Spherical Multipole Moments Research Articles

Biorthogonal harmonics for the decomposition of gravitational radiation. I. Angular modes, completeness, and the introduction of adjoint-spheroidal harmonics

The estimation of radiative modes is a central problem in gravitational wave theory, with essential applications in signal modeling and data analysis. This problem is complicated by most astrophysically relevant systems not having modes that are analytically tractable. A ubiquitous workaround is to use not modes, but multipole moments defined by spin-weighted spherical harmonics. However, spherical multipole moments are only related to the modes of systems without angular momentum. As a result, they can obscure the underlying physics of astrophysically relevant systems, such as binary black hole merger and ringdown. In such cases, spacetime angular momentum means that radiative modes are not spherical, but spheroidal in nature. Here, we work through various problems related to spheroidal harmonics. We show for the first time that spheroidal harmonics are not only capable of representing arbitrary gravitational wave signals, but that they also possess a kind of orthogonality not used before in general relativity theory. Along the way we present a new class of spin-weighted harmonic functions dubbed ``adjoint-spheroidal'' harmonics. These new functions may be used for the general estimation of spheroidal multipole moments via complete biorthogonal decomposition (in the angular domain). By construction, adjoint-spheroidal harmonics suppress mode-mixing effects known to plague spherical harmonic decomposition; as a result, they better approximate a system's true radiative modes. We discuss potential applications of these results. Lastly, we briefly comment on the challenges posed by the analogous problem with Teukolsky's radial functions.

Converting between the Cartesian tensor and spherical harmonic expansion of solutions to the Boltzmann equation

The Vlasov–Fokker–Planck (VFP) equation, a variant of the Boltzmann equation, is frequently used to model the dynamics of laboratory and astrophysical plasmas. A common approach in solving the VFP equation in numerical and analytic studies is to expand the momentum part of the distribution function $f$ (i.e. the particle density in phase-space) in terms of known functions. The Cartesian tensor and spherical harmonic expansion have been widely used, leading to the question of how to convert between the different coefficients of the two expansions. This problem is also familiar in multipole expansions of an electrostatic (or gravitational) potential. The coefficients of the Cartesian tensor expansion of the potential are called (Cartesian) multipole moments and the ones of the spherical harmonic expansion are called spherical multipole moments. In this paper, we investigate the relation between the two kinds of multipole moments and provide a general formalism to convert between them. We subsequently apply this formalism to the coefficients of the expansions of the distribution function $f$ . A free, open-source command-line tool which implements this formalism is provided.

A numerical study of vector resonant relaxation

Stars bound to a supermassive black hole interact gravitationally. Persistent torques acting between stellar orbits lead to the rapid resonant relaxation of the orbital orientation vectors ("vector" resonant relaxation) and slower relaxation of the eccentricities ("scalar" resonant relaxation), both at rates much faster than two-body or non-resonant relaxation. We describe a new parallel symplectic integrator, N-ring, which follows the dynamical evolution of a cluster of N stars through vector resonant relaxation, by averaging the pairwise interactions over the orbital period and periapsis-precession timescale. We use N-ring to follow the evolution of clusters containing over 10^4 stars for tens of relaxation times. Among other results, we find that the evolution is dominated by torques among stars with radially overlapping orbits, and that resonant relaxation can be modelled as a random walk of the orbit normals on the sphere, with angular step size ranging from 0.5-1 radian. The relaxation rate in a cluster with a fixed number of stars is proportional to the RMS mass of the stars. The RMS torque generated by the cluster stars is reduced below the torque between Kepler orbits due to apsidal precession and declines weakly with the eccentricity of the perturbed orbit. However since the angular momentum of an orbit also decreases with eccentricity, the relaxation rate is approximately eccentricity-independent for e<0.7 and grows rapidly with eccentricity for e>0.8. We quantify the relaxation using the autocorrelation function of the spherical multipole moments; this decays exponentially and the e-folding time may be identified with the vector resonant relaxation timescale.

