Innermost Stable Circular Orbit Research Articles

Orbital motion and quasi-periodic oscillations with periastron and Lense–Thirring precession of slowly rotating Einstein–Æther black hole

We study the orbital and oscillatory motion of test particles moving around slowly rotating first and second kinds of Einstein–Æther black holes. In relation to the black hole parameters, we find analytical solutions for the radial profiles of specific energy and specific angular momentum of the equatorial stable circular orbits. The properties of the co-rotating as well as contra-rotating innermost stable circular orbits are analyzed. We examine the radial profiles of the frequencies of latitudinal and radial harmonic oscillations as a function of the black hole mass and dimensionless coupling constants of the theory. The key features of quasi-periodic oscillations of test particles near the stable circular orbits in an equatorial plane of the black hole are discussed. We investigate the positions of resonant radii for high-frequency quasi-periodic oscillations models, namely epicyclic resonance and its variants, relativistic precession and its variants, tidal disruption, as well as warped disc models, considered in the background of slowly rotating first and second kinds of slowly rotating Einstein–Æther black holes. Furthermore, Periastron and Lense–Thirring precessions have been discussed. We demonstrate that the dimensionless coupling parameters of the theory have a strong influence on particle motion around Einstein–Æther black holes.

Quasiperiodic oscillations around hairy black holes in Horndeski gravity

Testing gravity theories and their parameters using observations is an important issue in relativistic astrophysics. In this context, we investigate the motion of test particles and their harmonic oscillations in the spacetime of non-rotating hairy black holes (BHs) in Hordeski gravity, together with astrophysical applications of quasiperiodic oscillations (QPOs). We show possible values of upper and lower frequencies of twin-peak QPOs which may occur in the orbits from innermost stable circular orbits to infinity for various values of the Horndeski parameter q in relativistic precession, warped disk models, and three different sub-models of the epicyclic resonant model. We also study the behaviour of the QPO orbits and their position relative to innermost stable circular orbits (ISCOs) with respect to different values of the parameter q. It is obtained that at a critical value of the Horndeski parameter ISCO radius takes 6M which has been in the pure Schwarzschild case. Finally, we obtain mass constraints of the central BH of microquasars GRS 1915+105 and XTE 1550-564 at the GR limit and the possible value of the Horndeski parameter in the frame of the above-mentioned QPO models. The analysis of orbits of twin peak QPOs with the ratio of upper and lower frequencies 3:2, around the BHs in the frame of relativistic precession (RP) and epicyclic resonance (ER4) QPO models have shown that the orbits locate close to the ISCO. It is obtained that the distance between QPO orbits and ISCO is less than the error of the observations.

High-density Reflection Spectroscopy of Black Hole X-Ray Binaries in the Hard State

We present a high-density relativistic reflection analysis of 21 spectra of six black hole X-ray binaries in the hard state with data from NuSTAR and Swift. We find that 76% of the observations in our sample require a disk density higher than the 1015 cm−3 assumed in the previous reflection analysis. Compared with the measurements from active galactic nuclei, stellar mass black holes have higher disk densities. Our fits indicate that the inner disk radius is close to the innermost stable circular orbit in the luminous hard state. The coronal temperatures are significantly lower than the prediction of a purely thermal plasma, which can be explained with a hybrid plasma model. If the disk density is fixed at 1015 cm−3, the disk ionization parameter is overestimated while the inner disk radius is unaffected.

Gravitational synchrotron radiation and Penrose process in STVG theory

The paper has explored analogue of gravitational synchrotron massive particle and Penrose process in MOdified Gravity (MOG) known as Scalar-Tensor-Vector-Gravity (STVG). Investigation of the gravitational field around Kerr-MOG black hole showed that it has strong gravitational field with large horizon and can rotate faster than Kerr black hole due to the effect of STVG. We have studied influence of STVG in circular motion of massive particle around Kerr-MOG black hole and discussed the Innermost Stable Circular Orbit (ISCO) of massive test particle. It is shown that STVG plays a crucial role in energy extraction from a rotating black hole, with an energy efficiency of more than 100% according to the Penrose process. Furthermore, we have explored the gravitational synchrotron radiation analogue produced by a massive particle orbiting around a Kerr-MOG black hole. It has been shown that the intensity of gravitational radiation from binary systems of stellar black holes (SBH) and supermassive black holes (SMBH).

