Mass Quadrupole Moment Research Articles

Axion Halo around a Binary System of Dwarf Stars

The gravitational field of a clump of ultralight axion like particles (ALPs) in its core with a rotating binary system of dwarf stars is computed. It is established that the induced quadrupole mass moment of the clump is controlled parametrically by the Ma/M mass ratio of the axion clump and the binary core.

An approximate Kerr–Newman-like metric endowed with a magnetic dipole and mass quadrupole

Approximate all-terrain spacetimes for astrophysical applications are presented. The metrics possess five relativistic multipole moments, namely, mass, rotation, mass quadrupole, charge, and magnetic dipole moment. All these spacetimes approximately satisfy the Einstein–Maxwell field equations. The first metric is generated using the Hoenselaers–Perjés method from given relativistic multipoles. The second metric is a perturbation of the Kerr–Newman metric, which makes it a relevant approximation for astrophysical calculations. The last metric is an extension of the Hartle–Thorne metric that is important for obtaining internal models of compact objects perturbatively. The electromagnetic field is calculated using Cartan forms for locally non-rotating observers. These spacetimes are relevant for inferring properties of compact objects from astrophysical observations. Furthermore, the numerical implementations of these metrics are straightforward, making them versatile for simulating potential astrophysical applications.

Neutron star accretion events in AGN discs: mutimessenger implications

ABSTRACT This paper investigates the accretion of neutron stars (NSs) in active galactic nucleus (AGN) accretion discs. We classify potential accretion modes of NSs in AGN discs, proposing a hierarchical model of NS accretion: accretion flow from the Bondi sphere to accretion columns. The accretion of NSs in AGN discs differs from that of BHs, especially within the scale of the NS’s magnetosphere due to its hard surface and magnetic field. As the accretion flow approaches the magnetosphere, the magnetic fields guide the accretion flow to form accretion columns, primarily dominated by neutrinos. While neutrinos generated from single NS accretion may not have observable effects, considering the all-sky background, they contribute to the neutrino background in the sub-MeV energy range comparable to that of supernova explosions. NS accretion may also lead to the generation of mass quadrupole moments, consequently generating gravitational waves (GWs). The GWs, which exhibit characteristic effects like periodic modulations and echoes, could be observed by third-generation GW detectors. The emission of neutrinos and GWs carries away energy and angular momentum brought by accretion, reducing the feedback effect on the AGN disc. This results in an exceptionally high NS accretion rate, leading to a collapse time-scale shorter than the migration-merge time-scale, making it less likely that binary NS mergers originate from AGN discs.

Rotating metrics and new multipole moments from scattering amplitudes in arbitrary dimensions

We compute the vacuum metric generated by a generic rotating object in arbitrary dimensions up to third post-Minkowskian order by computing the classical contribution of scattering amplitudes describing the graviton emission by massive spin-1 particles up to two loops. The solution depends on the mass, angular momenta, and on up to two parameters related to generic quadrupole moments. In D=4 spacetime dimensions, we recover the vacuum Hartle-Thorne solution describing a generic spinning object to second order in the angular momentum, of which the Kerr metric is a particular case obtained for a specific mass quadrupole moment dictated by the uniqueness theorem. At the level of the effective action, the case of minimal couplings corresponds to the Kerr black hole, while any other mass quadrupole moment requires nonminimal couplings. In D>4, the absence of black-hole uniqueness theorems implies that there are multiple spinning black hole solutions with different topology. Using scattering amplitudes, we find a generic solution depending on the mass, angular momenta, the mass quadrupole moment, and a new stress quadrupole moment which does not exist in D=4. As special cases, we recover the Myers-Perry and the single-angular-momentum black ring solutions, to third and first post-Minkowksian order, respectively. Interestingly, at variance with the four-dimensional case, none of these solutions corresponds to the minimal coupling in the effective action. This shows that, from the point of view of scattering amplitudes, black holes are the “simplest” general relativity vacuum solutions only in D=4. Published by the American Physical Society 2024

Nodal precession of a hot Jupiter transiting the edge of a late A-type star TOI-1518

Abstract TOI-1518b, a hot Jupiter around a late A-type star, is one of the few planetary systems that transit the edge of the stellar surface (the impact parameter b ∼ 0.9) among hot Jupiters around hot stars (Cabot et al. 2021, AJ, 162, 218). The high rotation speed of the host star (∼85 km s−1) and the nearly polar orbit of the planet (∼120○) may cause a nodal precession. In this study, we report the nodal precession undergone by TOI-1518 b. This system is the fourth planetary system in which nodal precession is detected. We investigate the time change in b from the photometric data of TOI-1518 acquired in 2019 and 2022 with TESS and from the spectral transit data of TOI-1518b obtained in 2020 with two high-dispersion spectrographs; CARMENES and EXPRES. We find that the value of b is decreasing with db$/$dt = −0.0116 ± 0.0036 yr−1, indicating that the transit trajectory is moving toward the center of the stellar surface. We also estimate the minimum value of the quadrupole mass moment of TOI-1518, J2,min = 4.41 × 10−5, and the logarithm of the Love number of TOI-1518, log k2 = −2.17 ± 0.33, from the nodal precession.

