Algorithms for Stellar Perturbation Computations on Oort Cloud Comets

We investigate different approximate methods of computing the perturbations on the orbits of Oort cloud comets caused by passing stars, by checking them against an accurate numerical integration using Everhart’s RA15 code. The scenario under study is the one relevant for long-term simulations of the cloud’s response to a predefined set of stellar passages. Our sample of stellar encounters simulates those experienced by the Solar System currently, but extrapolated over a time of 10¹⁰ years. We measure the errors of perihelion distance perturbations for high-eccentricity orbits introduced by several estimators – including the classical impulse approximation and Dybczyński’s (1994, Celest. Mech. Dynam. Astron. 58, 1330–1338) method – and we study how they depend on the encounter parameters (approach distance and relative velocity). We introduce a sequential variant of Dybczyński’s approach, cutting the encounter into several steps whereby the heliocentric motion of the comet is taken into account. For the scenario at hand this is found to offer an efficient means to obtain accurate results for practically any domain of the parameter space.     

Energization of charged particles in planetary magnetospheres

A model is presented to describe the energization of charged particles in planetary magnetospheres. The model is based on the stochastic acceleration produced by a random electric field that is induced by the magnetic field fluctuations measured within the magnetospheres. The stochastic behavior of the electric field is simulated through a Monte Carlo method. We solve the equation of motion for a single charged particle—which comprises the stochastic acceleration due to the stochastic electric field, the Lorentz acceleration (containing the local magnetic field and the corotational electric field) and the gravitational planetary acceleration of the particle—under several initial conditions. The initial conditions include the ion species and the velocity distribution of the particles which depends on the sources they come from (solar wind, ionospheres, rings and satellites). We applied this model to Saturn’s inner magnetosphere using a sample of particles (H⁺, H₂O⁺, N⁺, O⁺ and OH⁺) initially located on Saturn’s north pole, above the C-Ring, on the south pole of Enceladus, in the north pole of Dione and above the E-Ring. The results show that the particles tend to increase the value of their energy with time reaching several eV in a few seconds and the large energization is observed far from the planet. We can distinguish three main energization regions within Saturn’s inner magnetosphere: minimum (Saturn’s ionosphere), intermediate (Dione) and high-energy (Enceladus and the E-ring). The resulting energy spectrum follows a power-law distribution (>1 keV), a logistic, an exponential decay or an asymmetric sigmoidal (<1 keV).     

About the Hill stability of extrasolar planets in stellar binary systems

We use a three dimensional generalization of Szebehely’s invariant relation obtained by us (Makó and Szenkovits, Celest. Mech. Dyn. Astron. 90, 51, 2004) in the elliptic restricted three-body problem, to establish more accurate criterion of the Hill stability. By using this criterion, the Hill stability of four extrasolar planets (γ Cephei Ab, Gliese 86 Ab, HD 41004 Ab and HD 41004 Bb) is investigated.     

Numerical integration of dynamical systems with Lie series

The integration of the equations of motion in gravitational dynamical systems—either in our Solar System or for extra-solar planetary systems—being non integrable in the global case, is usually performed by means of numerical integration. Among the different numerical techniques available for solving ordinary differential equations, the numerical integration using Lie series has shown some advantages. In its original form (Hanslmeier and Dvorak, Astron Astrophys 132, 203 1984), it was limited to the N-body problem where only gravitational interactions are taken into account. We present in this paper a generalisation of the method by deriving an expression of the Lie terms when other major forces are considered. As a matter of fact, previous studies have been done but only for objects moving under gravitational attraction. If other perturbations are added, the Lie integrator has to be re-built. In the present work we consider two cases involving position and position-velocity dependent perturbations: relativistic acceleration in the framework of General Relativity and a simplified force for the Yarkovsky effect. A general iteration procedure is applied to derive the Lie series to any order and precision. We then give an application to the integration of the equation of motions for typical Near-Earth objects and planet Mercury.     

Canonical Modelling of Relative Spacecraft Motion Via Epicyclic Orbital Elements

This paper presents a Hamiltonian approach to modelling spacecraft motion relative to a circular reference orbit based on a derivation of canonical coordinates for the relative state-space dynamics. The Hamiltonian formulation facilitates the modelling of high-order terms and orbital perturbations within the context of the Clohessy–Wiltshire solution. First, the Hamiltonian is partitioned into a linear term and a high-order term. The Hamilton–Jacobi equations are solved for the linear part by separation, and new constants for the relative motions are obtained, called epicyclic elements. The influence of higher order terms and perturbations, such as Earth’s oblateness, are incorporated into the analysis by a variation of parameters procedure. As an example, closed-form solutions for J₂-invariant orbits are obtained.     

