Hydrostatic Equilibrium Equation Research Articles

Gravitational potential energy of a multi-component galactic disk

We calculate ab initio the gravitational potential energy per unit area for a gravitationally coupled multi-component galactic disk of stars and gas, which is given as the integration over vertical density distribution, vertical gravitational force, and vertical distance. This is based on the method proposed by Camm for a single-component disk, which we extend here for a multi-component disk by deriving the expression of the energy explicitly at any galactocentric radius R. For a self-consistent distribution, the density and force are obtained by jointly solving the equation of vertical hydrostatic equilibrium and the Poisson equation. Substituting the numerical values for the density distribution and force obtained for the coupled system, we find in the derived expression of the energy that the energy of each component remains unchanged compared to the energy for the corresponding single-component case. We explain this surprising result by simplifying the above expression for the energy of a component analytically, which turns out to be equal to the surface density times the squared vertical velocity dispersion of the component. However, the energy required to raise a unit test mass to a certain height z from the mid-plane is higher in the coupled case. The system is therefore more tightly bound closer to the mid-plane, and hence it is harder to disturb it due to an external tidal encounter.

Predictive model of fermionic dark matter halos with a quantum core and an isothermal atmosphere

We develop a thermodynamical model of fermionic dark matter halos at finite temperature. Statistical equilibrium states may be justified by a process of violent collisionless relaxation in the sense of Lynden-Bell or from a collisional relaxation of nongravitational origin if the fermions are self-interacting. The most probable state (maximum entropy state) generically has a "core-halo" structure with a quantum core (fermion ball) surrounded by an isothermal atmosphere. The quantum core is equivalent to a polytrope of index $n=3/2$. The Pauli exclusion principle creates a quantum pressure that prevents gravitational collapse and solves the core-cusp problem of the cold dark matter model. The isothermal atmosphere (which is similar to the NFW profile of cold dark matter) accounts for the flat rotation curves of the galaxies at large distances. We numerically solve the equation of hydrostatic equilibrium with the Fermi-Dirac equation of state and determine the density profiles and rotation curves of fermionic dark matter halos.

Universal Relations for the Increase in the Mass and Radius of a Rotating Neutron Star

Rotation causes an increase in a neutron star’s mass and equatorial radius. The mass and radius depend sensitively on the unknown equation of state (EOS) of cold, dense matter. However, the increases in mass and radius due to rotation are almost independent of the EOS. The EOS independence leads to the idea of neutron star universality. In this paper, we compute sequences of rotating neutron stars with constant central density. We use a collection of randomly generated EOSs to construct simple correction factors to the mass and radius computed from the equations of hydrostatic equilibrium for nonrotating neutron stars. The correction factors depend only on the nonrotating star’s mass and radius and are almost independent of the EOS. This makes it computationally inexpensive to include observations of rotating neutron stars in EOS inference codes. We also construct a mapping from the measured mass and radius of a rotating neutron star to a corresponding nonrotating star. The mapping makes it possible to construct a zero-spin mass–radius curve if the masses and radii of many neutron stars with different spins are measured. We show that the changes in polar and equatorial radii are symmetric, in that the polar radius shrinks at the same rate in which the equatorial radius grows. This symmetry is related to the observation that the equatorial compactness (the ratio of mass to radius) is almost constant on one of the constant-density sequences.

Development of local density perturbation scheme in f(R) gravity to identify cracking points

In this work, the extension of concept of cracking in modified f(R) theory of gravity is presented for spherically symmetric compact objects. We develop general framework to observe the instabilities in self-gravitating spherical system through cracking with anisotropic inner matter configuration. For this purpose, the local density perturbation is applied on the hydrostatic equilibrium equation to identify cracking points/intervals. The physical viability of developed technique is tested on the data of three different stars namely 4U 1820-30, Her X-1 and SAX J1808.4-3658, presented in f(R) model developed in Zubair and Abbas (Astrophys Space Sci 361:342, 2016). It is concluded that these objects exhibit cracking in different interior regions and identification of cracking points refine the stability analysis of the system by extracting instabilities.

