X-Ray Pulse-Profile Radius Inference

Pulse-profile inference turns the energy- and phase-resolved X-ray counts from a rotating neutron star into a likelihood for its mass and equatorial radius. The central observable is not a geometric radius alone: light bending constrains compactness, rotation supplies Doppler and time-delay structure, and the surface atmosphere and hot-region geometry set the emitted beam. A credible radius result must therefore carry its event selection, response calibration, forward model, priors, background treatment, and model alternatives. The evidence cutoff on this page is 2026-08-18.

Required background. Equivalence principles and gravitational measurement supplies local-frame emission and the distinction between coordinate quantities and measured photon energies.

Helpful background. Stress–energy and conservation supplies the material interpretation; the Einstein equation connects compactness to exterior geometry; relativistic matter and compact objects supplies stellar structure and atmosphere context; gravitational radiation and waveforms provides an independent radius-sensitive channel; and computational relativity supplies ray-tracing and sampling checks.

Relativistic pulse formation

Consider a surface element comoving with a star of gravitational mass $M$ and equatorial circumferential radius $R_{\rm eq}$. A local atmosphere emits specific intensity $I'_{E'}(\alpha')$, where $E'$ and $\alpha'$ are photon energy and emission angle in that frame. Liouville invariance gives $I_E/E^3=I'_{E'}/E'^3$, so the contribution received at infinity can be written schematically as

$$dF_E=g^3 I'_{E/g}(\alpha')\,d\Omega,$$

where $g=E/E'$ includes gravitational redshift and the rotational Doppler factor, and $d\Omega$ is the image-plane solid angle after ray tracing. In the nonrotating exterior, the surface redshift is

$$1+z=\left(1-\frac{2GM}{R}\right)^{-1/2}.$$

Increasing $GM/R$ bends more of the far side into view and suppresses pulse contrast. That is why the data first constrain compactness. The absolute radius enters through emitting area, rotational velocity, oblateness, time of flight, and the mapping between surface angles and image solid angle.

Millisecond rotation makes a spherical Schwarzschild surface insufficient at the desired precision. A common controlled approximation uses an oblate surface with an exterior metric accurate for the spin regime; faster rotation or a demanded sub-percent result may require additional multipoles. The model must specify which approximation is used, because “general-relativistic ray tracing” does not uniquely specify the shape, exterior spacetime, or arrival-time convention.

Surface physics is equally consequential. A magnetically weak hydrogen atmosphere beams differently from a blackbody, while helium composition, magnetic opacity, temperature gradients, and interstellar absorption change the energy dependence. Hot regions may be circular, crescent-shaped, overlapping, or temperature-structured. The independent J0030+0451 analyses illustrate why spot topology is inferred rather than assumed away: Riley et al. 2019, §§ 2–3 and 4.2 and Miller et al. 2019, §§ 2–4 use related physics but different parameterizations and computational implementations.

Counts, calibration, and likelihood

Let $n_{jk}$ be the counts in detector channel $j$ and rotational phase bin $k$. For independent Poisson bins, a forward model with expected counts $\mu_{jk}(\theta)$ has

$$\ln\mathcal L(\theta) =\sum_{j,k} \left[n_{jk}\ln\mu_{jk}(\theta)-\mu_{jk}(\theta)-\ln\Gamma(n_{jk}+1)\right].$$

The prediction is folded through the instrument response,

$$\mu_{jk} =T_k\int dE\,R_j(E)S_k(E;\theta)+b_{jk}(\beta),$$

with exposure $T_k$, redistribution and effective-area response $R_j$, phase-dependent incident photon spectrum $S_k$, and expected detector-background counts $b_{jk}$. The energy flux $F_E$ derived above must be divided by photon energy to obtain a photon spectrum; absorption is then included before response folding. An astrophysical sky background can enter the integral as an additional photon spectrum. Particle and instrumental backgrounds are modeled in detector space and must not be passed through the photon response. Calibration uncertainty may be represented by response nuisance parameters or by repeating the inference with alternative response realizations. Treating a nominal response as exact narrows the posterior by assumption.

For intuition, take a two-bin toy pulse with source expectations $s(1+a)$ and $s(1-a)$, where $0\le a\lt1$, and equal background $b$ in each bin. Its expected fractional modulation is

$$\frac{\mu_+-\mu_-}{\mu_++\mu_-} =\frac{sa}{s+b}.$$

With $s=100$, $a=0.4$ and $b=100$ counts per bin, the expectations are 240 and 160, and the modulation is 0.20 rather than the source value 0.40. A weaker observed pulse can therefore reflect stronger light bending, a different viewing geometry, or more background. Energy and phase information, along with independently constrained background, help distinguish those explanations. These numbers illustrate the forward model; they are not NICER measurements.

The public data boundary is the target-specific set of NICER observation IDs cited by each analysis, retrieved from the HEASARC NICER archive, together with the exact NICER CALDB, HEASoft/NICERDAS processing version, filter expressions, good-time intervals, clock-correction file, barycentric ephemeris, detector exclusions, energy and phase binning, response files, and background products. HEASARC 2025, NICER archive and calibration access identifies the authoritative archive and CALDB; HEASARC 2023, data-access workflow gives the machine retrieval route. The ObsID list in the primary analysis, not a hand-selected light curve, defines the sample.

