Star S2 Research Articles

Orbits of particles with magnetic dipole moment around magnetized Schwarzschild black holes: Applications to the S2 star orbit

This study provides a comprehensive analytical investigation of the bound and unbound motion of magnetized particles orbiting a Schwarzschild black hole immersed in an external asymptotically uniform magnetic field, which includes all conceivable types of bounded and unbounded orbits. In particular, for planetary orbits, we perform a comparative analysis of our findings with the observed position of the S2 star carrying magnetic dipole moment around Sagittarius A*. We found maximum and minimum values for the parameter of magnetic interaction between the magnetic dipole of the star and the external magnetic field, as well as the energy and angular momentum of the S2 star. As a result, we obtain estimations of the magnetic dipole of the star in the order of 106 G·cm3. Additionally, we explore deflecting trajectories akin to gravitational Rutherford scattering. In obtaining the solutions for the orbital equations, we articulate the elliptic integrals and Jacobi elliptic functions, and illustrative figures and simulations augment our study. Published by the American Physical Society 2024

Probing the regular black hole with an asymptotically Minkowski core by precessing motion of S2 star and OJ 287

We investigate the precessing motion around the the regular black hole with an asymptotically Minkowski core which has a suppression parameter ℓ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell $$\\end{document}. We obtain the third-order post-Newtonian quasi-Keplerian motion of a test particle in this spacetime and find its relativistic periastron advance. Then, we test this model with the precessing motion of the star S2 and of OJ 287 and determine the upper limits of the dimensionless suppression parameter as ℓ/m∼10-3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell /m \\sim 10^{-3}$$\\end{document}, where m is the mass of the black hole. Compared with the bound given by the shadow of Sgr A*, our bound on ℓ/m\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell /m$$\\end{document} is reduced by 2 orders of magnitude, although our upper limit of ℓ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell $$\\end{document} still needs further improvement.

Constraining a disformal Schwarzschild black hole in DHOST theories with the orbit of the S2 star

With the observed data of the S2 orbit around the black hole Sgr A∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$^*$$\\end{document} and the Markov Chain Monte Carlo method, we make a constraint on parameters of a disformal Schwarzschild black hole in quadratic degenerate higher-order scalar-tensor (DHOST) theories. This black hole belongs to a class of non-stealth solutions and owns an extra disformal parameter described the deviation from general relativity. Our results show that the best fit value of the disformal parameter is positive. However, in the range of 1σ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$1\\sigma $$\\end{document}, we also find that general relativity remains to be consistent with the observation of the S2 orbit.

Relativistic periastron advance beyond Einstein theory: analytical solution with applications

We find a new solution to calculate the orbital periastron advance of a test body subject to a central gravitational force field for relativistic theories and models beyond Einstein. This analitycal formula has general validity that includes all the post-Newtonian (PN) contributions to the dynamics and is useful for high-precision gravitational tests. The solution is directly applicable to corrective potentials of various forms, without the need for numerical integration. Later, we apply it to the scalar tensor fourth order gravity (STFOG) and noncommutative geometry, providing corrections to the Newtonian potential of Yukawa-like form V(r)=αe-βrr\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$V(r)=\\alpha \\frac{e^{-\\beta r}}{r}$$\\end{document}, and we conduct the first analysis involving all the PN terms for these theories. The same work is performed with a Schwarzschild geometry perturbed by a Quintessence Field, leading to a power-law potential V(r)=αqrq\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$V(r)=\\alpha _q {r}^q$$\\end{document}. Finally, by using astrometric data of the Solar System planetary precessions and those of the S2 star around Sgr A*, we infer new theoretical constraints and improvements in the bounds for β\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta $$\\end{document}. The resulting simulated orbits turn out to be compatible with general relativity.