Modeling overcontact binaries

Context. In the realm of massive stars, strong binary interactions are commonplace. One extreme case is that of overcontact systems, which are expected to be part of the evolution of all stars evolving towards a merger and hypothesized as playing a role in the formation of binary black holes. However, important simplifications are made to model the evolution of overcontact binaries. The deformation from tidal forces is almost always put aside, and even rotation is frequently ignored in such models. Yet, both observations and theory have shown that overcontact stars are tidally deformed to a great extent, leaving a potentially important effect on the outer layers unaccounted for in models. Furthermore, in eclipsing binaries where radii can be determined to high precision, the question of how large the effect of tidal deformation is on the inferred properties of stellar models is still uncertain. Aims. We aim to consistently model overcontact binary stars in a one-dimensional (1D) stellar evolution code. To that end, we developed the required methodology to represent tidally distorted stars in 1D evolution codes. Methods. Using numerical methods, we computed the structure correction factors to the 1D spherical stellar structure equations of hydrostatic equilibrium and radiative energy transfer due to the binary Roche potential. We then compared them to existing results and the structure corrections of single, rotating stars. We implemented the new structure correction factors in the stellar evolution code MESA and explored several case studies. We compared the differences between our simulations: when no rotation is included, when we treat rotation using single star corrections (only accounting for centrifugal deformation), or when we use tidal deformation. Results. We find that ignoring rotation in deformed detached eclipsing binaries can produce a radius discrepancy of up to 5%. The difference between tidal and single star centrifugal distortion models is more benign at 1%, showing that single rotating star models are a suitable approximation of tidally deformed stars in a binary system. In overcontact configurations, we find a similar 5% variation in surface properties as a result of tidal distortion with respect to non-rotating models, showing that it is inappropriate to model binary stars filling their Roche lobe significantly as non-rotating.

Neutron stars in mimetic gravity

In this paper, a modified version of the hydrostatic equilibrium equation based on the mimetic gravity in the presence of perfect fluid is revisited. By using the different known equation of states, the structural properties of neutron stars are investigated in general relativity and mimetic gravity. Comparing the obtained results, we show that, unlike general relativity, we can find the appropriate equation of states that support observational data in the context of mimetic gravity. We also find that the results of relativistic mean-field-based models of the equation of states are in better agreement with observational data than non-relativistic models.

Random models for exploring planet compositions I: Uranus as an example

Modeling the interior of a planet is difficult because the small number of measured parameters is insufficient to constrain the many variables involved in describing the interior structure and composition. One solution is to invoke additional constraints based on arguments about how the planet formed. However, a planet’s actual structure and composition may hold clues to its formation which would be lost if this structure were not allowed by the initial assumptions. It is therefore interesting to explore the space of allowable compositions and structures in order to better understand which cosmogonic constraints are absolutely necessary. To this end, we describe a code for generating random, monotonic, density distributions, ρ(r), that fit a given mass, radius, and moment of inertia. Integrating the equation of hydrostatic equilibrium gives the pressure, P(r), at each point in the body. We then provide three algorithms for generating a monotonic temperature distribution, T(r), and an associated composition that is consistent with the ρ−P relation, and realistic equations of state. We apply this code to Uranus as a proof of concept, and show that the ratio of rock to water cannot be much larger than 2.

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.

Electrically charged quark stars in 4D Einstein\u2013Gauss\u2013Bonnet gravity

In this work we study the properties of compact spheres made of a charged perfect fluid with a MIT bag model EoS for quark matter. Considering static spherically symmetric spacetime we derive the hydrostatic equilibrium equations in the recently formulated four dimensional Einstein–Gauss–Bonnet (4D EGB) gravity theory. In this setting, the modified TOV equations are solved numerically with the aim to investigate the impact of electric charge on the stellar structure. A nice feature of 4D EGB theory is that the Gauss–Bonnet term has a non-vanishing contribution to the gravitational dynamics in 4D spacetime. We therefore analyse the effects of Gauss–Bonnet coupling constant alpha and the charge fraction beta on the mass–radius (M{-}R) diagram and also the mass–central density (M{-}rho _c) relation of quark stars. Finally, we conclude that depending on the choice of coupling constant one could have larger mass and radius compared with GR and can also be relevant for more massive compact objects due to the effect of the repulsive Coulomb force.

Behavior of anisotropic fluids with Chaplygin equation of state in Buchdahl spacetime

In the present study we have proposed a new model of an anisotropic compact star which admits the Chaplygin equation of state. For this purpose, we consider Buchdahl ansatz. We obtain the solution of proposed model in closed form which is non-singular, regular and well-behaved. In addition to this, we show that the model satisfies all the energy conditions and maintains the hydrostatic equilibrium equation. This model represents compact stars like PSR B0943+10, Her X-1 and SAX J1808.4-3658 to a very good approximate.

