Neutron Star Radius Research Articles

Search for Continuous Gravitational Waves from Scorpius X-1 in LIGO O2 Data

Abstract We present the results of a search in LIGO O2 public data for continuous gravitational waves from the neutron star in the low-mass X-ray binary Scorpius X-1. We search for signals with ≈ constant frequency in the range 40–180 Hz. Thanks to the efficiency of our search pipeline we can use a long coherence time and achieve unprecedented sensitivity, significantly improving on existing results. This is the first search that has been able to probe gravitational wave amplitudes that could balance the accretion torque at the neutron star radius. Our search excludes emission at this level between 67.5 and 131.5 Hz, for an inclination angle 44° ± 6° derived from radio observations, and assuming that the spin axis is perpendicular to the orbital plane. If the torque arm is ≈26 km—a conservative estimate of the Alfvén radius—our results are more constraining than the indirect limit across the band. This allows us to exclude certain mass–radius combinations and to place upper limits on the strength of the star’s magnetic field with a different probe than ever used before. We also correct a mistake that appears in the literature in the equation that gives the gravitational wave amplitude at the torque balance and we re-interpret the associated latest LIGO/Virgo results in light of this.

Off-centred force-free neutron star magnetospheres

ABSTRACT Neutron star electromagnetic activity produces pairs that fill their magnetosphere represented to the zeroth order by the force-free approximation. Neither dissipation nor acceleration nor radiation from charged particles is expected from this simplified model. So far, only centred dipole magnetic fields have been studied in this limit. In this paper, we explore the consequences of a rotating off-centred dipole on the force-free magnetosphere, showing the new magnetic field geometry, its spin-down luminosity, and the electromagnetic kick and torque felt by the neutron star. Solutions are obtained by time-dependent numerical simulations of the force-free regime using our pseudo-spectral code written in spherical coordinates. Our results are also compared to known analytical expressions found for the off-centred vacuum dipole by an expansion to lowest order in the parameter ϵ = d/R, where d is the displacement of the dipole from the stellar centre and R the neutron star radius. The presence of a force-free plasma enhances the spin-down luminosity and the electromagnetic kick and torque with respect to a centred force-free dipole. The impact on isolated and binary neutron stars is revised in light of these new results.

A combined multimessenger analysis

Neutron Stars Neutron stars are stellar remnants with densities greater than that of an atomic nucleus. The properties of matter under such extreme conditions are poorly understood and inaccessible to terrestrial laboratories. Dietrich et al. developed a framework to combine multiple constraints on the masses and radii of neutron stars, including data from gravitational waves, electromagnetic observations, and theoretical nuclear physics calculations. They used this analysis to constrain the neutron-star equation of state and also improved the precision on the gravitational wave (standard siren) measurement of the Hubble constant—the expansion rate of the Universe. Science , this issue p. [1450][1] [1]: /lookup/doi/10.1126/science.abb4317

Multimessenger constraints on the neutron-star equation of state and the Hubble constant.

Observations of neutron-star mergers with distinct messengers, including gravitational waves and electromagnetic signals, can be used to study the behavior of matter denser than an atomic nucleus and to measure the expansion rate of the Universe as quantified by the Hubble constant. We performed a joint analysis of the gravitational-wave event GW170817 with its electromagnetic counterparts AT2017gfo and GRB170817A, and the gravitational-wave event GW190425, both originating from neutron-star mergers. We combined these with previous measurements of pulsars using x-ray and radio observations, and nuclear-theory computations using chiral effective field theory, to constrain the neutron-star equation of state. We found that the radius of a 1.4-solar mass neutron star is [Formula: see text] km at 90% confidence and the Hubble constant is [Formula: see text] at 1σ uncertainty.

