pith. machine review for the scientific record. sign in

arxiv: 2604.11621 · v1 · submitted 2026-04-13 · 🌌 astro-ph.HE · astro-ph.SR· nucl-th

Recognition: unknown

Combining the Mass--Radius Posteriors of J0030+0451 Allowing for Unknown Model Systematics

Ryan O'Connor , Chun Huang , Alexander Y. Chen
Authors on Pith no claims yet

Pith reviewed 2026-05-10 15:11 UTC · model grok-4.3

classification 🌌 astro-ph.HE astro-ph.SRnucl-th
keywords PSR J0030+0451neutron star mass-radiusNICERequation of stateBayesian combinationsystematic uncertaintieshotspot modeling
0
0 comments X

The pith

A Bayesian framework combines eight published mass-radius posteriors for PSR J0030+0451 while allowing for unknown model systematics.

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

The paper seeks to overcome inconsistencies among different hotspot-model analyses of the same NICER pulse-profile data for the millisecond pulsar PSR J0030+0451. These inconsistencies have created a practical barrier to using the observations for equation-of-state studies of dense matter. By adapting an existing Bayesian combination method, the authors fold the eight existing posteriors into one conservative posterior that treats the modeling differences as unknown systematics. The resulting constraint can be inserted directly into EoS inference pipelines and is shown to tighten joint constraints when combined with other neutron-star observations.

Core claim

We adapt a robust Bayesian combination framework to the published M–R posteriors of PSR J0030+0451 while allowing for unknown systematic uncertainties that might have led to the apparently divergent results. Using this technique, we combine eight existing M–R posteriors into a single conservative and reproducible posterior that incorporates unknown model systematics across the currently available analyses and is suitable for direct use in EoS studies. The resulting constraint is M = 1.46^{+0.09}_{-0.08} M_⊙, R = 12.69^{+0.64}_{-0.55} km, and C = 0.172^{+0.006}_{-0.007} (68% credible interval).

What carries the argument

Bayesian combination framework that introduces a set of unknown systematic parameters whose prior is chosen without reference to any specific equation of state to capture discrepancies between published posteriors.

If this is right

  • The combined posterior can be used directly in equation-of-state inference without choosing one analysis over the others.
  • Joint analysis with PSR J0437-4715 and GW170817 yields R_{1.4} = 11.98^{+0.58}_{-0.68} km and Lambda_{1.4} = 320^{+216}_{-138}.
  • The method supplies a reproducible way to fold cross-model uncertainty into future neutron-star equation-of-state pipelines.
  • The reported mass, radius, and compactness values at 68% credible interval are the direct output of the combination.

Where Pith is reading between the lines

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

  • The same combination procedure could be applied to other NICER pulsars that have multiple independent analyses.
  • If additional hotspot models are published later, they can be folded into an updated combined posterior using the identical framework.
  • The approach separates analysis-specific modeling choices from the underlying observational data, which may reduce the impact of any single modeling decision on EoS constraints.

Load-bearing premise

The discrepancies between the eight published posteriors can be adequately captured by a single set of unknown systematic parameters whose prior does not depend on the target equation of state.

What would settle it

Apply the same combination procedure to simulated posteriors generated from a known true mass-radius value with injected model differences; if the output posterior fails to contain the true value at the stated credible-interval rate, the method is miscalibrated.

Figures

Figures reproduced from arXiv: 2604.11621 by Alexander Y. Chen, Chun Huang, Ryan O'Connor.

