Polytropic Equation Of State Research Articles

Existence of slowly rotating bipolytropes with prolate cores

We report the existence of hydrostatic equilibrium states for a composite body made of two rigidly rotating, homogeneous layers bounded by spheroidal surfaces, where the core has a prolate shape. These new configurations require an oblate envelope that spins faster than the core. No solution with a prolate envelope is found. For some parameters, the prolate core can even be at rest. Numerical experiments based on the self-consistent field method support this result in the case of heterogeneous layers with polytropic equations of state. The possible cancellation of the first gravitational moment, $J_2$, is discussed.

Relativistic fluids interacting through gravitational decoupling in f(T,𝒯) theory

In this paper, we will manifest how polytropic fluids produce effects on an alternate gravitational source, regardless of its features, for spherically symmetric and static spacetimes. For this purpose, we use [Formula: see text] modified gravity, which is characterized by the trace of an energy–momentum tensor as [Formula: see text] and torsion scalar [Formula: see text], Additionally, using the polytropic equation of state, we will speculate about the distribution of energy among fluids within an astrophysical object. We offer a thorough methodology for identifying generic relativistic polytropes accompanied by the minimal geometric decoupling technique. Ultimately, a few tangible attributes of polytropes along with perfect fluids are observed using the Tolman IV solution, including total pressure, energy interchange, and thermal characteristics.

Stellar Modeling via the Tolman IV Solution: The Cases of the Massive Pulsar J0740+6620 and the HESS J1731-347 Compact Object

We model compact objects of known stellar mass and radius made of isotropic matter within Einstein’s gravity. The interior solution describing hydrostatic equilibrium we are using throughout the manuscript corresponds to the Tolman IV exact analytic solution obtained a long time ago. The three free parameters of the solutions are determined by imposing the matching conditions for objects of known stellar mass and radius. Finally, using well established criteria, it is shown that, contrary to the Kohler Chao solution, the Tolman IV solution is compatible with all requirements for well-behaved and realistic solutions, except for the relativistic adiabatic index that diverges at the surface of stars. The divergence of the index Γ may be resolved, including a thin crust assuming a polytropic equation of state, which is precisely the case seen in studies of neutron stars. To the best of our knowledge, we model here for the first time the recently discovered massive pulsar PSR J0740+6620 and the strangely light HESS compact object via the Tolman IV solution. The present work may be of interest to model builders as well as a useful reference for future research.

Semi-symmetric metric gravity: From the Friedmann–Schouten geometry with torsion to dynamical dark energy models

In this paper, we develop a geometric generalization of General Relativity based on the semi-symmetric metric connection introduced by Friedmann and Schouten in 1924. Although the mathematical properties of this connection, which allows for torsion, have been extensively studied, its physical implications remain underexplored. We provide a detailed exposition of the differential geometric aspects of semi-symmetric connections and formulate the corresponding field equations induced by the specific form of torsion we are investigating. We consider the cosmological applications of the theory by deriving the generalized Friedmann equations in a flat, homogeneous and isotropic geometry. The Friedmann equations also include some supplementary terms as compared to their general relativistic counterparts, which can be interpreted as a geometric type dark energy. To evaluate the proposed theory, we consider three cosmological models — the first with constant effective density and pressure, the second with the dark energy satisfying a linear equation of state, and a third one with a polytropic equation of state. We also compare the predictions of the semi-symmetric metric gravitational theory with the observational data for the Hubble function, and with the predictions of the ΛCDM model. Our findings indicate that the semi-symmetric metric cosmological models give a good description of the observational data, and for certain values of the model parameters, they can reproduce almost exactly the predictions of the ΛCDM paradigm. Consequently, Friedmann’s initially proposed geometry emerges as a credible alternative to standard general relativity, in which dark energy has a purely geometric origin.

A model for general relativistic magnetohydrodynamic spine jets

Context. We study jets using a semi-analytical model of the general relativistic magnetohydrodynamic (GRMHD) equations in the Kerr metric that describes them near the rotation axis, assuming a steady state, and axisymmetry. Aims. The goal is to model the inner spine of a relativistic jet in order to solve for the bulk acceleration and the shape of the jet and understand how these quantities depend on the enthalpy and the magnetic field. Methods. The model is constructed by expanding the rotating black hole metric and forces with respect to the polar angle about the rotation axis. This results in a system of ordinary differential equations that determine the dependence on the radial distance. The difference with previous semi-analytical models that expand the metric around the rotation axis is that the flow is governed by a polytropic equation of state. Results. The solutions in this work start from a stagnation surface very close to the event horizon and become highly relativistic, achieving large Lorentz factors at large distances.

