Radial dependence of solar energetic particle peak fluxes and fluences

(Solar Orbiter Nugget #55 by Yihang Cao¹, Yubao Wang¹, Jingnan Guo^1,2)

1. Introduction

As the core energy source of the solar system, the Sun continuously supplies energy to interplanetary space through persistent radiative and particle emissions. At the microscopic level, this energy propagates outward in the form of electromagnetic waves and particle streams of varying energies. Among these, Solar Energetic Particles (SEPs), originating from solar eruptions, may not only cause substantial interference with electronic equipment but also pose potential hazards for human space science endeavors due to their ionization process [1, 2]. Therefore, better understanding their distribution characteristics in the highly dynamic and complex interplanetary space, such as how their flux decay as they travel further from the Sun to the heliosphere, not only helps reveal the propagation mechanisms of SEPs but also provides theoretical support for space weather prediction and radiation mitigation strategies, especially for locations where SEP observations are lacking while forecasting is necessary.

Within the ecliptic plane, the propagation of SEP events can be simplified into two primary processes: outward radial propagation and longitudinal propagation to a wider coverage of longitudes (in comparison to the source). Separating these two processes for each individual event is rather challenging. Fortunately, studying the longitudinal distribution of SEP events has become straightforward with the data from the STEREO (Solar TErrestrial RElations Observatory launched in late 2006, see STEREO website) twin satellites, which were both positioned at approximately 1 AU from the Sun. Scientists discovered that the flux of SEP events typically exhibit a Gaussian distribution in longitude at 1 au [3, 4, 5]. However, due to the lack of data collected at solar distances other than 1 AU, it has been challenging to study the radial gradient of SEPs, although the theory of radial diffusion for SEP events was proposed decades ago [6]. A couple of studies had analyzed some specific SEP events observed by satellites at close longitudes but different radial positions [7, 8], while other studies explored the radial dependence using simulations of particle acceleration and diffusion processes [9].

In the last few years, with the successful launch of Solar Orbiter and Parker Solar Probe (PSP), we have accumulated many SEP events allowing detailed analysis of the radial distribution of SEP flux and its dependence on proton energies, offering new perspectives and theoretical references for SEP propagation processes. A couple of recent studies using the Solar Orbiter electron measurements have investigated the radial dependence of peak flux, anisotropy, and other related phenomena of solar energetic electrons in interplanetary space [10, 11]. Our work focuses on the distribution of energetic protons. 
 

2. Results (Statistical data obtained and SEP event selection)

This statistical analysis utilizes SEP data observed by three spacecraft: Solar and Heliospheric Observatory (SOHO), PSP, and Solar Orbiter. During the study period from 2021-01-01 to 2024-01-01, the SOHO spacecraft operated within a radial distance range of 0.97 to 1.01 AU from the Sun, while PSP and Solar Orbiter covered radial distances of 0.05 to 0.78 AU and 0.29 to 1.01 AU, respectively (see Figure 1 for overview of spacecraft solar distances and particle fluxes in 3 years).

Figure 1. An overview of multi-spacecraft SEP observations from January 1, 2021, to January 1, 2024. The pink arrows at the top represent the 42 SEP events simultaneously observed by at least two of the three spacecraft: SOHO, PSP, and Solar Orbiter.

Based on a simple Parker spiral and using observed solar wind speed at each spacecraft, we estimate the magnetic footpoint on the solar surface for each spacecraft during each SEP event. 

Subsequently, from the original list of 42 SEP events which have been simultaneously observed by at least two of the three spacecraft, we further selected 13 events that satisfy the following two criteria:
(1) the longitudinal separation between spacecraft observing the same SEP event is less than 30 degrees, and
(2) the SEP event observed by the spacecraft must exhibit a typical impulsive profile (i.e., the event reaches its peak within 6 hours after onset in the 10.5–15 MeV energy channel, and the peak intensity exceeds the background level by at least one order of magnitude).

This selection aims to ensure that the chosen events seen by different observers have minimized longitudinal effects, thereby enabling a better quantification of the radial dependence of SEP events across different energy ranges.

Figure 2. Radial dependence of the peak flux (top panels) and fluence (bottom ones) of the selected SEP events measured by SOHO and PSP (or Solar Orbiter) at different proton energies (different columns). The red diamonds represent the proton intensity measured by SOHO, and the blue triangles and green circles represent the measurements by the Solar Orbiter and PSP satellites, respectively. Different line colors denote the events selected following different selection criteria with the purple lines for the 13 events. The mean and medial values of -α in the radial dependence R^-α are shown in the legends. 
 

