A numerical-analytical method for studying the orbital evolution of distant planetary satellites

We describe an approximate numerical-analytical method for calculating the perturbations of the elements of distant satellite orbits. The model for the motion of a distant satellite includes the solar attraction and the eccentricity and ecliptic inclination of the orbit of the central planet. In addition, we take into account the variations in planetary orbital elements with time due to secular perturbations. Our work is based on Zeipel’s method for constructing the canonical transformations that relate osculating satellite orbital elements to the mean ones. The corresponding transformation of the Hamiltonian is used to construct an evolution system of equations for mean elements. The numerical solution of this system free from rapidly oscillating functions and the inverse transformation from the mean to osculating elements allows the evolution of distant satellite orbits to be studied on long time scales on the order of several hundred or thousand satellite orbital periods.     

Osculating and Superosculating Intermediate Orbits and their Applications

A method of construction of intermediate orbits for approximating the real motion of celestial bodies in the initial part of trajectory is proposed. The method is based on introducing a fictitious attracting centre with a time-variable gravitational parameter. The variation of thisparameter is assumed to obey the Eddington–Jeans mass-variationlaw. New classes of orbits having first-, second-, and third-order tangency to the perturbed trajectory at the initial instant of time are constructed. For planar motion, the tangency increases by one or two orders. The constructed intermediate orbits approximate the perturbed motion better than the osculating Keplerian orbit and analogous orbits of otherauthors. The applications of the orbits constructed in Encke's methodfor special perturbations and in the procedure for predicting themotion in which the perturbed trajectory is represented by a sequenceof short arcs of the intermediate orbits are suggested.The use of the constructed orbits is especially advantageous in the investigation of motion under the action of large perturbations.     

Special perturbations employing osculating reference states

The concept of employing osculating reference position and velocity vectors in the numerical integration of the equations of motion of a satellite is examined. The choice of the reference point is shown to have a significant effect upon numerical efficiency and the class of trajectories described by the differential equations of motion. For example, when the position and velocity vectors on the osculating orbit at a fixed reference time are chosen, a universal formulation is yielded. For elliptical orbits, however, this formulation is unattractive for numerical integration purposes due to Poisson terms (mixed secular) appearing in the equations of motion. Other choices for the reference point eliminate this problem but usually at the expense of universality. A number of these formulations, including a universal one, are considered here. Comparisons of the numerical characteristics of these techniques with those of the Encke method are presented.     

Lunar frozen orbits revisited

Lunar frozen orbits, characterized by constant orbital elements on average, have been previously found using various dynamical models, incorporating the gravitational field of the Moon and the third-body perturbation exerted by the Earth. The resulting mean orbital elements must be converted to osculating elements to initialize the orbiter position and velocity in the lunar frame. Thus far, however, there has not been an explicit transformation from mean to osculating elements, which includes the zonal harmonic \(J_2\), the sectorial harmonic \(C_{22}\), and the Earth third-body effect. In the current paper, we derive the dynamics of a lunar orbiter under the mentioned perturbations, which are shown to be dominant for the evolution of circumlunar orbits, and use von Zeipel’s method to obtain a transformation between mean and osculating elements. Whereas the dynamics of the mean elements do not include \(C_{22}\), and hence does not affect the equilibria leading to frozen orbits, \(C_{22}\) is present in the mean-to-osculating transformation, hence affecting the initialization of the physical circumlunar orbit. Simulations show that by using the newly-derived transformation, frozen orbits exhibit better behavior in terms of long-term stability about the mean values of eccentricity and argument of periapsis, especially for high orbits.     

Poisson equations of rotational motion for a rigid triaxial body with application to a tumbling artificial satellite

Differential equations are derived for studying the effects of either conservative or nonconservative torques on the attitude motion of a tumbling triaxial rigid satellite. These equations, which are analogous to the Lagrange planetary equations for osculating elements, are then used to study the attitude motions of a rapidly spinning, triaxial, rigid satellite about its center of mass, which, in turn, is constrained to move in an elliptic orbit about an attracting point mass. The only torques considered are the gravity-gradient torques associated with an inverse-square field. The effects of oblateness of the central body on the orbit are included, in that, the apsidal line of the orbit is permitted to rotate at a constant rate while the orbital plane is permitted to precess (either posigrade or retrograde) at a constant rate with constant inclination.A method of averaging is used to obtain an intermediate set of averaged differential equations for the nonresonant, secular behavior of the osculating elements which describe the complete rotational motions of the body about its center of mass. The averaged differential equations are then integrated to obtain long-term secular solutions for the osculating elements. These solutions may be used to predict both the orientation of the body with respect to a nonrotating coordinate system and the motion of the rotational angular momentum about the center of mass. The complete development is valid to first order in (n/w ₀)², wheren is the satellite's orbital mean motion andw ₀ its initial rotational angular speed.     