Figure 1
Figure 1. Figure 1: KDE of the compactness combined posterior for PSR J0030+0451, comparing individual hotspot-model inferences with the robust combined result. The colored curves correspond to the input posteriors derived from Riley et al. (2019), Miller et al. (2019), Vinciguerra et al. (2024), and Kini et al. (2026) illustrating the significant systematic scatter arising from different geometric assumptions. The thick blac… view at source ↗
Figure 2
Figure 2. Figure 2: Mass–radius posteriors for PSR J0030+0451 from individual NICER pulse-profile inferences compared to our Bayesian-combination posterior. In each of the first eight panels, the colored regions show the 68% (solid outline) and 95% (dashed outline) credible contours from a single published hotspot configuration (as labeled), while the gray/black contours show the corresponding 68% (solid black) and 95% (dashe… view at source ↗
Figure 3
Figure 3. Figure 3: Combined posterior distribution for the mass and radius of PSR J0030+0451. The central panel shows the joint two-dimensional posterior inferred by our Bayesian combination framework; the solid contour and dark-gray fill enclose the 68% credible region, while the outer dashed con￾tour marks the 95% credible region. The top and right panels show the marginalized one-dimensional posteriors for radius and mass… view at source ↗
Figure 4
Figure 4. Figure 4: Left: Mass–radius posteriors for GW170817 from the EoS-insensitive analysis, overlaid with the posterior for PSR J0437–4715 and our combined posterior for PSR J0030+0451. Dark (light) shading indicates the 68% (95%) credible region. Right: Posterior distribution of the canonical radius R1.4 obtained by combining our PSR J0030+0451 constraint with PSR J0437–4715; dashed vertical lines mark the 16%, 50%, and… view at source ↗
read the original abstract

The NASA Neutron star Interior Composition Explorer (\emph{NICER}) mission measures the X-ray pulse profiles of select millisecond pulsars and uses sophisticated pulse profile modeling (PPM) techniques to constrain their masses ($M$) and radii ($R$), in order to probe the state of matter in their interiors. One of the most studied pulsars, PSR J0030+0451, has been analyzed by multiple groups using different choices of hotspot models. The different choices of hotspot prescriptions to fit the same observational data led to different $M$--$R$ posteriors that do not completely agree with one another, resulting in a practical bottleneck for dense-matter equation-of-state (EoS) inference. In this paper, we adapt a robust Bayesian combination framework to the published $M$--$R$ posteriors of PSR J0030+0451 while allowing for unknown systematic uncertainties that might have led to the apparently divergent results. Using this technique, we combine eight existing $M$--$R$ posteriors into a single conservative and reproducible posterior that incorporates unknown model systematics across the currently available analyses and is suitable for direct use in EoS studies. The resulting constraint is $M = 1.46^{+0.09}_{-0.08}\,M_\odot$, $R = 12.69^{+0.64}_{-0.55}\,\mathrm{km}$, and compactness $C = 0.172^{+0.006}_{-0.007}$ (68\% credible interval). Incorporating this combined J0030+0451 constraint in an EoS-agnostic joint analysis with PSR~J0437--4715 and GW170817 yields $R_{1.4} = 11.98^{+0.58}_{-0.68}\,\mathrm{km}$ and $\Lambda_{1.4} = 320^{+216}_{-138}$. Our results provide a combined $M$--$R$ constraint for J0030+0451 and a practical framework for incorporating cross-model uncertainty into neutron star EoS inference pipelines.

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.

Referee Report

2 major / 1 minor

Summary. The manuscript adapts a Bayesian combination framework to merge eight published M–R posteriors for PSR J0030+0451 (from independent NICER PPM analyses using different hotspot models), explicitly marginalizing over unknown model systematics via additional parameters with an EoS-independent prior. The resulting combined posterior is reported as M = 1.46^{+0.09}_{-0.08} M_⊙, R = 12.69^{+0.64}_{-0.55} km, C = 0.172^{+0.006}_{-0.007} (68% CI) and is then used in an EoS-agnostic joint analysis with J0437–4715 and GW170817 to obtain R_{1.4} = 11.98^{+0.58}_{-0.68} km and Λ_{1.4} = 320^{+216}_{-138}. The central claim is that this yields a conservative, reproducible constraint suitable for direct EoS inference.

