Stellar Structure Equations Research Articles

Anisotropic interior models with Kohler–Chao–Tikekar-like complexity factor

This work explores the construction of spherically symmetric models of stellar interiors by incorporating the null complexity factor (CF) as an additional constraint. This supplementary condition helps us to close an array of stellar structure equations resulting from the process of gravitational decoupling. By making use of MGD-type gravitational decoupling we analyze the role of gravitational decoupling and its impact on the complexity of static, self-gravitational systems. We begin by considering an anisotropic seed solution described by the Kohler–Chao–Tikekar metric ansatz. We then apply the minimal geometric deformation technique to this seed solution, imposing the constraint that the effective anisotropic factor vanishes. This constraint leads to the generation of an isotropic stellar solution. Furthermore, we construct a second family of solutions in which the CF, remains the same for both the seed solution and its minimally deformed counterpart. Our analysis further investigated the influence of both the deformation parameter and the CF on the structural properties of the static and spherically symmetric stellar objects.

Non-radial oscillations in anisotropic dark energy stars

We study the non-radial f-mode oscillations of both isotropic and anisotropic dark energy stars by using the modified Chaplygin prescription of dark energy to model the stellar matter. The anisotropic pressure in the system is modeled with Bowers–Liang prescription. By solving the stellar structure equations in presence of anisotropy, we study the global properties of the dark energy star and compare the mass-radius profiles with data from GW events and milli-second pulsars. The stellar radial and anisotropic pressure profiles have also been investigated. We proceed to determine the prominent non-radial l=2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$l=2$$\\end{document}f-mode frequencies of the anisotropic dark energy star by employing the Cowling approximation and analyse and quantify the spectra by varying the anisotropic parameter. We report that f-mode spectra of dark energy star have distinctly different behaviour compared to neutron star and quark star, and this may possibly help in its future identification. Further, the tidal deformability factors of the anisotropic dark energy stars have also been analyzed.

Effects of anisotropic pressure on interacting quark star structure

Perturbative Quantum Chromodynamics (pQCD) corrections and color superconductivity predict that strongly interacting matter can reveal new physical phenomena under extreme conditions. Taking into account these interaction effects, we investigate the role of anisotropic pressure in quark stars composed of interacting quark matter. Adopting two physically well-motivated anisotropy profiles, we numerically solve the stellar structure equations in order to explore the consequences of anisotropic pressure on various macroscopic properties such as radius, gravitational mass, surface redshift, moment of inertia, tidal Love number and oscillation spectrum. Remarkably, for both anisotropy models, negative anisotropies increase the radial stability of interacting quark stars, while the opposite occurs for positive anisotropies. However, for the Bowers-Liang profile, the central density corresponding to the maximum-mass point does not coincide with the central density where the squared oscillation frequency vanishes, indicating that the existence of stable anisotropic interacting quark stars is possible beyond the maximum mass for negative anisotropies. Additionally, we compare our theoretical predictions with several observational mass-radius measurements and tidal deformability constraints, which suggest that both strong interaction effects and anisotropy effects play a crucial role in describing compact stars observed in the Universe.

Dark energy stars and quark stars within the context of [formula omitted] gravity

In this work, we investigate the relativistic structure of compact stars within the framework of f(Q) gravity, where Q is the nonmetricity scalar. In particular, we focus on the f(Q)=αQ+β gravity model, with α and β being free parameters of the theory. This study is split up into: 1. Models of dark energy stars by using metric potentials of Tolman–Kuchowicz type which are free of singularity. It considers the stellar fluid to be made up of both ordinary matter along with dark energy, where the constants in the metric are determined from observational measurements of some well-known compact stars. Part 2 deals with quark stars where their masses and radii are a consequence of integrating the stellar structure equations given a specific equation of state (EoS) for the dense matter involved. We use the pulsar SAX J1808.4-3658, which has a known mass and radius of M=0.9−0.3+0.3M⊙, and 7.95−1+1km respectively, to explain the physical characteristics of the dark energy star. For various values of α, the causality criteria and the model’s dynamical stability are discussed. In light of the discovery of gravitational waves GW190814, we also investigate the possibility of characterizing the secondary component of such an event as a stable dark energy star in the presence of anisotropy using the M−R relation, which characterizes a dark energy star with mass 2.57M⊙ and associated radius 9.3km. Furthermore, in the case of quark stars with MIT bag model EoS, we find that both the radius and the mass increase with increasing α for fixed β=0. Meanwhile, the effect of the parameter β is a substantial increase in the maximum-mass values as β becomes more negative for a fixed value of α.

