Polytropic Stars Research Articles

Compact stars with non-uniform relativistic polytrope

This paper presents new relativistic composite polytropic models for compact stars by simultaneously solving Einstein field equations with the polytropic state equation to simulate the spherically symmetric, static matter distribution. Using a non-uniform polytropic index, we get the Tolman–Oppenheimer–Volkoff equation for the relativistic composite polytrope (CTOV). To analyze the star's structure, we numerically solve the CTOV equation and compute the Emden and mass functions for various relativistic parameters and polytropic indices appropriate for neutron stars. The calculation results show that, as the relativistic parameter approaches zero, we recover the well-known Lane-Emden equation from the Newtonian theory of polytropic stars; thus, testing the computational code by comparing composite Newtonian models to those in the literature yields good agreement. We compute composite relativistic models for the neutron star candidates Cen X-3, SAXJ1808.4-3658, and PSR J1614-22304. We compare the findings with various existing models in the literature. Based on the accepted models for PSR J1614-22304 and Cen X-3, the star's core radius is predicted to be between 50 and 60% percent of its total radius, while we found that the radius of the core of star SAXJ1808.4-3658 is around 30% of the total radius. Our findings show that the neutron star structure may be approximated by a composite relativistic polytrope, resulting in masses and radii that are quite consistent with observation.

Generating exact polytropes in non-conservative unimodular geometries

The trace-free Einstein equations contain one equation less than the complete field equations. In a static and spherically symmetric spacetime, the number of field equations is thus reduced to two. The equation of pressure isotropy of general relativity, however, is preserved thus showing that any known perfect fluid spacetime is a suitable candidate for the trace-free scenario. The extra freedom in imposing two constraints may now be exploited to include polytopes, something that is difficult in general relativity. The point here is that using any known exact solution one can find a polytropic star for various values of the polytropic index. One arrives at Tolman–Oppenheimer–Volkoff type equations and can study their solutions explicitly. Two examples of well-known stellar distributions that generate polytropes with physically reasonable behaviour are discussed. These models are regular, exhibit a sound speed that is never superluminal and are adiabatically stable in the sense of Chandrasekhar. We investigate a compactness measure confirming that our results are consistent with some observational data.

Core-envelope polytropic star with distinct polytropic indexes

Core-envelope polytropic star with distinct polytropic indexes

Solving Tolman–Oppenheimer–Volkoff equations in f(T) gravity: a novel approach

The torsion models have stood out among the proposals for an alternative description of gravity. The simplest of them, the Teleparallel theory, is equivalent to general relativity and there are many studies dealing with its extension to more general functions of the torsion T. The purpose of our study is to consider a family of f(T) models and apply their corresponding Tolman–Oppenheimer–Volkoff equations to compact objects such as neutron stars.Thus, through a numerical analysis, calculate, among other things, the maximum mass allowed by the model for a neutron star, which also allows us to evaluate which models agree with the observations. In the present paper, the first in the series, we show explicitly the set of equations that must be solved, and how to solve it, in order to model compact stars in f(T) gravity without the need to adopt any particular form for the metric functions or consider any perturbative approach, as has been done in some works in the literature. Examples are given of how our approach works, modelling polytropic stars. We also show that some numerical instabilities reported in a previous study by other authors do not appear in our novel approach. This is an important advance, since it is possible to answer an issue not responded in a previous study, because numerical instabilities prevented proceeding with the calculations. Last but not least, we explicitly show the torsion behaviour inside and outside the star. This is an important question, because with this study we can understand the role of torsion in the structure of the star.

Gravitational Collapse for Polytropic Gaseous Stars: Self-Similar Solutions

In the supercritical range of the polytropic indices gamma in (1,frac{4}{3}) we show the existence of smooth radially symmetric self-similar solutions to the gravitational Euler–Poisson system. These solutions exhibit gravitational collapse in the sense that the density blows up in finite time. Some of these solutions were numerically found by Yahil in 1983 and they can be thought of as polytropic analogues of the Larson–Penston collapsing solutions in the isothermal case gamma =1. They each contain a sonic point, which leads to numerous mathematical difficulties in the existence proof.

