Cowling Approximation Research Articles

Quasinormal g modes of neutron stars with quarks

Quasinormal oscillation modes of neutron stars provide a means to probe their interior composition using gravitational wave astronomy. We compute the frequencies and damping times of composition-dependent core $g$-modes of neutron stars containing quark matter employing linearized perturbative equations of general relativity. We find that ignoring background metric perturbations due to the oscillating fluid, as in the Cowling approximation, underestimates the $g$-mode frequency by up to 10% for higher mass stars, depending on the parameters of the nuclear equation of state and how the mixed phase is constructed. The $g$-mode frequencies are well described by a linear scaling with the central lepton (or combined lepton and quark) fraction for nucleonic (hybrid) stars. Our findings suggest that neutron stars with and without quarks are manifestly different with regards to their quasinormal $g$-mode spectrum, and may thus be distinguished from one another in future observations of gravitational waves from merging neutron stars.

Gravitational wave asteroseismology on cooling neutron stars

We examine the gravitational wave frequencies from neutron stars during thermal evolution, adopting the relativistic Cowling approximation. We particularly focus on the neutron star models, in which the direct Urca (rapid cooling process) does not work, without the superfluidity and superconductivity. For such models, the cooling curve hardly depends on the equation of state (EOS) as well as the mass of neutron star, while we show that the gravitational wave frequencies strongly depend on the both properties. Then, we find that the frequencies of the fundamental and the 1st pressure mode multiplied with the stellar mass are well expressed as a function of the stellar compactness almost independently of the EOS. We also find that the frequency of the 1st gravity mode in later phase of the thermal evolution is strongly correlated with the stellar compactness. In addition, we derive the empirical formula estimating the threshold mass for the onset of the direct Urca inside the neutron star as a function of the nuclear saturation parameter. This formula will give us a constraint on the neutron star properties, if it would be observationally found that the direct Urca occurs (or does not work) inside the neutron star.

A spectral element approach to computing normal modes

SUMMARYWe introduce a new approach to the computation of gravito-elastic free oscillations or normal modes of spherically symmetric bodies based on a spectral element discretization of the radial ordinary differential equations. Our method avoids numerical instabilities often encountered in the classical method of radial integration and root finding of the characteristic function. To this end, the code is built around a sparse matrix formulation of the eigenvalue problem taking advantage of state-of-the-art parallel iterative solvers. We apply the method to toroidal, spheroidal and radial modes and we demonstrate its versatility in the presence of attenuation, fluid layers and gravity (including the purely elastic case, the Cowling approximation, and full gravity). We demonstrate higher-order convergence and verify the software by computing seismograms and comparing these to existing numerical solutions. Finally, to emphasize the general applicability of our code, we show spectra and eigenfunctions of Earth, Mars and Jupiter’s icy moon Europa and discuss the different types of modes that emerge.

Universal relation for supernova gravitational waves

Using the numerical simulation data for two-dimensional core-collapse supernova, we examine the protoneutron star (PNS) asteroseismology with the relativistic Cowling approximation. As shown in the previous study, e.g., [H. Sotani and T. Takiwaki, Mon. Not. R. Astron. Soc. 498, 3503 (2020).], the gravitational wave signals appearing in the numerical simulation can be well identified with the gravity (fundamental) oscillation in the early (later) phase before (after) the avoided crossing between the gravity and fundamental oscillations. On the other hand, the time evolution of supernova gravitational waves strongly depends on the PNS models, such as the progenitor mass and the equation of state for dense matter. Nevertheless, we find that the fundamental and gravity mode frequencies according to the gravitational wave signals appearing in the numerical simulations can be expressed as a function of the protoneutron star average density independently of the PNS models. Using the average density, we derive the empirical formula for supernova gravitational wave frequency. In addition, we confirm that the dependence of the PNS surface density on the PNS average density is almost independent of the PNS models, and we also discuss how the different treatment of the nonuniform matter in the equation of state affects the observables.

