Periodic orbits around an oblate spheroid

The general properties of certain differential systems are used to prove the existence of periodic orbits for a particle around an oblate spheroid.In a fixed frame, there are periodic orbits only fori=0 andi near /2. Furthermore, the generating orbits are circles.In a rotating frame, there are three families of orbits: first a family of periodic orbits in the vicinity of the critical inclination; secondly a family of periodic orbits in the equatorial plane with 0<e<1; thirdly a family of periodic orbits for any value of the inclination ife=0.     

Extension of the solution of Kepler's equation to high eccentricities

The classic Lagrange's expansion of the solutionE(e, M) of Kepler's equation in powers of eccentricity is extended to highly eccentric orbits, 0.6627 ... <e<1. The solutionE(e, M) is developed in powers of (e–e*), wheree* is a fixed value of the eccentricity. The coefficients of the expansion are given in terms of the derivatives of the Bessel functionsJ _n (ne). The expansion is convergent for values of the eccentricity such that |e–e*|<(e*), where the radius of convergence (e*) is a positive real number, which is calculated numerically.     

Expansions of (r/a)m cos jv and (r/a)m sin jv to high eccentricities

Expansions of the functions (r/a)^cos jv and (r/a)m ^sin jv of the elliptic motion 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 of these expansions are expressed in terms of the derivatives of Hansen's coefficients with respect to the eccentricity. The new expansions are convergent for values of the eccentricity such that |e–e*|<(e*), where the radius of convergence (e*) is the same of the extended solution of Kepler's equation. The new expansions are intrinsically related to Lagrange's series.     

The short-period eclipsing binary V1010 Ophiuchi does it have an eccentric orbit?

V1010 Oph is an unusually complicated close binary. Leung and Wilson (1977) found the system had an overcontact configuration. Subsequently, Margoniet al. (1981) claimed that the system had an eccentric orbit withe=0.25 based upon their spectroscopic study. We computed a theoretical radial velocity curve (accounting for tidal, reflection, and eclipse effects) based on the photometric parameters (e.g.,e=0) of Leung and Wilson. The computed curve fits the Asiago and Kitt Peak data very well. It is suggested that the asymmetry in the observed radial velocity arose from tidal, reflection, and eclipse effects rather than from orbital eccentricity. Other arguments against an eccentric orbit for V1010 Oph are also discussed. It is concluded that the eccentricity derived from the spectroscopic study may be spurious.On leave from Beijing University, China.     

A note on the elliptic restricted three-body problem

This work considers periodic solutions, arc-solutions (solutions with consecutive collisions) and double collision orbits of the plane elliptic restricted problem of three bodies for =0 when the eccentricity of the primaries,e _p , varies from 0 to 1. Characteristic curves of these three kinds of solutions are given.     

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.     

Rotationally-perturbed orbital elements of close binary systems

The aim of this investigation is to present the secular and periodic perturbations of the six orbital elements of a close binary system due to rotational distortion. In our study we consider very small inclinationst of the orbital plane of the system, whereas the eccentricity of the orbit may assume any value between 0<e<1. The final formulae for the various elements have been expressed by means of the unperturbed true anomaly measured from the ascending node.     

Interferometric observations of HCN and CN towards carbon stars

Using the IRAM interferometer we have observed four carbon stars (U Cam, CIT6, Y CVn, IRC+40540) in the HCN(J=1 0) and CN(N=1 0) lines. Here we present some results for CIT6 and U Cam.     

Encounter in the keplerian field: Analytical treatment

In extending the results of Henon and Petit (1986) an algorithm is suggested for constructing the series representing the general encounter-type solution of the spatial eccentric Hill's problem. The series are arranged in powers of the eccentricity E of Hill's problem and two integration constants e and k characterizing eccentricity and inclination of the relative motion. A particular non-periodic solution of Henon and Petit corresponding to E = e = k = 0 is taken as an intermediary. The perturbations to this solution are constructed similar to the lunar theory of Hill and Brown.     

