Hydrostatic Equilibrium Research Articles

Equilibrium Figures for a Rotating Compressible Capillary Two-Layer Liquid

The paper proves the existence of a family of axisymmetric equilibrium figures as solutions of a stationary problem with unknown boundaries for the Navier–Stokes equations corresponding to the slow rotation of a viscous compressible two-layer liquid mass about some axis. It is assumed that the liquids are barotropic and capillary, and have different viscosities, the internal fluid being bounded by a closed surface. This interface does not intersect with the external boundary of the cloud. The proof is based on implicit function theorem and carried out in the Hölder spaces.

Eco-synthesis, characterization and application of waste plastics pyrolysis char in arsenic removal from contaminated water: An integrated circular framework with parametric response surface methodology optimization-cum-artificial neural network model

Eco-synthesis of a novel, cost-effective activated adsorbent from waste low-density polyethylene plastic pyrolysis char at 550 °C and its arsenic adsorption potential in simulated aqueous media has been investigated. Characterization reveals good surface chemistry and evolved morphology of activated char adsorbent with carbon structures at high pyrolysis temperature, thereby facilitating high arsenic adsorption uptake. A unique integrated framework of kinetic equilibrium investigation, parametric optimization applying response surface methodology (RSM) with responsive arsenic adsorption efficiency and artificial neural network (ANN) model for response validation has been presented. There is reasonably good fitting of experimental equilibrium figures into the Langmuir isotherm model (R2 = 0.9920). The effects of adsorbent pyrochar dose, arsenic concentration, contact time and pH on the arsenic removal capacity have been depicted via conjugate RSM graphs. The ANN regression curves showed R2 value of 0.99436, validating the model with ANN architecture as 4–8-1. Iron oxide complexation with arsenic on adsorbent surface is reported to play a dominant role in the proposed adsorption mechanism. The proposed hybridized model can be applied to examine the adsorptive removal of various contaminants in water. Techno-economic analysis of scaled-up community arsenic filter showed high removal efficiency (> 90%) of arsenic by activated plastic pyro-char adsorbent from contaminated water at an affordable cost of 0.67 $/m3 of treated water. The activated plastic waste pyro-char with an adsorption capacity of 7.21 mg/g is, therefore, environmentally sustainable in plastic waste management in the post-pandemic era for a ‘waste to wealth’ circular economy with perspective arsenic removal from contaminated water.

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.

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.

Soft X-ray emission from warm gas in IllustrisTNG circum-cluster environments

Context. Whereas X-ray clusters are extensively used for cosmology, their idealised modelling, through the hypotheses of spherical symmetry and hydrostatic equilibrium, is being questioned more and more. The soft X-ray emission detected in tens of clusters with ROSAT was found to be higher than what was expected from the idealised hot gas modelling, pointing to our incomplete understanding of these objects. Aims. Given that cluster environments are at the interface between the hot intra-cluster medium (ICM), warm circum-galactic medium (WCGM), and warm-hot intergalactic medium (WHIM), we aim to explore the relative soft X-ray emission of different gas phases in circum-cluster environments. Method. By using the most massive halos in IllustrisTNG at z = 0, we have predicted the hydrodynamical properties of the gas from cluster centres to their outskirts (5 R200), and modelled their X-ray radiation for various plasma phases. Results. First, we found that the radial profile of temperature, density, metallicity and clumpiness of the ICM are in good agreement with recent X-ray observations of clusters. Secondly, we have developed a method to predict the radial profile of soft X-ray emission in different bands, the column density of ions, and the X-ray absorption lines (O VIII, O VII, Ne IX, and Ne IX) of warm-hot gas inside and around clusters. Conclusion. The warm gas (in the form of both WCGM and WHIM gas) is a strong emitter in soft X-ray bands and is qualitatively consistent with the observational measurements. Our results suggest that the cluster soft excess is induced by the thermal emission of warm gas in the circum-cluster environments.