Post-Newtonian Orbital Effects Induced by the Mass Quadrupole and Spin Octupole Moments of an Axisymmetric Body

The post-Newtonian orbital effects induced by the mass quadrupole and spin octupole moments of an isolated, oblate spheroid of constant density that is rigidly and uniformly rotating on the motion of a test particle are analytically worked out for an arbitrary orbital configuration and without any preferred orientation of the body’s spin axis. The resulting expressions are specialized to the cases of (a) equatorial and (b) polar orbits. The opportunity offered by a hypothetical new spacecraft moving around Jupiter along a Juno-like highly elliptical, polar orbit to measure them is preliminarily studied. Although more difficult to be practically implemented, also the case of a less elliptical orbit is considered since it yields much larger figures for the relativistic effects of interest. The possibility of using the S-stars orbiting the supermassive black hole in Sgr A* at the Galactic Center as probes to potentially constrain some parameters of the predicted extended mass distribution surrounding the hole by means of the aforementioned orbital effects is briefly examined.

Observable Properties of Thin Accretion Disk in the γ Spacetime

We study matter accretion in a static, axially symmetric and vacuum geometry describing the exterior gravitational field of a black hole mimicker called the γ metric. We evaluate the thermal and optical properties of thin accretion disks, including the emission rate, luminosity and shadow, in the gamma spacetime. Also, we explore the radial accretion of polytropic matter fields onto the central source and evaluate the thermal and optical properties of the infalling gas, such as temperature and luminosity. The results are discussed in the context of evaluating the possibility that the true nature of astrophysical black hole candidates may not be a black hole but some exotic compact object possessing a non-vanishing mass quadrupole moment.

Is It Possible to Measure the Lense–Thirring Orbital Shifts of the Short-period S Star S4716 Orbiting Sgr A*?

The maximal values of the general relativistic Lense–Thirring (LT) orbital shifts ΔI LT, ΔΩLT, and Δω LT of the inclination I, the longitude of the ascending node Ω, and the perinigricon ω, respectively, of the recently discovered star S4716, which has the shortest orbital period of all the S stars that orbit the supermassive black hole (SMBH) in Sgr A*, are of the order of ≃5–16 arcseconds per revolution . Given the current error σ ω = 0.°02 in determining ω, which is the most accurate orbital parameter of S4716 among all those affected by the SMBH's gravitomagnetic field through its angular momentum J •, about 48 yr would be needed to reduce σ ω to ≃10% of the cumulative LT perinigricon shift over the same time span. Measuring ΔI LT and ΔΩLT to the same level of accuracy would take much longer. Instead, after just 16 yr, a percent measurement of the larger gravitoelectric (GE) Schwarzschild-like perinigricon shift Δω GE, which depends only on the SMBH's mass M •, would be possible. On the other hand, the uncertainties in the physical and orbital parameters entering Δω GE would cause a huge systematic bias of Δω LT itself. The SMBH's quadrupole mass moment induces orbital shifts as little as ≃0.01–0.05″ rev−1.

Multipole expansion of gravitational waves: memory effects and Bondi aspects

In our previous work, we proposed an algorithm to transform the metric of an isolated matter source in the multipolar post-Minkowskian approximation in harmonic (de Donder) gauge to the Newman-Unti gauge. We then applied this algorithm at linear order and for specific quadratic interactions known as quadratic tail terms. In the present work, we extend this analysis to quadratic interactions associated with the coupling of two mass quadrupole moments, including both instantaneous and hereditary terms. Our main result is the derivation of the metric in Newman-Unti and Bondi gauges with complete quadrupole-quadrupole interactions. We rederive the displacement memory effect and provide expressions for all Bondi aspects and dressed Bondi aspects relevant to the study of leading and subleading memory effects. Then we obtain the Newman-Penrose charges, the BMS charges as well as the second and third order celestial charges defined from the known second order and novel third order dressed Bondi aspects for mass monopole-quadrupole and quadrupole-quadrupole interactions.

