Non-rotating Neutron Stars Research Articles

Asteroseismology: Radial oscillations of neutron stars with realistic equation of state

We study radial oscillations of non-rotating neutron stars (NSs) in four-dimensional General Relativity. The interior of the NS was modelled within a recently proposed multicomponent realistic equation of state (EoS) with the induced surface tension (IST). In particular, we considered the IST EoS with two sets of model parameters, that both reproduce all the known properties of normal nuclear matter, give a high quality description of the proton flow constraint, hadron multiplicities created in nuclear-nuclear collisions, consistent with astrophysical observations and the observational data from the NS-NS merger. We computed the 12 lowest radial oscillation modes, their frequencies and corresponding eigenfunctions, as well as the large frequency separation for six selected fiducial NSs (with different radii and masses of 1.2, 1.5 and 1.9 solar masses) of the two distinct model sets. The calculated frequencies show their continuous growth with an increase of the NS central baryon density. Moreover, we found correlations between the behaviour of first eigenfunction calculated for the fundamental mode, the adiabatic index and the speed of sound profile, which could be used to probe the internal structure of NSs with the asteroseismology data.

PSR J0030+0451, GW170817, and the Nuclear Data: Joint Constraints on Equation of State and Bulk Properties of Neutron Stars

Abstract Very recently the NICER collaboration published the first-ever accurate measurement of mass and radius together for PSR J0030+0451, a nearby isolated quickly rotating neutron star (NS). In this work we set the joint constraints on the equation of state (EoS) and some bulk properties of NSs with the data of PSR J0030+0451, GW170817, and some nuclear experiments. The piecewise polytropic expansion method and the spectral decomposition method have been adopted to parameterize the EoS. The resulting constraints are consistent with each other. Assuming the maximal gravitational mass of nonrotating NS M TOV lies between 2.04M ⊙ and 2.4M ⊙, with the piecewise method the pressure at twice nuclear saturation density is measured to be at the 90% level. For an NS with canonical mass of 1.4M ⊙, we have the moment of inertia , tidal deformability , radius , and binding energy at the 90% level, which are improved in comparison to the constraints with the sole data of GW170817. These conclusions are drawn for the mass/radius measurements of PSR J0030+0451 by Riley et al. For the measurements of Miller et al., the results are rather similar.

Gravitational waves or deconfined quarks: What causes the premature collapse of neutron stars born in short gamma-ray bursts?

We infer the collapse times of long-lived neutron stars into black holes using the X-ray afterglows of 18 short gamma-ray bursts. We then apply hierarchical inference to infer properties of the neutron star equation of state and dominant spin-down mechanism. We measure the maximum non-rotating neutron star mass $M_\mathrm{TOV} = 2.31 ^{+0.36}_{-0.21} M_{\odot}$ and constrain the fraction of remnants spinning down predominantly through gravitational-wave emission to $\eta = 0.69 ^{+0.21}_{-0.39}$ with $68 \%$ uncertainties. In principle, this method can determine the difference between hadronic and quark equation of states. In practice, however, the data is not yet informative with indications that these neutron stars do not have hadronic equation of states at the $1\sigma$ level. These inferences all depend on the underlying progenitor mass distribution for short gamma-ray bursts produced by binary neutron star mergers. The recently announced gravitational-wave detection of GW190425 suggests this underlying distribution is different from the locally-measured population of double neutron stars. We show that $M_\mathrm{TOV}$ and $\eta$ constraints depend on the fraction of binary mergers that form through a distribution consistent with the locally-measured population and a distribution that can explain GW190425. The more binaries that form from the latter distribution, the larger $M_\mathrm{TOV}$ needs to be to satisfy the X-ray observations. Our measurements above are marginalised over this unknown fraction. If instead, we assume GW190425 is not a binary neutron star merger, i.e the underlying mass distribution of double neutron stars is the same as observed locally, we measure $M_\mathrm{TOV} = 2.26 ^{+0.31}_{-0.17} M_{\odot}$.

Effects of the equation of state on the bulk properties of maximally rotating neutron stars

Neutron stars are among the densest known objects in the universe and an ideal laboratory for the strange physics of super-condensed matter. While the simultaneously measurements of mass and radius of non-rotating neutron stars may impose constraints on the properties of the dense nuclear matter, the observation and study of maximally-rotating ones, close to the mass-shedding limit, may lead to significantly further constraints. Theoretical predictions allow neutron stars to rotate extremely fast (even more than $2000 \ {\rm Hz}$). However, until this moment, the fastest observed rotating pulsar has a frequency of $716 \ {\rm Hz}$, much lower compared to the theoretical predictions. There are many suggestions for the mechanism which lead to this situation. In any case, the theoretical study of uniformly rotating neutron stars, along with the accurate measurements, may offer rich information concerning the high density part of the equation of state. In addition, neutron stars through their evolution, may provide us with a criteria to determine the final fate of a rotating compact star. Sensitivity of bulk neutron stars properties on the equation of state at the mass-shedding limit are the main subject of the present study.

