Compact Objects Research Articles

Astrophysical and relativistic modeling of the recoiling black hole candidate in quasar 3C 186

The compact object in quasar 3C 186 is one of the most promising recoiling black hole candidates, exhibiting both an astrometric displacement between the quasar and the host galaxy as well as a spectroscopic shift between broad and narrow lines. 3C 186 also presents a radio jet that, when projected onto the plane of the sky, appears to be perpendicular to the quasar-galaxy displacement. Assuming a gravitational-wave kick is indeed responsible for the properties of 3C 186 and using state-of-the-art relativistic modeling, we show that current observations allow for exquisite modeling of the recoiling black hole. Most notably, we find that the kick velocity and the black hole spin are almost collinear with the line of sight and the two former vectors appear perpendicular to each other only because of a strong projection effect. The targeted configuration requires substantial fine-tuning: while there is a region in the black hole binary parameter space that is compatible with 3C 186, the observed system appears to be a rare occurrence. Using archival radio observations, we explored different strategies that could potentially confirm or rule out our interpretation. In particular, we developed two observational tests that rely on the brightness ratio between the approaching and receding jet as well as the asymmetry of the jet lobes. While the available radio data provide loose constraints, deeper observations have the unique potential of unveiling the nature of 3C 186.

Physical properties and maximum allowable mass-radius relation of complexity-free compact stellar objects within modified $f(R,T)$ gravity formalism

Abstract This paper focuses on investigating the physical properties and predicted radii of compact stars generated by Tolman--IV complexity-free model within the background of modified gravity theory, especially $f(\mathcal{R},T)$-gravity theory, under complexity formalism for a spherically symmetric spacetime proposed by L. Herrera [Phys Rev D 97: 044010, 2018]. By solving the resulting set of differential equations, the explicit forms of the energy-momentum (EM) tensor components, including the density, radial pressure and tangential pressure, are obtained. The influence of the parameter $\chi$ on various physical properties of the star has been thoroughly investigated. The model undergoes a series of rigorous tests to determine its physical relevance. The findings indicate that the model exhibits regularity, stability and features a surface with vanishing pressure. The boundary of this surface is determined by carefully selecting the parameter space. The complexity method employed in $f(\mathcal{R},T)$ gravity offers an interesting approach for developing astrophysical models that are consistent with observable events as demonstrated by recent experiments. In this regard, we present a study that uses observational data from the GW190814 event, detected by the LIGO and Virgo observatories, to investigate the validity of the Tolman-IV model in $f(R, T)$ gravity. The analysis includes comparing the model's predictions with the observed characteristics of the compact object involved in the merger. In addition, data from two-millisecond pulsars, PSR J1614-2230 and PSR J0952-0607, are incorporated to further constrain theoretical theories. However, we present a diagram illustrating the relationship between the total mass and radius of the compact object candidates for different values of a parameter $\chi$.

Rotating regular black holes and other compact objects with a Tolman-type potential as a regular interior for the Kerr metric

In this paper, we obtain a new class of stationary axisymmetric spacetimes by using the Gürses–Gürsey metric with an appropriate mass function in order to generate a rotating core of matter that may be smoothly matched to the exterior Kerr metric. The same stationary spacetimes may be obtained by applying a slightly modified version of the Newman–Janis algorithm to a nonrotating spherically symmetric seed metric. The starting spherically symmetric configuration represents a nonisotropic de Sitter-type fluid whose radial pressure [Formula: see text] satisfies an state equation of the form [Formula: see text], where the energy density [Formula: see text] is chosen to be the Tolman-type-VII energy density [R. C. Tolman, Phys. Rev. 55, 364 (1939)]. The resulting rotating metric is then smoothly matched to the exterior Kerr metric, and the main properties of the obtained geometries are investigated. All the solutions considered in this study are regular in the sense they are free of curvature singularities. Depending on the relative values of the total mass m and rotation parameter a, the resulting stationary spacetimes represent different kinds of rotating compact objects such as regular black holes, extremal regular black holes, and regular starlike configurations.