Fitting Rotational Frequency of Polytropes as a Function of their Oblateness

Giving a simple functional relationship between the rotational frequency and the centrifugal deformation for compressible stellar models is useful both in observational and theoretical astrophysics. Here I present simple Padé approximants for the rotational frequency of polytropic stars (index N = 1, 1.5, 3) as a function of the oblateness, which are valid at the order of percent for the speed of rotation up to the mass-shedding limit.

Anisotropic solution for polytropic stars in 4D Einstein–Gauss–Bonnet gravity

In the present work we have investigated a new anisotropic solution for polytropic stars in the framework of 4D Einstein–Gauss–Bonnet (EGB) gravity. The possibility of determining the masses and radii of compact stars which puts some limitations on equation of state (EoS) above the nuclear saturation density. For this purpose, the 4D EGB field equations are solved by taking a generalized polytropic equation of state (EoS) with Finch–Skea ansatz. The generalized solution for anisotropic model has been tested for different values of Gauss–Bonnet constant alpha which satisfies all the physical criteria including causality with static stability via mass vs central mass density (M-rho _c), Bondi and Abreu criterion. The adiabatic index shows a minor influence of the GB coupling constant whereas the central and surface redshifts in the EGB gravity always remain lower than the GR. We present the possibility of fitting the mass and radius for some known compact star via M-R curve which satisfies the recent gravitational wave observations from GW 170817 event.

Generalized Oppenheimer–Snyder gravitational collapse into regular black holes

We shall study the formation of a particular class of regular black holes (BHs) from the gravitational collapse of a massive star. The inside geometry is described by spatially flat Friedmann–Robertson–Walker (FRW) metric and the stellar matter is distributed uniformly without any pre-assumption about its equation of state. Our model is a generalization of Oppenheimer–Snyder collapse for regular BHs. We have obtained the density and pressure of star by applying the condition of smooth joining of metrics at the freely falling surface of star. Specifying the regular BHs to Hayward and Bardeen cases, we see that the stellar matter is described by a polytropic equation of state and moreover, for the radius smaller than a certain value, the strong energy condition becomes invalid. Then for both BHs, the interior apparent and event horizons and also the stellar surface are obtained as functions of the proper time of star. At the end, we have constructed a new two parametric family of regular BHs jointed smoothly to the flat FRW interior metric of a polytropic star with an arbitrary index.

Study of anisotropic polytropes in f (, T) Theory

This paper examines the general formalism and applications of isotropic as well as anisotropic polytropic stars in curvature-matter coupled gravity. For this purpose, we consider static spherical and Schwarzschild spacetimes in the interior and exterior regions, respectively. We use two polytropic equations of state to obtain physically viable solutions of the field equations. The hydrostatic equilibrium and Lane-Emden equations are developed for both isotropic as well as anisotropic cases. We study the effects of anisotropic pressure on the stellar structure. Moreover, we graphically inspect the physical behavior of isotropic as well as anisotropic polytropes through energy conditions and stability criterion. Finally, we discuss Tolman mass to explore some characteristics of the models. It is concluded that more viable and stable polytropes are found in this theory as compared to general relativity.

Charged polytropic compact stars in [formula omitted] Einstein–Gauss–Bonnet gravity

We have investigated the possibility of existing a class of compact charged spheres made of a charged perfect fluid in the context of recently proposed 4D Einstein–Gauss–Bonnet (EGB) gravity theory. The main mechanism is to rescale the Gauss–Bonnet (GB) coupling constant α→α/(D−4) in D dimensions and redefining the 4D gravity in the limit D→4. In this way, the GB term yields a non-trivial contribution to the gravitational dynamics in 4D. Our analysis is based on the assumption of a polytropic equation of state (EoS) and the charge density is taken to be proportional to the energy density. More specifically, we have derived the hydrostatic equilibrium equations in 4D EGB, and we have solved them numerically to obtain mass–radius relations for charged compact stars. Eventually we have found the mass–radius relation depending on the values of the GB coupling constant α and the charge fraction ρch(r). In addition, we have studied the mass vs central mass density (M−εc) relation, which identifies the boundary separating the stable configuration region from the unstable one. We have also obtained a relation between the electric charge inside the stellar region and the maximum mass of compact star in this gravity theory. Finally, we conclude that in such a scenario charged stars may exist in Nature, and that such a deviation may be observable in future astrophysical probes.