Neutron stars in f(ℛ,T) gravity using realistic equations of state in the light of massive pulsars and GW170817

In this work we investigate neutron stars (NS) in f(ℛ,T) gravity for the case R+2λ\U0001d4af, ℛ is the Ricci scalar and \U0001d4af the trace of the energy-momentum tensor. The hydrostatic equilibrium equations are solved considering realistic equations of state (EsoS). The NS masses and radii obtained are subject to a joint constrain from massive pulsars and the event GW170817. The parameter λ needs to be negative as in previous NS studies, however we found a minimum value for it. The value should be |λ|≲0.02 and the reason for so small value in comparison with previous ones obtained with simpler EsoS is due to the existence of the NS crust. The pressure in theory of gravity depends on the inverse of the sound velocity vs. Since, vs is low in the crust, |λ| need to be very small. We found that the increment in the star mass is less than 1%, much smaller than previous ones obtained not considering the realistic stellar structure, and the star radius cannot become larger, its changes compared to GR is less than 3.6% in all cases. The finding that using several relativistic and non-relativistic models the variation on the NS mass and radius are almost the same for all the EsoS, manifests that our results are insensitive to the high density part of the EsoS. It confirms that stellar mass and radii changes depend only on crust, where the EoS is essentially the same for all the models. The NS crust effect implying very small values of |λ| does not depend on the theory's function chosen, since for any other one the hydrostatic equilibrium equation would always have the dependence 1/vs. Finally, we highlight that our results indicate that conclusions obtained from NS studies done in modified theories of gravity without using realistic EsoS that describe correctly the NS interior can be unreliable.

Fast Radio Burst Breakouts from Magnetar Burst Fireballs

Abstract The recent discovery of a Mega-Jansky radio burst occurring simultaneously with short X-ray bursts from the Galactic magnetar (strongly magnetized neutron star (NS)) SGR 1935+2154 is a smoking gun for the hypothesis that some cosmological fast radio bursts (FRBs) arise from magnetar bursts. We argue that the X-ray bursts with high temperature T ≳ 30 keV entail an electron–positron (e ±) outflow from a trapped–expanding fireball, polluting the NS magnetosphere before the FRB emission. The e ± outflow is opaque to induced Compton scatterings of FRB photons, and is strongly Compton-dragged by the X-ray bursts. Nevertheless, the FRB photons can break out of the e ± outflow with radiation forces if the FRB emission radius is larger than a few tens of NS radii. A FRB is choked if the FRB is weaker or the X-ray bursts are stronger, possibly explaining why there are no FRBs with giant flares and no detectable X-ray bursts with weak FRBs. We also speculate that the e ± outflow may be inevitable for FRBs, solving the problem of why the FRBs occur only with high-T X-ray bursts. The breakout physics is important for constraining the emission mechanism and electromagnetic counterparts to future FRBs.

Neutron Star Properties: Quantifying the Effect of the Crust–Core Matching Procedure

The impact of the equation of state (EoS) crust-core matching procedure on neutron star (NS) properties is analyzed within a meta-modeling approach. Using a Taylor expansion to parametrize the core equation of state (EoS) and the SLy4 crust EoS, we create two distinct EoS datasets employing two matching procedures. Each EoS describes cold NS matter in a β equilibrium that is thermodynamically stable and causal. It is shown that the crust-core matching procedure affects not only the crust-core transition but also the nuclear matter parameter space of the core EoS, and thus the most probable nuclear matter properties. An uncertainty of as much as 5% (8%) on the determination of low mass NS radii (tidal deformability) is attributed to the complete matching procedure, including the effect on core EoS. By restricting the analysis, imposing that the same set of core EoS is retained in both matching procedures, the uncertainty on the NS radius drops to 3.5% and below 1.5% for 1.9M⊙. Moreover, under these conditions, the crust-core matching procedure has a strong impact on the Love number k2, of almost 20% for 1.0M⊙ stars and 7% for 1.9M⊙ stars, but it shows a very small impact on the tidal deformability Λ, below 1%.

Estimating the nuclear saturation parameter via low-mass neutron star asteroseismology

We examine the fundamental ($f$) and the 1st pressure (${p}_{1}$) mode frequencies in gravitational waves from cold neutron stars constructed with various unified realistic equations of state. With the calculated frequencies, we derive the empirical formulas for the $f$- and ${p}_{1}$-mode frequencies, ${f}_{f}$ and ${f}_{{p}_{1}}$, as a function of the square root of the stellar average density and the parameter ($\ensuremath{\eta}$), which is a combination of the nuclear saturation parameters. With our empirical formulas, we show that by simultaneously observing the $f$- and ${p}_{1}$-mode gravitational waves, when $1.5\ensuremath{\lesssim}{f}_{{p}_{1}}/{f}_{f}\ensuremath{\lesssim}2.5$ (which corresponds to neutron star models with the mass of less than or approximately equal to $0.9\text{ }\text{ }{M}_{\ensuremath{\bigodot}}$), one could estimate the value of $\ensuremath{\eta}$ within approximately 10% accuracy, which makes a strong constraint on the equation of state for neutron star matter. In addition, we find that the maximum $f$-mode frequency is strongly associated with the minimum radius of neutron star. That is, if one would observe a larger frequency of the $f$ mode, one might constrain the upper limit of the minimum neutron star radius.