Observational test of ℛ2 spacetimes with the S2 star in the Milky Way galactic center

A novel class of vacuum metrics expressible in analytical form was recently found for pure ℛ2 gravity, based on a groundwork put forth by Buchdahl in 1962. These Buchdahl-inspired solutions offer a practical framework for testing ℛ2 gravity through empirical observations. Within a subclass of asymptotically flat Buchdahl-inspired vacuum spacetimes, we identified a parameter ϵ measuring the deviation from the classic Schwarzschild metric, which corresponds to ϵ=0. In this paper, we employ observational data from the S2 star's orbit around Sgr A* in the Milky Way galactic center and perform Monte Carlo Markov Chain simulations to probe the effects of the new metrics on the orbit of the S2 star. Our analysis presented herein reports a range at 95% confidence level on the deviation parameter as ϵ ∈ (-0.6690, 0.4452). While no decisive evidence either in favor or in disfavor of the asymptotically flat Buchdahl-inspired spacetimes has been achieved, the obtained bound is compatible with the tighter results using other data of different nature as recently reported in Eur. Phys. J. C 84 (2024) 330. As a meaningful test probing into a strong-field regime, our present study calls for further observations with prolonged period and improved accuracy in order to tighten the bound for ϵ using the S2 star orbit.

Timelike bound orbits and pericenter precession around black hole with conformally coupled scalar hair

We investigate the geodesic motions of timelike particles around a static hairy black hole with conformally coupled scalar field. We mainly focus on the effects of the scalar charge and electric charge on the marginally bound orbits (MBO), innermost stable circular orbits (ISCO) and on the precessing orbits around this black hole. Our results show that both the scalar and electric charges suppress the energy as well as the angular momentum of the particles in the bound orbits. Then, we study the relativistic periastron precessions of the particles and constrain the charge parameters by employing the observational result of the S2 star’s precession in SgrA*. It is found that the constraints on the charge parameters from S2 star’s motion are tighter than those from black hole shadow. Finally, we analyze the periodic motions of the particles and figure out samples of periodic orbits’ configurations around the hairy black hole.

Dark matter reconstruction from stellar orbits in the Galactic centre

Context. Current constraints on distributed matter in the innermost Galactic centre (such as a cluster of faint stars and stellar remnants, dark matter, or a combination thereof) based on the orbital dynamics of the visible stars closest to the central black hole typically assume simple functional forms for the distributions. Aims. We aim to take a general model-agnostic approach in which the form of the distribution is not constrained by prior assumptions on the physical composition of the matter. This approach yields unbiased, entirely observation-driven fits for the matter distribution and places constraints on our ability to discriminate between different density profiles (and consequently between physical compositions) of the distributed matter. Methods. We constructed a spherical shell model with the flexibility to fit a wide variety of physically reasonable density profiles by modelling the distribution as a series of concentric mass shells. We tested this approach in an analysis of mock observations of the star S2. Results. For a sufficiently large and precise data set, we find that it is possible to discriminate among several physically motivated density profiles. However, for data coming from current and expected next generation observational instruments, the potential for profile distinction will remain limited by the precision of the instruments. Future observations will still be able to constrain the overall enclosed distributed mass within the apocentre of the probing orbit in an unbiased manner. We interpret this in the theoretical context of constraining the secular versus non-secular orbital dynamics. Conclusions. Our results show that while stellar data over multiple orbits of currently known stars will eventually yield model-agnostic constraints for the overall amount of distributed matter within the probe’s apocentre in the innermost Galactic centre, an unbiased model distinction made by determining the radial density profile of the distribution is, in principle, out of the measurement accuracy of the current and next-generation instruments. Constraints on dark matter models will therefore remain subject to model assumptions and will not be able to significantly downsize the zoo of candidate models.

Precessing and periodic timelike orbits and their potential applications in Einsteinian cubic gravity

Einsteinian cubic gravity (ECG) is the most general theory up to cubic order in curvature, which has the same graviton spectrum as the Einstein theory. In this paper, we investigate the geodesic motions of timelike particles around the four dimensional asymptotically flat black holes in ECG, and discuss their potential applications when connecting them with recent observational results. We first explore the effects of the cubic couplings on the marginally bound orbits (MBO), innermost stable circular orbits (ISCO) and on the periodic orbits around the Einsteinian cubic black hole. We find that comparing to Schwarzschild black hole in general relativity, the cubic coupling enhances the energy as well as the angular momentum for all the bound orbits of the particles. Then, we derive the relativistic periastron precessions of the particles and give a preliminary bound on the cubic coupling employing the observational result of the S2 star’s pericenter precession in SgrA*. Finally, after calculating the periodic orbits’ configurations, we preliminarily evaluate the gravitational waveform radiated from several periodic orbits in one complete period of a test object which orbits a supermassive Einsteinian cubic black hole. Our studies could be helpful for us to better understand the gravitational structure of the theory with high curvatures.

