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Measuring pulsar profile variations with 2-D Gaussian process regression

T0 review · 4 major / 6 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read This paper builds a fast 2-D Gaussian-process model of pulsar pulse profiles over time and phase, detects significant shape variations in 21 of 26 pulsars, and reports spin-down-correlated changes in 7 pulsars for the first time.

desk verdict A clever and well-validated GP method for pulsar profile variability, but the seven new spin-down correlations are not all on solid statistical footing as presented. read the letter →

arxiv 2505.23413 v1 pith:43TJ2S26 submitted 2025-05-29 astro-ph.HE astro-ph.SR

Michael J. Keith , Renée Spiewak , Andrew G. Lyne , Patrick Weltevrede , Danai Antonopoulou , Ben Stappers This is my paper · ORCID
classification astro-ph.HEastro-ph.SR
keywords pulsarspulseprofilevariationsGaussianprocessregressionspin-downrateprincipalcomponentanalysismagnetosphericstateswitchingradioCelerite
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

Pulsar radio profiles are usually stable, but a small population shows time-correlated shape changes, a handful of which track the star's spin-down rate. This paper builds a fast two-dimensional Gaussian-process model that treats the pulse profile as a map over time and rotational phase, improving on earlier one-dimensional approaches that ignored phase correlations. Applied to 26 pulsars with long monitoring datasets, the model detects significant profile variations in 21, confirms spin-down-correlated changes in 11 previously known pulsars, and reports evidence for 7 new spin-down-correlated cases. For one pulsar, B0740-28, the correlation between profile shape and spin-down switches between quasi-stable states, and the profile shape associated with greater spin-down sometimes inverts, which complicates simple magnetospheric explanations.

What carries the argument

The central object is the 2-D Gaussian-process kernel $k(\delta x)=\Theta(\delta x)\,\tau(\delta x)$, the outer product of a periodic phase kernel $\Theta$ (approximated as a Fourier series of a Gaussian) and a time kernel $\tau$ (exponential or Matérn-3/2), rendered as a Celerite sum of complex-exponential terms, with Celerite being a fast GP solver whose cost grows linearly with data size. By flattening the pulse stack into a periodic 1-D series, the kernel borrows signal across both phase and observation epoch, recovering few-percent component-intensity variations from noise-dominated individual profiles. Principal-component analysis of the GP reconstruction then produces eigenprofiles and their time series, and the Pearson correlation of these with $\dot{\nu}$ is the paper's main statistic.

What would settle it

Run the pipeline on simulated pulse stacks with synthetic profile variations of the claimed amplitudes (1-2 percent of peak flux) and timescales injected into the real noise, and count how often the model reports significant variations and spin-down correlations; a false-positive rate above the nominal threshold, or Monte Carlo Pearson-$r$ errors wider than the reported $\pm 0.1$ to $\pm 0.2$ values, would show the formal covariance is too optimistic and weaken the new detections.

Watch

Extended reading notes

Core claim

The central claim is that pulse-profile variability can be separated from noise by a two-dimensional Gaussian process whose kernel is the product of a periodic phase kernel and a time kernel, implemented with the fast Celerite solver so the computation remains feasible. Analysing 26 pulsars with this model, the paper finds significant time-correlated profile shape variations in 21 of them. A principal-component analysis of the reconstructed profiles shows that the first principal component time series correlates with spin-down rate $\dot{\nu}$ in 18 pulsars, including 7 first-time detections (B0105+65, B0611+22, B0626+24, B1740-03, B1826-17, B1917+00, and B2148+63). The remaining 5 pulsars show no significant variations. In B0740-28 the correlation is not fixed: sliding-window PCA shows the eigenprofile itself changes with time, at some epochs inverting the shape associated with higher spin-down.

Load-bearing premise

The Gaussian-process reconstruction, including its formal covariance, faithfully represents the intrinsic profile variation; the paper itself cautions (Section 4) that the formal error is likely underestimated and that excess noise is visible on the leading and trailing edges, so unmodelled systematics or underestimated GP variance would inflate the significance of the 21 detections and the uncertainties on the new correlations.

