Particle In Circular Orbit Research Articles

Radiation fluxes of gravitational, electromagnetic, and scalar perturbations in type-D black holes: an exact approach

We present a novel method that solves Teukolsky equations with the source to calculate radiation fluxes at infinity and event horizon for any perturbation fields of type-D black holes. For the first time, we use the confluent Heun function to obtain the exact solutions of ingoing and outgoing waves for the Teukolsky equation. This benefits from our derivation of the asymptotic analytic expression of the confluent Heun function at infinity. It is interesting to note that these exact solutions are not subject to any constraints, such as low-frequency and weak-field. To illustrate the correctness, we apply these exact solutions to calculate the gravitational, electromagnetic, and scalar radiations emitted by a particle in circular orbits around a Schwarzschild black hole. Numerical results show that the proposed exact solution appreciably improves the computational accuracy and efficiency compared with the 23rd post-Newtonian order expansion and the Mano-Suzuki-Takasugi method.

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.

Constraints on the magnetized Ernst black hole spacetime through quasiperiodic oscillations

We study the dynamics of test particles around a magnetized Ernst black hole considering its magnetic field in the environment surrounding the black hole. We show how its magnetic field can influence the dynamics of particles and epicyclic motion around the black hole. Based on the analysis, we find that the radius of the innermost stable circular orbit (ISCO) for both neutral and charged test particles and epicyclic frequencies are strongly affected by the influence of the magnetic field. We also show that the ISCO radius of charged particles decreases rapidly. It turns out that the gravitational and Lorentz forces of the magnetic field are combined, thus strongly shrinking the values of the ISCO of charged test particles. Finally, we obtain the generic form for the epicyclic frequencies and select three microquasars with known astrophysical quasiperiodic oscillation (QPO) data to constrain the magnetic field. We show that the magnetic field is of the order of magnitude Bsim 10^{-7} Gauss, taking into account the motion of neutral particles in circular orbit about the black hole.

Stability criteria for wide binary stars harboring Oort Clouds

Context. In recent years, several numerical studies have been done in the field of the stability limit. Although, many of them included the analysis of asteroids or planets, is not possible to find in the literature any work on how the presence of a binary star could affect other possible configurations in a three-body problem. In order to develop this subject we consider other structures like Oort Clouds in wide binary systems. Regarding the existence of Oort Clouds in extrasolar systems there are recent works that do not reject its possible existence.Aim. The aim of this work is to obtain the stability limit for Oort Cloud objects considering different masses of the secondary star and zero and non-zero inclinations of the particles. We improve our numerical treatment getting a mathematical fit that allows us to find the limit and compare our results with other previous works in the field.Methods. We use a symplectic integrator to integrate binary systems where the primary star is m1 = 1 M⊙ and the secondary, m2, takes 0.25 M⊙ and 0.66 M⊙ in two sets of simulations S1 and S2. The orbital parameters of the secondary star were varied in order to study different scenarios. We also used two different integration times (one shorter than the other) and included the presence of 1000 to 10 000 massless particles in circular orbits to form the Oort Cloud. The particles were disposed in four different inclination planes to investigate how the presence of the binary companion could affect the stability limit.Results. Using the Maximum Eccentricity Method, emax, together with the critical semimajor axis acrit we found that the emax criteria could reduce the integration times to find the limit. For those cases where the particles were in inclined orbits we found that there are particle groups that survive the integration time with a high eccentricity. These particle groups are found for our two sets of simulations, meaning that they are independent of the secondary mass. We also find for the co-planar case that the numerical value of the stability limit for retrograde orbits is higher than those found for prograde orbits. These results are in agreement with several published studies. Finally, the results obtained in this work allow us to build a numerical expression depending of the mass ratio, e2 and ip to find acrit, which can be compared with other recent works in the field.

