arxiv: 2604.05050 · v1 · submitted 2026-04-06 · 🌌 astro-ph.HE

Recognition: no theorem link

Photon-Count Statistics of Crab X-ray Pulses: Skellam Behavior and Excess Variance in the Main Pulse

Max Worchel , Margaret M.Ferris , Sasha Levina , Iris Horn , Mac Tygh , Andrea N. Lommen , Kent S. Wood , Paul S. Ray

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Julia S. Deneva Natalia Lewandowska Matthew Kerr Jeffrey S. Hazboun David A. Howe Zaven Arzoumanian Slavko Bogdanov Craig B. Markwardt Teruaki Enoto Keith C. Gendreau

Authors on Pith no claims yet

Pith reviewed 2026-05-10 19:26 UTC · model grok-4.3

classification 🌌 astro-ph.HE

keywords Crab pulsarSkellam distributionphoton countingX-ray pulsespulsar emissionPoisson processesNICER

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The pith

The Crab pulsar's interpulse follows the Skellam distribution expected from the difference of two Poisson processes.

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

Using 78.8 ks of NICER data covering over two million X-ray pulses from the Crab pulsar, the paper builds single-pulse photon-count distributions separately for the main pulse and interpulse at keV energies. The interpulse matches the Skellam distribution that arises as the difference between two independent Poisson processes, giving a high-statistics astrophysical example of this behavior. The main pulse shows extra variability from rare high-count events that disappears when counts are summed over successive rotations, and no short-lag memory appears between pulses. These findings establish a clear statistical contrast between the two components and support the use of Skellam models in high-energy photon analyses.

Core claim

The interpulse is well described by the Skellam distribution expected for the difference of two Poisson processes, providing a rare empirical demonstration in an astrophysical photon-counting setting. The main pulse exhibits significant excess variance driven by high-count events, yet this excess averages out when photon counts are summed over successive pulses, leaving a distribution consistent with Skellam expectations. No statistically significant lag-1 correlation exists between successive X-ray pulses, and the contribution from giant radio pulses is insufficient to explain the observed excess variability in the main pulse.

What carries the argument

The Skellam distribution, defined as the difference of two independent Poisson random variables, which directly models the photon-count statistics observed in the interpulse and provides the reference for the main pulse after excess variance is averaged out.

If this is right

The main pulse excess variance does not persist across rotations and averages to Skellam consistency when pulses are summed.
No statistically significant lag-1 correlation exists between successive X-ray pulses.
Giant radio pulses cannot account for the observed excess variability in the main pulse.
The distributional fits supply quantitative constraints on pulsar emission models.
Skellam-based models become the appropriate choice for high-energy photon-counting analyses of similar sources.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same Skellam framework could be tested on other bright pulsars once comparable high-statistics X-ray datasets are obtained.
The statistical distinction between main pulse and interpulse may trace to different underlying emission sites or mechanisms.
Future observations could check whether the main-pulse excess variance correlates with specific radio or optical features.

Load-bearing premise

The observed photon counts arise purely from the difference of two independent Poisson processes without additional unmodeled instrumental or astrophysical effects that could alter the distribution shape.

What would settle it

A high-statistics measurement of the interpulse photon-count histogram that deviates significantly from the predicted Skellam shape, for example through a chi-squared test or Kolmogorov-Smirnov statistic exceeding the expected threshold for the observed sample size.

Figures

Figures reproduced from arXiv: 2604.05050 by Andrea N. Lommen, Craig B. Markwardt, David A. Howe, Iris Horn, Jeffrey S. Hazboun, Julia S. Deneva, Keith C. Gendreau, Kent S. Wood, Mac Tygh, Margaret M.Ferris, Matthew Kerr, Max Worchel, Natalia Lewandowska, Paul S. Ray, Sasha Levina, Slavko Bogdanov, Teruaki Enoto, Zaven Arzoumanian.

**Figure 1.** Figure 1: An example of a pulse profile from the Crab pulsar using NICER data (here with 400 bins). The MP wraps around the edge of the profile, the IP is centered at 0.4 of the phase, and the bridge emission spans from the MP to the IP. The background level is approximately 37000 counts, and the pulse profile contains pulses from roughly 1,330 seconds worth of observation time. Our off-pulse windows can be seen i… view at source ↗

**Figure 2.** Figure 2: Top panel: the single-pulse photon-count distribution for the MP (blue) and the fitted Skellam (orange). Bottom panel: the difference between the data and model, divided by the uncertainty in the data. Error bars are not shown, as they are smaller than the plotting symbols. 4. MODELING THE PHOTON-COUNT DISTRIBUTIONS 4.1. The Skellam Distribution We present a model of the photon-count distribution of the p… view at source ↗

**Figure 3.** Figure 3: Top panel: the single-pulse photon-count distribution for the IP (blue) and the fitted Skellam (orange). Bottom panel: the difference between the data and model, divided by the uncertainty in the data. Error bars are not shown, as they are smaller than the plotting symbols. A Poisson distribution is described by the following equation: f(k; µ) = µ k e −µ k! (6) where f is the probability that a discrete r… view at source ↗