Editorial extensions

If this is right

  • The number of pulsars known to have spin-down-correlated profile changes rises from roughly 11 to 18, making magnetospheric state switching appear to be a common rather than exceptional phenomenon.
  • The 2-D GP plus PCA pipeline can automate profile-variation searches, replacing hand-picked shape parameters and enabling surveys over the full catalogue of hundreds more pulsars.
  • The recurring 'M'-shaped eigenprofile, with shoulders growing relative to the centre, points toward a change in the relative strength of core and conal radio emission as the underlying physical pattern.
  • B0740-28 shows that no fixed shape-versus-spin-down mapping holds for all pulsars: the sign of the coupling can invert on timescales of years, so inferences about torque changes from profile data must allow for state changes.
  • For pulsars whose profile fluctuations are faster than the observing cadence, the measured spin-down rate may correlate with the time-averaged fraction spent in each state rather than with an instantaneous spin-down value, as suggested by B0950+08 and B1822-09.

Reading between the lines

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

  • Beyond this paper, the separable time-phase kernel construction could be applied to other quasi-periodic signals where correlations are smooth in both dimensions, such as frequency-resolved pulsar spectra or mode-switching analyses at other wavelengths.
  • The seven new $\dot{\nu}$ correlations include Pearson $r$ values as low as 0.13 with errors around 0.1-0.2, and the paper itself warns that formal GP errors are likely underestimated; an independent reanalysis with more conservative noise modelling would test whether the weakest new claims survive.
  • If B0740-28's sign inversions are physical, then pulsar emission models that derive magnetospheric geometry from a one-to-one relation between profile shape and spin-down rate need revision; a natural next check is whether polarisation position-angle sweeps change alongside the inversions.
  • A calibration experiment that injects synthetic profile variations of known amplitude and timescale into the same noise would quantify how faint a shape change the 2-D GP can recover, and would put the 21-of-26 detection rate on a firmer statistical footing.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 6 minor

Summary. The paper presents psrcelery, an efficient approximation to a 2-D Gaussian process for pulsar profile stacks, implemented by flattening the phase/time grid into a periodic 1-D coordinate and using Celerite. The GP reconstruction is followed by PCA to extract eigenprofiles and their time series, which are then correlated with independently derived spin-down-rate time series. Applied to 26 Jodrell Bank pulsars, the authors report significant profile variations in 21 pulsars, confirm spin-down-correlated shape changes in 11 previously known cases, and claim 7 new correlations (PSRs B0105+65, B0611+22, B0626+24, B1740-03, B1826-17, B1917+00, B2148+63). The paper also presents a detailed case study of PSR B0740-28, where the shape-spin-down correlation appears to switch sign between quasi-stable states. The manuscript includes open-source code and a data release.

Significance. If the statistical claims survive scrutiny, this is a valuable contribution: it offers a computationally tractable way to model phase- and time-correlated profile variability, and it would roughly double the number of pulsars with evidence for spin-down-correlated shape changes, strengthening the magnetospheric state-switching interpretation. The 11 confirmations of previously reported correlations provide important external validation, and the direct comparisons with the 1-D GP approach (Figs. 28-29) illustrate genuine gains in capturing phase-coherent features. The open-source package and data availability are strengths. However, the headline numbers rest on the GP covariance model being a faithful description of the intrinsic variability and on post-hoc PCA selections, and these foundations need stronger validation before the new detections can be considered secure.