A tale of two velocities: Threading versus slicing

One of the important quantities in cosmology and astrophysics is the 3-velocity of an object. Specifically, when the gravitational fields are strong, one should require the employment of general relativity both in its definition and measurement. Looking into the literature for GR-based definitions of 3-velocity, one usually finds different ad hoc definitions applied according to the case under consideration. Here, we introduce and analyze systematically the two principal definitions of 3-velocity assigned to a test particle following the timelike trajectories in stationary spacetimes. These definitions are based on the [Formula: see text] (threading) and [Formula: see text] (slicing) spacetime decomposition formalisms and defined relative to two different sets of observers. After showing that Synge’s definition of spatial distance and 3-velocity is equivalent to those defined in the [Formula: see text] (threading) formalism, we exemplify the differences between these two definitions by calculating them for particles in circular orbits in axially symmetric stationary spacetimes. Illustrating its geometric nature, the relative linear velocity between the corresponding observers is obtained in terms of the spacetime metric components. Circular particle orbits in the Kerr spacetime, as the prototype and the most well known of stationary spacetimes, are examined with respect to these definitions to highlight their observer-dependent nature. We also examine the Kerr-NUT spacetime in which the NUT parameter, contributing to the off-diagonal terms in the metric, is mainly interpreted not as a rotation parameter but as a gravitomagnetic monopole charge. Finally, in a specific astrophysical setup which includes rotating black holes, it is shown how the local velocity of an orbiting star could be related to its spectral line shifts measured by distant observers.

Linear-in-mass-ratio contribution to spin precession and tidal invariants in Schwarzschild spacetime at very high post-Newtonian order

Using black hole perturbation theory and arbitrary-precision computer algebra, we obtain the post-Newtonian (pN) expansions of the linear-in-mass-ratio corrections to the spin-precession angle and tidal invariants for a particle in circular orbit around a Schwarzschild black hole. We extract coefficients up to 20pN order from numerical results that are calculated with an accuracy greater than 1 part in $10^{500}$. These results can be used to calibrate parameters in effective-one-body models of compact binaries, specifically the spin-orbit part of the effective Hamiltonian and the dynamically significant tidal part of the main radial potential of the effective metric. Our calculations are performed in a radiation gauge, which is known to be singular away from the particle. To overcome this irregularity, we define suitable Detweiler-Whiting singular and regular fields in this gauge, and we devise a rigorous mode-sum regularization method to compute the invariants constructed from the regular field.

Gravitational waves from a particle in circular orbits around a rotating black hole to the 11th post-Newtonian order

We compute the energy flux of the gravitational waves radiated by a particle of mass |$\mu$| in circular orbits around a rotating black hole of mass |$M$| up to the 11th post-Newtonian order (11PN), i.e. |$v^{22}$| beyond the leading Newtonian approximation where |$v$| is the orbital velocity of the particle [L. Blanchet, Living Rev. Relativity 5, 3 (2002)]. By comparing the PN results for the energy flux with high-precision numerical results in black hole perturbation theory, we find the region of validity in the PN approximation becomes larger with increasing PN order. If one requires the relative error of the energy flux in the PN approximation to be less than |$10^{-5}$|⁠, the energy flux at 11PN (4PN) can be used for |$v\lessapprox 0.33$| (⁠|$v \lessapprox 0.13$|⁠). The region of validity can be further extended to |$v\lessapprox 0.4$| if one applies a resummation method to the energy flux at 11PN. We then compare the orbital phase during a two-year inspiral from the PN results with the high-precision numerical results. We find that, for late (early) inspirals when |$q\le 0.3$| (⁠|$q\le 0.9$|⁠), where |$q$| is the dimensionless spin parameter of the black hole, the difference in the phase is less than 1 (⁠|$10^{-4}$|⁠) rad and hence these inspirals may be detected in the data analysis for space detectors such as eLISA|$/$|New Gravitational wave Observatory by the PN templates. We also compute the energy flux radiated into the event horizon for a particle in circular orbits around a non-rotating black hole at 22.5PN, i.e. |$v^{45}$| beyond the leading Newtonian approximation, which is comparable to the PN order derived in our previous work for the energy flux to infinity at 22PN.

