Feasibility of ultra-high-energy cosmic ray backtracking through sparse local measurements of the Galactic magnetic field

M. Mastorakis; S. Romanopoulos; V. Pavlidou

REVIEW 3 major objections 5 minor 38 references

Feasibility of ultra-high-energy cosmic ray backtracking through sparse local measurements of the Galactic magnetic field

T0 review · 3 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read Sparse local measurements of the Galactic magnetic field can backtrack ultra-high-energy cosmic rays to a suspected source, until the ordered field is roughly ten times stronger than models predict.

desk verdict A clean closed-loop simulation showing that sparsity tolerance for UHECR backtracking depends strongly on ordered-field strength, but the headline thresholds are optimistic because the ground truth omits turbulence. read the letter →

arxiv 2507.21277 v1 pith:JRMUXMVA submitted 2025-07-28 astro-ph.HE

S. Romanopoulos , M. Mastorakis , V. Pavlidou This is my paper · ORCID

classification astro-ph.HE

keywords ultra-high-energycosmicraysGalacticmagneticfieldcosmic-raybacktrackingcharged-particleastronomyJF12modelM82magnetic-fieldsparsityTelescopeArrayhotspot

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper asks whether the sparse, cloud-by-cloud magnetic-field measurements now becoming available can actually be used to backtrack ultra-high-energy cosmic rays (UHECRs) to their source. Using the JF12 model as a stand-in for the true Galactic magnetic field and mock proton events aimed within 3 degrees of M82, it finds that when the ordered field has its nominal $\sim$1 $\mu$G average strength, backtracking stays accurate even with measurements spaced about 1600 pc apart. But when the ordered field is rescaled upward by a factor of 10, accuracy breaks down already at about 400 pc spacing. The authors conclude that sparsity is not automatically fatal for charged-particle astronomy, while high-strength field regions remain a serious obstacle, and they frame their no-interpolation results as a lower limit on what reconstruction techniques could achieve.

What carries the argument

The argument is carried by a mock-observation pipeline. The ground-truth field is the JF12 model with its ordered component multiplied by a factor $f$; the region toward M82 is divided into cubes of side length $L$, each cube assigned a constant field equal to the model value at its center, with Gaussian noise added to the field strength at the 25% or 50% level. Backtracking integrates the discretized Lorentz-force equation $\hat{v}_{\mathrm{prev}} = \hat{v}_{\mathrm{now}} - (Z e c^2 / E)(\hat{v} \times B)\,\delta t$, using at every step the field of the nearest cube. The control parameter is the slice length $L$—the spacing between measurements—and the rescalings $f=1,3,10$ set the field-strength regime; the output metrics are the mean and standard deviation of the angular distance of the 1000 backtracked events from M82.

What would settle it

Measure the local magnetic field along the M82 line of sight with mean spacing around 1.6 kpc, backtrack the actual Telescope Array hotspot events through that measured field, and compare the mean angular distance and spread to the near-2-degree level predicted for $f=1$; if residual structure on scales below 1.6 kpc produces larger deflections, or if the measured field strength is closer to $f=10$, the predicted tolerance to sparsity fails.

Watch

Extended reading notes

Core claim

The central claim is that the feasibility of UHECR backtracking through sparsely sampled local GMF measurements hinges more on the strength of the ordered Galactic field in the direction of the source than on the density of measurements. For mock proton events at $10^{20}$ eV traced from M82, a uniform grid of magnetic-field measurements with linear spacing up to about 1600 pc yields mean backtracked angular distances from M82 near the control-run level of about 2 degrees when the ordered component of the JF12 field is at its nominal strength (scaling factor $f=1$). Raising the ordered field by a factor of 10 ($f=10$) makes the mean angular distance grow to about 10 degrees once the spacing reaches a few hundred parsecs, with a comparable spread between events; $f=3$ gives intermediate behavior. Because the backtracking uses no interpolation—each step simply adopts the nearest measurement—the paper presents these numbers as a lower bound on achievable accuracy, with reconstruction methods expected to improve on them.

Load-bearing premise

The simulation assumes the smooth JF12 model is the true Galactic magnetic field, so the only field structure it contains is JF12's large-scale ordered component; if the real field toward M82 has strong small-scale structure, sparse measurements spaced about 1600 pc apart would miss it, and backtracking errors could be larger than the paper reports.