Significance. If validated, the work supplies a practical, reproducible way to handle discrepant M–R posteriors arising from discrete modeling choices in NICER analyses, which is a recurring bottleneck for neutron-star EoS studies. Explicitly parameterizing and marginalizing unknown systematics (rather than ad-hoc selection or simple averaging) is a constructive contribution, and releasing the combined posterior enhances immediate usability by the community.

major comments (2)
  1. [Methods (Bayesian combination)] Methods section on the Bayesian combination framework: no simulation-based calibration is presented in which synthetic pulse-profile data are generated from a known true M–R, refitted with the same eight hotspot models, and the recovered combined posterior checked for coverage, bias, or over-/under-conservatism. This test is load-bearing for the claim that the output posterior is conservative and properly calibrated when the true systematics are unknown.
  2. [Methods (systematic parameterization)] Description of the systematic-parameter prior (likely §3): the framework assumes that discrepancies among the eight input posteriors can be captured by a single set of unknown systematic parameters whose prior is chosen independently of the target EoS. If the actual modeling differences are correlated with the data in ways not spanned by this parameterization, the combined posterior will be miscalibrated; the manuscript provides no robustness checks or alternative prior specifications to quantify this risk.
minor comments (1)
  1. [Abstract] The abstract states the numerical results but does not explicitly note that eight posteriors are being combined until the body; adding this detail to the abstract would improve immediate clarity for readers.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thoughtful and constructive comments. We respond to each major comment below with the strongest honest defense of the manuscript, indicating where revisions have been made.

read point-by-point responses

  1. Referee: [Methods (Bayesian combination)] Methods section on the Bayesian combination framework: no simulation-based calibration is presented in which synthetic pulse-profile data are generated from a known true M–R, refitted with the same eight hotspot models, and the recovered combined posterior checked for coverage, bias, or over-/under-conservatism. This test is load-bearing for the claim that the output posterior is conservative and properly calibrated when the true systematics are unknown.

    Authors: We agree that an end-to-end simulation-based calibration would be the most direct way to verify coverage and calibration properties. However, performing this test requires generating synthetic NICER pulse-profile data under each of the eight distinct hotspot models and re-executing the full PPM analyses, which is computationally prohibitive and depends on proprietary details of the original pipelines and data reduction steps that are not accessible to us. The framework we adapt produces conservative posteriors by design through explicit marginalization over unknown systematics with a deliberately broad, EoS-independent prior; this construction prevents the combined posterior from being narrower than warranted by the input discrepancies. In the revised manuscript we have added a dedicated paragraph in the Methods section that explains this rationale, cites the conservative properties of the prior, and notes the practical barriers to full synthetic calibration as a limitation for future work. revision: partial

  2. Referee: [Methods (systematic parameterization)] Description of the systematic-parameter prior (likely §3): the framework assumes that discrepancies among the eight input posteriors can be captured by a single set of unknown systematic parameters whose prior is chosen independently of the target EoS. If the actual modeling differences are correlated with the data in ways not spanned by this parameterization, the combined posterior will be miscalibrated; the manuscript provides no robustness checks or alternative prior specifications to quantify this risk.

    Authors: The EoS-independent prior on the systematic parameters was chosen precisely to avoid injecting model-dependent bias into the combination. In the revised manuscript we have added an appendix containing explicit robustness checks: we re-ran the combination under alternative prior widths, different hyperparameter values, and a hierarchical variant of the systematic model. These tests show that the resulting M, R, and C posteriors for J0030+0451 shift by amounts well within the quoted 68% credible intervals. We have also expanded the main text to discuss the assumption that the chosen parameterization spans the dominant modeling differences and to note that any residual unmodeled correlations would be buffered by the breadth of the prior. These additions directly quantify the sensitivity the referee correctly identifies. revision: yes

Circularity Check

0 steps flagged

No significant circularity in the combination framework

full rationale

The paper takes eight independent, previously published M-R posteriors for PSR J0030+0451 as direct inputs and applies an adapted Bayesian combination procedure that introduces additional systematic parameters whose prior is chosen independently of the target EoS. The output posterior (M = 1.46^{+0.09}_{-0.08} M_⊙, R = 12.69^{+0.64}_{-0.55} km) is produced by marginalizing over those systematics; it is not equivalent to any input by definition, nor is any fitted parameter inside the paper renamed as a prediction. No self-definitional equations, fitted-input predictions, or load-bearing self-citations that reduce the central claim to an unverified prior result are present in the derivation chain. The method is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The method relies on an existing Bayesian combination framework whose internal hyperparameters for unknown systematics are not detailed in the abstract; these act as free parameters whose priors must be chosen. No new physical entities are postulated.

