Pulsation Equations Research Articles

Nonradial oscillation of strange stars in d dimensions

The influence of the dimensions on the f and p 1 pulsation modes from strange quark stars, in the Cowling approximation, are investigated. For that purpose, the d-dimensional nonradial pulsation equations (d > 4) are numerically integrated considering that the Schwarzschild-Tangherlini line element describes the spacetime outside the object. We found that the fluid pulsation modes could become larger than those obtained in four dimensions. In four dimensions, the f pulsation mode is nearly constant, and for high total masses, it increases monotonically and quickly with the total mass. In this mass interval, the f frequencies grow for the spacetime dimensions between 4 and 6 and decay for d larger than 7. Concerning the p 1 pulsation modes, we found that they increase with the spacetime dimension and decline with the increment of the total mass.

2D modelling of pulsating stars with rapid rotation

Rapid stellar rotation is an important phenomenon in stellar physics, particularly for massive and intermediate mass main-sequence stars. This affects all aspects of the star’s physics including its structure, evolution, and pulsations, and makes it necessary to use 2D numerical approaches rather than the 1D approaches typically used. In this contribution, we will review 2D numerical methods for modelling and interpreting pulsation modes in rapidly rotating stars. We will start by deriving the pulsation equations, both in an adiabatic and non-adiabatic setting, then provide a description of the 2D numerical implementation. We will then explain approximate implementations of the effects of rotation, namely first, second, and third order perturbative approaches, as well as the traditional approximation. This will then be followed by a description on how to calculate disk-integrated mode visibilities in various photometric bands, and how to apply this to mode identification in rapid rotators. Finally, we will review some of the recent works that interpret the pulsation spectra of various stars as viewed in either a single photometric band or in multiple bands, and including supplementary constraints from interferometry and spectroscopy.

General-relativistic treatment of tidal g-mode resonances in coalescing binaries of neutron stars – II. As triggers for precursor flares of short gamma-ray bursts

ABSTRACT In some short gamma-ray bursts, precursor flares occurring ∼ seconds prior to the main episode have been observed. These flares may then be associated with the last few cycles of the inspiral when the orbital frequency is a few hundred Hz. During these final cycles, tidal forces can resonantly excite quasi-normal modes in the inspiralling stars, leading to a rapid increase in their amplitude. It has been shown that these modes can exert sufficiently strong strains on to the neutron star crust to instigate yieldings. Due to the typical frequencies of g- modes being ∼100 Hz, their resonances with the orbital frequency match the precursor timings and warrant further investigation. Adopting realistic equations of state and solving the general-relativistic pulsation equations, we study g-mode resonances in coalescing quasi-circular binaries, where we consider various stellar rotation rates, degrees of stratification, and magnetic field structures. We show that for some combination of stellar parameters, the resonantly excited g1 and g2 modes may lead to crustal failure and trigger precursor flares.

On the linear stability of ℓ-boson stars with respect to radial perturbations

In previous work we constructed new boson star solutions consisting of a family of massive complex scalar fields minimally coupled to gravity in which the individual fields have angular momentum, yet the configuration as a whole is static and spherically symmetric. In the present article we study the linear stability of these ℓ-boson stars with respect to time-dependent, radial perturbations. The pulsation equations, governing the dynamics of such perturbations are derived, generalizing previous work initiated by M Gleiser, and shown to give rise to a two-channel Schrödinger operator. Using standard tools from the literature, we show that for each fixed value ℓ of the angular momentum number, there exists a family of ℓ-boson stars which are linearly stable with respect to radial fluctuations; in this case the perturbations oscillate in time with given characteristic frequencies which are computed and compared with the results from a nonlinear numerical simulation. Further, there is also a family of ℓ-boson stars which are linearly unstable. The two families are separated by the configuration with maximum mass. These results are qualitatively similar to the corresponding stability results of the standard boson stars with ℓ = 0, and they imply the existence of new stable configurations that are more massive and compact than usual boson stars.