Impacts of dark matter on the f -mode oscillation of hyperon star

We investigate the $f$-mode oscillation of the dark matter admixed hyperon star within the relativistic Cowling approximation. The macroscopic properties are calculated with the relativistic mean-field equation of states by assuming that the dark matter particles are inside the star. The $f$-mode oscillation frequencies (only for $l=2$) are calculated with four different neutron star equation of states. We also check the effects of hyperons/dark matter and hyperons with dark matter equation of states on the $f$-mode oscillations varying with different astrophysical quantities such as mass ($M$), radius ($R$), compactness ($M/R$), surface red-shift ($Z_s$), average density ($\bar{\rho}$), dimensionless tidal deformability ($\Lambda$) of the neutron star. Significant changes have been seen in the $f$-mode frequencies with and without hyperons/dark matter or hyperons+dark matter. Substantial correlations are observed between canonical frequencies and $\Lambda$ ($f_{1.4}-\Lambda_{1.4}$) and maximum frequencies and canonical $\Lambda$ ( $f_{max}-\Lambda_{1.4}$).

Stability of the protoneutron stars towards black hole formation

ABSTRACT We examine the protoneutron star (PNS) stability in this study by solving the radial oscillation equations. For this purpose, we adopt the numerical results of a massive PNS towards the black hole formation obtained by spherically symmetric numerical simulations for a core-collapse supernova with general relativistic neutrino-radiation hydrodynamics. We find that the PNSs are basically stable in their evolution against the radial perturbations, while the PNS finally becomes unstable before the apparent horizon appears inside the PNS. We also examine the gravitational wave frequencies from the PNS with the relativistic Cowling approximation. Then, we derive the empirical formula for the f-mode frequency, which weakly depends on the PNS models. This kind of universality tells us the PNS property, which is a combination of the PNS mass and radius in this study, once one would observe the f-mode gravitational waves.

An MHD spectral theory approach to Jeans’ magnetized gravitational instability

Abstract In this paper, we revisit the governing equations for linear magnetohydrodynamic (MHD) waves and instabilities existing within a magnetized, plane-parallel, self-gravitating slab. Our approach allows for fully non-uniformly magnetized slabs, which deviate from isothermal conditions, such that the well-known Alfvén and slow continuous spectra enter the description. We generalize modern MHD textbook treatments, by showing how self-gravity enters the MHD wave equation, beyond the frequently adopted Cowling approximation. This clarifies how Jeans’ instability generalizes from hydro to magnetohydrodynamic conditions without assuming the usual Jeans’ swindle approach. Our main contribution lies in reformulating the completely general governing wave equations in a number of mathematically equivalent forms, ranging from a coupled Sturm-Liouville formulation, to a Hamiltonian formulation linked to coupled harmonic oscillators, up to a convenient matrix differential form. The latter allows us to derive analytically the eigenfunctions of a magnetized, self-gravitating thin slab. In addition, as an example we give the exact closed form dispersion relations for the hydrodynamical p- and Jeans-unstable modes, with the latter demonstrating how the Cowling approximation modifies due to a proper treatment of self-gravity. The various reformulations of the MHD wave equation open up new avenues for future MHD spectral studies of instabilities as relevant for cosmic filament formation, which can e.g. use modern formal solution strategies tailored to solve coupled Sturm-Liouville or harmonic oscillator problems.

Oscillation dynamics of scalarized neutron stars

Scalar-tensor theories are well-studied extensions of general relativity that offer deviations which are yet within observational boundaries. We present the time evolution equations governing the perturbations of a nonrotating scalarized neutron star, including a dynamic spacetime as well as scalar field within the framework of such scalar-tensor theories. We employ a theory that allows for a massive scalar field or a self-interaction term and we study the impact of those parameters on the nonaxisymmetric $f$-mode. The time evolution approach allows for a comparatively simple implementation of the boundary conditions. We find that the $f$-mode frequency is no longer a simple function of the star's average density when a scalar field is present. We also evaluate the accuracy of different variants of the Cowling approximation commonly used in previous studies of neutron star oscillation modes in alternative theories of gravity and demonstrate that it can give us not only qualitatively correct results, but in some cases also good quantitative estimates of the oscillations frequencies.

Constraining Dense Matter Physics Using f-Mode Oscillations in Neutron Stars

In this paper, an investigation of the role of nuclear saturation parameters on f-mode oscillations in neutron stars is performed within the Cowling approximation. It is found that the uncertainty in the effective nucleon mass plays a dominant role in controlling the f-mode frequencies. The effect of the uncertainties in saturation parameters on previously-proposed empirical relations of the frequencies with astrophysical observables relevant for asteroseismology are also investigated. These results can serve as an important tool for constraining the nuclear parameter space and understand the behaviour of dense nuclear matter from the future detection of f-modes.

