State Of Hydrostatic Equilibrium Research Articles

Stability of a realistic astrophysical pulsar and its mass–radius relation in higher-order curvature gravity

The objective of this research is to explore compact celestial objects while considering the framework of an extended gravitational theory known as R+f(G) gravity. The notations R and G denote the Ricci scalar and the Gauss–Bonnet invariant, respectively. Radio pulsars, which are neutron stars with masses greater than 1.8 times that of the Sun (M⊙), provide exceptional opportunities for delving into fundamental physics in extraordinary environments unparalleled in the observable universe and surpassing the capabilities of experiments conducted on Earth. Through the utilization of both the linear and quadratic expressions of the function f(G)=α1G2, where α1 (with dimensional units of [length6]) are incorporated, we have achieved an accurate analytical solution for anisotropic perfect-fluid spheres in a state of hydrostatic equilibrium. By integrating the dimensional parameters α1 and the compactness factor, defined as C=2GMRc2, we showcase our capacity to encompass and depict all physical characteristics within the stellar structure. We illustrate that the model is capable of producing a stable arrangement for the pulsar PSR J0740+6620, encompassing its geometric and physical properties. We illustrate that by utilizing the quadratic form of G in the R+f(G) framework, the Krori-Barua ansatz establishes semi-analytical relationships between radial pressure (pr), tangential pressure (pt), and density (ρ). Remarkably, in the context of the positive/negative α1 quadratic R+f(G) gravity framework, the maximum compactness is inherently restricted to values fail below the Buchdahl limit when the surface density exceeds 4×1014 g/cm3. This differs from General Relativity (GR) in which the compactness is unlikely to approach the black hole limit. The model predicts a core density that is multiple times greater than the saturation nuclear density, represented as ρnuc=4×1014 g/cm3, and a surface density ρs that surpasses the nuclear saturation density. We offer the mass–radius diagram associated with the boundary density we have determined, and this diagram has been shown to be consistent with other observational discoveries.

Anisotropic quintessence compact star in f(T) gravity with Tolman–Kuchowicz metric potentials

To obtain analytically relativistic quintessence anisotropic spherical solutions in the f(T) paradigm is the primary objective of this paper. To do this, the pressure anisotropy condition is imposed, and we employ a metric potential of the Tolman–Kuchowicz (TK) type. We also suppose that our current model incorporates a quintessence field characterized by a parameter ω q , in addition to the anisotropic matter distribution. In the presence of the parameter α, the field equations are modified by the choice of the f(T) function. The f(T) gravity parameter α adds new components to the basic physical characteristics, such as density, pressure, subliminal sound velocity, surface redshift, etc, of the present model. By selecting the compact star Her X-1 and varying α from 0.5 to 2.5, we examined all the physical characteristics of the model parameter of the configuration. The graphical process demonstrates that a more compact item is produced with greater values of α. The hydrostatic equilibrium condition of the model is discussed, as well as the mass-radius relationship for our current model is obtained.

Anisotropies and modified gravity theories in stellar and substellar objects

In several classes of modified gravity theories, extra degrees of freedom are not completely screened in the interiors of stellar and substellar objects. In such theories, the hydrostatic equilibrium condition inside these objects is altered. Moreover, the interior structures of these objects might have a small pressure anisotropy induced by several physical phenomena, including rotation and magnetic fields. All these effects, both individually and collectively, induce changes in predicted stellar observables. Such changes have an impact on different phases of the stellar life cycle, starting from its birth to its death, covering almost all the branches of the Hertzsprung–Russell diagram. The aim of this work is to systematically review the current literature on the topic. We discuss the main results and constraints obtained on a class of modified gravity theories.

Representation of three-dimensional mass distribution of the Earth's interior by biorthogonal series and its use for studying internal structure of the planet

This paper utilizes the mathematical concept of approximation within an ellipsoid from a single viewpoint to present the spatial mass distribution function of the Earth's interior and its internal potential. The primary focus lies in constructing the volume distribution of masses in the planet's interior, with the expansion coefficients being linear combinations of the Stokes constants. Several possible approaches are suggested for determining accurately these coefficients employing three-dimensional (biorthogonal) polynomials. By expressing the mass distribution function of the Earth's interior and its internal potential as a series, an algorithm is introduced for the calculation of gravitational energy. It allows us to estimate fluctuations in gravitational energy. The implementation of this algorithm offers the means of establishing the extent to which the Earth deviates from a state of hydrostatic equilibrium as a celestial body. Due to the aforementioned method, calculations have been conducted to validate its effectiveness and reliability. This example is given as an illustration of a given method for studying the internal structure of planets.