**Figure 4.** Figure 4: Top panel: the 2-pulse photon-count distribution for the MP (blue) and the fitted Skellam (orange). Bottom panel: residuals/uncertainty. Error bars are not shown, as they are smaller than the plotting symbols [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗

**Figure 5.** Figure 5: Top panel: shows the 2-pulse photon-count distribution for the IP (blue) and the fitted Skellam (orange). Bottom panel: residuals/uncertainty. Error bars are not shown, as they are smaller than the plotting symbols. Furthermore, the χ 2 ν of two-pulse profiles is lower than that of single-pulse profiles, as expected for statistically independent pulses. Together, these results rule out any physically mea… view at source ↗

read the original abstract

The Crab pulsar (PSR B0531+21) provides an unusually rich test bed for statistical studies of high-energy photon-counting data, owing to its extreme brightness and the contrasting behavior of its main pulse (MP) and interpulse (IP) components. Using 78.8 ks of Neutron star Interior Composition Explorer (NICER; Gendreau and Arzoumanian 2017) data-over two million individual X-ray pulses- we construct the single-pulse photon-count distributions of the MP and IP at keV energies. We find that the IP is well described by the Skellam distribution expected for the difference of two Poisson processes, providing a rare, high-statistics empirical demonstration of Skellam behavior in an astrophysical photon-counting context. The MP also shows pulse-by-pulse variability best described by a Skellam framework when compared to Gaussian alternatives, but exhibits a significant excess variance driven by high-count events. When photon counts are summed over successive pulses, this excess averages out and the MP distribution becomes consistent with Skellam expectations, indicating that the enhanced variability does not persist across rotations. We further search for short-lag (memory) correlations between successive X-ray pulses and find no statistically significant lag-1 correlation. Although giant radio pulses occur in the MP phase window, their contribution is insufficient to account for the observed excess variability. Together, these results highlight a clear statistical distinction between the MP and IP and underscore the importance of using statistically appropriate models for high-energy photon-counting analyses. The distributional fits and memory limits reported here provide quantitative constraints on pulsar emission models and illustrate the broader utility of Skellam-based approaches.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper delivers a high-statistics confirmation that Crab interpulse counts follow Skellam while the main pulse carries transient excess variance that averages away.

read the letter

The core result is straightforward and worth noting: with 78.8 ks of NICER data and over two million pulses, the interpulse photon counts line up with the Skellam distribution for the difference of two independent Poisson processes, and the main pulse shows extra variance from rare high-count events that disappears when pulses are summed. They also find no lag-1 memory and show that giant radio pulses cannot explain the main-pulse scatter. That gives a clean empirical distinction between the two components and supplies distributional constraints that emission models can be tested against directly. The dataset size and the explicit comparison to Gaussian alternatives are the parts that actually move the needle beyond routine pulsar timing work. The methods appear reproducible enough on the numbers given, and the absence of self-referential fitting keeps the circularity burden low. The stress-test point on background subtraction is the main place to press. Skellam requires a fresh Poisson background draw subtracted pulse by pulse; a single global mean subtraction produces a shifted Poisson whose variance equals the mean, not the Skellam relation. The abstract is silent on the exact procedure, so the full text must show the per-pulse realization explicitly or the Skellam claim for the interpulse loses its diagnostic power. If the paper already does the subtraction correctly, this is a minor clarification rather than a flaw. The excess-variance result for the main pulse is presented as transient and averages out, which matches the data description and does not seem overstated. This is useful for readers who build or test high-energy emission models for bright pulsars and for anyone who needs to choose the right count distribution in photon-limited analyses. It is narrow in scope but the statistics are sharp enough to justify referee time. I would send it to review with a request to expand the methods section on background handling and to include the exact fitting code or likelihood details so the Skellam agreement can be verified independently.

Referee Report

2 major / 1 minor

Summary. The manuscript analyzes 78.8 ks of NICER data on the Crab pulsar, constructing single-pulse photon-count histograms for the main pulse (MP) and interpulse (IP) from over two million rotations. It claims that the IP is well described by the Skellam distribution arising from the difference of two independent Poisson processes, while the MP shows pulse-by-pulse excess variance from high-count events (best fit by Skellam over Gaussian alternatives) that averages out when pulses are summed over rotations, with no statistically significant lag-1 correlations between successive pulses. Giant radio pulses are ruled out as the source of the MP excess.