major comments (4)
  1. [§2.1-2.2, Eq. (1) and Eq. (3)] The kernel in Eq. (3) is not a genuine 2-D product kernel once the flattening of Eq. (1) is used. With x = φ/δφ + T/days, the lag is δx = δφ + δT (with δT an integer), so the covariance is Θ(δφ)·τ(δφ + δT). The time kernel is therefore evaluated at a lag that includes the phase separation: points in different phase bins are smoothed with different effective time length scales, and the covariance is not the outer product of a phase-only kernel and a time-only kernel as claimed in §2.2. The admitted excess noise at the leading and trailing profile edges (§4) is the kind of artifact this phase-time entanglement could produce. Because the 3-σ masks in the figures and the 1000-realisation errors on r (§2.4) use the formal covariance of this misspecified model, the significance of the 21 detections and of the new correlations is not established by the current analysis. I would like to see either a proper 2-D product kernel or a simulation study (injecting known profile variations and recovering them) demonstrating that the flattened-kernel approximation does not bias the detections or the reported r uncertainties.
  2. [§4, Table 1] The paper never defines what makes a detection of profile-shape variation 'significant'. The reconstructed maps are shown with 3-σ masks based on the formal GP uncertainty, but §4 cautions that 'this formal error is likely underestimated'. Without a quantitative detection criterion, such as a likelihood-ratio test against a constant-profile model or a cross-validated prediction score, the statement that 21 of 26 pulsars show significant variations is not falsifiable from the text. Please add a defined threshold and report which pulsars pass it, ideally using a null-hypothesis test that does not rely solely on the GP's own covariance.
  3. [§2.4 and Table 1] The Pearson r uncertainties are computed from realisations of the maximum-likelihood GP model, so they inherit any model misspecification and do not include systematic errors from template alignment, scaling, backend changes, or interstellar scintillation. Several of the newly claimed correlations are individually consistent with zero at the quoted level: B0626+24 has r=0.13±0.14, B1740-03 has r=0.14±0.10, and B2148+63 has r=0.37±0.18 and is described in §4.1.11 as 'barely significant' with 'firm conclusions hard to draw'. In addition, B1917+00 has r=0.73±0.19, but §4.2.9 notes that this is 'surprisingly strong relative to the impression of the plot' and is partly driven by two early observations, with r≈0.4 after excluding them. At minimum these three or four sources should be reclassified as candidates, or the uncertainties should be propagated through an explicit model of systematic errors.
  4. [Table 1 footnote and §4.1.2, §4.1.8, §4.2.3] The eigenprofile used for the correlation is selected post hoc: PSR B1826−17 uses the fourth eigenprofile, and PSRs B0611+22, B0919+06 and J2043+2740 use the sum of the first two eigenprofiles. These choices are made after inspecting the correlation with ν-dot, so the reported r values are maxima over a set of candidate shape parameters. The 1000-realisation error in §2.4 does not account for this selection. Please report the correlations for all eigenprofiles, apply a multiple-comparison correction, or define the shape parameter independently of the ν-dot comparison.
minor comments (6)
  1. [§2.4] 'Principle component analysis' should be 'principal component analysis'; the same typo appears elsewhere in the text.
  2. [§4.1.1] 'Eignenvalues' should be 'eigenvalues'.
  3. [§4.2.3] 'Busrts' should be 'bursts'.
  4. [Figure 4 caption and §4.2.7] There are pulsar-name inconsistencies: Figure 4 is captioned 'PSR B1740−28' where the text and Table 1 refer to B0740−28, and §4.2.7 refers to 'PSR B1824+14' where the heading and Table 1 give B1842+14.
  5. [Eq. (10)] The Fourier coefficient θ_n = √(2πσ²) exp(−2πσ² n²) appears to lack a factor of π in the exponent for the standard Fourier transform of a Gaussian under ω0=2π; please verify the coefficient convention or state the normalization used.
  6. [Figure 3 caption] The caption says the colour map fades to white where the uncertainty 'drops below 3-σ'; this seems likely to be 'exceeds 3-σ' and should be clarified.

Circularity Check

2 steps flagged · score 5.0 of 10

Post-hoc eigenprofile selection and self-referential GP error bars make the claimed new spin-down correlations partly selected.

  1. fitted input called prediction [Section 4.1.8 (PSR B1826−17) and Table 1 footnote]
    "However, it is notable that the fourth eigenprofile, which represents only a very subtle change in the leading shoulder and trailing peak, correlates very well with the ν̇ time series. ... The first eigenprofile is used for the correlation, except for those marked with *, specifically PSR B1826−17 uses the fourth eigenprofile and B0611+22, B0919+06 and J2043+2740 use the sum of the first two eigenprofiles."

    The eigenprofile (or linear combination of eigenprofiles) is selected after examining which component best correlates with ν̇ on the full dataset, and the same correlation is then reported as evidence of a spin-down–shape link. Because the choice is data-dependent and no correction is made for the number of eigenprofiles tested, the reported r values for B1826−17 (fourth eigenprofile) and B0611+22 (sum of first two) are the outcome of a search, not a pre-specified prediction. The 'new correlations' are therefore statistically forced by the selection.