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.

Teaching ideal gas in a uniform field: exploring student preferences

An ideal classical gas under uniform gravity is a commonly discussed problem in statistical thermodynamics. At an introductory level, the condition of hydrostatic equilibrium gives rise to the barometric formula, which describes the variation of gas pressure with height. At an advanced level, the partition function can be used to find the density and the internal energy of the gas. These methods rely heavily on mathematical concepts, which may pose a difficulty to some students. This article presents teaching the problem via the virial theorem, emphasising the physical picture of the particle distribution. The virial theorem allows the internal energy to be expressed as an integral over the surface of the container. For the pedagogical purpose, visualisation of how the particles distribute themselves at extreme temperatures helps determine the internal energy of the gas. Student feedback is used as a basis for evaluating different approaches to the problem.

Relativistic anisotropic solution satisfying vanishing complexity condition

In this paper, we investigated anisotropic solution for a self-bound compact star model. The solution is obtained via Vaidya–Tikekar metric function [Formula: see text] and the vanishing complexity factor condition. The physical analysis of the solution has been discussed in detail which includes mass-radius ratio, and stability analysis via adiabatic index for a secondary component of the GW190814 event for different values [Formula: see text]. Furthermore, we predicted the radii and obtained the compactness factor of different self-bound compact objects for different [Formula: see text]. Moreover, the hydrostatic equilibrium condition of the solution is also checked. The physical result shows that the solution so obtained is well behaved and may be useful for modeling the self-bound compact objects as far as observational data is concerned.

A comprehensive discussion for the identification of cracking points in f(R) theories of gravity

This paper investigates the concept of cracking and overturning to analyze the impact of local density perturbations on the stability of self-gravitating compact objects in the framework of f(R,phi , X) theory of gravity, where R, phi , and X denote the Ricci scalar, scalar potential, and kinetic term, respectively. In this context, we developed the hydrostatic equilibrium equation for spherically symmetric spacetime with anisotropic matter configuration and subsequently employed the Krori Barua technique. We then perturb the hydrostatic equilibrium state of the configuration by employing the local density perturbation technique, while taking into account the barotropic equation of state. To validate this technique, we employed it on different compact stars namely, Her X-1, SAX J1808.4-3658, 4U 1820-30, PSR J1614-2230, Vela X-1, and Cen X-3, and found that all stars exhibit cracking or overturning for a specific range of model parameters. Conclusively, this study emphasizes that the proposed cracking technique provides significant insights into the stability analysis of self-gravitating compact objects.

On high order positivity-preserving well-balanced finite volume methods for the Euler equations with gravitation

This paper studies three high-order structure-preserving finite volume weighted essentially non-oscillatory (WENO) methods, which are not only well balanced (WB) for a general known hydrostatic equilibrium state but also preserve the positivity of density and pressure, for the compressible Euler equations under gravitational fields. These methods are built on a simple local scaling positivity-preserving (PP) limiter and a modified WENO-ZQ reconstruction exactly preserving the cell average value and scaling invariance. The WB properties of these three methods are achieved based on suitable numerical fluxes and approximation to the gravitational source terms. Based on some convex decomposition techniques as well as several critical properties of the admissible states and numerical flux, we carry out rigorous positivity-preserving analyses for these three WB schemes. We rigorously prove that the three WB methods, coupled with the PP limiter and a strong-stability-preserving time discretization, are always PP under suitable Courant-Friedrichs-Lewy conditions. Extensive numerical examples are provided to confirm WB and PP properties of three methods.