Analytic expressions for gravitational inner multipole moments of elementary solids and for the force between two rectangular solids

We give analytic expressions for the gravitational inner spherical multipole moments, qlm with l ⩽ 5, for 11 elementary solid shapes. These moments, in conjunction with their known rotational and translational properties, can be used to calculate precisely the moments of complex objects that may be assembled from the elementary shapes. We also give an analytic expression for the gravitational force between two rectangular solids at arbitrary separations. These expressions are useful for computing the gravitational properties of complex instruments, such as those used in equivalence principle tests, and in the gravitational balancing of drag-free spacecraft.

Spherical magnetic multipole moments system of currents

Concepts of spherical magnetic multipoles that represent distributions of electric currents over a spherical surface are introduced. Vector potentials of magnetic multipoles meet solenoidal- and harmonic-field conditions outside of the spherical surface and are continuous on it. Within the sphere, the vector potential of currents flowing outside of it is represented by the sum of vector potentials of basis magnetic multipoles with coefficients expressed by spherical multipole moments of system of currents. This expansion of the vector potential is in many respects analogous to the multipole expansion known from electrostatics. The first three terms of the expansion represent components of the well-known magnetic moment, the next five terms represent components of the magnetic quadrupole moment, etc. Possible applications of the magnetic spherical multipole technique are discussed.

Variational principle, characteristic electric multipoles, and higher polarizing moments in field theory

Based on the variational principle, we introduce a new notion: the characteristic electric multipoles constituting a system of basic distributions of charge on the boundary of a spatial domain. Inside the domain, potentials of the characteristic multipoles are harmonic polynomials whose orders determine the minimum orders of nonzero spherical multipole moments of the characteristic multipoles. Using the characteristic multipole formalism, we solve the moment problem in electrostatics and construct the superconductor Lagrangian in an electrostatic field. We express the empty-space Green's function for the Laplace equation using the characteristic multipole potentials.

The Electrostatic Interaction of Rigid, Globular Proteins with Arbitrary Charge Distributions

We use the linearized Poisson–Boltzmann equation to construct an analytical estimate of the electrostatic interaction energy for two rigid, globular proteins with arbitrary charge distributions represented by spherical multipole moments. Our analysis avoids invoking the superposition approximation and therefore is not limited to weakly interacting double layers. The interaction energy is a function of the protein properties (radii, multipole moments, and dielectric constant); the geometry (protein separation and orientations); and the solution properties (Debye screening length and dielectric constant). The proteins may be different. This is an extension of previous treatments for constant surface charge or for constant surface potential particles that are more applicable to isotropic colloids. We give example calculations for the interactions of Ribonuclease A molecules interacting in an electrolyte. These results suggest that electrostatic effects may be largely responsible for attractions between proteins, even if they are like charged. A set of routines using MATHEMATICA is available for performing the interaction energy and electrostatic potential calculations.

Higher-order interaction energies for a system of N arbitrary molecules in the light of spherical tensor theory

The Rayleigh–Schrödinger perturbation treatment accompanied by a multipole expansion of the interaction potential can be used in the quantum-mechanical studies of long-range intermolecular forces. The total third-order interaction energy in a system of N molecules separates into several induction and dispersion categories. Only some of them are purely pairwise additive. In the present paper the closed expressions for all possible categories of the third-order anisotropic interaction energy in a collection of N arbitrary molecules are derived, in which the orientational dependence is pushed to its limits. The derivation is based on the spherical tensor formalism. The formulas for the induction energies are directly related to the spherical multipole moments and irreducible (hyper)polarizabilities of the interacting molecules in body-fixed frames. The same is achieved for the dispersion categories after employing some simple approximations. The present paper can be treated as an extension of the spherical tensor description of the two-body long-range molecular interactions for the most important quantum-mechanical pairwise nonadditive forces.