  2. self definitional [Section 4 and Section 2.4]
    "The formal error in the GP reconstruction is used as a transparency mask on the figures such that colours reach full intensity where the GP reconstruction is more than 3-σ from zero. However, we caution that this formal error is likely underestimated, and excess noise can be seen in the leading and trailing edges of the profile. ... To account for this, we estimate the error on r by drawing 1000 realisations of the maximum likelihood GP model and computing the variance of the resulting correlation coefficients of these with ν̇."

    The statistical significance of the detected profile variations and the uncertainties on the correlations are computed from the GP model's own fitted covariance. A 'detection' is judged by whether the GP reconstruction exceeds 3σ of the GP's own formal error, and the r error is the spread of correlations of realisations drawn from the same maximum-likelihood GP. This is a self-referential validation: the model's fitted covariance certifies the model's output. The paper concedes the formal error is likely underestimated, so the quoted significances are not independent evidence.

full rationale

The paper's core cross-correlation is not circular in origin: the ν̇ time series is produced by an independent timing fit (Keith & Niţu 2023), and the GP is fit to the pulse profiles only, so the correlation is not forced by construction in the timing domain. However, the presentation of new spin-down correlations is undermined by post-hoc eigenprofile selection, and the statistical significance of both the detections and the correlations is computed from the GP model's own fitted covariance, which the paper itself flags as likely underestimated. These issues make several of the headline claims partly selected rather than independently predicted, though the underlying methodology, the confirmation of previously known correlations, and the independent ν̇ measurements retain substantial independent content.

Assumptions & free parameters 6 free parameters · 5 assumptions · 0 invented entities

The central claims rest on fitted GP hyperparameters and several domain assumptions, most notably that the flattened 1-D kernel approximates a 2-D model and that the GP formal errors are trustworthy. No new physical entities are introduced.

free parameters (6)
  • GP time-kernel amplitude A = per pulsar, maximum likelihood
    Amplitude of the Matérn-3/2 time covariance in psrcelery; fitted to each pulsar's profile stack.
  • GP time-kernel length scale λ = per pulsar, maximum likelihood
    Correlation timescale of profile variations; fitted, controls smoothing in time.
  • Phase-kernel Gaussian width σ = per pulsar, maximum likelihood
    Width of phase correlation in the Fourier-approximated Gaussian kernel; fitted.
  • Additive white noise (jitter) amplitude = per pulsar, maximum likelihood
    Additional white noise term added to the GP diagonal to account for pulse-to-pulse variability; fitted.
  • Number of Fourier terms N = chosen per pulsar
    Truncation order for the periodic phase kernel; trade-off between approximation quality and computational cost, chosen by hand.
  • Eigenprofile choice for correlation = first, fourth, or sum of first two depending on pulsar
    The principal component used as the shape parameter is selected after inspecting correlations with ν-dot for some pulsars (B1826-17 uses e3; B0611+22, B0919+06 and J2043+2740 use e0+e1).
assumptions (5)
  • domain assumption The flattened 1-D kernel k(δx)=Θ(δx)τ(δx) adequately represents the 2-D covariance of the pulse profile stack.
    Section 2.2 introduces the kernel as the product of a periodic phase kernel and a time kernel on the scalar distance x; this is an approximation to a true 2-D separable GP and is not proven, only asserted to be sufficient in practice.
  • domain assumption Pulse profile variations in the on-pulse window are a stationary Gaussian process; off-pulse data are excluded because they are not.
    Section 2.1 explicitly discards off-pulse data and assumes stationarity within the on-pulse window.
  • domain assumption The GP formal covariance, used for error bars and significance masks, is a reliable estimate of the uncertainty.
    Section 2.4 uses 1000 realisations of the maximum-likelihood GP model to estimate errors on r, but Section 4 admits the formal error is likely underestimated.
  • domain assumption The timing-model GP for spin noise (Keith & Niţu 2023) provides an unbiased ν-dot time series.
    Section 3.1 derives ν-dot from a simultaneous timing plus Fourier-domain GP fit; the paper relies on that earlier method without independent verification here.
  • domain assumption Template alignment and flux scaling do not introduce time-correlated artifacts that masquerade as profile shape variations.
    Section 3 describes alignment and scaling by cross-matching with a noise-free template; no null test for systematics is reported.

