Velocity dispersion of Solar Energetic Particles in turbulent heliosphere

arXiv:2603.06433 · astro-ph.SR, physics.space-ph · Submitted 2026-03-06 · Read on arXiv

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Introduction to the show: ident: Astrophysics Radio. Generated commentary on the latest astrophysics papers.

Vera: Today's paper: "Velocity dispersion of Solar Energetic Particles in turbulent heliosphere".

Jocelyn: The gist The Velocity Dispersion Analysis (VDA) method, when applied to full-orbit simulations in a novel model of interplanetary magnetic turbulence superposed on a Parker Spiral magnetic field,

Vera: First, who's behind it and why it matters.

Title and authors: Vera: We started by looking at the title and authors, and it’s about Velocity dispersion of Solar Energetic Particles in turbulent heliosphere. It sounds pretty technical, but essentially it’s about using velocity dispersion analysis to study those solar particles in a messy magnetic environment.

Jocelyn: That’s right. The paper is by Laitinen and Dalla, and they are tackling the fundamental question of how the turbulence in space changes our view of when those energetic protons are actually being released from the Sun.

Subrahmanyan: The authors set up their simulation to test this idea by superimposing a novel model of interplanetary magnetic turbulence onto a Parker Spiral field, which is the standard magnetic background we usually assume exists in our solar system.

Vera: So, they're not just using one fixed field; they’re creating a more realistic picture where the turbulence can be described as dominant transverse fluctuations with 2D structure and some smaller slab-like components at different distances from the Sun <ref:2603.06433#pg1>.

Jocelyn: That level of detail in describing the magnetic field, especially those specific turbulence strengths—the zero point two, zero point six, and two for dB2/B2 at one au—is what allows them to systematically test how much that complexity changes their results <ref:2603.06433#pg3>.

Subrahmanyan: It’s important to remember that the turbulence model they use is tied back to established theory, specifically relating those dB2/B2 values at one au to the parallel scattering mean free paths of zero point eight au for the weak case and zero point three seven au for the moderate case <ref:2603.06433#pg3>.

Vera: So it’s not just a theoretical exercise; they’re using these specific turbulence levels to see if VDA can still give them reliable answers under different conditions <ref:2603.06433#pg1>.

Jocelyn: They are specifically interested in the reliability of the Velocity Dispersion Analysis method, since it’s often the primary tool we have for determining that solar injection time tsun <ref:2603.06433#pg2>.

Subrahmanyan: And they are explicitly asking if there's a link between the apparent path lengths they get from VDA and the standard Parker Spiral path length of one point one to one point two au that we typically assume <ref:2603.06433#pg2>.

Vera: They’re trying to determine if that discrepancy between their calculated long path lengths and the expected Parker Spiral length implies a real problem with the solar injection time estimate itself <ref:2603.06433#pg2>.

Jocclusion: So, they are testing if those extra distances mean the timing might be wrong too <ref:2603.06433#pg2>.

The paper's summary: Vera: Moving on to the summary of this paper, the main point is that when you factor in this complex turbulence and the pre-event background, VDA estimates for solar injection time tsun often include significant contributions from those things and aren't a good estimate of the actual acceleration time <ref:2603.06433#pg1>.

Jocelyn: It means that just trusting a simple VDA result might lead you to be off by several minutes, depending on the turbulence level you are dealing with <ref:2603.06433#pg1>.

Subrahmanyan: The simulation results confirm this: for weak and moderate turbulence, the VDA estimates of tsun are two to sixteen minutes after the actual solar injection time <ref:2603.06433#pg2>. They also get path lengths that are longer by about zero point two to zero point three au compared to the Parker Spiral <ref:2603.06433#pg2>.

Vera: That’s a concrete error margin we can work with when looking at these simulations, which is a deviation of about that magnitude from the expected baseline <ref:2603.06433#pg2>.

Jocelyn: But then they show the strong turbulence case is much more extreme; the path lengths become greater than five au, which is way longer than what we typically observe in SEP events <ref:2603.06433#pg2>.