Editorial extensions

If this is right

Sparse local GMF measurements with about 1.6 kpc spacing are enough to backtrack $10^{20}$ eV protons to M82 with mean angular errors near the 2-degree control level when the ordered field is at JF12 strength, so sparsity alone does not rule out charged-particle astronomy.
If the ordered field toward M82 is ten times stronger than JF12 predicts, mean backtracked angles reach about 10 degrees at few-hundred-parsec spacings, so source localization in strong-field directions will need dense, accurate measurements.
Measurement error in the field strength is tolerable (up to 50%) when the field is weak, but error tolerance shrinks sharply at high field strength, so the observational priority shifts from quantity to accuracy in strong-field regions.
These results are obtained with no interpolation or field reconstruction—just nearest-neighbor assignment—so they set a lower limit; reconstruction techniques should push the achievable accuracy beyond what is reported.
A successful backtracking erases the linear correlation between the angular distance from M82 and $E^{-1}$ seen in the mock arrival directions, providing a data-driven check of whether the magnetic field model used for backtracking is adequate.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The paper's $f=1$ versus $f=10$ asymmetry suggests a staged strategy that the authors only gesture at: first use sparse local measurements to determine which field-strength regime the Galaxy is in, then decide whether to invest in dense, high-accuracy sampling; that explicit decision rule is our inference.
The relevant tolerance parameter is the path length a particle spends in the Galactic field, so sources at higher Galactic latitude or with shorter in-galaxy path lengths should tolerate sparser sampling than M82; testing this by repeating the experiment for different sky directions is a natural extension.
Because the mock data contain no small-scale ordered structure beyond JF12's parameterization, injecting coherent field patches below the grid scale $L$ would quantify how much tolerable spacing shrinks when the real field is clumpy—a testable extension the paper does not run.
The thresholds are derived for protons; if the M82 events include heavier nuclei, their lower rigidity at fixed energy means the same backtracking accuracy would require denser measurements, so composition uncertainty should widen the quoted error bars.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

You should know two things about this paper before reading it. First, it gives a genuinely new quantitative map: backtracking through a nearest-neighbor-sampled GMF keeps mean angular errors near the ~2-degree floor even at 1600 pc spacing when the ordered field is ~1 µG, but at 10× that strength errors climb to ~10 degrees already at 400 pc. That is exactly the kind of number the PASIPHAE and SOUTH POL communities need when planning survey strategies. Second, the thresholds are computed in a smooth, ordered-only GMF. There is no turbulent component in the ground truth, so the results are optimistic for the real Galaxy, not feasibility statements.

What the paper does well: it isolates sparsity from reconstruction by deliberately avoiding any interpolation and using the nearest measurement at each step. That makes the results a conservative lower bound on what a simple method can achieve. The mock-event generation is careful—energies from a shifted-exponential fit to the Telescope Array hotspot events, events selected only if they backtrack within 3 degrees of M82, 1000 events per field-strength case, and a control run at L=1 pc. The scan over f=1,3,10 and 25/50% strength errors is sensible, and the qualitative conclusion that ordered-field strength matters more than sparsity is likely robust.

The soft spots are real but not fatal. The biggest is the missing turbulence. The real GMF has a random component with coherence scales of tens of parsecs and strength comparable to the ordered field. Sparse measurements at L≥400 pc cannot sample that structure, and the backtracking code therefore cannot correct turbulent deflections. For protons at 40–100 EeV crossing several kpc, turbulent deflections are typically several degrees—more than the reported mean errors. So the reported angular distances are lower bounds, and the "satisfactory at 1600 pc" claim is untested for realistic fields. The paper does note in Section 6 that results are lower limits without sophisticated reconstruction, but the abstract states the thresholds as feasibility statements, which oversells it. Also, "satisfactory" and "breaks down" are never given quantitative definitions, the Monte Carlo averages are shown without error bars (minor with 1000 events, but still), and the abstract's "breaks down" at f=10 is softer in the discussion, which says "more challenging." The all-three-components-sampled assumption is stated and defended, so that one is fine.

Who it is for: anyone planning sparse local GMF surveys or interpreting their data for UHECR backtracking. The qualitative message—that sparsity is not automatically a deal-breaker if the ordered field is modest—is valuable. The specific thresholds need a follow-up with a turbulent field before being used for design decisions. It deserves a serious referee, but a referee who will push for a turbulence-inclusive version or a clearly framed "ordered-field-only" scope. I would read the revised version, but I would not cite these thresholds as physical limits.