Stars disformally coupled to a shift-symmetric scalar field

We investigate static and spherically symmetric stars disformally coupled to a scalar field. The scalar field is assumed to be shift symmetric, and hence the conformal and disformal factors of the metric coupled to matter are dependent only on the kinetic term of the scalar field. Assuming that the scalar field is linearly dependent on time, we consider a general shift-symmetric scalar-tensor theory and a general form of the matter energy-momentum tensor that allows for the anisotropic pressure and the heat flux in the radial direction. This is a natural starting point in light of how the gravitational field equations and the energy-momentum tensor transform under a disformal transformation. By inspecting the structure of the hydrostatic equilibrium equation in the presence of the derivative-dependent conformal and disformal factors, we show that the energy density and tangential pressure must vanish at the surface of a star. This fact is used to prove the disformal invariance of the surface of a star, which was previously subtle and unclear. We then focus on the shift-symmetric k-essence disformally coupled to matter, and study the interior and exterior metric functions and scalar-field profile in more detail. It is found that there are two branches of the solution depending on the velocity of the scalar field. The disformally-related metric functions of the exterior spacetime are also discussed.

Neutron stars in f(mathtt {R,L_m}) gravity with realistic equations of state: joint-constrains with GW170817, massive pulsars, and the PSR J0030+0451 mass-radius from NICER data

In this work, we investigate neutron stars (NS) in f(mathtt {R,L_m}) theory of gravity for the case f(mathtt {R,L_m})= mathtt {R}+ mathtt {L_m}+ sigma mathtt {R}mathtt {L_m}, where mathtt {R} is the Ricci scalar and mathtt {L_m} the Lagrangian matter density. In the term sigma mathtt {R}mathtt {L_m}, sigma represents the coupling between the gravitational and particles fields. For the first time the hydrostatic equilibrium equations in the theory are solved considering realistic equations of state and NS masses and radii obtained are subject to joint constrains from massive pulsars, the gravitational wave event GW170817 and from the PSR J0030+0451 mass-radius from NASA’s Neutron Star Interior Composition Explorer (NICER) data. We show that in this theory of gravity, the mass-radius results can accommodate massive pulsars, while the general theory of relativity can hardly do it. The theory also can explain the observed NS within the radius region constrained by the GW170817 and PSR J0030+0451 observations for masses around 1.4~M_{odot }.

Ricci-Determinant gravity: Dynamical aspects and astrophysical implications

The Palatini gravitational action is enlarged by an arbitrary function $f(\mathbit{X})$ of the determinants of the Ricci tensor and the metric, $\mathbit{X}=|\mathbf{det}.R|/|\mathbf{det}.g|$. The resulting Ricci-determinant theory exhibits novel deviations from general relativity. We study a particular realization where the extension is characterized by the square-root of the Ricci determinant, $f(\mathbit{X})={\ensuremath{\lambda}}_{\mathrm{Edd}}\sqrt{\mathbit{X}}$, which corresponds to the famous Eddington action. We analyze the obtained equations for perfect fluid source and show that the affine connection can be solved in terms of the energy density and pressure of the fluid through an obtained disformal metric. As an application, we derive the hydrostatic equilibrium equations for relativistic stars and inspect the significant effects induced by the square-root of the Ricci tensor. We find that an upper bound on ${\ensuremath{\lambda}}_{\text{Edd}}$, at which deviations from the predictions of general relativity on neutron stars become prominent, corresponds to the hierarchy between the Planck and the vacuum mass scales. The Ricci-determinant gravity that we propose here is expected to have interesting implications in other cosmological domains.

One-dimensional prominence threads

Context. Threads are the building blocks of solar prominences and very often show longitudinal oscillatory motions that are strongly attenuated with time. The damping mechanism responsible for the reported oscillations is not fully understood yet. Aims. To understand the oscillations and damping of prominence threads we must first investigate the nature of the equilibrium solutions that arise under static conditions and under the presence of radiative losses, thermal conduction, and background heating. This provides the basis to calculate the eigenmodes of the thread models. Methods. The non-linear ordinary differential equations for hydrostatic and thermal equilibrium under the presence of gravity are solved using standard numerical techniques and simple analytical expressions are derived under certain approximations. The solutions to the equations represent a prominence thread, a dense and cold plasma region of a certain length that connects with the corona through a prominence corona transition region (PCTR). The solutions can also match with a chromospheric-like layer if a spatially dependent heating function localised around the footpoints is considered. Results. We have obtained static solutions representing prominence threads and have investigated in detail the dependence of these solutions on the different parameters of the model. Among other results, we show that multiple condensations along a magnetic field line are possible, and that the effect of partial ionisation in the model can significantly modify the thermal balance in the thread, and therefore their length. This last parameter is also shown to be comparable to that reported in the observations when the radiative losses are reduced for typical thread temperatures.

