The equation of state of cold supra-nuclear-density matter, such as in neutron stars, is an open question in astrophysics. A promising method for constraining the neutron star equation of state is modelling pulse profiles of thermonuclear X-ray burst oscillations from hotspots on accreting neutron stars. The pulse profiles, constructed using spherical and oblate neutron star models, are comparable to what would be observed by a next-generation X-ray timing instrument like ASTROSAT, NICER, or LOFT. In this paper, we showcase the use of an evolutionary optimization algorithm to fit pulse profiles to determine the best-fit masses and radii. By fitting synthetic data, we assess how well the optimization algorithm can recover the input parameters. Multiple Poisson realizations of the synthetic pulse profiles, constructed with 1.6 million counts and no background, were fitted with the Ferret algorithm to analyze both statistical and degeneracy-related uncertainty, and to explore how the goodness-of-fit depends on the input parameters. For the regions of parameter space sampled by our tests, the best-determined parameter is the projected velocity of the spot along the observer's line-of-sight, with an accuracy of $\le3$% compared to the true value and with $\le5$\% statistical uncertainty. The next best-determined are the mass and radius; for a neutron star with a spin frequency of 600\,Hz, the best-fit mass and radius are accurate to $\le5$%, with respective uncertainties of $\le7$% and $\le10$%. The accuracy and precision depend on the observer inclination and spot co-latitude, with values of $\sim1$ % achievable in mass and radius if both the inclination and co-latitude are > 60 degrees.
This article gives a very brief introduction about measuring the mass and radius of neutron star from X-ray observations. The masses and radii of neutron stars can be determined from photospheric radius expansion bursts in low-mass X-ray binaries, X-ray pulse profile modeling in accreting X-ray pulsars, gravitational redshift measurement in low-mass X-ray binaries and thermal X-ray spectral fitting in quiescent low-mass X-ray binaries.
Millisecond pulsars are rapidly rotating neutron stars where general relativity plays a strong role in the propagation of light from the neutron star to observer. The observed X-ray pulse shapes carry information on the mass, radius and surface shape of the neutron star. Comparison of theoretical calculations of pulse shapes with observed pulse shapes can give useful constraints on neutron star properties. Then comparison with calculated properties giving an assumed equation of state (EOS) can confirm or rule out the assumed EOS.
We present a new method for determining masses and radii of neutron stars residing in thermo-nuclear X-ray burst sources. To illustrate this method we apply it to a burst from the source 4U 1820-30 recorded by the Rossi X-Ray Timing Explorer. Fits of the observed X-ray spectra to grids of Comptonised model atmospheres yield estimates for the mass and radius of the neutron star, M=1.3 \pm 0.6 M_sol and R=11^+3_-2 km, respectively.
X-ray spectra of quiescent low-mass X-ray binaries containing neutron stars can be fit with atmosphere models to constrain the mass and the radius. Mass-radius constraints can be used to place limits on the equation of state of dense matter. We perform fits to the X-ray spectrum of a quiescent neutron star in the globular cluster M13, utilizing data from ROSAT, Chandra and XMM-Newton, and constrain the mass-radius relation. Assuming an atmosphere composed of hydrogen and a 1.4${\rm M}_{\odot}$ neutron star, we find the radius to be $R_{\rm NS}=12.2^{+1.5}_{-1.1}$ km, a significant improvement …
We calculate the effects of spot size on pulse profiles of moderately rotating neutron stars. Specifically, we quantify the bias introduced in radius measurements from the common assumption that spots are infinitesimally small. We find that this assumption is reasonable for spots smaller than 10$^\circ$-18$^\circ$ and leads to errors that are $\le$10% in the radius measurement, depending on the location of the spot and the inclination of the observer. We consider the implications of our results for neutron star radius measurements with the upcoming and planned X-ray missions NICER and LOFT. We…
We use time resolved spectroscopy of thermonuclear X-ray bursts observed from SAX J1748.9-2021 to infer the mass and the radius of the neutron star in the binary. Four X-ray bursts observed from the source with RXTE enable us to measure the angular size and the Eddington limit on the neutron star surface. Combined with a distance measurement to the globular cluster NGC 6440, in which SAX J1748.9-2021 resides, we obtain two solutions for the neutron star radius and mass, R = 8.18 +/- 1.62 km and M = 1.78 +/- 0.3 M_\sun or R = 10.93 +/- 2.09 km and M = 1.33 +/- 0.33 M_\sun.
