We study self-gravitating stars in the bootstrapped Newtonian picture for polytropic equations of state. We consider stars that span a wide range of compactness values. Both matter density and pressure are sources of the gravitational potential. Numerical solutions show that the density profiles can be well approximated by Gaussian functions. Later we assume Gaussian density profiles to investigate the interplay between the compactness of the source, the width of the Gaussian density profile and the polytropic index. We also dedicate a section to comparing the pressure and density profiles of the bootstrapped Newtonian stars to the corresponding General Relativistic solutions. We also point out that no Buchdahl limit is found, which means that the pressure can in principle support a star of arbitrarily large compactness. In fact, we find solutions representing polytropic stars with compactness above the Buchdhal limit.
We consider compact objects in a classical and non-relativistic generalisation of Newtonian gravity, dubbed bootstrapped Newtonian theory, which includes higher-order derivative interaction terms of the kind generically present in the strong-field regime of gravity. By means of a field redefinition, the original bootstrapped Newtonian action is written in a canonical Newtonian form with non-linear source terms. Exact analytic solutions remain unattainable, but we show that perturbative solutions of the canonical theory can be efficiently used to derive approximate descriptions of compact objec…
We study the bootstrapped Newtonian potential generated by a localised source in one and two spatial dimensions, and show that both cases naturally lead to finite spatial extensions of the outer vacuum. We speculate that this implies the necessary existence of a cosmological (particle) horizon associated with compact sources. In view of the possible dimensional reduction occurring in ultra-high energy processes - like scatterings at Planckian energies, the gravitational collapse of compact objects or the end-point of black hole evaporation - one can consider such lower-dimensional "bubbles" im…
Bootstrapped Newtonian gravity is a non-linear version of Newton's law which can be lifted to a fully geometric theory of gravity starting from a modified potential. Here, we study geodesics in the bootstrapped Newtonian effective metric in vacuum and obtain bounds on a free parameter from Solar System data and S-star orbits near our Galaxy centre. These bounds make vacuum bootstrapped Newtonian gravity experimentally indistinguishable from General Relativity.
Bootstrapped Newtonian gravity was developed with the purpose of estimating the impact of quantum physics in the nonlinear regime of the gravitational interaction, akin to corpuscular models of black holes and inflation. In this work, we set the ground for extending the bootstrapped Newtonian picture to cosmological spaces. We further discuss how such models of quantum cosmology can lead to a natural solution of the cosmological constant problem.
We study binary mergers in bootstrapped Newtonian gravity, where higher-order couplings are added to the non-relativistic Lagrangian for the Newtonian potential. In this theory, the Arnowitt-Deser-Misner (ADM) mass differs from both the proper mass of Newtonian gravity and the proper mass of general relativity, which affects the interpretation of astrophysical and cosmological events. The aforementioned difference particularly provides important phenomenological constraints for the mass of the emitted matter and the compactness of the final object after the merger. The interpretation of the GW…
We show that the bootstrapped Newtonian potential generated by a uniform and isotropic source does not depend on the one-loop correction for the matter coupling to gravity. The latter however affects the relation between the proper mass and the ADM mass and, consequently, the pressure needed to keep the configuration stable.
We develop a "hybrid approximative scheme" in the framework of the post-Newtonian approximation for computing general-relativistic polytropic models simulating neutron stars in critical rigid rotation. We treat the differential equations governing such a model as a "complex initial value problem", and we solve it by using the so-called "complex-plane strategy". We incorporate into the computations the complete solution for the relativistic effects, this issue representing a significant improvement with regard to the classical post-Newtonian approximation, as verified by extended comparisons of…
We compute quantum corrections for the gravitational potential obtained by including a derivative self-coupling in its classical dynamics as a toy model for analysing quantum gravity in the strong field regime. In particular, we focus on quantum corrections to the classical solutions in the vacuum outside localised matter sources.