Charged particles and Penrose process near charged black holes in Einstein–Maxwell-scalar theory

We study the dynamics of charged test particles around an electrically charged black hole in Einstein–Maxwell-scalar (EMS) gravity. The event horizon properties of the spacetime around the black hole are explored and the upper limit for the EMS theory parameters corresponding to extreme charge and minimal value of the event horizon are found. The effective potential for the radial motion of the charged particles at the equatorial plane is investigated. Specific energy and angular momentum of the particles corresponding to circular stable orbits are also studied. We also investigate the effects of the EMS parameter and the black hole charge on innermost stable circular orbits (ISCOs). We also investigate synchrotron radiation of charged particles in the spacetime of the charged black hole in EMS gravity. We also explore electric Penrose and Bañados–Silk–West processes near the black hole horizon, where we analyse in detail the effects of EMS parameters on energy efficiency in the Penrose process and critical angular momentum that allows colliding particles near the horizon, together with the center of mass energy in charged particles collisions.

Light ring behind wormhole throat: Geodesics, images, and shadows

The geodesics of the Ellis-Bronnikov wormhole with two parameters are studied. The asymmetric wormhole has only one light ring and one innermost stable circular orbit located on one side of the wormhole throat. Consequently, certain light rays can be reflected back by the wormhole. Additionally, the same wormhole can have different appearances on both sides of the throat. We present novel images of the wormhole with a light ring behind the throat in a scenario with an accretion disk as the light source and in a backlit wormhole scenario, which are distinct from the images of other compact objects and have the potential to be observed.

Accretion onto a static spherically symmetric regular MOG dark compact object

In astrophysics, the process of a massive body acquiring matter is referred to as accretion. The extraction of gravitational energy occurs as a result of the infall. Since it converts gravitational energy into radiation, accretion onto dark compact objects, e.g. black holes, neutron stars, and white dwarfs is an extremely significant process in the astrophysical context. Accretion process is a fruitful way to explore the features of modified gravity (MOG) theories by testing the behavior of their solutions associated with dark compact objects. In this paper, we study the motion of electrically neutral and charged particles moving in around a regular spherically symmetric MOG dark compact object to explore their related innermost stable circular orbit (ISCO) and energy flux. Then, we turn to investigate the accretion of perfect fluid onto the regular spherically symmetric MOG dark compact object. We obtain analytical expressions for four-velocity and proper energy density of the accreting fluid. We see that the MOG parameter increases the ISCO radius of either electrically neutral or charged test particles while it decreases the corresponding energy flux. Moreover, the energy density and the radial component of the four-velocity of the infalling fluid decrease by increasing the MOG parameter near the central source.

Accretion disc around black hole in Einstein-SU(N) non-linear sigma model

The accretion of matter onto celestial bodies like black holes and neutron stars is a natural phenomenon that releases up to 40% of the matter’s rest-mass energy, which is considered a source of radiation. In active galactic nuclei and X-ray binaries, huge luminosities are observed as a result of accretion. Using isothermal fluid, we examine the accretion and geodesic motion of particles in the vicinity of a spherically symmetric black hole spacetime in the Einstein-SU(N) non-linear sigma model. In the accretion process, the disk-like structure is produced by the geodesic motion of particles near the black hole. We determine the innermost stable circular orbit, energy flux, radiation temperature, and radioactive efficiency numerically. In the equatorial plane, we investigate the mobility of particles with stabilities that form circular orbits. We examine perturbations of a test particle by using restoring forces and particle oscillations in the vicinity of the black hole. We analyze the maximum accretion rate and critical flow of the fluid. Our findings demonstrate how parameter N influences the circular motion of a test particle as well as the maximum accretion rate of the black hole in the Einstein-SU(N) non-linear sigma model.