Close-to-fission dumbbell Jupiter-Trojan (17365) Thymbraeus

Context. Every population of small bodies in the Solar System contains a sizable fraction of multiple systems. Of these, the Jupiter Trojans have the lowest number of known binary systems and they are the least well characterized. Aims. We aim to characterize the reported binary system (17365) Thymbraeus, one of only seven multiple systems of Jupiter Trojans known. Methods. We conducted light curve observing campaigns in 2013, 2015, and 2021 with ground-based telescopes. We modeled these light curves using dumbbell figures of equilibrium. Results. We show that Thymbraeus is unlikely a binary system. Its light curves are fully consistent with a bilobated shape: a dumbbell equilibrium figure. We determine a low density of 830 ± 50 kg m−3, consistent with the reported density of other Jupiter-Trojan asteroids and small Kuiper belt objects. The angular velocity of Thymbraeus is close to fission. If separated, its components would become a similarly sized double asteroid, like the Jupiter-Trojan (617) Patroclus.

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

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

Effect of rainbow function on the structural properties of dark energy star

Confirming the existence of compact objects with a mass greater than 2.5M⊙ by observational results such as GW190814 makes that is possible to provide theories to justify these observational results using modified gravity. This motivates us to use gravity's rainbow, which is the appropriate case for dense objects, to investigate the dark energy star structure as a suggested alternative case to the mass gap between neutron stars and black holes in the perspective of quantum gravity. Hence, in the present work, we derive the modified hydrostatic equilibrium equation for an anisotropic fluid, represented by the extended Chaplygin equation of state in gravity's rainbow. Then, for two isotropic and anisotropic cases, using the numerical solution, we obtain energy-dependent maximum mass and its corresponding radius, and the other properties of the dark energy star including the pressure, energy density, stability, etc. In the following, using the observational data, we compare the obtained results in two frameworks of general relativity and gravity's rainbow.

The Hydrostatic Mass of A478: Discrepant Results from Chandra, NuSTAR, and XMM-Newton

Galaxy clusters are the most recently formed and most massive, gravitationally bound structures in the Universe. The number of galaxy clusters formed is highly dependent on cosmological parameters, such as the dark matter density, σ 8, and Ω m . The number density is a function of the cluster mass, which can be estimated from the density and temperature profiles of the intracluster medium under the assumption of hydrostatic equilibrium. The temperature of the plasma, hence its mass, is calculated from the X-ray spectra. However, effective-area calibration uncertainties in the soft band result in significantly different temperature measurements from various space-based X-ray telescopes. NuSTAR is potentially less susceptible to these issues than Chandra and XMM-Newton, having larger effective area, particularly at higher energies, enabling high-precision temperature measurements. In this work, we present analyses of Chandra, NuSTAR, and XMM-Newton data of A478 to investigate the nature of this calibration discrepancy. We find that NuSTAR temperatures are on average ∼11% lower than those of Chandra, and XMM-Newton temperatures are on average ∼5% lower than those of NuSTAR. This results in a NuSTAR mass at r 2500,Chandra of M ⊙, which is ∼10% lower than that of M 2500,Chandra and ∼4% higher than M 2500,XMM−Newton.

New mass limit of a strange star admitting a colour flavor locked equation of state

A class of strange stars is analysed in the present article in hydrostatic equilibrium, whose state is defined by a CFL phase equation of state. We have compared our results with those obtained from the MIT equation of state for strange quark matter, which is regarded as free particles. We have noted that if we consider quarks to form a cooper pair and if their description is made by the CFL equation of state, the maximum mass of strange star reaches a value as high as 3.61 M⊙\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$M_{\\odot }$$\\end{document}. This value is well above the value of 2.03 M⊙\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$M_{\\odot }$$\\end{document} obtained by using the MIT bag equation of state for massless free quarks. Both the maximum masses are determined by solving the TOV equation for different values of the strange quark mass ms\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$m_s$$\\end{document}. Thus, the inclusion of the possibility of quark pair formation in the theory permits us to accommodate a wider class of compact objects such as PSR J1614-2230, PSR J0740+6620, PSR J0952-0607 etc. and the mass of the companion star in the GW190814 event in our model. The consideration of such a high value of mass is hardly theoretically obtainable from normal strange star models in general relativity even with a fast rotation effect. The object PSR J0952-0607 is found to be the fastest and heaviest pulsar in the disk of Milky Way Galaxy, having a mass of 2.35 M⊙\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$M_{\\odot }$$\\end{document}, which may be predicted in our model, as observational evidence supports the existence of strange quark matter in its composition.