We determine the complete space-time metric from the bootstrapped Newtonian potential generated by a static spherically symmetric source in the surrounding vacuum. This metric contains post-Newtonian parameters which can be further used to constrain the complete underlying dynamical theory. For values of the post-Newtonian parameters within experimental bounds, the reconstructed metric appears very close to the Schwarzschild solution of General Relativity in the whole region outside the event horizon. The latter is however larger in size for the same value of the mass compared to the Schwarzsc…
We study equilibrium configurations of a homogenous ball of matter in a bootstrapped description of gravity which includes a gravitational self-interaction term beyond the Newtonian coupling. Both matter density and pressure are accounted for as sources of the gravitational potential for test particles. Unlike the general relativistic case, no Buchdahl limit is found and the pressure can in principle support a star of arbitrarily large compactness. By defining the horizon as the location where the escape velocity of test particles equals the speed of light, like in Newtonian gravity, we find a…
We analyse the classical configurations of a bootstrapped Newtonian potential generated by homogeneous spherically symmetric sources in terms of a quantum coherent state. We first compute how the mass and mean wavelength of these solutions scale in terms of the number of quanta in the coherent state. We then note that the classical relation between the ADM mass and the proper mass of the source naturally gives rise to a Generalised Uncertainty Principle for the size of the gravitational radius in the quantum theory. Consistency of the mass and wavelength scalings with this GUP requires the com…
A general formalism developed few years ago to model polytropic Newtonian stars with anisotropic pressure is applied to model stars for which, both, radial and tangential pressure satisfy polytropic equations of state. We obtain the corresponding Lane--Emden equation, and integrate it for a wide range of values of the parameters, thereby illustrating the effects of the pressure anisotropy on stellar objects. We calculate Chandrasekhar mass for a white dwarf and compare it with previous results. Finally, prospective applications to some astrophysical scenarios are discussed.
An iterative method is presented for solving the problem of a uniformly rotating, self-gravitating ring without a central body in Newtonian gravity by expanding about the thin ring limit. Using this method, a simple formula relating mass to the integrated pressure is derived to the leading order for a general equation of state. For polytropes with the index n=1, analytic coefficients of the iterative approach are determined up to the third order. Analogous coefficients are computed numerically for other polytropes. Our solutions are compared with those generated by highly accurate numerical me…
An analytical method is presented for treating the problem of a uniformly rotating, self-gravitating ring without a central body in Newtonian gravity. The method is based on an expansion about the thin ring limit, where the cross-section of the ring tends to a circle. The iterative scheme developed here is applied to homogeneous rings up to the 20th order and to polytropes with the index n=1 up to the third order. For other polytropic indices no analytic solutions are obtainable, but one can apply the method numerically. However, it is possible to derive a simple formula relating mass to the i…
For the analysis of the r-mode oscillation of hot young neutron stars, it is necessary to consider the effect of it differential rotation, because viscosity is not strong enough for differentially rotating young neutron stars to be lead to uniformly rotating configurations on a very short time scale after their birth. In this paper, we have developed a numerical scheme to solve r-mode oscillations of differentially rotating polytropic inviscid stars. This is the extended version of the method which was applied to compute r-mode oscillations of uniformly rotating Newtonian polytropic stars. By …
In this paper, the gravitational field equations for static spherically symmetric perfect fluid models with a polytropic equation of state, $p=kρ^{1+1/n}$, are recast into two complementary 3-dimensional {\it regular} systems of ordinary differential equations on compact state spaces. The systems are analyzed numerically and qualitatively, using the theory of dynamical systems. Certain key solutions are shown to form building blocks which, to a large extent, determine the remaining solution structure. In one formulation, there exists a monotone function that forces the general relativistic sol…
We determine the onset point of secular instability for the nonaxisymmetric bar mode in rigidly rotating equilibrium configurations in the Post-Newtonian approximation, in order to apply it to neutron stars. The treatment is based on a precedent Newtonian analytic energy variational method which we have extended to the Post-Newtonian case. This method, based upon Landau's theory of second-order phase transitions, provides the critical value of the ellipsoid polar eccentricity e at the onset of viscosity-driven instability and it is valid for any equation of state. The extension of this method …
In this work we evaluate the physical acceptability of relativistic anisotropic spheres modeled by two polytropic equations of state -- with the same newtonian limit -- commonly used to describe compact objects in General Relativity. We integrate numerically the corresponding Lane-Emden equation in order to get density, mass and pressure profiles. An ansatz is used for the anisotropic pressure allowing us to have material configurations slightly deviated from isotropic condition. Numerical models are classified in a parameter space according to the number of physical acceptability conditions t…
Using the ellipsoidal model for the density configuration, we calculate the equilibrium sequence of the corotating binary stars of the polytropic equation of state in the first post-Newtonian approximation of general relativity. After we calibrate this model by comparing with previous numerical results, we perform the stability analysis by calculating the energy and the angular momentum of the system as a function of the orbital separation. We find that the orbital angular velocity at the energy and/or momentum minimum increases with the increase of the compactness of each star, and this fact …
Solutions to equilibrium sequences of irrotational binary polytropic stars in Newtonian gravity are expanded in a power of $ε=a_0/R$, where R and $a_0$ are the orbital separation of the binary system and the radius of each star for $R=\infty$. For each order of $ε$, we should solve ordinary differential equations for arbitrary polytropic indices n. We show solutions for polytropic indices n= 0.5, 1, 1.5 and 2 up to $ε^6$ orders. Our semi-analytic solutions can be used to check the validity of numerical solutions.
