Octupole Term Research Articles

Hierarchical Three-Body Problem at High Eccentricities = Simple Pendulum II: Octupole including Brown’s Hamiltonian

Abstract The very long-term evolution of the hierarchical restricted three-body problem with a massive perturber is analyzed analytically in the high eccentricity regime. Perturbations on the time scale of the outer orbit can accumulate over long timescales and be comparable to the effect of the octupole term. These perturbations are described by Brown’s Hamiltonian - having different forms in the literature. We show that at the high eccentricity regime - the effect of Brown’s Hamiltonian is an azimuthal precesssion of the eccentricity vector and can be solved analytically. In fact, the dynamics are equivalent to a simple pendulum model allowing an explicit flip criterion.

Hierarchical three-body problem at high eccentricities = simple pendulum I: octupole

ABSTRACT The gradual evolution of the restricted hierarchical three body problem is analysed analytically, focusing on conditions of Kozai–Lidov cycles that may lead to orbital flips from prograde to retrograde motion due to the octupole (third order) term which are associated with extremely high eccentricities. We revisit the approach described by Katz, Dong and Malhotra (2011) and show that for most initial conditions, to an excellent approximation, the analytical derivation can be greatly simplified and reduces to a simple pendulum model allowing an explicit flip criterion. The resulting flip criterion is much simpler than the previous one but the latter is still needed in a small fraction of phase space. We identify a logical error in the earlier derivation but clarify why it does not affect the final results.

Stability of hierarchical triples – II. The inclination ι = 140° resonance in the stability surface

ABSTRACT The stability of a hierarchical triple star may be decided by a simple criterion, which was derived in Paper I. However, there is a region in the phase space where the stability limit Qmax is raised by a factor of two in a small region of the phase space with respect to the surrounding phase space. The phase space is defined by the inner and outer eccentricities ein and eout, respectively, as well as by the inclination ιtot between the inner and outer orbits. Additional parameters of the phase space are the masses of the three bodies. We study by numerical integration the orbits of over 100 000 triple systems in the resonance region. We find that the instability that causes the high value of Qmax arises from the octupole Kozai–Lidov resonance. This resonance region has rather equal contributions from the quadrupole and octupole terms and leads to secular evolution with an amplitude larger than either of the two oscillations in isolation. The conditions for this situation are best satisfied near the relative inclination ι = 140°. Additionally, the relative orientation of the two orbits plays a decisive role: the resonance is found only at certain values of the orbit’s node line longitude Ω. An analytical approximation of the energy change in a single close encounter between the inner and outer systems suggests a cos 2Ω dependence of Qmax on Ω, which seems to be qualitatively valid. We model Qmax as a function of cos ι and ein by a Gaussian function.

Hyperfine coupling constants of the cesium 7D5/2 state measured up to the octupole term.

We report the measurement of hyperfine splitting (HFS) in the 7D5/2 state of 133Cs using high-resolution Doppler-free two-photon spectroscopy enabled by precise frequency scans using an acousto-optic modulator (AOM). All six hyperfine levels are resolved in our spectra. We determine the hyperfine coupling constants A = -1.70867(62) MHz and B = 0.050(14) MHz which represent an over 20-times improvement in the precision of both A and B. Moreover, our measurement is sufficiently precise to put bounds on the value of the magnetic octupole coupling constant C = 0.4(1.4) kHz for the 7D5/2 state. We additionally report the measurement of the ac Stark shift [-46 ± 4 Hz/(W/cm2)], collisional shift, and pressure broadening which are important for optical frequency standards based on the 6S1/2 → 7D5/2 two-photon transition.

The equation of state for metal-doped ferroelectrics within the Weiss model

This paper presents a theoretical model for describing the thermodynamic properties of doped ferroelectric crystals based on a modified Weiss mean-field approach. Accounting for quadrupole and octupole terms in the expression for the effective field within the Weiss model makes it possible to move from the Langevin equation to the Landau–Ginzburg equation. Furthermore, the coefficients of the Landau–Ginzburg equation can be expressed in terms of the physical parameters of the crystal lattice. For these parameters, analytical expressions are proposed that describe their change when adding dopants in ceramic matrix composites. Perovskite barium titanate ceramics with a variety of inclusions is considered as an application example of the developed method. The obtained agreement between the analytical and experimental results for barium titanate ceramics with lanthanum/magnesium/zirconium dopants gives us hope of the applicability of the present theory to the calculation of other doped ferroelectrics as well.

