Relativistic Orbits Research Articles

Tidal heating as a discriminator for horizons in equatorial eccentric extreme mass ratio inspirals

Tidal heating in a binary black hole system is driven by the absorption of energy and angular momentum by the black hole’s horizon. Previous works have shown that this phenomenon becomes particularly significant during the late stages of an extreme mass ratio inspiral (EMRI) into a rapidly spinning massive black hole, a key focus for future low-frequency gravitational-wave observations by (for instance) the Laser Interferometer Space Antenna mission. Past analyses have largely focused on quasicircular inspiral geometry, with some of the most detailed studies looking at equatorial cases. Though useful for illustrating the physical principles, this limit is not very realistic astrophysically, since the population of EMRI events is expected to arise from compact objects scattered onto relativistic orbits in galactic centers through many-body events. In this work, we extend those results by studying the importance of tidal heating in equatorial EMRIs with generic eccentricities. Our results suggest that accurate modeling of tidal heating is crucial to prevent significant dephasing and systematic errors in EMRI parameter estimation. We examine a phenomenological model for EMRIs around exotic compact objects by parametrizing deviations from the black hole (BH) picture in terms of the fraction of radiation absorbed compared to the BH case. Based on a mismatch calculation, we find that reflectivities as small as |R|2∼O(10−5) are distinguishable from the BH case, irrespective of the value of the eccentricity. We stress, however, that this finding should be corroborated by future parameter estimation studies. Published by the American Physical Society 2024

Note on black holes with kilometer-scale ultraviolet regulators

Regular black hole metrics involve a universal, mass-independent regulator that can be up to O(700km) while remaining consistent with terrestrial tests of Newtonian gravity and astrophysical tests of general relativistic orbits. However, for such large values of the regulator scale, the metric describes a compact, astrophysical-mass object with no horizon rather than a black hole. We note that allowing the regulator to have a nontrivial mass dependence preserves the horizon, while allowing large, percent-level effects in black hole observables. By considering the deflection angle of light and the black hole shadow, we demonstrate this possibility explicitly.

Short and long time-scales variability of the H α double-peaked profile of Pictor A

ABSTRACT We present the results of the analysis of 18 optical spectra covering the H α profile region of the active nucleus of the galaxy Pictor A obtained in different epochs along 12 yr. With these data, we model the variability of the double-peaked H α emission of the nucleus of this galaxy. We find small variations in the integrated flux of the broad component of H α, occurring on short time-scales, of the order of 40 d. In addition, significant variations in the relative intensities of the red and blue sides of the broad profile occur on time-scales of years. In order to explain the observations, we propose a scenario in which the double-peak profile is produced by a flattened distribution of clouds surrounding the accretion disc with Keplerian and relativistic orbits, inclined in relation to the sky plane, and with axially asymmetric surface emissivity with the radius of maximum emissivity. We find that, for the theoretical time-scales of the disc-like BLR to conform the time-scales of variation observed for the double-peak profile, it is necessary for the gas in the disc to orbit a SMBH with a mass of M• $\approx \, 2\times$ $10^{8}\, \mbox{M}_\odot$. In this scenario, the physical extension of the line-emitting region extends by 6–48 light-days from the SMBH.

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.

An analytical, fully relativistic framework for tidal disruption event streams in Schwarzschild geometry