Primary superconductivity in regular rotating electrically charged compact objects

In this paper, we refer to the early hypotheses on superconductivity in the models of the electron as an extended particle, and show how Nonlinear Electrodynamics minimally coupled to Gravity (NED-GR) predicts, without additional assumptions, the existence of primary superconductivity as the universal property of regular electrically charged rotating black holes and electromagnetic spinning solitons, replacing naked singularities and including that with the parameters of the electron. NED-GR dynamical equations describe basic generic properties of these objects in a self-consistent and the model-independent way, and establish the existence of the superconducting ring current in their deep interiors, which presents the non-dissipative source of the electromagnetic field of an object and provides its practically unlimited life time.

Simple numerical X-ray polarization models of reflecting axially symmetric structures around accreting compact objects

Abstract We present a series of numerical models suitable for X-ray polarimetry of accreting systems. Firstly, we provide a spectropolarimetric routine that integrates reflection from inner optically thick walls of a geometrical torus of arbitrary size viewed under general inclination. In the studied example, the equatorial torus is illuminated by a central isotropic source of X-ray power-law emission, representing a hot corona. Nearly neutral reprocessing inside the walls is precomputed by Monte Carlo code STOKES that incorporates both line and continuum processes, including multiple scatterings and absorption. We created a new XSPEC model, called xsstokes, which in this version enables efficient X-ray polarimetric fitting of the torus parameters, observer’s inclination and primary emission properties, interpolating for arbitrary state of primary polarization. Comparison of the results to a Monte Carlo simulation allowing partial transparency shows that the no-transparency condition may induce different polarization by tens of %. Allowing partial transparency leads to lower/higher polarization fraction, if the resulting polarization orientation is perpendicular/parallel to the rotation axis. We provide another version of xsstokes that is suitable for approximating nearly neutral reflection from a distant optically thick disc of small geometrical thickness. It assumes local illumination averaged for a selected range of incident angles, representing a toy model of a diffuse corona of various physical extent. Assessing both xsstokes variants, we conclude that the resulting polarization can be tens of % and perpendicularly/parallelly oriented towards the rotation axis, if the reflecting medium is rather vertically/equatorially distributed with respect to a compact central source.

Realistic compact objects in the f(R, T) gravity in the background of polytropic and barotropic gas models

The f(R, T) gravity in the background of the polytropic and barotropic fluid has been investigated in this work. We have selected the TOV equation to determine the internal spacetime of a spherically symmetric galactic object. With the use of the Einstein equation, we have selected KB-spacetime to calculate the mass, compactness, and surface redshift of a spherically symmetric body. Explicit conditions for model parameters have been constructed for the boundary conditions of the interior and exterior spacetime, and the Schwarzschild solution has been employed in the modified f(R, T) gravity theory to evaluate different matching criteria. An increasing pattern in compactness with respect to the different radii is evident in the graphical representation of the compactness evolution for each of the individual star models. After selecting a non-vacuum field equation for higher order curvature, we reformulated f(R, T) for R and T. As a result, the tangential pressure, radial pressure, and matter density have all been calculated. According to the study, as the radius goes to infinity, the tangential and radial pressures display asymptotic flatness and converge to zero. Polytropic and barotropic gas EoS have been adopted since the star model confronts the presence of an isotropic fluid backdrop. It has been noted that in a polytropic background, density and pressure increase with distance from the star’s core, but in a barotropic background, the pressure exhibits an ascending pattern as a function of radius.