pith-pipeline@v0.9.0 · 5695 in / 1368 out tokens · 50789 ms · 2026-05-10T15:11:50.730012+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

65 extracted references · 58 canonical work pages · 2 internal anchors

  1. [1]

    P., et al

    Abbott, B. P., et al. 2017, PhRvL, 119, 161101, doi: 10.1103/PhysRevLett.119.161101

    work page doi:10.1103/physrevlett.119.161101 2017

  2. [2]

    , keywords =

    Abbott, B. P., Abbott, R., Abbott, T. D., et al. 2018, PhRvL, 121, 161101, doi: 10.1103/PhysRevLett.121.161101

    work page doi:10.1103/physrevlett.121.161101 2018

  3. [3]

    Generic conditions for stable hybrid stars.Phys

    Alford, M. G., Han, S., & Prakash, M. 2013, Physical Review D, 88, doi: 10.1103/physrevd.88.083013

    work page doi:10.1103/physrevd.88.083013 2013

  4. [4]

    AlGendy, M., & Morsink, S. M. 2014, ApJ, 791, 78, doi: 10.1088/0004-637X/791/2/78

    work page doi:10.1088/0004-637x/791/2/78 2014

  5. [5]

    Beloborodov, A. M. 2002, ApJL, 566, L85, doi: 10.1086/339511

    work page doi:10.1086/339511 2002

  6. [6]

    L., & Peacock, J

    Bernal, J. L., & Peacock, J. A. 2018, Journal of Cosmology and Astroparticle Physics, 2018, 002

    2018

  7. [7]

    V., Watts, A

    Bilous, A. V., Watts, A. L., Harding, A. K., et al. 2019, The Astrophysical Journal Letters, 887, L23, doi: 10.3847/2041-8213/ab53e7

    work page doi:10.3847/2041-8213/ab53e7 2019

  8. [8]

    Bogdanov, S., & Grindlay, J. E. 2009, ApJ, 703, 1557, doi: 10.1088/0004-637X/703/2/1557

    work page doi:10.1088/0004-637x/703/2/1557 2009

  9. [9]

    2021, The Journal of Open Source Software, 6, 3001, doi: 10.21105/joss.03001 22Rackham et al

    Buchner, J. 2021, The Journal of Open Source Software, 6, 3001, doi: 10.21105/joss.03001

    work page doi:10.21105/joss.03001 2021

  10. [10]

    M., Leahy, D., & Campbell, S

    Cadeau, C., Morsink, S. M., Leahy, D., & Campbell, S. S. 2007, ApJ, 654, 458, doi: 10.1086/509103

    work page doi:10.1086/509103 2007

  11. [11]

    Y., Yuan, Y., & Vasilopoulos, G

    Chen, A. Y., Yuan, Y., & Vasilopoulos, G. 2020, The Astrophysical Journal Letters, 893, L38, doi: 10.3847/2041-8213/ab85c5

    work page doi:10.3847/2041-8213/ab85c5 2020

  12. [12]

    Choudhury et al., Astrophys

    Choudhury, D., Salmi, T., Vinciguerra, S., et al. 2024, arXiv e-prints, arXiv:2407.06789, doi: 10.48550/arXiv.2407.06789

    work page doi:10.48550/arxiv.2407.06789 2024

  13. [13]

    L., Dittmann, A

    Choudhury, D., Watts, A. L., Dittmann, A. J., et al. 2024, The Astrophysical Journal, 975, 202, doi: 10.3847/1538-4357/ad7255

    work page doi:10.3847/1538-4357/ad7255 2024

  14. [14]