Liquid core dynamical stability and the Earth's rotational modes

The Earth's interior, especially the mantle (MT), is believed to deviate from the state of hydrostatic equilibrium. This is mainly because the customary approaches in computing the period of the free-core nutation (FCN) for Earth models in hydrostatic equilibrium yield predicted values larger than the observed value by an average of about 30 sidereal days (sd). However, results from alternative computational approaches yield predicted periods of this mode that are close to the observed value. This suggests that the Earth's interior maybe closer to hydrostatic equilibrium than previously envisioned. In this work I study the dynamics of a compressible liquid core (LC) model, bounded by the rigid MT, and vary the square of the Brunt Väsälä frequency, within its known limits, such that the LC density profile is stably, neutrally, and unstably stratified, and compute the periods of the CW and FCN. I show that the period of the CW is unaffected, as expected, by the variations of the Brunt Väsälä frequency. Unless the LC is moderately unstably stratified, for which the computed period of this mode does not converge to a single value, the period of the FCN is also not significantly affected by these variations except for a small range of Brunt Väsälä frequency for which the core is unstably stratified. I conclude that either the LC is moderately-unstably stratified, or the discrepancy between the observed and computed period of the FCN, once the effects of the viscous and electromagnetic torques on the inner-core boundary (ICB) and core-mantle boundary (CMB) are considered, may be in the theoretical/computational treatment of the dynamics of the elastic MT.

Wormhole formation in massive gravity: an analytic description

The present study analyses the wormhole solution both in the dRGT-f(R, T) massive gravity and Einstein massive gravity. In both the models, the anisotropic pressure solution in ultrastatic wormhole geometry gives rise to the shape function that involves massive gravity parameters gamma and Lambda . However, the terms consisting of gamma and Lambda acts in such a way that the spacetime loses asymptotic flatness. Similar to the black hole solution in massive gravity, this inconsistency arises due to the repulsive effect of gravity which can be represented by the photon deflection angle that goes negative after a certain radial distance. It is investigated that the repulsive effect induced in the massive gravitons push the spacetime geometry so strongly that the asymptotic flatness is effected. On the other hand, in this model, one can have a wormhole with ordinary matter at the throat that satisfies all the energy conditions while the negative energy density is sourced by massive gravitons. Finally, using the TOV equation, it is found that the model is stable under the hydrostatic equilibrium condition.

Accreting neutron stars: composition of the upper layers of the inner crust

Abstract We model the nuclear evolution of an accreted matter as it sinks toward the stellar center, in order to find its composition and equation of state. To this aim, we developed a simplified reaction network that allows for redistribution of free neutrons in the inner crust to satisfy the recently suggested neutron hydrostatic and diffusion equilibrium condition. We analyse the main reaction pathways for the three representative thermonuclear ash compositions: Superburst, Kepler, and Extreme rp. In contrast to the previous results, which neglect redistribution of free (unbound) neutrons in the inner crust, the most significant reactions in our calculations are neutron captures and electron emissions. The pycnonuclear fusion plays some role only for Kepler ashes. For the direct application of our results in astrophysical codes we present profiles of the average charge, 〈Z〉, impurity parameter, Qimp and equation of state for a set of models, parametrized by the pressure at the outer-inner crust interface. Typically, for Superburst ashes Qimp ≈ 1 − 4, while for Kepler ashes Qimp decreases from ≈23 at the outer-inner crust interface to ≈5 at the end of our simulation (the corresponding density equals ρdc ≈ 2 × 1012 g cm−3). At the same time, for Extreme rp ashes Qimp remains large ≈30 − 35 in the considered inner crust region. Our results are important for modeling the thermal relaxation of transiently accreting neutron stars after the end of the outburst.

Topographical analysis of a candidate subglacial water region in Ultimi Scopuli, Mars

Radar signals of the MARSIS (Mars Advanced Radar for Subsurface and Ionosphere Sounding) instrument at the base of the South Polar Layered Deposits (SPLD) in the region known as Ultimi Scopuli, on Mars, have been interpreted as reflections from putative subglacial standing bodies of liquid water. The interpretation of the MARSIS radar signals is not unique and different explanations have been proposed for the anomalous bright radar reflection observed in Ultimi Scopuli, including the presence of CO2 deposits within the ice, or clays and metal-bearing minerals. We analysed the surface topography of Ultimi Scopuli with Digital Elevations Models (DEMs) derived from MOLA (Mars Orbiter Laser Altimeter), photogrammetric processing of the High-Resolution Stereo Camera (HRSC) images and SHARAD radargrams. In each case, we found an anomalous flat and smooth area in correspondence to the potential subglacial water zone, indicating a condition of hydrostatic equilibrium of the ice, similar to the surface observations found above large terrestrial subglacial lakes. We found that the surface topography, in addition to MARSIS radar sounder signals, are compatible with the presence of subglacial standing bodies of liquid water at the base of Ultimi Scopuli.