Referee Report

3 major / 5 minor

Summary. The paper simulates UHECR backtracking through sparse local measurements of the Galactic magnetic field, using the JF12 ordered-field model as ground truth with its strength rescaled by a factor f = 1, 3, or 10. Mock proton events are generated with energies drawn from a shifted exponential fit to 74 Telescope Array hotspot events and with arrival directions that backtrack to within 3 degrees of M82 under the exact ground-truth field. The authors then backtrack these events through a cubic grid of field measurements of spacing L, using the nearest-neighbor value and adding 0%, 25%, or 50% Gaussian errors on field strength. The main results, shown in Figures 5 and 6, are that mean angular distance from M82 remains small for f=1 and f=3 even at L ~ 1 kpc, while f=10 degrades sharply for L of a few hundred parsecs. The authors conclude that sparse local measurements are not automatically a deal-breaker for backtracking in low-field directions, and that their results should be read as a lower limit on achievable quality.

Significance. If the quantitative thresholds in the abstract are correct, the paper provides a useful, concrete motivation for local GMF surveys such as PASIPHAE and SOUTH POL: even very sparse sampling could support UHECR source localization in regions of modest ordered-field strength, while strong-field regions require denser sampling. The study has several genuine strengths: the nearest-neighbor sampling and the explicit refusal to interpolate make the reported accuracies a conservative lower limit relative to any reconstruction method; the control run at L=1 pc cleanly isolates the residual 3-degree source scatter; and the energy distribution is taken from external Telescope Array data rather than fitted to the simulation output. The closed-loop design is a standard and appropriate test of reconstruction accuracy, not a circular derivation. The significance is, however, limited by the smoothness of the ground truth, as discussed below: the paper does not actually demonstrate feasibility in a field containing the small-scale and turbulent structure known to exist in the real interstellar medium.

major comments (3)

[Section 2, Section 4, Section 5] The quantitative thresholds in the abstract (satisfactory at L ~ 1600 pc for 1 microgauss field; breakdown at L ~ 400 pc for f=10) are derived from a ground truth that contains only the smooth, analytic JF12 ordered component, rescaled by f. The mock event generation in Section 4 adds only a 3-degree scatter meant to represent experimental and intergalactic deflections; no turbulent or small-scale ordered Galactic field is present. In the real Galaxy, the random component of the GMF has coherence scales of order 10-100 pc and strength comparable to the ordered component, and it would be completely unresolved by measurements spaced 400-1600 pc apart. The reported mean angular distances are therefore optimistic lower bounds, and the specific 1600 pc / 400 pc thresholds are not validated against realistic field structure. I request either an explicit turbulence-inclusive simulation (e.g., adding a random component with 10-100 pc coherence to the ground truth) or a substantial rewording of the abstract and conclusions so that the quantitative thresholds are presented as conditional on a smooth ordered field only.
[Section 5, Figures 5 and 6] The central claims of 'satisfactory' backtracking and 'breakdown' are not quantitatively defined, and no statistical uncertainties accompany the Monte Carlo averages. Figure 5 reports the mean angular distance for N=1000 events, but there are no error bars, confidence intervals, or a criterion such as 'the fraction of events within X degrees of M82'. Without such a criterion, the abstract statement that L ~ 1600 pc yields 'satisfactory' results cannot be independently evaluated, and it is unclear whether the difference between f=1 and f=3 curves is significant given finite sample noise. Please define quantitative success metrics (e.g., median angular error, fraction of events within 3 or 5 degrees, barycenter offset) and provide corresponding uncertainties, at least for the headline configurations.
[Section 5, Section 6] The discussion in Section 6 states that sparsity has 'only a modest result' on accuracy for f=1 and f=3, but this is based on visual inspection of Figure 5 rather than any formal comparison. The mean angular distance at L ~ 1600 pc appears to be several times larger than the control value of about 2 degrees, even for f=1 with 25% error. Whether a degradation from ~2 degrees to, say, ~4-5 degrees is 'satisfactory' depends entirely on the scientific goal (source identification versus low-energy counterpart localization), which is never specified. Please tie the success criterion to the intended downstream use and report results against that criterion.

minor comments (5)