Relativistic structure of charged quark stars in energy–momentum squared gravity

Within the context of energy–momentum squared gravity (EMSG), where non-linear matter contributions appear in the gravitational action, we derive the modified TOV equations describing the hydrostatic equilibrium of charged compact stars. We adopt two different choices for the matter Lagrangian density (Lm=p versus Lm=−ρ) and investigate the impact of each one on stellar structure. Furthermore, considering a charge profile where the electric charge density ρch is proportional to the standard energy density ρ, we solve numerically the stellar structure equations in order to obtain the mass–radius diagrams for the MIT bag model equation of state (EoS). For Lm=p and given a specific value of β (including the uncharged case when β=0), the maximum-mass values increase (decrease) substantially as the gravity model parameter α becomes more negative (positive). However, for uncharged configurations and considering Lm=−ρ, our numerical results reveal that when we increase α (from a negative value) the maximum mass first increases and after reaching a maximum value it starts to decrease. Remarkably, this makes it a less trivial behavior than that caused by the first choice when we take into account the presence of electric charge (β≠0).

Predicting convective blueshift and radial-velocity dispersion due to granulation for FGK stars

ABSTRACT To detect Earth-mass planets using the Doppler method, a major obstacle is to differentiate the planetary signal from intrinsic stellar variability (e.g. pulsations, granulation, spots, and plages). Convective blueshift, which results from small-scale convection at the surface of Sun-like stars, is relevant for Earth-twin detections as it exhibits Doppler noise of the order of 1 $\rm m\, s^{-1}$. Here, we present a simple model for convective blueshift based on fundamental equations of stellar structure. Our model successfully matches observations of convective blueshift for FGK stars. Based on our model, we also compute the intrinsic noise floor for stellar granulation in the radial-velocity observations. We find that for a given mass range, stars with higher metallicities display lower radial-velocity dispersion due to granulation, in agreement with magnetohydrodynamic simulations. We also provide a set of formulae to predict the amplitude of radial-velocity dispersion due to granulation as a function of stellar parameters. Our work is vital in identifying the most amenable stellar targets for Extreme Precision Radial Velocity surveys and radial velocity follow-up programmes for TESS, CHEOPS, and the upcoming PLATO mission.

A Review of the Mixing Length Theory of Convection in 1D Stellar Modeling

We review the application of the one-dimensional Mixing Length Theory (MLT) model of convection in stellar interiors and low-mass stellar evolution. We summarize the history of MLT, present a derivation of MLT in the context of 1D stellar structure equations, and discuss the physical regimes in which MLT is relevant. We review attempts to improve and extend the formalism, including to higher dimensions. We discuss the interactions of MLT with other modeling physics, and demonstrate the impact of introducing variations in the convective mixing length, αMLT, on stellar tracks and isochrones. We summarize the process of performing a solar calibration of αMLT and state-of-the-art on calibrations to non-solar targets. We discuss the scientific implications of changing the mixing length, using recent analyses for demonstration. We review the most prominent successes of MLT, and the remaining challenges, and we conclude by speculating on the future of this treatment of convection.

Probing strong field f(R) gravity and ultradense matter with the structure and thermal evolution of neutron stars

Thermal evolution of neutron stars is studied in the $f(R)=R+\ensuremath{\alpha}{R}^{2}$ theory of gravity. We first review the equations of stellar structure and evolution for a spherically symmetric spacetime plus a perfect fluid at rest. We then present numerical results for the structure of neutron stars using four nucleonic dense matter equations of state and a series of gravity theories for $\ensuremath{\alpha}$ ranging from zero, i.e., general relativity, up to $\ensuremath{\alpha}\ensuremath{\approx}{10}^{16}\text{ }\text{ }{\mathrm{cm}}^{2}$. We emphasize properties of these neutron star models that are of relevance for their thermal evolution as the threshold masses for enhanced neutrino emission by the direct Urca process, the proper volume of the stellar cores where this neutrino emission is allowed, the crust thickness, and the surface gravitational acceleration that directly impact the observable effective temperature. Finally, we numerically solve the equations of thermal evolution and explicitly analyze the effects of altering gravity. We find that uncertainties in the dense matter microphysics, such as the core chemical composition and superfluidity/superconductivity properties, as well as the astrophysical uncertainties on the chemical composition of the surface layers, have a much stronger impact than possible modifications of gravity within the studied family of $f(R)$ theories. We conclude that within this family of gravity theories, conclusions from previous studies of neutron star thermal evolution are not significantly altered by modification of gravity theory. Conversely, this implies that neutron star cooling modeling may not be a useful tool to constrain deviations of gravity from Einstein theory unless these are much more radical than in the $f(R)=R+\ensuremath{\alpha}{R}^{2}$ framework.

Building post-Newtonian neutron stars

Owed to their compactness, neutron stars involve strong gravity and extreme density physics. Nevertheless, at present, there are a variety of problems where progress (at least conceptually) can be made in the context of weak gravity. Motivated by this we examine how accurately one can model neutron stars using the post-Newtonian (pN) approximation to general relativity. In general, we find there is a significant degree of freedom in how the pN equations of stellar structure can be formulated. We discuss this flexibility in the formulation and provide examples to demonstrate the impact on stellar models. We also consider the (closely related) problem of building neutron stars using isotropic coordinates. In this context, we provide a new strategy for solving the equations (based on a scaling argument) which significantly simplifies the problem.