Subrahmanyan: And for that strong turbulence scenario, the solar release time error can be as large as about one hundred minutes <ref:2603.06433#pg2>. That’s a huge uncertainty compared to what we usually see in observations.

Vera: So the core message is that you have to account for the fact that turbulence and the pre-event background really do inject a lot of error into our VDA estimates <ref:2603.06433#pg1>. It's not just a small correction; it’s often a substantial one.

Jocelyn: So, for us on the ground, it means we need to treat those VDA results as conditional—they depend heavily on the turbulence level in that heliospheric region <ref:2603.06433#pg1>.

The paper's improvements: Vera: Now let’s talk about what the authors suggest to improve this analysis, because they point out a few ways we can get a better handle on these uncertainties <ref:2603.06433#pg1>. They focus on how the turbulence level affects the heliolongitudinal extent and the path length and timing of the event <ref:2603.06433#pg1>.

Jocelyn: They highlight that for low-turbulence conditions, VDA results can be fairly accurate, with tsun being within ten minutes from the actual solar injection time and path length s being close to the Parker Spiral length <ref:2603.06433#pg2>.

Subrahmanyan: But as you move into moderate turbulence, they find that the VDA solar injection time tsun ranges between five and fifteen minutes later than the actual solar injection time <ref:2603.06433#pg2>. The apparent path length is also about fifteen to forty percent longer than the Parker spiral length near the best connection <ref:2603.06433#pg2>.

Vera: That’s a clear quantification of how much that moderate turbulence pushes the timing and distance away from our baseline expectations <ref:2603.06433#pg2>. It shows the effect is measurable even in these moderate conditions.

Jocelyn: And they also look at energy dependence in the onset threshold, which mimics the pre-event background, and this has a significant effect on path lengths, causing a five to ten percent difference <ref:2603.06433#pg2>.

Subrahmanyan: And that same energy dependence also causes a five to twenty minute difference in VDA solar injection times between using an energy-dependent and an energy-independent onset threshold <ref:2603.06433#pg2>. That’s a clear trade-off they found.

Vera: It seems the authors are suggesting that we can use these results to differentiate between the effects of field line meandering versus particle scattering, especially when the observer isn't well connected to the source along that nominal Parker Spiral <ref:2603.06433#pg2>.

Jocelyn: So, they’re giving us ways to separate those different physical processes influencing the SEP event structure improvement one <ref:2603.06433#pg1>. It helps isolate which effect is causing the shift in timing or path length improvement one <ref:2603.06433#pg1>.

Conclusion: Vera: To wrap things up on this Velocity dispersion of Solar Energetic Particles in turbulent heliosphere study, it’s clear that VDA isn't a perfectly accurate method when turbulence levels aren't particularly low <ref:2603.06433#pg1>.

Jocelyn: In moderate to strong turbulence, the path lengthening due to field line meandering and particle scattering really reduces the accuracy considerably, especially if we aren’t well connected magnetically to the source along the nominal Parker Spiral <ref:2603.06433#pg2>.

Subrahmanyan: And they emphasize that high and energy-dependent pre-event cosmic ray background can significantly increase this uncertainty for both the solar release timing and the apparent path length <ref:2603.06433#pg2>. That’s a major factor we have to keep in mind.

Vera: So, for anyone using VDA, if you’re seeing strong turbulence or not being well connected magnetically, you should expect much larger uncertainties in your solar injection time estimates <ref:2603.06433#pg1>.

Jocelyn: This paper is a strong reminder that the level and spectrum of the pre-event background really do inject uncertainty into VDA results <ref:2603.06433#pg2>. It’s not just a small correction; it’s often substantial.

Subrahmanyan: Ultimately, this work offers ways to constrain interplanetary turbulence parameters using these VDA results as a way to test them improvement two, and it shows how turbulence variance directly translates into path length differences of three to fifteen minutes in solar injection time between low and moderate cases improvement three.

