Body Spin Research Articles

Precisely computing bound orbits of spinning bodies around black holes. I. General framework and results for nearly equatorial orbits

Very large mass ratio binary black hole systems are of interest both as a clean limit of the two-body problem in general relativity, as well as for their importance as sources of low-frequency gravitational waves. At lowest order, the smaller body moves along a geodesic of the larger black hole's spacetime. Accurate models of such systems require postgeodesic corrections to this motion. Postgeodesic effects that drive the small body away from the geodesic include the gravitational self-force, which incorporates the backreaction of gravitational-wave emission, and the spin-curvature force, which arises from coupling of the small body's spin to the black hole's spacetime curvature. In this paper, we describe a method for precisely computing bound orbits of spinning bodies about black holes. Our analysis builds off of pioneering work by Witzany which demonstrated how to describe the motion of a spinning body to linear order in the small body's spin. Exploiting the fact that in the large mass-ratio limit spinning-body orbits are close to geodesics (in a sense that can be made precise) and using closed-form results due to van de Meent describing precession of the small body's spin along black hole orbits, we develop a frequency-domain formulation of the motion which can be solved very precisely. We examine a range of orbits with this formulation, focusing in this paper on orbits which are eccentric and nearly equatorial (i.e., the orbit's motion is $\mathcal{O}(S)$ out of the equatorial plane), but for which the small body's spin is arbitrarily oriented. We discuss generic orbits with general small-body spin orientation in a companion paper. We characterize the behavior of these orbits, contrasting them with geodesics, and show how the small body's spin shifts the frequencies ${\mathrm{\ensuremath{\Omega}}}_{r}$ and ${\mathrm{\ensuremath{\Omega}}}_{\ensuremath{\phi}}$ which affect orbital motion. These frequency shifts change accumulated phases which are direct gravitational-wave observables, illustrating the importance of precisely characterizing these quantities for gravitational-wave observations.

Precisely computing bound orbits of spinning bodies around black holes. II. Generic orbits

In this paper, we continue our study of the motion of spinning test bodies orbiting Kerr black holes. Non-spinning test bodies follow geodesics of the spacetime in which they move. A test body's spin couples to the curvature of that spacetime, introducing a "spin-curvature force" which pushes the body's worldline away from a geodesic trajectory. The spin-curvature force is an important example of a post-geodesic effect which must be modeled carefully in order to accurately characterize the motion of bodies orbiting black holes. One motivation for this work is to understand how to include such effects in models of gravitational waves produced from the inspiral of stellar mass bodies into massive black holes. In this paper's predecessor, we describe a technique for computing bound orbits of spinning bodies around black holes with a frequency-domain description which can be solved very precisely. In that paper, we present an overview of our methods, as well as present results for orbits which are eccentric and nearly equatorial (i.e., the orbit's motion is no more than $\mathcal{O}(S)$ out of the equatorial plane). In this paper, we apply this formulation to the fully generic case -- orbits which are inclined and eccentric, with the small body's spin arbitrarily oriented. We compute the trajectories which such orbits follow, and compute how the small body's spin affects important quantities such as the observable orbital frequencies $\Omega_r$, $\Omega_\theta$ and $\Omega_\phi$.

Particle Ejection Contributions to the Rotational Acceleration and Orbit Evolution of Asteroid (101955) Bennu.

This paper explores the implications of the observed Bennu particle ejection events for that asteroid's spin rate and orbit evolution, which could complicate interpretation of the Yarkovsky‐O'Keefe‐Radzievskii‐Paddack (YORP) and Yarkovsky effects on this body's spin rate and orbital evolution. Based on current estimates of particle ejection rates, we find that the overall contribution to Bennu's spin and orbital drift is small or negligible as compared to the Yarkovsky and YORP effects. However, if there is a large unseen component of smaller mass ejections or a strong directionality in the ejection events, it could constitute a significant contribution that could mask the overall YORP effect. This means that the YORP effect may be stronger than currently assumed. The analysis is generalized so that the particle ejection effect can be assessed for other bodies that may be subject to similar mass loss events. Further, our model can be modified to address different potential mechanisms of particle ejection, which are a topic of ongoing study.