Scalarized hybrid neutron stars in scalar tensor gravity

Hybrid neutron stars, the compact objects consisting hadronic matter and strange quark matter, can be considered as the probes for the scalar tensor gravity. In this work, we explore the scalarization of hybrid neutron stars in the scalar tensor gravity. For the hadronic phase, we apply a piecewise polytropic equation of state constrained by the observational data of GW170817 and the data of six low-mass X-ray binaries with thermonuclear burst or the symmetry energy of the nuclear interaction. In addition, to describe the strange quark matter inside the hybrid neutron star, different MIT bag models are employed. We study the effects of the value of bag constant, the mass of s quark, the perturbative quantum chromodynamics correction parameter, and the density jump at the surface of quark-hadronic phase transition on the scalarization of hybrid neutron stars. Our results confirm that the scalarization is more sensitive to the value of bag constant, the mass of s quark, and the density jump compared to the perturbative quantum chromodynamics correction parameter.

Stability analysis of fractional relativistic polytropes

In astrophysics, the gravitational stability of a self-gravitating polytropic fluid sphere is an intriguing subject, especially when trying to comprehend the genesis and development of celestial bodies like planets and stars. This stability is the sphere’s capacity to stay in balance in the face of disruptions. We utilize fractional calculus to explore self-gravitating, hydrostatic spheres governed by a polytropic equation of state P=Kρ1+1/n. We focus on structures with polytropic indices ranging from 1 to 3 and consider relativistic and fractional parameters, denoted by σ and α, respectively. The stability of these relativistic polytropes is evaluated using the critical point method, which is associated with the energetic principles developed in 1964 by Tooper. This approach enables us to pinpoint the critical mass and radius at which polytropic spheres shift from stable to unstable states. The results highlight the critical relativistic parameter ( σCR ) where the polytrope’s mass peaks, signaling the onset of radial instability. For polytropic indices of n = 1, 1.5, 2, and 3 with a fractional parameter α=1, we observe stable relativistic polytropes for σ values below the critical thresholds of σCR= 0.42, 0.20, 0.10, and 0.0, respectively. Conversely, instability emerges as σ surpasses these critical values. Our comprehensive calculations reveal that the critical relativistic value ( σCR ) for the onset of instability tends to increase as the fractional parameter α decreases.

The role of heating in the formation and the dynamics of YSO jets

Context. Theoretical arguments as well as observations of young stellar objects (YSOs) support the presence of a diversified circumstellar environment. A stellar jet is thought to account for most of the stellar spin down and disk wind outflow for the observed high mass-loss rate, thus playing a major role in the launching of powerful jets. RY Tau, for instance, is an extensively studied intermediate mass pre-main sequence star. Observational data reveal a small-scale jet called micro-jet. Nevertheless, it is not clear how the micro-jet shapes the jet observed at a large scale. Aims. The goal is to investigate the spatial stability and structure of the central jet at a large scale by mixing the stellar and disk components. Methods. Two existing analytical self-similar models for the disk and the stellar winds to build the initial setups. Instead of using a polytropic equation of state, we mapped the heating and cooling sources from the analytical solutions. The heating exchange rate was controlled by two parameters, its spatial extent and its intensity. Results. The central jet and the surrounding disk are strongly affected by these two parameters. We separate the results into three categories, which show different emissivity, temperature, and velocity maps. We reached this categorization by looking at the opening angle of the stellar solution. For cylindrically, well-collimated jets, we have opening angles as low as 10° between 8 − 10 au, and for the wider jets, we can reach 30° with a morphology closer to radial solar winds. Conclusions. Our parametric study shows that the less heated the outflow is, the more collimated it appears. We also show that recollimation shocks appear consistently with UV observations in terms of temperature but not density.