Vera: This Velocity dispersion of Solar Energetic Particles in turbulent heliosphere paper lays out exactly where the limitations are, especially concerning strong turbulence and background effects on timing estimates.

Jocelyn: It’s a solid piece for anyone trying to understand the practical impact of magnetic complexity on SEP studies <ref:2603.06433#pg2>. We’re wrapping up this discussion here for now, and we have some new topics coming up next week.

Jeremiah Horrocks Institute, University of Lancashire

astro-ph.SR, physics.space-ph

Submitted: 2026-03-06

Updated: 2026-10-08

Comments: 9 pages, 4 figures. Accepted to be published at Astrophysical Journal Letters. Data to be made available at publication

License: http://creativecommons.org/licenses/by/4.0/

Importance score: 61/100

The gist: The gist The Velocity Dispersion Analysis (VDA) method, when applied to full-orbit simulations in a novel model of interplanetary magnetic turbulence superposed on a Parker Spiral magnetic field,

Key concepts

Velocity Dispersion Analysis (VDA): VDA is a method used to derive the solar injection time (tsun) and apparent path length (s) of SEPs by fitting onset times and particle velocities to a specific equation. It attempts to estimate when particles were actually released from the Sun.
Parker Spiral Magnetic Field: This is a model of the large-scale magnetic field in the heliosphere that resembles a spiral. The study superimposes turbulence onto this background field to simulate real solar conditions and how they affect particle travel.
Turbulence Variance (dB2/B^2): This parameter quantifies the strength of magnetic turbulence at 1 au, representing different levels of disorder in the magnetic field. The study tested weak (0.2), moderate (0.6), and strong (2) turbulence to see how these changes impact VDA accuracy.
Pre-event Background: This refers to the cosmic ray background present before a solar event occurs, which is energy-dependent. This background influences the onset threshold for particle detection, causing measurable differences in the calculated injection times and path lengths in VDA.

Terminology

Velocity Dispersion — 15× in this paper · explained in 2 episodes
Velocity Dispersion Analysis — 6× in this paper
Accuracy — 5× in this paper · explained in 1 episode
Core — 1× in this paper · explained in 2 episodes
UK Science and Technology Facilities Council — 1× in this paper

Summary

The gist The Velocity Dispersion Analysis (VDA) method, when applied to full-orbit simulations in a novel model of interplanetary magnetic turbulence superposed on a Parker Spiral magnetic field, reveals that VDA-derived injection times often include significant contributions from turbulence and/or the pre-event background and are not an accurate estimate of the acceleration time<ref:2603.06433#pg2>

The Methodology

The study employs full 3D equation of motion simulations for Solar Energetic Particles (SEPs) to record their time and heliospheric location as they pass through a spherical surface at 1 au heliocentric distance<ref:2603.06433#pg3> The heliospheric magnetic field is constructed by superposing the Parker spiral with an analytic model of turbulence described in detail in Laitinen et al. (2023b)<ref:2603.06433#pg2> The turbulence model includes a main 2D wave mode field transverse to the Parker spiral direction, combined with a slab-like component that is asymptotically radial and azimuthal at distances smaller and much larger than 1 au, respectively<ref:2603.06433#pg3> For the turbulence model, relative turbulence variance at 1 au, dB2/B squared having values of 0.2, 0.6 and 2 for the weak, moderate and strong turbulence cases, respectively<ref:2603.06433#pg4> The simulation uses a spectrum at injection proportional to E-1 for protons with energies E = 1–100 MeV<ref:2603.06433#pg5> The SEP source function is defined as Qs(r, θ, ϕ, t) = δ[r − r0] × g(t) × (H[ϕ − ϕ−] − H[ϕ − ϕ+]) (H[(θ − θ−] − H[θ − θ+])<ref:2603.06433#pg5>