Influence of charge on anisotropic class-one solution in non-minimally coupled gravity

This paper studies charged star models associated with anisotropic matter distribution in theory, where . For this purpose, we take a linear model of this gravity as , where ζ represents a coupling constant. We consider a self-gravitating spherical geometry in the presence of electromagnetic field and generate solution to the modified field equations by using the ‘embedding class-one’ condition and bag model equation of state. The observational data (masses and radii) of four different stellar models like 4U 1820-30, SAX J 1808.4-3658, SMC X-4 and Her X-I is employed to analyze the effects of charge on their physical properties. Finally, the effect of the coupling constant is checked on the viability, hydrostatic equilibrium condition and stability of the resulting solution. We conclude that the considered models show viable and stable behavior for all the considered values of charge and ζ.

High order well-balanced conservative finite difference AWENO scheme with hydrostatic reconstruction for the Euler equations under gravitational fields

The system of compressible Euler equations under gravitational fields drew much attention during the past few decades, due to its broad applications in astrophysics and atmospheric sciences. A critical feature of the system is that it admits a hydrostatic equilibrium state when the gradient of fluid flux is exactly balanced by the source term brought by gravitational fields, which is referred to as well-balancedness. In this paper, a well-balanced high order conservative finite difference alternative WENO (AWENO) scheme is proposed for the system. The finite difference AWENO scheme allows us to use an arbitrary consistent Riemann solver, such as LF, HLLC, etc. To preserve hydrostatic equilibrium, the hydrostatic reconstruction technique is adopted. We slightly modify the original hydrostatic reconstruction procedure to achieve the fifth order of accuracy in the framework of finite difference methods, without introducing spurious oscillations. Besides, the discretization is fully conservative without the source term, which is not the case for the methods with original hydrostatic reconstructions. Moreover, we will show that the well-balancedness is exactly guaranteed by our approach at the discrete level. Finally, numerical examples illustrate the effectiveness and robustness of the designed approach.

Study of anisotropic stellar objects, a revisit to Buchdahl metric potential

In this paper, we explore anisotropic compact stellar models in the framework of general theory of relativity with [Formula: see text] component governed by Buchdahl ansatz. We choose the anisotropic factor such as to render the solution physically compatible for stellar modeling. We have analyzed the solution for different values of [Formula: see text] to demonstrate the profiles of thermodynamical variables. The stability of the proposed models has been verified using different stability criteria. The hydrostatic equilibrium condition is well retained by the models studied in this work. The Buchdahl limit is well obeyed by the proposed models. The models of four compact stars viz., PSR J1614-2230, Vela X-1, Cen X-3 and SMC X-4 are extensively analyzed in close agreement with their observational details.

The possible equation of state of dark matter in low surface brightness galaxies * *Supported by the National Natural Science Foundation (NSF) of China (11973081, 11573062, 11403092, 11390374, 11521303), the YIPACAS Foundation (2012048), the Chinese Academy of Sciences (CAS, KJZD-EW-M06-01), the NSF of Yunnan Province (2019FB006) and the Youth Project of Western Light of CAS

The observed rotation curves of low surface brightness (LSB) galaxies play an essential role in studying dark matter, and indicate the existence of a central constant density dark matter core. However, the cosmological N-body simulations of cold dark matter predict an inner cusped halo with a power-law mass density distribution, and cannot reproduce a central constant-density core. This phenomenon is called cusp-core problem. When dark matter is quiescent and satisfies the condition for hydrostatic equilibrium, the equation of state can be adopted to obtain the density profile in the static and spherically symmetric space-time. To address the cusp-core problem, we assume that the equation of state is independent of the scaling transformation. Its lower order approximation for this type of equation of state can naturally lead to a special case, i.e., , where p and represent the pressure and density, respectively, depicts the rotation velocity of galaxy, and and are positive constants. It can obtain a density profile that is similar to the pseudo-isothermal halo model when is approximately 0.15. To obtain a more universally used model, let the equation of state include the polytropic model, i.e. , from which we can obtain other types of density profiles, such as the profile that is nearly same as the Burkert profile, where s and are positive constants.

