X-ray Pulse Profiles Research Articles

The Neutron Star Mass, Distance, and Inclination from Precision Timing of the Brilliant Millisecond Pulsar J0437-4715

The observation of neutron stars enables the otherwise impossible study of fundamental physical processes. The timing of binary radio pulsars is particularly powerful, as it enables precise characterization of their (three-dimensional) positions and orbits. PSR J0437–4715 is an important millisecond pulsar for timing array experiments and is also a primary target for the Neutron Star Interior Composition Explorer (NICER). The main aim of the NICER mission is to constrain the neutron star equation of state by inferring the compactness (M p /R) of the star. Direct measurements of the mass M p from pulsar timing therefore substantially improve constraints on the radius R and the equation of state. Here we use observations spanning 26 yr from Murriyang, the 64 m Parkes radio telescope, to improve the timing model for this pulsar. Among the new precise measurements are the pulsar mass M p = 1.418 ± 0.044 M ⊙, distance D = 156.96 ± 0.11 pc, and orbital inclination angle i = 137.°506 ± 0.°016, which can be used to inform the X-ray pulse profile models inferred from NICER observations. We demonstrate that these results are consistent between multiple data sets from the Parkes Pulsar Timing Array (PPTA), each modeled with different noise assumptions. Using the longest available PPTA data set, we measure an apparent second derivative of the pulsar spin frequency and discuss how this can be explained either by kinematic effects due to the proper motion and radial velocity of the pulsar or excess low-frequency noise such as a gravitational-wave background.

Localizing the non-thermal X-ray emission of PSR J2229+6114 from its multi-wavelength pulse profiles

Context. Pulsars are detected over the whole electromagnetic spectrum, from radio wavelengths up to very high energies, in the GeV-TeV range. While the radio emission site for young pulsars is well constrained to occur at altitudes about several percent of the light-cylinder radius and γ-ray emission is believed to be produced in the striped wind, outside the light cylinder, the non-thermal X-ray production site remains unknown. Aims. The aim of this Letter is to localize the non-thermal X-ray emission region based on a multi-wavelength pulse profile fitting for PSR J2229+6114, which stands as a particularly good candidate given its high X-ray brightness. Methods. Based on the geometry deduced from the joint radio and γ-ray pulse profiles, we fixed the magnetic axis inclination angle and the line-of-sight (LoS) inclination angle. However, we left the region of X-ray emission unlocalized, setting it somewhere between the surface and the light cylinder. We localized this region and its extension by fitting the X-ray pulse profile as observed by the NICER, NuSTAR, and RXTE telescopes in the ranges of 2–7 keV, 3–10 keV, and 9.4–22.4 keV, respectively. Results. We constrained the non-thermal X-ray emission to arise from altitudes between 0.2 rL and 0.55 rL where rL is the light-cylinder radius. The magnetic obliquity is approximately α ≈ 45° −50° and the LoS inclination angle is ζ ≈ 32° −48°. Conclusions. This Letter is among the first works to offer a tight constraint on the location of non-thermal X-ray emission from pulsars. We plan to apply this procedure to several other promising candidates to confirm this new result.

The “Magnificent Seven” X-Ray Isolated Neutron Stars Revisited. I. Improved Timing Solutions and Pulse Profile Analysis

We present the first systematic X-ray pulse timing analysis of the six members of the so-called “Magnificent Seven” nearby thermally emitting isolated neutron stars (XINS) with detected pulsations. Using the extensive collection of archival XMM-Newton, Chandra, and NICER observations spanning over two decades, we obtain the first firm measurement of the spin-down rate for RX J2143.0+0654, while for the rest we improve upon previously published spin ephemerides and extend them by up to an additional decade. Five of the XINS follow steady spin-down with no indication of major anomalies in their long-term timing behavior; the notable exception is RX J0720.4−3125, for which, in addition to confirming the previously identified glitch, we detect a second spin derivative. The high-quality folded X-ray pulse profiles produced with the updated timing solutions exhibit diverse and complex morphologies, as well as striking energy dependence. These peculiarities cannot be readily explained by blackbody-like isotropic emission and simple hot-spot configurations, hinting at the presence of complex multitemperature surface heat distributions and highly anisotropic radiation patterns, such as may arise from a strongly magnetized atmospheric layer.

