Relativistic Stellar Models Research Articles

Electromagnetic field on the complexity of minimally deformed compact stars

In the context of this endeavor, we establish a simple protocol for formulating interior stellar solutions that exhibit spherically symmetric configurations against the backdrop of relativistic gravitational decoupling through radial metric deformation (minimal geometric deformation scheme). In this pursuit, we make use of the vanishing complexity factor (Y~TF\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\widetilde{Y}_{TF}$$\\end{document}) condition, based on Herrera’s (Phys Rev D 97, 044010, 2018) innovative concept regarding the complexity of static or slowly evolving spherical matter configurations. The idea of a complexity factor emerges as the outcome of the orthogonal splitting of the Riemann–Christoffel tensor, which yields different scalar functions, known as structure scalars. The protocol is demonstrated by employing the Buchdahl and Tolman relativistic stellar ansatzes as isotropic seeds. Both of these ansatzes exhibit similar physical features, with a minor variation in their magnitudes in the case of Y~TF≠0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\widetilde{Y}_{TF}\ e 0$$\\end{document}, where 0≤α<1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$0\\le \\alpha <1$$\\end{document}, and α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document} represents a coupling parameter. However, when Y~TF=0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\widetilde{Y}_{TF}=0$$\\end{document}, the Buchdahl stellar ansatz exhibits a uniform density matter configuration, while the Tolman model features an increasing pressure profile. The obtained relativistic stellar models satisfy the basic viability constraints required for the physically realistic configurations.

Optimal design of relativistic stellar models

A new approach to construct relativistic stellar models is presented. The problem is formulated as an optimal control problem, with an objective function to be optimized. As illustration, the problem is formulated and solved, with the rest mass as the objective function to be minimized, in three cases. First, if the density is piece-wise continuous the optimal solution is the spherical shell. Second, if the density gradient is piece-wise continuous the optimal solution has constant density. Third, if the speed of sound does not exceed that of light, the optimal solution has a linear equation of state. Thus, using this new approach, the equation of state is determined by a physical motivation. Results support the suggestion that the linear equation of state could represent a relativistic incompressible fluid. Extensions to some other configurations are explored.

A study on the effect of anisotropy under Finch–Skea geometry

The popularity of the Finch-Skea ansatz to describe relativistic stellar model have encouraged us to study the analytic solutions of the Einstein field equation. We have presented a class of exact solutions to the field equations after considering the corresponding two cases: (i) positive value of anisotropic parameter, and (ii) absence of any anisotropy. Smooth matching of the interior solutions with the Schwarzschild exterior solution helped us to determine constants. The physical features of the solutions thus obtained have been studied both graphically and numerically with the specific pulsar $4U~1608-52$ (Mass = $1.57^{+0.30}_{-0.29}~M_\odot$ and radius = $9.8\pm 1.8~km$). The stability conditions for the model have also been discussed, however the model is found to be stable for zero anisotropy.

Static spherical perfect fluid stars with finite radius in general relativity: a review

In this article, we provide a pedagogical review of the Tolman-Oppenheimer-Volkoff (TOV) equation and its solutions which describe static, spherically symmetric gaseous stars in general relativity. Our discussion starts with a systematic derivation of the TOV equation from the Einstein field equations and the relativistic Euler equations. Next, we give a proof for the existence and uniqueness of solutions of the TOV equation describing a star of finite radius, assuming suitable conditions on the equation of state characterizing the gas. We also prove that the compactness of the gas contained inside a sphere centered at the origin satisfies the well-known Buchdahl bound, independent of the radius of the sphere. Further, we derive the equation of state for an ideal, classical monoatomic relativistic gas from statistical mechanics considerations and show that it satisfies our assumptions for the existence of a unique solution describing a finite radius star. Although none of the results discussed in this article are new, they are usually scattered in different articles and books in the literature; hence it is our hope that this article will provide a self-contained and useful introduction to the topic of relativistic stellar models.

Higher dimensional charged compact objects in Finch–Skea geometry

We obtain relativistic solutions of anisotropic charged compact objects in hydrodynamical equilibrium with Finch–Skea geometry in the usual four and in higher dimensions. The relativistic solutions are employed to construct physically viable stellar models. The radial variations of density, pressure and other different physical features inside the stars are studied in the relativistic stellar models. It is noted that a compact star in four-dimensional Finch–Skea geometry describes an isotropic uncharged star always which however predicts existence of an anisotropic star in higher dimensional spacetime. The plausibility of such stars are studied here for a given mass and radius. Considering known compact objects we construct stellar models satisfying all the criteria of a physically realistic star. The results obtained here may be important to understand some of the physical properties of known stars including predictions of equations of states at extreme conditions.