Model comparison from LIGO–Virgo data on GW170817’s binary components and consequences for the merger remnant

GW170817 is the very first observation of gravitational waves originating from the coalescence of two compact objects in the mass range of neutron stars, accompanied by electromagnetic counterparts, and offers an opportunity to directly probe the internal structure of neutron stars. We perform Bayesian model selection on a wide range of theoretical predictions for the neutron star equation of state. For the binary neutron star hypothesis, we find that we cannot rule out the majority of theoretical models considered. In addition, the gravitational-wave data alone does not rule out the possibility that one or both objects were low-mass black holes. We discuss the possible outcomes in the case of a binary neutron star merger, finding that all scenarios from prompt collapse to long-lived or even stable remnants are possible. For long-lived remnants, we place an upper limit of 1.9 kHz on the rotation rate. If a black hole was formed any time after merger and the coalescing stars were slowly rotating, then the maximum baryonic mass of non-rotating neutron stars is at most , and three equations of state considered here can be ruled out. We obtain a tighter limit of for the case that the merger results in a hypermassive neutron star.

Relation between gravitational mass and baryonic mass for non-rotating and rapidly rotating neutron stars

With a selected sample of neutron star (NS) equation-of-states (EOSs) that are consistent with the current observations and have a range of maximum masses, we investigate the relations between NS gravitational mass $M_g$ and baryonic mass $M_b$, and the relations between the maximum NS mass supported through uniform rotation ($M_{\rm max}$) and that of nonrotating NSs ($M_{\rm TOV}$). We find that if one intends to apply an EOS-independent quadratic, universal transformation formula ($M_b=M_g+A\times M_{g}^2$) to all EOSs, the best fit $A$ value is 0.080 for non-rotating NSs only and 0.073 when different spin periods are considered. The residual error of the transformation is as large as $\sim0.1M_{\odot}$. For different EOSs, we find that the parameter $A$ for non-rotating NSs is proportional to $R_{1.4}^{-1}$ (where $R_{1.4}$ is NS radius for 1.4$M_\odot$ in unit of km). For a particular EOS, if one adopts the best-fit parameters for different spin periods, the residual error of the transformation is smaller, which is of the order of 0.01$M_\odot$ for the quadratic form and less than 0.01$M_\odot$ for the cubic form ($M_b=M_g+A_1\times M_{g}^2+A_2\times M_{g}^3$). We also find a very tight and general correlation between the normalized mass gain due to spin $\Delta m\equiv(M_{\rm max}-M_{\rm TOV})/M_{\rm TOV}$ and the spin period normalized to the Keplerian period ${\cal P}$, i.e. ${\rm log_{10}}\Delta m = (-2.74\pm0.05){\rm log_{10}}{\cal P}+{\rm log_{10}}(0.20\pm 0.01)$, which is independent of EOS models. Applications of our results to GW170817 is discussed.

Kilohertz gravitational waves from binary neutron star remnants: Time-domain model and constraints on extreme matter

The remnant star of a neutron star merger is an anticipated loud source of kiloHertz gravitational waves that conveys unique information on the equation of state of hot matter at extreme densities. Observations of such signals are hampered by the photon shot noise of ground-based interferometers and pose a challenge for gravitational-wave astronomy. We develop an analytical time-domain waveform model for postmerger signals informed by numerical relativity simulations. The model completes effective-one-body waveforms for quasi-circular nonspinning binaries in the kiloHertz regime. We show that a template-based analysis can detect postmerger signals with a minimal signal-to-noise ratios (SNR) of 8, corresponding to GW170817-like events for third-generation interferometers. Using Bayesian model selection and the complete inspiral-merger-postmerger waveform model it is possible to infer whether the merger outcome is a prompt collapse to a black hole or a remnant star. In the latter case, the radius of the maximum mass (most compact) nonrotating neutron star can be determined to kilometer precision. We demonstrate the feasibility of inferring the stiffness of the equation of state at extreme densities using the quasiuniversal relations deduced from numerical-relativity simulations.