Gravitational-wave flux for a particle orbiting a Kerr black hole to 20th post-Newtonian order: A numerical approach

In this article we present the post-Newtonian (pN) coefficients of the energy flux (and angular momentum flux) at infinity and event horizon for a particle in circular, equatorial orbits about a Kerr black hole (of mass $M$ and spin-parameter $a$) up to 20-pN order. When a pN term is not a polynomial in $a/M$ and includes irrational functions (like polygamma functions), it is written as a power series of $a/M$. This is achieved by calculating the fluxes numerically with an accuracy greater than 1 part in $10^{600}$. Such high accuracy allows us to extract analytical values of pN coefficients that are linear combinations of transcendentals like the Euler constant, logarithms of prime numbers and powers of $\pi$. We also present the 22-pN expansion (spin-independent pN expansion) of the ingoing energy flux at the event horizon for a particle in circular orbit about a Schwarzschild black hole.

The thermal expansion of particles and their secular orbital evolution as they circle a planet

The thermal expansion and contraction of particles orbiting a planet can cause secular orbit evolution. This effect, called here the thermal expansion effect, depends on particles entering and exiting the shadow of the body they orbit. A particle cools off in the shadow and heats up again in the sunshine, suffering thermal contraction and expansion. The changing cross-section that the particle presents to solar radiation pressure, plus a time lag due to thermal inertia, lead to a net along-track force. The effect causes outward drift for rocky particles in circular orbits. For particles in the size range ∼0.002–0.02m orbiting the inner planets, particle orbits can outwardly evolve at the rate of ∼0.1RPlan per million years for Mars to ∼1RPlan per million years for Mercury for distances ∼2RPlan from the body, where RPlan is the planet’s radius. Poynting–Robertson dominates thermal expansion beyond a few RPlan for the inner planets. Hence there are distances from a planet where the effects balance, depending on particle size. Orbits evolving outward from the thermal expansion effect would stop there, as well as those inwardly evolving from Poynting–Robertson. Thus particles would accumulate in these places, assuming the absence of other forces. Mars appears to be the best candidate for the operation of the thermal expansion effect. Particles in the size range considered here and orbiting in the Phobos–Deimos region would tend to be collected by the moons, sweeping the particles up and perhaps helping keep the region free of dust. The thermal expansion effect is overwhelmed by Poynting–Robertson for rocky particles orbiting Jupiter and Saturn and thus is unimportant; these planets are not considered here. For particles orbiting small asteroids, the thermal expansion effect is much larger than the Poynting–Robertson effect, but both are overwhelmed by ordinary solar radiation pressure, which increases orbital eccentricities rapidly. Meteoroids in eccentric orbits about the Sun also suffer the thermal expansion effect, but with only ∼0.0003e2AU change in semimajor axis over a million years for a 2m meteoroid orbiting between Mercury and Earth.

Analytic determination of the eight-and-a-half post-Newtonian self-force contributions to the two-body gravitational interaction potential

We {\it analytically} compute, to the eight-and-a-half post-Newtonian order, and to linear order in the mass ratio, the radial potential describing (within the effective one-body formalism) the gravitational interaction of two bodies, thereby extending previous analytic results. These results are obtained by applying analytical gravitational self-force theory (for a particle in circular orbit around a Schwarzschild black hole) to Detweiler's gauge-invariant redshift variable. We emphasize the increase in \lq\lq transcendentality" of the numbers entering the post-Newtonian expansion coefficients as the order increases, in particular we note the appearance of $\zeta(3)$ (as well as the square of Euler's constant $\gamma$) starting at the seventh post-Newtonian order. We study the convergence of the post-Newtonian expansion as the expansion parameter $u=GM/(c^2r)$ leaves the weak-field domain $u\ll 1$ to enter the strong field domain $u=O(1)$.

High-order post-Newtonian contributions to the two-body gravitational interaction potential from analytical gravitational self-force calculations

We extend the analytical determination of the main radial potential describing (within the effective one-body formalism) the gravitational interaction of two bodies beyond the 4th post-Newtonian approximation recently obtained by us. This extension is done to linear order in the mass ratio by applying analytical gravitational self-force theory (for a particle in circular orbit around a Schwarzschild black hole) to Detweiler's gauge-invariant redshift variable. By using the version of black hole perturbation theory developed by Mano, Suzuki and Takasugi, we have pushed the analytical determination of the (linear in mass ratio) radial potential to the 6th post-Newtonian order (passing through 5 and 5.5 post-Newtonian terms). In principle, our analytical method can be extended to arbitrarily high post-Newtonian orders.

