The motion of a satellite in resonance with the second-degree sectorial harmonic

The solution to the motion of a satellite in an eccentric orbit and in resonance with the second-degree sectorial harmonic of the potential field is developed. The method of solution used parallels the well known von Zeipel method of general perturbations. The solution consists of expressions for the variations of the Delaunay variables. These expressions are composed of the perturbations developed by Brouwer in 1959 for the motion of an artificial satellite plus first-order perturbations due to the second-degree sectorial harmonic (in terms of the Legendre normal elliptic integrals of the first and second kind).This paper presents the results of one phase of research carried out at the Jet Propulsion Laboratory, California Institute of Technology, under Contract No. NAS 7-100, sponsored by the National Aeronautics and Space Administration.     

Lunar perturbations of artificial satellites of the earth

Lunisolar perturbations of an artificial satellite for general terms of the disturbing function were derived by Kaula (1962). However, his formulas use equatorial elements for the Moon and do not give a definite algorithm for computational procedures. As Kozai (1966, 1973) noted, both inclination and node of the Moon's orbit with respect to the equator of the Earth are not simple functions of time, while the same elements with respect to the ecliptic are well approximated by a constant and a linear function of time, respectively. In the present work, we obtain the disturbing function for the Lunar perturbations using ecliptic elements for the Moon and equatorial elements for the satellite. Secular, long-period, and short-period perturbations are then computed, with the expressions kept in closed form in both inclination and eccentricity of the satellite. Alternative expressions for short-period perturbations of high satellites are also given, assuming small values of the eccentricity. The Moon's position is specified by the inclination, node, argument of perigee, true (or mean) longitude, and its radius vector from the center of the Earth. We can then apply the results to numerical integration by using coordinates of the Moon from ephemeris tapes or to analytical representation by using results from lunar theory, with the Moon's motion represented by a precessing and rotating elliptical orbit.     

On error bounds and initialization in satellite orbit theories

The order of magnitude of the error is investigated for a first-order von Zeipel theory of satellite orbits in an axisymmetric force field, i.e., first-order long period and short-period effects are included along with second order secular rates. The treatment is valid for zero eccentricity and/or inclination. In the case where initial position and velocity vectors are known, the in-track position error over time intervals of order 1/J ₂ is kept at 0(J ₂ ²), like the other position errors and velocity errors, by calibration of the mean motion with the aid of the energy integral. The results are specifically applicable to accuracy comparisons of the Brouwer orbit prediction method with numerical integration. A modified calibration is presented for the general asymmetric force field which includes tesseral harmonics.     

Analytical method for perturbed frozen orbit around an asteroid in highly inhomogeneous gravitational fields: a first approach

This article provides a method for finding initial conditions for perturbed frozen orbits around inhomogeneous fast rotating asteroids. These orbits can be used as reference trajectories in missions that require close inspection of any rigid body. The generalized perturbative procedure followed exploits the analytical methods of relegation of the argument of node and Delaunay normalisation to arbitrary order. These analytical methods are extremely powerful but highly computational. The gravitational potential of the heterogeneous body is firstly stated, in polar-nodal coordinates, which takes into account the coefficients of the spherical harmonics up to an arbitrary order. Through the relegation of the argument of node and the Delaunay normalization, a series of canonical transformations of coordinates is found, which reduces the Hamiltonian describing the system to a integrable, two degrees of freedom Hamiltonian plus a truncated reminder of higher order. Setting eccentricity, argument of pericenter and inclination of the orbit of the truncated system to be constant, initial conditions are found, which evolve into frozen orbits for the truncated system. Using the same initial conditions yields perturbed frozen orbits for the full system, whose perturbation decreases with the consideration of arbitrary homologic equations in the relegation and normalization procedures. Such procedure can be automated for the first homologic equation up to the consideration of any arbitrary number of spherical harmonics coefficients. The project has been developed in collaboration with the European Space Agency (ESA).     

A symplectic mapping approach to the study of the stochasticity in asteroidal resonances

In the case of the 2:1 and 3:2 resonances with Jupiter, it has not been yet possible to have a complete identification of all chaotic diffusion processes at work, mainly because the time scale of some of them are of an order still out of the reach of precise integrations. A planar Hadjidemetriou's mapping, using expansions valid for high eccentricities and scaled in order to accelerate the diffusion processes, was derived. The solutions obtained with the mapping show huge eccentricity variations in all orbits starting in the middle of the 2:I resonance, when the main short-period perturbations of Jupiter's orbit are considered. The solutions starting in the middle of the 3:2 resonance do not show any important diffusion.     