Non-Local Cosmology: From Theory to Observations

We examine the key aspects of gravitational theories that incorporate non-local terms, particularly in the context of cosmology and spherical symmetry. We thus explore various extensions of General Relativity, including non-local effects in the action through the function F(R,□−1R), where R denotes the Ricci curvature scalar and the operator □−1 introduces non-locality. By selecting the functional forms using Noether Symmetries, we identify exact solutions within a cosmological framework. We can thus reduce the dynamics of these chosen models and obtain analytical solutions for the equations of motion. Therefore, we study the capability of the selected models in matching cosmological observations by evaluating the phase space and the fixed points; this allows one to further constrain the non-local model selected by symmetry considerations. Furthermore, we also investigate gravitational non-local effects on astrophysical scales. In this context, we seek symmetries within the framework of f(R,□−1R) gravity and place constraints on the free parameters. Specifically, we analyze the impact of non-locality on the orbits of the S2 star orbiting SgrA*.

Using the motion of S2 to constrain vector clouds around Sgr A*

ABSTRACT The dark compact object at the centre of the Milky Way is well established to be a supermassive black hole with mass $M_{\bullet } \sim 4.3 \times 10^6 \, {\rm M}_{\odot }$, but the nature of its environment is still under debate. In this work, we used astrometric and spectroscopic measurements of the motion of the star S2, one of the closest stars to the massive black hole, to determine an upper limit on an extended mass composed of a massive vector field around Sagittarius A*. For a vector with effective mass $10^{-19} \lesssim m_\mathrm{ s} \lesssim 10^{-18} \, \rm eV$, our Markov chain Monte Carlo analysis shows no evidence for such a cloud, placing an upper bound $M_{\rm cloud} \lesssim 0.1 \% \, M_{\bullet }$ at 3σ confidence level. We show that dynamical friction exerted by the medium on S2 motion plays no role in the analysis performed in this and previous works, and can be neglected thus.

Observational tests of asymptotically flat R2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathcal {R}}}^{2}$$\\end{document} spacetimes

A novel class of Buchdahl-inspired metrics with closed-form expressions was recently obtained based on Buchdahl’s seminal work on searching for static, spherically symmetric metrics in R2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathcal {R}}}^{2}$$\\end{document} gravity in vacuo. Buchdahl-inspired spacetimes provide an interesting framework for testing predictions of R2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathcal {R}}}^{2}$$\\end{document} gravity models against observations. To test these Buchdahl-inspired spacetimes, we consider observational constraints imposed on the deviation parameter, which characterizes the deviation of the asymptotically flat Buchdahl-inspired metric from the Schwarzschild spacetime. We utilize several recent solar system experiments and observations of the S2 star in the galactic center and the black hole shadow. By calculating the effects of Buchdahl-inspired spacetimes on astronomical observations both within and outside of the solar system, including the deflection angle of light by the Sun, gravitational time delay, perihelion advance, shadow, and geodetic precession, we determine observational constraints on the corresponding deviation parameters by comparing theoretical predictions with the most recent observations. Among these constraints, we find that the tightest one comes from the Cassini mission’s measurement of gravitational time delay.

A Proposal for Enhancing Technosignature Search toward the Galactic Center

We discuss the possibility of enhancing intelligent life searches toward the Galactic center. From the clockwork orbital motions of stars around the Sgr A* black hole, we can determine the distance to the Galactic center at an exceptional accuracy, despite its remoteness ∼8.3 kpc. In addition, we can define precise reference epochs by selecting a prominent object such as the bright B-type star S2. These properties have a particular affinity for the coordinated signaling scheme that was hypothesized by Seto for systematically connecting intentional senders to searchers without a prior communication. If S2 is actually being used as a common reference clock, we can compress the search directions around the Galactic center by more than 2 orders of magnitude, with the scanning interval of ∼16 yr.