[Abstract] The abstract says 'charge-particle astronomy'; this should be 'charged-particle astronomy'.
[Section 5] The text contains the typo 'sparce' in the paragraph after Figure 5; it should be 'sparse'.
[Equation (7)] The phrase 'Glactic coordinates' should be 'Galactic coordinates'.
[Section 2] The assumption that all three magnetic-field components are sampled is stated clearly, but the two justifications given are qualitative. Since the claim that line-of-sight sensitivity is small is central to this assumption, a quantitative estimate (or a reference to the explicit demonstration in Tsouros et al. 2024b) would strengthen the argument.
[Figures 5 and 6] The figure captions do not state the number of Monte Carlo events or the definition of the plotted quantities beyond the axis labels; adding N=1000 and a reference to Equations (9) and (10) in the captions would improve clarity.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the central sparsity result is produced by direct ray-tracing through an external GMF model with openly labeled mock-event construction; the only minor self-citations are not load-bearing.

full rationale

The paper's quantitative claims (Section 5 and abstract) are obtained by a closed-loop simulation: mock UHECR events are generated by backtracking through the JF12 ground-truth field and keeping events whose post-backtracking direction lies within 3 degrees of M82, and mock sparse measurements are obtained by sampling that same field on a grid of spacing L. This is a standard reconstruction-accuracy test, not a derivation that reduces to its input. The paper explicitly labels the round-trip property as 'by design' (Section 4, Figure 3), so the control run recovering the input scatter is not presented as a prediction. The energy spectrum is fitted to external Telescope Array data reported by Abbasi et al. (2020), and the GMF ground truth is the external Jansson & Farrar (2012) model; neither is derived from the paper's own results. The f rescaling is a parameter study motivated by Tritsis et al. (2019), a self-citation, but the sparsity thresholds are computed by direct integration of the Lorentz-force equations (Eqs. 3-4) rather than assumed from that citation. The paper also cites Tsouros et al. (2024b) for the claim that approximate LOS magnetic-field measurements suffice, but it supplies an independent physical argument (small LOS sensitivity for small deflections; Faraday-rotation supplements), so this self-citation is not load-bearing. Section 6 explicitly limits the results to a lower limit on backtracking quality given no interpolation or reconstruction; Section 2 states the ground truth is the smooth ordered JF12 component, and Section 4 states the 3-degree scatter is an input assumption. These are limitations on realism, not circularity. The only reason the score is not 0 is the presence of a few self-citations by overlapping authors (Tritsis et al. 2019; Tsouros et al. 2024b) used for context or auxiliary justification; none of them forces the central sparsity conclusion.

Assumptions & free parameters 4 free parameters · 6 assumptions · 0 invented entities

The central claim rests on the choice of JF12 as ground truth, the uniform-grid sampling with all three field components measured, the nearest-neighbor backtracking rule, and the Monte Carlo construction of mock events. None of these are independently verified; they are assumptions of the study. The fitted energy spectrum comes from external Telescope Array data, not from the authors' own previous results.

free parameters (4)

f (GMF ordered-field rescaling factor) = 1, 3, 10
Hand-chosen to bracket the uncertainty in GMF strength in the M82 direction (Section 2); the central conclusions depend strongly on f.
lambda (exponential energy spectrum rate) = 0.53 EeV^-1
Fitted to the 74 Telescope Array hotspot energies from Abbasi et al. (2020); used to generate mock event energies (Section 4).
k (energy spectrum shift) = 34.96 EeV
Fitted to the same 74 Telescope Array energies; shifts the exponential distribution (Section 4).
Mock event angular scatter radius = 3 degrees
Hand-chosen estimate of combined experimental and extragalactic deflection uncertainty; defines the ground-truth event distribution around M82 (Section 4).

assumptions (6)

