this pathlength, favoring eastward over westward sources for a given separation and leading to the widely observed statistical excess of east-source events (He et al., 2016). These effects are encapsulated by the power-law radial scaling $I_{\max}(r) \propto r^{-1.7}$ for peak intensity, mapping intensity differences between geometrically inequivalent (east/west) locations at constant radial distance.

4. Impact of CME-Driven Shock Properties, Source Geometry, and Magnetic Connectivity

The 3D structure, temporal evolution, and varying strength of the CME-driven shock have direct control over the longitude profile of SEP intensities (Zhou et al., 20 Jan 2026, Reames, 2022). Key findings include:

• Shock nose regions, characterized by higher shock speeds ($v_n$), compression ratios ($r$), and Alfven Mach numbers ($M_A$), dominate the early, rapid intensity rises at longitudes magnetically connected to the nose.
• Flanks show delayed acceleration, lower efficiency, and softer spectra.
• Magnetic connectivity timing (the cobpoint, or instant when a field line connects to the shock front) determines the onset and peak SEP flux at different longitudes.
• In situ measurements of SEP spectral indices match predictions from relativistic diffusive shock acceleration, i.e., $s_{\mathrm{DSA}} = (r+2)/(r-1)$, confirming the theoretical connection between shock compression and SEP energy spectra at all observed longitudes (Zhou et al., 20 Jan 2026).

Empirical models corroborate this picture, with peak and fluence SEP intensity distributions at 1 AU modeled as energy-dependent Gaussians centered westward of the CME's longitude and with characteristic widths of $\sigma_\phi \sim 35^\circ$–$45^\circ$ (Bruno et al., 2021).

5. Corotation Effects on Longitudinal Dependence

The corotation of particle-filled Parker-spiral flux tubes with the Sun is a critical modifier of SEP temporal profiles across longitude. When corotation is included in full-orbit test-particle simulations:

• Western observers rapidly lose connection to the SEP source as their flux tubes are swept away, dramatically shortening SEP event durations and steepening decay phases.
• Eastern observers benefit from corotation, as SEP-filled field lines drift into their longitude, resulting in enhanced intensities and prolonged events.
• Including corotation strengthens the classic east-west asymmetry (shifting Gaussian centroid of $I_{\mathrm{peak}}(\Delta\phi)$ another $\sim5^\circ$ eastward), while suppressing the dependence of decay constants on the scattering mean free path, rendering decay primarily controlled by the rotation rate (14° per day) rather than diffusive leakage (Hutchinson et al., 2022).

6. Synthesis: Observational Signatures and Forecasting Implications

Modern space missions and case studies (e.g., the 2024 superstorm, multipoint events with STEREO, ACE, and PSP) consistently reveal:

• SEP onset times, rise rates, and energy spectra at widely separated longitudes are predominantly determined by evolving connectivity to the shock front, field-line meandering, and the merged effects of sequential CMEs (Muro et al., 5 Nov 2025).
• Multi-spacecraft observations demonstrate impulsive, hard-spectrum events at well-connected longitudes and delayed, softer, more gradual rises at remote longitudes, in quantitative agreement with models featuring moderate to strong turbulence, efficient perpendicular diffusion, and strong shock compressions (Young et al., 19 Sep 2025, Zhou et al., 20 Jan 2026).
• Energy-dependent Fe/O ratios and spectral slopes varying with longitude further indicate that both seed population composition and species-dependent transport (rigidity scaling of $\lambda_\parallel$ and $\kappa_\perp$) modulate the observed distribution, especially during extreme superstorms (Muro et al., 5 Nov 2025).
• Empirical and physics-based models provide practical “knobs” (notably $\lambda_\parallel^*$, the turbulence spectrum, $\kappa_\perp/\kappa_\parallel$, and shock geometry) for forecasting longitudinal SEP flux extents and asymmetries (Laitinen et al., 2018, Bruno et al., 2021).

7. Quantitative Summary of Key Parameters Influencing Longitudinal SEP Distributions

Model/Study	Key Parameter(s)	Typical Value / Scaling	Gaussian Width or Shift
FP+FLRW (Laitinen+)	$\lambda_\parallel^*$	0.1–1 AU; $(1/\lambda_\parallel^*)^{1/4}$	$\sigma_\phi$ = 23–41°
Empirical model	CME speed, $\sigma_\phi(E)$	$\sigma_{0}$–$\sigma_{1} \log E$ ($\sigma_\phi\sim35$–45° at 10–130 MeV)	centroid off CME by $\gtrsim$35°
EPREM	$\kappa_\perp/\kappa_\parallel$	$\sim10^{-2}$ (baseline, critical for spread $>90^\circ$)	—
SDE focus-transp.	$\lambda_\parallel$, $\lambda_\perp$	$\lambda_\perp/\lambda_\parallel$ = 0.01–0.02	$\sim 1.3\times$ for east vs west
Corotation	$\Omega$ (solar rotation)	$14.2^\circ$/day drift, event durations/fits	centroid west shift $>$5° (w/ corotation)

These results collectively establish that the observed longitudinal dependence of SEP intensities is a multi-factorial outcome—with transport regime transitions, turbulence amplitude, field-line topology, injection geometry, shock evolution, and large-scale solar rotation all contributing in quantifiable ways to the detectable 1 AU phenomenology (Laitinen et al., 2018, Hutchinson et al., 2022, Hutchinson et al., 2022, Bruno et al., 2021, Young et al., 19 Sep 2025, He et al., 2015, 2610.13692, Muro et al., 5 Nov 2025, Zhou et al., 20 Jan 2026, Reames, 2022).

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References:

(Laitinen et al., 2018, Strauss et al., 2018, He et al., 2015, He et al., 2016, Laitinen et al., 2023, Zhou et al., 20 Jan 2026, Hutchinson et al., 2022, Bruno et al., 2021, Muro et al., 5 Nov 2025, Hutchinson et al., 2022, Reames, 2022, Young et al., 19 Sep 2025, Strauss et al., 2018)