The generalized Vaidya spacetime with polytropic equation of state

The generalized Vaidya spacetime with polytropic equation of state

Accretion flows around spinning compact objects in the post-Newtonian regime

We present the structure of a low angular momentum accretion flows around rotating compact objects incorporating relativistic corrections up to the leading post-Newtonian order. To begin with, we formulate the governing post-Newtonian hydrodynamic equations for the mass and energy-momentum flux without imposing any symmetries. However, for the sake of simplicity, we consider the flow to be stationary, axisymmetric, and inviscid. Toward this, we adapt the polytropic equation of state (EoS) and analyze the vertically integrated accretion flow confined to the equatorial plane. It is shown that the spin-orbit effects manifest themselves in the accretion dynamics. In the present analysis, we focus on global transonic accretion solutions, where a subsonic flow enters far away from the compact object and gradually gains radial velocity as it moves inwards. Thus, the flow becomes supersonic after reaching a certain radius, known as the critical point. To better understand the transonic solutions and examine the effect of post-Newtonian corrections, we classify the post-Newtonian equations into semi-relativistic (SR), semi-Newtonian (SN), and non-relativistic (NR) limits and compare the accretion solutions and their corresponding flow variables. With these, we find that SR and SN flow are in good agreement all throughout, although they deviate largely from the NR ones. Interestingly, the density profile seems to follow the profile ρ ∝ r -3/2 in the post-Newtonian regime. The present study has the potential to connect Newtonian and GR descriptions of accretion dynamics.

Comparison of magnetized thick disks around black holes and boson stars

Boson stars are considered as promising candidates for black hole mimickers. Similar to other compact objects they can form accretion disks around them. The properties of these disks could possibly distinguish them from other compact objects like black holes in future observations. Retrograde thick disks around boson stars and the influence of strong magnetic fields on them were already studied and it was shown that they can harbor very distinct features compared to black hole disks. However, the case of prograde thick disks is mostly unexplored, since they may appear much more similar to black hole disks. In this work we will investigate similarities and differences regarding prograde thick disks around non-selfinteracting rotating boson stars and rotating black holes. We assume thereby a polytropic equation of state and a constant specific angular momentum distribution of the disks. We classify the various conceivable boson star and black hole solutions by a dimensionless spin parameter a and compare their corresponding disk solutions. The influence of toroidal magnetic fields on the disks is analyzed by selected disk properties, as the rest-mass density distribution and the Bernoulli parameter. Disk solutions are characterized by their degree of magnetization represented by the magnetization parameter βmc\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta _{mc}$$\\end{document}. We found that strong magnetic fields can strengthen the differences of disk solutions or oppositely even lead to a correlation in disk properties, depending on the spin parameter of the boson star and black hole solutions. We identify the vertical thickness of the boson star disks as the main differentiating factor, since for most solutions the vertical density distribution is far more outreaching for boson star disks compared to black hole disks.

Ambipolar Diffusion with a Polytropic Equation of State

Ambipolar diffusion is the mechanism believed to be responsible for the loss of magnetic support in dense molecular cloud cores, and is therefore likely to play a key role in the star formation process. As such, this mechanism has been studied extensively both semianalytically and numerically. We build upon this existing body of work by considering a one-dimensional self-gravitating gas with a polytropic equation of state (P ∝ ρ ϵ ), and consider cases that range from softer (ϵ < 1) to stiffer (ϵ > 1) than isothermal. Our results indicate that the diffusion time is not very sensitive to the polytropic exponent ϵ when stiffer than isothermal, but is sensitive to the exponent when softer than isothermal. Additionally, the presence of magnetic and density fluctuations causes the ambipolar diffusion process to speed up, with the shortest diffusion times obtained for gases with large initial magnetic to gas pressure ratios and fairly soft equations of state. However, the diffusion time starts to increase significantly for ϵ ≲ 0.5, indicating that such soft equations of state are inconsistent with observations.

Possible existence of Bose–Einstein condensate compact stars

In this paper, we have model a compact star model composed of Bose–Einstein Condensate (BEC) matter that satisfy a polytropic equation of state (EoS) with γ=2 and a specific density profile. Then we tested the model through various physical criteria like non-singular pressure and density, energy conditions, causality condition etc. Further, we have checked the stability through Bondi criterion, hydrostatic equilibrium and static stability criterion. The stiffness of the EoS, stability, adiabatic index and Mmax depends upon a single parameter k. As k increases, the central Γ decreases near 4/3 and the compactness increases closer to 4/9 while the mass that can hold when increasing central density due to radial perturbation decreases, implies that the system will undergo gravitational collapse sooner if k increases. We have also found that this new category of compact star also follows that constraints provided by GW 170817 event. Hence, this work put forward the possibility of different category of compact star composed of BEC matters that can mimic neutron stars.