At the cutoff, peer-reviewed public pulse-profile analyses exist for J0030+0451, J0740+6620, J0437−4715, J1231−1411, and J0614−3329; a public preprint also analyzes J2124−3358. They do not carry equal evidential weight. The J1231 result is explicitly sensitive to the radius prior and to finding the preferred hot-region solution Salmi et al. 2024, abstract and §§ 3–5. The J0614 analysis combines NICER and XMM-Newton data with informative radio mass and inclination constraints Mauviard et al. 2025, §§ 2–6. The J2124 preprint reports broad posteriors and a strong hydrogen-versus-helium atmosphere dependence, so it must remain labeled as a preprint rather than silently pooled with the peer-reviewed set González-Caniulef et al. 2026, abstract and §§ 2–5.

Updated J0740 work provides a particularly useful reproducibility model because Salmi et al. 2024, § II, especially §§ II.1–II.2 records the event selection, X-PSI analysis products, and repository. Choudhury et al. 2024, §§ 2–4 exposes how background information and radio mass information enter the J0437 likelihood. These are distinct target likelihoods; their published posteriors cannot simply be multiplied without undoing interim priors.

Degeneracies, priors, and systematic alternatives

The dominant degeneracies are physical. Compactness trades against spot colatitude and observer inclination through pulse amplitude. Radius trades against distance, emitting area, and temperature through flux normalization. Doppler asymmetry trades against spot longitude and spin geometry. Background counts can dilute modulation; atmosphere beaming can restore it. A radio mass prior can sharply reduce the $M$–$R$ direction but does not turn the radius into a prior-free measurement.

The statistical model must list priors for $M$, $R_{\rm eq}$, distance, inclination, spot number and topology, temperatures, atmosphere composition, column density, background, effective area, and cross-instrument normalization. Model comparison across different hot-region dimensions requires properly normalized evidences or predictive performance; comparing maximum likelihoods rewards flexible models. Sampler adequacy is checked with repeated runs, effective sample size, mode discovery, and, where nested sampling is used, stability against live-point and termination settings.

Systematic uncertainty is assessed by an explicit alternative matrix: hydrogen versus other admissible atmospheres; two versus more hot regions; simple versus temperature-structured regions; nominal versus perturbed calibration; competing background prescriptions; spherical versus oblate surface; independent ray-tracing codes; and the inclusion or removal of external mass, distance, or inclination information. Posterior-predictive phase–energy residuals should reveal coherent structure that a scalar goodness-of-fit can hide. Synthetic pulse injection and recovery tests whether the inference can recover known $M$ and $R$ without forcing the injected geometry into an incompatible spot family.

Reproduction boundary and interpretation

A bounded reproduction starts from the archived ObsIDs, runs the declared screening and calibration pipeline, verifies the phase assignment against the radio ephemeris, recreates the channel–phase count array and response/background products, and evaluates at least one published forward-model configuration. It then repeats the result with one calibration, one background, and one spot-topology alternative. This page documents that route; it does not claim to have rerun NICER data or to reproduce a published radius posterior.

The scientific conclusion should be stated at the level supported by this hierarchy. Pulse profiles constrain an $M$–$R$ likelihood conditional on an exterior spacetime, atmosphere, surface map, instrument model, and priors. Mapping that likelihood to pressure or composition adds a common-EOS model and the stellar-structure equations. Agreement among independent targets and analysis pipelines is more informative than a visually narrow one-dimensional radius interval.

Check your understanding

Two nonrotating stars have the same compactness $C=GM/R=0.20$ but radii $R_1=10\,\mathrm{km}$ and $R_2=15\,\mathrm{km}$. What surface redshift does each have, and why can rotation help distinguish them?

Solution

Both have

$$1+z=(1-2C)^{-1/2}=0.6^{-1/2}\simeq1.291, \qquad z\simeq0.291.$$

Their static light-bending and redshift signatures are therefore identical at fixed emission geometry. At a known angular frequency $\Omega$, however, the surface speed scales approximately as $v\sim\Omega R$. The larger star has stronger Doppler boosting, aberration, oblateness, and propagation-time differences. Energy-resolved rotational structure can consequently break part of the compactness degeneracy, provided atmosphere and spot geometry are inferred consistently.

References

• Choudhury, Devarshi, et al. “A NICER View of the Nearest and Brightest Millisecond Pulsar: PSR J0437−4715.” Astrophysical Journal Letters 971 (2024): L20. DOI.
• González-Caniulef, Denis, et al. “A NICER View of the Millisecond Pulsar PSR J2124−3358: Evidence for a Helium Atmosphere.” arXiv:2607.03721 [astro-ph.HE] (2026), version 1 submitted July 4, 2026. arXiv.
• HEASARC. “NICER Archive” and “NICER Data Analysis Threads: Accessing Data.” NASA Goddard Space Flight Center (updated 2025 and 2023). Archive; access workflow.
• Mauviard, Lucien, et al. “A NICER View of the 1.4 $M_\odot$ Edge-on Pulsar PSR J0614−3329.” Astrophysical Journal 995 (2025): 60. DOI.
• Miller, M. Coleman, et al. “PSR J0030+0451 Mass and Radius from NICER Data and Implications for the Properties of Neutron Star Matter.” Astrophysical Journal Letters 887 (2019): L24. DOI.
• Riley, Thomas E., et al. “A NICER View of PSR J0030+0451: Millisecond Pulsar Parameter Estimation.” Astrophysical Journal Letters 887 (2019): L21. arXiv:1912.05702.
• Salmi, Tuomo, et al. “A NICER View of PSR J1231−1411: A Complex Case.” Astrophysical Journal 976 (2024): 58. DOI.
• Salmi, Tuomo, et al. “The Radius of the High-Mass Pulsar PSR J0740+6620 with 3.6 yr of NICER Data.” Astrophysical Journal 974 (2024): 294. DOI.