Gravitational waves from the coalescence of two neutron stars were recently detected for the first time by the LIGO-Virgo collaboration, in event GW170817. This detection placed an upper limit on the effective tidal deformability of the two neutron stars and tightly constrained the chirp mass of the system. We report here on a new simplification that arises in the effective tidal deformability of the binary, when the chirp mass is specified. We find that, in this case, the effective tidal deformability of the binary is surprisingly independent of the component masses of the individual neutron …
The mass and radius of the neutron star (NS) in low-mass X-ray binaries can be obtained by fitting the X-ray spectrum of the NS in quiescence, and the mass and radius constrains the properties of dense matter in NS cores. A critical ingredient for spectral fits is the composition of the NS atmosphere: hydrogen atmospheres are assumed in most prior work, but helium atmospheres are possible if the donor star is a helium white dwarf. Here we perform spectral fits to XMM, Chandra, and ROSAT data of a quiescent NS in the globular cluster M13. This NS has the smallest inferred radius from previous s…
In this paper we show that X-ray spectral observations of the ATHENA mission, which is planned to launch in 2031, can constrain the equation of state of superdense matter. We use our well-constrained continuum fitting method for mass and radius determination of the neutron star. Model spectra of the emission from a neutron star were calculated using the atmosphere code ATM24. In the next step, those models were fitted to a simulated spectra of the neutron star calculated for ATHENA's WFI detector, using the satellite calibration files. To simulate the spectra we assumed three different values …
We describe the model of surface emission from a rapidly rotating neutron star that is applied to Neutron Star Interior Composition Explorer X-ray data of millisecond pulsars in order to statistically constrain the neutron star mass-radius relation and dense matter equation of state. To ensure that the associated calculations are both accurate and precise, we conduct an extensive suite of verification tests between our numerical codes for both the Schwarzschild + Doppler and Oblate Schwarzschild approximations, and compare both approximations against exact numerical calculations. We find super…
The accreting millisecond pulsar XTE J1807-294 is studied through a pulse shape modeling analysis. The model includes blackbody and Comptonized emission from the one visible hot spot and makes use of the Oblate Schwarzschild approximation for ray-tracing. We include a scattered light contribution, which accounts for flux scattered off an equatorial accretion disk to the observer including time delays in the scattered light. We give limits to mass and radius for XTE J1807-294 and compare to limits determined for SAX J1808-3658 and XTE J1814-334 previously determined using similar methods. The r…
We interpret recent observations of high-to-low frequency drifting features in the spectra of the repeating FRBs as evidence of sharply changing plasma properties in the emission region, presumably the neutron stars magnetospheres. The drifts are then FRBs' analogues of radius-to-frequency mapping in pulsars and Solar type-III radio burst (but not in a sense of a particular emission mechanism). The drifts rates of $\sim 100$ MHz ms$^{-1}$ at frequencies $\sim$ GHz translate to physical size of $\sim {c ω}/{\dotω} \sim$ few $\times 10^8$ cm, matching the hypothesis of the FRB origin in the magn…
Methods. We use both a Monte Carlo (MC) sampling and a Bayesian analysis to find the effects of the photospheric composition and the touchdown radius. We apply these two methods to six LMXBs that show PRE XRBs. In both methods, we solve the Eddington flux equation and the apparent angular area equation both of which include the correction terms. For the MC sampling, we have developed an iterative method in order to solve these two equations more efficiently. Results. We confirm that the effects of the photospheric composition and the touchdown radius are similar in the statistical and analytic…
We analyse four XMM-Newton observations of the neutron-star low-mass X-ray binary EXO 0748$-$676 in quiescence. We fit the spectra with an absorbed neutron-star atmosphere model, without the need for a high-energy (power-law) component; with a 95 per cent confidence the power-law contributes less than 1 per cent to the total flux of the source in $0.5-10.0$ keV. The fits show significant residuals at around 0.5 keV which can be explained by either a hot gas component around the neutron star or a moderately broad emission line from a residual accretion disc. The temperature of the neutron-star …