Non-dipole anisotropy parameters in the photoionization of Kr in the region of deep inner shell excitations

The photoelectron angular distribution (PAD) of the Kr 4p1/2 and 4p3/2 fine structure states was measured with linearly polarized synchrotron radiation in the vicinity of the resonant excitations of the 3p subshell. Experimental dipole and non-dipole anisotropy parameters were determined from the measured angular differential cross sections. In order to interpret our experimental results we have used a theoretical model going beyond the dipole approximation considering the quadrupole and octupole terms for the direct photoionization. We have taken into account several autoionization channels relevant for the studied photon energy range. The photon energy dependence of the measured anisotropy parameters puts in evidence the importance of the channel interactions. Moreover, unexpectedly large non-dipole contribution have been observed.

Structures of secular resonances for inner test particles in hierarchical planetary systems

In this study, dynamics of secular resonances for inner test particles are investigated under the octupole-level approximation by taking non-perturbative approaches. In practice, webs of the major secular resonances are produced by identifying families of stable periodic orbits and the associated stable libration zones are obtained by analysing Poincaré surfaces of section. By taking different values of the factor \(\epsilon \) (\(\epsilon \) measures the contribution of octupole terms), the influences of the octupole-order terms upon the dynamical structures are evaluated. Under the condition of \(\epsilon = 0\) (without octupole-order contribution), the dynamical model is totally integrable and there is only Kozai resonance arising in the phase space. When the factor \(\epsilon \) is different from zero, the dynamical structure in the phase space becomes complicated due to varieties of secular resonances. Numerical results further indicate that (a) distributions of libration centres and stable libration zones remain qualitatively similar with different values of \(\epsilon \), (b) Kozai resonance disappears in the low-eccentricity region due to the chaotic motion and (c) the chaotic area arising in the low-eccentricity region increases with the factor \(\epsilon \). Secular resonances are the source of many important dynamical phenomena, such as chaos, orbit alignment and orbit flipping, and thus, the results presented in this work could be useful to understand the secular dynamics for those high-eccentricity and/or high-inclination objects in hierarchical planetary systems.

Characteristic periods of the paleosecular variation of the Earth's magnetic field during the Holocene from global paleoreconstructions

The knowledge of the secular variation of the geomagnetic field at different time scales is important to determine the mechanisms that maintain the geomagnetic field and can help to establish constraints in dynamo theories. We have focused our study on the secular variation at millennial and centennial time scale searching for characteristic periods during the last 10 kyr. The frequency study was performed using four recent updated global paleomagnetic field reconstructions (SHA.DIF.14k, CALS10k.2, BIGMUDI4k and SHAWQ2k) by applying three techniques commonly used in signal analysis: the Fourier transform, the Empirical Mode Decomposition, and the wavelet analysis.Short-term variability of the geomagnetic field energy shows recurrent periods of around 2000, 1000–1400, and 600–800 and 250–400 years. The characteristic time around 600–800 years is well determined in all paleomagnetic reconstructions and it is mostly related to the axial dipole and axial octupole terms, but also observable in the equatorial dipole. In addition to this period, longer characteristic times of around 1000–1400 years are found particularly in the equatorial dipole and quadrupole terms in SHA.DIF.14k, CALS10k.2 and BIGMUDI4k while the 2000 year period is only well determined in the total geomagnetic field energy of SHA.DIF.14k and CALS10k.2. The most detailed paleoreconstructions for younger times also detect shortest characteristic times of around 250–400 years.The long-term variation of the geomagnetic energy is only observable in the axial dipole. A characteristic period of around 7000 years in both SHA.DIF.14k and CALS10k.2 has been found. This long period is related to two decays in the dipole field and a period of increasing intensity. The oldest decay took place between 7000 BCE and 4500 BCE and the present decay that started around 100 BCE. We have modeled the 4500 BCE up to present variation as a combination of a continuous decay, representing the diffusion term of the geomagnetic field, and one pulse that reinforces the strength of the field. Results show a characteristic diffusion time of around 11,000–15,000 years, which is compatible with the diffusion times of the dipole field used in geodynamo theories.