Improvement of graviton mass constraints using GRAVITY’s detection of Schwarzschild precession in the orbit of S2 star around the Galactic Center

Improvement of graviton mass constraints using GRAVITY’s detection of Schwarzschild precession in the orbit of S2 star around the Galactic Center

Relativistic orbits of S2 star in the presence of scalar field

The general theory of relativity predicts the relativistic effect in the orbital motions of S-stars which are orbiting around our Milky-way Galactic Center. The post-Newtonian or higher-order approximated Schwarzschild black hole models have been used by GRAVITY and UCLA Galactic Center groups to carefully investigate the S2 star’s periastron precession. In this paper, we investigate the scalar field effect on the orbital dynamics of S2 star. Hence, we consider a spacetime, namely Janis-Newman-Winicour (JNW) spacetime which is seeded by a minimally coupled, mass-less scalar field. The novel feature of this spacetime is that one can retain the Schwarzschild spacetime from JNW spacetime considering zero scalar charge. We constrain the scalar charge of JNW spacetime by best fitting the astrometric data of S2 star using the Monte-Carlo–Markov-Chain (MCMC) technique assuming the charge to be positive. Our best-fitted result implies that similar to the Schwarzschild black hole spacetime, the JNW naked singularity spacetime with an appropriate scalar charge also offers a satisfactory fitting to the observed data for S2 star. Therefore, the JNW naked singularity could be a contender for explaining the nature of Sgr A* through the orbital motions of the S2 star.

On the Parameters of the Spherically Symmetric Parameterized Rezzolla–Zhidenko Spacetime through Solar System Tests, the Orbit of the S2 Star about Sgr A*, and Quasiperiodic Oscillations

In this paper, we find the higher-order expansion parameters α and λ of spherically symmetric parameterized Rezzolla–Zhidenko (PRZ) spacetime by using its functions of the radial coordinate. We subject the parameters of this spacetime to classical tests, including weak gravitational field effects in the solar system, observations of the S2 star that is located in the star cluster close to the Sgr A⋆, and of the frequencies of selected microquasars. Based on this spherically symmetric spacetime, we perform the analytic calculations for solar system effects such as perihelion shift, light deflection, and gravitational time delay to determine limits on the parameters by using observational data. We restrict our attention to the limits on the two higher-order expansion parameters α and λ that survive at the horizon or near the horizon of spherically symmetric metrics. The properties of the expansion of these two small parameters in PRZ parameterization are discussed. We further apply Markov Chain Monte Carlo simulations to analyze and obtain the limits on the expansion parameters by using observations of the phenomena of the S2 star. Finally, we consider the epicyclic motions and derive analytic expressions of the epicyclic frequencies. Applying these expressions to the quasiperiodic oscillations of selected microquasars allows us to set further limits on the parameters of the PRZ spacetime. Our results demonstrate that the higher-order expansion parameters can be given in the range α, λ = (−0.09, 0.09) and of order ∼10−2 as a consequence of three different tests and observations.

The q-metric naked singularity: a viable explanation for the nature of the central object in the Milky Way

Abstract In this work, we investigate whether the compact object at the center of the Milky Way is a naked singularity described by the q-metric spacetime. Our fitting of the astrometric and spectroscopic data for the S2 star implies that similarly to the Schwarzschild black hole, the q-metric naked singularity offers a satisfactory fit to the observed measurements. Additionally, it is shown that the shadow produced by the naked singularity is consistent with the shadow observed by the Event Horizon Telescope collaboration for Sgr-A*. It is worth mentioning that the spatial distribution of the S-stars favors the notion that the compact object at the center of our Galaxy can be described by an almost static spacetime. Based on these findings, the q-metric naked singularity turns up as a compelling candidate for further investigation.