Third post-Newtonian effective-one-body Hamiltonian in scalar-tensor and Einstein-scalar-Gauss-Bonnet gravity

We build an effective-one-body (EOB) Hamiltonian at third post-Newtonian (3PN) order in scalar-tensor (ST) and Einstein-scalar-Gauss-Bonnet (ESGB) theories of gravity. The latter is an extension of general relativity that predicts scalar hair for black holes. We start from the known two-body Lagrangian at 3PN order, and use order-reduction methods to construct its ordinary Hamiltonian counterpart. We then reduce the conservative two-body dynamics to the (nongeodesic) motion of a test particle in an effective metric by means of canonical transformations. The resulting EOB Hamiltonian is a modification of the general relativistic Hamiltonian, and already at 3PN order, it must account for nonlocal-in-time tail contributions. We include the latter beyond circular orbits and up to sixth order in the binary's orbital eccentricity. We finally calculate the orbital frequency at the innermost stable circular orbit (ISCO) of binary black holes in the shift-symmetric ESGB model. Our work extends F.L. Juli\'e and N. Deruelle [Phys. Rev. D 95, 124054 (2017)], and it is an essential step toward the accurate modeling of gravitational waveforms beyond general relativity.

Properties of Spherically Symmetric Black Holes in the Generalized Brans-Dicke Modified Gravitational Theory.

The many problems faced by the theory of general relativity (GR) have always motivated us to explore the modified theory of GR. Considering the importance of studying the black hole (BH) entropy and its correction in gravity physics, we study the correction of thermodynamic entropy for a kind of spherically symmetric black hole under the generalized Brans-Dicke (GBD) theory of modified gravity. We derive and calculate the entropy and heat capacity. It is found that when the value of event horizon radius r+ is small, the effect of the entropy-correction term on the entropy is very obvious, while for larger values r+, the contribution of the correction term on entropy can be almost ignored. In addition, we can observe that as the radius of the event horizon increases, the heat capacity of BH in GBD theory will change from a negative value to a positive value, indicating that there is a phase transition in black holes. Given that studying the structure of geodesic lines is important for exploring the physical characteristics of a strong gravitational field, we also investigate the stability of particles' circular orbits in static spherically symmetric BHs within the framework of GBD theory. Concretely, we analyze the dependence of the innermost stable circular orbit on model parameters. In addition, the geodesic deviation equation is also applied to investigate the stable circular orbit of particles in GBD theory. The conditions for the stability of the BH solution and the limited range of radial coordinates required to achieve stable circular orbit motion are given. Finally, we show the locations of stable circular orbits, and obtain the angular velocity, specific energy, and angular momentum of the particles which move in circular orbits.

Probing Hořava-Lifshitz gravity using particle and photon dynamics in the presence of plasma* *This study is partly supported by Grants F-FA-2021-432, F-FA-2021-510, and MRB-2021-527 of the Ministry of Higher Education, Science and Innovations of the Republic of Uzbekistan

We study the particle motion around a black hole (BH) in Hořava-Lifshitz (HL) gravity with the Kehagias-Sfetsos (KS) parameter. First, the innermost stable circular orbit (ISCO) is obtained for massive particles around the BH in HL gravity. We find that the radii of the ISCOs decrease as the KS parameter decreases, meaning that the parameter causes the orbits of particles to move inward with respect to that of the Schwarzschild BH case. Then, the optical properties of a KS BH are studied in detail, that is, the BH shadow and gravitational weak lensing. We demonstrate that the size of the BH shadow decreases under the influence of the KS parameter.

Slowly rotating black holes in quartic generalized quasi-topological gravity

We study slowly rotating black hole solutions in the six independent theories of Quartic Generalized Quasi-topological Gravity in four dimensions. Unlike in the static case for which all six theories yield the same solution, for rotating black holes we obtain distinct results for five out of the six theories. Working to leading order in the rotation parameter, we find that the equations characterizing these black holes can be reduced to second order for each theory, similar to what has already been done for Einstein Cubic Gravity. We construct approximate and numerical solutions to these equations, and study how physical properties of the solutions such as the angular velocity, photon sphere, black hole shadow, and innermost stable circular orbit are modified, working to leading order in the coupling constant.