Gravitational Magnus effect

It is well known that a spinning body moving in a fluid suffers a force orthogonal to its velocity and rotation axis --- it is called the Magnus effect. Recent simulations of spinning black holes and (indirect) theoretical predictions, suggest that a somewhat analogous effect may occur for purely gravitational phenomena. The magnitude and precise direction of this "gravitational Magnus effect" is still the subject of debate. Starting from the rigorous equations of motion for spinning bodies in General Relativity (Mathisson-Papapetrou equations), we show that indeed such an effect takes place and is a fundamental part of the spin-curvature force. The effect arises whenever there is a current of mass/energy, non-parallel to a body's spin. We compute the effect explicitly for some astrophysical systems of interest: a galactic dark matter halo, a black hole accretion disk, and the Friedmann-Lema\^itre-Robertson-Walker (FLRW) spacetime. It is seen to lead to secular orbital precessions potentially observable by future astrometric experiments and gravitational-wave detectors. Finally, we consider also the reciprocal problem: the "force" exerted by the body on the surrounding matter, and show that (from this perspective) the effect is due to the body's gravitomagnetic field. We compute it rigorously, showing the matching with its reciprocal, and clarifying common misconceptions in the literature regarding the action-reaction law in post-Newtonian gravity.

Tidal deformation of a slowly rotating material body: Interior metric and Love numbers

The metric outside a compact body deformed by a quadrupolar tidal field is universal up to its Love numbers, constants which encode the tidal response's dependence on the body's internal structure. For a non-rotating body, the deformed external geometry is characterized by the familiar gravitational Love numbers $K_2^{\text{el}}$ and $K_2^{\text{mag}}$. For a slowly rotating body, these must be supplemented by rotational-tidal Love numbers, which measure the response to couplings between the body's spin and the external tidal field. By integrating the interior field equations, I find that the response of a barotropic perfect fluid to spin-coupled tidal perturbations is described by two rotational-tidal Love numbers, which I calculate explicitly for polytropes. Two other rotational-tidal Love numbers identified in prior work are found to have a fixed, universal value for all barotropes. Equipped with the complete interior solution, I calculate the amplitude of the time-varying internal currents induced by the gravitomagnetic part of the tidal field. For a typical neutron star in an equal-mass binary system, the size of the equatorial velocity perturbation is on the order of kilometers per second.

Gyroscopes orbiting black holes: A frequency-domain approach to precession and spin-curvature coupling for spinning bodies on generic Kerr orbits

A small body orbiting a black hole follows a trajectory that, at leading order, is a geodesic of the black hole spacetime. Much effort has gone into computing "self force" corrections to this motion, arising from the small body's own contributions to the system's spacetime. Another correction to the motion arises from coupling of the small body's spin to the black hole's spacetime curvature. Spin-curvature coupling drives a precession of the small body, and introduces a "force" (relative to the geodesic) which shifts the small body's worldline. These effects scale with the small body's spin at leading order. In this paper, we show that the equations which govern spin-curvature coupling can be analyzed with a frequency-domain decomposition, at least to leading order in the small body's spin. We show how to compute the frequency of precession along generic orbits, and how to describe the small body's precession and motion in the frequency domain. We illustrate this approach with a number of examples. This approach is likely to be useful for understanding spin coupling effects in the extreme mass ratio limit, and may provide insight into modeling spin effects in the strong field for non-extreme mass ratios.

Nonlinear gravitational self-force: Field outside a small body

A small extended body moving through an external spacetime $g_{\alpha\beta}$ creates a metric perturbation $h_{\alpha\beta}$, which forces the body away from geodesic motion in $g_{\alpha\beta}$. The foundations of this effect, called the gravitational self-force, are now well established, but concrete results have mostly been limited to linear order. Accurately modeling the dynamics of compact binaries requires proceeding to nonlinear orders. To that end, I show how to obtain the metric perturbation outside the body at all orders in a class of generalized wave gauges. In a small buffer region surrounding the body, the form of the perturbation can be found analytically as an expansion for small distances $r$ from a representative worldline. Given only a specification of the body's multipole moments, the field obtained in the buffer region suffices to find the metric everywhere outside the body via a numerical puncture scheme. Following this procedure at first and second order, I calculate the field in the buffer region around an arbitrarily structured compact body at sufficiently high order in $r$ to numerically implement a second-order puncture scheme, including effects of the body's spin. I also define $n$th-order (local) generalizations of the Detweiler-Whiting singular and regular fields and show that in a certain sense, the body can be viewed as a skeleton of multipole moments.

