Anisotropic Polytropes Research Articles

Complexity factor of spherically anisotropic polytropes from gravitational decoupling

In this work we will analyse the complexity factor, proposed by L. Herrera, for spherically symmetric static matter distributions satisfying a polytropic equation through the gravitational decoupling method. Specifically, we will use the 2-step GD, which is a particular case of the extended geometric deformation (EGD), to obtain analytic polytropic solutions of Einstein's equations. In order to give an example, we construct a model satisfying a polytropic equation of state using Tolman IV as a seed solution.

Integration of the Lane–Emden equation for relativistic anisotropic polytropes through gravitational decoupling: a novel approach

In this work we propose a novel approach to integrate the Lane–Emden equations for relativistic anisotropic polytropes. We take advantage of the fact that Gravitational Decoupling allows to decrease the number of degrees of freedom once a well known solution of the Einstein field equations is provided as a seed so after demanding the polytropic equation for the radial pressure the system is automatically closed. The approach not only allows to extend both isotropic or anisotropic known solutions but simplifies the computation of the Tolman mass whenever the Minimal Geometric Deformation is considered given that the g_{tt} component of the metric remains unchanged. We illustrate how the the method works by analyzing the solutions obtained from Tolman IV, Durgapal IV and Wymann IIa isotropic systems as a seed for the integration.

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.

Class I polytropes for anisotropic matter

In this work we study class I interior solutions supported by anisotropic polytropes. The generalized Lane–Emden equation compatible with the embedding condition is obtained and solved for a different set of parameters in both the isothermal and non-isothermal regimes. For completeness, the Tolman mass is computed and analysed to some extend. As a complementary study we consider the impact of the Karmarkar condition on the mass and the Tolman mass functions respectively. Comparison with other results in literature are discussed.

Deep-MOND polytropes

Working within the deep-MOND limit (DML), I describe spherical, self-gravitating systems governed by a polytropic equation of state, $P=\mathcal{K}\rho^\gamma$. As self-consistent structures, such systems can serve as heuristic models for DML, astronomical systems, such as dwarf spheroidal galaxies, low-surface-density elliptical galaxies and star clusters, and diffuse galaxy groups. They can also serve as testing ground for various theoretical MOND inferences. In dimensionless form, the equation satisfied by the radial density profile $\zeta(y)$ is (for $\gamma\not=1$) $[\int_0^y \zeta \bar y^2 d\bar y]^{1/2}=-yd(\zeta^{\gamma-1})/dy$. Or, $\theta^n(y)=y^{-2}[(y\theta')^2]'$, where $\theta=\zeta^{\gamma-1}$, and $n\equiv (\gamma-1)^{-1}$. I discuss properties of the solutions, contrasting them with those of their Newtonian analogues -- the Lane-Emden polytropes. Due to the stronger MOND gravity, all DML polytropes have a finite mass, and for $n<\infty$ ($\gamma>1$) all have a finite radius. (Lane-Emden spheres have a finite mass only for $n\le 5$.) I use the DML polytropes to study DML scaling relations. For example, they satisfy a universal relation (for all $\mathcal{K}$ and $\gamma$) between the total mass, $M$, and the mass-average velocity dispersion $\sigma$: $MGa_0=(9/4)\sigma^4$. However, the relation between $M$ and other measures of the velocity dispersion, such as the central, projected one, $\bar\sigma$, does depend on $n$ (but not $\mathcal{K}$), defining a `fundamental surface' in the $[M,~\bar\sigma,~n]$ space. I also describe the generalization to anisotropic polytropes, which also all have a finite radius (for $\gamma>1$), and all satisfy the above universal $M-\sigma$ relation. This more extended class of models exhibits the central-surface-densities relation: a tight relation between the baryonic and the dynamical central surface densities predicted by MOND.

Impact of generalized polytropic equation of state on charged anisotropic polytropes

In this paper, generalized polytropic equation of state is used to get new classes of polytropic models from the solution of Einstein-Maxwell field equations for charged anisotropic fluid configuration. The models are developed for different values of polytropic index n=1,~frac{1}{2},~2. Masses and radii of eight different stars have been regained with the help of developed models. The speed of sound technique and graphical analysis of model parameters is used for the viability of developed models. The analysis of models indicates they are well behaved and physically viable.

New models of charged anisotropic polytropes with radiation density

In this manuscript, new classes of polytropic models have been developed by using polytropic equation of state (PEoS) for spherically symmetric gravitating sources in isotropic coordinates. The inner fluid configuration is charged anisotropic and models are developed for different values of polytropic index $$n=1,~\frac{1}{2},~2,~\frac{2}{3}$$. Mass and radii of eight stars 4U 1820-30, Cen X-3, EXO 1785-248, SMC X-4, LMC X-4, SAX J1808.4-3658, 4U 1538-52 and Her X-1 have been regained with the help of developed models. The stability of models is discussed by using speed of sound technique and graphical analysis of model parameters. It is concluded that all models are well behaved and physically acceptable.