Stability and physical properties of spherical excited scalar boson stars

We study the time evolution of spherical, excited -- with $n$ radial nodes -- scalar boson stars in General Relativity minimally coupled to a complex massive scalar field with quartic self-interactions. We report that these stars, with up to $n=10$, can be made dynamically stable, up to timescales of $t\sim\frac{10^{4}}{c\mu}$, where $\mu$ is the inverse Compton wavelength of the scalar particle, for sufficiently large values of the self-interactions coupling constant $\lambda$, which depend on $n$. We observe that the compactness of these solutions is rather insensitive to $n$, for large $\lambda$ and fixed frequency. Generically, along the branches where stability was studied, these excited boson stars are not compact enough to allow for innermost stable circular orbits or light rings. Finally, we discuss the angular velocity of particles along timelike circular orbits, suggesting an application, for solutions in the Newtonian limit, to galactic rotation curves.

Effective-one-body Hamiltonian in scalar-tensor gravity at third post-Newtonian order

We determine the general local-in-time effective-one-body (EOB) Hamiltonian for massless Scalar-Tensor (ST) theories at third post-Newtonian (PN) order. Starting from the Lagrangian derived in [Phys. Rev. D 99, 044047 (2019)], we map it to the corresponding ordinary Hamiltonian describing the two-body interaction in ST theories at 3PN level. Using a canonical transformation, we then map this onto an EOB Hamiltonian so as to determine the ST corrections to the 3PN-accurate EOB potentials $(A,B,Q_e)$ at 3PN. We then focus on circular orbits and compare the effect of the newly computed 3PN terms, also completed with finite-size and nonlocal-in-time contributions, on predictions for the frequency at the innermost stable circular orbit. Our results will be useful to build high-accuracy waveform models in ST theory, which could be used to perform precise tests against General Relativity using gravitational wave data from coalescing compact binaries.

Particle motion around Schwarzschild-MOG black hole

In this paper, we present an analysis of the circular motion of test particles around a Schwarzschild-MOG black hole. Initially, our focus lies on studying the shadow cast by the spherically symmetric black hole within the framework of MOG gravity. Notably, we observe that the presence of MOG influences both the photon-sphere and the black hole’s shadow, causing them to increase in size. Furthermore, our research reveals that the characteristic radii of massive particles in circular orbits around the Schwarzschild-MOG black hole, specifically the innermost stable circular orbits (ISCO) and marginally bound orbits, are greater than those observed in the Schwarzschild metric alone. Additionally, we examine the electromagnetic field structure when a black hole is subjected to an external uniform magnetic field. Our findings demonstrate that in the vicinity of the Schwarzschild-MOG black hole, the magnetic field exhibits non-uniform behavior, with field lines becoming more densely distributed. Lastly, we delve into the motion of charged particles around the Schwarzschild-MOG black hole in the presence of an external magnetic field. Our investigation highlights that the ISCO position for charged particles is consistently less than that for neutral particles, indicating a significant distinction between the two scenarios.

Tidal deformations of a binary system induced by an external Kerr black hole

The dynamics of a binary system moving in the background of a black hole is affected by tidal forces. In this work, for the Kerr black hole, we derive the electric and magnetic tidal moments at quadrupole order, where the latter are computed for the first time in full generality. We make use of these moments in the scenario of a hierarchical triple system made of a Kerr black hole and an extreme-mass ratio binary system consisting of a Schwarzschild black hole and a test particle. We study how the secular dynamics of the test particle in the binary system is distorted by the presence of tidal forces from a much larger Kerr black hole. Our treatment includes strong gravitational effects beyond the post-Newtonian approximation both for the binary system and for the tidal forces since the binary system is allowed to be close to the event horizon of the Kerr black hole. We compute the shifts in the physical quantities for the secular dynamics of the test particle and show that they are gauge invariant. In particular, we apply our formalism to the innermost stable circular orbit for the test particle and to the case of the photon sphere. Our results are relevant for the astrophysical situation in which the binary system is in the vicinity of a supermassive black hole.

Particle motions around regular black holes

We investigate the bound orbits of massive/massless, neutral particles and photons moving around regular black holes of Fan and Wang. For massive particles, we show the existence of stable/unstable circular orbits and the charge dependence of the radius of the innermost stable circular orbit. Remarkably, we find an unstable circular orbit of photons inside the event horizon. For massless particles and photons, we show that both stable and unstable circular orbits can exist in a regular and horizonless spacetime with a slight overcharge. Then, we also discuss the periapsis shift of massive neutral particles orbiting around the black hole, and show that the charge gives a negative correction to the shift for black holes with small nonlinearity of electrodynamics.