Optimization of pumping and injection regimes for mitigation of seawater intrusion

Seawater intrusion (SWI) stands as a significant challenge impacting coastal aquifers worldwide. The emergence of SWI is a natural phenomenon owing to the density difference between seawater and freshwater. Even without any pumping, the seawater naturally advances inland until a hydrostatic equilibrium between the two water sources is reached. However, human interventions, such as excessive groundwater extraction, coastal developments, and land use as well as climate change intensify the SWI predicament. Efficient solutions to counteract the SWI problem involve the implementation of hydraulic barriers. By employing the Henry problem as a benchmark, the Finite Element Simulator (FEFLOW), which is a numerical modeling software, was used for simulating the flow and transport processes. An integrated FEFLOW-Python code was developed to optimize hydraulic barriers for mitigating SWI. The methodology aims to achieve the maximum possible reduction in SWI by iteratively adjusting the locations and rates of pumping and injection wells. Unlike other investigations, this study explores the utilization of negative SWI barriers to extract brackish water from the dispersion zone and potentially offer a cost-effective alternative water source for desalination plants’ intakes. For the Henry Problem, to maximize the reduction in of SWI, brackish water extraction wells should be situated near the center-bottom of the intrusion zone, with a pumping rate of 2.5 m3/day. For injection wells, the optimal location varies depending on the injection rate (I). Achieving the highest retardation (61%) occurs when the injection rate is 1 m3/day, positioned at the aquifer’s bottom and close to the coastal boundary. However, for injection rates below 0.7 m3/day, the injection well’s optimal location shifts towards the seawater intrusion toe. This study presents a novel approach by establishing explicit correlations for optimizing hydraulic barriers to effectively manage and mitigate SWI.

AGN radiation imprints on the circumgalactic medium of massive galaxies

ABSTRACT Active galactic nuclei (AGNs) in cosmological simulations generate explosive feedback that regulates star formation in massive galaxies, modifying the gas phase structure out to large distances. Here, we explore the direct effects that AGN radiation has on gas heating and cooling within one high-resolution z = 3 dark matter halo as massive as a quasar host (Mh = 1012.5M⊙), run without AGN feedback. We assume AGN radiation to impact the circumgalactic medium (CGM) anisotropically, within a bi-cone of angle α. We find that even a relatively weak AGN (black hole mass M• = 108M⊙ with an Eddington ratio λ = 0.1) can significantly lower the fraction of halo gas that is catastrophically cooling compared to the case of gas photoionized only by the ultraviolet background (UVB). Varying M•, λ, and α, we study their effects on observables. A 109M⊙ AGN with λ = 0.1 and $\alpha \approxeq 60^{^{\rm o}}$ reproduces the average surface brightness (SB) profiles of Ly α, He ii, and C iv, and results in a covering fraction of optically thick absorbers within observational estimates. The simulated SB$_{\rm C\, \rm{\small IV}}$ profile is steeper than observed, indicating that not enough metals are pushed beyond the very inner CGM. For this combination of parameters, the CGM mass catastrophically cooling is reduced by half with respect to the UVB-only case, with roughly same mass out of hydrostatic equilibrium heating up and cooling down, hinting to the importance of self-regulation around AGNs. This study showcases how CGM observations can constrain not only the properties of the CGM itself, but also those of the AGN engine.