Small-Body Postrendezvous Characterization via Slow Hyperbolic Flybys

DOI: 10.2514/1.53722 The most crucial task for a spacecraft upon arriving at a small body is to determine the strength of the body's gravity field. This research proposes and models this initial characterization via a series of slow flybys and analyzes how rapidly the gravity field can be estimated and the precision to which it can be determined. Two analytical issues areaddressedinthispaperthatarepertinenttothedesignofthischaracterizationprocessandcanbeusedtoevaluate whether thereis aneed for lidarmeasurements. A new operational procedure calledV rangingis proposed, which can eliminate the need for lidar during the initial characterization phase. Following this, the characterization of the gravity field is addressed by performing a covariance analysis around three asteroids: Itokawa, Didymos, and Eros, representing a two-order-of-magnitude difference in size. I. Introduction T HEREisanincreasinginterestinstudyingsmall-bodysystems, such as asteroids and comets, with spacecraft rendezvous missions. On arrival to a small body, one of the most important tasks is for the spacecraft to characterize the body's environment to determine the gravitational parameterand the gravity field of the body. This is the major source of spacecraft perturbation when in proximity to the body, and lack of knowledge of thegravity field can delay the start of the main scientific mission. Thus, it is of interest to carry out this initial characterization of the body in as rapid amanner as is feasible and with minimum risk to the spacecraft. Estimation of thegravitationalparameterandgravitational fieldofabodygenerally uses a least-squares filter, most commonly a batch square-root infor- mation filter (SRIF) due to its numerical stability (1-3), assuming that the deviation from the true dynamics can be approximated within the linear range of the modeled nonlinear dynamics of the spacecraft (4). In addition to the gravity terms, other commonly estimated parameters in the filter are the spacecraft state, body's spin state, and navigationV maneuvers. The estimated parameters can be time-dependent as in spacecraft position and velocity, or time- independent (i.e., global) as in the higher-degree and higher-order gravity field. As will be shown later, a method for carrying out this initial estimation task throughaseriesof relativelyslow flybys ofthe body is proposed. We probe the level of characterization that can occur using this approach, and whether such an approach could eliminate the need for a dedicated orbital phase for some mission types. This topic is studied using both analytic and numerical methods. For definiteness, we assume that the target body is a near- Earth asteroid, with additional details on its size, shape, spin state, and heliocentric orbit given later. This paper consists of an analytical section and a numerical section, bothof whichwill discussthe estimation ofthesmall body's gravitational environment. The results presented here combine two previous conference papers (5,6). The first (analytical) part of the paperanalyzestheeffectivenessofperformingaprecisemaneuverto break the scale invariance in a hyperbolic flyby with optical-only observations. The goal of this analysis is to characterize the need for carrying a laser or radar ranging instrument to sense distance to the asteroid, and replaces this with a concept calledV ranging. A similar concept using optical-based navigation is also discussed by Alonso et al. (7) to estimate the relative position and attitude in the contextofformation flyingandValaseketal.(8)fortheapplicationof autonomous air refueling. Following this, we provideV ranging, which is a derivation of a new method to analytically estimate the precision of the gravitational parameter determination based on the geometry and time of the spacecraft trajectory around an asteroid. Specifically, wederiveanalytical expressionsfor the uncertainty of a small-body gravity field and spacecraft relative trajectory using Doppler tracking and optical navigation. The conventional method derivesby leveraging the property that the inbound and outbound hyperbolicvelocitiesV1 areequalinmagnitudeandseparatedbythe hyperbolic turn angle� . This method rendersas a function ofb1 (impact radius),V1, and change in hyperbolic velocityVflyby. Our new analytical estimate is based on the measured time of flight of a flyby trajectory to derive an expression for� . These two analytical expressionsforarecomparedwiththefullleast-squarescovariance discussedinthesecondpartofthispaperinordertodeterminewhich effect actually controls the full estimation.