Radial oscillations of relativistic stars (original) (raw)

Fundamental oscillation modes of neutron stars: Validity of universal relations

Physical Review D, 2015

We study the f-mode frequencies and damping times of nonrotating neutron stars (NS) in general relativity (GR) by solving the linearized perturbation equations, with the aim to establish "universal" relations that depend only weakly on the equations of state (EOS). Using a more comprehensive set of EOSs, we reexamine some proposed empirical relations that describe the f-mode parameters in terms of mass and radius of the neutron star (NS), and we test a more recent proposal for expressing the f-mode parameters as quadratic functions of the effective compactness. Our extensive results for each equation of state considered allow us to study the accuracy of each proposal. In particular, the empirical relation proposed in the literature for the damping time in terms of the mass and radius deviates considerably from our results. We introduce a new universal relation for the product of the f-mode frequency and damping time as a function of the (ordinary) compactness, which proved to be more accurate. The more recently proposed relations using the effective compactness, on the other hand, also fit our data accurately. Our results show that the maximum oscillation frequency depends strongly on the EOS, such that the measurement of a high oscillation frequency would rule out several EOSs. Lastly, we compare the exact mode frequencies to those obtained in the Cowling approximation, and also to results obtained with a nonlinear evolution code, validating the implementations of the different approaches.

Coupling of Radial and Non-radial Oscillations of Neutron Stars

2005

Linear perturbation theory is appropriate to describe small oscillations of stars, while a mild non-linearity is still tractable perturbatively but requires to consider mode coupling, i.e. to take into account second order effects. It is natural to start to look at this problem by considering the coupling between linear radial and non-radial modes. A radial pulsation may be thought of as an important component of an overall mildly non-linear oscillation, e.g. of a proto-neutron star. Radial pulsations of a spherical compact objects do not per se emit gravitational waves but, if the coupling between the existing first order radial and non-radial modes is efficient in driving and possibly amplifying the non-radial oscillations, one may expect the appearance of non-linear harmonics, and gravitational radiation could then be produced to a significant level. More in general, mode coupling typically leads to an interesting phenomenology, thus it is worth investigating it in the context of star perturbations. In this paper we develop the relativistic formalism to study the coupling of radial and non-radial first order perturbations of a compact spherical star. From a mathematical point of view, it is convenient to treat the two sets of perturbations as separately parametrized, using a 2-parameter perturbative expansion of the metric, the energy-momentum tensor and Einstein equations in which λ is associated with the radial modes, ǫ with the non-radial perturbations, and the λǫ terms describe the coupling. This approach provides a well-defined framework to consider the gauge dependence of perturbations, allowing us to use ǫ order gauge-invariant non-radial variables on the static background and to define new second order λǫ gauge-invariant variables representing the result of the non-linear coupling. We present the evolution and constraint equations for our variables outlining the setup for numerical computations, and briefly discuss the surface boundary conditions in terms of the second order λǫ Lagrangian pressure perturbation.

Radial pulsation frequencies of slowly rotating neutron stars

We study the radial pulsation frequencies of slowly rotating neutron stars in general relativistic formalism using realistic equations of state. It is found that the pulsation frequencies are always an increasing function of rotation rate. The increasing rate of frequency depends on the nature of equations of state. Cameron [1] suggested that the vibration of neutron stars might excite motions that might have interesting astrophysical applications, which lead to a series of investigations of the vibrational properties of neutron stars. The earliest detailed calculations were done by Meltzer and Thorne [2] and Thorne [3], where they investigated the radial as well as nonradial oscillations using available equation of state, such as the Harrison-Walker-Wheeler equation of state. These and other early studies by Wheeler [4], Chau [5] and Occhionero [6] indicated that the majority

Exploring Radial Oscillations in Slow Stable and Hybrid Neutron Stars

arXiv (Cornell University), 2024

In the era of gravitational wave astronomy, radial oscillations hold significant potential for not only uncovering the microphysics behind the internal structure but also investigating the stability of neutron stars (NSs). We start by constructing families of static NSs following nucleonic, quarkyonic, and hybrid equations of state and then subject them to radial perturbations in order to explore the stability of these stars. Unlike other literature where the fluid elements are assumed to be in chemical equilibrium, we consider the out-of-equilibrium effects on the chemical composition of fluid elements for the calculation of radial modes. Taking these considerations into account, we observe that the sound speed (c 2 s) and adiabatic index (γ) avoid singularities and discontinuities over the equilibrium case. We elucidate the response of the fundamental radial modes by examining the out-ofequilibrium matter distribution scenario, offering insights into its dynamic variations. We also demonstrate that this approach extends the stable branches of stellar models, enabling stars to sustain stable higher-order mass doublets, shedding some light on observation and existence of PSR J0740+6620.