We have investigated the critical conditions required for a steady propeller effect for magnetized neutron stars with optically thick, geometrically thin accretion disks. We have shown through simple analytical calculations that a steady-state propeller mechanism cannot be sustained at an inner disk radius where the viscous and magnetic stresses are balanced. The radius calculated by equating these stresses is usually found to be close to the conventional Alfven radius for spherical accretion, r_A. Our results show that: (1) a steady propeller phase can be established with a maximum inner disk…
The aim of this work is to study the possible effects and biases on the radius constraints for rotation-powered millisecond pulsars when using Thomson approximation to describe electron scattering in the atmosphere models, instead of using exact formulation for Compton scattering. We compare the differences between the two models in the energy spectrum and angular distribution of the emitted radiation. We also analyse a self-generated synthetic phase-resolved energy spectrum, based on Compton atmosphere and the most X-ray luminous rotation-powered millisecond pulsars observed by the Neutron st…
We make extensive numerical studies of masses and radii of proto-neutron stars during the first second after their birth in core-collapse supernova events. We use a quasi-static approach for the computation of proto-neutron star structure, built on parameterized entropy and electron fraction profiles, that are then evolved with neutrino cooling processes. We vary the equation of state of nuclear matter, the proto-neutron star mass and the parameters of the initial profiles, to take into account our ignorance of the supernova progenitor properties. We show that if masses and radii of a proto-ne…
Neutron stars with near-Eddington observable luminosities were shown to harbor levitating atmospheres, suspended above their surface. We report a new method to simultaneously measure the mass and radius of a neutron star based on oscillations of such atmospheres. In this paper, we present an analytic derivation of a family of relativistic, oscillatory, spherically symmetric eigenmodes of the optically and geometrically thin levitating atmospheres, including the damping effects induced by the radiation drag. We discover characteristic maxima in the frequencies of the damped oscillations and sho…
Connecting radio pulsars, magnetars, and XDINSs in a unified evolutionary framework using simulation-based inference
ASTRON, the Netherlands Institute for Radio Astronomy（荷兰射电天文研究所）；Institute of Space Sciences (CSIC-ICE)（空间科学研究所）；Institut d’Estudis Espacials de Catalunya (IEEC)（加泰罗尼亚太空研究学院）；Department of Physics, Royal Holloway, University of London（伦敦大学皇家霍洛威学院物理系）；Departament de Física, Universitat d’Alacant（阿利坎特大学物理系）
We analyze the latest \emph{Suzaku} observation of the bright neutron star low-mass X-ray binary Serpens X-1 taken in 2013 October and 2014 April. The observation was taken using the burst mode and only suffered mild pile-up effects. A broad iron line is clearly detected in the X-ray spectrum. We test different models and find that the iron line is asymmetric and best interpreted by relativistic reflection. The relativistically broadened iron line is generally believed to originate from the innermost regions of the accretion disk, where strong gravity causes a series of special and general rel…
Rotation causes an increase in a neutron star's mass and equatorial radius. The mass and radius depend sensitively on the unknown equation of state (EOS) of cold, dense matter. However, the increases in mass and radius due to rotation are almost independent of the EOS. The EOS independence leads to the idea of neutron star universality. In this paper, we compute sequences of rotating neutron stars with constant central density. We use a collection of randomly generated EOS to construct simple correction factors to the mass and radius computed from the equations of hydrostatic equilibrium for n…
The measurement of neutron star mass and radius is one of the most direct way to distinguish between various dense matter equations of state. The mass and radius of accreting neutron stars hosted in low mass X-ray binaries can be constrained by several methods, including photospheric radius expansion from type-I X-ray bursts and from quiescent spectra. In this paper, we apply for the first time these two methods simultaneously to constrain the mass and radius of Aql X-1, as a reliable distance estimation, high signal-to-noise ratio quiescent spectra from Chandra and XMM-Newton, and photospheri…