Atmospheric Effects on Neutron Star Parameter Constraints with NICER

We present an analysis of the effects of uncertainties in the atmosphere models on the radius, mass, and other neutron star parameter constraints for the NICER observations of rotation-powered millisecond pulsars. To date, NICER has applied the X-ray pulse profile modeling technique to two millisecond-period pulsars: PSR J0030+0451 and the high-mass pulsar PSR J0740+6620. These studies have commonly assumed a deep-heated, fully ionized hydrogen atmosphere model, although they have explored the effects of partial ionization and helium composition in some cases. Here, we extend that exploration and also include new models with partially ionized carbon composition, externally heated hydrogen, and an empirical atmospheric beaming parameterization to explore deviations in the expected anisotropy of the emitted radiation. None of the studied atmosphere cases have any significant influence on the inferred radius of PSR J0740+6620, possibly due to its X-ray faintness, tighter external constraints, and/or viewing geometry. In the case of PSR J0030+0451, both the composition and ionization state could significantly alter the inferred radius. However, based on the evidence (prior predictive probability of the data), partially ionized hydrogen and carbon atmospheres are disfavored. The difference in the evidence for ionized hydrogen and helium atmospheres is too small to be decisive for most cases, but the inferred radius for helium models trends to larger sizes around or above 14–15 km. External heating or deviations in the beaming that are less than 5% at emission angles smaller than 60°, on the other hand, have no significant effect on the inferred radius.

Observational properties of accreting neutron stars with an optically thick envelope

ABSTRACT Pulsating ultraluminous X-ray sources are thought to be X-ray bright, accreting, magnetized neutron stars. Their measured soft (<20 keV) X-ray luminosity can exceed the Eddington luminosity for a neutron star by a few orders of magnitude. Although several scenarios have been proposed to explain the different components observed in the X-ray spectrum and the characteristics of the X-ray pulse profile of these systems, detailed quantitative calculations are still missing. In particular, the observed soft X-ray pulse profiles are almost sinusoidal and show an increase in the pulsed fraction (from 8 up to even 30 per cent) with increasing energy. In this work, we discuss how emission originating from an optically thick envelope, expected to be formed during super-Eddington accretion, can match the observed spectra and pulse profiles.

Multiwavelength observations of PSR J2021+4026 across a mode change reveal a phase shift in its X-ray emission

Context. We have investigated the multiwavelength emission of PSR J2021+4026, the only isolated γ-ray pulsar known to be variable, which in October 2011 underwent a simultaneous change in γ-ray flux and spin-down rate, followed by a second mode change in February 2018. Multiwavelength monitoring is crucial to understand the physics behind these events and how they may have affected the structure of the magnetosphere. Aims. The monitoring of pulse profile alignment is a powerful diagnostic tool for constraining magnetospheric reconfiguration. We aim to investigate timing or flux changes related to the variability of PSR J2021+4026 via multiwavelength observations, including γ-ray observations from Fermi-LAT, X-ray observations from XMM-Newton, and a deep optical observation with the Gran Telescopio Canarias. Methods. We performed a detailed comparison of the timing features of the pulsar in γ and X-rays and searched for any change in phase lag between the phaseogram peaks in these two energy bands. Although previous observations did not detect a counterpart in visible light, we also searched for optical emission that might have increased due to the mode change, making this pulsar detectable in the optical. Results. We have found a change in the γ-to X-ray pulse profile alignment by 0.21 ± 0.02 in phase, which indicates that the first mode change affected different regions of the pulsar magnetosphere. No optical counterpart was detected down to g′ = 26.1 and r′ = 25.3. Conclusions. We suggest that the observed phase shift could be related to a reconfiguration of the connection between the quadrupole magnetic field near the stellar surface and the dipole field that dominates at larger distances. This is consistent with the picture of X-ray emission coming from the heated polar cap and with the simultaneous flux and frequency derivative change observed during the mode changes.