VDA Analysis and Results

The VDA method is used to derive the solar injection time tsun and the apparent path length s by fitting onset times and particle velocities to Equation (1), tonset(E) = tsun + s/v<ref:2603.06433#pg2> The study investigates three different turbulence amplitudes corresponding to the Palmer consensus range with λ∥ between 0.08 and 0.3 au<ref:2603.06433#pg2> For weak and moderate turbulence, VDA estimates of tsun are 2–16 minutes after the actual solar injection time, and the path lengths are 0.2–0.3 au longer than the Parker spiral<ref:2603.06433#pg2> For strong turbulence, the path lengths are > 5 au, considerably longer than those typically obtained from SEP observations<ref:2603.06433#pg2> The effect of energy-dependence of the pre-event proton background results in 5–20-minute difference in VDA injection times depending on the heliolongitude<ref:2603.06433#pg2>

Impact of Turbulence and Background

The turbulence amplitude significantly affects the heliolongitudinal extent of the analyzed SEP event, the path length, and the obtained solar release time tsun<ref:2603.06433#pg2> For low-turbulence conditions (dB2/B squared = 0.2), VDA results can be fairly accurate, with tsun within 10 minutes from the beginning of the injection of the simulated SEPs and path length s being of the order of sParker<ref:2603.06433#pg2> In moderate turbulence (dB2/B squared = 0.6), VDA solar injection time tsun ranges between 5–15 minutes later than the actual solar injection time, and the apparent path length is about 15–40% longer than the Parker spiral length near the best connection<ref:2603.06433#pg2> Strong turbulence (dB2/B squared = 2) produces s > 5 au, rarely seen in VDA analysis of SEP events, and a solar release time error of order 100 minutes<ref:2603.06433#pg2> The details of energy-dependence of the onset threshold affect the VDA results significantly, with 5–10% difference in VDA path lengths, and 5–20-minute difference in VDA solar injection times between energy-dependent and energyindependent onset thresholds<ref:2603.06433#pg2>

Conclusion

The study concludes that in low-turbulence conditions the VDA results can be fairly accurate, whereas in moderate to strong turbulence both the path lengthening likely due to meandering of field lines, and scattering of particles even at the start of the SEP event, reduce the accuracy considerably particularly when the observer is not well connected to the source along the nominal Parker spiral<ref:2603.06433#pg2> These effects of turbulence are naturally exacerbated by high and energy-dependent pre-event cosmic ray background, which can affect both the VDA solar release timing and the apparent path length<ref:2603.06433#pg2> Thus, in many situations VDA is not an accurate method for estimating the SEP pathlengths and the solar injection times<ref:2603.06433#pg2>

How it works

The study solves the full 3D equation of motion for SEPs, recording their time and heliospheric location as they pass through a spherical surface at 1 au heliocentric distance<ref:2603.06433#pg3> The analysis involves binning simulated particles on 10 logarithmically equispaced bins with boundaries log E i[MeV] = 2i/10, and then deriving synthetic intensities I(E, t, ∆ϕ) as a function of time at observer location ∆ϕ and energy channel E<ref:2603.06433#pg5> The onset times tonset(E, ∆ϕ) are determined using a criterion where the proton intensity increases above a given threshold Ionset(E, ∆ϕ) = f(E) max[I(E, t, ∆ϕ)]<ref:2603.06433#pg2> This onset time is then used in Equation (1), tonset(E) = tsun + s/v to obtain the solar injection time tsun and the apparent path length s<ref:2603.06433#pg2>

Key Findings Summary

** The VDA-derived solar release times are within 16 minutes from the release time of the simulated particles for the low and moderate turbulence cases, with dB2/B squared = 0.2 and 0.6 at 1 au, respectively<ref:2603.06433#pg2> **

** The difference in the turbulence variance between the low and moderate turbulence cases results in 3 to 15 minute ∆ϕ-dependent difference in the derived VDA solar injection time, and 0.2–0.3 au difference in the VD path length<ref:2603.06433#pg2> **

** Strong turbulence, with dB2/B squared = 2 at 1 au, corresponding to the low scattering mean free path limit of the Palmer consensus, produces s > 5 au, rarely seen in VDA analysis of SEP events (e.g., Paassilta et al. 2017), and solar release time error of order 100 minutes<ref:2603.06433#pg2> **