Multipole interaction in spectral line broadening by plasmas

A long-standing ambiguity in spectral line broadening theory related to the electric field divergence in the quadrupole term of the screened Coulomb radiator-plasma interaction is examined. This is accomplished with multipole expansions from a Taylor series as well as a partial wave analysis of the screened Coulomb interaction. It is explicitly shown that the two approaches agree up to the octupole term for an effective radiator size small compared to the plasma screening length. Furthermore, the ambigous corrections are confirmed appearing naturally in both methods. For higher multipoles the agreement is demonstrated numerically with a computer program using exact arithmetic.

Implicit Solvation Using the Superposition Approximation (IS-SPA): Extension to Peptides in a Polar Solvent.

Efficient, accurate, and adaptable implicit solvent models remain a significant challenge in the field of molecular simulation. A recent implicit solvent model, IS-SPA, based on approximating the mean solvent force using the superposition approximation, provides a platform to achieve these goals. IS-SPA was originally developed to handle nonpolar solutes in a polar solvent and did not accurately capture polar solvation. Here, we demonstrate that IS-SPA can accurately capture polar solvation by incorporating solvent orientation and accounting for the contributions from long ranged electrostatics. Solvent orientation is approximated as that of an ideal dipole aligned in a mean electrostatic field and an analytic form of the long ranged electrostatics is derived. Parameters for the model are calculated from explicit solvent simulations of an isolated atom or molecule and include atom-based solvent densities, mean electric field functions, radially symmetric averaged Lennard-Jones forces, and multipoles of the explicit solvent model. Using these parameters, IS-SPA accounts for asymmetry of charge solvation and reproduces the explicit solvent potential of mean force of dimerization of two oppositely charged Lennard-Jones spheres in chloroform with high fidelity. Additionally, the model more accurately captures the effect of explicit solvent on the monomer and dimer configurations of alanine dipeptide in chloroform than a generalized Born or constant density dielectric model. The current version of the algorithm is expected to outperform explicit solvent simulations for aggregation of small peptides at concentrations below 150 mM, well above the typical experimental concentrations for these materials.

Secular Resonances in Nonrestricted Hierarchical Triple Systems

Abstract In this paper, the averaged Hamiltonian of a nonrestricted hierarchical triple system truncated at the third order is investigated. First, each secular resonant term is studied. For the well-studied secular quadrupole theory, it is analytically reformulated in a different manner in our work. The resonance width is numerically determined and displayed on the plane (also denoted as the plane). In terms of the octupole terms, we show that for a near-planar configuration of the system, considerable variations of both the eccentricities of the inner and outer orbits can be generated by a single resonant term. The resonance width for every secular resonant angle from the octupole terms is also numerically determined and displayed on the plane. The results show that an orbit flip with a near-perpendicular initial mutual inclination is possible for each secular resonance. By displaying the resonance widths of different resonant terms on the same plane, we intuitively show the overlap of different secular resonances. Then, the full averaged Hamiltonian with both quadrupole and octupole terms is investigated using the Poincaré surface of section, with a special focus on the orbit flip. For the cases we exploited, we find that the near-planar flip of the inner orbit can be either regular or chaotic while the outer orbit flip is generally chaotic.

Description of heterogeneous plasma microfield and optical properties of plasma by the QUIP model

Optical properties of plasmas are determined by presence of fluctuating microscopic electric field. The present work provides thorough analysis of contemporary models and points out their shortcomings. To overcome the latter, we take the QUIP (QUasi-Independent Particles) model derived ab initio. We provide generalization of the model allowing to account for microfield heterogeneity up to octupole term. We investigate convergence of the multipole series and show that higher order terms can be neglected. The model does not require laborious computations because all formulae are rather simple and are given in explicit form. This fact is an advantage of the proposed model compared to other contemporary models. We perform verification of the model via comparison with experiments. We emphasize that the comparison should be made with respect to the number of observed lines because this number strongly depends on the selected model. We outline experiments suitable for such testing. These are the experiments on Ar+Kr radiating plasma heated by laser radiation. In these experiments, the entire Ar+16 spectral series is observed. The QUIP model correctly describes the number of observed lines of the series, so its adequateness is justified.

Simulation of heterogeneous plasma microfield

Optical properties of plasma are determined by presence of fluctuating micro-scopic electric field. In the present work, we construct a simple ab initio model of plasma microfield accounting for its heterogeneity up to octupole term for the first time. Comparison with experiments shows that only this model describes the observed number of spectral lines.