As shown in Figure 2, the analysis of the radial distribution of SEP peak flux and fluence (flux integrated over the event duration) reveals that the radial dependence can be approximated by a power-law relationship with heliocentric distance, R^-α, with the exponent varying within the range of R^-3.7 to R^-2 for peak flux and R^-2.7 to R^-1.4 for fluence, respectively. Notably, both distributions exhibit a pronounced energy dependence, α=α(E), with α decreasing as particle energy increases. In other words, the radial gradient is slightly larger for particles with lower energies and the radial dependence becomes weaker as particle energy increases.
 

3. Discussion（Energy dependence of power-law exponents)

For the radial dependence of peak fluxes in SEP events, our results are consistent with previous theoretical predictions. For example, Hamilton (1988) used a spherically symmetric transport model to calculate that α for peak fluxes in pure diffusion process is -3 [6]. Lario (2007) conducted simulations under various parameter conditions and found that α for proton flux and fluence ranged from -3 to -1.3 and -1.5 to -0.7, respectively [8]. Additionally, Verkhoglyadova (2012) modeled the acceleration and transport of protons and iron ions and discovered that the α for peak flux ranging from -2.9 to -1.8 [9]. 

Furthermore, we provide an in-depth analysis of the key parameters influencing α and their specific mechanisms of influence. According to the particle transport equation, various factors may influence the SEP intensity so as the radial dependence of SEP flux, including cross-field diffusion, pitch-angle scattering, gradient or curvature drifts, magnetic focusing, and adiabatic cooling terms [12, 13]. Among these, radial diffusion and adiabatic cooling may be critical factors for radial dependence. Based on the proton peak spectrum model [14], we derive the radial dependence of particle peak flux which show that α(E) is related to the power-law spectral index of the particle source (parameter γ defined in the original flux ∝E^-γ), solar wind speed and the diffusion coefficient which depends on the particle energy. Figure 3 illustrates the consistency between observational results in white lines and the theoretical results as background colors.

During SEP events, a larger diffusion coefficient leads to an increase in the mean free path of particles, thereby reducing the probability of interaction between SEPs and the solar wind plasma. This process leads to a smaller radial gradient of the SEP flux. Additionally, due to adiabatic cooling effects, particles of the same energy detected at different radial positions, say R₁ and R₂  , may originate from source regions with different initial energies, say E_s1 and E_s2  . This energy difference causes the radial dependence to also reflect the initial flux gradient between E_s1 and E_s2. Consequently, a harder SEP source spectrum (i.e., a smaller spectral index γ associated with the original flux ∝E^-γ) leads to a smaller gradient and smaller α (Figure 3).

Figure 3. Relation between power-law index of radial dependence α(E) for the peak flux (shown in color for theoretical predictions and white lines for observational results) and two parameters, power-law index of source particle (y-axis) and diffusion coefficient related parameter(x-axis). 

Additionally, another factor influencing the radial dependence α is the magnetic connectivity of the observing satellites which affects the longitudinal distribution. In Figure 4(a), all satellites had direct magnetic connection with the SEP source (the effective connection region/ECR is marked in red). Based on pure diffusive transport model, the radial dependence of peak flux satisfies |α|∼3. Panels (b) and (c) illustrate scenarios where an observer is located outside the ECR. Considering that SEPs need to undergo sufficient cross-field transport to be detected by satellites outside ECR, the actual flux measurement there is typically lower than that at the same radial distance inside the ECR. This discrepancy may lead to variations in the radial dependence of peak flux and fluence.

Figure 4. Panels (a) to (c) illustrate three different connection scenarios, demonstrating how magnetic connectivity affects the radial dependence parameter α of the peak flux. More explanations in the text.

4. Conclusion

In summary, we found that SEP proton flux/fluence has a radial dependence that can be approximated by R^-α, with α between 2 and 3.7 for peak flux and between 1.4 and 2.7 for fluence, respectively. We also found that the radial dependence differs for different particle energies, i.e., α=α(E), where the radial gradient is slightly larger for particles with lower energies.

Our results indicate that a simple model of the SEP flux peak energy spectrum, based on diffusion and adiabatic cooling, can effectively characterize the energy dependence of SEP radial distribution. However, more measurements and more efforts are still needed to better quantify the spatial distribution of SEPs. Our current results are mostly based on the measurements provided by two spacecraft (i.e., SOHO together with either Solar Orbiter or PSP). We need more joint-spacecraft observations of the same SEP, ideally at least three observers detecting the same event within close longitudes to better quantify the possible change of the radial gradient on the radial distance. With more precise quantifications of the SEP distribution characteristics, we hope to make data-based predictions of the global heliospheric distribution of the SEP flux which is important for understanding their large-scale impact, especially for future deep-space exploration missions. 

For further details, see Cao et al., A&A, 965, A25 (2025)
 

Affiliations

(1) Deep space Exploration Laboratory/School of Earth and Space Sciences, University of Science and Technology of China, Hefei, PR China
(2) CAS Center for Excellence in Comparative Planetology, University of Science and Technology of China, Hefei 230026 , China

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