Long-term evolution of a planetesimal swarm in the vicinity of a protoplanet

Many models of planet formation involve scenarios in which one or a few large protoplanets interact with a swarm of much smaller planetesimals. In such scenarios, three-body perturbations by the protoplanet as well as mutual collisions and gravitational interactions between the swarm bodies are important in determining the velocity distribution of the swarm. We are developing a model to examine the effects of these processes on the evolution of a planetesimal swarm. The model consists of a combination of numerical integrations of the gravitational influence of one (or a few) massive protoplanets on swarm bodies together with a statistical treatment of the interactions between the planetesimals. Integrating the planetesimal orbits allows us to take into account effects that are difficult to model analytically or statistically, such as three-body collision cross-sections and resonant perturbations by the protoplanet, while using a statistical treatment for the particle-particle interactions allows us to use a large enough sample to obtain meaningful results

Formation of Kuiper Belt Binaries by Gravitational Collapse

A large fraction of 100-km-class low-inclination objects in the classical Kuiper Belt (KB) are binaries with comparable mass and wide separation of components. A favored model for their formation was capture during the coagulation growth of bodies in the early KB. Instead, recent studies suggested that large objects can rapidly form in the protoplanetary disks when swarms of locally concentrated solids collapse under their own gravity. Here we examine the possibility that KB binaries formed during gravitational collapse when the excess of angular momentum prevented the agglomeration of available mass into a solitary object. We find that this new mechanism provides a robust path toward the formation of KB binaries with observed properties, and can explain wide systems such as 2001 QW322 and multiples such as (47171) 1999 TC36. Notably, the gravitational collapse is capable of producing 100% binary fraction for a wide range of the swarm's initial angular momentum values. The binary components have similar masses (80% have the secondary-over-primary radius ratio >0.7) and their separation ranges from ~1,000 to ~100,000 km. The binary orbits have eccentricities from e=0 to ~1, with the majority having e<0.6. The binary orbit inclinations with respect to the initial angular momentum of the swarm range from i=0 to ~90 deg, with most cases having i<50 deg. Our binary formation mechanism implies that the primary and secondary components in each binary pair should have identical bulk composition, which is consistent with the current photometric data. We discuss the applicability of our results to the Pluto-Charon, Orcus-Vanth, (617) Patroclus-Menoetius and (90) Antiope binary systems.Comment: Astronomical Journal, in pres

A coherent method for the detection and estimation of continuous gravitational wave signals using a pulsar timing array

The use of a high precision pulsar timing array is a promising approach to detecting gravitational waves in the very low frequency regime ($10^{-6} -10^{-9}$ Hz) that is complementary to the ground-based efforts (e.g., LIGO, Virgo) at high frequencies ($\sim 10 -10^3$ Hz) and space-based ones (e.g., LISA) at low frequencies ($10^{-4} -10^{-1}$ Hz). One of the target sources for pulsar timing arrays are individual supermassive black hole binaries that are expected to form in galactic mergers. In this paper, a likelihood based method for detection and estimation is presented for a monochromatic continuous gravitational wave signal emitted by such a source. The so-called pulsar terms in the signal that arise due to the breakdown of the long-wavelength approximation are explicitly taken into account in this method. In addition, the method accounts for equality and inequality constraints involved in the semi-analytical maximization of the likelihood over a subset of the parameters. The remaining parameters are maximized over numerically using Particle Swarm Optimization. Thus, the method presented here solves the monochromatic continuous wave detection and estimation problem without invoking some of the approximations that have been used in earlier studies.Comment: 33 pages, 10 figures, submitted to Ap

A primer of swarm equilibria

We study equilibrium configurations of swarming biological organisms subject to exogenous and pairwise endogenous forces. Beginning with a discrete dynamical model, we derive a variational description of the corresponding continuum population density. Equilibrium solutions are extrema of an energy functional, and satisfy a Fredholm integral equation. We find conditions for the extrema to be local minimizers, global minimizers, and minimizers with respect to infinitesimal Lagrangian displacements of mass. In one spatial dimension, for a variety of exogenous forces, endogenous forces, and domain configurations, we find exact analytical expressions for the equilibria. These agree closely with numerical simulations of the underlying discrete model.The exact solutions provide a sampling of the wide variety of equilibrium configurations possible within our general swarm modeling framework. The equilibria typically are compactly supported and may contain $\delta$-concentrations or jump discontinuities at the edge of the support. We apply our methods to a model of locust swarms, which are observed in nature to consist of a concentrated population on the ground separated from an airborne group. Our model can reproduce this configuration; quasi-two-dimensionality of the model plays a critical role.Comment: 38 pages, submitted to SIAM J. Appl. Dyn. Sy

Binary-induced collapse of a compact, collisionless cluster

We improve and extend Shapiro's model of a relativistic, compact object which is stable in isolation but is driven dynamically unstable by the tidal field of a binary companion. Our compact object consists of a dense swarm of test particles moving in randomly-oriented, initially circular, relativistic orbits about a nonrotating black hole. The binary companion is a distant, slowly inspiraling point mass. The tidal field of the companion is treated as a small perturbation on the background Schwarzschild geometry near the hole; the resulting metric is determined by solving the perturbation equations of Regge and Wheeler and Zerilli in the quasi-static limit. The perturbed spacetime supports Bekenstein's conjecture that the horizon area of a near-equilibrium black hole is an adiabatic invariant. We follow the evolution of the system and confirm that gravitational collapse can be induced in a compact collisionless cluster by the tidal field of a binary companion.Comment: 9 Latex pages, 14 postscript figure