Neutron stars in f(R,T) gravity with conserved energy-momentum tensor: Hydrostatic equilibrium and asteroseismology

We investigate the equilibrium and radial stability of spherically symmetric relativistic stars, considering a polytropic equation of state (EoS), within the framework of f(R,T) gravity with a conservative energy-momentum tensor. Both modified stellar structure equations and Chandrasekhar's pulsation equations are derived for the f(R,T)= R+ h(T) gravity model, where the function h(T) assumes a specific form in order to safeguard the conservation equation for the energy-momentum tensor. The neutron star properties, such as radius, mass, binding energy and oscillation spectrum are studied in detail. Our results show that a cusp — which signals the appearance of instability — is formed when the binding energy is plotted as a function of the compact star proper mass. We find that the squared frequency of the fundamental vibration mode passes through zero at the central-density value corresponding to such a cusp where the binding energy is a minimum.

Boson stars and their radial oscillations

We review the derivation of the pulsations equations for spherically symmetric boson stars and then make a thorough study of the radial oscillation frequencies for the fundamental and first excited modes. We do this for self-interacting boson stars and consider a range of values for the self-coupling constant. We also numerically evolve boson stars and Fourier transform the dynamic solutions. The Fourier transform gives an independent computation of the radial oscillation frequencies and allows us to verify our results obtained from the pulsation equations. We find excellent agreement between the two methods.

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.

Radial oscillations and stability of multiple-fluid compact stars

I derive a system of pulsation equations for compact stars made up of an arbitrary number of perfect fluids that can be used to study radial oscillations and stability with respect to small perturbations. I assume spherical symmetry and that the only inter-fluid interactions are gravitational. My derivation is in line with Chandrasekhar's original derivation for the pulsation equation of a single-fluid compact star and keeps the contributions from the individual fluids manifest. I illustrate solutions to the system of pulsations equations with one-, two-, and three-fluid examples.

Radial pulsation of a compact object in d dimensions

The influence of the extra dimensions on the equilibrium and radial pulsation of a compact object is investigated. For such purpose, we solve the stellar structure equations and radial pulsation equations, both modified from their original version to include the extra dimensions (d ≥ 4) taking into account that spacetime outside the object is depicted by a Schwarzschild-Tangherlini metric. In addition, we consider that the pressure and the energy density are connected by a linear relation. Some properties of compact objects are analyzed, such as mass and period of the fundamental mode and their dependencies with the spacetime dimensions. We found that the maximum mass marks the begining of the instability, indicating that in a sequence of equilibrium configurations, the regions constitute by stable and unstable compact objects are distinguished by the relations and , respectively.

Oscillations in g-mode period spacings in red giants as a way to determine their state of evolution

In this work we consider the sensitivity of gravity-mode period spacings to sharp changes in the inner structure of red giant stars, more specifically in the buoyancy frequency inside the g-mode propagation cavity. Based on a comparison between the solutions to the linear pulsation equations in the Cowling approximation for pure g-modes with results obtained with a full oscillation code we identify and correctly interpret the signature of the above-mentioned sharp variations in the period spacings. Two examples, of red giant models in different evolutionary phases, are discussed. Detection of these signatures in CoRoT, Kepler or future PLATO red-giant stars would pin down their evolutionary state in an unprecedented way.

BLUE STRAGGLER MASSES FROM PULSATION PROPERTIES. I. THE CASE OF NGC 6541

We used high-spatial resolution images acquired with the WFC3 on board HST to probe the population of variable blue straggler stars in the central region of the poorly studied Galactic globular cluster NGC 6541. The time sampling of the acquired multi wavelength (F390W, F555W and F814W) data allowed us to discover three WUma stars and nine SX Phoenicis. Periods, mean magnitudes and pulsation modes have been derived for the nine SX Phoenicis and their masses have been estimated by using pulsation equations obtained from linear non adiabatic models. We found masses in the range 1.0-1.1Mo, with an average value of 1.06+-0.09 Mo (sigma = 0.04), significantly in excess of the cluster Main Sequence Turn Off mass (~ 0.75Mo). A mild trend between mass and luminosity seems also to be present. The computed pulsation masses turn out to be in very good agreement with the predictions of evolutionary tracks for single stars, indicating values in the range ~ 1.0-1.2 Mo for most of the BSS population, in agreement with what discussed in a number of previous studies.