What to expect from dynamical modelling of cluster haloes – I. The information content of different dynamical tracers

ABSTRACT Using hydrodynamical simulations, we study how well the underlying gravitational potential of a galaxy cluster can be modelled dynamically with different types of tracers. In order to segregate different systematics and the effects of varying estimator performances, we first focus on applying a generic minimal assumption method (oPDF) to model the simulated haloes using the full 6D phase-space information. We show that the halo mass and concentration can be recovered in an ensemble unbiased way, with a stochastic bias that varies from halo to halo, mostly reflecting deviations from steady state in the tracer distribution. The typical systematic uncertainty is ∼0.17 dex in the virial mass and ∼0.17 dex in the concentration as well when dark matter (DM) particles are used as tracers. The dynamical state of satellite galaxies are close to that of DM particles, while intracluster stars are less in a steady state, resulting in an ∼0.26-dex systematic uncertainty in mass. Compared with galactic haloes hosting Milky-Way-like galaxies, cluster haloes show a larger stochastic bias in the recovered mass profiles. We also test the accuracy of using intracluster gas as a dynamical tracer modelled through a generalized hydrostatic equilibrium equation, and find a comparable systematic uncertainty in the estimated mass to that using DM. Lastly, we demonstrate that our conclusions are largely applicable to other steady-state dynamical models including the spherical Jeans equation, by quantitatively segregating their statistical efficiencies and robustness to systematics. We also estimate the limiting number of tracers that leads to the systematics-dominated regime in each case.

Invariant quantities of scalar\u2013tensor theories for stellar structure

We present the relativistic hydrostatic equilibrium equations for a wide class of gravitational theories possessing a scalar–tensor representation. It turns out that the stellar structure equations can be written with respect to the scalar–tensor invariants, allowing to interpret their physical role.

Radial oscillations and stability of compact stars in f(R, T) = R+ 2β T gravity

We examine the static structure configurations and radial stability of compact stars within the context of f(R, T) gravity, with R and T standing for the Ricci scalar and trace of the energy-momentum tensor, respectively. Considering the f(R, T)=R+2β T functional form, with β being a constant, we derive the corresponding hydrostatic equilibrium equation and the modified Chandrasekhar's pulsation equation. The mass-radius relations and radial mode frequencies are obtained for some realistic equations of state. Our results show that the traditional stellar stability criteria, namely, the necessary condition d M/dρc >0 and sufficient condition ω2 >0, still hold in this theory of gravity.

Charged strange star in $f(R,T)$ gravity with linear equation of state

Our present study involves the strange stars model in the framework of $f(R,T)$ theory of gravitation. We have taken a linear function of the Ricci scalar $R$ and the trace $T$ of the stress-energy tensor $T_{\mu \nu}$ for the expression of $f(R,T)$, i.e., $f(R,T)=R+ 2 \gamma T $ to obtain the proposed model, where $\gamma$ is a coupling constant. Moreover, to solve the hydrostatic equilibrium equations, we consider a linear equation of state between the radial pressure $p_r$ and matter density $\rho$ as $p_r=\alpha \rho-\beta$, where $\alpha$ and $\beta$ are some positive constants, Both $\alpha,\,\beta$ depend on coupling constant $\gamma$ which have been also depicted in this paper. By employing the Krori-Barua {\em ansatz} already reported in the literature [J. Phys. A, Math. Gen. 8:508, 1975] we have found the solutions of the field equations in $f (R, T )$ gravity. The effect of coupling constant $\gamma$ have been studied on the model parameters like density, pressures, anisotropic factor, compactness, surface redshift, etc. both numerically and graphically. A suitable range for $\gamma$ is also obtained. The physical acceptability and stability of the stellar system have been tested by different physical tests, e.g., the causality condition, Herrera cracking concept, relativistic adiabatic index, energy conditions, etc. One can regain the solutions in Einstein gravity when $\gamma\rightarrow 0$

A test of Alzain’s modified inertia model for MOND using galaxy cluster observations

Recently, Alzain proposed a modified inertial formalism of modified Newtonian dynamics (MOND), wherein Galiliean invariance is violated and new Lorentz transformation properties from an inertial to an accelerated frames are posited. Milgrom’s acceleration constant is then found to be invariant under the aegis of these new space-time coordinate transformations. In this model, a modified equation of hydrostatic equilibrium is proposed, which can be applied to relaxed galaxy clusters. However, when we apply this equation to a Chandra sample of 12 clusters, we find that the total dynamical mass obtained using Alzain’s model is much smaller than the baryonic mass for almost all the clusters, thereby implying that this original model is not viable. A variant of this model with a value of $$a_0$$ about two times smaller than that which fits galaxy data, is necessary in order to reconcile the missing mass problem in clusters, without the need for dark matter.