The effect of orbital element variations on the mean seasonal daily insolation on Mars

In this paper we briefly study changes in the mean seasonal insolations on the planet Mars caused by significant large-scale variations in the following orbital elements: the eccentricity (e), the obliquity (ε) and the longitude of perihelion (λ _p ). Three orbital configurations have been investigated. In the first, the eccentricity equals successively 0, 0.075, and 0.15, whereas for the obliquity and the longitude of perihelion we took the present values which amount, respectively to 25° and 250°. In the second situation, ε=15, 25, and 35° for a circular orbit (e=0) and with λ _p =250°. In the last model we have sete=0.075 and ε=25° for λ _p =?90,0, and 90°. Although long-term periodic oscillations ofe (first case) and λ _p (third case) produce, respectively, very small or no variations in the average yearly insolation, fluctuations of the above mentioned planetary data strongly effect the mean summer and winter daily insolations. Indeed, the calculations reveal that between the two extreme values of the orbital elements used, the seasonal insolations exhibit a change in amplitude of about 15 to 20% difference over the entire latitude interval. Considering more particularly the second case it is found that the summertime insolation experiences a nearly similar variation as the mean annual daily insolation — i.e., a decrease of about 7% at the equator and a more than twofold increase at the poles. The corresponding mean winter daily insolation varies maximally by approximately 60% in the 60–80° latitude range.     

A general algorithm for the solution of Kepler's equation for elliptic orbits

An efficient algorithm is presented for the solution of Kepler's equationf(E)=E–M–e sinE=0, wheree is the eccentricity,M the mean anomaly andE the eccentric anomaly. This algorithm is based on simple initial approximations that are cubics inM, and an iterative scheme that is a slight generalization of the Newton-Raphson method. Extensive testing of this algorithm has been performed on the UNIVAC 1108 computer. Solutions for 20 000 pairs of values ofe andM show that for single precision (10^–8) 42.0% of the cases require one iteration, 57.8% two and 0.2% three. For double precision (10^–18) one additional iteration is required. Single- and double-precision FORTRAN subroutines are available from the author.     

Irregular behaviour and eccentricity increase in the trajectories of a dynamical system modelling asteroidal motions

Many trajectories of the third body are integrated numerically in a modified elliptical restricted three body problem (ERTBP), in which the eccentricity, e, of the orbit of the second primary varies sinusoidally with time. It is found that, in the case of the 2:1 resonance, the introduction of the time variability of e modifies significantly the behaviour of the trajectories of the third body. In particular their osculating eccentricity e, present the following two notable features: (a) In all cases it shows a definite chaotic variation, which appears at significantly shorter time-scales than the one found by Wisdom in the e = constant case. (b) In many cases it shows a significant increase, up and beyond the (critical) value e _crit = 0.52. As a result the third body approaches the first primary at distances smaller than 0.29 (where by we denote the semi-major axis of the trajectory of the second primary around the first), which in the actual Sun-Jupiter-asteroid problem corresponds to the semi-major axis of Mars. Our result might be of interest in the context of explaining the Kirkwood gaps at the resonances where the osculating eccentricity of asteroid trajectories calculated in the classical (e = constant) ERTBP does not reach Mars crosser values.     

Long-term orbit computations with KS uniformly regular canonical elements with oblateness

This paper concerns with the study of KS uniformly regular canonical elements with Earth's oblateness. These elements, ten in number, are all constant in the unperturbed motion and even in the perturbed motion, the substitution is straightforward and elementary due to the transformation laws being explicit and closed expression. By utilizing the recursion formulas of Legendre's polynomials, we are able to include any number of Earth's zonal harmonics J _n in the package and also economize the computations. A fixed step-size fourth-order Runge-Kutta-Gill method is employed for numerical integration of the canonical equations.Utilizing 5 test cases covering a large range of semimajor axis and eccentricity, we have carried out computations to study the effects of Earth's zonal harmonics (up to J ₃₆) and integration step-size variation. Bilinear relations and energy equation are used for checking the accuracies of numerical integration. From the application point of view, the package is utilized to study the behaviour of 900 km height near-circular sun-synchronous satellite orbit over a longer duration of 220 days time (nearly 3078 revolutions) and the necessity of including more number of Earth's zonal harmonic terms is noticed. The package is also used to study the effect of higher zonal harmonics on three 900 km height near-circular orbits with inclinations of 60, 63.2, and 65 degrees, by including Earth's zonal harmonics up to J ₂₄. The mean eccentricity (e _m) is found to have long-periods of 459.6, 6925.1 and 1077.6 days, respectively. Sharp changes in the variation of _m near the minima to e_m are noticed. The values of _m are found to be very near to +-90 degrees at the extrema of e_m. The same orbit is employed to study the effect of variation of inclination from 0 to 180 degrees on long-period (T) of eccentricity with J ₂ to J ₂₄ terms. T is found to increase rapidly as we proceed towards the critical inclinations.     