On constructing the general Earth’s rotation theory

In the present paper the equations of the orbital motion of the major planets and the Moon and the equations of the three–axial rigid Earth’s rotation in Euler parameters are reduced to the secular system describing the evolution of the planetary and lunar orbits (independent of the Earth’s rotation) and the evolution of the Earth’s rotation (depending on the planetary and lunar evolution). Hence, the theory of the Earth’s rotation can be presented by means of the series in powers of the evolutionary variables with quasi-periodic coefficients with respect to the planetary–lunar mean longitudes. This form of the Earth’s rotation problem is compatible with the general planetary theory involving the separation of the short–period and long–period variables and avoiding the appearance of the non–physical secular terms.     

Precession/nutation solution consistent with the general planetary theory

In the present paper the equations of the translatory motion of the major planets and the Moon and the Poisson equations of the Earth’s rotation in Euler parameters are reduced to the secular system describing the evolution of the planetary and lunar orbits (independent of the Earth’s rotation) and the evolution of the Earth’s rotation (depending on the planetary and lunar evolution). Hence, the theory of the Earth’s rotation is presented by means of the series in powers of the evolutionary variables with quasi-periodic coefficients.     

Complicated magnetic field structure of the star HD 45583

Preliminary data on the magnetic field structure of the unique magnetic star HD 45583 are obtained. The observational data are well described by a configuration of two magnetic dipoles located on opposite sides relative to the star’s center, with their axes directed roughly in a radial direction. The positive monopoles are closer to the surface and the negative, closer to the star’s center. For this reason, there appear to be two positive magnetic poles on the star’s surface but no negative poles. The need for further observations of this unique object is pointed out. Translated from Astrofizika, Vol. 52, No. 1, pp. 127–133 (February 2009).     

MSFC Stream Model Preliminary Results: Modeling Recent Leonid and Perseid Encounters

The cometary meteoroid ejection model of Jones and Brown [Physics, Chemistry, and Dynamics of Interplanetary Dust, ASP Conference Series 104 (1996b) 137] was used to simulate ejection from comets 55P/Tempel-Tuttle during the last 12 revolutions, and the last 9 apparitions of 109P/Swift-Tuttle. Using cometary ephemerides generated by the Jet Propulsion Laboratory’s (JPL) HORIZONS Solar System Data and Ephemeris Computation Service, two independent ejection schemes were simulated. In the first case, ejection was simulated in 1 h time steps along the comet’s orbit while it was within 2.5 AU of the Sun. In the second case, ejection was simulated to occur at the hour the comet reached perihelion. A 4th order variable step-size Runge–Kutta integrator was then used to integrate meteoroid position and velocity forward in time, accounting for the effects of radiation pressure, Poynting–Robertson drag, and the gravitational forces of the planets, which were computed using JPL’s DE406 planetary ephemerides. An impact parameter (IP) was computed for each particle approaching the Earth to create a flux profile, and the results compared to observations of the 1998 and 1999 Leonid showers, and the 1993 and 2004 Perseids.     

Long-term evolution of orbits about a precessing oblate planet. 2. The case of variable precession

We continue the study undertaken in Efroimsky [Celest. Mech. Dyn. Astron. 91, 75–108 (2005a)] where we explored the influence of spin-axis variations of an oblate planet on satellite orbits. Near-equatorial satellites had long been believed to keep up with the oblate primary’s equator in the cause of its spin-axis variations. As demonstrated by Efroimsky and Goldreich [Astron. Astrophys. 415, 1187–1199 (2004)], this opinion had stemmed from an inexact interpretation of a correct result by Goldreich [Astron. J. 70, 5–9 (1965)]. Although Goldreich [Astron. J. 70, 5–9 (1965)] mentioned that his result (preservation of the initial inclination, up to small oscillations about the moving equatorial plane) was obtained for non-osculating inclination, his admonition had been persistently ignored for forty years. It was explained in Efroimsky and Goldreich [Astron. Astrophys. 415, 1187–1199 (2004)] that the equator precession influences the osculating inclination of a satellite orbit already in the first order over the perturbation caused by a transition from an inertial to an equatorial coordinate system. It was later shown in Efroimsky [Celest. Mech. Dyn. Astron. 91, 75–108 (2005a)] that the secular part of the inclination is affected only in the second order. This fact, anticipated by Goldreich [Astron. J. 70, 5–9 (1965)], remains valid for a constant rate of the precession. It turns out that non-uniform variations of the planetary spin state generate changes in the osculating elements, that are linear in , where is the planetary equator’s total precession rate that includes the equinoctial precession, nutation, the Chandler wobble, and the polar wander. We work out a formalism which will help us to determine if these factors cause a drift of a satellite orbit away from the evolving planetary equator.By “precession,” in its most general sense, we mean any change of the direction of the spin axis of the planet—from its long-term variations down to nutations down to the Chandler wobble and polar wander.     