ABSTRACT We present an analytical and fully relativistic framework for studying the self-intersection of tidal disruption event (TDE) streams, restricting ourselves to the Schwarzschild spacetime. By taking advantage of the closed-form solution to the geodesic equations in the Schwarzschild metric, we calculate properties of the self-intersection without numerically evaluating the geodesic equations or making any post-Newtonian approximations. Our analytical treatment also facilitates geometric definitions of the orbital semimajor axis and eccentricity, as opposed to Newtonian formulas which lead to unphysical results for highly relativistic orbits. Combined with assumptions about energy dissipation during the self-intersection shock, our framework enables the calculation of quantities such as the fraction of material unbound during the self-intersection shock, and the characteristic semimajor axes and eccentricities of the material that remains in orbit after the collision. As an example, we calculate grids of post-intersection properties in stellar and supermassive black hole (SMBH) masses for disruptions of main-sequence stars, identifying regions where no material is ejected during self-intersection (e.g. SMBH mass $\lesssim 5\times 10^6\, {\rm M_\odot }$ for $1\, {\rm M_\odot }$ stars disrupted at the tidal radius), potentially explaining the TDEs observed by SGR/eROSITA that are visible in X-rays but not optical wavelengths. We also identify parameters for which the post-intersection accretion flow has low eccentricity (e ≲ 0.6), and find that the luminosity generated by self-intersection shocks only agrees with observed trends in the relationship between light curve decay time-scales and peak luminosities over a narrow range of SMBH masses.

Effective two-body approach to the hierarchical three-body problem

The motion of three bodies can be solved perturbatively when a tightly bound inner binary is orbited by a distant perturber, giving rise for example to the well-known Kozai-Lidov oscillations. We propose to study the relativistic hierarchical three-body orbits by adapting the Effective Field Theory techniques used in the two-body problem. This allows us to conveniently treat the inner binary as an effective point-particle, thus reducing the complexity of the three-body problem to a simpler spinning two-body motion. We present in details the mapping between the inner binary osculating elements and the resulting spin of the effective point-particle. Our study builds towards a derivation of three-body analytic waveforms.

Improved gravitational radiation time-scales II: Spin–orbit contributions and environmental perturbations

ABSTRACT Peters’ formula is an analytical estimate of the time-scale of gravitational wave (GW)-induced coalescence of binary systems. It is used in countless applications, where the convenience of a simple formula outweighs the need for precision. However, many promising sources of the Laser Interferometer Space Antenna (LISA), such as supermassive black hole binaries and extreme mass-ratio inspirals (EMRIs), are expected to enter the LISA band with highly eccentric (e ≳ 0.9) and highly relativistic orbits. These are exactly the two limits in which Peters’ estimate performs the worst. In this work, we expand upon previous results and give simple analytical fits to quantify how the inspiral time-scale is affected by the relative 1.5 post-Newtonian (PN) hereditary fluxes and spin–orbit couplings. We discuss several cases that demand a more accurate GW time-scale. We show how this can have a major influence on quantities that are relevant for LISA event-rate estimates, such as the EMRI critical semimajor axis. We further discuss two types of environmental perturbations that can play a role in the inspiral phase: the gravitational interaction with a third massive body and the energy loss due to dynamical friction and torques from a surrounding gas medium ubiquitous in galactic nuclei. With the aid of PN corrections to the time-scale in vacuum, we find simple analytical expressions for the regions of phase space in which environmental perturbations are of comparable strength to the effects of any particular PN order, being able to qualitatively reproduce the results of much more sophisticated analyses.

Constraining the dense matter equation-of-state with radio pulsars

ABSTRACT Radio pulsars provide some of the most important constraints for our understanding of matter at supranuclear densities. So far, these constraints are mostly given by precision mass measurements of neutron stars (NS). By combining single measurements of the two most massive pulsars, J0348+0432 and J0740+6620, the resulting lower limit of 1.98 M⊙ (99 per cent confidence) of the maximum NS mass, excludes a large number of equations of state (EOSs). Further EOS constraints, complementary to other methods, are likely to come from the measurement of the moment of inertia (MOI) of binary pulsars in relativistic orbits. The Double Pulsar, PSR J0737−3039A/B, is the most promising system for the first measurement of the MOI via pulsar timing. Reviewing this method, based in particular on the first MeerKAT observations of the Double Pulsar, we provide well-founded projections into the future by simulating timing observations with MeerKAT and the SKA. For the first time, we account for the spin-down mass-loss in the analysis. Our results suggest that an MOI measurement with 11 per cent accuracy (68 per cent confidence) is possible by 2030. If by 2030 the EOS is sufficiently well known, however, we find that the Double Pulsar will allow for a 7 per cent test of Lense–Thirring precession, or alternatively provide a ∼3σ-measurement of the next-to-leading order gravitational wave damping in GR. Finally, we demonstrate that potential new discoveries of double NS systems with orbital periods shorter than that of the Double Pulsar promise significant improvements in these measurements and the constraints on NS matter.