Trajectories of Bright Stars and Shadows around Supermassive Black Holes as Tests of Gravity Theories

General relativity (GR), created more than a century ago, has been checked in various experimental and observational tests. At an early stage of its development, GR predictions were tested in problems where the gravitational field is weak and relativistic corrections can be considered as small perturbations of the Newtonian theory of gravity. However, in recent years due to the progress of new technologies it turned out to be possible to verify the predictions of GR in the limit of a strong gravitational field, as it was done to verify predictions about the profile of the X-ray line of iron Kalpha , estimates of the gravitational wave signal during the mergers of binary black holes and/or neutron stars and during the reconstruction of the shadows of black holes in Sgr A* and M87*. Groups of astronomers using the Keck and VLT (GRAVITY) telescopes confirmed the GR predictions for the redshift of the spectral lines of the S2 star near the passage of its pericenter (these predictions were done in the first post-Newtonian approximation). It is expected that in the near future, observations of bright stars using large telescopes VLT (GRAVITY), Keck, E-ELT and TMT will allow us to verify the predictions of GR in the strong gravitational field of supermassive black holes. Observations of bright stars in the vicinity of the Galactic Center and reconstructions of the shadows of black holes allow not only to verify the predictions of the GR, but also to obtain restrictions on alternative theories of gravity.

Constraining an Einstein-Maxwell-dilaton-axion black hole at the Galactic Center with the orbit of the S2 star

We derive new constraints on the dilaton parameter appearing in the spherically-symmetric black hole solution of Einstein-Maxwell-dilaton-axion gravity, by studying the geodesic motion of the S2 star in the Galactic Center. Einstein-Maxwell-dilaton-axion black holes represent a compelling alternative to the standard black hole paradigm in General Relativity. This theory emerges from the low energy effective action of the heterotic string theory and has been proven to predict peculiar observational features from the direct imaging of black hole shadows. At a fundamental level, Einstein-Maxwell-dilaton-axion includes additional electromagnetic, dilatonic and axionic fields coupled to the space-time metric. When considering charged non-rotating black hole solutions, the additional fields endow the metric with one extra parameter b, called dilaton parameter, that is theoretically bound to 0 < b < M. Using publicly available astrometric data for S2 we derive an upper bound on b ≲ 12M at 95% confidence level and we demonstrate that only including the measurement of the relativistic orbital precession for S2 is sufficient to reduce this bound to b ≲ 1.4M at the same confidence level. Additionally, using a mock data mimicking future observations of S2 with the GRAVITY interferometer, we show that improved astrometric precision can help further narrow down the allowed dilaton parameter range to b ≲ 0.033M after monitoring the S2 orbit for one and a half period.

Nonsingular black holes from conformal symmetries

We derive the form of the metric for static, nonsingular black holes with a de Sitter core, representing a deformation of the Schwarzschild solution, by assuming that the gravitational sources describe a flow between two conformal points, at small and great distances. The resulting black-hole metric turns out to be a particular case of the Fan & Wang metric, whose parameters have been recently constrained by using the data of the S2 star orbits around the galactic center SgrA.

Orbital precession of stars in the Galactic Centre

ABSTRACT The region around the centre of our Galaxy is very dense of stars. The kinematics of inner moving stars in the Galaxy (the so-called S-stars) has been deeply studied by different research groups leading to the conclusion of the existence of a very compact object (Sgr A*, likely a supermassive black hole) responsible for their high speed. Here, we start from the observational evidence of orbital apsidal line precession for the S2 (also called S0-2) star to investigate on a theoretical side what level of quality in such regime of relatively strong gravitational field is reached in the orbit angular precession determination when using a direct orbital integration of the star motion subjected to an acceleration computed in the post-Newtonian (PN) scheme up to different orders. This approach, although approximated and limited to particle speed not exceeding ∼ 0.3c, allows the inclusion of various effects, like that of a possible spin of the central massive object. Our results show that the inclusion of PN terms above the standard 1PN term (the one corresponding to the classic Einstein–Schwarzschild estimate of pericenter advance) is compulsory to determine angular precession at sufficient level of accuracy for those penetrating stars that would allow to pick contemporary the value of the mass and of the spin of a rotating (Kerr-like) supermassive black hole (SMBH). We discuss how future observational data, together with a proper modelization, could allow the determination of both mass and spin of the SMBH of our Galaxy.