3-D Rpic Simulations of Relativistic Jets: Particle Acceleration, Magnetic Field Generation, and Emission

We have applied numerical simulations and modeling to the particle acceleration, magnetic field generation, and emission from relativistic shocks. We investigate the nonlinear stage of theWeibel instability and compare our simulations with the observed gamma-ray burst emission. In collisionless shocks, plasma waves and their associated instabilities (e.g., the Weibel, Buneman and other two-stream instabilities) are responsible for particle (electron, positron, and ion) acceleration and magnetic field generation. 3-D relativistic electromagnetic particle (REMP) simulations with three different electron-positron jet velocity distributions and also with an electron-ion plasma have been performed and show shock processes including spatial and temporal evolution of shocks in unmagnetized ambient plasmas. The growth time and nonlinear saturation levels depend on the initial jet parallel velocity distributions. Simulations show that the Weibel instability created in the collisionless shocks accelerates jet and ambient particles both perpendicular and parallel to the jet propagation direction. The nonlinear fluctuation amplitude of densities, currents, electric, and magnetic fields in the electron-positron shocks are larger for smaller jet Lorentz factor. This comes from the fact that the growth time of the Weibel instability is proportional to the square of the jet Lorentz factor. We have performed simulations with broad Lorentz factor distribution of jet electrons and positrons, which is assumed to be created by photon annihilation. Simulation results with this broad distribution show that the Weibel instability is excited continuously by the wide-range of jet Lorentz factor from lower to higher values. In all simulations the Weibel instability is responsible for generating and amplifying magnetic fields perpendicular to the jet propagation direction, and contributes to the electron’s (positron’s) transverse deflection behind the jet head. This small scale magnetic field structure contributes to the generation of “jitter” radiation from deflected electrons (positrons), which is different from synchrotron radiation in uniform magnetic fields. The jitter radiation resulting from small scale magnetic field structures may be important for understanding the complex time structure and spectral evolution observed in gamma-ray bursts or other astrophysical sources containing relativistic jets and relativistic collisionless shocks. The detailed studies of shock microscopic process evolution may provide some insights into early and later GRB afterglows.     

Long-Term Evolution of Orbits About A Precessing Oblate Planet: 1. The Case of Uniform Precession

It was believed until very recently that a near-equatorial satellite would always keep up with the planet’s equator (with oscillations in inclination, but without a secular drift). As explained in Efroimsky and Goldreich [Astronomy & Astrophysics (2004) Vol. 415, pp. 1187–1199], this misconception originated from a wrong interpretation of a (mathematically correct) result obtained in terms of non-osculating orbital elements. A similar analysis carried out in the language of osculating elements will endow the planetary equations with some extra terms caused by the planet’s obliquity change. Some of these terms will be non-trivial, in that they will not be amendments to the disturbing function. Due to the extra terms, the variations of a planet’s obliquity may cause a secular drift of its satellite orbit inclination. In this article we set out the analytical formalism for our study of this drift. We demonstrate that, in the case of uniform precession, the drift will be extremely slow, because the first-order terms responsible for the drift will be short-period and, thus, will have vanishing orbital averages (as anticipated 40 years ago by Peter Goldreich), while the secular terms will be of the second order only. However, it turns out that variations of the planetary precession make the first-order terms secular. For example, the planetary nutations will resonate with the satellite’s orbital frequency and, thereby, may instigate a secular drift. A detailed study of this process will be offered in a subsequent publication, while here we work out the required mathematical formalism and point out the key aspects of the dynamics. In this article, as well as in (Efroimsky 2004), we use the word ‘‘precession’’ in its most general sense which embraces the entire spectrum of changes of the spin-axis orientation -- from the long-term variations down to the Chandler Wobble down to nutations and to the polar wonder.     

A second order analytical atmospheric drag theory based on the TD88 thermospheric density model

A second order atmospheric drag theory based on the usage of TD88 model is constructed. It is developed to the second order in terms of TD88 small parameters K _n,j . The short periodic perturbations, of all orbital elements, are evaluated. The secular perturbations of the semi-major axis and of the eccentricity are obtained. The theory is applied to determine the lifetime of the satellites ROHINI (1980 62A), and to predict the lifetime of the microsatellite MIMOSA. The secular perturbations of the nodal longitude and of the argument of perigee due to the Earth’s gravity are taken into account up to the second order in Earth’s oblateness.     