Electromagnetic Radiation and Motion of a Particle

We consider the motion of uncharged dust grains of arbitrary shape including the effects of electromagnetic radiation and thermal emission. The resulting relativistically covariant equation of motion is expressed in terms of standard optical parameters. Explicit expressions for secular changes of osculating orbital elements are derived in detail for the special case of the Poynting-Robertson effect. Two subcases are considered: (i) central acceleration due to gravity and the radial component of radiation pressure independent of the particle velocity, (ii) central acceleration given by gravity and the radiation force as the disturbing force. The latter case yields results which may be compared with secular orbital evolution in terms of orbital elements for an arbitrarily shaped dust particle. The effects of solar wind are also presented. This revised version was published online in July 2006 with corrections to the Cover Date.     

Computation of Partial Derivatives of Perturbed Motion Using the Arcs of Osculating and Superosculating Orbits

The methods for analytical determination of partial derivatives of the current parameters of motion with respect to their initial values are described. The methods take into account principal perturbations and are based on the use of the osculating and superosculating intermediate orbits constructed earlier by the author. These orbits ensure the first-, second-, and third-order contact to the real trajectory at the initial time. The solution for parameters of the intermediate motion and partial derivatives of these parameters is given in a universal closed form. The partial derivatives on long time intervals are computed using a step-by-step procedure combined with the Encke method of special perturbations, in which the intermediate orbits are used as the reference. The numerical results show that the new approach can be efficiently used for solving the problem of differential correction of orbits of asteroids and comets on the basis of observational data.     

Poynting-Robertson drag and orbital resonance

Both the Poynting-Robertson drag and resonant orbits appear to be very important for the motion of small grains in the early solar system. While orbital resonances are very often stable and tend to force bodies into noncircular orbits, the Poynting-Robertson drag produces secular variations in the semimajor axis and tends to circularize the orbits. We study numerically the competition between the Poynting-Robertson drag and the gravitational interaction of grains with Jupiter near the 2/1 resonance. Computations are based on the plane-restricted problem. Numerical investigations show that the grains always cross the resonance region without any oscillation, except in the special case where the grains were initially inside the resonance. In both cases the variations of the osculating elements exhibit a drastic step, which can be explained by Greenberg's and Schubart's theories.     

The influence of direct solar radiation pressure on the nodal period of artificial earth satellites

Approximate formulae for the evaluation of the effect of direct solar radiation pressure (neglecting the POYNTING -ROBERTSON effect and shadow effect) on the nodal period of artificial Earth satellites with quasi-circular orbits are derived. The cases of osculating elements considered in the ascending node and of circular orbits are studied.     

Superosculating intermediate orbits: Fourth-and fifth-order tangencies to trajectories of perturbed motion

The theory of superosculating intermediate orbits previously suggested by the author is developed. A new class of orbits with a fourth-order tangency to the actual trajectory of a celestial body at the initial time is constructed. Orbits with a fifth-order tangency have been constructed for the first time. The motion in the constructed orbits is represented as a combination of two motions: the motion of a fictitious attracting center with a variable mass and the motion relative to this center. The first motion is generally parabolic, while the second motion is described by the equations of the Gylden—Mestschersky problem. The variation in the mass of the fictitious center obeys Mestschersky’s first and combined laws. The new orbits represent more accurately the actual motion in the initial segment of the trajectory than an osculating Keplerian orbit and other existing analogues. Encke’s generalized methods of special perturbations in which the constructed intermediate orbits are used as reference orbits are presented. Numerical simulations using the approximations of the motions of Asteroid Toutatis and Comet P/Honda—Mrkos—Pajdu?áková as examples confirm that the constructed orbits are highly efficient. Their application is particularly beneficial in investigating strongly perturbed motion.     

Nonlinear analytic solution for perturbed relative motion using differential equinoctial elements

This paper develops a nonlinear analytic solution for satellite relative motion in J₂-perturbed elliptic orbits by using the geometric method that can avoid directly solving the complex differential equations. The differential equinoctial elements (DEEs) are used to remove any singularities for zero-eccentricity or zero-inclination orbits. Based on the relationship between the relative states and the DEEs, state transition tensors (STTs) for transforming the osculating DEEs and propagating the mean DEEs have been derived. The formulation of these STTs has been split into a set of vector and matrix operations, which avoids directly expanding the complex second-order terms, and thus, the obtained STTs could be easy-to-understand and easy-to-code. Numerical results show that the proposed nonlinear solution is valid for zero-eccentricity and zero-inclination reference orbit and is more accurate than the previous linear or nonlinear methods for the long-term prediction of satellite relative motion.     

Model orbits of asteroid 3040 Kozai

The orbital evolutions of the asteroid 3040 Kozai and model asteroids with similar orbits have been investigated. Their osculating orbits for an epoch 1991 December 10 were numerically integrated forward within the interval of 20,000 years, using a dynamical model of the solar system consisting of all inner planets, Jupiter, and Saturn.The orbit of the asteroid Kozai is stable. Its motion is affected only by long-period perturbations of planets. With change of the argument of perihelion of the asteroid Kozai, the evolution of the model asteroid orbits changes essentially, too. The model orbits with the argument of perihelion changed by the order of 10% show that asteroids with such orbital parameters may approach the Earth orbit, while asteroids with larger changes may even cross it, at least after 10,000 years. Long-term orbital evolution of asteroids with these orbital parameters is very sensitive on their angular elements.     