domain assumption JF12 is a faithful large-scale representation of the Galactic magnetic field in the M82 direction, apart from the ordered-component rescaling.
Section 2 adopts JF12 as ground truth; if the real field contains small-scale structure absent from JF12, the sparsity thresholds would be optimistic.
domain assumption Cosmic rays are protons, propagate ultrarelativistically with v approximately c, and experience no electric field in interstellar space.
Section 3 derives Eqs. (1)-(4) under these assumptions; composition is fixed to Z=1 throughout.
domain assumption All three components of the magnetic field are measured at each cloud, with uncertainty only in field strength, not direction.
Section 2 states this explicitly and argues the LOS component has small effect; if direction errors are significant, the reported tolerances could change.
domain assumption Backtracking uses the nearest magnetic field measurement at each step, with no interpolation or field reconstruction.
Section 2 and Section 6 describe this choice; it makes the results a conservative lower limit on achievable accuracy.
domain assumption Mock events are generated so that, under the ground-truth field, they backtrack to within 3 degrees of M82, with no energy dependence in this scatter.
Section 4 describes the Monte Carlo acceptance procedure; the fixed scatter radius is an input assumption, not derived from data.
domain assumption The shifted exponential energy distribution of Eq. (8) is a good approximation to the energies of the 74 Telescope Array events.
Section 4 and Figure 1 support this visually, but it is a fitted model used for mock generation, not an independently verified law.

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Cite this review

Pith. "Pith review of Feasibility of ultra-high-energy cosmic ray backtracking through sparse local measurements of the Galactic magnetic field." pith.science (2026). https://pith.science/paper/JRMUXMVA

@misc{pith2026250721277,
  author       = {Pith},
  title        = {Pith review of: Feasibility of ultra-high-energy cosmic ray backtracking through sparse local measurements of the Galactic magnetic field},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JRMUXMVA}},
  note         = {Machine review of arXiv:2507.21277}
}

abstract

Planned and ongoing campaigns for the acquisition of high-quality local measurements of the Galactic magnetic field (GMF) at interstellar cloud locations have generated intense interest in the use of such measurements to accurately backtrack Ultra High-Energy Cosmic Rays (UHECR) through the Milky Way, a crucial aspect of charged-particle astronomy. However, the inherent sparsity of these measurements raises concerns regarding the feasibility of this approach. We assessed the achievable accuracy of UHECR backtracking using mock sparse local GMF data derived from the Jansson & Farrar 2012 (JF12) GMF model and mock UHECR events. We created mock UHECR datasets that trace back within a 3 degree angular range from the galaxy M82 (a hypothesized UHECR source), and we investigated the impact on such backtracking attempts of varying GMF measurement sparsity and of varying GMF strength, which we emulated by rescaling the strength of the ordered components of the JF12 model. We found that: (a) for an average GMF strength of $1\mu G$, satisfactory backtracking results for magnetic rigidities of $10^{20}$ eV can be obtained even with very sparse measurements ($ \sim 1600$ pc); (b) when the average GMF strength is significantly increased ($\sim$ factor of 10) the accuracy of backtracking breaks down at measurement spacings of 400 pc. These findings emphasize on one hand that sparsity is not an automatic deal-breaker for the utility of local GMF measurements in UHECR backtracking. On the other hand, we also confirm that important challenges remain on the path from sparse local GMF measurements to precise charge-particle astronomy, especially in directions of high-strength ordered magnetic fields. This underscores the importance of using all available complementary magnetic field measurements and sophisticated reconstruction techniques to enable accurate backtracking of UHECR.

Figures

Figures reproduced from arXiv: 2507.21277 by the authors.

**Figure 2.** Sky distribution of simulated dataset generated using f = 3 as the amplification factor for ordered component of the GMF, before the backtracking process (as they would appear when detected). The colorscale indicates the event energy. M82 is marked by a red star. Deflections are systematically larger for lower energy events. Due to a strong ordered component of the GMF, events are deflected systematically to one s… view at source ↗

**Figure 3.** Sky distribution of simulated events generated f = 3 as the amplification factor for ordered component of the GMF, after their backtracking through the ground-truth GMF. Symbols as in [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

**Figure 4.** Angular distance from M82 plotted against E −1 for the f = 3 simulated dataset before the backtracking process (as they would appear when detected). A clear linear correlation with energy is apparent. the backtracking, especially as far as the ordered component is concerned. 5. Results We backtracked each of the 3 sets of 1000 simulated events (corresponding to different values of the GMF scaling factor f) through t… view at source ↗

Figures from the paper (2 more)

**Figure 5.** Figure 5: Mean angular position of our simulated events after the backtracking, as a function of the slice length of the cubic grid, L, of magnetic field measurements. 6. Conclusions and Discussion In this work, we have performed an extensive parameter study examining the comb…

**Figure 6.** Figure 6: Standard deviation of the angular position of our simulated events after the backtracking, as a function of the slice length of the cubic grid, L, of magnetic field measurements. (2019), they will establish the regime in which we have to operate (in this work’s termin…

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