Effect of the crust on neutron star empirical relations

We analyze how the crust equation of state affects several neutron star properties and how it impacts on possible constraints inferred from astrophysical observations. Using three distinct crusts, we generate three sets of model-independent equations of state describing stellar matter from a Taylor expansion around saturation density. The equations of state are thermodynamically consistent, causal, and compatible with astrophysical observations. The relations between the tidal deformability $\Lambda$ and compactness $C$, Love number $k_2$ and radius of neutron star with mass $M$ are studied, and the effect of the crust equation of state on these relations analyzed. In most of the relations, the impact of the crust equation of state is not larger that 2\%. If, however, a fixed neutron star mass is considered, the relation between the tidal deformability and the radius depends on the crust. We have found that the relation $\Lambda_{M_i} = \alpha R_{M_i}^{\beta}$ becomes almost exact and crust independent for massive neutron stars. It is shown that it is possible to determine the tidal deformability of an 1.4$M_\odot$ star from the GW179817 effective tidal deformability $\tilde\Lambda$ with an accuracy of at least $\approx 10\%$. A high correlation between $\tilde\Lambda$ and the radius of the most massive star of the neutron star binary was confirmed, however, it was demonstrated that the crust has an effect of $\approx 14\%$ on this relation. We have found that the relation $\Lambda_1/\Lambda_2=q^a$ depends on $M_{\text{chirp}}$ as $a\sim \sqrt{M_{\text{chirp}}}$.

Neutron-star tidal deformability and equation-of-state constraints

Despite their long history and astrophysical importance, some of the key properties of neutron stars are still uncertain. The extreme conditions encountered in their interiors, involving matter of uncertain composition at extreme density and isospin asymmetry, uniquely determine the stars' macroscopic properties within General Relativity. Astrophysical constraints on those macroscopic properties, such as neutron star masses and radii, have long been used to understand the microscopic properties of the matter that forms them. In this article we discuss another astrophysically observable macroscopic property of neutron stars that can be used to study their interiors: their tidal deformation. Neutron stars, much like any other extended object with structure, are tidally deformed when under the influence of an external tidal field. In the context of coalescences of neutron stars observed through their gravitational wave emission, this deformation, quantified through a parameter termed the \emph{tidal deformability}, can be measured. We discuss the role of the tidal deformability in observations of coalescing neutron stars with gravitational waves and how it can be used to probe the internal structure of Nature's most compact matter objects. Perhaps inevitably, a large portion of the discussion will be dictated by GW170817, the most informative confirmed detection of a binary neutron star coalescence with gravitational waves as of the time of writing.

Spin-polarized β -stable neutron star matter: The nuclear symmetry energy and GW170817 constraint

Magnetic field of rotating pulsar might be so strong that the equation of state (EOS) of neutron star (NS) matter is significantly affected by the spin polarization of baryons. In the present work, the EOS of the spin-polarized nuclear matter is investigated in the nonrelativistic Hartree-Fock formalism, using a realistic density dependent nucleon-nucleon interaction with its spin and spin-isospin dependence accurately adjusted to the Brueckner-Hartree-Fock results for the spin-polarized nuclear matter. The nuclear symmetry energy and proton fraction are found to increase significantly with the increasing spin polarization of baryons, leading to a larger probability of the direct Urca process in the cooling of magnetar. The EOS of the $\beta$-stable np$e\mu$ matter obtained at different spin polarizations of baryons is used as the input for the Tolman-Oppenheimer-Volkov equations to determine the hydrostatic configuration of NS. Based on the GW170817 constraint on the radius $R_{1.4}$ of NS with $M\approx 1.4_\odot$, our mean-field results show that up to $60~\%$ of baryons in the NS merger might be spin-polarized. This result supports the magnetar origin of the blue kilonova ejecta of GW170817 suggested by Metzger et al., and the spin polarization of baryons needs, therefore, to be properly treated in the many-body calculation of the EOS of NS matter before comparing the calculated NS mass and radius with those constrained by the multi-messenger GW170817 observation.