Equation of state effects in the core collapse of a 20−M⊙ star

Uncertainties in our knowledge of the properties of dense matter near and above nuclear saturation density are among the main sources of variations in multi-messenger signatures predicted for core-collapse supernovae (CCSNe) and the properties of neutron stars (NSs). We construct 97 new finite-temperature equations of state (EOSs) of dense matter that obey current experimental, observational, and theoretical constraints and discuss how systematic variations in the EOS parameters affect the properties of cold nonrotating NSs and the core collapse of a $20\,M_\odot$ progenitor star. The core collapse of the $20\,M_\odot$ progenitor star is simulated in spherical symmetry using the general-relativistic radiation-hydrodynamics code GR1D where neutrino interactions are computed for each EOS using the NuLib library. We conclude that the effective mass of nucleons at densities above nuclear saturation density is the largest source of uncertainty in the CCSN neutrino signal and dynamics even though it plays a subdominant role in most properties of cold NS matter. Meanwhile, changes in other observables affect the properties of cold NSs, while having little effect in CCSNe. To strengthen our conclusions, we perform six octant three-dimensional CCSN simulations varying the effective mass of nucleons at nuclear saturation density. We conclude that neutrino heating and, thus, the likelihood of explosion is significantly increased for EOSs where the effective mass of nucleons at nuclear saturation density is large.

The Equation of State and Some Key Parameters of Neutron Stars: Constraints from GW170817, the Nuclear Data, and the Low-mass X-Ray Binary Data

Abstract In this work we parameterize the equation of state of dense neutron star (NS) matter with four pressure parameters of and then set the combined constraints with the data of GW170817 and the data of six low-mass X-ray binaries (LMXBs) with thermonuclear burst or alternatively the symmetry energy of the nuclear interaction. We find that the nuclear data effectively narrow down the possible range of , the gravitational-wave data plays the leading role in bounding , and the LMXB data as well as the lower bound on the maximal gravitational mass of non-rotating NSs govern the constraints on and . Using posterior samples of pressure parameters and some universal relations, we further investigate how the current data sets can advance our understanding of tidal deformability (Λ), moment of inertia (I), and binding energy (BE) of NSs. For a canonical mass of 1.4 M ⊙, we have , , , and if the constraints from the nuclear data and the gravitational-wave data have been jointly applied. For the joint analysis of gravitational-wave data and the LMXB data, we have , , , and . These results suggest that the current constraints on Λ and R still suffer from significant systematic uncertainties, while I 1.4 and BE1.4 are better constrained.

Effect of viscosity and thermal conductivity on the radial oscillation and relaxation of relativistic stars

In this paper we present a generic formulation of the linearized dynamical equations governing small adiabatic radial oscillations of relativistic stars. The dynamical equations are derived by taking into consideration those effects of viscosity and thermal conductivity of neutron-star matter which directly determine the minimum period of observable pulsars. A variational principle is applied to determine a discrete set of eigenfunctions with complex eigenvalues. The real and imaginary parts of eigenvalues represent the squared natural frequencies and relaxation time of radial oscillations of non-rotating neutron stars, respectively. We provide a suitable framework which may be supplemented with various potential species of cold-nuclear-matter models to compute the spectra of the normalized eigenfrequencies with a certain numerical precision. In the last section, we provide a qualitative estimation of the rate at which viscosity and thermal conductivity drain the kinetic energy of radial oscillation mode in reasonably uniform neutron stars, without relying on explicit numerical computations.

Argonne family potentials and neutron star matter equation of state

Properties of neutron star matter are studied using different microscopic equation of states, derived from the lowest order constrained variational (LOCV) many-body theory using the two-body nuclear force AV8' , and AV6' and also implementing the three-nucleon interaction in AV18, AV8' , and AV6' . The maximum mass configuration of the neutron star is obtained, \( M_{\max}= 2.11, 2.03, 2.12 M_{\odot}\), with AV18+TNI, AV8' +TNI and AV6' +TNI, respectively, which is an excellent agreement with the accepted maximum mass of non-rotating neutron star range \( 2.01 M_{\odot} < M_{\max} < 2.16 M_{\odot}\).

Parameter optimization for the latest quark-meson coupling energy-density functional

The Quark--Meson--Coupling (QMC) model self-consistently relates the dynamics of the internal quark structure of a hadron to the relativistic mean fields arising in nuclear matter. It offers a natural explanation to some open questions in nuclear theory, including the origin of many-body nuclear forces and their saturation, the spin-orbit interaction and properties of hadronic matter at a wide range of densities. The QMC energy density functionals QMC-I and QMC$\pi$-I have been successfully applied to calculate ground state observables of finite nuclei in the Hartree-Fock + BCS approximation, as well as to predict properties of dense nuclear matter and cold non-rotating neutron stars. Here we report the latest development of the model, QMC$\pi$-II, extended to include higher order self-interaction of the $\sigma$ meson. A derivative-free optimization algorithm has been employed to determine a new set of the model parameters and their statistics, including errors and correlations. QMC$\pi$-II predictions for a wide range of properties of even-even nuclei across the nuclear chart, with fewer adjustable parameters, are comparable with other models. Predictions of ground state binding energies of even-even isotopes of superheavy elements with Z$>$96 are particularly encouraging.

