A Note on a General Algorithm for Two-Body Expansions

This note describes briefly a technique, easily implemented with most computer algebra systems, for the purpose of computing the two-body expansions (in powers of the eccentricity and in Fourier series of the mean-anomaly) of a large class of functions of the distance, the true anomaly and/or the eccentric anomaly. This revised version was published online in July 2006 with corrections to the Cover Date.     

General planetary theory in elliptic functions

It is shown that the first-order general planetary theory, i.e. the theory without secular terms, developed in (Brumberg and Chapront, 1973) may be re-constructed and presented by the series in powers of the eccentricity and inclination variables with the closed form coefficients expressed in terms of elliptic functions. The intermediate solution of the zero degree in eccentricities and inclinations has been given explicitly with the aid of elliptic functions and the Hansen type quadratures with trigonometric function kernels. In determining the first and higher degree terms in eccentricities and inclinations one meets the Hansen type quadratures with elliptic function kernels. The secular evolution is described by the autonomous polynomial differential system.     

Mean values of particular functions in the elliptic motion

The mean values % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaamaalaaabaGaaGymaaqaaiaaikdacqaHapaCaaWaa8qCaeaacaGG% OaacbaGaa8NKbiabgkHiTiaadYgacaGGPaGaa8hiaiGacogacaGGVb% Gaai4Caiaa-bcacaWGRbGaa8NKbiaa-bcacaWGKbGaamiBaaWcbaGa% aGimaaqaaiaaikdacqaHapaCa0Gaey4kIipakiaa-bcacaqGHbGaae% OBaiaabsgacaWFGaWaaSaaaeaacaaIXaaabaGaaGOmaiabec8aWbaa% daWdXbqaaiaacIcacaWFsgGaeyOeI0IaamiBaiaacMcacaWFGaGaci% 4CaiaacMgacaGGUbGaa8hiaiaadUgacaWFsgGaa8hiaiaadsgacaWG% SbaaleaacaaIWaaabaGaaGOmaiabec8aWbqdcqGHRiI8aaaa!6BC2!\[\frac{1}{{2\pi }}\int\limits_0^{2\pi } {(f - l) \cos kf dl} {\rm{and}} \frac{1}{{2\pi }}\int\limits_0^{2\pi } {(f - l) \sin kf dl}\] (where f and l are respectively the true anomaly and the mean anomaly in the elliptic motion and k is an integer) are given in closed form.     

Planetary theories with the aid of the expansions of elliptic functions

For coplanar circular orbits, the mutual perturbations between two bodies can be expressed in term of the argument of Jacobian elliptic functions instead of the difference of the mean longitudes. For a given pair of planets, such a change of time variable improves the convergence of the developments. At the first order of planetary masses an integration of Lagrange's equations for the osculating elements is performed. When compared to classical developments the results are reduced by an important factor. The method is then extended to the mutual perturbations of Jupiter and Saturn, at any order of planetary masses, either with Fourier series with two arguments, or with one argument solely, taking advantage of the close commensurability of the mean motions.     

Some expansions of the elliptic motion to high eccentricities

Some classic expansions of the elliptic motion — cosmE and sinmE — in powers of the eccentricity are extended to highly eccentric orbits, 0.6627...<e<1. The new expansions are developed in powers of (e–e*), wheree* is a fixed value of the eccentricity. The coefficients are given in terms of the derivatives of Bessel functions with respect to the eccentricity. The expansions have the same radius of convergence (e*) of the extended solution of Kepler's equation, previously derived by the author. Some other simple expansions — (a/r), (r/a), (r/a) sinv, ..., — derived straightforward from the expansions ofE, cosE and sinE are also presented.     

The main mean motion commensurabilities in the planar circular and elliptic problem

We study the motion of asteroids in the main mean motion commensurabilities in the frame of the planar restricted three-body problem. No assumption is made about the size of the eccentricity of the asteroid. At small to moderate eccentricity, we recover existing results (shape of the phase space and location of secondary resonances). We also provide global pictures of the dynamics in the region of secondary resonances. At high eccentricity, the phase space portraits of the integrable part of the Hamiltonian show new families of stable orbits for the 3:2 and 2:1 cases and the secular resonances ₅ and ₆ are located.     

On the expansions of intermediate orbit for general planetary theory

Intermediate orbit for general planetary theory is constructed in the form of multivariate Fourier series with numerical coefficients. The structure and efficiency of the derived series are illustrated by giving various statistical properties of the coefficients.The ability of the recently proposed elliptic function approach to compress the Fourier series representing the intermediate orbit is investigated. Our results confirm that when mutual perturbations of a pair of planets are considered the elliptic function approach is quite efficient and allows one to compress the series substantially. However, when perturbations of three or more planets are under study the elliptic function approach does not give any advantages.     