Pulsar timing array signals induced by black hole binaries in relativistic eccentric orbits

Individual supermassive black hole binaries in non-circular orbits are possible nanohertz gravitational wave sources for the rapidly maturing Pulsar Timing Array experiments. We develop an accurate and efficient approach to compute Pulsar Timing Array signals due to gravitational waves from inspiraling supermassive black hole binaries in relativistic eccentric orbits. Our approach employs a Keplerian-type parametric solution to model third post-Newtonian accurate precessing eccentric orbits while a novel semi-analytic prescription is provided to model the effects of quadrupolar order gravitational wave emission. These inputs lead to a semi-analytic prescription to model such signals, induced by non-spinning black hole binaries inspiralling along arbitrary eccentricity orbits. Additionally, we provide a fully analytic prescription to model Pulsar Timing Array signals from black hole binaries inspiraling along moderately eccentric orbits, influenced by Boetzelet al.[Phys. Rev. D 96,044011(2017)]. These approaches are being incorporated into Enterprise and TEMPO2 for searching the presence of such binaries in Pulsar Timing Array datasets.

Phase-plane analysis of test particle orbits in regular black holes

We investigate phase-plane analysis of general relativistic orbits in a gravitational field of the Reissner–Nordström-type regular black hole spacetime. We employ phase-plane analysis to obtain different phase-plane diagrams of the test particle orbits by varying charge q and dimensionless parameter β, where β contains angular momentum of the test particle. We compute numerical values of radii for the innermost stable orbits and corresponding values of energy required to place the test particle in orbits. Later on, we employ similar analysis on an Ayón–Beato–García (ABG) regular black hole and a comparison regarding key results is also included.

Perihelion precession in binary systems: higher order corrections

Higher order corrections (up to n-th order) are obtained for the perihelion precession in binary systems like OJ287 using the Schwarzschild metric and complex integration. The corrections are performed considering the third root of the motion equation and developing the expansion in terms of $r_{s}/ (a(1-e^{2}) )$ .The results are compared with other expansions that appear in the literature giving corrections to second and third order. Finally, we simulate the shape of relativistic orbits for binary systems with different masses.

Lagrangian vs Hamiltonian: The best approach to relativistic orbits

In introductory general relativity courses, free particle trajectories, such as astronomical orbits, are generally developed via a Lagrangian and variational calculus, so that physical examples can precede the mathematics of tensor calculus. The use of a Hamiltonian is viewed as more advanced and typically comes later if at all. We suggest here that this might not be the optimal order in a first course in general relativity, especially if orbits are to be solved with numerical methods. We discuss some of the issues that arise in both the Lagrangian and Hamiltonian approaches.

A stellar fly‐by close to the Galactic center: Can we detect stars on highly relativistic orbits?

The Galactic center Nuclear Star Cluster is one of the densest stellar clusters in the Galaxy. The stars in its inner portions orbit the supermassive black hole associated with the compact radio source Sgr A* at the orbital speeds of several thousand kilometers per second. The B‐type star S2 is currently the best case to test the general relativity as well as other theories of gravity, based on its stellar orbit. Yet, its orbital period of ∼16 years and the eccentricity of ∼0.88 yields the relativistic pericenter shift of ∼11′, which is observationally still difficult to reliably measure due to possible Newtonian perturbations as well as reference‐frame uncertainties. A naive way to solve this problem is to find stars with smaller pericenter distances, rp ≲ 1529 Schwarzschild radii (120 AU), and thus with more prominent relativistic effects. In this paper, we show that to detect stars on relativistic orbits is progressively less likely, given the volume shrinkage and the expected stellar density distributions. Finally, one arrives at a sparse region where the total number of bright stars is expected to fall below 1. One can, however, still potentially detect stars crossing this region. In this paper, we provide a simple formula for the detection probability of a star crossing a sparse region. We also examine an approximate timescale in which the star reappears in the sparse region, i.e., a “waiting” timescale for observers.