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Cite this review

Pith. "Pith review of Measuring pulsar profile variations with 2-D Gaussian process regression." pith.science (2026). https://pith.science/paper/43TJ2S26

@misc{pith2026250523413,
  author       = {Pith},
  title        = {Pith review of: Measuring pulsar profile variations with 2-D Gaussian process regression},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/43TJ2S26}},
  note         = {Machine review of arXiv:2505.23413}
}
read the original abstract

Time-correlated variations in the pulse profiles of radio pulsars provide insights into changes in their magnetospheres. For a small number of pulsars (~20), these variations have been shown to correlate with spin-down rate. Many of these profile changes involve small (few percent) variations in the relative intensity of different profile components, and hence tools such as Gaussian process regression have been employed to separate the time-correlated profile variation from intrinsic noise. In this paper, we present a computationally efficient approximation of a 2-D Gaussian process model that enhances sensitivity by simultaneously tracking time- and phase-correlated signals. Applying this model to 26 pulsars observed at the Jodrell Bank Observatory, we detect significant profile shape variations in 21 pulsars. Using principal component analysis, we confirm spin-down correlated shape variations in 11 pulsars where this had been previously observed. Additionally, we find evidence of spin-down correlated shape changes in 7 pulsars for the first time (PSRs B0105+65, B0611+22, B0626+24, B1740-03, B1826-17, B1917+00, and B2148+63). We look in greater detail at PSR B0740-28, where the correlation between profile shape and spin-down itself seems to switch between quasi-stable states. Notably the profile shape associated with greater spin-down seems to invert at times, presenting a challenge to our understanding of the physical processes at work.

Figures

Figures reproduced from arXiv: 2505.23413 by the authors.