Quantum entanglement in Plebański–Demiański spacetimes

For an Einstein–Podolsky–Rosen pair of spin-1/2 particles in circular orbits in a general axially symmetric spacetime, the spin precession angle is obtained. Hovering observers are introduced for ensuring fixed reference frames to perform suitable reliable measurements. Frame-dragging of spinning holes is explicitly incorporated relative to hovering observers. The spin-singlet state is found to be mixed with the spin-triplet by acceleration and gravity effects, which deteriorate the perfect anti-correlation of an entangled pair of spins measured by hovering observers. Finally, an algorithm to calculate spin precession for a general axially symmetric spacetime is proposed. This algorithm is applied to study the complete list of expanding and twisting Type-D Plebański–Demiański black holes and their descendent limiting solutions with lower parameters.

Coronal ejection and heating in variable X-ray sources

A sudden increase in stellar luminosity may lead to the ejection of a large part of any optically thin gas orbiting the star. Test particles in circular orbits will become unbound and will escape to infinity (if radiation drag is neglected) when the luminosity changes from zero to at least one half the Eddington value, or more generally, from L to (LEdd + L)/2 or more. Conversely, a decrease in luminosity will lead to the tightening of orbits of optically thin fluid. Even a modest fluctuation of luminosity of accreting neutron stars or black holes is expected to lead to substantial coronal heating. Luminosity fluctuations may thus account for the high temperatures of the X-ray corona in accreting black hole and neutron star systems.

Gravitational Waves from a Particle in Circular Orbits around a Schwarzschild Black Hole to the 22nd Post-Newtonian Order

We extend our previous results of the 14th post-Newtonian (PN) order expansion of gravitational waves for a test particle in circular orbits around a Schwarzschild black hole to the 22PN order, i.e. $v^{44}$ beyond the leading Newtonian approximation where $v$ is the orbital velocity of a test particle. Comparing our 22PN formula for the energy flux with high precision numerical results, we find that the relative error of the 22PN flux at the innermost stable circular orbit is about $10^{-5}$. We also estimate the phase difference between the 22PN waveforms and numerical waveforms after a two-year inspiral. We find that the dephase is about $10^{-9}$ for $\mu/M=10^{-4}$ and $10^{-2}$ for $\mu/M=10^{-5}$ where $\mu$ is the mass of the compact object and $M$ the mass of the central supermassive black hole. Finally, we construct a hybrid formula of the energy flux by supplementing the 4PN formula of the energy flux for circular and equatorial orbits around a Kerr black hole with all the present 22PN terms for the case of a Schwarzschild black hole. Comparing the hybrid formula with the the full numerical results, we examine the performance of the hybrid formula for the case of Kerr black hole.

Coronal ejection and heating in variable-luminosity X-ray sources

AbstractA sudden increase in stellar luminosity may lead to the ejection of a large part of any optically thin gas orbiting the star. Test particles in circular orbits will become unbound, and will escape to infinity (if radiation drag is neglected), when the luminosity changes from zero to at least one half the Eddington value, or more generally, from L to (LEdd+L)/2 or more. Conversely, a decrease in luminosity will lead to the tightening of orbits of optically thin fluid. Even a modest fluctuation of luminosity of accreting neutron stars or black holes is expected to lead to substantial coronal heating. Luminosity fluctuations may thus account for the high temperatures of the X-ray corona in accreting black hole and neutron star systems.