Approximate equation relevant to axial oscillations on slowly rotating relativistic stars

Physical Review D, 2000

Axial oscillations relevant to the r-mode instability are studied with the slow rotation formalism in general relativity. The approximate equation governing the oscillations is derived with second-order rotational corrections. The equation contains an effective ''viscositylike'' term, which originates from coupling to the polar g-mode displacements. The term plays a crucial role on the resonance point, where the disturbance on the rotating stars satisfies a certain condition at the lowest order equation. The effect is significant for newly born hot neutron stars, which are expected to be subject to the gravitational radiation driven instability of the r mode.

Coupling of radial and nonradial oscillations of relativistic stars: Gauge-invariant formalism

Physical Review D, 2005

This is a progress report on our study of the coupling of first-order radial and non-radial relativistic perturbations of a static spherical star. Our goal is to investigate the effects of this coupling on the gravitational wave signal of neutron stars. In particular, we are looking for the existence of resonances and parametric amplifications, changes in the damping time of non-radial oscillations, etc. To that end, we have developed a formalism that introduces gauge invariant quantities to describe the coupling. Their equations have the same structure as the equations for first-order non-radial perturbations plus some source terms, which makes them very appealing for time domain studies.

Nonlinear Evolution of the r-Modes in Neutron Stars

Physical Review Letters, 2001

The evolution of a neutron-star r-mode driven unstable by gravitational radiation (GR) is studied here using numerical solutions of the full nonlinear fluid equations. The dimensionless amplitude of the mode grows to order unity before strong shocks develop which quickly damp the mode. In this simulation the star loses about 40% of its initial angular momentum and 50% of its rotational kinetic energy before the mode is damped. The nonlinear evolution causes the fluid to develop strong differential rotation which is concentrated near the surface and poles of the star.

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

Physical Review D, 2020

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.

Non-linear evolution of rotating relativistic stars

We present first results of the non-linear evolution of rotating relativistic stars obtained with an axisymmetric relativistic hydrodynamics code in a fixed spacetime. As initial data we use stationary axisymmetric and perturbed configurations. We find that, in order to prevent (numerical) angular momentum loss at the surface layers of the star a high-resolution grid (or a numerical scheme that retains high order at local extrema) is needed. For non-rotating stars, we compute frequencies of radial and non-radial small-amplitude oscillations, which are in excellent agreement with linear normal mode frequencies computed in the Cowling approximation. As a first application of our code, quasi-radial modes of rapidly rotating relativistic stars are computed. By generalizing our numerical code to 3-D, we plan to study the evolution and non-linear dynamics of toroidal oscillations (rmodes) of rapidly rotating neutron stars, which are a promising source of gravitational waves.

Quasi‐normal Modes of Rotating Relativistic Stars: Neutral Modes for Realistic Equations of State

The Astrophysical Journal, 1999

We compute zero-frequency (neutral) quasi-normal f -modes of fully relativistic and rapidly rotating neutron stars, using several realistic equations of state (EOSs) for neutron star matter. The zero-frequency modes signal the onset of the gravitational radiation-driven instability. We find that the l = m = 2 (bar) f -mode is unstable for stars with gravitational mass as low as 1.0 − 1.2M ⊙ , depending on the EOS. For 1.4M ⊙ neutron stars, the bar mode becomes unstable at 83% − 93% of the maximum allowed rotation rate. For a wide range of EOSs, the bar mode becomes unstable at a ratio of rotational to gravitational energies T /W ∼ 0.07 − 0.09 for 1.4M ⊙ stars and T /W ∼ 0.06 for maximum mass stars. This is to be contrasted with the Newtonian value of T /W ∼ 0.14. We construct the following empirical formula for the critical value of T /W for the bar mode, (T /W ) 2 = 0.115 − 0.048 M/M sph max , which is insensitive to the EOS to within 4 − 6%. This formula yields an estimate for the neutral mode sequence of the bar mode as a function only of the star's mass, M, given the maximum allowed mass, M sph max , of a nonrotating neutron star. The recent discovery of the fast millisecond pulsar in the supernova remnant N157B, supports the suggestion that a fraction of proto-neutron stars are born in a supernova collapse with very large initial angular momentum. If some neutron stars are born in an accretion-induced-collapse of a white dwarf, then they will also have very large angular momentum at birth. Thus, in a fraction of newly born neutron stars the instability is a promising source of continuous gravitational waves. It could also play a major role in the rotational evolution (through the emission of angular momentum) of merged binary neutron stars, if their post-merger angular momentum exceeds the maximum allowed to form a Kerr black hole.