Relativistic satellite orbits: central body with higher zonal harmonics

Satellite orbits around a central body with arbitrary zonal harmonics are considered in a relativistic framework. Our starting point is the relativistic Celestial Mechanics based upon the first post-Newtonian approximation to Einstein’s theory of gravity as it has been formulated by Damour et al. (Phys Rev D 43:3273–3307, 1991; 45:1017–1044, 1992; 47:3124–3135, 1993; 49:618–635, 1994). Since effects of order \((\mathrm{GM}/c^2R) \times J_k\) with \(k \ge 2\) for the Earth are very small (of order \( 7 \times 10^{-10}\,\times \,J_k\)) we consider an axially symmetric body with arbitrary zonal harmonics and a static external gravitational field. In such a field the explicit \(J_k/c^2\)-terms (direct terms) in the equations of motion for the coordinate acceleration of a satellite are treated first with first-order perturbation theory. The derived perturbation theoretical results of first order have been checked by purely numerical integrations of the equations of motion. Additional terms of the same order result from the interaction of the Newtonian \(J_k\)-terms with the post-Newtonian Schwarzschild terms (relativistic terms related to the mass of the central body). These ‘mixed terms’ are treated by means of second-order perturbation theory based on the Lie-series method (Hori–Deprit method). Here we concentrate on the secular drifts of the ascending node \( \) and argument of the pericenter \( \). Finally orders of magnitude are given and discussed.

Modeling approaches for precise relativistic orbits: Analytical, Lie-series, and pN approximation

Accurate orbit modeling plays a key role in contemporary and future space missions such as GRACE and its successor GRACE-FO, GNSS, and altimetry missions. To fully exploit the technological capabilities and correctly interpret measurements, relativistic orbital effects need to be taken into account.Within the theory of General Relativity, equations of motion for freely falling test objects, such as satellites orbiting the Earth, are given by the geodesic equation. We analyze and compare different solution methods in a spherically symmetric background, i.e.for the Schwarzschild spacetime, as a test bed. We investigate satellite orbits and use direct numerical orbit integration as well as the semi-analytical Lie-series approach. The results are compared to the exact analytical reference solution in terms of elliptic functions. For a set of exemplary orbits, we determine the respective accuracy of the different methods.Within the post-Newtonian approximation of General Relativity, modified orbital equations are obtained by adding relativistic corrections to the Newtonian equations of motion. We analyze the accuracy of this approximation with respect to the general relativistic setting. Therefore, we solve the post-Newtonian equation of motion using the eXtended High Performance Satellite dynamics Simulator. For corresponding initial conditions, we compare orbits in the Schwarzschild spacetime to those in its post-Newtonian approximation. Moreover, we compare the magnitude of relativistic contributions to several typical perturbations of satellite orbits due to, e.g., solar radiation pressure, Earth’s albedo, and atmospheric drag. This comparison is done for our test scenarios and for a real GRACE orbit to highlight the importance of relativistic effects in geodetic space missions. For the considered orbits, first-order relativistic contributions give accelerations of about 20 nm/s2 and are dominant in the radial direction.

The stepwise path integral of the relativistic point particle

In this paper we present a stepwise construction of the path integral over relativistic orbits in Euclidean spacetime. It is shown that the apparent problems of this path integral, like the breakdown of the naive Chapman–Kolmogorov relation, can be solved by a careful analysis of the overcounting associated with local and global symmetries. Based on this, the direct calculation of the quantum propagator of the relativistic point particle in the path integral formulation results from a simple and purely geometric construction.