Compact dark objects in neutron star mergers

We estimate the long-lasting gravitational wave (GW) emission of compact dark objects following a binary neutron-star (NS) merger. We consider compact dark objects, which initially reside in the centers of NSs and which may consist of self-interacting dark matter (DM). By approximating the compact dark objects as test particles, we model the merging of NS binaries hosting DM components with three-dimensional relativistic simulations. Our simulation results suggest that the DM components remain gravitationally bound and orbit inside the merger remnant with orbital separations of typically a few km. The subsequent orbital motion of the DM components generates a GW signal with frequencies in the range of a few kHz. When considering a range of different binary masses and high-density equations of state (EOS), we find that the GW frequency of the orbiting DM components scales with the compactness of NSs. Similarly, we find relations between the DM GW frequency and the dominant postmerger GW frequency of the stellar fluid or the tidal deformability, which quantifies EOS effects during the binary inspiral. Hence, a measurement of these quantities can be used to specify the frequency range of the GW emission by DM. Under the assumption that GW back reaction is the only relevant dissipative process, the GW signal may last between seconds and years depending on the mass of the DM component. We estimate the detectibility of the GW signals and find that DM components in NS mergers may only be detectable with existing and projected GW instruments if the dark objects are as massive as about 0.01 to $0.1{M}_{\ensuremath{\bigodot}}$. We emphasize that the GW emission is limited by the lifetime of the remnant. A forming black hole will immediately swallow the DM objects because their orbits are smaller than the innermost stable circular orbit of the black hole.

The Spin of a Newborn Black Hole: Swift J1728.9-3613

The origin and distribution of stellar-mass black hole spins are a rare window into the progenitor stars and supernova events that create them. Swift J1728.9-3613 is an X-ray binary, likely associated with the supernova remnant (SNR) G351.9-0.9. An NuSTAR X-ray spectrum of this source during its 2019 outburst reveals reflection from an accretion disk extending to the innermost stable circular orbit. Modeling of the relativistic Doppler shifts and gravitational redshifts imprinted on the spectrum measures a dimensionless spin parameter of a = 0.86 ± 0.02 (1σ confidence), a small inclination angle of the inner accretion disk θ < 10°, and a subsolar iron abundance in the disk A Fe < 0.84. This high spin value rules out a neutron star primary at the 5σ level of confidence. If the black hole is located in a still visible SNR, it must be young. Therefore, we place a lower limit on the natal black hole spin of a > 0.82, concluding that the black hole must have formed with a high spin. This demonstrates that black hole formation channels that leave an SNR, and those that do not (e.g., Cyg X-1), can both lead to high natal spin with no requirement for subsequent accretion within the binary system. Emerging disparities between the population of high-spin black holes in X-ray binaries and the low-spin black holes that merge in gravitational wave events may therefore be explained in terms of different stellar conditions prior to collapse, rather than different environmental factors after formation.

Precession and Lense-Thirring effect of hairy Kerr spacetimes

We investigate the frame-dragging effect of the hairy Kerr spacetimes on the spin of a test gyro and accretion disk. First, we analyze Lense-Thirring (LT) precession frequency, geodetic precession frequency, and the general spin precession frequency of a test gyro attached to a stationary observer in the spacetime. We find that the black hole hair suppresses those precession frequencies in comparison with that which occurs in Kerr spacetime in general relativity. Moreover, using those frequencies as a probe, we differentiate the hairy Kerr black hole (BH) from naked singularity (NS). Specifically, as the observer approaches the central source along any direction, the frequencies grow sharply for the hairy Kerr BH, while for the hairy NS they are finite except at the ring singularity. Then, we investigate the quasiperiodic oscillations (QPOs) phenomena as the accretion disk approaches the hairy Kerr BH or NS. To this end, we analyze the bound circular orbits and their perturbations. We find that as the orbits approach the corresponding innermost stable circular orbit (ISCO), both LT precession frequency and periastron precession frequency behave differently in the hairy Kerr BH and NS. Additionally, the hairy parameters have significant effects on the two frequencies. We expect that our theoretical studies could shed light on astrophysical observations in distinguishing hairy theories from Einstein's gravity, and also in distinguishing BH from NS in spacetime with hair.