A comparative study on the maximum mass and radius of anisotropic compact stars from Heintzmann geometry and the TOV approach

In this paper, a class of anisotropic compact stars is analyzed in Heintzmann geometry. The Einstein field equations (EFEs) have been solved to obtain the stellar model in presence of pressure anisotropy. We have considered the [Formula: see text] metric component as proposed by Heintzmann, and by solving the EFEs, the [Formula: see text] metric component is evaluated in the presence of pressure anisotropy. It is noted that for an isotropic star ([Formula: see text]), the maximum mass lies within the range 1.87–3.04[Formula: see text] for radii ranging between 8–13[Formula: see text]km. For anisotropic compact stars, the maximum mass increases with [Formula: see text] and lies within the range 1.99–3.23 [Formula: see text] for anisotropy parameter [Formula: see text]. The physical viability of the model is examined by applying our model to study the properties of a few known compact objects. All the stability conditions are fulfilled in the proposed model. It is also interesting to note that the maximum mass calculated from our model from geometrical consideration and solving the TOV equation is approximately equal, and the radii predicted from the present model comply with the estimated radius from observations of recently observed pulsars and lighter compact objects of GW events such as GW 190814 and GW 170817.

Reconstructing the near- to mid-infrared environment in the stellar merger remnant V838 Monocerotis

V838 Mon is a stellar merger remnant that erupted in a luminous red nova event in 2002. Although it has been well studied in the optical, near-infrared, and submillimeter regimes, its structure in the mid-infrared wavelengths remains elusive. Over the past two decades, only a handful of infrared interferometric studies have been performed, suggesting the presence of an elongated structure at multiple wavelengths. However, given the limited nature of these observations, the true morphology of the source has not yet been conclusively determined. By performing image reconstruction using observations taken at the VLTI and CHARA, we aim to map out the circumstellar environment in V838 Mon. We observed V838 Mon with the MATISSE ($LMN$ bands) and GRAVITY ($K$ band) instruments at the VLTI as well as the MIRCX/MYSTIC ($HK$ bands) instruments at the CHARA array. We geometrically modelled the squared visibilities and the closure phases in each of the bands to obtain the constraints on the physical parameters. Furthermore, we constructed high-resolution images of V838 Mon in the $HK$ bands using the MIRA and SQUEEZE algorithms to study the immediate surroundings of the star. Lastly, we also modelled the spectral features seen in the $K$ and $M$ bands at various temperatures. The image reconstructions show a bipolar structure that surrounds the central star in the post-merger remnant. In the $K$ band, the super-resolved images show an extended structure (uniform disk diameter $ 1.94$ mas) with a clumpy morphology that is aligned along a north-west position angle (PA) of --$40 On the other hand, in the $H$ band, the extended structure (uniform disk diameter $ 1.18$ mas) lies roughly along the same PA. Yet the northern lobe is slightly misaligned with respect to the southern lobe, which results in the closure phase deviations. The VLTI and CHARA imaging results show that V838 Mon is surrounded by features resembling jets that are intrinsically asymmetric. This is further confirmed by the closure phase modelling. Further observations with VLTI can help to determine whether this structure shows any variations over time and also if such bi-polar structures are commonly formed in other stellar merger remnants.

Formation of long-period post-common-envelope binaries. II. Explaining the self-lensing binary KOI 3278

The vast majority of close binaries containing a compact object, including the progenitors of supernovae\,Ia and at least a substantial fraction of all accreting black holes in the Galaxy, form through common-envelope (CE) evolution. Despite this importance, we struggle to even understand the energy budget of CE evolution. For decades, observed long-period post-CE binaries have been interpreted as evidence of additional energies contributing during CE evolution. We have recently shown that this argument is based on simplified assumptions for all long-period post-CE binaries containing massive white dwarfs (WDs). The only remaining post-CE binary star that has been claimed to require contributions from additional energy sources to understand its formation is KOI 3278. Here, we address in detail the potential evolutionary history of KOI 3278. In particular, we investigate whether extra energy sources, such as recombination energy, are indeed required to explain its existence. We used the 1D stellar evolution code MESA to carry out binary evolution simulations and searched for potential formation pathways for KOI 3278 that are able to explain its observed properties. We find that KOI 3278 can be explained if the WD progenitor filled its Roche lobe during a helium shell flash. In this case, the orbital period of KOI 3278 can be reproduced if the CE binding energy is calculated taking into account gravitational energy and thermodynamic internal energy. While the CE evolution that led to the formation of KOI 3278 must have been efficient -- that is, most of the available orbital energy must have been used to unbind the CE -- recombination energy is not required. We conclude that currently not a single observed post-CE binary requires one to assume that energy sources other than gravitational and thermodynamic energy are contributing to CE evolution. KOI 3278, however, remains an intriguing post-CE binary as, unlike its siblings, understanding its existence requires highly efficient CE ejection.