GYRE: an open-source stellar oscillation code based on a new Magnus Multiple Shooting scheme

We present a new oscillation code, GYRE, which solves the stellar pulsation equations (both adiabatic and non-adiabatic) using a novel Magnus Multiple Shooting numerical scheme devised to overcome certain weaknesses of the usual relaxation and shooting schemes appearing in the literature. The code is accurate (up to 6th order in the number of grid points), robust, efficiently makes use of multiple processor cores and/or nodes, and is freely available in source form for use and distribution. We verify the code against analytic solutions and results from other oscillation codes, in all cases finding good agreement. Then, we use the code to explore how the asteroseismic observables of a 1.5 M_sun star change as it evolves through the red-giant bump.

GYRE: A new open-source stellar oscillation code

AbstractWe introduce GYRE, a new open-source stellar oscillation code which solves the adiabatic/non-adiabatic pulsation equations using a novel Magnus Multiple Shooting (MMS) numerical scheme. The code has a global error scaling of up to 6th order in the grid spacing, and can therefore achieve high accuracy with few grid points. It is moreover robust and efficiently makes use of multiple processor cores and/or nodes. We present an example calculation using GYRE, and discuss recent work to integrate GYRE into the asteroseismic optimization module of the MESA stellar evolution code.

Radial and Nonradial Oscillation Modes in Rapidly Rotating Stars

We use two-dimensional stellar models and a two-dimensional finite difference integration of the linearized pulsation equations to calculate nonradial oscillations. This approach allows us to directly calculate the pulsation modes for a distorted rotating star without treating the rotation as a perturbation. We are also able to express the finite difference solution in the horizontal direction as a sum of multiple spherical harmonics for any given mode. Using these methods, we have investigated the effects of increasing rotation and the number of spherical harmonics on the calculated eigenfrequencies and eigenfunctions and compared the results to perturbation theory. We use 10 M☉ models with velocities ranging from 0 to 420 km s−1 (0.89Ωc) and examine low-order p-modes. We find that one spherical harmonic remains reasonable up to a rotation rate around 300 km s−1 (0.69Ωc) for the radial fundamental mode, but can fail at rotation rates as low as 90 km s−1 (0.23Ωc) for the l = 2 p2 mode, based on the eigenfrequencies alone. Depending on the mode in question, a single spherical harmonic may fail at lower rotation rates if the shape of the eigenfunction is taken into consideration. Perturbation theory, in contrast, remains valid up to relatively high rotation rates for most modes. We find the lowest failure surface equatorial velocity is 120 km s−1 (0.30Ωc) for the l = 2 p2 mode, but failure velocities between 240 and 300 km s−1 (0.58Ωc–0.69Ωc) are more typical.

PULSE: a finite element code for solving adiabatic nonradial pulsation equations

We briefly present the nonradial adiabatic pulsation code PULSE first developped for white dwarf asteroseismology and now used to compute adiabatic oscillation properties for various types of stellar objects. Numerical tests show that the code is able to provide the accuracy (for a given stellar model) required to deal with the precision in frequency expected from the COROT long runs. While the ultimate objective is to compare the output of various pulsation codes (see these proceedings), we already emphasize problems that need to be addressed concerning, in particular, the mesh resolution of the input stellar models and its impact on the accuracy at which frequencies can be computed.

Synthetic photometry for non-radial pulsations in subdwarf B stars

We describe a method for computing theoretical photometric amplitude ratios for a number of modes of nonradially pulsating subdwarf B stars in both SDSS and UBVR systems. In order to avoid costly solutions of the non-adiabatic non- radial pulsation equations, we have adopted the adiabatic approximation. We argue that this is a valid approach, at least for the V361 Hya stars, because observations show that the temperature perturbations dominate the radius perturbations in the flux variation. We find that for V361 Hya stars, low-degree (� = 0, 1, 2) modes may be difficult to distinguish using optical photometry. However, the high degree modes (� = 3, 4) are relatively well separated and may be distinguished more easily. We have also computed the amplitude ratios for a number of modes in PG 1716+426 stars. For these stars, the amplitude ratios for low degree modes (� = 0, 1) are well resolved. For oscillations with periods ∼40 min, higher-degree modes (� = 2−4) may also be identified easily from their amplitude ratios. However for longer period oscillations, the � = 3a nd the� = 2, 4 modes approach the � = 0a nd� = 1 modes respectively.