Resonance suppression of the r -mode instability in superfluid neutron stars: Accounting for muons and entrainment

We calculate the finite-temperature r-mode spectrum of a superfluid neutron star accounting for both muons in the core and the entrainment between neutrons and protons. We show that the standard perturbation scheme, considering the rotation rate as an expansion parameter, breaks down in this case. We develop an original perturbation scheme which circumvents this problem by treating both the perturbations due to rotation and (weak) entrainment simultaneously. Applying this scheme, we propose a simple method for calculating the superfluid r-mode eigenfrequency in the limit of vanishing rotation rate. We also calculate the r-mode spectrum at finite rotation rate for realistic microphysics input (adopting, however, the Newtonian framework and Cowling approximation when considering perturbed oscillation equations) and show that the normal r-mode exhibits resonances with superfluid r-modes at certain values of temperatures and rotation frequencies in the parameter range relevant to neutron stars in low-mass X-ray binaries (LMXBs). This turns the recently suggested phenomenological model of resonance r-mode stabilization into a quantitative theory, capable of explaining observations. A strong dependence of resonance rotation rates and temperatures on the neutron superfluidity model allows us to constrain the latter by confronting our calculations with the observations of neutron stars in LMXBs.

Discontinuity g‐modes of hybrid stars: Effect of sequential phase transitions

AbstractThe detection of event GW170817 and its electromagnetic counterpart started the era of multi‐messenger astronomy with gravitational waves. The observational constraints from this event and the estimation of mass and radius of PSR J0030 + 0451 is being used to get insights related to the nature of matter inside these compact objects. In this work, we construct hybrid equations of state using modern parametrizations of the relativistic mean field model for the outer core and a generalization of the Constant Speed of Sound parametrization that allow to study sharp sequential phase transitions in the inner core. Moreover, we study non‐radial perturbations of these stellar configurations using the relativistic Cowling approximation. We focus our attention on the g‐modes that arise due to the sharp phase transitions and show that, due to the second phase transition, a new g‐mode appears, g2‐mode, with frequencies smaller that ∼0.5 kHz.

Fluid pulsation modes from strange stars in a higher-dimensional spacetime

In this work, we make the first step to derive non-radial pulsation equations in extra dimensions and investigate how the $f$- and $p_1$-mode frequencies of strange quark stars, within the Cowling approximation, change with the number of dimensions. In this regard, the study is performed by solving numerically the non-radial pulsation equations, adjusted for a $d$-dimensional space-time $(d\geq4)$. We connect the interior to a Schwarzschild-Tangherlini exterior metric and analyze the $f$- and $p_1$- mode frequencies. We found that the frequencies could become higher than those found in four-dimensional space-time. The $f$-mode frequency is essentially constant and only for large gravitational radius values grows monotonically and fast with the gravitational radius. In a gravitational radius range, where $f$-mode frequencies are constant, they increase for space-time dimensions $4\leq d\leq6$ and decrease for $d\geq7$. Regarding $p_1$-mode frequencies they are always larger for higher dimensions and decay monotonically with the increase of the gravitational radius. In extra dimensions, as it happens for four-dimensional space-time, we found $p_1$-mode frequencies are always larger than the $f$-modes ones. In the Newtonian gravity, for a homogeneous star in $d$ dimensions, we observe that the $f$-mode eigenfrequencies are constant and given by the relation $\omega^2=l\, M\, G_d/R^{d-1}$; where $l$ represents the spherical harmonic index, $M\,G_d$ being the total star mass and $R$ the stellar radius.

Accuracy of the relativistic Cowling approximation in protoneutron star asteroseismology

The relativistic Cowling approximation, where the metric perturbations are neglected during the fluid oscillations, is often adopted for considering the gravitational waves from the protoneutron stars (PNSs) provided via core-collapse supernova explosions. In this study, we evaluate how the Cowling approximation works well by comparing the frequencies with the Cowling approximation to those without the approximation. Then, we find that the behavior of the frequencies with the approximation is qualitatively the same way as that without the approximation, where the frequencies with the approximation can totally be determined within $\sim 20\%$ accuracy. In particular, the fundamental mode with the Cowling approximation is overestimated. In addition, we also discuss the damping time of various eigenmodes in gravitational waves from the PNSs, where the damping time for the PNSs before the avoided crossing between the $f$- and $g_1$-modes, is quite different from that for cold neutron stars, but it is more or less similar to that for cold neutron stars in the later phase. The damping time is long enough compared to the typical time interval of short-Fourier transformation that often used in the analysis, and that ideally guarantees the validity of the transformation.