We analyzed the thermal emission from the entire surface of the millisecond pulsar PSR J0437-4715 observed in the ultraviolet and soft X-ray bands. For this, we calculated non-magnetized, partially ionized atmosphere models of hydrogen, helium, and iron compositions and included plasma frequency effects that may affect the emergent spectrum. This is particularly true for the coldest atmospheres composed of iron (up to a few percent changes in the soft X-ray flux). Employing a Markov chain Monte Carlo method, we found that the spectral fits favour a hydrogen atmosphere, disfavour a helium compo…
The dependence of the spin frequency derivative $\dotν$ of accreting neutron stars with a strong magnetic field (X-ray pulsars) on the mass accretion rate (bolometric luminosity, $L_{bol}$) has been investigated for eight transient pulsars in binary systems with Be stars. Using data from the Fermi/GBM and Swift/BAT telescopes, we have shown that for seven of the eight systems the dependence $\dotν$ can be fitted by the model of angular momentum transfer through an accretion disk, which predicts the relation $\dotν\sim L^{6/7}_{bol}$. Hysteresis in the dependence $\dotν(L_{bol})$ has been confi…
Inspired by the excluded volume model for isospin symmetric quarkyonic matter, we construct an `excluded volume' model for a charge neutral quarkyonic phase whose hadronic sector contains only neutrons. We refer to this model as quarkyonic neutron matter. We compute the equation of state for this model and solve the Tolman-Oppenhermer-Volkoff equations to obtain mass and radius relations relevant for neutron stars. The most straightforward extension of the model for symmetric quarkyonic matter to quarkyonic neutron matter does not satisfy the mass radius constraints from neutron star measureme…
We report on the measurement of the mass and radius of the neutron star in the low-mass X-ray binary 4U 1820-30. The analysis of the spectroscopic data on multiple thermonuclear bursts yields well-constrained values for the apparent emitting area and the Eddington flux, both of which depend in a distinct way on the mass and radius of the neutron star. The distance to the source is that of the globular cluster NGC 6624, where the source resides. Combining these measurements, we uniquely determine the probability density over the stellar mass and radius. We find the mass to be M = 1.58 +/- 0.06 …
Thermonuclear runaway burning of carbon is in rare cases observed from accreting neutron stars as day-long X-ray flares called superbursts. In the few cases where the onset is observed, superbursts exhibit a short precursor burst at the start. In each instance, however, the data was of insufficient quality for spectral analysis of the precursor. Using data from the propane anti-coincidence detector of the PCA instrument on RXTE, we perform the first detailed time resolved spectroscopy of precursors. For a superburst from 4U 1820-30 we demonstrate the presence of photospheric radius expansion. …
X-ray spectral analysis of quiescent low-mass X-ray binaries (LMXBs) has been one of the most common tools to measure the radius of neutron stars (NSs) for over a decade. So far, this method has been mainly applied to NSs in globular clusters, primarily because of their well-constrained distances. Here, we study Chandra data of seven transient LMXBs in the Galactic plane in quiescence to investigate the potential of constraining the radius (and mass) of the NSs inhabiting these systems. We find that only two of these objects had X-ray spectra of sufficient quality to obtain reasonable constrai…
We calculate the rotational broadening in the observed thermal spectra of neutron stars spinning at moderate rates in the Hartle-Thorne approximation. These calculations accurately account for the effects of the second-order Doppler boosts as well as for the oblate shapes and the quadrupole moments of the neutron stars. We find that fitting the spectra and inferring the bolometric fluxes under the assumption that a star is not rotating causes an underestimate of the inferred fluxes and, thus, radii. The correction depends on the stellar spin, radius, and observer's inclination. For a 10 km neu…
Upcoming observing campaigns with improved detectors will yield numerous detections of gravitational waves from neutron star binary inspirals. Rare loud signals together with numerous signals of moderate strength promise stringent constraints on the properties of neutron star matter, with a projected radius statistical uncertainty of $50-200$m with ${\cal{O}}(2000)$ sources. Given this precision we revisit all analysis assumptions and identify sources of systematic errors, quantify their impact on radius extraction, and discuss their relative importance and ways to mitigate them.