Search for Extreme Mass Ratio Inspirals Using Particle Swarm Optimization and Reduced Dimensionality Likelihoods

Extreme-mass-ratio inspirals (EMRIs) are significant observational targets for spaceborne gravitational wave detectors, namely, LISA, Taiji, and Tianqin, which involve the inspiral of stellar-mass compact objects into massive black holes (MBHs) with a mass range of approximately 104∼107M⊙. EMRIs are estimated to produce long-lived gravitational wave signals with more than 105 cycles before plunge, making them an ideal laboratory for exploring the strong-gravity properties of the spacetimes around the MBHs, stellar dynamics in galactic nuclei, and properties of the MBHs itself. However, the complexity of the waveform model, which involves the superposition of multiple harmonics, as well as the high-dimensional and large-volume parameter space, make the fully coherent search challenging. In our previous work, we proposed a 10-dimensional search using Particle Swarm Optimization (PSO) with local maximization over the three initial angles. In this study, we extend the search to an 8-dimensional PSO with local maximization over both the three initial angles and the angles of spin direction of the MBH, where the latter contribute a time-independent amplitude to the waveforms. Additionally, we propose a 7-dimensional PSO search by using a fiducial value for the initial orbital frequency and shifting the corresponding 8-dimensional Time Delay Interferometry responses until a certain lag returns the corresponding 8-dimensional log-likelihood ratio’s maximum. The reduced dimensionality likelihoods enable us to successfully search for EMRI signals with a duration of 0.5 years and signal-to-noise ratio of 50 within a wider search range than our previous study. However, the ranges used by both the LISA Data Challenge (LDC) and Mock LISA Data Challenge (MLDC) to generate their simulated signals are still wider than the those we currently employ in our direct searches. Consequently, we discuss further developments, such as using a hierarchical search to narrow down the search ranges of certain parameters and applying Graphics Processing Units to speed up the code. These advances aim to improve the efficiency, accuracy, and generality of the EMRI search algorithm.

Fast Radio Bursts in the disks of Active Galactic Nuclei

Abstract Fast radio bursts (FRBs) are luminous millisecond-duration radio pulses with extragalactic origin, which were discovered more than a decade ago. Despite the numerous samples, the physical origin of FRBs remains poorly understood. FRBs have been thought to originate from young magnetars or accreting compact objects (COs). Massive stars or COs are predicted to be embedded in the accretion disks of active galactic nuclei (AGNs). The dense disk absorbs FRBs severely, making them difficult to observe. However, progenitors’ ejecta or outflow feedback from the accreting COs interact with the disk material to form a cavity. The existence of the cavity can reduce the absorption by the dense disk materials, making FRBs escape. Here we investigate the production and propagation of FRBs in AGN disks and find that the AGN environments lead to the following unique observational properties, which can be verified in future observation. First, the dense material in the disk can cause large dispersion measure (DM) and rotation measure (RM). Second, the toroidal magnetic field in the AGN disk can cause Faraday conversion. Third, during the shock breakout, DM and RM show non-power-law evolution patterns over time. Fourth, for accreting-powered models, higher accretion rates lead to more bright bursts in AGN disks, accounting for up to 1 % of total bright repeating FRBs.