Three-layered relativistic stellar model endowed with distinct equation of states

In this paper, we amalgamate three-layer hypothesis of compact star models given by Ginzburg (Usp Fiz Nauk 103:393, 1971) with our recent works on core-envelope compact star models (Gedela et al. in Eur Phys J C 79:566, 2019; Pant et al. in Eur Phys J C 79:602, 2019) and propose a three-layered relativistic stellar model endowed with distinct equation of states (EoSs) in the realm of general relativity. Instead of considering core and envelope of a neutron star, we insert an extra layer, namely, an intermediate layer in between the core and envelope to enhance a new conceptual aspect of realistic and accurate modeling of internal structure of stellar objects. The quark core is assumed to satisfy MIT-bag EoS, an intermediate region and a thin envelope of matter are considered with modified Bose–Einstein condensate (BEC) and quadratic EoSs, respectively. Further, respective pairs of interfaces in between stellar objects are matched with Dormois–Israel conditions. We verify almost all physical and geometrical parameters, including anisotropic, radial pressure, compactification factor, energy conditions, and stability conditions using TOV equation are with viable trends and continuous with the smooth variations throughout interior of the stars.

Generalized Durgapal–Fuloria relativistic stellar models

We present a new class of closed-form solutions to the Einstein–Maxwell system of equations for a static spherically symmetric anisotropic star in the presence of an electric field by generalizing earlier approaches. The field equations are integrated by specifying the form of the electric field, anisotropic factor, and one of the gravitational potentials which are physically reasonable. We regain some of the earlier solutions of Durgapal and Fuloria (1985), Gupta and Maurya (2011) and Maurya et al. (2015) as special cases. A detailed physical analysis of the solution indicates the physical viability for modelling anisotropic charged superdense stars.

Charged radiating stars with Lie symmetries

We consider the general model of an accelerating, expanding and shearing radiating star in the presence of charge. Using a new set of variables arising from the Lie symmetries of differential equations we transform the boundary equation into ordinary differential equations. We present several new exact models for a charged gravitating sphere. A particular family of solution may be interpreted as a generalised Euclidean star in the presence of the electromagnetic field. This family admits a linear barotropic equation of state. In the uncharged limit, we regain general relativistic stellar models where proper and areal radii are equal, and its generalisations. Our group theoretical approach selects the physically important cases of Euclidean stars and equations of state.

A new relativistic stellar model with anisotropic fluid in Karmarkar space–time

We are presenting a new class of well-behaved solutions in embedding class-I. We proceed our calculations by assuming a new type of grr metric potential and solved for the other metric gtt using Karmarkar condition. The necessary condition that any solutions of Einstein’s field equations to be class-I is to satisfy Karmarkar condition and sufficient condition is to satisfy Pandey–Sharma condition i.e. R2323≠0. The solution also satisfies strong energy condition, null energy condition, dominant energy condition and weak energy condition. The obtained compactness parameter is within Buchdahl limit i.e. 2M/R≤8/9. The solution also satisfies the causality condition and can represent stable stellar fluid system as the adiabatic index Γ>4/3 and the stability factor holds −1≤vt2−vr2≤0 good. Finally, we have tuned our solution for two compact stars PSR J1614-2230 and 4U1608-52 which are well-behaved in all respects.

Stable hybrid stars within a SU(3) quark-meson-model

The inner regions of the most massive compact stellar objects might be occupied by a phase of quarks. Since the observations of the massive pulsars PSR J1614-2230 and PSR $\mathrm{J}0348+0432$ with about two solar masses, the equations of state constructing relativistic stellar models have to be constrained respecting these new limits. We discuss stable hybrid stars, i.e. compact objects with an outer layer composed of nuclear matter and with a core consisting of quark matter (QM). For the outer nuclear layer we utilize a density dependent nuclear equation of state and we use a chiral SU(3) quark-meson model with a vacuum energy pressure to describe the object's core. The appearance of a disconnected mass-radius branch emerging from the hybrid star branch implies the existence of a third family of compact stars, so-called twin stars. Twin stars did not emerge as the transition pressure has to be relatively small with a large jump in energy density, which could not be satisfied within our approach. This is, among other reasons, due to the fact that the speed of sound in QM has to be relatively high, which can be accomplished by an increase of the repulsive coupling. This increase on the other hand yields transition pressures that are too high for twins stars to appear.