We consider both mode calculations and time evolutions of axial r-modes for relativistic uniformly rotating non-barotropic neutron stars, using the slow-rotation formalism, in which rotational corrections are considered up to linear order in the angular velocity Ω. We study various stellar models, such as uniform density models, polytropic models with different polytropic indices n, and some models based on realistic equations of state. For weakly relativistic uniform density models, and polytropes with small values of n, we can recover the growth times predicted from Newtonian theory when sta…
A solution to an equilibrium of irrotational binary polytropic stars in Newtonian gravity is expanded in a power of ε=a_0/R, where R and a_0 are the separation of the binary system and the radius of each star for R=\infty. For the polytropic index n=1, the solutions are given almost analytically up to order ε^6. We have found that in general an equilibrium solution should have the velocity component along the orbital axis and that the central density should decrease when R decreases. Our almost analytic solutions can be used to check the validity of numerical solutions.
In this paper we will discuss charged stars with polytropic equation of state, where we will try to derive an equation analogous to the Lane-Emden equation. We will assume that these stars are spherically symmetric, and the electric field have only the radial component. First we will review the field equations for such stars and then we will proceed with the analog of the Lane-Emden equation for a polytropic Newtonian fluid and their relativistic equivalent. These kind of equations are very interesting because they transform all the structure equations of the stars in a group of differential e…
Circularly orbiting black hole-gaseous star close binary systems are examined by using numerically exact stationary configurations in the framework of Newtonian gravity. We have chosen a polytropic star for the fluid component of the binary system and considered two ideal situations: 1) a synchronously rotating star and 2) an irrotationally rotating star. They correspond to a rotating star under the influence of viscosity and to that in the inviscid limit, respectively. By analyzing the stationary sequences of binary systems with small separations, we can discuss the final stages of black hole…
We set up in detail the general formalism to model polytropic general relativistic stars with anisotropic pressure. We shall consider two different possible polytropic equations, all of which yield the same Lane-Emden equation in the Newtonian limit. A heuristic model based on an ansatz to obtain anisotropic matter solutions from known solutions for isotropic matter is adopted to illustrate the effects of the pressure anisotropy on the structure of the star. In this context, the Tolman mass, which is a measure of the active gravitational mass, is invoked to explain some features of the models.…
As a preliminary step towards simulating binary neutron star coalescing problem, we test a post-Newtonian approach by constructing a single neutron star model. We expand the Tolman-Oppenheimer-Volkov equation of hydrostatic equilibrium by the power of $c^{-2}$, where $c$ is the speed of light, and truncate at the various order. We solve the system using the polytropic equation of state with index $Γ=5/3, 2$ and 3, and show how this approximation converges together with mass-radius relations. Next, we solve the Hamiltonian constraint equation with these density profiles as trial functions, and …
Using an energy variational method, we calculate quasi-equilibrium configurations of binary neutron stars modeled as compressible triaxial ellipsoids obeying a polytropic equation of state. Our energy functional includes terms both for the internal hydrodynamics of the stars and for the external orbital motion. We add the leading post-Newtonian (PN) corrections to the internal and gravitational energies of the stars, and adopt hybrid orbital terms which are fully relativistic in the test-mass limit and always accurate to PN order. The total energy functional is varied to find quasi-equilibrium…
The equation of state inside very compact objects like neutron stars is still largely unkown. Even though a lot progress has been made in recent years to develop the so-called realistic equations of state, a lot of insight can be gained by using polytropic equations of state to integrate the stellar equations of structure. In this work we provide a brief review of the Newtonian and relativistic equations of structure and present some numerical results, which we believe that can be useful for students starting to work on this field. The internal structure of the Newtonian polytropes is obtained…