Constraining axion-like particles dark matter in Coma Berenices with FAST

Axions and axion-like particles (ALPs) appear in many extensions of the Standard Model and are being investigated as promising dark matter (DM) candidates. One viable methodology for their detection involves the investigation of the line-like radio emissions from the dwarf spheroidal galaxy, potentially originating from the radiative decay of ALPs or the conversion of ALPs in the magnetic field. In this work, we constrain the properties of ALPs using the 2-hour radio observation of Coma Berenices through the Five-hundred-meter Aperture Spherical radio Telescope (FAST). The 95% upper limits of the ALP-photon coupling are calculated for the ALP decay and conversion scenarios, respectively. Note that the sensitive ALP masses for FAST range from ∼μeV to tens of μeV, where ALP can explain the DM abundance naturally. However, our limits are weaker than those of the CAST helioscope, which can provide an independent and complementary check on the ALP non-detection for ground experiments. Furthermore, we evaluate the expected sensitivity on the ALP of FAST with its full designed bandwidth (70 MHz - 3 GHz) for 100 hours of observation time. Our results indicate that, even with the exceptional sensitivity of the FAST, it is challenging to surpass the existing experimental constraints on ALP DM using radio observation of dSphs, unless the possible enhancements of ALP signals by compact stars in dSphs are considered.

GYOTO 2.0: a polarized relativistic ray-tracing code

Polarized general-relativistic radiative transfer in the vicinity of black holes and other compact objects has become a crucial tool for probing the properties of relativistic astrophysics plasmas. Instruments like GRAVITY, the Event Horizon telescope, Atacama Large Millimeter/submillimeter Array, or Imaging x-ray Polarimetry Explorer make it very timely to develop such numerical frameworks. In this article, we present the polarized extension of the public ray-tracing code Gyoto, and offer a python notebook allowing to easily perform a first realistic computation. The code is very modular and allows to conveniently add extensions for the specific needs of the user. It is agnostic about the spacetime and can be used for arbitrary compact objects. We demonstrate the validity of the code by providing tests, and show in particular a perfect agreement with the ipole code. Our article also aims at pedagogically introducing all the relevant formalism in a self-contained manner.

Correction to: X-ray polarization properties of partially ionized equatorial obscurers around accreting compact objects

Correction to: X-ray polarization properties of partially ionized equatorial obscurers around accreting compact objects

Pulsar glitches from quantum vortex networks

Neutron stars or pulsars are very rapidly rotating compact stars with extremely high density. One of the unsolved long-standing problems of these enigmatic celestial bodies is the origin of pulsars’ glitches, i.e., the sudden rapid deceleration in the rotation speed of neutron stars. Although many glitch events have been reported, there is no consensus on the microscopic mechanism responsible for them. One of the important characterizations of the glitches is the scaling law P(E)∼E-α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$P(E) \\sim E^{-\\alpha }$$\\end{document} of the probability distribution for a glitch with energy E. Here, we reanalyse the accumulated up-to-date observation data to obtain the exponent α≈0.88\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha \\approx 0.88$$\\end{document} for the scaling law, and propose a simple microscopic model that naturally deduces this scaling law without any free parameters. Our model explains the appearance of these glitches in terms of the presence of quantum vortex networks arising at the interface of two different kinds of superfluids in the core of neutron stars; a p-wave neutron superfluid in the inner core which interfaces with the s-wave neutron superfluid in the outer core, where each integer vortex in the s-wave superfluid connects to two half-quantized vortices in the p-wave superfluid through structures called “boojums”.