Magnetic field distribution in magnetars

Using an axisymmetric numerical code, we perform an extensive study of the magnetic field configurations in non-rotating neutron stars, varying the mass, magnetic field strength and the equation of state. We find that the monopolar (spherically symmetric) part of the norm of the magnetic field can be described by a single profile, that we fit by a simple eighth-order polynomial, as a function of the star's radius. This new generic profile applies remarkably well to all magnetized neutron star configurations built on hadronic equations of state. We then apply this profile to build magnetized neutron stars in spherical symmetry, using a modified Tolman-Oppenheimer-Volkov (TOV) system of equations. This new formalism produces slightly better results in terms of mass-radius diagrams than previous attempts to add magnetic terms to these equations. However, we show that such approaches are less accurate than usual, non-magnetized TOV models, and that consistent models must depart from spherical symmetry. Thus, our `universal' magnetic field profile is intended to serve as a tool for nuclear physicists to obtain estimates of magnetic field inside neutron stars, as a function of radial depth, in order to deduce its influence on composition and related properties. It possesses the advantage of being based on magnetic field distributions from realistic self-consistent computations, which are solutions of Maxwell's equations.

On the inverse spectrum problem of neutron stars

In this work we revisit axial perturbations of spherically symmetric and non-rotating neutron stars. Although it has been object of many studies, it still offers new insights that are of potential interest for more realistic scenarios or in the study exotic compact objects, which have drawn much attention recently. By using WKB theory, we first derive a new Bohr–Sommerfeld rule that allows to investigate the quasi-normal mode spectrum and address the inverse spectrum problem. The pure analytical treatment of the wave equation is rather involved, because it requires the solution of the TOV equations and the non-trivial tortoise coordinate transformation depending on the underlying space-time. Therefore we provide an easy way to construct potentials that simplifies the analytical treatment, but still captures the relevant physics. The approximated potential can be used for calculations of the axial perturbation spectrum. These results are also useful in the treatment of the inverse problem. We demonstrate this by reconstructing the time–time component of the metric throughout the star and constraining the equation of state in the central region. Our method also provides an analytical explanation of the empirically known asteroseismology relation that connects the fundamental QNM and radius of a neutron star with its compactness.

Significance of Deformity on the Mass Quadrupole Moment of Non-Rotating Neutron Stars

Generally, when modeling non-rotating neutron stars, assumptions of perfect spherical symmetry is made; however, this assumption is not correct if the interior composition of the star is described by an anisotropic equation of state. Due to extremely high magnetic fields, magnetars and/or neutron stars that contain color-superconducting quark matter cores can exhibit such anisotropies. Recent publications on models of the global structure of highly magnetized neutron stars indicate that these objects can be deformed making them either oblate or prolate spheroids. Due to these deformations, the gravitational quadrupole moment is expected to be non-zero resulting in a non-homogeneous mass distribution in either the equatorial or polar directions, thus resulting in different masses from the spherical case. In this work, we examine this inhomogeneity by calculating the gravitational mass quadrupole moment of non-rotating deformed neutron stars in the framework of general relativity and investigate any changes from traditional spherical models.

Universal relations for the Keplerian sequence of rotating neutron stars

We investigate the Keplerian (mass-shedding) sequence of rotating neutron stars. Twelve different equations of state are used to describe the nuclear structure. We find four fitting relations which connect the rotating frequency, mass and radius of stars in the mass-shedding limit to the mass and radius of stars in the static sequence. We show the breakdown of approximate relation for the Keplerian frequency derived by Lattimer and Prakash [Science, 304, 536, (2004)] and then we present a new, EOS-independent and more accurate relation. This relation fits the Keplerian frequency of rotating neutron stars to about $2\%$ for a large range of the compactness $M_{S}/R_{S}$ of the reference non-rotating neutron star, namely the static star with the same central density as the rotating one. The performance of the fitting formula is close to $4\%$ for $M_{S}/R_{S}\leq 0.05~M_{\odot}$/km ($f_{K}\leq 350$~Hz). We present additional EOS-independent relations for the Keplerian sequence including relations for $M_{K}f_{K}$ and $R_{K}f_{K}$ in terms of $M_{S}f_{S}$ and $R_{S}f_{S}$, respectively, one of $M_K/R_K$ as a function of $f_{K}/f_{S}$ and $M_S/R_S$, and a relation between the $M_K$, $R_K$ and $f_K$. These new fitting relations are approximately EOS-independent with an error in the worst case of $8\%$. The universality of the Keplerian sequence properties presented here add to the set of other neutron star universal relations in the literature such as the $I$-Love-$Q$ relation, the gravitational binding energy and the energy, angular momentum and radius of the last circular orbit of a test-particle around rotating neutron stars. This set of universal, analytic formulas, facilitate the inclusion of general relativistic effects in the description of relativistic astrophysical systems involving fast rotating neutron stars.

