Polytropic Stars Research Articles

Anomalous diffusion and collapse of self-gravitating Langevin particles in D dimensions.

We address the generalized thermodynamics and the collapse of a system of self-gravitating Langevin particles exhibiting anomalous diffusion in a space of dimension D. This is a basic model of stochastic particles in interaction. The equilibrium states correspond to polytropic configurations similar to stellar polytropes and polytropic stars. The index n of the polytrope is related to the exponent of anomalous diffusion. We consider a high-friction limit and reduce the problem to the study of the nonlinear Smoluchowacute;ski-Poisson system. We show that the associated Lyapunov functional is the Tsallis free energy. We discuss in detail the equilibrium phase diagram of self-gravitating polytropes as a function of D and n, and determine their stability by using turning point arguments and analytical methods. When no equilibrium state exists, we investigate self-similar solutions of the nonlinear Smoluchowski-Poisson system describing the collapse. Our stability analysis of polytropic spheres can be used to settle the generalized thermodynamical stability of self-gravitating Langevin particles as well as the nonlinear dynamical stability of stellar polytropes, polytropic stars and polytropic vortices. Our study also has applications concerning the chemotactic aggregation of bacterial populations.

Oscillations of Fast Rotating Stars: p-Modes in Centrifugally Flattened Polytropes

Oscillation modes of rapidly rotating stars have not yet been calculated with precision, rotational effects being generally approximated by perturbation methods. We developed a mathematical formalism and a numerical method which fully account for the deformation of the star by the centrifugal force. The method has been first tested in the case of Maclaurin spheroids and then applied to uniformly rotating polytropic stars.

Highly accurate calculation of rotating neutron stars

We give a detailed description of the recently developed multi-domain spectral method for constructing highly accurate general-relativistic models of rapidly rotating stars. For both "ordinary" and "critical" configurations, it is exhibited by means of representative examples, how the accuracy improves as the order of the approximation increases. Apart from homogeneous fluid bodies, we also discuss models of polytropic and strange stars.

Inertial modes of slowly rotating relativistic stars in the Cowling approximation

We study oscillations of slowly rotating relativistic barotropic as well as non-barotropic polytropic stars in the Cowling approximation, including first order rotational corrections. By taking into account the coupling between the polar and axial equations, we find that, in contrast to previous results, the $m=2$ $r$ modes are essentially unaffected by the continuous spectrum and exist even for very relativistic stellar models. We perform our calculations both in the time and frequency domain. In order to numerically solve the infinite system of coupled equations, we truncate it at some value $l_{\rm max}$. Although the time dependent equations can be numerically evolved without any problems, the eigenvalue equations possess a singular structure, which is related to the existence of a continuous spectrum. This prevents the numerical computation of an eigenmode if its eigenfrequency falls inside the continuous spectrum. The properties of the latter depend strongly on the cut-off value $l_{\rm max}$ and it can consist of several either disconnected or overlapping patches, which are the broader the more relativistic the stellar model is. By discussing the dependence of the continuous spectrum as a function of both the cut-off value $l_{\rm max}$ and the compactness $M/R$, we demonstrate how it affects the inertial modes. Through the time evolutions we are able to show that some of the inertial modes can actually exist inside the continuous spectrum, but some cannot. For more compact and therefore more relativistic stellar models, the width of the continuous spectrum strongly increases and as a consequence, some of the inertial modes, which exist in less relativistic stars, disappear.

Gravitational Radiation Reaction Driven Secular Instability off‐Mode Oscillations in Differentially Rotating Newtonian Stars

We have carried out the linear stability analysis of equilibrium sequences of differentially rotating Newtonian polytropic stars. In particular, we have found critical points along the equilibrium sequences for the gravitational radiation reaction driven secular instability of f-mode oscillations with azimuthal wavenumbers m = 2, 3, and 4. We show that the critical values of T/|W| where the instability sets in strongly depend on the degree of differential rotation of the models. Here, T and W are the rotational kinetic energy and the gravitational energy of the star, respectively. We also find that as the degree of differential rotation is increased, the dependence of the critical value of T/|W| on the equations of state becomes weaker.