Computation of elliptic Hansen coefficients and their derivatives

The problem of computation of elliptic Hansen coefficients and their derivatives is considered for constructing a motion theory of an artificial Earth satellite with large eccentricity. An algorithm for analytical and numerical computation of these coefficients and their derivatives is described. The recurrence relations for derivatives of the first and second order and initial values for recurrences are obtained. As an example, numerical values of some elliptic Hansen coefficients are given for the orbit with eccentricityk=0.74.     

The effect of solar radiation pressure on the Lagrangian points in the elliptic restricted three-body problem

In this paper the effect of solar radiation pressure on the location and stability of the five Lagrangian points is studied, within the frame of elliptic restricted three-body problem, where the primaries are the Sun and Jupiter acting on a particle of negligible mass. We found that the radiation pressure plays the rule of slightly reducing the effective mass of the Sun and changes the location of the Lagrangian points. New formulas for the location of the collinear libration points were derived. For large values of the force ratio β, we found that at β=0.12, the collinear point L₃ is stable and some families of periodic orbits can be drawn around it.     

Elements of spin motion

For use in numerical studies of rotational motion, a set of elements is introduced for the torque-free rotational motion of a rigid body around its barycenter. The elements are defined as the initial values of a modification of the Andoyer canonical variables. A computational procedure is obtained for determining these elements from the combination of the spin angular momentum vector and a triad defining the orientation of the rigid body. A numerical experiment shows that the errors of transformation between the elements and variables are sufficiently small. The errors increase linearly with time for some elements and quadratically for some others.     

Motion of dust in mean motion resonances with planets

Effect of stellar electromagnetic radiation on the motion of spherical dust particle in mean motion orbital resonances with a planet is investigated. Planar circular restricted three-body problem with the Poynting–Robertson (P–R) effect yields monotonic secular evolution of eccentricity when the particle is trapped in the resonance. Planar elliptic restricted three-body problem with the P–R effect enables nonmonotonous secular evolution of eccentricity and the evolution of eccentricity is qualitatively consistent with the published results for the complicated case of interaction of electromagnetic radiation with nonspherical dust grain. Thus, it is sufficient to allow either nonzero eccentricity of the planet or nonsphericity of the grain and the orbital evolutions in the resonances are qualitatively equal for the two cases. This holds both for exterior and interior mean motion orbital resonances. Evolutions of argument of perihelion in the planar circular and elliptical restricted three-body problems are shown. Numerical integrations show that an analytic expression for the secular time derivative of the particle’s argument of perihelion does not exist, if only dependence on semimajor axis, eccentricity and argument of perihelion is admitted. Connection between the shift of perihelion and oscillations in secular eccentricity is presented for the planar elliptic restricted three-body problem with the P–R effect. Period of the oscillations corresponds to the period of one revolution of perihelion. Change of optical properties of the spherical grain with the heliocentric distance is also considered. The change of the optical properties: (i) does not have any significant influence on the secular evolution of eccentricity, (ii) causes that the shift of perihelion is mainly in the same direction/orientation as the particle motion around the Sun. The statements hold both for circular and noncircular planetary orbits.     

On a global expansion of the disturbing function in the planar elliptic restricted three-body problem

Starting with a simple Taylor-based expansion of the inverse of the distance between two bodies, we are able to obtain a series expansion of the disturbing function of the three-body problem (planar elliptic case) which is valid for all points of the phase space outside the immediate vicinity of the collision points. In particular, the expansion is valid for very high values of the eccentricity of the perturbed body. Furthermore, in the case of an interior mean-motion resonant configuration, the above-mentioned expression is easily averaged with respect to the synodic period, yielding once again a global expansion of (R) valid for very high eccentricities.Comparisons between these results and the numerically computed exact function are presented for various resonances and values of the eccentricity. Maximum errors are determined in each case and their origin is established. Lastly, we discuss the applicability of the present expansion to practical problems.     

Dynamical model of motion for asteroids based on the DE405 theory

A numerical model of motion for asteroids was developed on the basis of the DE405 theory. The positions of main-belt asteroids are calculated accurate to 0.03 mas when the integration is taken over a 50-year interval. For the model to be computationally stable, the local truncation error of integration should be equal to 10^?14 and the double precision of the Standard for Binary Floating-Point Arithmetic IEEE 754-1985 should be used.     