Figure 1
Figure 1. Left: The arrangement of three pulse profiles in a ‘pulse stack’ in terms of phase, 𝜙, and time, 𝑇. Right: The arrangement of pulse profile data after converting to our new combined coordinate, 𝑥 (𝜙, 𝑇) (See Equation 1 for details. . first derive the kernel for general 𝜔0. The complete covariance kernel is the product 𝑘 (𝛿𝑥) = Θ(𝛿𝑥)𝜏(𝛿𝑥). An example of such a kernel can be seen in the top panel of [PITH_FULL_IMAGE:… view at source ↗
Figure 2
Figure 2. An example of the simple kernel. Left, the 1-D kernel as a function of 𝛿 𝑥. Right: The kernel ‘folded’ into 2-D. the Celerite coefficients are: 𝑎𝑛 = 2𝐴 √︁ 2𝜋𝜎2𝑒 −2𝜋 𝜎2𝑛 2 , 𝑏𝑛 = 0, 𝑐𝑛 = −1/𝜆, 𝑑𝑛 = 2𝜋𝑛, (10) for 1 ≤ 𝑛 ≤ 𝑁, and a purely real-valued term, 𝑎𝑁+1 = 𝐴 √︁ 2𝜋𝜎2𝑒 −2𝜋 𝜎2𝑛 2 , 𝑐𝑁+1 = −1/𝜆. (11) An example of this kernel is shown in [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 4
Figure 4. Upper panel: The first three eigenprofiles for PSR B1740−28. Middle panel: Correlation of the first three eigenprofiles (square, star and cross markers) of PSR B1740−28 with 𝜈¤. Correlations were computed over sliding 400-day windows, stepping by 200 days each time. Vertical dashed lines indicate the different regions analysed in [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figures from the paper (17 more)
Figure 3
Figure 3. Figure 3: The output of psrcelery for PSR B0740−28. The central panel shows the GP reconstructed deviation from the median profile as a function of MJD and phase. Short ticks mark the epoch of each observation. The colour map is adjusted in each row to fade to white where the un…
Figure 5
Figure 5. Figure 5: PCA analysis performed on short segments of the Gaussian process model for PSR B0740−28. Upper three rows show AFB data and lower three rows show DFB data. The upper row shows the median profile within each segment, and the first eigenprofile from the PCA analysis. The…
Figure 6
Figure 6. Figure 6: The output of psrcelery for PSR B0919+06. As [PITH_FULL_IMAGE:figures/full_fig_p008_6.png]
Figure 7
Figure 7. Figure 7: The output of psrcelery for combined AFB and DFB datasets for the main pulse of PSR B0950+08. Otherwise as [PITH_FULL_IMAGE:figures/full_fig_p009_7.png]
Figure 8
Figure 8. Figure 8: The 𝜈¤ and PCTS of the first eigenprofile (𝜆0) for PSR B0950+08. 𝜈¤ is shown as the black dashed line. The PCTS shown as points is computed directly at each observation epoch to more clearly show the rapid variability and the solid line shows the running mean over 200 …
Figure 10
Figure 10. Figure 10: The output of psrcelery for PSR B1642−03. Otherwise as Fig￾ure 3. B1822−09 is relatively stable for long periods, with sudden short in￾creases, three of which are seen in [PITH_FULL_IMAGE:figures/full_fig_p010_10.png]
Figure 12
Figure 12. Figure 12: The output of psrcelery for PSR B1822−09 (Main pulse only). Otherwise as [PITH_FULL_IMAGE:figures/full_fig_p011_12.png]
Figure 13
Figure 13. Figure 13: The 𝜈¤ and PCTS of the first eigenprofile (𝜆0) timeseries for PSR B1822−09 between MJD 55000 and MJD 56000. 𝜈¤ is shown as the black dashed line. The PCTS shown as points is computed directly at each observation epoch to more clearly show the rapid variability and the…
Figure 15
Figure 15. Figure 15: The output of psrcelery for PSR B1828−11. Otherwise as Fig￾ure 3. cycles. Nevertheless, the profile shape variations still seem to track the 𝜈¤ well over this period. 4.1.10 PSR B2035+36 PSR B2035+36 showed a large step-change in 𝜈¤ around MJD 53000, from −1.17 × 10−1…
Figure 17
Figure 17. Figure 17: The output of psrcelery for PSR B2148+63. Otherwise as Fig￾ure 3. shape and the 𝜈¤, but firm conclusions are hard to draw given the low significance of the profile shape fluctuations. 4.1.12 PSR J2043+2740 Both L10 and S22 report long-term profile changes in PSR J2043…
Figure 19
Figure 19. Figure 19: The output of psrcelery for the main pulse of PSR B0105+65. Otherwise as [PITH_FULL_IMAGE:figures/full_fig_p014_19.png]
Figure 21
Figure 21. Figure 21: The output of psrcelery for PSR B0611+22. Otherwise as Fig￾ure 3, except an additional leftmost panel shows the correlation of the PCTS computed from the sum of the first two eigenprofiles. 4.2.3 PSR B0611+22 PSR B0611+22 has been shown to have quasi-periodic variatio…
Figure 23
Figure 23. Figure 23: The output of psrcelery for PSR B1740−03. This includes both AFB and DFB data. Otherwise as [PITH_FULL_IMAGE:figures/full_fig_p016_23.png]
Figure 25
Figure 25. Figure 25: The output of psrcelery for PSR B1842+14. Otherwise as Fig￾ure 3. a correlation analysis. B24 find that increasing 𝜈¤ is associated with a strengthening of the leading component, and we also find this during the period MJD 56000 to 57000, suggesting that the correlati…
Figure 28
Figure 28. Figure 28: Comparison of the input dataset (interpolated), 1-D GP analysis and 2-D GP analysis for PSR B1642−03. 0 10 Phase (deg) 1D GP 0 10 Phase (deg) 56000 57000 58000 59000 60000 MJD 2D GP 0 10 Phase (deg) 56000 57000 58000 59000 60000 MJD Input [PITH_FULL_IMAGE:figures/ful…
Figure 27
Figure 27. Figure 27: The output of psrcelery for DFB observations of PSR B1917+00. Otherwise as [PITH_FULL_IMAGE:figures/full_fig_p018_27.png]
Figure 29
Figure 29. Figure 29: Comparison of the input dataset (interpolated), 1-D GP analysis and 2-D GP analysis for PSR B1842+14. MNRAS 000, 1–20 (2024) [PITH_FULL_IMAGE:figures/full_fig_p018_29.png]

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.