    2024, Reproduction package for: ‘A NICER View of the Nearest and Brightest Millisecond Pulsar: PSR J0437–4715’, Zenodo, doi: 10.5281/zenodo.13766753

    Choudhury, D., Salmi, T., Vinciguerra, S., et al. 2024, Reproduction package for: ‘A NICER View of the Nearest and Brightest Millisecond Pulsar: PSR J0437–4715’, Zenodo, doi: 10.5281/zenodo.13766753

    work page doi:10.5281/zenodo.13766753 2024

  15. [15]

    (NANOGrav), 2019

    Cromartie, H. T., Fonseca, E., Ransom, S. M., et al. 2019, Nature Astronomy, 4, 72–76, doi: 10.1038/s41550-019-0880-2

    work page doi:10.1038/s41550-019-0880-2 2019

  16. [16]

    B., Pennucci, T., Ransom, S

    Demorest, P. B., Pennucci, T., Ransom, S. M., Roberts, M. S. E., & Hessels, J. W. T. 2010, Nature, 467, 1081–1083, doi: 10.1038/nature09466

    work page doi:10.1038/nature09466 2010

  17. [17]

    J., Miller, M

    Dittmann, A. J., Miller, M. C., Lamb, F. K., et al. 2024, arXiv e-prints, arXiv:2406.14467, doi: 10.48550/arXiv.2406.14467

    work page doi:10.48550/arxiv.2406.14467 2024

  18. [18]

    2017, Nature Astronomy, 1, 895, doi: 10.1038/s41550-017-0301-3

    Gendreau, K., & Arzoumanian, Z. 2017, Nature Astronomy, 1, 895, doi: 10.1038/s41550-017-0301-3

    work page doi:10.1038/s41550-017-0301-3 2017

  19. [19]

    Space Telescopes and Instrumentation 2016: Ultraviolet to Gamma Ray , year = 2016, editor =

    Gendreau, K. C., Arzoumanian, Z., Adkins, P. W., et al. 2016, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 9905, Space Telescopes and Instrumentation 2016: Ultraviolet to Gamma Ray, ed. J.-W. A. den Herder, T. Takahashi, & M. Bautz, 99051H, doi: 10.1117/12.2231304

    work page doi:10.1117/12.2231304 2016

  20. [20]

    Ho, W. C. G., & Lai, D. 2001, MNRAS, 327, 1081, doi: 10.1046/j.1365-8711.2001.04801.x —. 2003, MNRAS, 338, 233, doi: 10.1046/j.1365-8711.2003.06047.x

    work page doi:10.1046/j.1365-8711.2001.04801.x 2001

  21. [21]

    2002, Monthly Notices of the Royal Astronomical Society, 335, 377

    Hobson, M., Bridle, S., & Lahav, O. 2002, Monthly Notices of the Royal Astronomical Society, 335, 377

    2002

  22. [22]

    2025a, Astrophys

    Huang, C. 2025a, Astrophys. J. Lett., 978, L14, doi: 10.3847/2041-8213/ad9f3c —. 2025b, Astrophys. J., 985, 216, doi: 10.3847/1538-4357/add5ef

    work page doi:10.3847/2041-8213/ad9f3c 2041

  23. [23]

    Huang, C., & Chen, A. Y. 2025, Astrophys. J., 991, 90, doi: 10.3847/1538-4357/adf747

    work page doi:10.3847/1538-4357/adf747 2025

  24. [24]

    2025, Astrophys

    Huang, C., & Sourav, S. 2025, Astrophys. J., 983, 17, doi: 10.3847/1538-4357/adbb67

    work page doi:10.3847/1538-4357/adbb67 2025

  25. [25]

    2024, CompactObject: An open-source Python package for full-scope neutron star equation of state inference

    Huang, C., Malik, T., Cartaxo, J., et al. 2024, CompactObject: An open-source Python package for full-scope neutron star equation of state inference. https://arxiv.org/abs/2411.14615 14

    work page arXiv 2024

  26. [26]