A NuSTAR and Chandra Investigation of the Misaligned Outflow of PSR J1101–6101 and the Lighthouse Pulsar Wind Nebula

PSR J1101–6101 is an energetic young pulsar that powers the remarkable Lighthouse pulsar wind nebula (PWN). The pulsar belongs to the rare type of radio- and gamma-ray-quiet pulsars that are bright in hard X-rays. Moreover, the Lighthouse PWN is remarkable for its misaligned outflow (which gave rise to the PWN’s nickname). Also known as “pulsar filaments,” these collimated parsec-scale X-ray structures have been recently discovered in the vicinity of a handful of fast-moving pulsars, and they appear unaffected by the ram pressure that confines pulsar tails. We report on NuSTAR observations of PSR J1101–6101 and its misaligned outflow—the first observation of such a structure above ∼10 keV. We detect the outflow up to 25 keV, spatially resolve its spectral evolution with distance from the pulsar, find unambiguous evidence of spectral cooling with distance from the pulsar, and infer physical properties of the particles and magnetic field in the outflow. We also reanalzye archival Chandra data and discuss the outflow’s small-scale structure. We detect pulsations from PSR J1101–6101 up to 20 keV, present the X-ray pulse profile, confirm its period derivative, and perform phase-resolved spectroscopy. Finally, we discuss the X-ray source 2CXO J110158.4–605649 = 2XMM J110158.5–605651 (a serendipitously observed blazar) and suggest it may be the X-ray counterpart to the GeV source 4FGL J1102.0–6054.

Simultaneous radio and X-ray observations of the magnetar Swift J1818.0−1607

ABSTRACT Swift J1818.0−1607 is a radio-emitting magnetar that was discovered in X-ray outburst in 2020 March. Starting 4 d after this outburst, we began a nearly 5-month multifrequency observing campaign at 2.2, 8.4, and 32 GHz using telescopes in the NASA Deep Space Network. Using a dual-frequency observing mode, we were able to observe Swift J1818.0−1607 simultaneously at either 2.2 and 8.4 GHz or 8.4 and 32 GHz. Over the course of the campaign, we find that the flux density increases substantially and the spectrum changes from uncharacteristically steep (α < −2.2) to the essentially flat (α ≈ 0) spectrum typical of radio-emitting magnetars. In addition to the expected profile evolution on time-scales of days to months, we find that Swift J1818.0−1607 also exhibits mode switching where the pulse profile changes between two distinct shapes on time-scales of seconds to minutes. For two of the radio observations, we also had accompanying X-ray observations using the Neutron Star Interior Composition Explorer telescope that occurred on the same day. We find a near anti-alignment (0.40 phase cycles) between the peaks of the radio and X-ray pulse profiles, which is most likely explained by an intrinsic misalignment between the X-ray- and radio-emitting regions.

Machine-learning calibration of intense x-ray free-electron-laser pulses using Bayesian optimization

X-ray free-electron lasers (XFELs) have brought new ways to probe and manipulate atomic and molecular dynamics with unprecedented spatial and temporal resolutions. A quantitative comparison of experimental results with their simulated theoretical counterpart, however, generally requires a precise characterization of the spatial and temporal x-ray pulse profile, providing a nonuniform photon distribution. The determination of the pulse profile constitutes a major, yet inevitable, challenge. Here, we propose a calibration scheme for intense XFEL pulses utilizing a set of experimental charge-state distributions of light noble gas atoms at a series of pulse energies in combination with first-principles simulations of the underlying atomic x-ray multiphoton ionization dynamics. The calibration builds on Bayesian optimization, which is a powerful, machine-learning-based tool particularly well suited for computationally expensive numerical optimization. We demonstrate the presented scheme to calibrate the pulse duration as well as the spatial fluence distribution profile of XFEL pulses. Our proposed method can serve as a comprehensive tool for characterizing ultraintense and ultrafast x-ray pulses.

X-Ray Studies of the Pulsar PSR J1420–6048 and Its TeV Pulsar Wind Nebula in the Kookaburra Region

We present a detailed analysis of broadband X-ray observations of the pulsar PSR J1420−6048 and its wind nebula (PWN) in the Kookaburra region with Chandra, XMM-Newton, and NuSTAR. Using the archival XMM-Newton and new NuSTAR data, we detected 68 ms pulsations of the pulsar and characterized its X-ray pulse profile, which exhibits a sharp spike and a broad bump separated by ∼0.5 in phase. A high-resolution Chandra image revealed a complex morphology of the PWN: a torus-jet structure, a few knots around the torus, one long (∼7′) and two short tails extending in the northwest direction, and a bright diffuse emission region to the south. Spatially integrated Chandra and NuSTAR spectra of the PWN out to 2.′5 are well-described by a power-law model with a photon index Γ ≈ 2. A spatially resolved spectroscopic study, as well as NuSTAR radial profiles of the 3–7 keV and 7–20 keV brightness, showed a hint of spectral softening with increasing distance from the pulsar. A multiwavelength spectral energy distribution (SED) of the source was then obtained by supplementing our X-ray measurements with published radio, Fermi-LAT, and H.E.S.S. data. The SED and radial variations of the X-ray spectrum were fit with a leptonic multizone emission model. Our detailed study of the PWN may be suggestive of (1) particle transport dominated by advection, (2) a low magnetic-field strength (B ∼ 5 μG), and (3) electron acceleration to ∼PeV energies.