The physically realizable anisotropic strange star models

In the present work, we search a new exact solution of Einstein’s field equations using Karmarkar condition of embedded class one space time. The new solution is analyzed graphically as well as analytically to check its viability for compact star modeling. The different parameters of the solution are ascertained by matching the interior space time metric with the Schwarzschild’s exterior space time metric. We tune our solution for the modeling of strange stars EXO1785-248, SAXJ1808.4-3658 and HER X-1 and found their masses and radii as 8.849 km, 1.3 $$M_{\Theta }$$ : 7.951 km, 0.9 $$M_{\Theta }$$ : 8.1 km, 0.85 $$M_{\Theta }$$ , respectively. The physical reliability of the solution depends on the values of the independent parameters b and c. The solution is well behaved for the range of the values $$0.000009\le b\le 0.0039$$ and $$0.0000005\le c\le 0.0000182$$ . Our models satisfy the causality condition and adiabatic index is well behaved everywhere inside the fluid sphere. The compactness parameter is well defined as it does not cross the Buchdahl limit. All the energy conditions hold good inside the compact fluid spheres. The stability of models is assessed via Herrera’s cracking concept. The hydrostatic equilibrium condition is well maintained by our models as gravitational force is counterbalanced by the combined effects of anisotropic force and hydrostatic force. The proper graphical analysis is provided to authenticate the physically admissible character of proposed models.

Simple high order well-balanced finite difference WENO schemes for the Euler equations under gravitational fields

The compressible Euler equations coupled with the gravitational source terms admit a hydrostatic equilibrium state where the gradients of the flux terms can be exactly balanced by those in the source terms. This property of exact preservation of the equilibrium is highly desirable at the discrete level when they are numerically solved. In this study, we design the simple high order well-balanced finite difference weighted essentially non-oscillatory (WENO-R/I) schemes, which base on the WENO reconstruction and interpolation procedures respectively, for this system with an ideal gas equation of state and a prescribed gravitational field. The main idea in achieving the well-balanced property is to rewrite the source terms in a special way and discretize them by using the nonlinear WENO differential operators with the homogenization condition to those for the flux terms. The proposed well-balanced schemes can be proved mathematically to preserve the hydrostatic isothermal and polytropic equilibria states exactly and at the same time maintain genuine high order accuracy. Moreover, the resulting schemes are high efficiency of computation and can be implemented straightforwardly into the existing finite difference WENO code with the Lax-Friedrichs numerical flux for solving the compressible Euler equations. Last but not least, the proposed well-balanced framework also works for a class of moving equilibrium state. Extensive one- and two-dimensional numerical examples are carried out to investigate the performance of the proposed schemes in term of high order accuracy, well-balanced property, shock capturing essentially non-oscillatory and resolving the small perturbation on the coarse mesh resolution.

Uniformly High-Order Structure-Preserving Discontinuous Galerkin Methods for Euler Equations with Gravitation: Positivity and Well-Balancedness

This paper presents a class of novel high-order accurate discontinuous Galerkin (DG) schemes for the compressible Euler equations under gravitational fields. A notable feature of these schemes is that they are well-balanced for a general hydrostatic equilibrium state, and at the same time, provably preserve the positivity of density and pressure. In order to achieve the well-balanced and positivity-preserving properties simultaneously, a novel DG spatial discretization is carefully designed with suitable source term reformulation and a properly modified Harten-Lax-van Leer contact (HLLC) flux. Based on some technical decompositions as well as several key properties of the admissible states and HLLC flux, rigorous positivity-preserving analyses are carried out. It is proven that the resulting well-balanced DG schemes, coupled with strong stability preserving time discretizations, satisfy a weak positivity property, which implies that one can apply a simple existing limiter to effectively enforce the positivity-preserving property, without losing high-order accuracy and conservation. The proposed methods and analyses are applicable to the Euler system with general equation of state. Extensive one- and two-dimensional numerical tests demonstrate the desired properties of these schemes, including the exact preservation of the equilibrium state, the ability to capture small perturbation of such state, the robustness for solving problems involving low density and/or low pressure, and good resolution for smooth and discontinuous solutions.