Structure and evolution of ultra-massive white dwarfs in general relativity

Context. Ultra-massive white dwarfs (M⋆ ≳ 1.05 M⊙) are of utmost importance in view of the role they play in type Ia supernovae explosions, merger events, the existence of high-magnetic -field white dwarfs, and the physical processes in the super asymptotic giant branch phase. Aims. We aim to present the first set of constant rest-mass ultra-massive oxygen-neon white dwarf cooling tracks with masses of M⋆ > 1.29 M⊙ which fully take into account the effects of general relativity on their structural and evolutionary properties. Methods. We computed the full evolution sequences of 1.29, 1.31, 1.33, 1.35, and 1.369 M⊙ white dwarfs with the La Plata stellar evolution code, LPCODE. For this work, the standard equations of stellar structure and evolution have been modified to include the effects of general relativity. Specifically, the fully general relativistic partial differential equations governing the evolution of a spherically symmetric star are solved in a way so that they resemble the standard Newtonian equations of stellar structure. For comparison purposes, the same sequences have been computed for the Newtonian case. Results. According to our calculations, the evolutionary properties of the most massive white dwarfs are strongly modified by general relativity effects. In particular, the resulting stellar radius is markedly smaller in the general relativistic case, being up to 25% smaller than predicted by the Newtonian treatment for the more massive ones. We find that oxygen-neon white dwarfs more massive than 1.369 M⊙ become gravitationally unstable with respect to general relativity effects. When core chemical distribution due to phase separation on crystallization is considered, such instability occurs at somewhat lower stellar masses, ≳1.36 M⊙. In addition, cooling times for the most massive white dwarf sequences are about a factor of two smaller than in the Newtonian case at advanced stages of evolution. Finally, a sample of white dwarfs have been identified as ideal candidates to test these general relativistic effects. Conclusions. We conclude that the general relativity effects should be taken into account for an accurate assessment of the structural and evolutionary properties of the most massive white dwarfs. These new ultra-massive white dwarf models constitute a considerable improvement over those computed in the framework of the standard Newtonian theory of stellar interiors.

Stellar Structure Model in the Post-Newtonian Approximation

In this work the influence of the post-Newtonian corrections to the equations of stellar structure are analyzed. 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–Emden equations in the Newtonian and post-Newtonian theories are determined and the physical quantities for the Sun, the white dwarf Sirius B and neutron stars with masses M ≃ 1.4 M ⊙, 1.8 M ⊙ and 2.0 M ⊙ are calculated. It is shown that the post-Newtonian corrections to the fields of mass density, pressure and temperature are negligible for the Sun and Sirius B, but for stars with strong fields the differences become important. For the neutron stars analyzed here the central pressure and the central temperature which follow from the post-Newtonian Lane–Emden equation are about fifty to sixty percent greater than those of the Newtonian theory and the central mass density is about three to four percent smaller.

Stability Analysis of Isotropic Spheres in Einstein Gauss–Bonnet Gravity

AbstractThis work investigates the effect of electromagnetic field on relativistic compact stellar structures coupled with anisotropic collapsing fluid under the framework of string‐motivated Gauss–Bonnet‐powered gravity. Using the power‐law form of Maxwell‐GB corrections, the stellar structure equations dubbed as Tolman–Oppenheimer–Volkoff (TOV) equations are developed. In the presence of a power‐law form of ‐corrections, an evolution equation for the Weyl scalar is obtained, which plays a central role in the investigation of the examined fluid configuration. It is observed that the existence of shear in the fluid flow, irregularities in the distribution of energy density, dissipative fluxes, and modified Gauss–Bonnet corrections are the certain factors that are accountable for generating anisotropies in an originally isotropic fluid distribution. Some of the interesting consequences of formulated results are examined.

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.

A comment on the properties of the matter flow through the first Langrangian point

Abstract We analyse properties of the mass outflow from the Roche lobe filling component of a semi-detached binary system. We follow the approaches published by Paczyński & Sienkiewicz and by Lubow & Shu, which we compare with other simplified approaches. We find that the density of the flow at L1 is orders of magnitude lower than the density on the same equipotential but away from L1. Furthermore, the effective cross-section of the flow, after averaging over its profile of the momentum density, is much lower than some published estimates done without accounting for the averaging. Thus, the use of some simplified formulae for the density and the flow cross-section can lead to overestimates of the accretion rate and of the mass contained in the L1 regions by very large factors unless they are supported by simultaneous integrations of the equations of stellar structure for the outer layers of the donor.