Planetary distance law and resonance

The relation between the planetary distance law and the resonant structures in the solar system and in the satellite systems is shown, in that, the resonance relation has been expressed in terms of Roche’s (Bode’s) constant defined in the text. This brings forth a coherent, elegant and unified picture of the formation and structure of the solar system and the satellite systems. The Roche’s (Bode’s) constant is seen to play a central role in this unified picture, in that, it also appears to govern the resonance phenomenon in the systems     

Planetary perturbations of the Moon in ELP 2000

A new solution for the planetary perturbations of the Moon is being built in the frame of ELP 2000, using Bretagnon's planetary theories, and achieved at the first order. It contains the two actions commonly distinguished: direct and indirect. The internal precision of computation is 2×10^–6 arcsec. First-order planetary perturbations, in the direct case (Venus & Mars), have been compared to Standaert's solution. The major discrepancy reaches 70 cm in the longitude of Venus. Perturbations of the second order with respect to planetary masses, have been undertaken and illustrations are given. Finally, new values for the perigee and node motions are proposed.Proceedings of the Conference on Analytical Methods and Ephemerides: Theory and Observations of the Moon and Planets. Facultés universitaires Notre Dame de la Paix, Namur, Belgium, 28–31 July, 1980.     

The accelerated Kepler problem

The accelerated Kepler problem (AKP) is obtained by adding a constant acceleration to the classical two-body Kepler problem. This setting models the dynamics of a jet-sustaining accretion disk and its content of forming planets as the disk loses linear momentum through the asymmetric jet-counterjet system it powers. The dynamics of the accelerated Kepler problem is analyzed using physical as well as parabolic coordinates. The latter naturally separate the problem’s Hamiltonian into two unidimensional Hamiltonians. In particular, we identify the origin of the secular resonance in the AKP and determine analytically the radius of stability boundary of initially circular orbits that are of particular interest to the problem of radial migration in binary systems as well as to the truncation of accretion disks through stellar jet acceleration.     

Deprit’s reduction of the nodes revisited

We revisit a set of symplectic variables introduced by Andre Deprit (Celest Mech 30, 181–195, 1983), which allows for a complete symplectic reduction in rotation invariant Hamiltonian systems, generalizing to arbitrary dimension Jacobi’s reduction of the nodes. In particular, we introduce an action-angle version of Deprit’s variables, connected to the Delaunay variables, and give a new hierarchical proof of the symplectic character of Deprit’s variables.     

Recent Experimental Results and Modelling of High-Mach-Number Jets and the Transition to Turbulence

In recent years, we have carried out experiments at the University of Rochester’s Omega laser in which supersonic, dense-plasma jets are formed by the interaction of strong shocks in a complex target assembly (Foster et al., Phys. Plasmas 9 (2002) 2251). We describe recent, significant extensions to this work, in which we consider scaling of the experiment, the transition to turbulence, and astrophysical analogues. In new work at the Omega laser, we are developing an experiment in which a jet is formed by laser ablation of a titanium foil mounted over a titanium washer with a central, cylindrical hole. Some of the resulting shocked titanium expands, cools, and accelerates through the vacuum region (the hole in the washer) and then enters a cylinder of low-density foam as a jet. We discuss the design of this new experiment and present preliminary experimental data and results of simulations using AWE hydrocodes. In each case, the high Reynolds number of the jet suggests that turbulence should develop, although this behaviour cannot be reliably modelled by present, resolution-limited simulations (because of their low-numerical Reynolds number).     

A New Way that Planets Can Affect the Sun

We derive a perturbation inside a rotating star that occurs when the star is accelerated by orbiting bodies. If a fluid element has rotational and orbital components of angular momentum with respect to the inertially fixed point of a planetary system that are of opposite sign, then the element may have potential energy that could be released by a suitable flow. We demonstrate the energy with a very simple model in which two fluid elements of equal mass exchange positions, calling to mind a turbulent field or natural convection. The exchange releases potential energy that, with a minor exception, is available only in the hemisphere facing the barycenter of the planetary system. We calculate its strength and spatial distribution for the strongest case (“vertical”) and for weaker horizontal cases whose motions are all perpendicular to gravity. The vertical cases can raise the kinetic energy of a few well positioned convecting elements in the Sun’s envelope by a factor ≤7. This is the first physical mechanism by which planets can have a nontrivial effect on internal solar motions. Occasional small mass exchanges near the solar center and in a recently proposed mixed shell centered at 0.16R _s would carry fresh fuel to deeper levels. This would cause stars like the Sun with appropriate planetary systems to burn somewhat more brightly and have shorter lifetimes than identical stars without planets. The helioseismic sound speed and the long record of sunspot activity offer several bits of evidence that the effect may have been active in the Sun’s core, its envelope, and in some vertically stable layers. Additional proof will require direct evidence from helioseismology or from transient waves on the solar surface.