Role of σR2 + γRμνTμν model on anisotropic polytropes

This paper analyzes the anisotropic stellar evolution governed by polytropic equation-of-state in the framework of [Formula: see text] gravity, where [Formula: see text]. We construct the field equations, hydrostatic equilibrium equation and trace equation to obtain their solutions numerically under the influence of [Formula: see text] gravity model, where [Formula: see text] and [Formula: see text] are arbitrary constants. We examine the dependence of various physical characteristics such as radial/tangential pressure, energy density, anisotropic factor, total mass and surface redshift for specific values of the model parameters. The physical acceptability of the considered model is discussed by verifying the validity of energy conditions, causality condition and adiabatic index. We also study the effects arising due to strong nonminimal matter-curvature coupling on anisotropic polytropes. It is found that the polytropic stars are stable and their maximum mass point lies within the required observational Chandrasekhar limit.

Cracking of charged polytropes with generalized polytropic equation of state

We discuss the occurrence of cracking in charged anisotropic polytropes with generalized polytropic equation of state through two different assumptions; (i) by carrying out local density perturbations under a conformally flat condition (ii) by perturbing anisotropy, polytropic index and charge parameters. For this purpose, we consider two different definitions of polytropes that exist in literature. We conclude that under local density perturbations scheme cracking does not appear in both types of polytropes and stable configuration is observed, while with the second type of perturbation cracking appears in both types of polytropes under certain conditions.

On cracking of charged anisotropic polytropes

Recently in [1], the role of electromagnetic field on the cracking of spherical polytropes has been investigated without perturbing charge parameter explicitly. In this study, we have examined the occurrence of cracking of anisotropic spherical polytropes through perturbing parameters like anisotropic pressure, energy density and charge. We consider two different types of polytropes in this study. We discuss the occurrence of cracking in two different ways (i) by perturbing polytropic constant, anisotropy and charge parameter (ii) by perturbing polytropic index, anisotropy and charge parameter for each case. We conclude that cracking appears for a wide range of parameters in both cases. Also, our results are reduced to [2] in the absence of charge.

Analytical families of two-component anisotropic polytropes and their relativistic extensions

In this paper, we study a family of two-component anisotropic polytropes which model a wide range of spherically symmetric astrophysical systems such as early-type baryonic galaxies. This family is found to contain a large class of models such as the hypervirial family (which satisfy the virial theorem locally), the Plummer and Hernquist models and Navarro-Frenk-White (NFW) like models. The potential–density pair for these models are derived, as well as their velocity dispersions and anisotropy parameters. The projected quantities are computed and found to reduce to analytical expressions in some cases. The first section of this paper presents an extension of the two-term anisotropic polytropes to encompass a very wide range of potential–density pairs. In the next section, we present the general relativistic extension of the potential–density pair and calculate the stress–energy tensor, the relativistic anisotropy parameter, the velocity of circular orbits and the angular momentum. Remarkably, for the case of the hypervirial family, the relativistic pressure in the Newtonian limit and the relativistic anisotropy parameter are found to coincide with the corresponding Newtonian expressions. The weak, dominant and strong energy conditions are found to be satisfied only for a certain range of the free parameters. We show that the relativistic hypervirial family also has a finite total mass like its Newtonian counterpart. In the first appendix, a relativistic extension of a different hypervirial family of models is studied and the relativistic anisotropy parameter is found to coincide with the Newtonian one. Finally, we present a family of models obtained from our distribution function that are similar to the Ossipkov–Merritt models; by computing their anisotropy parameters, we show that they model systems with isotropic cores and radially anisotropic exteriors.

Notes on the stability threshold for radially anisotropic polytropes

We discuss some contradictions found in the literature concerning the problem of stability of collisionless spherical stellar systems which are the simplest anisotropic generalization of the well-known polytrope models. Their distribution function $F(E,L)$ is a product of power-low functions of the energy $E$ and the angular momentum $L$, i.e. $F\propto L^{-s}(-E)^q$. On the one hand, calculation of the growth rates in the framework of linear stability theory and N-body simulations show that these systems become stable when the parameter $s$ characterizing the velocity anisotropy of the stellar distribution is lower than some finite threshold value, $s<s_\textrm{crit}$. On the other hand Palmer & Papaloizou (1987) showed that the instability remained up to the isotropic limit $s=0$. Using our method of determining the eigenmodes for stellar systems, we show that the growth rates in weakly radially-anisotropic systems are indeed positive, but decrease exponentially as the parameter $s$ approaches zero, i.e. $\gamma\propto \exp(-s_{\ast}/s)$. In fact, for the systems with finite lifetime this means stability.

Models for spherical stellar systems with isotropic cores and anisotropic haloes

We discuss one particular class of anisotropic distribution functions for spherical stellar systems. These are separable functions of the binding energy |$\epsilon\equiv \psi(r)-\upsilon^2/2$|⁠, where the positive potential |$\psi(r)$| is defined to be zero at the outer boundary, and the dimensionless quantity |$\kappa\equiv L^2/2r^2_a\epsilon$|⁠, where L is the angular momentum and the parameter ra is the anisotropy radius. In general, these models have regular cores, are isotropic for r ≪ ra, become increasingly anisotropic for r ≫ ra and reach maximum, but finite, anisotropy at the outer boundary. We study in detail self-consistent models with distribution functions of the form |$f=\epsilon^{\upsilon-3/2}(1+\kappa)^{\mu}$|⁠. These models resemble isotropic polytropes in the inner parts and anisotropic polytropes in the outer parts. The three parameters υ, μ, and ra which determine the central concentration, the degree of anisotropy and the transition region, respectively, can be chosen arbitrarily within certain bounds. The density, velocity dispersions, anisotropy etc. can be worked out analytically and are separable functions of potential and radius. Some applications of these models are briefly discussed.