A Broadband X-Ray Study of the Rabbit Pulsar Wind Nebula Powered by PSR J1418-6058

We report on broadband X-ray properties of the Rabbit pulsar wind nebula (PWN) associated with the pulsar PSR J1418−6058 using archival Chandra and XMM-Newton data, as well as a new NuSTAR observation. NuSTAR data above 10 keV allowed us to detect the 110 ms spin period of the pulsar, characterize its hard X-ray pulse profile, and resolve hard X-ray emission from the PWN after removing contamination from the pulsar and other overlapping point sources. The extended PWN was detected up to ∼20 keV and is described well by a power-law model with a photon index Γ ≈ 2. The PWN shape does not vary significantly with energy, and its X-ray spectrum shows no clear evidence of softening away from the pulsar. We modeled the spatial profile of X-ray spectra and broadband spectral energy distribution in the radio to TeV band to infer the physical properties of the PWN. We found that a model with low magnetic field strength (B ∼ 10 μG) and efficient diffusion (D ∼ 1027 cm2 s−1) fits the PWN data well. The extended hard X-ray and TeV emission, associated respectively with synchrotron radiation and inverse Compton scattering by relativistic electrons, suggest that particles are accelerated to very high energies (≳500 TeV), indicating that the Rabbit PWN is a Galactic PeVatron candidate.

Common patterns in pulse profiles of high-mass X-ray binaries

Context. X-ray pulsars are binary systems which consist of a neutron star in orbit with a mass donor (companion). In these systems the neutron accretes matter from the companion star, which creates accretion columns or hot spots on the neutron star surface and gives rise to pulsations in the X-ray light curve. The pulse profiles carry information about the accretion and magnetic field geometry. Here we present a study and classification of energy-resolved pulse profiles of a sample of X-ray pulsars, focusing on high-mass X-ray binaries. Aims. Our goal is to perform a classification of X-ray pulsars based on their observed pulse profiles and look for correlations between this classification and their principle physical observables. The analysis pipeline is available online. Methods. We analysed the pulse profiles of a sample of X-ray pulsars using data obtained with the X-ray Multi-Mirror Mission (XMM-Newton) and the Nuclear Spectroscopic Telescope Array (NuSTAR). We fitted the energy-resolved pulse profiles with a Fourier series of up to five harmonics. We then used the energy evolution of the different Fourier components to classify the pulse profiles into groups. We investigated relationships between the pulse profile properties and other observables of the systems (e.g. orbital period, magnetic field strength, and luminosity) to study the extreme physics of these systems. Results. The sources were divided into three groups using a classification based on the shape, the dominance of the fitted Fourier harmonics, and their respective evolution with energy. We do not find a conclusive correlation between the pulse profile shapes or groups and other parameters of the systems. However, a weak trend was found when comparing our classification to the sources’ locations in the spin period-orbital period diagram. Further studies are required to confirm this trend. Conclusions. Despite the large variety of pulse profiles of the X-ray pulsars, we found that with our approach clear categories emerge which we use to classify their behaviour as a function of energy. As we do not find a clear relationship between our classification scheme and other parameters, like the luminosity, the magnetic field strength, or the orbital and spin periods, we conclude that X-ray pulse profiles are influenced by other hidden variables.

A Study on the X-Ray Pulse Profile and Spectrum of the Crab Pulsar Using NICER and Insight-HXMT's Observations

Abstract We analyze the energy dependence of the X-ray pulse profile and the phase-resolved spectra (PRS) of the Crab pulsar using observations from the Neutron star Interior Composition Explorer (NICER) and the Hard X-ray Modulation Telescope (Insight-HXMT). We parameterize the pulse profiles and quantify the evolution of these parameters in the broad energy band of 0.4–250 keV. A log-parabola function is used to fit the PRS in 2–250 keV, and the curvature of the spectrum, i.e., the evolution of the photon index with energy, as represented by the parameter β of the log-parabola model, also changes with phase. The relation of β and phase has two turning points slightly later than those of the pulse intensity profile, where the values of β are the lowest, suggesting that the energy-loss rate of the particles is the lowest in the corresponding regions. A three-segment broken-power-law model is also used to fit those PRS. The differences between the hard spectral index and the soft ones have a distribution similar to that of β, confirming the fitting results of the log-parabola model, while the broken energies are generally higher in the region bridging the two pulses. We find anticorrelations between the spectral indices and the curvature of the log-parabola model fitting and a similar anticorrelation between the spectral indices and broken energies of the broken-power-law model fitting, suggesting a scenario where the highest-energy particles are produced in regions where radiation energy loss is strongest.