    2021, The Astrophysical Journal, 907, 63, doi: 10.3847/1538-4357/abcec0

    Kazanas, D. 2021, The Astrophysical Journal, 907, 63, doi: 10.3847/1538-4357/abcec0

    work page doi:10.3847/1538-4357/abcec0 2021

  27. [27]

    2026, arXiv preprint arXiv:2602.23743

    Kini, Y., Mauviard, L., Salmi, T., et al. 2026, arXiv preprint arXiv:2602.23743

    work page arXiv 2026

  28. [28]

    2026,A NICER View of PSR J0030+0451: Updated Constraints from Six Years of NICER Observations, Zenodo, doi: 10.5281/zenodo.18741941

    Kini, Y., Mauviard, L., Salmi, T., et al. 2026,A NICER View of PSR J0030+0451: Updated Constraints from Six Years of NICER Observations, Zenodo, doi: 10.5281/zenodo.18741941

    work page doi:10.5281/zenodo.18741941 2026

  29. [29]

    2000, Monthly Notices of the Royal Astronomical Society, 315, L45

    Lahav, O., Bridle, S., Hobson, M., Lasenby, A., & Sodre Jr, L. 2000, Monthly Notices of the Royal Astronomical Society, 315, L45

    2000

  30. [30]

    Lattimer, J. M. 2021, Annual Review of Nuclear and Particle Science, 71, 433, doi: 10.1146/annurev-nucl-102419-124827

    work page doi:10.1146/annurev-nucl-102419-124827 2021

  31. [31]

    M., & Prakash, M

    Lattimer, J. M., & Prakash, M. 2001, ApJ, 550, 426, doi: 10.1086/319702 —. 2007, PhR, 442, 109, doi: 10.1016/j.physrep.2007.02.003

    work page doi:10.1086/319702 2001

  32. [32]

    , keywords =

    Lattimer, J. M., & Prakash, M. 2016, Physics Reports, 621, 127–164, doi: 10.1016/j.physrep.2015.12.005

    work page doi:10.1016/j.physrep.2015.12.005 2016

  33. [33]

    H., Miller, M

    Lo, K. H., Miller, M. C., Bhattacharyya, S., & Lamb, F. K. 2013, ApJ, 776, 19, doi: 10.1088/0004-637X/776/1/19

    work page doi:10.1088/0004-637x/776/1/19 2013

  34. [34]

    E., ¨Ozel, F., & Psaltis, D

    Lockhart, W., Gralla, S. E., ¨Ozel, F., & Psaltis, D. 2019, MNRAS, 490, 1774, doi: 10.1093/mnras/stz2524

    work page doi:10.1093/mnras/stz2524 2019

  35. [35]

    N., Kipphorn, R

    Lommen, A. N., Kipphorn, R. A., Nice, D. J., et al. 2006, ApJ, 642, 1012, doi: 10.1086/501067

    work page doi:10.1086/501067 2006

  36. [36]

    C., & Lamb, F

    Miller, M. C., & Lamb, F. K. 2015, ApJ, 808, 31, doi: 10.1088/0004-637X/808/1/31 —. 2016, European Physical Journal A, 52, 63, doi: 10.1140/epja/i2016-16063-8

    work page doi:10.1088/0004-637x/808/1/31 2015

  37. [37]

    C., Lamb, F

    Miller, M. C., Lamb, F. K., Dittmann, A., et al. 2019, The Astrophysical Journal Letters, 887, L24

    2019

  38. [38]

    C., Lamb, F

    Miller, M. C., Lamb, F. K., Dittmann, A. J., et al. 2019, NICER PSR J0030+0451 Illinois-Maryland MCMC Samples, 1.0.0, Zenodo, doi: 10.5281/zenodo.3473466 —. 2021, The Astrophysical Journal Letters, 918, L28, doi: 10.3847/2041-8213/ac089b Monta˜ na, G., Tol´ os, L., Hanauske, M., & Rezzolla, L. 2019, Physical Review D, 99, doi: 10.1103/physrevd.99.103009

    work page doi:10.5281/zenodo.3473466 2019

  39. [39]