Post‐Newtonian Treatment of Bar Mode Instability in Rigidly Rotating Equilibrium Configurations for Polytropic Stars

In this paper we determine the onset point of secular instability for the nonaxisymmetric bar mode in rigidly rotating equilibrium configurations in the post-Newtonian approximation, in order to apply it to neutron stars. The treatment is based on a precedent Newtonian analytic energy variational method that we have extended to the post-Newtonian case. This method, based on Landau's theory of second-order phase transitions, provides the critical value of the ellipsoid polar eccentricity e at the onset of instability, i.e., at the bifurcation point from the axisymmetric Maclaurin to the triaxial Jacobi ellipsoids, and it is valid for any equation of state. The extension of this method to post-Newtonian fluid configurations has been accomplished by combining two earlier orthogonal works, specialized, respectively, to slow rotating configurations but with arbitrary density profile and to constant mass density but arbitrarily fast rotating ellipsoids. We also determine the explicit expressions for the density functionals that allow the generalization of the physical quantities involved in our treatment from the constant mass density to an arbitrary density profile form. We find that, considering homogeneous ellipsoids, the value of the critical eccentricity increases as the stars become more relativistic, in qualitative agreement with previous investigations but with a less marked amount of such an increase. Then we have studied the dependence of this critical value on the configuration equation of state. Considering polytropic matter distributions, we find that the increase in the eccentricity at the onset of instability with the star compactness is confirmed for softer equations of state (with respect to the incompressible case). The amount of this stabilizing effect is nearly independent of the polytropic index.

Distortion of secondaries in semi-detached binaries and the cataclysmic variable period minimum

Based on SPH simulations, we quantify the geometrical distortion eect due to tidal and rotational forces on polytropic secondaries in semi-detached binaries. The main eect is an expansion of the polytropic star, with an eect on the radius of5%{12%, depending on the polytropic index and the mass ratio. We apply such distortion eects to the secular evolution of secondaries in cataclysmic variable systems. We focus on systems below the 2{3 h period gap and that approach the minimum period. We nd a signicant increase of the predicted minimum period (4% if changes in the secondary's thermal relaxation are approximately taken into account). Though an improvement, the eect is not big enough to solve the mismatch between predicted and observed minimum period at 80 min.

The new model of a tidally disrupted star: further development and relativistic calculations

In this paper we develop the new semi-analitical model of a tidally perturbed or tidally disrupted star proposed recently by two of us. This model is effectively a one dimensional Lagrangian model and it can be evolved numerically much faster that the conventional 3D models. A self-consistent derivation of the dynamical equations of the model is performed and several important theorems about the dynamics of the model are proved without any particular assumption about the equation of state of the stellar gas. The dynamical equations are solved numerically for the case of $n=1.5$ polytropic star evolving in the relativistic field of a $10^7M_{\odot}$ Kerr black hole. Some results of these calculations are compared with the results of calculations based on finite-difference 3D simulations. The comparison shows a very good agreement between both approaches to the problem. Then we show that the strength of the tidal encounter depends significantly on the relative orientation of the orbital angular momentum of the star and the spin of the black hole.

General relativistic electromagnetic fields of a slowly rotating magnetized neutron star - II. Solution of the induction equations

We have solved numerically the general relativistic induction equations in the interior background space-time of a slowly rotating magnetized neutron star. The analytic form of these equations was discussed recently (Paper I), where corrections due to both the space-time curvature and the dragging of reference frames were shown to be present. Through a number of calculations we have investigated the evolution of the magnetic field with different rates of stellar rotation, different inclination angles between the magnetic moment and the rotation axis, as well as different values of the electrical conductivity. All of these calculations have been performed for a constant-temperature relativistic polytropic star and make use of a consistent solution of the initial-value problem which avoids the use of artificial analytic functions. Our results show that there exist general relativistic effects introduced by the rotation of the space-time which tend to decrease the decay rate of the magnetic field. The rotation-induced corrections are however generally hidden by the high electrical conductivity of the neutron star matter, and when realistic values for the electrical conductivity are considered, these corrections become negligible even for the fastest known pulsar.

Functional approach to the problem of self-gravitating systems: Conditions of integrability

Using a functional method based on the introduction of a velocity potential to solve the Euler, continuity and Poisson equations, a new analytic study of the equilibrium of self-gravitating rotating systems with a polytropic equation of state has permitted the formulation of the conditions of integrability. For the polytropic index $n=1$ in the incompressible case $(\ensuremath{\nabla}\ensuremath{\cdot}\stackrel{\ensuremath{\rightarrow}}{v}=0),$ we are able to find the conditions for solving the problem of the equilibrium of polytropic self-gravitating systems that rotate and have nonuniform vorticity. This work contains the conditions which give analytic and quasi-analytic solutions for the equilibrium of polytropic stars and galactic systems in Newtonian gravity. In special cases, explicit analytic solutions are presented.