Simulation of an Inhomogeneous Plasma Microfield

The optical properties of plasma are determined by the presence of a fluctuating microscopic electric field. A simple ab initio model of a plasma microfield accounting for its inhomogeneity up to an octupole term has been constructed for the first time. A comparison with experiments has shown that only this model correctly describes the observed number of spectral lines.

The approximate calculation of 2D field strength with a Coulomb potential by using multipole expansion

The necessary formulae are derived for approximate computation of the Coulomb-type potential and the field strength, generated by system of particles in 2D case. On the one hand, these formulae represent the multipole expansion in which monopole, dipole, quadrupole and octupole terms are taken into account that allows for computing the potential and the field strength, generated by set of the charges with rather high accuracy. On the other hand, polynomial expansions of the obtained formulae are derived up to the cubic terms with respect to the observation point coordinates, which make it possible to compute the field strength at closely located points approximately. The expressions are derived that permit one to compute the multipole moments for the system of charges, which consists of smaller subsystems. Such formulae are required in tree-codes when the hierarchical (tree-type) structure of geometrical regions is introduced in the region of the initial system of the charges. In particular case these expressions allows for changing of the reference point of multipole moments computation (multipole moments shifting). The test problem is considered for rather small number of the charges, which shows the dependency of the relative error of the field strength reconstruction on the type of multipole expansion and number of considered terms in the Taylor expansion.

Analytic computation of the secular effects of encounters on a binary: third-order perturbation, octupole, and post-Newtonian terms; steady-state distribution

ABSTRACTDense stellar systems such as globular clusters are believed to harbour merging binary black holes (BHs). The evolution of such binaries is driven by interactions with other stars, most notably, binary-single interactions. Traditionally, so-called ‘strong’ interactions are believed to be the driving force in this evolution. However, we recently showed that more distant, i.e. ‘weak’ or ‘secular’ encounters, can have important implications for the properties of merging BH binaries in globular clusters. This motivates more detailed understanding of the effects of secular encounters on a binary. In another previous paper, we analytically calculated expressions for the changes of the eccentricity and angular-momentum vectors taking into account second-order perturbation theory, and showed that, for highly eccentric binaries, the new expressions give rise to behaviour that is not captured by first-order theory. Here, we extend our previous work to third-order perturbation theory. We also include terms up to and including octupole order. The latter are non-zero for binaries with unequal component masses. In addition, we consider the effects of post-Newtonian terms, and we determine the steady-state distribution due to the cumulative effect of secular encounters by computing the associated angular-momentum diffusion coefficients, and applying the Fokker–Planck equation. Together with our previous work, the results in this paper provide a framework for incorporating the effects of distant encounters on binaries in models of cluster evolution, such as Monte Carlo codes.

Analysis of the hypothesis of a giant gaussian process as a means for describing secular variations of the geomagnetic field vector

The consistency of the empirical data on paleointensity and paleoinclinations contained in the BOROKPINT paleointensity world database with the Geocentric Axial Dipole (GAD) hypothesis and Giant Gaussian Process (GGP) model describing the geomagnetic field variations in the Brunhes epoch is tested. The calculations are based on the geomagnetic field potential representation by the sum of spherical functions of spatial coordinates with random coefficients, enabling computer simulation of the data with the given statistical characteristics of the coefficients. The estimates show that the Kolmogorov–Smirnov and Anderson–Darling tests reject the GAD hypothesis in its canonical form. The extension of GAD to GGD with nonzero time-average quadrupole and octupole terms makes the paleointensity and paleoinclination data to comply with GGP model realizations; however, these models are mutually exclusive because of mutual inconsistency of their parameters. Testing the paleoinclination data against GDP model shows that a small correction to the purely dipole component of the geomagnetic field should be introduced. At the same time, the paleointensity data analysis suggests that these data highly probably agree with GGP models with a high quadrupole contribution making up 1/3 of the dipole coefficient, which is strongly at odds with the parameters of the model corresponding to paleoinclination data. This inconsistency is most likely to be due to the artifacts associated with incorrect paleointensity determinations; however, this interpretation does not explain the causes of the strong latitudinal dependence of the intensity of the Virtual Axial Dipole Moment (VADM) which follows from the empirical data.