Slowly rotating Tolman VII solution

We present a model of a slowly rotating Tolman VII (T-VII) fluid sphere, at second order in the angular velocity. The structure of this configuration is obtained by integrating numerically the Hartle–Thorne equations for slowly rotating relativistic masses. We consider a sequence of models where we vary the parameter , where R is the radius of the configuration and is its Schwarzschild radius, representing an adiabatic and quasi-stationary contraction by progressively reducing the radius while keeping the angular momentum and gravitational mass constant. We determined the moment of inertia I, mass quadrupole moment Q, and the ellipticity ɛ, for various configurations. Similarly to previous results for Maclaurin and polytropic spheroids, in slow rotation, we found a change in the behavior of the ellipticity when reaches a certain critical value. Based on our analysis for the T-VII solution, we found variations of in the and relations, and variation in the I − Q relation, with respect to the universal fittings proposed for realistic neutron stars (NSs). Our results suggest that the T-VII solution can be considered a rather good approximation for the description of the interior of NSs.

The radio shut-off, glitch, and X-ray burst in 1E 1547.0−5408 interpreted through magnetic reconfiguration

ABSTRACT A short X-ray burst was observed from the radio-loud magnetar 1E 1547.0–5408 in 2022 April. Unusually however, the source stopped showing radio pulsations $\gtrsim 3\,$ weeks prior to the burst. After recovery, radio timing revealed that the object had also undergone a modest glitch. A model for the overall event is constructed where an initially mild perturbation adjusts the magnetic geometry near the polar caps, leading to shallow fractures. Crustal ejecta or particles leaking from a pair-plasma fireball pollute the magnetospheric gaps, shutting off the pulsar mechanism, but the energy release is not yet large enough to noticeably enhance the X-ray flux. This perturbation gradually ramps, eventuating in a large-scale energy redistribution which fuels the burst. The star’s mass quadrupole moment changes in tandem, issuing a glitch. Some quantitative estimates for the magnetic reconfiguration under this interpretation are provided, based on a quasi-static model where the fluid evolves through a sequence of hydromagnetic equilibria.

Onset of chaotic gravitational lensing in non-Kerr rotating black holes with quadrupole mass moment* *Supported by the National Natural Science Foundation of China (NNSFC, 12005077) and Guangdong Basic and Applied Basic Research Foundation (2021A1515012374). We also gratefully acknowledge the financial support from Brazilian agencies Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP), Fundação de Amparo à Pesquisa

In the electromagnetic channel, chaotic gravitational lensing is a peculiar phenomenon in strong gravitational lensing. In this study, we analyze the properties and emergence of chaotic gravitational lensing in the Manko-Novikov black hole spacetime. Aiming to better understand the underlying physics, we elaborate on the boundaries of the accessible region through analyses of the contours of the effective potentials. The latter is associated with the two roots of a quadratic equation. In particular, we explore its interplay with an ergoregion, which leads to specific features of the effective potentials, such as the emergence of a cuspy edge and the formation of a pocket, which serve as static constraints on the geodesics. Additionally, we investigate the properties of the radial and angular accelerations at the turning points in photon trajectories. The accelerations are further examined and may provide kinematic constraints on the geodesics, as argued herein. It is concluded that the onset of the chaotic lensing is significantly related to both constraints; as a result, an arbitrary slight deviation in the incident photon is significantly amplified during evolution through an extensive period, demonstrating the complexity in the highly nonlinear deterministic gravitational system.

Limitations in Testing the Lense–Thirring Effect with LAGEOS and the Newly Launched Geodetic Satellite LARES 2

The new geodetic satellite LARES 2, cousin of LAGEOS and sharing with it almost the same orbital parameters apart from the inclination, displaced by 180 deg, was launched last year. Its proponents suggest using the sum of the nodes of LAGEOS and of LARES 2 to measure the sum of the Lense–Thirring node precessions independently of the systematic bias caused by the even zonal harmonics of the geopotential, claiming a final ≃0.2 percent total accuracy. In fact, the actual orbital configurations of the two satellites do not allow one to attain the sought for mutual cancellation of their classical node precessions due to the Earth’s quadrupole mass moment, as their sum is still ≃5000 times larger than the added general relativistic rates. This has important consequences. One is that the current uncertainties in the eccentricities and the inclinations of both satellites do not presently allow the stated accuracy goal to be met, needing improvements of 3–4 orders of magnitude. Furthermore, the imperfect knowledge of the Earth’s angular momentum S impacts the uncancelled sum of the node precessions, from 150 to 4900 percent of the relativistic signal depending on the uncertainty assumed in S. It is finally remarked that the real breakthrough in reliably testing the gravitomagnetic field of the Earth would consist in modeling it and simultaneously estimating one or more dedicated parameter(s) along with other ones characterising the geopotential, as is customarily performed for any other dynamical feature.