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.

A linear low effort stabilization method for the Euler equations using discontinuous Galerkin methods

SummaryWe present a novel and simple yet intuitive approach to the stabilization problem for the numerically solved Euler equations with gravity source term relying on a low‐order nodal Discontinuous Galerkin Method (DGM). Instead of assuming isothermal or polytropic solutions, we only take a hydrostatic balance as a given property of the flow and use the hydrostatic equation to calculate a hydrostatic pressure reconstruction that replaces the gravity source term. We compare two environments that both solve the Euler equations using the DGM: deal.II and StormFlash. We utilize StormFlash as it allows for the use of the novel stabilization method. Without stabilization, StormFlash does not yield results that resemble correct physical behavior while the results with stabilization for StormFlash, as well as deal.II model the fluid flow more accurately. Convergence rates for deal.II do not match the expected order while the convergence rates for StormFlash with the stabilization scheme (with the exceptions for the L errors for momentum) meet the expectation. The results from StormFlash with stabilization also fit reference solutions from the literature much better than those from deal.II. We conclude that this novel scheme is a low cost approach to stabilize the Euler equations while not limiting the flow in any way other than it being in hydrostatic balance.

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.

Anisotropic quark stars in energy-momentum squared gravity

At super-saturation densities, quarks in the core of neutron stars (NSs) have been a subject of intense investigation for last few decades. In particular, an intriguing conjecture has emerged suggesting that the true ground state of hadronic matter may be strange quark matter (SQM). In the present manuscript, we study internal structure and the physical properties of quark stars (QSs) in energy-momentum squared gravity (EMSG) with MIT bag model equations of state (EoS). An important feature of this theory is the addition of the term f(TμνTμν) to the Einstein-Hilbert action. This modification in the action lead to changes in the right-hand side of Einstein's field equations (EFE). We obtain mass-radius relations for a spherically symmetric configuration supported by the anisotropic fluid matter distribution and solved the hydrostatic equilibrium equations numerically in the framework of EMSG. In addition, we present and discuss the stability of stellar structure depending on the model parameters.

A new step forward in realistic cluster lens mass modelling: analysis of Hubble Frontier Field Cluster Abell S1063 from joint lensing, X-ray, and galaxy kinematics data

ABSTRACT We present a new method to simultaneously and self-consistently model the mass distribution of galaxy clusters that combines constraints from strong lensing features, X-ray emission, and galaxy kinematics measurements. We are able to successfully decompose clusters into their collisionless and collisional mass components thanks to the X-ray surface brightness, as well as use the dynamics of cluster members, to obtain more accurate masses exploiting the fundamental plane of elliptical galaxies. Knowledge from all observables is included through a consistent Bayesian approach in the likelihood or in physically motivated priors. We apply this method to the galaxy cluster Abell S1063 and produce a mass model that we publicly release with this paper. The resulting mass distribution presents different ellipticities for the intra-cluster gas and the other large-scale mass components as well as deviation from elliptical symmetry in the main halo. We assess the ability of our method to recover the masses of the different elements of the cluster using a mock cluster based on a simplified version of our Abell S1063 model. Thanks to the wealth of mutliwavelength information provided by the mass model and the detected X-ray emission, we also found evidence for an ongoing merger event with gas sloshing from a smaller infalling structure into the main cluster. In agreement with previous findings, the total mass, gas profile, and gas mass fraction are all consistent with small deviations from the hydrostatic equilibrium. This new mass model for Abell S1063 is publicly available, as the lenstool extension used to construct it.