Solution of canonical equations which result from elimination of short-periodic terms in second-order planetary theory

We solve the first order non-linear differential equation and we calculate the two quadratures to which are reduced the canonical differential equations resulting from the elimination of the short period terms in a second order planetary theory carried out through Hori's method and slow Delaunay canonical variables when powers of eccentricities and the sines of semi-inclinations which are >3 are neglected and the eccentricity of the disturbing planet is identically equal to zero. The procedure can be extended to the case when the eccentricity of the disturbing planet is not identically equal to zero. In this latter general case, we calculatedthe two quadratures expressing angular slow Delaunay canonical variable ₁ of the disturbed planet and angular slow Delaunay canonical variable ₂ of the disturbing planet in terms of timet.     

Long period variations of the motion of a satellite due to non-resonant tesseral harmonics of a gravity potential

The main effects of tesseral harmonics of a gravity potential expansion on the motion of a satellite, are short period variations as well as long period variations due to resonances. However, other smaller long period and secular variations can arise from interactions between tesseral terms of the same order. The analytical integration of these effects is developed, using numerical evaluation of Kaula eccentricity and inclination functions. Examples for some Earth's geodetic satellites show that secular effects can reach a few decameters per year. The secular variations can even reach several hundred of meters per year for the Mars natural satellite Phobos.     

Some orbital characteristics of lunar artificial satellites

In this paper we present an analytical theory with numerical simulations to study the orbital motion of lunar artificial satellites. We consider the problem of an artificial satellite perturbed by the non-uniform distribution of mass of the Moon and by a third-body in elliptical orbit (Earth is considered). Legendre polynomials are expanded in powers of the eccentricity up to the degree four and are used for the disturbing potential due to the third-body. We show a new approximated equation to compute the critical semi-major axis for the orbit of the satellite. Lie-Hori perturbation method up to the second-order is applied to eliminate the terms of short-period of the disturbing potential. Coupling terms are analyzed. Emphasis is given to the case of frozen orbits and critical inclination. Numerical simulations for hypothetical lunar artificial satellites are performed, considering that the perturbations are acting together or one at a time.     

Long-term motion of resonant satellites with arbitrary eccentricity and inclination

A first-order, semi-analytical method for the long-term motion of resonant satellites is introduced. The method provides long-term solutions, valid for nearly all eccentricities and inclinations, and for all commensurability ratios. The method allows the inclusion of all zonal and tesseral harmonics of a nonspherical planet.We present here an application of the method to a synchronous satellite includingonly theJ ₂ andJ ₂₂ harmonics. Global, long-term solutions for this problem are given for arbitrary values of eccentricity, argument of perigee and inclination.     

The Use of Lie Series in the Construction of a Perturbation Theory and Some Recent Results in the Theory of the Motion of Hyperion

This paper begins with a brief review of a form of the Lie series transformation, and then reports some new results in the study, using Lie series methods, of the orbit of Saturn's satellite Hyperion. In particular, improved expressions are given for the long-period perturbations of the orbital elements which describe the motion in the orbit plane, and also first results for expressions for the short-period perturbations in the apse longitude, derived from the Lie series generating function. This revised version was published online in July 2006 with corrections to the Cover Date.     

Tesseral harmonic perturbations in radial transverse and binormal components

The formulae for the perturbations in radial, transverse and binormal components of the Earth artificial satellite motion have been derived. Perturbations due to the tesseral part of the geopotential are considered. The geopotential expressed in terms of the orbital elements has the form proposed by Wnuk (1988). The formulae for the perturbations have been obtained using the Hori (1966) method. They can be effectively applied in calculation of the perturbations in the components including the coefficients of the high order and degree tesseral harmonics. The derived formulae reveal no singularities at zero eccentricity.     

Application of Hamiltonian structure-preserving control to formation flying on a J 2-perturbed mean circular orbit

The bounded quasi-periodic relative trajectories are investigated in this paper for on-orbit surveillance, inspection or repair, which requires rapid changes in formation configuration for full three-dimensional imaging and unpredictable evolutions of relative trajectories for non-allied spacecraft. A linearized differential equation for modeling J ₂ perturbed relative dynamics is derived without any simplified treatment of full short-period effects. The equation serves as a nominal reference model for stationkeeping controller to generate the quasi-periodic trajectories near the equilibrium, i.e., the location of the chief. The developed model exhibits good numerical accuracy and is applicable to an elliptic orbit with small eccentricity inheriting from the osculating conversion of orbital elements. A Hamiltonian structure-preserving controller is derived for the three-dimensional time-periodic system that models the J ₂-perturbed relative dynamics on a mean circular orbit. The equilibrium of the system has time-varying topological types and no fixed-dimensional unstable/stable/center manifolds, which are quite different from the two-dimensional time-independent system with a permanent pair of hyperbolic eigenvalues and fixed-dimensions of unstable/stable/ center manifolds. The unstable and stable manifolds are employed to change the hyperbolic equilibrium to elliptic one with the poles assigned on the imaginary axis. The detailed investigations are conducted on the critical controller gain for Floquet stability and the optimal gain for the fuel cost, respectively. Any initial relative position and velocity leads to a bounded trajectory around the controlled elliptic equilibrium. The numerical simulation indicates that the controller effectively stabilizes motions relative to the perturbed elliptic orbit with small eccentricity and unperturbed elliptic orbit with arbitrary eccentricity. The developed controller stabilizes the quasi-periodic relative trajectories involved in six foundational motions with different frequencies generated by the eigenvectors of the Floquet multipliers, rather than to track a reference relative configuration. Only the relative positions are employed for the feedback without the information from the direct measurement or the filter estimation of relative velocity. So the current controller has potential applications in formation flying for its less computation overload for on-board computer, less constraint on the measurements, and easily-achievable quasi-periodic relative trajectories.     