Quasiequilibrium sequences of synchronized and irrotational binary neutron stars in general relativity. II. Newtonian limits

We study equilibrium sequences of close binary systems composed of identical polytropic stars in Newtonian gravity. The solving method is a multidomain spectral method which we have recently developed. An improvement is introduced here for accurate computations of binary systems with a stiff equation of state $(\ensuremath{\gamma}>2).$ The computations are performed for both cases of synchronized and irrotational binary systems with adiabatic indices $\ensuremath{\gamma}=3, 2.5, 2.25, 2,$ and $1.8.$ It is found that the turning points of total energy along a constant-mass sequence appear only for $\ensuremath{\gamma}>~1.8$ for synchronized binary systems and $\ensuremath{\gamma}>~2.3$ for irrotational ones. In the synchronized case, the equilibrium sequences terminate by the contact between the two stars. On the other hand, for irrotational binaries, it is found that the sequences terminate at a mass shedding limit which corresponds to a detached configuration.

R-mode oscillations of differentially and rapidly rotating Newtonian polytropic stars

For analysis of the r-mode oscillation of hot young neutron stars, it is necessary to consider the effect of differential rotation, because viscosity is not strong enough for differentially rotating young neutron stars to be led to uniformly rotating configurations on a very short time scale after their birth. In this paper, we have developed a numerical scheme to solve the r-mode oscillations of differentially rotating polytropic inviscid stars. This is the extended version of the method which was applied to compute the r-mode oscillations of uniformly rotating Newtonian polytropic stars. By using this new method, we have succeeded in obtaining eigenvalues and eigenfunctions of the r-mode oscillations of differentially rotating polytropic stars. Our numerical results show that as the degree of differential rotation is increased, it becomes more difficult to solve the r-mode oscillations for slightly deformed configurations from a sphere compared to solving the r-mode oscillations of considerably deformed stars. One reason for this seems that for slightly deformed stars a corotation cylinder appears near the stellar surface region if the degree of differential rotation is large enough. This is similar to the situation that the perturbational approach of low-frequency r-mode oscillations for slowly rotating stars in general relativity results in a singular eigenvalue problem.

Physical characteristics of critically distorted or almost spherically shaped rotating magnetic polytropic models

We use the so-called complex plane iterative technique (CIT) to the computation of polytropic stars distorted by rotation (either rigid or differential) and magnetic fields (both toroidal and poloidal). We give emphasis on computing(i) critically rotating configurations, and (ii) configurations that – dueto the counterbalancing of the effects of rotation and poloidal magnetic field with the effects of toroidal magnetic field – obtain an almost spherical shape.

R-mode oscillations of rapidly rotating Newtonian stars: A new numerical scheme and its application to the spin evolution of neutron stars

We have developed a new numerical scheme to solve r-mode oscillations of rapidly rotating polytropic stars in Newtonian gravity. In this scheme, Euler perturbations of the density, three components of the velocity are treated as four unknown quantities together with the oscillation frequency. For the basic equations of oscillations, the compatibility equations are used instead of the linearized equations of motion. By using this scheme, we have solved the classical r-mode oscillations of rotational equilibrium sequences of polytropes with the polytropic indices $N=0.5,$ 1.0, and 1.5 for $m=2,3,$ and 4 modes. Here m is the rank of the spherical harmonics ${Y}_{l}^{m}.$ These results have been applied to investigate the evolution of uniformly rotating hot young neutron stars by considering the effect of gravitational radiation and viscosity. We have found that the maximum angular velocities of neutron stars are around 10--20 % of the Keplerian angular velocity irrespective of the softness of matter. This confirms the results obtained from the analysis of r-modes with the slow rotation approximation employed by many authors.

Equilibrium sequences of irrotational binary polytropic stars: The case of double polytropic stars

Solutions to equilibrium sequences of irrotational binary polytropic stars in Newtonian gravity are expanded in a power of $\ensuremath{\epsilon}{=a}_{0}/R,$ where R and ${a}_{0}$ are the orbital separation of the binary system and the radius of each star for $R=\ensuremath{\infty}.$ For each order of $\ensuremath{\epsilon},$ we should solve ordinary differential equations for arbitrary polytropic indices n. We show solutions for polytropic indices $n=0.5,1,1.5,$ and $2$ up to ${\ensuremath{\epsilon}}^{6}$ orders. Our semianalytic solutions can be used to check the validity of numerical solutions.