Parametrized post-Einsteinian framework for precessing binaries

In general relativity, isolated black holes obey the no hair theorems, which fix the multipolar structure of their exterior spacetime. However, in modified gravity, or when the compact objects are not black holes, the exterior spacetime may have a different multipolar structure. When two black holes are in a binary, this multipolar structure determines the morphology of the dynamics of orbital and spin precession. In turn, the precession dynamics imprint onto the gravitational waves emitted by an inspiraling compact binary through specific amplitude and phase modulations. The detection and characterization of these amplitude and phase modulations can therefore lead to improved constraints on fundamental physics with gravitational waves. Recently, analytic precessing waveforms were calculated in two scenarios: (i) dynamical Chern-Simons gravity, where the no-hair theorems are violated, and (ii) deformed compact objects with generic mass quadrupole moments. In this work, we use these two examples to propose an extension of the parameterized post-Einsteinian~(ppE) framework to include precession effects. The new framework contains $2n$ ppE parameters $(\mathscr{b}^{\rm ppE}_{(m',n)}, b^{\rm ppE}_{(m',n)})$ for the waveform phase, and $2n$ ppE parameters $(\mathscr{a}^{\rm ppE}_{(m',n)}, a^{\rm ppE}_{(m',n)})$ for the waveform amplitudes. The number of ppE corrections $n$ corresponds to the minimum number of harmonics necessary to achieve a given likelihood threshold when comparing the truncated ppE waveform with the exact one, and $(m',n)$ corresponds to the harmonic numbers of the harmonics containing ppE parameters. We show explicitly how these ppE parameters map to the specific example waveforms discussed above. The proposed ppE framework can serve as a basis for future tests of general relativity with gravitational waves from precessing binaries.

The quadrupole moment of compact binaries to the fourth post-Newtonian order: relating the harmonic and radiative metrics

Motivated by the completion of the fourth post-Newtonian (4PN) gravitational-wave generation from compact binary systems, we analyse and contrast different constructions of the metric outside an isolated system, using post-Minkowskian (PM) expansions. The metric in ‘harmonic’ coordinates has been investigated previously, in particular to compute tails and memory effects. However, it is plagued by powers of the logarithm of the radial distance r when (with const). As a result, the tedious computation of the ‘tail-of-memory’ effect, which enters the gravitational-wave flux at 4PN order, is more efficiently performed in the so-called ‘radiative’ coordinates, which admit a (Bondi-type) expansion at infinity in simple powers of r −1, without any logarithms. Here we consider a particular construction, performed order by order in the PM expansion, which directly yields a metric in radiative coordinates. We relate both constructions, and prove that they are physically equivalent as soon as a relation between the ‘canonical’ moments which parametrize the radiative metric, and those parametrizing the harmonic metric, is verified. We provide the appropriate relation for the mass quadrupole moment at 4PN order, which will be crucial when deriving the ‘tail-of-memory’ contribution to the gravitational flux.

Erratum: Post-Newtonian direct and mixed orbital effects due to the oblateness of the central body

The dimensional quadrupole mass moment [Formula: see text] of a black hole is negative; thus, Eq. (98) of Ref. 1 must be corrected to [Formula: see text].

Post-Newtonian effects on some characteristic time-scales of transiting exoplanets

ABSTRACT Some measurable characteristic time-scales {ttrn} of transiting exoplanets are investigated in order to check preliminarily if their cumulative shifts over the years induced by the post-Newtonian (pN) gravitoelectric (Schwarzschild) and gravitomagnetic (Lense–Thirring) components of the stellar gravitational field are, at least in principle, measurable. Both the primary (planet in front of the star) and the secondary (planet behind the star) transits are considered along with their associated characteristic time intervals: the total transit duration tD, the ingress/egress transit duration $\tau$, the full-width at half maximum primary transit duration tH, and also the time of conjunction tcj. For each of them, the net changes per orbit $\left\langle \Delta t_D\right\rangle ,\, \left\langle \Delta \tau\right\rangle ,\, \left\langle \Delta t_H\right\rangle ,\, \left\langle \Delta t_\mathrm{cj}\right\rangle$ induced by the aforementioned pN accelerations are analytically obtained; also the Newtonian effect of the star’s quadrupole mass moment $J_2^\star$ is worked out. They are calculated for a fictitious Sun-Jupiter system in an edge-on elliptical orbit, and the results are compared with the present-day experimental accuracies for the HD 286123 b exoplanet. Its pN gravitoelectric shift $\left\langle \Delta t_\mathrm{cj}^\mathrm{1pN}\right\rangle$ may become measurable, at least in principle, at a ≃8 × 10−5 level of (formal) relative accuracy after about 30 yr of continuous monitoring corresponding to about 1000 transits. Systematics like, e.g. confusing time standards, neglecting star-spots, neglecting clouds, additional planets in the system, etc. would likely deteriorate the actual accuracy. The method presented is general enough to be applied also to modified models of gravity.