    M., Leahy, D

    Morsink, S. M., Leahy, D. A., Cadeau, C., & Braga, J. 2007, ApJ, 663, 1244, doi: 10.1086/518648 N¨ attil¨ a, J., & Pihajoki, P. 2018, A&A, 615, A50, doi: 10.1051/0004-6361/201630261

    work page doi:10.1086/518648 2007

  40. [40]

    2017, Rev

    Oertel, M., Hempel, M., Kl¨ ahn, T., & Typel, S. 2017, Reviews of Modern Physics, 89, 015007, doi: 10.1103/RevModPhys.89.015007 ¨Ozel, F., & Freire, P. 2016, ARA&A, 54, 401, doi: 10.1146/annurev-astro-081915-023322

    work page doi:10.1103/revmodphys.89.015007 2017

  41. [41]

    R., Ftaclas, C., & Cohen, J

    Pechenick, K. R., Ftaclas, C., & Cohen, J. M. 1983, ApJ, 274, 846, doi: 10.1086/161498

    work page doi:10.1086/161498 1983

  42. [42]

    2025, Monthly Notices of the Royal Astronomical Society, 539, 2608

    Phillips, A., & Kochanek, C. 2025, Monthly Notices of the Royal Astronomical Society, 539, 2608

    2025

  43. [43]

    Accurate analytic formula for light bending in Schwarzschild metric.Astron

    Poutanen, J. 2020, A&A, 640, A24, doi: 10.1051/0004-6361/202037471

    work page doi:10.1051/0004-6361/202037471 2020

  44. [44]

    Poutanen, J., & Beloborodov, A. M. 2006, MNRAS, 373, 836, doi: 10.1111/j.1365-2966.2006.11088.x

    work page doi:10.1111/j.1365-2966.2006.11088.x 2006

  45. [45]

    Press, W. H. 1996, arXiv preprint astro-ph/9604126

    work page internal anchor Pith review arXiv 1996

  46. [46]

    P., et al

    Prigozhin, G., Gendreau, K., Doty, J. P., et al. 2016, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 9905, Space Telescopes and Instrumentation 2016: Ultraviolet to Gamma Ray, ed. J.-W. A. den Herder, T. Takahashi, & M. Bautz, 99051I, doi: 10.1117/12.2231718

    work page doi:10.1117/12.2231718 2016

  47. [47]

    K., Hebeler, K., Hinderer, T., et al

    Raaijmakers, G., Greif, S. K., Hebeler, K., Hinderer, T., et al. 2021, The Astrophysical Journal Letters, 918, L29, doi: 10.3847/2041-8213/ac089a

    work page doi:10.3847/2041-8213/ac089a 2021

  48. [48]

    E., Watts, A

    Raaijmakers, G., Riley, T. E., Watts, A. L., et al. 2019, The Astrophysical Journal Letters, 887, L22, doi: 10.3847/2041-8213/ab451a

    work page doi:10.3847/2041-8213/ab451a 2019

  49. [49]

    J., Bailes, M., Shannon, R

    Reardon, D. J., Bailes, M., Shannon, R. M., et al. 2024, The Astrophysical Journal Letters, 971, L18, doi: 10.3847/2041-8213/ad614a

    work page doi:10.3847/2041-8213/ad614a 2024

  50. [50]

    E., Watts, A

    Riley, T. E., Watts, A. L., Bogdanov, S., et al. 2019, The Astrophysical Journal Letters, 887, L21

    2019

  51. [51]

    E., Watts, A

    Riley, T. E., Watts, A. L., Bogdanov, S., et al. 2019,A NICER View of PSR J0030+0451: Nested Samples for Millisecond Pulsar Parameter Estimation, Zenodo, doi: 10.5281/zenodo.3386449

    work page doi:10.5281/zenodo.3386449 2019

  52. [52]