Nested spheroidal figures of equilibrium − IV. On heterogeneous configurations

ABSTRACT The theory of nested figures of equilibrium, expanded in Papers I and II, is investigated in the limit where the number of layers of the rotating body is infinite, enabling to reach full heterogeneity. In the asymptotic process, the discrete set of equations becomes a differential equation for the rotation rate. In the special case of rigid rotation (from centre to surface), we are led to an integro-differential equation (IDE) linking the ellipticity of isopycnic surfaces to the equatorial mass-density profile. In contrast with most studies, these equations are not restricted to small flattenings, but are valid for fast rotators as well. We use numerical solutions obtained from the self-consistent-field method to validate this approach. At small ellipticities (slow rotation), we fully recover Clairaut’s equation. Comparisons with Chandrasekhar’s perturbative approach and with Roberts’ work based on virial equations are successful. We derive a criterion to characterize the transition from slow to fast rotators. The treatment of heterogeneous structures containing mass-density jumps is proposed through a modified IDE.

Controversial Issues of the Mechanism of Crystalline Lens Accommodation and the Rationale the Hydraulic Component in its Implementation

Purpose: To consider controversial issues of the mechanism of crystalline lens accommodation and to justify the hydraulic component in its implementation.
 Materials and Methods: Theories of the mechanism of accommodation and its assessment according to ultrasound biomicroscopy, magnetic resonance imaging and optical coherence tomography were analyzed. For the first time, the features of accommodative activation of intraocular fluid exchange in the closed hydrostatic system of the lens with the participation of mechanosensitive aquaporins were considered. When substantiating the hydraulic component in the mechanism of the crystalline lens accommodation, special emphasis was placed on the rapid decrease in pressure in the anterior and posterior chambers of the eye during contraction of the meridional portion of the ciliary muscle.
 Results: Analysis of various theories of accommodation has shown that mechanism of the crystalline lens its implementation continues to be discussed to this day. For the first time, the lens was considered as a unique closed hydrostatic system in which the pressure level is established through ultrafiltration and diffusion of intraocular fluid with the participation of aquaporins. Aquaporins form ion channels in the capsule, cuboidal epithelial cells, lens fibers and are mechanosensitive receptor proteins. The opening and closing of ion channels regulates the potassium-sodium pump, directed transport and exchange of intraocular fluid in the lens. The hydrostatic balance between the pressure in the crystalline lens and the anterior and posterior chambers of the eye is ensured by the crystalline lens capsule. The capsular bag of the crystalline lens can be considered as a curved diaphragm that separates two hydrostatic systems with different levels of pressure. Due to the hydrostatic buffering effect, the IOP level does not affect the crystalline lens, but it responds to a rapid decrease. This decrease in pressure in the anterior and posterior chambers is realized through the tension of the scleral spur by the meridional portion of the ciliary muscle and the activation of the valve mechanism of the scleral sinus. The greater the decrease in pressure, the more convex the crystalline lens takes on and increases its refraction.
 Conclusion: The presence of a hydraulic component in the mechanism of crystalline lens accommodation allows us to understand how the contraction of the small ciliary muscle can change the shape and refractive power of the large crystalline lens.

Analysis of complexity on the anisotropic charged fluid in modified teleparallel gravity

This article is about the impact of complexity on the anisotropic charged fluid in f(T) gravity, where accountability of the torsion scalar T is offered by the gravitational influence. Our focus is on the anisotropic charged fluid. We took into account the static spherical distribution for the interior region. Foremost, we present the basic formalism of f(T) theory, which comprises the electric charge, and then evaluate the field equations along with the hydrostatic equilibrium equation in this framework. The criterion of complexity is taken up from Herrera’s design (Herrera, 2018), where the complexity factor is attained from the division of the Riemann curvature tensor in an orthogonal direction. We set up the relation between the Misner–Sharp mass function and the hydrostatic equilibrium equation comprising the electric charge in f(T) theory. We match the interior region with the Reissner-Nordström geometry in the exterior direction and investigate the role of the Tolman mass in this background. We carry out the analysis of the structure scalars for charged anisotropic fluid within f(T) gravity and study the impact of the density homogeneity and inhomogeneity along with local anisotropy on the complexity factor YTF. The manuscript concluded with the defined concepts that satisfy the constraint of diminishing complexity in f(T) theory, which comprises the electric charge.