A non singular solution for spherical configuration with infinite central density

A non-singular exact solution with an infinite central density is obtained for the interior of spherically symmetric and static structures. Both the energy density and the pressure are infinite at the center but we have e ^λ(0)=1 and e ^ν(0)≠0. The solution admits the possibility of receiving signals from the region of infinite pressure.     

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.     

On the orbital eccentricity on an eclipsing binary system

A new method is presented to precisely deduce the orbital eccentricity of an eclipsing binary from observed epochs of its light minima. Application to the system V526 Sagittarii givese=0.2220±0.0016.     

Stability of planar oscillations of a satellite in an elliptic orbit

A problem of stability of odd 2-periodic oscillations of a satellite in the plane of an elliptic orbit of arbitrary eccentricity is considered. The motion is supposed to be only under the influence of gravitational torques.Stability of plane oscillations was investigated earlier (Zlatoustovet al., 1964) in linear approximation. In the present paper a problem of stability is solved in the non-linear mode. Terms up to the forth order inclusive are taken into consideration in expansion of Hamiltonian in a series.It is shown that necessary conditions of stability obtained in linear approximation coincide with sufficient conditions for almost all values of parameters ande (inertial characteristics of the satellite and eccentricity of the orbit). Exceptions represent either values of the parameters ,e when a problem of stability cannot be solved in a strict manner by non-linear approximation under consideration, or values of the parameters which correspond to resonances of the third and fourth orders. At the resonance of the third order oscillations are unstable, but at the resonance of the fourth order both unstability and stability of the satellite's oscillations take place depending on the values of the parameters ,e.     

A simple analytical model for the secular resonance ν6 in the asteroidal belt

When asteroids are in the secular resonance ₆, the variation of the eccentricity becomes very large. In this paper, the dynamics of this secular resonance ₆ is investigated by a simple analytical model, in which the third degree terms of the eccentricity and inclination are taken into account. The eccentricity variations of asteroids located near this resonance are represented clearly by the diagrams of equi-Hamiltonian curves on the plane of versuse ( the longitude of perihelion of asteroids and Saturn,e: the eccentricity of asteroids). These diagrams predict that the eccentricity of these asteroids suffers a large increase or decrease, and that the secular resonance argument librates about 0° and 180°. In order to confirm these predictions, numerical integrations are carried out over one million years. By these integrations, it is found that the eccentricity of secular resonant asteroids becomes more than 0.8, and that the libration about 0° also exists, as well as the libration about 180°. The strongly depopulated region in the asteroidal belt, which corresponds to the position of the secular resonance ₆, is also explained well by this analytical model.     

Variations of perturbations in perigee height with eccentricity for artificial Earth's satellites due to air drag

The variations of perturbations in perigee distance for different values of the orbital eccentricity for artificial Earth's satellites due to air drag have been studied. The analytical solution of deriving these perturbations, using the TD model (Total Density) have been applied, Helali (1987). The Theory is valid for altitudes ranging from 200 to 500 km above the Earth's surface and for solar 10.7 cm flux. Numerical examples are given to illustrate the variations of the perturbations in perigee distance with changing eccentricity (e < 0.2). A stronge perturbations in the perigee distance have been shown when the eccentricity in the range 0.001 <e < 0.05, especially for perigee distance 200 km.