    E., Watts, A

    Riley, T. E., Watts, A. L., Ray, P. S., et al. 2021, The Astrophysical Journal Letters, 918, L27, doi: 10.3847/2041-8213/ac0a81

    work page doi:10.3847/2041-8213/ac0a81 2021

  53. [53]

    2024, ApJL, 971, L19, doi: 10.3847/2041-8213/ad5f02

    Rutherford, N., Mendes, M., Svensson, I., et al. 2024, ApJL, 971, L19, doi: 10.3847/2041-8213/ad5f02

    work page doi:10.3847/2041-8213/ad5f02 2024

  54. [54]

    Salmi et al., Astrophys

    Salmi, T., Choudhury, D., Kini, Y., et al. 2024, arXiv e-prints, arXiv:2406.14466, doi: 10.48550/arXiv.2406.14466

    work page doi:10.48550/arxiv.2406.14466 2024

  55. [55]

    1986, Density Estimation for Statistics and Data Analysis, Chapman & Hall/CRC Monographs on Statistics & Applied Probability (Taylor & Francis)

    Silverman, B. 1986, Density Estimation for Statistics and Data Analysis, Chapman & Hall/CRC Monographs on Statistics & Applied Probability (Taylor & Francis). https://books.google.com/books?id=e-xsrjsL7WkC

    1986

  56. [56]

    2023, Physical Review C, 107, doi: 10.1103/physrevc.107.025801

    Somasundaram, R., Tews, I., & Margueron, J. 2023, Physical Review C, 107, doi: 10.1103/physrevc.107.025801

    work page doi:10.1103/physrevc.107.025801 2023

  57. [57]

    Stairs, I. H. 2003, Living Reviews in Relativity, 6, doi: 10.12942/lrr-2003-5

    work page doi:10.12942/lrr-2003-5 2003

  58. [58]

    L., et al

    Vinciguerra, S., Salmi, T., Watts, A. L., et al. 2023, An updated mass-radius analysis of the 2017-2018 NICER data set of PSR J0030+0451, Zenodo, doi: 10.5281/zenodo.8239000 15

    work page doi:10.5281/zenodo.8239000 2023

  59. [59]

    L., et al

    Vinciguerra, S., Salmi, T., Watts, A. L., et al. 2024, ApJ, 961, 62, doi: 10.3847/1538-4357/acfb83

    work page doi:10.3847/1538-4357/acfb83 2024

  60. [60]

    E., et al

    Virtanen, P., Gommers, R., Oliphant, T. E., et al. 2020, Nature Methods, 17, 261, doi: 10.1038/s41592-019-0686-2

    work page doi:10.1038/s41592-019-0686-2 2020

  61. [61]

    Waskom, M. L. 2021, Journal of Open Source Software, 6, 3021, doi: 10.21105/joss.03021

    work page doi:10.21105/joss.03021 2021

  62. [62]

    Watts, A. L. 2019, in American Institute of Physics Conference Series, Vol. 2127, Xiamen-CUSTIPEN Workshop on the Equation of State of Dense Neutron-Rich Matter in the Era of Gravitational Wave Astronomy (AIP), 020008, doi: 10.1063/1.5117798

    work page doi:10.1063/1.5117798 2019

  63. [63]

    L., Andersson, N., Chakrabarty, D., et al

    Watts, A. L., Andersson, N., Chakrabarty, D., et al. 2016, Reviews of Modern Physics, 88, 021001, doi: 10.1103/RevModPhys.88.021001

    work page doi:10.1103/revmodphys.88.021001 2016

  64. [64]

    2025, arXiv e-prints, arXiv:2510.07764, doi: 10.48550/arXiv.2510.07764

    Zhou, T., & Huang, C. 2025, arXiv e-prints, arXiv:2510.07764, doi: 10.48550/arXiv.2510.07764

    work page internal anchor Pith review doi:10.48550/arxiv.2510.07764 2025

  65. [65]

    Zhou and C

    Zhou, T., & Huang, C. 2025, arXiv e-prints. https://arxiv.org/abs/2504.08662

    work page arXiv 2025