Selection Effects in Periodic X-Ray Data from Maximizing Detection Statistics

Abstract The Neutron Star Interior Composition Explorer (NICER) records exceptional data on pulsars’ energy-dependent X-ray pulse profiles. However, in searching for evidence of pulsations, Guillot et al. (2019) introduce a procedure to select an ordered subset of data that maximizes a detection statistic (the H-test). I show that this can degrade subsequent analyses using an idealized model with stationary expected count rates from both noise and signal. Specifically, the data-selection procedure biases the inferred mean count rate to be too low and the inferred pulsation amplitude to be too high, and the size of these biases scales strongly with the amount of data that is rejected and the true signal amplitude. The procedure also alters the H-test’s null distribution, rendering nominal significance estimates overly optimistic. While the idealized model does not capture all the complexities of real NICER data, it suggests that these biases could be important for NICER’s observations of J0740+6620 and other faint pulsars (observations of J0030+0451 are likely less affected). I estimate that these effects may introduce a bias of  ( 10 % ) on average in the inferred modulation depth of lightcurves like J0740+6620's, and may be as large as  ( 50 % ) for fainter pulsars. However, the change for a single data set like J0740+6620 is expected to be a shift between −5% and +20%. This could imply that the lower limit on J0740+6620's radius is slightly larger than it should be, although preliminary investigations suggest the radius constraints shift to larger radii by  ( 1 % ) with the same overall statistical precision using real J0740+6620 data.

Pulse Peak Migration during the Outburst Decay of the Magnetar SGR 1830-0645: Crustal Motion and Magnetospheric Untwisting

Abstract Magnetars, isolated neutron stars with magnetic-field strengths typically ≳1014 G, exhibit distinctive months-long outburst epochs during which strong evolution of soft X-ray pulse profiles, along with nonthermal magnetospheric emission components, is often observed. Using near-daily NICER observations of the magnetar SGR 1830-0645 during the first 37 days of a recent outburst decay, a pulse peak migration in phase is clearly observed, transforming the pulse shape from an initially triple-peaked to a single-peaked profile. Such peak merging has not been seen before for a magnetar. Our high-resolution phase-resolved spectroscopic analysis reveals no significant evolution of temperature despite the complex initial pulse shape, yet the inferred surface hot spots shrink during peak migration and outburst decay. We suggest two possible origins for this evolution. For internal heating of the surface, tectonic motion of the crust may be its underlying cause. The inferred speed of this crustal motion is ≲100 m day−1, constraining the density of the driving region to ρ ∼ 1010 g cm−3, at a depth of ∼200 m. Alternatively, the hot spots could be heated by particle bombardment from a twisted magnetosphere possessing flux tubes or ropes, somewhat resembling solar coronal loops, that untwist and dissipate on the 30–40 day timescale. The peak migration may then be due to a combination of field-line footpoint motion (necessarily driven by crustal motion) and evolving surface radiation beaming. This novel data set paints a vivid picture of the dynamics associated with magnetar outbursts, yet it also highlights the need for a more generic theoretical picture where magnetosphere and crust are considered in tandem.

Scalarized neutron stars in massive scalar-tensor gravity: X-ray pulsars and tidal deformability

Neutron stars (NSs) in scalar-tensor theories of gravitation with the phenomenon of spontaneous scalarization can develop significant deviations from general relativity. Cases with a massless scalar have been studied widely. We compare the NS scalarizations in Damour--Esposito-Far\`ese theory, Mendes-Ortiz theory, and $\ensuremath{\xi}$ theory with a massive scalar field. Numerical solutions for slowly rotating NSs are obtained. They are used to construct the x-ray pulse profiles of a pair of extended hot spots on the surface of NSs. We also calculate the tidal deformability for NSs with spontaneous scalarization, which is done for the first time with a massive scalar field. We show the universal relation between the moment of inertia and the tidal deformability. The x-ray pulse profiles, the tidal deformability, and the universal relation may help us to constrain the massive scalar-tensor theories in x-ray and gravitational-wave observations of NSs, including the Neutron star Interior Composition Explorer satellite, the Square Kilometre Array telescope, and LIGO/Virgo/KAGRA laser interferometers.