Curvature-slope correlation of nuclear symmetry energy and its imprints on the crust-core transition, radius, and tidal deformability of canonical neutron stars

Background: The nuclear symmetry energy $E_{sym}(\rho)$ encodes information about the energy necessary to make nuclear systems more neutron-rich. While its slope parameter L at the saturation density $\rho_0$ of nuclear matter has been relatively well constrained by recent astrophysical observations and terrestrial nuclear experiments, its curvature $K_{\rm{sym}}$ characterizing the $E_{sym}(\rho)$ around $2\rho_0$ remains largely unconstrained. Over 520 calculations for $E_{sym}(\rho)$ using various nuclear theories and interactions in the literature have predicted several significantly different $K_{\rm{sym}}-L$ correlations. Purpose: If a unique $K_{\rm{sym}}-L$ correlation of $E_{sym}(\rho)$ can be firmly established, it will enable us to progressively better constrain the high-density behavior of $E_{sym}(\rho)$ using the available constraints on its slope parameter L. We investigate if and by how much the different $K_{\rm{sym}}-L$ correlations may affect neutron star observables. Method: A meta-model of nuclear Equation of States (EOSs) with three representative $K_{\rm{sym}}-L$ correlation functions is used to generate multiple EOSs for neutron stars. We then examine effects of the $K_{\rm{sym}}-L$ correlation on the crust-core transition density and pressure as well as the radius and tidal deformation of canonical neutron stars. Results:The $K_{\rm{sym}}-L$ correlation affects significantly both the crust-core transition density and pressure. It also has strong imprints on the radius and tidal deformability of canonical neutron stars especially at small L values. The available data from LIGO/VIRGO and NICER set some useful limits for the slope L but can not distinguish the three representative $K_{\rm{sym}}-L$ correlations considered.

Bayesian inference of nuclear symmetry energy from measured and imagined neutron skin thickness in Sn116,118,120,122,124,130,132, Pb208 , and Ca48

The neutron skin thickness $\Delta r_{np}$ in heavy nuclei has been known as one of the most sensitive terrestrial probes of the nuclear symmetry energy around $\frac{2}{3}$ of the saturation density $\rho_0$ of nuclear matter. Existing neutron skin data mostly from hadronic observables suffer from large uncertainties and their extraction from experiments are often strongly model dependent. While waiting eagerly for the promised model-independent and high-precision neutron skin data for $^{208}$Pb and $^{48}$Ca from the parity-violating electron scattering experiments (PREX-II and CREX at JLab as well as MREX at MESA), within the Bayesian statistical framework using the Skyrme-Hartree-Fock model we infer the posterior probability distribution functions (PDFs) of the slope parameter $L$ of the nuclear symmetry energy at $\rho_0$ from imagined $\Delta r_{np}(^{208}$Pb$)=0.15$, 0.20, and 0.30 fm as well as $\Delta r_{np}(^{48}$Ca$)=0.12$, 0.15, and 0.25 fm, with different $1\sigma$ error bars. The results are compared with the PDFs of $L$ inferred using the same approach from the available $\Delta r_{np}$ data for Sn isotopes from hadronic probes. They are also compared with results from a recent Bayesian analysis of the radius and tidal deformability data of canonical neutron stars from GW170817 and NICER. The neutron skin data for Sn isotopes gives $L=45.5^{+26.5}_{-21.6}$ MeV surrounding its mean value or $L=53.4^{+18.6}_{-29.5}$ MeV surrounding its maximum {\it a posteriori} value, respectively, with the latter smaller than but consistent with the $L=66_{-20}^{+12}$ MeV from the neutron star data within their 68\% confidence intervals. In order to provide additionally useful information on $L$ extracted from the $\Delta r_{np}$ of Sn isotopes, the experimental error bar of $\Delta r_{np}(^{208}$Pb) should be at least smaller than 0.06 fm aimed by some current experiments.