Stars Crushed by Black Holes. II. A Physical Model of Adiabatic Compression and Shock Formation in Tidal Disruption Events

Abstract We develop a Newtonian model of a deep tidal disruption event (TDE), for which the pericenter distance of the star, r p, is well within the tidal radius of the black hole, r t, i.e., when β ≡ r t/r p ≫ 1. We find that shocks form for β ≳ 3, but they are weak (with Mach numbers ∼1) for all β, and that they reach the center of the star prior to the time of maximum adiabatic compression for β ≳ 10. The maximum density and temperature reached during the TDE follow much shallower relations with β than the previously predicted ρ max ∝ β 3 and T max ∝ β 2 scalings. Below β ≃ 10, this shallower dependence occurs because the pressure gradient is dynamically significant before the pressure is comparable to the ram pressure of the free-falling gas, while above β ≃ 10, we find that shocks prematurely halt the compression and yield the scalings ρ max ∝ β 1.62 and T max ∝ β 1.12 . We find excellent agreement between our results and high-resolution simulations. Our results demonstrate that, in the Newtonian limit, the compression experienced by the star is completely independent of the mass of the black hole. We discuss our results in the context of existing (affine) models, polytropic versus non-polytropic stars, and general relativistic effects, which become important when the pericenter of the star nears the direct capture radius, at β ∼ 12.5 (2.7) for a solar-like star disrupted by a 106 M ⊙ (107 M ⊙) supermassive black hole.

Gravitational decoupling of generalized Horndeski hybrid stars

Gravitational decoupled compact polytropic hybrid stars are here addressed in generalized Horndeski scalar-tensor gravity. Additional physical properties of hybrid stars are scrutinized and discussed in the gravitational decoupling setup. The asymptotic value of the mass function, the compactness, and the effective radius of gravitational decoupled hybrid stars are studied for both cases of a bosonic and a fermionic prevalent core. These quantities are presented and discussed as functions of Horndeski parameters, the decoupling parameter, the adiabatic index, and the polytropic constant. Important corrections to general relativity and generalized Horndeski scalar-tensor gravity, induced by the gravitational decoupling, comply with available observational data. Particular cases involving white dwarfs, boson stellar configurations, neutron stars, and Einstein–Klein–Gordon solutions, formulated in the gravitational decoupling context, are also scrutinized.

A class of polytropic stars in Einstein–Gauss–Bonnet gravity

A class of polytropic stars in Einstein–Gauss–Bonnet gravity

Rapidly rotating compact stars in Rastall’s gravity

In this work we study rapidly rotating stars by considering the Rastall theory of gravity. We obtain and solve the equations by numerical methods for two usual parametrization of polytropic stars. Then the mass–radius relations, moments of inertia and other results of interest are obtained and compared with the ones for non-rotating stars.

Polytropic stars in bootstrapped Newtonian gravity

We study self-gravitating stars in the bootstrapped Newtonian picture for polytropic equations of state. We consider stars that span a wide range of compactness values. Both matter density and pressure are sources of the gravitational potential. Numerical solutions show that the density profiles can be well approximated by Gaussian functions. Later we assume Gaussian density profiles to investigate the interplay between the compactness of the source, the width of the Gaussian density profile and the polytropic index. We also dedicate a section to comparing the pressure and density profiles of the bootstrapped Newtonian stars to the corresponding General Relativistic solutions. We also point out that no Buchdahl limit is found, which means that the pressure can in principle support a star of arbitrarily large compactness. In fact, we find solutions representing polytropic stars with compactness above the Buchdhal limit.