Amplification and Saturation of Turbulent Magnetic Fields in Collapsing Primordial Gas Clouds

Recent numerical studies suggest that magnetic fields play an important role in primordial star formation in the early Universe. However, the detailed evolution of the magnetic field in the collapse phase still has uncertainties because of the complicated physics associated with turbulence in a collapsing magnetized system. Here, we perform a suite of numerical MHD simulations that follow the collapse of magnetized, turbulent primordial gas clouds to investigate the evolution of the magnetic field associated with the turbulence, assuming a polytropic equation of state with exponent γ eff and with various numerical resolutions. In addition, we generalize the analytic theory of magnetic field growth/saturation so that it can deal with various exponents γ eff and turbulence energy spectra. We find that the numerical results are well reproduced by the theory for various γ eff through the collapse phase during the formation of the first stars. The magnetic field is eventually amplified by a factor of 1012–1015 due to kinematic and nonlinear turbulent dynamo effects and reaches 3%–100% of the equipartition level, depending on γ eff. We also find that the transition between the kinematic and nonlinear stages can be analytically estimated. These results indicate that the strong magnetic field accompanied by supersonic turbulence is a general property and suggest that it can play a crucial role in the formation of the first stars.

Exploring $$f\left(R,T\right)$$ Theory with Polytropic Equation of State

Exploring $$f\left(R,T\right)$$ Theory with Polytropic Equation of State

Compact stars in scalar–tensor theories with a single-well potential and the corresponding [formula omitted] theory

The macroscopic properties of compact stars in modified gravity theories can be significantly different from the general relativistic (GR) predictions. Within the gravitational context of scalar–tensor theories, with a scalar field ϕ and coupling function Φ(ϕ)=exp[2ϕ/3], we investigate the hydrostatic equilibrium structure of neutron stars for the simple potential V(ϕ)=ωϕ2/2 defined in the Einstein frame (EF). From the scalar field in the EF, we also interpret such theories as f(R) gravity in the corresponding Jordan frame (JF). The mass–radius relations, proper mass, and binding energy are obtained for a polytropic equation of state (EoS) in the JF. Our results reveal that the maximum-mass values increase substantially as ω gets smaller, while the radius and mass decrease in the low-central-density region as we move further away from the pure GR scenario. Furthermore, a cusp is formed when the binding energy is plotted as a function of the proper mass, which indicates the appearance of instability. Specifically, we find that the central-density value where the binding energy is a minimum corresponds precisely to dM/dρcJ=0 on the M(ρcJ)-curve.

Physical analysis of Rastall PFRF black hole through accretion process

The adiabatic accretion onto the charged black hole surrounded by perfect fluid radiation field (PFRF) in Rastall gravity is addressed in this manuscript. For this purpose, mass accretion rate M˙\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\dot{M}$$\\end{document}, critical horizon radius and some other flow parameters are being determined in the presence of polytropic fluid. Overall the process is being done analytically. The location of critical points, polytropic gas compression ratios and temperature profiles are also being investigated for different versions of polytropic equation of state. We also give the comparison of the location of critical points with case of Schwarzschild black hole in which critical points lies out side the horizon. Through above scenario, it is found that charge Q and Rastall parameter Nr\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$N_{r}$$\\end{document} have deep effects on the accretion process. It is also mentioned here that under some constraints on parameters, our results reduce to Schwarzschild and charged black holes results.

Equilibrium spherical shell of condensed matter around a scalar naked singularity

From a observational point of view, the central regions of scalar naked singularities possess a very special spacetime geometry, which in turn leads to existence of unusual orbital motion near the centers. We consider naked singularities in the model of static, spherically symmetric, asymptotically flat configurations of a self-gravitating scalar field minimally coupled to gravity. In this case, the effective potential of test particles orbiting around a scalar naked singularity has a minimum even at zero specific angular momentum. This means that the baryonic matter, captured by the naked singularity in the region of gravitational potential well near the center, will eventually be concentrated in some spherical shell, which will be in hydrostatic equilibrium after cooling. We find the conditions of hydrostatic equilibrium of the shell for the polytropic equation of state. In order to show the observational consequences of the possible existence of such configurations as real astrophysical objects, in particular in relation to tidal disruption events, we consider a simple illustrative example.

Neutron stars with polytropic equation of state

In this work, the neutron star model is considered in the framework of general relativity with the polytropic equation of state p=ργ, where γ=1+1k and k is known as a polytropic index. The homotopy perturbation method is used to solve Tolman–Oppenheimer–Volkoff (TOV) equation which gives the mass of the stellar structure. The solution of Einstein’s field equations is obtained and stability, causality, and physical viability of the considered model are examined. Using the observational results of mass and radius of known stars 4U 1608-52 RX J185635-3754, SAX J1748.9-2021 and 4U 1728-34, the model parameters are estimated which led to the non-singular solutions satisfying the energy conditions.

Thermodynamically adaptive piecewise polytropic equation of state for neutron stars

Thermodynamically adaptive piecewise polytropic equation of state for neutron stars