Rigidity-Dependent Diffusion Model for SEPs

• The rigidity-dependent diffusion model is a transport framework that explains the formation of double-power-law energy spectra in SEPs accelerated by CME-driven shocks.
• It explicitly incorporates a diffusion coefficient that depends on particle rigidity and charge-to-mass ratio, accounting for spatial variations along inhomogeneous shock fronts.
• The model’s quantitative predictions, including spectral break scaling and differing acceleration regions, align with both remote and in-situ observations from missions such as Parker Solar Probe.

A rigidity-dependent double-power-law diffusion model describes the formation of double power-law energy spectra of solar energetic particles (SEPs) accelerated by coronal shocks, particularly in events associated with coronal mass ejections (CMEs) propagating through streamer-like magnetic structures. This model incorporates the transport and acceleration of particles, with an explicit dependence of the diffusion coefficient on particle rigidity and charge-to-mass ratio, and accounts for the superposition of particle populations accelerated at different regions along an inhomogeneous shock front. This approach provides quantitative predictions for the charge-to-mass scaling of the spectral break energy, the presence of double-power-law spectral shapes, and their variation along the shock, consistent with remote and in-situ observations of SEP events (Yu et al., 2022).

1. Fundamental Transport Equation and Shock Geometry

The evolution of the gyrotropic phase-space density $f(\vec{x},p,t)$ of energetic particles in the proximity of CME-driven shocks is governed by the Parker transport equation: $$\frac{\partial f}{\partial t} = -(\vec{u} \cdot \nabla) f + \nabla \cdot (\kappa \cdot \nabla f) + \frac{1}{3} (\nabla \cdot \vec{u}) \frac{\partial f}{\partial \ln p} + Q(\vec{x}, p, t)$$ where $f$ is the isotropic particle distribution function, $\vec{u}(\vec{x})$ is the background solar wind and shock velocity, $\kappa$ is the (anisotropic) spatial diffusion tensor with parallel ($\kappa_\parallel$) and perpendicular ($\kappa_\perp$) components relative to the local magnetic field, and $Q$ accounts for particle injection upstream of the shock. In the modeling framework of Yu et al. (2021), the Parker equation is solved numerically in realistic streamer-like coronal magnetic fields, explicitly capturing spatial and momentum diffusion, convection, and adiabatic energy changes (Yu et al., 2022).

The CME-driven shock is highly structured, exhibiting regions of quasi-perpendicular geometry near the streamer axis, where closed magnetic fields trap particles more efficiently, and open-field regions along the flanks. This geometrical inhomogeneity is central to the model's prediction of spatially varying particle acceleration efficiency and spectral features.

2. Rigidity-Dependent Diffusion Coefficient Construction

Particle diffusion is described using quasi-linear theory, relating the spatial diffusion coefficient to the local magnetic turbulence spectrum. The turbulence power spectral density as a function of wavenumber $k$ is parameterized as

$P(k) \propto \left[1 + (k L_c)^\Gamma\right]^{-1}$

where $L_c$ is the coherence scale and $\Gamma$ is the inertial-range spectral index.

For a particle with momentum $p$, Lorentz factor $\gamma$, and charge-to-mass ratio $Q/A$, the parallel diffusion coefficient is

$$\kappa_\parallel(p) = \kappa_{\parallel,0} \left(\frac{\gamma_0}{\gamma}\right) \left(\frac{p}{p_0}\right)^{3-\Gamma} \left(\frac{Q}{A}\right)^{\Gamma-2}$$

with reference values $\kappa_{\parallel,0}$, $p_0$, and $\gamma_0$ at the prescribed injection momentum ($\sim100$ keV/nuc). In terms of particle rigidity $R \propto p/(Qe)$, this becomes

$\kappa_\parallel(R) \propto R^{3-\Gamma}$

In all cases, $\kappa_\perp \approx 0.01 \kappa_\parallel$ is adopted for perpendicular transport. The explicit $Q/A$ dependence enables direct predictions of spectral differences for different ion species (Yu et al., 2022).

3. Superposition and Formation of Double Power Laws

A classical diffusive shock acceleration (DSA) solution at a planar shock with compression ratio $X$ yields a single power-law in particle momentum, $f(p) \propto p^{-3X/(X-1)}$, translating to an energy spectrum $dJ/dE \propto E^{-\gamma}$, with $\gamma \approx (X+2)/(X-1)$. For $X=3$, Yu et al. (2021) recover a low-energy slope $\gamma_1 \simeq 1.25$, in exact agreement with the DSA solution.