Gravitational Radiation for Extreme Mass Ratio Inspirals to the 14th Post-Newtonian Order

We derive gravitational waveforms needed to compute the 14th post-Newtonian (14PN) order energy flux for a particle in circular orbit around a Schwarzschild black hole, i.e. $v^{28}$ beyond the leading Newtonian approximation where $v$ is the orbital velocity of a test particle. We investigate the convergence of the energy flux in the PN expansion and suggest a fitting formula which can be used to extract unknown higher order PN coefficients from accurate numerical data for more general orbits around a Kerr black hole. The phase difference between the 14PN waveforms and numerical waveforms after two years inspiral is shown to be about $10^{-7}$ for $\mu/M=10^{-4}$ and $10^{-3}$ for $\mu/M=10^{-5}$ where $\mu$ is the mass of a compact object and $M$ the mass of the central supermassive black hole. In first order black hole perturbation theory, for extreme mass ratio inspirals which are one of the main targets of Laser Interferometer Space Antenna, the 14PN expressions will lead to the data analysis accuracies comparable to the ones resulting from high precision numerical waveforms.

Dilatonic effects on a falling test mass in scalar-tensor theory

Effects of a 4d dilaton field on a falling test mass are examined from the Einstein frame perspective of scalar-tensor theory. Results are obtained for the centripetal acceleration of particles in circular orbits, and the radial acceleration for particles with pure radial motion. These results are applied to the specific case of nonrelativistic motion in the weak field approximation of Brans-Dicke theory, employing the exact Xanthopoulos-Zannias solutions. For a given parameter range, the results obtained from Brans-Dicke theory are qualitatively dramatically different from those of general relativity. Comments are made concerning a comparison with the general relativistic results in the limit of an infinite Brans-Dicke parameter.

Conservative, gravitational self-force for a particle in circular orbit around a Schwarzschild black hole in a radiation gauge

This is the second of two companion papers on computing the self-force in a radiation gauge; more precisely, the method uses a radiation gauge for the radiative part of the metric perturbation, together with an arbitrarily chosen gauge for the parts of the perturbation associated with changes in black-hole mass and spin and with a shift in the center of mass. In a test of the method delineated in the first paper, we compute the conservative part of the self-force for a particle in circular orbit around a Schwarzschild black hole. The gauge vector relating our radiation gauge to a Lorenz gauge is helically symmetric, implying that the quantity ${h}_{\ensuremath{\alpha}\ensuremath{\beta}}{u}^{\ensuremath{\alpha}}{u}^{\ensuremath{\beta}}$ must have the same value for our radiation gauge as for a Lorenz gauge; and we confirm this numerically to one part in ${10}^{14}$. As outlined in the first paper, the perturbed metric is constructed from a Hertz potential that is in a term obtained algebraically from the retarded perturbed spin-2 Weyl scalar, ${\ensuremath{\psi}}_{0}^{\mathrm{ret}}$. We use a mode-sum renormalization and find the renormalization coefficients by matching a series in $L=\ensuremath{\ell}+1/2$ to the large-$L$ behavior of the expression for the self-force in terms of the retarded field ${h}_{\ensuremath{\alpha}\ensuremath{\beta}}^{\mathrm{ret}}$; we similarly find the leading renormalization coefficients of ${h}_{\ensuremath{\alpha}\ensuremath{\beta}}{u}^{\ensuremath{\alpha}}{u}^{\ensuremath{\beta}}$ and the related change in the angular velocity of the particle due to its self-force. We show numerically that the singular part of the self-force has the form ${f}_{\ensuremath{\alpha}}^{\mathrm{S}}=⟨{\ensuremath{\nabla}}_{\ensuremath{\alpha}}{\ensuremath{\rho}}^{\ensuremath{-}1}⟩$, the part of ${\ensuremath{\nabla}}_{\ensuremath{\alpha}}{\ensuremath{\rho}}^{\ensuremath{-}1}$ that is axisymmetric about a radial line through the particle. This differs only by a constant from its form for a Lorenz gauge. It is because we do not use a radiation gauge to describe the change in black-hole mass that the singular part of the self-force has no singularity along a radial line through the particle and, at least in this example, is spherically symmetric to subleading order in $\ensuremath{\rho}$.