We present numerical results on the hydrodynamic stability of coalescing binary stars in the first post Newtonian(1PN) approximation of general relativity. We pay particular attention to the hydrodynamical instability of corotating binary stars in equilibrium states assuming the stiff polytropic equation of state with the adiabatic constant $Γ=3$. In previous 1PN numerical studies on corotating binary stars in equilibrium states, it was found that along the sequence of binary stars as a function of the orbital separation, they have the energy and/or angular momentum minima where the secular in…
In this work the influence of the post-Newtonian corrections to the equations of stellar structure is analysed. The post-Newtonian Lane-Emden equation follows from the corresponding momentum density balance equation. From a polytropic equation of state the solutions of the Lane-Endem equations in the Newtonian and post-Newtonian theories are determined and the physical quantities for the \textit{Sun}, for the white dwarf \textit{Sirius B} and for neutron stars with masses $M\simeq1.4M_\odot, 1.8M_\odot$ and $2.0M_\odot$ are calculated. It is shown that the post-Newtonian corrections to the fie…
We set up the general formalism to model polytropic Newtonian stars with anisotropic pressure. We obtain the corresponding Lane-Emden equation. A heuristic model based on an ansatz to obtain anisotropic matter solutions from known solutions for isotropic matter is adopted to illustrate the effects of the pressure anisotropy on the structure of the star. In particular, we calculate the Chandrasekhar mass for a white dwarf. It is clearly displayed how the Chandrasekhar mass limit changes depending on the anisotropy. Prospective astrophysical applications of the proposed approach are discussed.
Triaxial neutron stars may be important sources of gravitational radiation for the forthcoming generation of interferometric gravitational wave detectors such as LIGO, VIRGO, and GEO600. We investigate the viscosity triggered bar mode secular instability of rapidly rotating neutron stars by means of a perturbation analysis of numerically constructed ``exact'' general relativistic axisymmetric star models. In the theoretical approach, only the dominant parts of the nonaxisymmetric terms of the 3D-Einstein equations are taken into account. A comparison of our results with previous studies of New…
We investigate static spherically symmetric perfect fluid models in Newtonian gravity for barotropic equations of state that are asymptotically polytropic at low and high pressures. This is done by casting the equations into a 3-dimensional regular dynamical system with bounded dependent variables. The low and high central pressure limits correspond to two 2-dimensional boundary subsets, described by homology invariant equations for exact polytropes. Thus the formulation naturally places work about polytropes in a more general context. The introduced framework yields a visual aid for obtaining…
We study equilibrium sequences of close binary systems composed of identical polytropic stars in Newtonian gravity. The solving method is a multi-domain spectral method which we have recently developed. An improvement is introduced here for accurate computations of binary systems with stiff equation of state ($γ> 2$). The computations are performed for both cases of synchronized and irrotational binary systems with adiabatic indices $γ=3,~2.5,~2.25,~2$ and 1.8. It is found that the turning points of total energy along a constant-mass sequence appear only for $γ\ge 1.8$ for synchronized binary …
Spherical systems with polytropic equations of state are of great interest in astrophysics. They are widely used to describe neutron stars, red giants, white dwarfs, brown dwarfs, main sequence stars, galactic halos, and globular clusters of diverse sizes. In this paper we construct analytically a family of self-gravitating spherical models in the post-Newtonian approximation of general relativity. These models present interesting cusps in their density profiles which are appropriate for the modeling of galaxies and dark matter halos. The systems described here are anisotropic in the sense tha…
We study equilibrium sequences of close binary systems on circular orbits and composed of different mass stars with polytropic equation of state in Newtonian gravity. The solving method is a multidomain spectral method which we have recently developed. The computations are performed for both cases of synchronized and irrotational binary systems with adiabatic indices $γ= 3, 2.5, 2.25, 2$ and 1.8, and for three mass ratios: $M_1/M_2 =0.5, 0.2$ and 0.1. It is found that the equilibrium sequences always terminate at a mass shedding limit (appearance of a cusp on the surface of the less massive st…