The equation of state (EOS) of dense matter has been a long-sought goal of nuclear physics. Equations of state generate unique mass versus radius (M-R) relations for neutron stars, the ultra-dense remnants of stellar evolution. In this work, we determine the neutron star mass-radius relation and, based on recent observations of both transiently accreting and bursting sources, we show that the radius of a 1.4 solar mass neutron star lies between 10.4 and 12.9 km, independent of assumptions about the composition of the core. We show, for the first time, that these constraints remain valid upon r…
We calculate the equation of state of neutron matter with realistic two- and three-nucleon interactions using quantum Monte Carlo techniques, and illustrate that the short-range three-neutron interaction determines the correlation between neutron matter energy at nuclear saturation density and higher densities relevant to neutron stars. Our model also makes an experimentally testable prediction for the correlation between the nuclear symmetry energy and its density dependence -- determined solely by the strength of the short-range terms in the three neutron force. The same force provides a sig…
We investigate theoretical and observational constraints on the mass-radius relations for neutron stars. For that purpose we consider the model of neutron stars taking into considerations strong, weak, electromagnetic and gravitational interactions in the equation of state and integrate the structure equations within the Hartle-Thorne formalism for rotating configurations. On the basis of the theoretical restrictions imposed by general relativity, mass-shedding and axisymmetric secular instabilities we calculate the upper and lower bounds for the parameters of neutron stars. Our theoretical ca…
We show how gravitational-wave observations with advanced detectors of tens to several tens of neutron-star binaries can measure the neutron-star radius with an accuracy of several to a few percent, for mass and spatial distributions that are realistic, and with none of the sources located within 100 Mpc. We achieve such an accuracy by combining measurements of the total mass from the inspiral phase with those of the compactness from the postmerger oscillation frequencies. For estimating the measurement errors of these frequencies we utilize analytical fits to postmerger numerical-relativity w…
Neutron stars, produced at the death of massive stars, are often regarded as giant neutron-rich nuclei. This picture is especially relevant for low-mass (below about solar mass) neutron stars, where non-nucleonic components are not expected to occur. Due to the saturation property of nucleonic matter, leading to the celebrated liquid-drop picture of atomic nuclei, empirical nuclear masses and radii can be approximately expressed as function of atomic mass number. It is, however, not straightforward to express masses and radii of neutron stars even in the low-mass range where the structure is d…
We present Chandra ACIS-S sub-array observations of the quiescent neutron star low-mass X-ray binaries X7 and X5 in the globular cluster 47 Tuc. The large reduction in photon pile-up compared to previous deep exposures enables a substantial improvement in the spectroscopic determination of the neutron star radius and mass of these neutron stars. Modeling the thermal emission from the neutron star surface with a non-magnetized hydrogen atmosphere and accounting for numerous sources of uncertainties, we obtain for the neutron star in X7 a radius of $R=11.1^{+0.8}_{-0.7}$ km for an assumed stella…
Motivated by the unknown nature of the $2.50-2.67\,M_\odot$ compact object in the binary merger event GW190814, we study the maximum neutron star mass based on constraints from low-energy nuclear physics, neutron star tidal deformabilities from GW170817, and simultaneous mass-radius measurements of PSR J0030+045 from NICER. Our prior distribution is based on a combination of nuclear modeling valid in the vicinity of normal nuclear densities together with the assumption of a maximally stiff equation of state at high densities, a choice that enables us to probe the connection between observed he…
The neutron star maximum mass and the radius are investigated within the framework of the relativistic mean-field (RMF) model. The variation in the radius at the canonical mass, $R_{1.4}$, using different inner crust equation of state (EoS) with different symmetry energy slope parameter is studied. It is found that although the NS maximum mass and the corresponding radius do not vary much with different inner crust EoSs, the radius and the tidal deformability at 1.4$M_{\odot}$ vary with the different choice of crust EoS and variation of about 1-2 km is seen in the radius at the canonical mass.…
We derive the empirical formulas for the neutron star mass and gravitational redshift as a function of the central density and specific combination of the nuclear saturation parameters, which are applicable to the stellar models constructed with the central density up to threefold nuclear saturation density. Combining the both empirical formulas, one also estimates the neutron star radius. In practice, we find that the neutron star mass (radius) can be estimated within $\sim 10\%$ (a few percent) accuracy by comparing the mass and radius evaluated with our empirical formulas to those determine…