Rotational Modes of Nonisentropic Stars and the Gravitational Radiation‐driven Instability

We investigate the properties of r-mode and inertial mode of slowly rotating, non-isentropic, Newtonian stars, by taking account of the effects of the Coriolis force and the centrifugal force. For the non-isentropic models we consider only two cases, that is, the models with the stable fluid stratification in the whole interior and the models that are fully convective. For simplicity we call these two kinds of models "radiative" and "convective" models in this paper. For both cases, we assume the deviation of the models from isentropic structure is small. Examining the dissipation timescales due to the gravitational radiation and several viscous processes for the polytropic neutron star models, we find that the gravitational radiation driven instability of the r-modes remains strong even in the non-isentropic models. Calculating the rotational modes of the radiative models as functions of the angular rotation frequency $\Omega$, we find that the inertial modes are strongly modified by the buoyant force at small $\Omega$, where the buoyant force as a dominant restoring force becomes comparable with or stronger than the Coriolis force. Because of this property we obtain the mode sequences in which the inertial modes at large $\Omega$ are identified as g-modes or the r-modes with l=|m| at small $\Omega$. We also note that as $\Omega$ increases from $\Omega=0$ the retrograde g-modes become retrograde inertial modes, which are unstable against the gravitational radiation reaction.

Almost analytic solutions to equilibrium sequences of irrotational binary polytropic stars for n = 1

A solution to an equilibrium of irrotational binary polytropic stars in Newtonian gravity is expanded in a power of epsilon = a(0)/R, where R and a(0) are the separation of the binary system and the radius of each star for R = infinity. For the polytropic index n = 1, the solutions are given almost analytically up to order epsilon(6). We have found that in general an equilibrium solution should have the velocity component along the orbital axis and that the central density should decrease when R decreases. Our almost analytic solutions can be used to check the validity of numerical solutions.

New numerical scheme to compute three-dimensional configurations of quasiequilibrium compact stars in general relativity: Application to synchronously rotating binary star systems

We have developed a new numerical scheme to obtain quasiequilibrium structures of nonaxisymmetric compact stars such as binary neutron star systems as well as the spacetime around those systems in general relativity. Concerning quasiequilibrium states of binary systems in general relativity, several investigations have been already carried out by assuming conformal flatness of the spatial part of the metric. However, the validity of the conformally flat treatment has not been fully analyzed except for axisymmetric configurations. Therefore it is desirable to solve quasiequilibrium states by developing totally different methods from the conformally flat scheme. In this paper we present a new numerical scheme to solve directly the Einstein equations for 3D configurations without assuming conformal flatness, although we make use of the simplified metric for the spacetime. This new formulation is the extension of the scheme which has been successfully applied for structures of axisymmetric rotating compact stars in general relativity. It is based on the integral representation of the Einstein equations by taking the boundary conditions at infinity into account. We have checked our numerical scheme by computing equilibrium sequences of binary polytropic star systems in Newtonian gravity and those of axisymmetric polytropic stars in general relativity. We have applied this numerical code to binary star systems in general relativity and have succeeded in obtaining several equilibrium sequences of synchronously rotating binary polytropes with the polytropic indices N = 0.0, 0.5 and 1.0.

Polytropic stars in three-dimensional spacetime

We investigate three-dimensional perfect fluid stars with polytropic equation of state, matched to the exterior three-dimensional black hole geometry of Bañados, Teitelboim and Zanelli. A new class of exact solutions for a generic polytropic index is found and analysed.

Inertial Modes of Slowly Rotating Isentropic Stars

We investigate inertial mode oscillations of slowly and uniformly rotating, isentropic, Newtonian stars. Inertial mode oscillations are induced by the Coriolis force due to the star's rotation, and their characteristic frequencies are comparable with the rotation frequency $\Omega$ of the star. So called r-mode oscillations form a sub-class of the inertial modes. In this paper, we use the term ``r-modes'' to denote the inertial modes for which the toroidal motion dominates the spheroidal motion, and the term ``inertial modes'' to denote the inertial modes for which the toroidal and spheroidal motions have comparable amplitude to each other. Using the slow rotation approximation consistent up to the order of $\Omega^3$, we study the properties of the inertial modes and r-modes, by taking account of the effect of the rotational deformation of the equilibrium on the eigenfrequencies and eigenfunctions. The eigenfrequencies of the r-modes and inertial modes calculated in this paper are in excellent agreement with those obtained by Lindblom et al (1999) and Lockitch & Friedman (1998). We also estimate the dissipation timescales due to the gravitational radiation and several viscous processes for polytropic neutron star models. We find that for the inertial modes, the mass multipole gravitational radiation dominates the current multipole radiation, which is dominating in the case of the r-modes. It is also found that the inertial mode instability is more unstable than previously reported by Lockitch & Friedman (1998), and survives the viscous damping processes relevant in neutron stars.