Einstein–Maxwell Field Equation in Isotropic Coordinates: An Application to Modeling Superdense Star

We present a charged analogue of Pant et al. (2010, Astrophys. Space Sci., 330, 353) solution of the general relativistic field equations in isotropic coordinates by using simple form of electric intensity E that involve charge parameter K. Our solution is well behaved in all respects for all values of X lying in the range 0 <X≤ 0.11, K lying in the range 4 <K≤ 6.2 and Schwarzschild compactness parameter u lying in the range 0 <u≤ 0.247. Since our solution is well behaved for wide ranges of the parameters, we can model many different types of ultra-cold compact stars like quark stars and neutron stars. We have shown that corresponding to X = 0.077 and K = 6.13 for which u = 0.2051 and by assuming surface density ρ b =4.6888×1014 g cm −3 the mass and radius are found to be 1.509M ⊙, 10.906 km respectively which match with the observed values of mass 1.51M ⊙ and radius 10.90 km of the quark star XTE J1739-217. The well behaved class of relativistic stellar models obtained in this work might have astrophysical significance in the study of more realistic internal structures of compact stars.

Magnetized relativistic stellar models in Eddington-inspired Born-Infeld gravity

We consider the structure of the magnetic fields inside the neutron stars in Eddington-inspired Born-Infeld (EiBI) gravity. In order to construct the magnetic fields, we derive the relativistic Grad-Shafranov equation in EiBI, and numerically determine the magnetic distribution in such a way that the interior magnetic fields should be connected to the exterior distribution. Then, we find that the magnetic distribution inside the neutron stars in EiBI is qualitatively similar to that in general relativity, where the deviation of magnetic distribution in EiBI from that in general relativity is almost comparable to uncertainty due to the equation of state (EOS) for the neutron star matter. However, we also find that the magnetic fields in the crust region are almost independent of the coupling constant in EiBI, which suggests a possibility to obtain the information about the crust EOS independently of the gravitational theory via the observations of the phenomena associated with the crust region. In any case, since the imprint of EiBI gravity on the magnetic fields is weak, the magnetic fields could be a poor probe of gravitational theories, considering many magnetic uncertainties.

Anisotropic charged analogue of Heintzmann’s solution

We present an anisotropic charged analogue of Heintzmann’s (Z. Phys. 228:489, 1969) solution of the general relativistic field equations in curvature coordinates by using simple form of electric intensity E and pressure anisotropy factor Δ that involve charge parameter K and anisotropy parameter α respectively. Our solution is well behaved in all respects for all values of X lying in the range 0<X≤1.1, α lying in the range 0≤α≤6.2, K lying in the range 0<K≤9.7 and Schwarzschild compactness parameter “u” lying in the range 0<u≤0.391. Since our solution is well behaved for a wide ranges of the parameters, we can model many different types of ultra-cold compact stars like quark stars and neutron stars. We present some models of super dense quark star and neutron stars corresponding to X=0.1, α=1 and K=3. By assuming surface density of quark star, ρ b =4.6888×1014 g cm−3 the mass and radius are 1.271M ⊙, 10.09 km respectively. For ρ b =2.7×1014 g cm−3 the mass and radius of neutron star are 1.675M ⊙, 13.297 km respectively. The well behaved class of relativistic stellar models obtained in this work might have astrophysical significance in the study of more realistic internal structure of compact stars.

A Class of Super Dense Stars Models Using Charged Analogues of Hajj-Boutros Type Relativistic Fluid Solutions

We present a spherically symmetric solution of the general relativistic field equations in isotropic coordinates for perfect charged fluid, compatible with a super dense star modeling. The solution is well behaved for all the values of Schwarzschild parameter u lying in the range 0 < u < 0.1727 for the maximum value of charge parameter K = 0.08163. The maximum mass of the fluid distribution is calculated by using stellar surface density as ρ b = 4.6888×1014g cm−3. Corresponding to K = 0.08 and u max = 0.1732, the resulting well behaved solution has a maximum mass M = 0.9324M ⊙ and radius R = 8.00 and by assuming ρ b = 2×1014g cm−3 the solution results a stellar configuration with maximum mass M = 1.43M ⊙ and radius R b = 12.25 km. The maximum mass is found increasing with increasing K up to 0.08. The well behaved class of relativistic stellar models obtained in this work might has astrophysical significance in the study of internal structure of compact star such as neutron star or self-bound strange quark star like Her X-1.

Some electrically charged relativistic stellar models in general relativity

Some new families of electrically charged stellar models of ultra-compact star have been studied. With the help of particular form of one of the metric potentials the Einstein–Maxwell field equations in general relativity have been transformed to a system of ordinary differential equations. The interior matter pressure, energy–density, and the adiabatic sound speed are expressed in terms of simple algebraic functions. The constant parameters involved in the solution have been set so that certain physical criteria satisfied. Based on the analytic model developed in the present work, the values of the relevant physical quantities have been calculated by assuming the estimated masses and radii of some well known potential strange star candidates like X-ray pulsar Her X-1, millisecond X-ray pulsar SAX J 1808.4-3658, and 4U 1820-30. The analytical equations of state of the charged matter distribution may play a significant role in the study of the internal structure of highly compact charged stellar objects in general relativity.