As an extension of our previous work, we investigate the dynamical instability against nonaxisymmetric bar-mode deformations of differentially rotating stars in Newtonian gravity varying the equations of state and velocity profiles. We performed the numerical simulation and the followup linear stability analysis adopting polytropic equations of state with the polytropic indices n=1, 3/2, and 5/2 and with two types of angular velocity profiles (the so-called j-constant-like and Kepler-like laws). It is confirmed that rotating stars of a high degree of differential rotation are dynamically unsta…
In this work we present a stellar structure model from $f(\mathcal{R})$-gravity point of view capable to describe some classes of stars (White Dwarfs, Brown Dwarfs, Neutron stars, Red Giants and the Sun). This model was based on $f(\mathcal{R})$-gravity field equations for $f(\mathcal{R})= \mathcal{R}+ f_2 \mathcal{R}^2$, hydrostatic equilibrium equation and a polytropic equation of state. We compared the results obtained with those found by the Newtonian theory. It has been observed that in these systems, where high curvature regimes emerge, stellar structure equations undergo modifications. …
We consider the density profile of pressureless dark matter in Eddington-inspired Born-Infeld (EiBI) gravity. The gravitational field equations are investigated for a spherically symmetric dark matter galactic halo, by adopting a phenomenological tangential velocity profile for test particles moving in stable circular orbits around the galactic center. The density profile and the mass distribution, as well as the general form of the metric tensor is obtained by numerically integrating the gravitational field equations, and in an approximate analytical form by using the Newtonian limit of the t…
Quasi-equilibrium models of rapidly rotating triaxially deformed stars are computed in general relativistic gravity, assuming a conformally flat spatial geometry (Isenberg-Wilson-Mathews formulation) and a polytropic equation of state. Highly deformed solutions are calculated on the initial slice covered by spherical coordinate grids, centered at the source, in all angular directions up to a large truncation radius. Constant rest mass sequences are calculated from nearly axisymmetric to maximally deformed triaxial configurations. Selected parameters are to model (proto-) neutron stars; the com…
In the weak field limit, analytic $f(R)$ models of gravity introduce a Yukawa-like correction to the Newtonian gravitational potential. These models have been widely tested at galactic scales and provide an alternative explanation to the dynamics of galaxies without Dark Matter. We study if the temperature anisotropies due to the thermal Sunyaev-Zeldovich effect are compatible with these Extended Theories of Gravity. We assume that the gas is in hydrostatic equilibrium within the modified Newtonian potential and it is well described by a polytropic equation of state. We particularize the model…
We investigate the viscosity driven instability in rotating relativistic stars by means of an iterative approach. We focus on polytropic rotating equilibrium stars and impose an m=2 perturbationin the lapse. We vary both the stiffness of the equation of state and the compactness of the star to study those effects on the value of the threshold. For a uniformly rotating star, the criterion T/W, where T is the rotational kinetic energy and W is the gravitational binding energy, mainly depends on the compactness of the star and takes values around 0.13 ~ 0.16, which differ slightly from that of Ne…
We study both analytically and numerically the gravitational fields of stars in f(R) gravity theories. We derive the generalized Tolman-Oppenheimer-Volkov equations for these theories and show that in metric f(R) models the Parameterized Post-Newtonian parameter $γ_{\rm PPN} = 1/2$ is a robust outcome for a large class of boundary conditions set at the center of the star. This result is also unchanged by introduction of dark matter in the Solar System. We find also a class of solutions with $γ_{\rm PPN} \approx 1$ in the metric $f(R)=R-μ^4/R$ model, but these solutions turn out to be unstable …
Starting from the equations of modified gravity hydrodynamics, we derive the equ tions of motion governing linear, adiabatic, radial perturbations of stars in scalar-tensor theories. There are two new features: first, the eigenvalue equation for the period of stellar oscillations is modified such that the eigenfrequencies are always larger than predicted by General Relativity. Second, the General Relativity condition for stellar instability is altered so that the adiabatic index can fall below 4/3 before unstable modes appear. Stars are more stable in modified gravity theories. Specialising to…