Determination of properties of protoneutron stars toward black hole formation via gravitational wave observations

We examine the frequencies of the fundamental ($f$-) and the gravity ($g_i$-) mode in gravitational waves from accreting protoneutron stars (PNSs) toward black hole formation. For this purpose, we analyze numerical results of gravitational collapse of massive stars for two different progenitors with three equations of state. We adopt profiles of central objects obtained from the numerical simulations by solving general relativistic neutrino-radiation hydrodynamics under the spherical symmetry. Using a series of snapshots as a static configuration at each time step, we solve the eigenvalue problem to determine the specific frequencies of gravitational waves from the evolving PNSs with accretion from core bounce to black hole formation by the relativistic Cowling approximation. We find that the frequency of the $f$-mode gravitational waves can be expressed as a function of the average density of the PNS at each time step almost independently of the progenitor models in accord with the previous studies on ordinary PNSs. We also find that the ratio of the $g_1$-mode frequency to the $f$-mode frequency is insensitive to the progenitor, but strongly depends on the compactness of the PNSs. Having a simultaneous observation of the $f$- and $g_1$-mode gravitational waves, it would provide the information of the average density and compactness of PNS, which can reconstruct the evolution of mass and radius of the PNSs as a constraint on the equation of state at high densities. Furthermore, we show that the highest average density of the PNS with the maximum mass can be estimated with the detection of the $f$-mode gravitational waves together with the help of the neutrino observations.

Thermal Conductivity via Entropy Scaling: An Approach That Captures the Effect of Intramolecular Degrees of Freedom

The thermal conductivity of gases depends strongly on the vibrational and rotational degrees of freedom of the molecule under consideration. Entropy scaling is based on the residual entropy, which does not capture the intramolecular and rotational contributions. This study proposes a model for the thermal conductivity that accounts for these degrees of freedom. We use the Chapman–Cowling approximation, where contributions of internal degrees of freedom to the thermal conductivity of an ideal gas are related to the self-diffusion coefficient. A resulting expression for the thermal conductivity is used as a reference in entropy scaling. We find experimental values for thermal conductivities in the entire fluid range to be (to good approximation) a function of residual entropy only. This study shows that entropy scaling is a strong approximation also for thermal conductivity, provided a suitable expression is chosen for the reference thermal conductivity.

Torsional oscillations of magnetized neutron stars with mixed poloidal-toroidal fields

The quasiperiodic oscillations found in the three giant flares of soft gamma-ray repeaters observed to date have been interpreted as torsional oscillations caused by a starquake related to a magnetospheric reconnection event. Motivated by these observations, we study the influence of the magnetic field geometry in the frequencies of the torsional oscillations of magnetized neutron stars. We use realistic tabulated equations of state for the core and crust of the stars and model their magnetic field as a dipole plus a toroidal component, using the relativistic Grad-Shafranov equation. The frequencies of the torsional modes are obtained by the numerical solution of the eigenvalue problem posed by the linear perturbation equations in the Cowling approximation. Our results show how the asteroseismology of these stars becomes complicated by the degeneracy in the frequencies due to the large relevant parameter space. However, we are able to propose approximately equation-of-state--independent relations that parametrize the influence of the magnetic field in the torsional oscillations, as well as a testable scenario in which the rearrangement of the magnetic field causes an evolution of the frequencies. Finally, we show that there is a magnetic field configuration that maximizes the energy in the perturbation at linear order, which could be related to the trigger of the giant flare.