X-Ray Afterglows of Short Gamma-Ray Bursts: Magnetar or Fireball?

Abstract The origin of the X-ray afterglows of gamma-ray bursts has regularly been debated. We fit both the fireball-shock and millisecond-magnetar models of gamma-ray bursts to the X-ray data of GRB 130603B and 140903A. We use Bayesian model selection to answer the question of which model best explains the data. This is dependent on the maximum allowed non-rotating neutron star mass M TOV, which depends solely on the unknown nuclear equation of state. We show that the data for GRB 140903A favors the millisecond-magnetar model for all possible equations of state, while the data for GRB 130603B favors the millisecond-magnetar model if M TOV ≳ 2.3 M ⊙. If M TOV ≲ 2.3 M ⊙, the data for GRB 130603B supports the fireball-shock model. We discuss implications of this result in regards to the nuclear equation of state and the prospect of gravitational-wave emission from newly born millisecond magnetars.

Effective field theory with genuine many‐body forces and tidal effects on neutron stars

Neutron stars are characterized by their masses, radii, spinning speeds (associated with their moments of inertia), and degree of deformation. Under the action of an external static tidal quadrupole field ℰij, the neutron star undergoes a deformation measured in terms of the ratio between the quadrupole moment induced in the star and the field ℰij, which defines, in turn, the tidal deformability parameter. In this contribution, we combine our recent predictions for the neutron stars' tidal parameter with recent gravitational wave observations of merging the binary system GW170817 with quasiuniversal relations between the maximum mass of rotating and nonrotating neutron stars. Our results indicate that predictions of the tidal parameter represent a useful constraint of the Equation of State (EoS) of neutron stars.

Simulating the Collapse of a Thick Accretion Disk due to a Type I X-Ray Burst from a Neutron Star

Abstract We use two-dimensional, general relativistic, viscous, radiation hydrodynamic simulations to study the impact of a Type I X-ray burst on a hot and geometrically thick accretion disk surrounding an unmagnetized, non-rotating neutron star. The disk is initially consistent with a system in its low/hard spectral state, and is subject to a burst that rises to a peak luminosity of 1038 erg s−1 in 2.05 s. At the peak of the burst, the temperature of the disk has dropped by more than three orders of magnitude and its scale height has gone down by more than one order of magnitude. The simulations show that these effects predominantly happen due to Compton cooling of the hot plasma, and clearly illustrate the potential cooling effects of bursts on accretion disk coronae. In addition, we demonstrate the presence of Poynting–Robertson drag, though it only enhances the mass accretion rate onto the neutron star by a factor of ∼3–4 compared to a simulation with no burst. Simulations such as these are important for building a general understanding of the response of an accretion disk to an intense X-ray impulse, which, in turn, will be crucial for deciphering burst spectra. Detailed analysis of such spectra offers the potential to measure neutron star radii, and hence constrain the neutron star equation of state, but only if the contributions coming from the impacted disk and its associated corona can be understood.

Tidal Love numbers of neutron stars in f(R) gravity

The recent detection of gravitational waves from a neutron star merger was a significant step towards constraining the nuclear matter equation of state by using the tidal Love numbers (TLNs) of the merging neutron stars. Measuring or constraining the neutron star TLNs allows us in principle to exclude or constraint many equations of state. This approach, however, has the drawback that many modified theories of gravity could produce deviations from General Relativity similar to the deviations coming from the uncertainties in the equation of state. The first and the most natural step in resolving the mentioned problem is to quantify the effects on the TLNs from the modifications of General Relativity. With this motivation in mind, in the present paper we calculate the TLNs of (non-rotating) neutron stars in R^2-gravity. More precisely, by solving numerically the perturbation equations, we calculate explicitly the polar and the axial l=2 TLNs for three characteristic realistic equations of state and compare the results to General Relativity. Our results show that while the polar TLNs are slightly influenced by the R^2 modification of General Relativity, the axial TLNs can be several times larger (in terms of the absolute value) compared to the general relativistic case.