Neutron stars in Palatini R+alpha R^2 and R+alpha R^2+beta Q theories

We study solutions of the stellar structure equations for spherically symmetric objects in modified theories of gravity, where the Einstein-Hilbert Lagrangian is replaced by f(R)=R+alpha R^2 and f(R,Q)=R+alpha R^2+beta Q, with R being the Ricci scalar curvature, Q=R_{mu nu }R^{mu nu } and R_{mu nu } the Ricci tensor. We work in the Palatini formalism, where the metric and the connection are assumed to be independent dynamical variables. We focus on stellar solutions in the mass-radius region associated to neutron stars. We illustrate the potential impact of the R^2 and Q terms by studying a range of viable values of alpha and beta . Similarly, we use different equations of state (SLy, FPS, HS(DD2) and HS(TMA)) as a simple way to account for the equation of state uncertainty. Our results show that for certain combinations of the alpha and beta parameters and equation of state, the effect of modifications of general relativity on the properties of stars is sizeable. Therefore, with increasing accuracy in the determination of the equation of state for neutron stars, astrophysical observations may serve as discriminators of modifications of General Relativity.

An approximation for the capture radius of gaseous protoplanets

ABSTRACT Determining the heavy-element accretion rate of growing giant planets is crucial for understanding their formation and bulk composition. The solid (heavy-element) accretion rate should be carefully modelled during the various stages of giant planet formation and therefore the planetary capture radius must be determined. In some simulations that model the heavy-element accretion rate, such as in N-body simulations, the presence of the gaseous envelope is either neglected or treated in an oversimplified manner. In this paper, we present an approximation for the capture radius that does not require the numerical solution of the stellar structure equations. Our approximation for the capture radius works extremely well for various planetesimal sizes and compositions. We show that the commonly assumed constant density assumption for inferring the capture radius leads to a large error in the calculated capture radius and we therefore suggest that our approximation should be implemented in future simulations.

Neutron stars in f(R,T) gravity with conserved energy-momentum tensor: Hydrostatic equilibrium and asteroseismology

We investigate the equilibrium and radial stability of spherically symmetric relativistic stars, considering a polytropic equation of state (EoS), within the framework of f(R,T) gravity with a conservative energy-momentum tensor. Both modified stellar structure equations and Chandrasekhar's pulsation equations are derived for the f(R,T)= R+ h(T) gravity model, where the function h(T) assumes a specific form in order to safeguard the conservation equation for the energy-momentum tensor. The neutron star properties, such as radius, mass, binding energy and oscillation spectrum are studied in detail. Our results show that a cusp — which signals the appearance of instability — is formed when the binding energy is plotted as a function of the compact star proper mass. We find that the squared frequency of the fundamental vibration mode passes through zero at the central-density value corresponding to such a cusp where the binding energy is a minimum.

Parameterized nonrelativistic limit of stellar structure equations in Ricci-based gravity theories

We present the nonrelativistic limit of the stellar structure equations of Ricci-based gravities, a family of metric-affine theories whose Lagrangian is built via contractions of the metric with the Ricci tensor of an a priori independent connection. We find that this limit is characterized by four parameters that arise in the expansion of several geometric quantities in powers of the stress-energy tensor of the matter fields. We discuss the relevance of this result for the phenomenology of nonrelativistic stars, such as main-sequence stars as well as several substellar objects.

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.

How planets grow by pebble accretion

During their formation, planets form large, hot atmospheres due to the ongoing accretion of solids. It has been customary to assume that all solids end up at the center, constituting a “core” of refractory materials, whereas the envelope remains metal-free. However, recent work, as well as observations by the Juno mission, indicate that the distinction may not be so clear cut. Indeed, small silicate, pebble-sized particles will sublimate in the atmosphere when they hit the sublimation temperature (T ~ 2000 K). In this paper we extend previous analytical work to compute the properties of planets within such a pebble accretion scenario. We conduct 1D numerical calculations of the atmosphere of an accreting planet, solving the stellar structure equations, augmented by a nonideal equation of state that describes a hydrogen and helium-silicate vapor mixture. Calculations terminate at the point where the total mass in metal is equal to that of the H+He gas, which we numerically confirm as the onset of runaway gas accretion. When pebbles sublimate before reaching the core, insufficient (accretion) energy is available to mix dense, vapor-rich lower layers with the higher layers of lower metallicity. A gradual structure in which Z decreases with radius is therefore a natural outcome of planet formation by pebble accretion. We highlight, furthermore, that (small) pebbles can act as the dominant source of opacity, preventing rapid cooling and presenting a channel for (mini-)Neptunes to survive in gas-rich disks. Nevertheless, once pebble accretion subsides, the atmosphere rapidly clears followed by runaway gas accretion. We consider atmospheric recycling to be the most probable mechanism to have stalled the growth of the envelopes of these planets.