Boson stars in f(T) extended theory of gravity

Spherically symmetric configurations of the non-interacting massive complex scalar field, representing non-rotating boson stars, are considered within the framework of the modified torsion based $f(T)$ gravity, with $f(T) = T + \alpha \, T^2/2$. We find that with sufficiently large negative value of $\alpha$ the mass of the boson stars can be made arbitrarily large. This is in contrast to General Relativity where an upper bound, $M_{max} \sim M_{Planck}^2/m$, to the mass of the boson stars built from the non-interacting scalar field exists and where the masses of boson stars in the astrophysical regime can be obtained only with the introduction of the scalar field self-interaction. With sufficiently large negative $\alpha$ we also find negative gravitational binding energy for all masses, which can be seen as an indication of the stability of such configurations. In its positive regime, $\alpha$ can not be made arbitrarily large as a phase transition in the stress--energy components of the $f(T)$-fluid develops. This phenomenon has already been reported to occur in polytropic stars constructed within the $f(T)$ gravity theory.

The double polytrope for anisotropic matter: Newtonian case

A general formalism developed few years ago to model polytropic Newtonian stars with anisotropic pressure is applied to model stars for which, both, radial and tangential pressure satisfy polytropic equations of state. We obtain the corresponding Lane--Emden equation, and integrate it for a wide range of values of the parameters, thereby illustrating the effects of the pressure anisotropy on stellar objects. We calculate Chandrasekhar mass for a white dwarf and compare it with previous results. Finally, prospective applications to some astrophysical scenarios are discussed.

Stability of Polytropic Stars in Palatini Gravity

We will briefly discuss the necessary conditions for stability of polytropies in $f(\hat R)$ Palatini gravity and the differences with the General Relativity ones.

Nonaxisymmetric instabilities and star-disk coupling I. Moderate mass disks

We present results of an investigation of the effects massive circumstellar disks have on the stability properties of polytropic stars. We consider systems with moderate mass disks, $M_{d}/M_{*} = 0.2$ and 0.25 where $M_{d}$ and $M_{*}$ are the masses of the disk and star respectively, and with central stars in slow uniform rotation and central stars in rapid differential rotation. We study systems where the central stars are above the barlike mode secular and dynamic instability thresholds for rotating solitary stars and systems where the stars are below the barlike mode secular instability threshold for rotating solitary stars. Equilibrium systems composed of stars with massive circumstellar disks are calculated, and their stability properties determined using linear and nonlinear techniques. For the uniform rotation case, the central stars are in slow rotation in the sense that they are below barlike instability thresholds for rotating single stars. For the differential rotation case, the central stars span the range from slow rotation to rapid rotation where rapid rotation means the stars are unstable to dynamic barlike instabilities. Gravitational coupling between the star and disk drives one-armed instabilities in the slowly rotating central star case. The couple primarily forces orbital motion rather than spin-up of the star. For systems with differentially rotating central stars with slow rotation, star-disk coupling drives instability in one-armed and two-armed modes. At the fastest rotation considered, the central star becomes unstable to the barlike instability evolving in a manner similar to that found for single rapidly rotating stars. The central stars did not fission. Star-disk coupling does not strongly influence the development of the barlike mode for this case.

Structure of polytropic stars in General Relativity

The inner structure of a star or a primordial interstellar cloud is a major topic in classical and relativistic physics. The impact that General Relativistic principles have on this structure has been the subject of many research papers. In this paper we consider within the context of General Relativity a prototype model for this problem by assuming that a star consists of polytropic gas. To justify this assumption we observe that stars undergo thermodynamically irreversible processes and emit heat and radiation to their surroundings. Due to the emission of this energy it is worthwhile to consider an idealized model in which the gas is polytropic. To find interior solutions to the Einstein equations of General Relativity in this setting we derive a single equation for the cumulative mass distribution of the star and use Tolman-Oppenheimer-Volkoff equation to derive formulas for the isentropic index and coefficient. Using these formulas we present analytic and numerical solutions for the polytropic structure of self-gravitating stars and examine their stability. We prove also that when the thermodynamics of a star as represented by the isentropic index and coefficient is known, the corresponding matter density within the star is uniquely determined.