Neutron stars are formed in core-collapse supernova explosions, where a large number of neutrinos are emitted. In this paper, supernova neutrino light curves are computed for the cooling phase of protoneutron stars, which lasts a few minutes. In the numerical simulations, 90 models of the phenomenological equation of state with different incompressibilities, symmetry energies, and nucleon effective masses are employed for a comprehensive study. It is found that the cooling timescale is longer for a model with a larger neutron star mass and a smaller neutron star radius. Furthermore, a theoreti…
We analyze observations of eight quiescent low-mass X-ray binaries in globular clusters and combine them to determine the neutron star mass-radius curve and the equation of state of dense matter. We determine the effect that several uncertainties may have on our results, including uncertainties in the distance, the atmosphere composition, the neutron star maximum mass, the neutron star mass distribution, the possible presence of a hotspot on the neutron star surface, and the prior choice for the equation of state of dense matter. We find that the radius of a 1.4 solar mass neutron star is most…
The compact neutron star in 4U 1746-37 revisited: Reassessing the mass and radius
Ulsan National Institute of Science and Technology（蔚山科学技术院）；Institute for Basic Science（基础科学研究院）；Korea Astronomy and Space Science Institute（韩国天体宇宙科学研究所）；University of Science and Technology（科学技术大学）；Pusan National University（釜山国立大学）
We construct parameter sets of the relativistic mean-field model fitted to the recent constraints on the asymmetry energy $J$ and the slope parameter $L$ for pure neutron matter. We find cases of unphysical behaviour, i.e.\ the appearance of negative pressures, for stiff parameter sets with low values of the effective mass $m^*/m$. In some cases the equation of state of pure neutron matter turns out to be outside the allowed band given by chiral effective field theory. The mass-radius relations of neutron stars for all acceptable parameter sets shows a maximum mass in excess of $2M_\odot$ bein…
We study the effects of free space nucleon radius on nuclear matter and neutron stars within the framework of quark mean field model. The nucleon radius is treated self-consistently with this model, where quark confinement is adjusted to fit different values of nucleon radius. Corrections due to center-of-mass motion, quark-pion coupling, and one gluon exchange are included to obtain the nucleon mass in vacuum. The meson coupling constants that describe the behavior of many-body nucleonic system are newly-constructed by reproducing the empirical saturation properties of nuclear matter, includi…
Relativistic spectral lines from the accretion disc of a neutron star low-mass X-ray binary can be modelled to infer the disc inner edge radius. A small value of this radius tentatively implies that the disc terminates either at the neutron star hard surface, or at the innermost stable circular orbit (ISCO). Therefore an inferred disc inner edge radius either provides the stellar radius, or can directly constrain stellar equation of state (EoS) models using the theoretically computed ISCO radius for the spacetime of a rapidly spinning neutron star. However, this procedure requires numerical co…
The post-merger gravitational wave (GW) emission from a binary neutron star merger is expected to provide exciting new constraints on the dense-matter equation of state (EoS). Such constraints rely, by and large, on the existence of quasi-universal relations, which relate the peak frequencies of the post-merger GW spectrum to properties of the neutron star structure in a model-independent way. In this work, we report on violations of existing quasi-universal relations between the peak spectral frequency, f_2, and the stellar radius, for EoSs models with backwards-bending slopes in their mass-r…
Coalescing neutron star (NS)-black hole (BH) binaries are promising sources of gravitational-waves (GWs) to be detected within the next few years by current GW observatories. If the NS is tidally disrupted outside the BH innermost stable circular orbit, an accretion torus may form, and this could eventually power a short gamma-ray burst (SGRB). The observation of an SGRB in coincidence with gravitational radiation from an NS-BH coalescence would confirm the association between the two phenomena and also give us new insights on NS physics. We present here a new method to measure NS radii and th…
The luminosity of accreting magnetised neutron stars can largely exceed the Eddington value due to appearance of accretion columns. The height of the columns can be comparable to the neutron star radius. The columns produce the X-rays detected by the observer directly and illuminate the stellar surface, which reprocesses the X-rays and causes additional component of the observed flux. The geometry of the column and the illuminated part of the surface determines the radiation beaming. Curved space-time affects the angular flux distribution. We construct a simple model of the beam patterns forme…