Neural network reconstruction of the dense matter equation of state derived from the parameters of neutron stars

Context. Neutron stars are currently studied with an rising number of electromagnetic and gravitational-wave observations, which will ultimately allow us to constrain the dense matter equation of state and understand the physical processes at work within these compact objects. Neutron star global parameters, such as the mass and radius, can be used to obtain the equation of state by directly inverting the Tolman-Oppenheimer-Volkoff equations. Here, we investigate an alternative approach to this procedure. Aims. The aim of this work is to study the application of the artificial neural networks guided by the autoencoder architecture as a method for precisely reconstructing the neutron star equation of state, using their observable parameters: masses, radii, and tidal deformabilities. In addition, we study how well the neutron star radius can be reconstructed using only the gravitational-wave observations of tidal deformability, that is, using quantities that are not related in any straightforward way. Methods. The application of an artificial neural network in the equation-of-state reconstruction exploits the non-linear potential of this machine learning model. Since each neuron in the network is basically a non-linear function, it is possible to create a complex mapping between the input sets of observations and the output equation-of-state table. Within the supervised training paradigm, we construct a few hidden-layer deep neural networks on a generated data set, consisting of a realistic equation of state for the neutron star crust connected with a piecewise relativistic polytropes dense core, with its parameters representative of state-of-the art realistic equations of state. Results. We demonstrate the performance of our machine-learning implementation with respect to the simulated cases with a varying number of observations and measurement uncertainties. Furthermore, we study the impact of the neutron star mass distributions on the results. Finally, we test the reconstruction of the equation of state trained on parametric polytropic training set using the simulated mass–radius and mass–tidal-deformability sequences based on realistic equations of state. Neural networks trained with a limited data set are capable of generalising the mapping between global parameters and equation-of-state input tables for realistic models.

Delta baryons and diquark formation in the cores of neutron stars

We investigate the hadron-quark phase transition in cold neutron stars in light of (i) the observed limits on the maximum-mass of heavy pulsars, (ii) constraints on the tidal properties inferred from the gravitational waves emitted in binary neutron-star mergers, and (iii) mass and radius constraints derived from the observation of hot spots on neutron star observed with NICER. Special attention is directed to the possible presence of $\Delta(1232)$ baryons in neutron star matter. Our results indicate that this particle could make up a large fraction of the baryons in neutron stars and thus have a significant effect on the properties of such objects, particularly on their radii. This is partially caused by the low density appearance of $\Delta$s for a wide range of theoretically defensible sets of meson-hyperon, SU(3) ESC08 model, and meson-$\Delta$ coupling constants. The transition of hadronic matter to quark matter, treated in the 2SC+s condensation phase, is found to occur only in neutron stars very close to the mass peak. Nevertheless, quark matter may still constitute an appreciable fraction of the stars' total matter if the phase transition is treated as Maxwell-like (sharp), in which case the neutron stars located beyond the gravitational mass peak would remain stable against gravitational collapse. In this case, the instability against gravitational collapse is shifted to a new (terminal) mass different from the maximum-mass of the stellar sequence, giving rise to stable compact objects with the same gravitational masses as those of the neutron stars on the traditional branch, but whose radii are smaller by up to 1 km. All models for the equation of state of our study fall comfortably within the bound established very recently by Annala {\it et al.} (Nature Physics, 2020)

Fast Rotating Relativistic Stars: Spectra and Stability without Approximation.

We study oscillations and instabilities of relativistic stars using perturbation theory in general relativity and take into account the contribution of a dynamic spacetime. We present the oscillation spectrum as well as the critical values for the onset of the secular CFS instability of neutron stars, and propose universal relations for gravitational wave asteroseismology, which may help constrain the neutron star radius and/or the nuclear equation of state. The results are relevant for all stages during a neutron star's life but especially to nascent or remnant objects following a binary merger.

Searching optimum equations of state of neutron star matter in strong magnetic fields with rotation

Abstract Masses and radii of neutron stars are obtained in the presence of strong magnetic fields together with rotation. Mass-radius relations are calculated using 11 equations of state (EoSs: GM1, TM1-a, TM1-b, TM2$\omega\rho$-a, TM2$\omega\rho$-b, NL3-a, NL3-b, NL3$\omega\rho$-a, NL3$\omega\rho$-b, DDME2-a and DDME2-b) in relativistic mean field (RMF) theory. Obtained masses are over and around twice the solar mass ($M_\odot$) for all EoSs in the presence of strong magnetic fields of $3 \times 10^{18}$ G at the center. For NL3$\omega\rho$-a and NL3$\omega\rho$-b EoSs, masses are more than $M=2.17\,M_\odot$(observed maximum mass: $2.14\,M_\odot$) even without magnetic fields. Rotational effects are found to be insignificant in any case, at least up to the Kepler frequency. Suitable EoSs are also selected concerning the constraint on the radius of a neutron star.