Effects of Hadron-Quark Phase Transitions in Hybrid Stars within the NJL Model

We study local and non-local Polyakov Nambu-Jona-Lasinio models and analyze their respective phase transition diagram. We construct hybrid stars using the zero temperature limit of the local and non-local versions of Nambu-Jona-Lasinio model for quark matter and the modern GM1(L) parametrization of the non-linear relativistic mean field model for hadronic matter. We compare our models with data from PSR J1614-2230 and PSR J0343+0432 and also from GW170817 and its electromagnetic counterpart GRB170817A and AT2017gfo. We study observational signatures of the appearance of a mixed phase as a result of modeling a phase transition that mimics the Gibbs formalism and compare the results with the sharp first-order phase transition obtained using the Maxwell construction. We also study in detail the g-mode associated with discontinuities in the equation of state, and calculate non-radial oscillation modes using relativistic Cowling approximation.

Incompressible analytical models for spinning-down pulsars

Abstract We study a class of Newtonian models for the deformations of non-magnetised neutron stars during their spin-down. All the models have an analytical solution which allows to easily grasp the dependence of the strain on the star’s main physical quantities, such as radius, mass, and crust thickness. We first use the model proposed by Franco, Link, and Epstein that depicts the star as made of a fluid core and an elastic crust with the same density, to compare the response to a decreasing centrifugal force on stars having different masses and equations of state. We find that the strain angle is peaked at the equator and its maximum value decreases as a function of the mass. Afterwards, we introduce a second, more refined, model in which the core and the crust have different densities, and the gravitational potential of the deformed body is self-consistently accounted for. The strain angle is still a decreasing function of the stellar mass, but now its maximum value is typically peaked at the poles and is larger (by a factor of four) than the corresponding value in the one-density model. Finally, within the present analytic approach, we evaluate the impact of the Cowling approximation: when the perturbations of the gravitational potential are neglected, we find an underestimation of the centrifugal effect on the star, since the strain angle is about 40% of the one obtained with the complete model.

Oscillation modes of hybrid stars within the relativistic Cowling approximation

The first direct detection of gravitational waves has opened a new window to study the Universe and would probably start a new era: the gravitational wave Astronomy. Gravitational waves emitted by compact objects like neutron stars could provide significant information about their structure, composition and evolution. In this paper we calculate, using the relativistic Cowling approximation, the oscillations of compact stars focusing on hybrid stars, with and without a mixed phase in their cores. We study the existence of a possible hadron-quark phase transition in the central regions of neutron stars and the changes it produces on the gravitational modes frequencies emitted by these stars. We pay particular attention to the g-modes, which are extremely important as they could signal the existence of pure quark matter inside neutron stars. Our results show a relationship between the frequency of the g-modes and the constant speed of sound parametrization for the quark matter phase. We also show that the inclusion of color superconductivity produces an increase on the oscillation frequencies. We propose that observations of g-modes with frequencies fg between 1 kHz and 1.5 kHz should be interpreted as an evidence of a sharp hadron-quark phase transition in the core of a compact object.

Compressional modes in two-superfluid neutron stars with leptonic buoyancy

We investigate the compressional modes of cold neutron stars with cores consisting of superfluid neutrons, superconducting protons and normal fluid electrons and muons, and crusts that contain superfluid neutrons plus a normal fluid of (spherical) nuclei and electrons. We develop a two-fluid formalism for the core that accounts for leptonic buoyancy, and an analogous treatment for the crust. We adopt the Cowling approximation, neglecting gravitational perturbations, but include all effects of the background space-time. We introduce a phenomenological, easily-modified nuclear equation of state which contains all of the thermodynamic information required to compute the coupled fluid oscillations, with parameters that are constrained by nuclear physics and the requirement that the maximum mass of a neutron star is $\geq 2M_{\odot}$. Using four parametrizations of this equation of state with nuclear compressibilities $K=230$-$280$ MeV, we calculate the Brunt-V\"{a}is\"{a}l\"{a} frequency due to leptonic buoyancy, and find the corresponding $g$-mode frequencies and eigenfunctions. We find that the WKB approximation reproduces $g$-mode frequencies closely. We examine the dependence of $g$-mode frequencies on stellar mass, nuclear compressibility and strength of neutron-proton entrainment, and compare to previous calculations of $g$-mode frequencies due to leptonic buoyancy. We also compute the $p$-mode spectra, confirming previous findings that the two fluids behave as if uncoupled except the case of large entrainment, and show the existence of nearly resonant mode pairs which could lead to nonlinear $p$-$g$ instabilities even at zero temperature.