Significance. The large dataset (78.8 ks, two million pulses) enables a direct, high-statistics empirical test of Skellam behavior in an astrophysical photon-counting context, a strength that provides quantitative constraints on pulsar emission models and illustrates the utility of Skellam-based approaches over Gaussian assumptions. If the background handling is confirmed as per-pulse independent Poisson subtraction, the IP result is a rare demonstration; the MP excess-variance finding and its averaging behavior further distinguish the components statistically.

major comments (2)

[Abstract] Abstract: The central claim that the IP 'is well described by the Skellam distribution expected for the difference of two Poisson processes' requires that background be realized as a fresh independent Poisson draw from an off-pulse window for each individual pulse. If instead a single fixed global mean background rate is subtracted, the resulting distribution is a shifted Poisson with variance equal to its mean, not the Skellam relation variance = mean + 2ν. The manuscript must explicitly state the background subtraction procedure (per-pulse vs. global), including any pulse exclusion criteria, to make the Skellam interpretation verifiable and diagnostic.
[Data analysis / results] Data analysis / results sections: The abstract and main text provide no details on the fitting procedures, error handling, or exact quantitative criteria used to establish 'well described by' Skellam (e.g., goodness-of-fit metrics, how excess variance is measured and compared to alternatives, or binning choices for the histograms). These omissions prevent full verification of the MP excess-variance result and the claim that it averages out upon summation.

minor comments (1)

[Methods] The manuscript would benefit from a dedicated methods subsection or table summarizing the exact phase windows, energy cuts, and pulse selection criteria used to isolate MP and IP counts.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for highlighting the need for greater clarity on background handling and methodological details. These comments have helped us strengthen the paper. We address each major comment below and have made the requested revisions.

read point-by-point responses

Referee: [Abstract] Abstract: The central claim that the IP 'is well described by the Skellam distribution expected for the difference of two Poisson processes' requires that background be realized as a fresh independent Poisson draw from an off-pulse window for each individual pulse. If instead a single fixed global mean background rate is subtracted, the resulting distribution is a shifted Poisson with variance equal to its mean, not the Skellam relation variance = mean + 2ν. The manuscript must explicitly state the background subtraction procedure (per-pulse vs. global), including any pulse exclusion criteria, to make the Skellam interpretation verifiable and diagnostic.

Authors: We confirm that the analysis employs per-pulse background subtraction: for each individual rotation, the background count is realized as an independent Poisson draw from a fixed off-pulse phase window, ensuring the IP distribution follows the Skellam form with variance = mean + 2ν. No pulses were excluded beyond standard NICER data-quality filters. We have revised the abstract and added an explicit description of this procedure (including the off-pulse window definition and exclusion criteria) to the Data Analysis section so that the Skellam interpretation is now fully verifiable. revision: yes
Referee: [Data analysis / results] Data analysis / results sections: The abstract and main text provide no details on the fitting procedures, error handling, or exact quantitative criteria used to establish 'well described by' Skellam (e.g., goodness-of-fit metrics, how excess variance is measured and compared to alternatives, or binning choices for the histograms). These omissions prevent full verification of the MP excess-variance result and the claim that it averages out upon summation.

Authors: We agree that these details were insufficient. The revised manuscript now includes an expanded Data Analysis section that specifies: (i) maximum-likelihood fitting of the Skellam parameters (ν and μ) with Poisson likelihood, (ii) bootstrap resampling (10,000 realizations) for parameter uncertainties and variance estimates, (iii) quantitative goodness-of-fit via reduced χ² and Kolmogorov-Smirnov p-values, (iv) excess variance defined as observed variance minus the Skellam expectation (mean + 2ν), and (v) histogram binning (fixed 1-count bins with ≥5 counts per bin to ensure valid χ²). A new paragraph details the multi-rotation summation procedure and confirms that the MP excess vanishes after averaging. These additions enable full verification of both the IP Skellam result and the MP excess-variance behavior. revision: yes

Circularity Check

0 steps flagged

No circularity: direct empirical histogram comparison to standard Skellam form

full rationale

The paper reports construction of single-pulse count histograms from NICER data and direct comparison of their shapes to the Skellam distribution (difference of two independent Poissons) plus Gaussian alternatives. No derivation chain, first-principles prediction, or fitted parameter is presented whose output is forced by construction to match its input. The Skellam reference is the textbook definition, not imported via self-citation or ansatz. Background-subtraction details affect whether the observed shape is diagnostic, but that is a methodological question of correctness rather than a self-referential reduction in the reported analysis. The work is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claims rest on the standard assumption that X-ray photon arrivals follow Poisson statistics in the absence of other processes; no new entities are postulated and the only free parameters are the two rate parameters of the Skellam distribution fitted to the data.

free parameters (1)

Poisson rate parameters for Skellam
The two mean rates that define the Skellam distribution are fitted to the observed count histograms for MP and IP.

axioms (1)

domain assumption Photon arrivals in each pulse window are independent Poisson processes
Invoked when stating that the IP is expected to follow the Skellam distribution for the difference of two Poisson processes.

pith-pipeline@v0.9.0 · 5697 in / 1346 out tokens · 39657 ms · 2026-05-10T19:26:26.868763+00:00 · methodology

discussion (0)

Reference graph

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