** The details of energy-dependence of the onset threshold, mimicking the effect of the pre-event background in observed SEP events, affect the VDA results significantly, with 5–10% difference in VDA path lengths, and 5–20-minute difference in VDA solar injection times between energy-dependent and energyindependent onset thresholds<ref:2603.06433#pg2> **

The authors suggest that VDA is inaccurate when the turbulence levels are not particularly low and when the observer is not well magnetically connected, and the level and spectrum of the pre-event background can significantly increase the uncertainty of the VDA results<ref:2603.06433#pg2>

Acknowledgements

TL and SD acknowledge support from the UK Science and Technology Facilities Council (STFC) through grants ST/V000934/1 and ST/Y002725/1<ref:2603.06433#pg6>

Data Availability

Data used in this study will be made available after acceptance of the manuscript<ref:2603.06433#pg6>

References

Bieber, J. W., Wanner, W., & Matthaeus, W. H. 1996, J. Geophys. Res., 101, 2511, doi: 10.1029/95JA02588

Chhiber, R.

Improvements for AI systems

textbf Processing VDA Uncertainty Quantification for SEP Onset Times in Turbulent Heliospheres: The improved AI system can now accurately estimate the uncertainty of order 10-20 minutes for tsun for 1-100 MeV protons by incorporating the effects of interplanetary turbulence and the pre-event background.
textbf Constraining Interplanetary Turbulence Parameters from VDA Results: The improved AI system can now use VDA results as a constraint, as indicated by the finding that the VDA analysis record can be used as a constraint for the interplanetary turbulence parameters.
textbf Differentiating Turbulence Impact on Path Lengths and Timing: The system can now distinguish between turbulence effects, noting that "the difference in the turbulence variance between the low and moderate turbulence cases results in 3 to 15 minute ∆ϕ-dependent difference in the derived VDA solar injection time, and 0.2-0.3 au difference in the VD path length."
textbf Robustness Assessment of VDA Method: The improved system can assess when VDA is not an accurate method for estimating the SEP pathlengths and the solar injection times, specifically when "the turbulence levels are not particularly low and when the observer is not well magnetically connected, and the level and spectrum of the pre-event background can significantly increase the uncertainty of the VDA results."
textbf Energy-Dependence Modeling of Thresholds: The system can accurately model how the details of energy-dependence of the onset threshold, mimicking the effect of the pre-event background in observed SEP events, affect the VDA results, yielding a 5-20-minute difference in VDA solar injection times between energy-dependent and energyindependent onset thresholds.

Abstract

Solar Energetic Particles (SEPs) are a signature of solar eruptions, and to link them to acceleration mechanisms many studies investigate their injection time at the Sun, t sun. We assess velocity dispersion analysis (VDA), an often-used method to derive t sun. We use full-orbit simulations of 1--100 MeV SEP protons in a novel model of the interplanetary magnetic turbulence superposed on a Parker Spiral magnetic field. The turbulence is described analytically as dominant transverse fluctuations that are 2D with respect to the mean field, supplemented with a minor contribution of asymptotically slab turbulence modes. We determine simulated SEP intensities for three turbulence strengths and use VDA to obtain t sun and the apparent path length s of the SEPs, employing an SEP onset threshold to mimic a realistic energetic proton background before the SEP event. We find that turbulence strongly affects t sun and s. For weak and moderate turbulence, VDA estimates of t sun are 2-16 minutes after the actual solar injection time, and the path lengths are 0.2-0.3 au longer than the Parker spiral. For strong turbulence, the path lengths are >5 au, considerably longer than those typically obtained from SEP observations. We also investigate the effect of energy-dependence of the pre-event proton background, and find that different background spectra result in 5-20-minute difference in VDA injection times, depending on the heliolongitude. We conclude that in many cases VDA-derived injection times include a significant contribution from turbulence and/or the pre-event background and are not an accurate estimate of the acceleration time.