Asymptotic expressions for the angular dependence of low-frequency pulsation modes in rotating stars

Through the solution of Laplace's tidal equations, approximated to describe equatorially trapped wave propagation, analytical expressions are obtained for the angular dependence of pulsation modes in uniformly rotating stars. As the ratio between rotation and pulsation frequencies becomes large, these expressions approach the exact solutions of the governing low-frequency pulsation equations. Four classes of asymptotic solution are found, corresponding to g (gravito-inertial), r (Rossby), Kelvin and Yanai modes. The Kelvin modes arise through the conservation of specific vorticity, much like the r modes, but propagate in the same sense as the rotation; they are found to be the equivalents of prograde sectoral modes. The prograde Yanai modes behave like g modes, as do the retrograde ones if the rotation is sufficiently rapid; otherwise, the latter exhibit the character of r modes. Comparison between asymptotic and numerical solutions to the tidal equations reveals that the former converge rapidly towards the latter, for g and Yanai modes. The convergence is slower for Kelvin and r modes, as these become equatorially trapped only when the rotation is very rapid. It is argued that the utility of the asymptotic solutions does not rest on their accuracy alone, but also on the valuable physical insights that they are capable of providing.

Applications of Non-Adiabatic Radial Pulsation Equations to roAp Stars

AbstractWe apply a code that solves grey radial non-adiabatic pulsation equations with consistent treatment of radiative transfer to the roAp frequency regime. We find evidence for the existence of chromospheric modes at frequencies 2.5 mHz, 3.7 mHz, 4.6 mHz and 5.7 mHz in the equilibrium model we used. We also find that, since the oscillations are not adiabatic at the surfaces of roAp stars, better surface boundary conditions need to be considered in future studies.

On the double-mode RR Lyrae variables of the Sculptor dwarf galaxy

Frequency analysis of more than 300 stars of the ogle database on Sculptor galaxy has led to the discovery of 18 double-mode RR Lyrae (RRd) variables. This yields a 20% incidence rate for double-mode pulsation among the variables previously classified as first overtone RR Lyrae stars in this galaxy. Most of the RRd stars cover the period range of but there are two stars with longer periods of . All variables fit well in the pattern of the diagram, spanned by the RRd stars of the Galactic globular clusters and those of the Large Magellanic Cloud (LMC). It follows from our previous investigations that the luminosities and masses of the RRd stars in Galactic globular clusters and in the LMC are almost independent of metallicity. By assuming that the Sculptor RRd variables also obey this rule, with the aid of the pulsation equations we estimate their metallicities. For most of the stars we get , which is the same value as that obtained from a semi-empirical method for the average metallicity of the fundamental mode (RRab) stars. Two RRd stars have considerably lower metallicities, but even those are within the range corresponding to the RRab stars. The narrower metallicity range of the RRd stars is in agreement with their observed luminosity range, which is about three times smaller than that of the RRab stars.

Self-adjoint wave equations for dynamical perturbations of self-gravitating fields

It is shown that the dynamical evolution of linear perturbations on a static space-time is governed by a constrained wave equation for the extrinsic curvature tensor. The spatial part of the wave operator is manifestly elliptic and self-adjoint. In contrast to metric formulations, the curvature-based approach to gravitational perturbation theory generalizes in a natural way to self-gravitating matter fields. It is also demonstrated how to obtain symmetric pulsation equations for self-gravitating non-Abelian gauge fields, Higgs fields and perfect fluids. For vacuum fluctuations on a vacuum space-time, the Regge-Wheeler and Zerilli equations are rederived.