Test particles dynamics in the JOREK 3D non-linear MHD code and application to electron transport in a disruption simulation

In order to contribute to the understanding of runaway electron generation mechanisms during tokamak disruptions, a test particle tracker is introduced in the JOREK 3D non-linear MHD code, able to compute both full and guiding center relativistic orbits. Tests of the module show good conservation of the invariants of motion and consistency between full orbit and guiding center solutions. A first application is presented where test electron confinement properties are investigated in a massive gas injection-triggered disruption simulation in JET-like geometry. It is found that electron populations initialised before the thermal quench (TQ) are typically not fully deconfined in spite of the global stochasticity of the magnetic field during the TQ. The fraction of ‘survivors’ decreases from a few tens down to a few tenths of percent as the electron energy varies from 1 keV to 10 MeV. The underlying mechanism for electron ‘survival’ is the prompt reformation of closed magnetic surfaces at the plasma core and, to a smaller extent, the subsequent reappearance of a magnetic surface at the edge. It is also found that electrons are less deconfined at 10 MeV than at 1 MeV, which appears consistent with a phase averaging effect due to orbit shifts at high energy.

Upper-twin-peak quasiperiodic oscillation in x-ray binaries and the energy from tidal circularization of relativistic orbits

High frequency quasiperiodic oscillations (HF QPOs) detected in the power spectra of low mass x-ray binaries (LMXBs) could unveil the fingerprints of gravitation in strong field regime. Using the energy-momentum relation we calculate the energy a clump of plasma orbiting in the accretion disk releases during circularization of its slightly eccentric relativistic orbit. Following previous works, we highlight the strong tidal force as mechanism to dissipate such energy. We show that tides acting on the clump are able to reproduce the observed coherence of the upper HF QPO seen in LMXBs with a neutron star (NS). The quantity of energy released by the clump and relativistic boosting might give a modulation amplitude in agreement with that observed in the upper HF QPO. Both the amplitude and coherence of the upper HF QPO in NS LMXBs could allow us to disclose, for the first time, the tidal circularization of relativistic orbits occurring around a neutron star.

Highly relativistic spin-gravity- Λ coupling

The effects of highly relativistic spin-gravity coupling in the Schwarzschild-de Sitter background which follow from the Mathisson-Papapetrou equations are investigated. The dependence of gravitoelectric and gravitomagnetic components of gravitational field on the velocity of an observer which is moving in Schwarzschild-de Sitter's background is estimated. The action of gravitomagnetic components on a fast moving spinning particle is considered. Different cases of the highly relativistic circular orbits of a spinning particle which essentially differ from the corresponding geodesic orbits are described.

Investigating the Relativistic Motion of the Stars Near the Supermassive Black Hole in the Galactic Center

Abstract The S-star cluster in the Galactic center allows us to study the physics close to a supermassive black hole, including distinctive dynamical tests of general relativity. Our best estimates for the mass of and the distance to Sgr A* using the three stars with the shortest period (S2, S38, and S55/S0-102) and Newtonian models are M BH = (4.15 ± 0.13 ± 0.57) × 106 M ⊙ and R 0 = 8.19 ± 0.11 ± 0.34 kpc. Additionally, we aim at a new and practical method to investigate the relativistic orbits of stars in the gravitational field near Sgr A*. We use a first-order post-Newtonian approximation to calculate the stellar orbits with a broad range of periapse distance r p . We present a method that employs the changes in orbital elements derived from elliptical fits to different sections of the orbit. These changes are correlated with the relativistic parameter defined as ϒ ≡ r s /r p (with r s being the Schwarzschild radius) and can be used to derive ϒ from observational data. For S2 we find a value of ϒ = 0.00088 ± 0.00080, which is consistent, within the uncertainty, with the expected value of ϒ = 0.00065 derived from M BH and the orbit of S2. We argue that the derived quantity is unlikely to be dominated by perturbing influences such as noise on the derived stellar positions, field rotation, and drifts in black hole mass.