Relativistic rotating vector model for X-ray millisecond pulsars

The X-ray radiation produced on the surface of accreting magnetised neutron stars is expected to be strongly polarised. A swing of the polarisation vector with the pulsar phase gives a direct measure of the source inclination and magnetic obliquity. In the case of rapidly rotating millisecond pulsars, the relativistic motion of the emission region causes additional rotation of the polarisation plane. Here, we develop a relativistic rotating vector model, where we derive analytical expression for the polarisation angle as a function of the pulsar phase accounting for relativistic aberration and gravitational light bending in the Schwarzschild metric. We show that in the case of fast pulsars the rotation of the polarisation plane can reach tens of degrees, strongly influencing the observed shape of the polarisation angle’s phase dependence. The rotation angle grows nearly linearly with the spin rate but it is less sensitive to the neutron star radius. Overall, this angle is large even for large spots. Our results have implications with regard to the modelling of X-ray polarisation from accreting millisecond pulsars that are to be observed with the upcoming Imaging X-ray Polarimeter Explorer and the enhanced X-ray Timing and Polarimetry mission. The X-ray polarisation may improve constraints on the neutron star mass and radius coming from the pulse profile modelling.

Astrophysical implications of neutron star inspiral and coalescence

The first inspiral of two neutron stars observed in gravitational waves was remarkably close, allowing the kind of simultaneous gravitational wave and electromagnetic observation that had not been expected for several years. Their merger, followed by a gamma-ray burst and a kilonova, was observed across the spectral bands of electromagnetic telescopes. These GW and electromagnetic observations have led to dramatic advances in understanding short gamma-ray bursts; determining the origin of the heaviest elements; and determining the maximum mass of neutron stars. From the imprint of tides on the gravitational waveforms and from observations of X-ray binaries, one can extract the radius and deformability of inspiraling neutron stars. Together, the radius, maximum mass, and causality constrain the neutron-star equation of state, and future constraints can come from observations of post-merger oscillations. We selectively review these results, filling in some of the physics with derivations and estimates.

Observational appearance of rapidly rotating neutron stars

Neutron stars (NSs) in low-mass X-ray binaries rotate at frequencies high enough to significantly deviate from sphericity (ν*∼ 200–600 Hz). First, we investigate the effects of rapid rotation on the observational appearance of a NS. We propose analytical formulae relating gravitational mass and equatorial radius of the rapidly rotating NS to the massMand radiusRof a non-rotating NS of the same baryonic mass using accurate fully relativistic computations. We assume that the NS surface emission is described by the Planck function with two different emission patterns: the isotropic intensity and that corresponding to the electron-scattering dominated atmosphere. For these two cases we compute spectra from an oblate rotating NS observed at different inclination angles using the modified oblate Schwarzschild approximation, where light bending is computed in Schwarzschild metric, but frame dragging and quadrupole moment of a NS are approximately accounted for in the photon redshift calculations. In particular, we determine the solid angle at which a rotating NS is seen by a distant observer, the observed colour temperature and the blackbody normalization. Then, we investigate how rapid rotation affects the results of NS radius determination using the cooling tail method applied to the X-ray burst spectral evolution. We approximate the local spectra from the NS surface by a diluted blackbody with the luminosity-dependent dilution factor using previously computed NS atmosphere models. We then generalize the cooling tail method to the case of a rapidly rotating NS to obtain the most probable values ofMandRof the corresponding non-rotating NS with the same baryonic mass. We show that the NS radius could be overestimated by 3–3.5 km for face-on stars ofR ≈ 11 km rotating atν*= 700 Hz if the version of the cooling tail method for a non-rotating NS is used. We apply the method to an X-ray burst observed from the NS rotating atν* ≈ 532 Hz in SAX J1810.8−2609. The resulting radius of the non-rotating NS (assumingM = 1.5M⊙) becomes 11.8 ± 0.5 km if it is viewed at inclinationi = 60° andR = 11.2 ± 0.5 km for a face-on view, which are smaller by 0.6 and 1.2 km than the radius obtained using standard cooling tail method ignoring rotation. The corresponding equatorial radii of these rapidly rotating NSs are 12.3 ± 0.6 km (fori = 60°) and 11.6 ± 0.6 km (fori = 0°).