Chaotic behavior of geodesics in Kerr-like spacetime

In recent years, experimental and theoretical research on the nature of supermassive bodies that inhabit the central regions of galaxies has grown significantly. In this contribution, we study the motion of unitary mass test particles in a perturbed Kerr-like metric. The metric represents the approximate exterior spacetime of a massive rotating body with mass quadrupole moment $q$. Through extensive study of the geodesics, it is found that the chaotic behavior proliferates as the $q$ parameter increases. Several structures are found in the phase space, such as Birkhoff chains of islands and hyperbolic points if $q\ensuremath{\ne}0$. Additionally, chaotic regions crossing the null radial momentum axis (${p}_{r}=0$) are studied by analyzing the rotation number, and several islands of stability are found that are preserved as the deformation of the rotating body increases, while others are destroyed. Furthermore, chaotic behavior is found to be dictated by stickiness, chaotic orbits that remain attached to stable ones for a long time before they become divergent and are attracted to the event horizon.

Searches for Gravitational Waves from Known Pulsars at Two Harmonics in the Second and Third LIGO-Virgo Observing Runs

We present a targeted search for continuous gravitational waves (GWs) from 236 pulsars using data from the third observing run of LIGO and Virgo (O3) combined with data from the second observing run (O2). Searches were for emission from the l = m = 2 mass quadrupole mode with a frequency at only twice the pulsar rotation frequency (single harmonic) and the l = 2, m = 1, 2 modes with a frequency of both once and twice the rotation frequency (dual harmonic). No evidence of GWs was found, so we present 95% credible upper limits on the strain amplitudes h 0 for the single-harmonic search along with limits on the pulsars’ mass quadrupole moments Q 22 and ellipticities ε. Of the pulsars studied, 23 have strain amplitudes that are lower than the limits calculated from their electromagnetically measured spin-down rates. These pulsars include the millisecond pulsars J0437−4715 and J0711−6830, which have spin-down ratios of 0.87 and 0.57, respectively. For nine pulsars, their spin-down limits have been surpassed for the first time. For the Crab and Vela pulsars, our limits are factors of ∼100 and ∼20 more constraining than their spin-down limits, respectively. For the dual-harmonic searches, new limits are placed on the strain amplitudes C 21 and C 22. For 23 pulsars, we also present limits on the emission amplitude assuming dipole radiation as predicted by Brans-Dicke theory.

Inspiraling compact objects with generic deformations

Self-gravitating bodies can have an arbitrarily complex shape, which implies a much richer multipolar structure than that of a black hole in General Relativity. With this motivation, we study the corrections to the dynamics of a binary system due to generic, nonaxisymmetric mass quadrupole moments to leading post-Newtonian (PN) order. Utilizing the method of osculating orbits and a multiple scale analysis, we find analytic solutions to the precession and orbital dynamics of a (generically eccentric) binary in terms of the dimensionless modulus parameters $\epsilon_{m}$, corresponding to axial $m=1$ and polar $m=2$ corrections from oblateness/prolateness. The solutions to the precession dynamics are exact for $0 \le \epsilon_{2} < 1$, and perturbative in $\epsilon_{1} \ll 1$. We further compute the leading order corrections to the gravitational wave amplitude and phase for a quasi-circular binary due to mass quadrupole effects. Making use of the stationary phase approximation and shifted uniform asymptotics (SUA), the corrections to the phase enter at relative 2PN order, while the amplitude modulations enter at -0.5PN order with a SUA amplitude correction at 3.25PN order, relative 2PN order to the leading order SUA correction. By investigating the dephasing due to generic quadrupole moments, we find that a phase difference $\gtrsim 0.1$~radians is achievable for $\epsilon_{m} \gtrsim 10^{-3}$, which suggests that constraints with current and future ground-based gravitational wave detectors are possible. Our results can be implemented in parameter estimation studies to constrain generic multipolar deformations of the Kerr geometry and of neutron stars.