Analysis of the Hypothesis of a Giant Gaussian Process as a Means for Describing Secular Variations of the Geomagnetic Field Vector

The consistency of the empirical data on the paleointensity and paleoinclinations contained in the BOROKPINT paleointensity world database with the Geocentric Axial Dipole (GAD) hypothesis and the Giant Gaussian Process (GGP) model describing the geomagnetic field variations in the Brunhes epoch is tested. The calculation procedure is based on the representation of the geomagnetic field potential by the sum of the spherical functions of a spatial coordinate with random coefficients, which enables the computer simulation of the data that correspond to the given statistical characteristics of the coefficients. The estimation shows that according to the Kolmogorov–Smirnov and Anderson–Darling tests, the GAD hypothesis in its canonical form should be rejected. The extension of GAD to GGD with nonzero time-average quadrupole and octupole terms brings the paleointensity and paleoinclination data into agreement with the GGP model realizations; however, these models turn out to be mutually exclusive because their parameters are inconsistent with each other. Testing the paleoinclination data against GDP model demonstrates the necessity of introducing a small correction to the purely dipole component of the geomagnetic field. At the same time, the analysis of the paleointensity data shows that these data with a very high probability agree with the GGP models with a high quadrupole contribution making up 1/3 of the dipole coefficient, which is strongly at odds with the parameters of the model corresponding to the paleoinclination data. This inconsistency is most probably caused by the artifacts due to the incorrect determinations of paleointensity; however, this interpretation leaves unsolved the question about the causes of such a strong latitudinal dependence of the intensity of the Virtual Axial Dipole Moment (VADM) what follows from the empirical data.

Monopolar and quadrupolar gravitational radiation from magnetically deformed neutron stars in modified gravity

Some modified theories of gravity are known to predict monopolar, in addition to the usual quadrupolar and beyond, gravitational radiation in the form of `breathing' modes. For the same reason that octupole and higher-multipole terms often contribute negligibly to the overall wave strain, monopole terms tend to dominate. We investigate both monopolar and quadrupolar continuous gravitational radiation from neutron stars deformed through internal magnetic stresses. We adopt the Parameterised-Post-Newtonian formalism to write down equations describing the leading-order stellar properties in a theory-independent way, and derive some exact solutions for stars with mixed poloidal-toroidal magnetic fields. We then turn to the specific case of scalar-tensor theories to demonstrate how observational upper limits on the gravitational-wave luminosity of certain neutron stars may be used to place constraints on modified gravity parameters, most notably the Eddington parameter $\gamma$. For conservative, purely poloidal models with characteristic field strength given by the spindown minimum, upper limits for the Vela pulsar yield $1-\gamma \lesssim 4.2 \times 10^{-3}$. For models containing a strong toroidal field housing $\sim 99\%$ of the internal magnetic energy, we obtain the bound $1- \gamma \lesssim 8.0 \times 10^{-7}$. This latter bound is an order of magnitude tighter than those obtained from current Solar system experiments, though applies to the strong-field regime.

The eccentric Kozai–Lidov effect as a resonance phenomenon

Exploring weakly perturbed Keplerian motion within the restricted three-body problem, Lidov (1962) and, independently, Kozai (1962) discovered coupled oscillations of eccentricity and inclination (the KL-cycles). Their classical studies were based on an integrable model of the secular evolution, obtained by double averaging of the disturbing function approximated with its first non-trivial term. This was the quadrupole term in the series expansion with respect to the ratio of the semimajor axis of the disturbed body to that of the disturbing body. If the next (octupole) term is kept in the expression for the disturbing function, long-term modulation of the KL-cycles can established (Ford et al., 2000, Naoz et al., 2011, Katz et al., 2011). Specifically, flips between the prograde and retrograde orbits become possible. Since such flips are observed only when the perturber has a non-zero eccentricity, the term "Eccentric Kozai-Lidov Effect" (or EKL-effect) was proposed by Lithwick and Naoz (2011) to specify such behaviour. We demonstrate that the EKL-effect can be interpreted as a resonance phenomenon. To this end, we write down the equations of motion in terms of "action-angle" variables emerging in the integrable Kozai-Lidov model. It turns out that for some initial values the resonance is degenerate and the usual "pendulum" approximation is insufficient to describe the evolution of the resonance phase. Analysis of the related bifurcations allows us to estimate the typical time between the successive flips for different parts of the phase space.