Gravitational self-force in a radiation gauge

In this, the first of two companion papers, we present a method for finding the gravitational self-force in a modified radiation gauge for a particle moving on a geodesic in a Schwarzschild or Kerr spacetime. An extension of an earlier result by Wald is used to show the spin weight $\ifmmode\pm\else\textpm\fi{}2$ perturbed Weyl scalar (${\ensuremath{\psi}}_{0}$ or ${\ensuremath{\psi}}_{4}$) determines the metric perturbation outside the particle up to a gauge transformation and an infinitesimal change in mass and angular momentum. A Hertz potential is used to construct the part of the retarded metric perturbation that involves no change in mass or angular momentum from ${\ensuremath{\psi}}_{0}$ in a radiation gauge. The metric perturbation is completed by adding changes in the mass and angular momentum of the background spacetime outside the radial coordinate ${r}_{0}$ of the particle in any convenient gauge. The resulting metric perturbation is singular only on the trajectory of the particle. A mode-sum method is then used to renormalize the self-force. Gralla shows that the renormalized self-force can be used to find the correction to a geodesic orbit in a gauge for which the leading, $O({\ensuremath{\rho}}^{\ensuremath{-}1})$, term in the metric perturbation has spatial components even under a parity transformation orthogonal to the particle trajectory, and we verify that the metric perturbation in a radiation gauge satisfies that condition. We show that the singular behavior of the metric perturbation and the expression for the bare self-force have the same power-law behavior in $L=\ensuremath{\ell}+1/2$ as in a Lorenz gauge (with different coefficients). We explicitly compute the singular Weyl scalar and its mode-sum decomposition to subleading order in $L$ for a particle in circular orbit in a Schwarzschild geometry and obtain the renormalized field. Because the singular field can be defined as this mode sum, the coefficients of each angular harmonic in the sum must agree with the large $L$ limit of the corresponding coefficients of the retarded field. One may therefore compute the singular field by numerically matching the retarded field to a power series in $L$. To check the accuracy of the numerical method, we analytically compute leading and subleading terms in the singular expansion of ${\ensuremath{\psi}}_{0}$ and compare the numerical and analytic values of the renormalization constants, finding agreement to high precision. Details of the numerical computation of the perturbed metric, the self-force, and the quantity ${h}_{\ensuremath{\alpha}\ensuremath{\beta}}{u}^{\ensuremath{\alpha}}{u}^{\ensuremath{\beta}}$ (gauge invariant under helically symmetric gauge transformations) are presented for this test case in the companion paper.

Spherical harmonic modes of 5.5 post-Newtonian gravitational wave polarizations and associated factorized resummed waveforms for a particle in circular orbit around a Schwarzschild black hole

Recent breakthroughs in numerical relativity enable one to examine the validity of the post-Newtonian expansion in the late stages of inspiral. For the comparison between post-Newtonian (PN) expansion and numerical simulations, the waveforms in terms of the spin-weighted spherical harmonics are more useful than the plus and cross polarizations, which are used for data analysis of gravitational waves. Factorized resummed waveforms achieve better agreement with numerical results than the conventional Taylor expanded post-Newtonian waveforms. In this paper, we revisit the post-Newtonian expansion of gravitational waves for a test-particle of mass $\m$ in circular orbit of radius $r_0$ around a Schwarzschild black hole of mass $M$ and derive the spherical harmonic components associated with the gravitational wave polarizations up to order $v^{11}$ beyond Newtonian. Using the more accurate $h_{\ell m}$'s computed in this work, we provide the more complete set of associated $\rho_{\ell m}$'s and $\delta_{\ell m}$'s that form important bricks in the factorized resummation of waveforms with potential applications for the construction of further improved waveforms for prototypical compact binary sources in the future. We also provide ready-to-use expressions of the 5.5PN gravitational waves polarizations $h_+$ and $h_\times$ in the test-particle limit for gravitational wave data analysis applications. Additionally, we provide closed analytical expressions for 2.5PN $h_{\ell m}$, 2PN $\rho_{\ell m}$ and 3PN $\delta_{\ell m}$, for general multipolar orders $\ell$ and $m$ in the test-particle limit. Finally, we also examine the implications of the present analysis for compact binary sources in Laser Interferometer Space Antenna.