However, when accounting for the finite spatial extent and the varying acceleration/escape efficiency along the shock, high-energy particles are subject to exponential rollovers at energies characteristic of each local region: $$\frac{dJ_i}{dE} \propto E^{-\gamma_1} \exp(-E/E_{B,i})$$ with region-specific break energy $E_{B,i}$. Integration across the shock or across a transition between regions of differing acceleration efficiency leads to a composite spectrum. When the spectrum is summed over both streamer-trapped (high-$E_B$) and open-region (low-$E_B$) components, the resulting domain-integrated spectrum approximates a double power law over a finite energy range: $$J_\mathrm{total}(E) \simeq A\,E^{-\gamma_1} \exp(-E/E_{B,low}) + B\,E^{-\gamma_1} \exp(-E/E_{B,high})$$ For $E \ll E_{B,low}$, the spectrum is dominated by the primary power law. In the intermediary regime $E_{B,low} \ll E \ll E_{B,high}$, the spectrum mimics a second, steeper power law, with index $\gamma_2 > \gamma_1$. Beyond $E_{B,high}$, the total flux exhibits an exponential cutoff. In simulations for $\Gamma=5/3$, $\gamma_2 \approx 2.4$ is found (Yu et al., 2022).

4. Charge-to-Mass Ratio Scaling of Spectral Break

The break energy $E_B$ at which the spectrum steepens is set by the condition that the particle diffusion length, $\sim \kappa/V_{sh}$ ($V_{sh}$ being the shock speed), equals the characteristic size of the acceleration zone. Fixing the effective $\kappa_\parallel(E_B)$ for different species gives the scaling: $$E_B \propto (Q/A)^\alpha, \qquad \alpha = \frac{2(2-\Gamma)}{3-\Gamma}$$ This relation predicts that the spectral break shifts to higher energies for species with greater $Q/A$. The value of $\alpha$ is controlled by the turbulence spectral index $\Gamma$. Simulation results yield

• $\Gamma = 1.9 \implies \alpha \approx 0.35$
• $\Gamma = 5/3 \implies \alpha \approx 0.51$
• $\Gamma = 1.1 \implies \alpha \approx 0.84$
• $\Gamma = 0.5 \implies \alpha \approx 1.10$

As turbulence becomes flatter (lower $\Gamma$), the $Q/A$ dependence strengthens, with $\alpha \to 2$ in the self-generated wave regime. This scaling matches observed variations in break energies across different ion species in SEP events (Yu et al., 2022).

5. Spatial Diffusion, Mixing, and Cross-Field Transport

In streamer-influenced CME shocks, closed-field regions confine particles and enable acceleration to higher maximum energies ($E_{B,II} \gg E_{B,I}$) compared to open-field flanks. Perpendicular diffusion and magnetic field-line wandering facilitate cross-field leakage of high-energy particles from streamer regions into neighboring zones. An observer with magnetic connection to such a transition region detects a particle fluence spectrum composed of superposed local (low $E_B$) and remote (high $E_B$) spectra: $$f = f_{I}(E) + \varepsilon f_{II}(E), \quad \varepsilon \ll 1$$ with $f_I(E) \propto E^{-\gamma_1} \exp(-E/E_{B,I})$ and $f_{II}(E) \propto E^{-\gamma_1} \exp(-E/E_{B,II})$. In the intermediate energy range $E_{B,I} \ll E \ll E_{B,II}$, the sum mimics an effective second, steeper power law ($\gamma_2 > \gamma_1$), with simulations yielding $\gamma_2 \approx 2.4$ for typical parameter choices. This process quantitatively accounts for observed region-to-region spectral diversity and double-power-law features (Yu et al., 2022).

6. Observational Consequences and Validation

The model predicts:

• A low-energy power-law index ($\gamma_1$) governed by shock compression ratio.
• A pronounced spectral break at $E_B \sim (Q/A)^\alpha$ with $0.2 \lesssim \alpha \lesssim 1.2$, matching the observed range in SEP events.
• An intermediate-energy regime with a steeper power-law index ($\gamma_2$), reflecting integrated contributions from regions of different acceleration efficiency.
• An eventual exponential cutoff at the highest energies.

SEPs sampled near the Sun—by missions such as Parker Solar Probe and Solar Orbiter—are expected to directly reflect these predictions, particularly the spatial and $Q/A$ dependence of $E_B$, as well as variation in spectral form along different magnetic footpoints. For example, observers connected to streamer-flank field lines should measure higher $E_B$ and a single-rollover spectrum. Observers on open-field flanks should register a single power law with low $E_B$, while those at magnetic transitions witness the characteristic double-power-law structure (Yu et al., 2022).

7. Broader Implications and Future Directions

The rigidity-dependent double-power-law diffusion model reconciles key observational features of SEP spectra with first-principles transport modeling, offering a self-consistent explanation for the Q/A dependence of spectral breaks and double-power-law structure. This framework motivates further observational tests using high-cadence and multi-point measurements from next-generation solar and heliospheric missions, and may be applicable for interpreting heavy-ion abundances and compositional anomalies (such as Fe/O enhancement at high energies) in SEP populations. The precise mapping between turbulence properties (via $\Gamma$), magnetic topology, and observed spectra remains an important direction for refinement and further study (Yu et al., 2022).