Analytical external spherical solutions in entangled relativity

In this manuscript, we present analytical external spherical solutions of entangled relativity, which we compare to numerical solutions obtained in a Tolman–Oppenheimer–Volkoff framework. Analytical and numerical solutions match perfectly well outside spherical compact objects, therefore validating both types of solutions at the same time. The analytical external (hairy) solutions – which depend on two parameters only – may be used in order to easily compute observables – such as X-ray pulse profiles – without having to rely on an unknown equation of state for matter inside the compact object.

Impact of the PSR J0740+6620 radius constraint on the properties of high-density matter

X-ray pulse profile modeling of PSR J0740+6620, the most massive known pulsar, with data from the NICER and XMM-Newton observatories recently led to a measurement of its radius. We investigate this measurement's implications for the neutron star equation of state (EoS), employing a nonparametric EoS model based on Gaussian processes and combining information from other x-ray, radio and gravitational-wave observations of neutron stars. Our analysis mildly disfavors EoSs that support a disconnected hybrid star branch in the mass-radius relation, a proxy for strong phase transitions, with a Bayes factor of $6.9$. For such EoSs, the transition mass from the hadronic to the hybrid branch is constrained to lie outside ($1,2$) $M_{\odot}$. We also find that the conformal sound-speed bound is violated inside neutron star cores, which implies that the core matter is strongly interacting. The squared sound speed reaches a maximum of $0.75^{+0.25}_{-0.24}\, c^2$ at $3.60^{+2.25}_{-1.89}$ times nuclear saturation density at 90% credibility. Since all but the gravitational-wave observations prefer a relatively stiff EoS, PSR J0740+6620's central density is only $3.57^{+1.3}_{-1.3}$ times nuclear saturation, limiting the density range probed by observations of cold, nonrotating neutron stars in $\beta$-equilibrium.

AstroSat-CZTI as a hard X-ray pulsar monitor

The Cadmium Zinc Telluride Imager (CZTI) is an imaging instrument onboard AstroSat. This instrument operates as a nearly open all-sky detector above ~60 keV, making possible long integrations irrespective of the spacecraft pointing. We present a technique based on the AstroSat-CZTI data to explore the hard X-ray characteristics of the $\gamma$-ray pulsar population. We report highly significant ($\sim 30\sigma$) detection of hard X-ray (60--380 keV) pulse profile of the Crab pulsar using $\sim$5000 ks of CZTI observations within 5 to 70 degrees of Crab position in the sky, using a custom algorithm developed by us. Using Crab as our test source, we estimate the off-axis sensitivity of the instrument and establish AstroSat-CZTI as a prospective tool in investigating hard X-ray characteristics of $\gamma$-ray pulsars as faint as 10 mCrab.

Constraining the Neutron Star Mass–Radius Relation and Dense Matter Equation of State with NICER. III. Model Description and Verification of Parameter Estimation Codes

Abstract We describe the X-ray pulse profile models we use and how we use them to analyze Neutron Star Interior Composition Explorer (NICER) observations of rotation-powered millisecond pulsars to obtain information about the mass–radius relation of neutron stars and the equation of state of the dense matter in their cores. Here we detail our modeling of the observed profile of PSR J0030+0451 that we analyzed in Miller et al. and Riley et al. and describe a cross-verification of computations of the pulse profiles of a star with R/M 3, in case stars this compact need to be considered in future analyses. We also present our early cross-verification efforts of the parameter estimation procedures used by Miller et al. and Riley et al. by analyzing two distinct synthetic data sets. Both codes yielded credible regions in the mass–radius plane that are statistically consistent with one another, and both gave posterior distributions for model parameter values consistent with the values that were used to generate the data. We also summarize the additional tests of the parameter estimation procedure of Miller et al. that used synthetic pulse profiles and the NICER pulse profile of PSR J0030+0451. We then illustrate how the precision of mass and radius estimates depends on the pulsar’s spin rate and the size of its hot spot by analyzing four different synthetic pulse profiles. Finally, we assess possible sources of systematic error in the estimates made using this technique, some of which may warrant further investigation.