Analytical Solutions of the Riccati Equation with Coefficients Satisfying Integral or Differential Conditions with Arbitrary Functions

Ten new exact solutions of the Riccati equation dy/dx = a(x) + b(x)y + c(x)y² are presented. The solutions are obtained by assuming certain relations among the coefficients a(x), b(x) and c(x) of the Riccati equation, in the form of some integral or differential expressions, also involving some arbitrary functions. By appropriately choosing the form of the coefficients of the Riccati equation, with the help of the conditions imposed on the coefficients, we obtain ten new integrability cases for the Riccati equation. For each case the general solution of the Riccati equation is also presented. The possibility of the application of the obtained mathematical results for the study of anisotropic general relativistic stellar models is also briefly considered.

A class of exact isotropic solutions of Einstein’s equations and relativistic stellar models in general relativity

In this paper we have studied a particular class of exact solutions of Einstein’s gravitational field equations for spherically symmetric and static perfect fluid distribution in isotropic coordinates. The Schwarzschild compactness parameter, GM/c 2 R, can attain the maximum value 0.1956 up to which the solution satisfies the elementary tests of physical relevance. The solution also found to have monotonic decreasing adiabatic sound speed from the centre to the boundary of the fluid sphere. A wide range of fluid spheres of different mass and radius for a given compactness is possible. The maximum mass of the fluid distribution is calculated by using stellar surface density as parameter. The values of different physical variables obtained for some potential strange star candidates like Her X-1, 4U 1538–52, LMC X-4, SAX J1808.4−3658 given by our analytical model demonstrate the astrophysical significance of our class of relativistic stellar models in the study of internal structure of compact star such as self-bound strange quark star.

Some Exact Relativistic Models of Electrically Charged Self-bound Stars

In continuation of recent work done by the present authors (Int. J. Theor. Phys. 2013, doi: 10.1007/s10773-013-1538-y , hereafter paper I) some new exact families of static spherically symmetric perfect fluid solution of Einstein–Maxwell gravitational field equations are presented. These solutions and the corresponding equations of state, presented in parametric form, may be astrophysically significant in constructing relativistic stellar models of electrically charged self-bound stars.

Constraining neutron-star tidal Love numbers with gravitational-wave detectors

Ground-based gravitational wave detectors may be able to constrain the nuclear equation of state using the early, low frequency portion of the signal of detected neutron star--neutron star inspirals. In this early adiabatic regime, the influence of a neutron star's internal structure on the phase of the waveform depends only on a single parameter $\ensuremath{\lambda}$ of the star related to its tidal Love number, namely, the ratio of the induced quadrupole moment to the perturbing tidal gravitational field. We analyze the information obtainable from gravitational wave frequencies smaller than a cutoff frequency of 400 Hz, where corrections to the internal-structure signal are less than 10%. For an inspiral of two nonspinning $1.4{M}_{\ensuremath{\bigodot}}$ neutron stars at a distance of 50 Megaparsecs, LIGO II detectors will be able to constrain $\ensuremath{\lambda}$ to $\ensuremath{\lambda}\ensuremath{\le}2.0\ifmmode\times\else\texttimes\fi{}{10}^{37}\text{ }\text{ }\mathrm{g}\text{ }{\mathrm{cm}}^{2}\text{ }{\mathrm{s}}^{2}$ with 90% confidence. Fully relativistic stellar models show that the corresponding constraint on radius $R$ for $1.4{M}_{\ensuremath{\bigodot}}$ neutron stars would be $R\ensuremath{\le}13.6\text{ }\text{ }\mathrm{km}$ (15.3 km) for a $n=0.5$ ($n=1.0$) polytrope with equation of state $p\ensuremath{\propto}{\ensuremath{\rho}}^{1+1/n}$.

Relativistic Stars with Poloidal and Toroidal Magnetic Fields and Meridional Flow

We investigate stationary axisymmetric configurations of magnetized stars in the framework of general relativistic ideal magnetohydrodynamics. Our relativistic stellar model incorporates a toroidal magnetic field and meridional flow in addition to a poloidal magnetic field for the first time. The magnetic field and meridional flow are treated as perturbations, but no other approximation is made. We find that the stellar shape can be prolate rather than oblate when a toroidal field exists. We also find that, for fixed baryonic mass and total magnetic helicity, the more spherical the star is, the lower the energy it has. Further, we find two new types of the frame-dragging effect that differ from the standard one in a rotating star or Kerr geometry. They may violate the reflection symmetry about the equatorial plane.