About averaging procedures in the problem of asteroid motion

A new mathematically correct approach to construct an averaging procedure for the motion of a massless body around the central body perturbed by fully interacting planets is developed and the errors of the standard solution are discussed. The new technique allows to combine the advantages of the Hamiltonian representation with the usage of standard osculating elements in combination with all the standard expansions of the perturbing functions. The main idea is to introduce new additional variables conjugate to all the standard elements and to work in a corresponding super phase space. In this way, the number of variables is doubled at first, but one has to deal with only one Hamiltonian. The artificially introduced variables disappear from the final averaged equations as well as from the transformation formulae connecting the osculating and the mean elements.     

Trajectories of satellites under the combined influences of earth oblateness and air drag

Trajectories of satellites under the influences of earth oblateness and air drag are derived by the asymptotic method in nonlinear mechanics. Based on the assumptions: (1) the dominant oblateness factor of the earth is the second harmonic (J ₂), (2) a non-rotating, spherically symmetric atmosphere and an exponential distribution of atmospheric density, (3) original elliptical orbits being of small eccentricity, closed-form solutions for the improved first order approximation are obtained. After finding the osculating orbital elements of the resulting trajectories, we expose the behavior of osculating orbits at various inclinations.     

A semi-analytical method to study perturbed rotational motion

A semi-analytical method is presented to study the system of differential equations governing the rotational motion of an artificial satellite. Gravity gradient and non gravitational torques are considered. Operations with trigonometric series were performed using an algebraic manipulator. Andoyer's variables are used to describe the rotational motion. The osculating elements are transformed analytically into a mean set of elements. As the differential equations in the mean elements are free of fast frequency terms, their numerical integration can be performed using a large step size.     

On linking coefficient of two Keplerian orbits

We define a function of the set of pairs of Keplerian ellipses so that the sign of the function will be a topological invariant of their configuration. The sign is negative if and only if the related ellipses are linked. Two modifications of the coefficient which are more reliable in the case of closed to coplanar orbits are proposed. Explicit formulae representing the linking coefficients as functions of orbital elements are deduced. Extension in the case of unbounded orbits is obtained. We suggest different ways to use these coefficients for determining intersections of pairs of osculating Keplerian orbits. If we study dynamical behaviour of geometric configuration of pairs of Keplerian orbits, we can fix the moments of their intersections. These moments correspond exactly to the vanishing of linking coefficients. This revised version was published online in July 2006 with corrections to the Cover Date.     

Controlled Solar Sail Transfers into Near-Sun Regions Combined with Planetary Gravity-Assist Flybys

Results of numerical simulations of 'local-optimal' (or 'instantaneously optimal') trajectories of a space probe with a flat solar sail which moves from the circular Earth orbit to near-Sun regions are presented. We examine planar (ecliptic) solar sail transfer with gravity-assist flybys of Earth, Venus and Mercury. Several complex control modes of the sail tilt orientation angle for near-Sun orbits and for some 'falling onto the Sun' trajectories are investigated. The numerical simulations are used to examine the flight duration of some sail missions and to investigate the evolution of osculating elliptical orbits.     

Almost rectilinear halo orbits

Numerical studies over the entire range of mass-ratios in the circular restricted 3-body problem have revealed the existence of families of three-dimensional halo periodic orbits emanating from the general vicinity of any of the 3 collinear Lagrangian libration points. Following a family towards the nearer primary leads, in 2 different cases, to thin, almost rectilinear, orbits aligned essentially perpendicular to the plane of motion of the primaries. (i) If the nearer primary is much more massive than the further, these thin L₃-family halo orbits are analyzed by looking at the in-plane components of the small osculating angular momentum relative to the larger primary and at the small in-plane components of the osculating Laplace eccentricity vector. The analysis is carried either to 1st or 2nd order in these 4 small quantities, and the resulting orbits and their stability are compared with those obtained by a regularized numerical integration. (ii) If the nearer primary is much less massive than the further, the thin L₁-family and L₂-family halo orbits are analyzed to 1st order in these same 4 small quantities with an independent variable related to the one-dimensional approximate motion. The resulting orbits and their stability are again compared with those obtained by numerical integration.     

Accurate integration of geostationary orbits with Burdet's focal elements

The theory of Burdet's focal elements is outlined. The differential equations are presented, and the initial value problem is described together with the transformation to rectangular coordinates and classical elements. The focal elements are well defined for zero eccentricity and inclination. They can be adopted for the computation of elliptic, parabolic and hyperbolic motion. For the numerical integration of near-geostationary orbits a comparison of the efficiency is made between focal elements, KS theory and rectangular coordinates. For this class of orbits, a higher accuracy has been obtained by integrating elements than integrating rectangular coordinates.