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

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

Charged anisotropic fluid sphere in [formula omitted] gravity satisfying Vaidya - Tikekar metric

In this paper, we study the properties of electrically charged anisotropic and spherically symmetric compact stars in f(Q) gravity theory. The field equations have been solved using the MIT-Bag Equation of State (EoS) along with the Vaidya-Tikekar potential. However, the specific form of charge E2(r)=q0L2r2(1+Lr2)2 has been used to study the charged compact star model. The interior charge anisotropic solution has been matched with the charged Schwarzchild-(Anti)-de-Sitter metric. In particular, this study has been conducted on different compact stars viz, EXO 1785-248, PSR J1614-2230, PSR J0740+620 and SMC X-1. The mass-radius ratio of these compact stars has been predicted, and the physical viability of the model has been checked through the behavior of causality condition, matter variables, energy constraints, modified TOV equation, and adiabatic index of the star. In conclusion, we observe a well-behaved model of a compact star in f(Q) gravity with charged matter distribution.

Flare Hunting in Hot Subdwarf and White Dwarf Stars from Cycles 1–5 of TESS Photometry

Stellar flares are critical phenomena on stellar surfaces, which are closely tied to stellar magnetism. While extensively studied in main-sequence (MS) stars, their occurrence in evolved compact stars, specifically hot subdwarfs and white dwarfs (WDs), remains scarcely explored. Based on Cycles 1–5 of TESS photometry, we conducted a pioneering survey of flare events in ∼12,000 compact stars, corresponding to ∼38,000 light curves with a 2 minute cadence. Through dedicated techniques for detrending light curves, identifying preliminary flare candidates, and validating them via machine learning, we established a catalog of 1016 flares from 193 compact stars, including 182 from 58 sdB/sdO stars and 834 from 135 WDs, respectively. However, all flaring compact stars showed signs of contamination from nearby objects or companion stars, preventing sole attribution of the detected flares. For WDs, it is highly probable that the flares originated from their cool MS companions. In contrast, the higher luminosities of sdB/sdO stars diminish companion contributions, suggesting that detected flares originated from sdB/sdO stars themselves or through close magnetic interactions with companions. Focusing on a refined sample of 23 flares from 13 sdB/sdO stars, we found their flare frequency distributions were slightly divergent from those of cool MS stars; instead, they resemble those of hot B/A-type MS stars having radiative envelopes. This similarity implies that the flares on sdB/sdO stars, if these flares did originate from them, may share underlying mechanisms with hot MS stars, which warrants further investigation.

Investigating X-ray emission in the GeV-emitting compact symmetric objects PKS 1718–649 and TXS 1146+596

Aims. Compact symmetric objects (CSOs) are thought to represent the first step in the evolutionary path of radio galaxies. In the present study, we investigated the X-ray emission of two CSOs confirmed to emit at GeV energies: PKS 1718–649 and TXS 1146+596. Unveiling the origin of their observed high-energy emission is crucial to establishing the physical parameters of the radio source and understanding how CSOs interact with the surrounding medium. Methods. We combined archival and new NuSTAR observations of PKS 1718–649 and TXS 1146+596 to have broadband X-ray coverage. For both sources, we modeled the broadband spectral energy distribution (SED) from the radio band up to γ-rays in order to derive their physical parameters. We also discuss the role of the ambient medium in confining the source expansion, which we investigate using X-ray obscuration. Results. For the first time, we report X-ray detections of PKS 1718–649 and 1146+596 with NuSTAR at energies higher than 10 keV. Combining Chandra and NuSTAR observations of TXS 1146+596, we reveal the presence of a multitemperature thermal component dominating the soft X-ray spectrum, and we interpret this finding as indicative of an AGN feedback process in action in this source. In addition, we show that two emitting electron populations are necessary to reproduce the observed broadband SED of TXS 1146+596: in our models, the X-ray emission could either be produced by synchrotron radiation or by a weak X-ray corona, or could be an ADAF-type emission. Interestingly, an additional X-ray component, namely a weak corona, is also required for PKS 1718–649. Moreover, we argue that heavily obscured and possibly frustrated sources tend to show different radio sizes with respect to those that are unobscured and free to expand.

Compact stars in Rastall gravity: hydrostatic equilibrium and radial pulsations

Compact stars in Rastall gravity: hydrostatic equilibrium and radial pulsations