Analytic properties of Hansen coefficients

Hansen’s coefficients in the theory of elliptic motion with eccentricity e are studied as functions of the parameter η = (1 − e ²)^1/2. Their analytic behavior in the complex η plane is described and some symmetry relations are derived. In particular, for every Hansen coefficient, multiplication by suitable powers of e and η results in an entire analytic function of η. Consequently, Hansen’s coefficients can be in principle computed by means of rapidly convergent series in powers of η. A representation of Hansen’s coefficients in terms of two entire functions of e ² follows.      

Effect of radiation on the stability of a retrograde particle orbit in different stellar systems

We have numerically investigated the stability of retrograde orbits/trajectories around Jupiter and the smaller of the primaries in binary systems RW-Monocerotis (RW-Mon) and Krüger-60 in the presence of radiation. A trajectory is considered as stable if it remains around the smaller mass for at least few hundred binary periods. In case of circular binary orbit, we find that the third order resonance provides the basis for reduction of stability region of retrograde motion of particle in RW-Mon and Sun-Jupiter system both in the presence and absence of radiation. Considering finite ellipticity in Sun-Jupiter system we find that for distant retrograde orbits, radiation from the Sun increases the width of the stable region and covers a significant portion of the region obtained in the absence of solar radiation. Further, due to solar radiation pressure, the stable region in the neighborhood of Jupiter has been found to shift much below the characteristic asymptotic line for the periodic retrograde orbits. In case of Krüger-60 we observe the distant retrograde orbits around the smaller of the primaries get affected considerably with increase in radiation parameter β₁. Further the range of velocities for which stable motion may persist narrows down for distant retrograde orbits in this system.     

Effect of Planetary Eccentricity on Ballistic Capture in the Solar System

Ballistic capture of spacecraft and celestial bodies by planets of the solar system is studied considering the elliptic restricted three body model. A preferential region, due to the eccentricity of the planet and the Sun-gravity-gradient effect is found for the capture phenomenon. An analytical formula is derived which determines the limiting value of the satellite capture eccentricity e_c as a function of the pericenter distance x_p and planet’s true anomaly. The analytic values e_c are tested by a numerical propagator, which makes use of planetary ephemeris, and only a small difference with respect to numerical integration is found. It turns out that lower values of e_c occur when the planet anomaly is close to zero; that is, capture is easier when the planet is at its perihelion. This fact is confirmed by the capture of celestial bodies. It is shown that Jupiter comets are generally captured when Jupiter is in its perihelion region. Ballistic capture is also important in interplanetary missions. The propellant saved using the minimum ballistic capture eccentricity is evaluated for different missions and compared with respect to the case in which the insertion orbit is a parabola: a significant saving can be accomplished.     

A note on lower bounds for relative equilibria in the main problem of artificial satellite theory

In the analytical approach to the main problem in satellite theory, the consideration of the physical parameters imposes a lower bound for normalized Hamiltonian. We show that there is no elliptic frozen orbits, at critical inclination, when we consider small values of H, the third component of the angular momentum. The argument used suggests that it might be applied also to more realistic zonal and tesseral models. Moreover, for almost polar orbits, when H may be taken as another small parameter, a different approach that will simplify the ephemerides generators is proposed.     

System for normalization of a hamiltonian function based on lie series

A software system for normalization of a Hamiltonian function is described. A few examples of its applications are given. It is written in PASCAL and runs on an IBM XT/AT with 640 KB memory.     

Luminosity function of contact binaries based on the All Sky Automated Survey (ASAS)

The luminosity function for contact binary stars of the W UMa type is evaluated on the basis of the All Sky Automated Survey (ASAS) photometric project covering all stars south of  δ=+ 28°  within a magnitude range  8 < V < 13  . Lack of colour indices enforced a limitation to 3374 systems with   P < 0.562 d  (i.e. 73 per cent of all systems with   P < 1 d  ) where a simplified M_V (log  P ) calibration could be used. The spatial density relative to the main-sequence FGK stars of 0.2 per cent, as established previously from the Hipparcos sample to   V = 7.5  , is confirmed. While the numbers of contact binaries in the ASAS are large and thus the statistical uncertainties small, derivation of the luminosity function required a correction for missed systems with small amplitudes and with orbital periods longer than 0.562 d; the correction, by a factor of 3, carries an uncertainty of about 30 per cent.     

Efficient Method for Computing with Certainty Periodic Orbits on a Surface of Section

We present an improved method for locating periodic orbits of a dynamical system of arbitrary dimension. The method first employs the characteristic bisection method (CBM) to roughly locate a periodic orbit, followed by the quadratically convergent Newton method to rapidly refine its position. The method is applied to the physically interesting example of the two degrees of freedom photogravitational problem, and shown to surpass the CBM algorithm and Newton's method alone.