Hyperboloidal precession of a dynamically symmetric satellite. Construction of normal forms of the Hamiltonian

The Norma specialized program package, intended for normalization of autonomous Hamiltonian systems by means of computer algebra, is used in studies of small-amplitude periodic motions in the neighbourhood of regular precessions of a dynamically symmetric satellite on a circular orbit. The case of hyperboloidal precession is considered. Analytical expressions for normal forms and generating functions depending on frequencies of the system as on parameters are derived. Possible resonances are considered in particular. The 6th order of normalization is achieved. Though the intermediate analytical expressions occupy megabytes of computer's main memory, final ones are quite compact. Obtained analytical expressions are applied to the analysis of stability of small-amplitude periodic motions in the neighbourhood of hyperboloidal precession.     

The Parabolic Three-Body Problem

In this communication we present an analytical model for the restricted three-body problem, in the case where the perturber is in a parabolic orbit with respect to the central mass. The equations of motion are derived explicitly using the so-called Global Expansion of the disturbing function, and are valid for any eccentricity of the massless body, as well as in the case where both secondary masses have crossing orbits. Integrating the equations of motion over the complete passage of the perturber through the system, we are then able to construct a first-order algebraic mapping for the change in semimajor axis, eccentricity and inclination of the perturbed body.Comparisons with numerical solutions of the exact equations show that the map yields precise results, as long as the minimum distance between both bodies is not too small. Finally, we discuss several possible applications of this model, including the evolution of asteroidal satellites due to background bodies, and simulations of passing stars on extra-solar planets.     

On the choice of reference orbit,canonical variables,and perturbation method in satellite theory

The author's second-order artificial satellite theory (Aksnes, 1970) is reviewed and compared with that of Kozai (1962). These theories differ in that the former makes use of: (1) an intermediate orbit, being a rotating ellipse instead of a fixed ellipse, (2) Hill variables instead of Delaunay variables, and (3) Hori's perturbation method in Lie series rather than Von Zeipel's method in Taylor series.It is demonstrated that because of these differences, the former theory enjoys a greater simplicity and compactness, it is non-singular at zero eccentricity, and the process of deriving the perturbations is considerably simplified (Aksnes, 1972). For example, the number of second-order short-period terms due to the planet's oblateness (J ₂) is reduced by a factor of about three (Hori, 1970). The intermediate orbit and Hori's perturbation method contribute about equally to this reduction.Presented at the Conference on Celestial Mechanics, Oberwolfach, Germany, August 27–September 2, 1972.     

Eccentricity evolution in hierarchical triple systems with eccentric outer binaries

We develop a technique for estimating the inner eccentricity in hierarchical triple systems, with the inner orbit being initially circular, while the outer one is eccentric. We consider coplanar systems with well-separated components and comparable masses. The derivation of short-period terms is based on an expansion of the rate of change of the Runge–Lenz vector. Then, the short-period terms are combined with secular terms, obtained by means of canonical perturbation theory. The validity of the theoretical equations is tested by numerical integrations of the full equations of motion.     

On the stability of dust orbits in mean-motion resonances perturbed by from an interstellar wind

Circumstellar dust particles can be captured in a mean-motion resonance (MMR) with a planet and simultaneously be affected by non-gravitational effects. It is possible to describe the secular variations of a particle orbit in the MMR analytically using averaged resonant equations. We derive the averaged resonant equations from the equations of motion in near-canonical form. The secular variations of the particle orbit depending on the orientation of the orbit in space are taken into account. The averaged resonant equations can be derived/confirmed also from Lagrange’s planetary equations. We apply the derived theory to the case when the non-gravitational effects are the Poynting–Robertson effect, the radial stellar wind, and an interstellar wind. The analytical and numerical results obtained are in excellent agreement. We found that the types of orbits correspond to libration centers of the conservative problem. The averaged resonant equations can lead to a system of equations which holds for stationary points in a subset of resonant variables. Using this system we show analytically that for the considered non-gravitational effects, all stationary points should correspond to orbits which are stationary in interplanetary space after an averaging over a synodic period. In an exact resonance, the stationary orbits are stable. The stability is achieved by a periodic repetition of the evolution during the synodic period. Numerical solutions of this system show that there are no stationary orbits for either the exact or non-exact resonances.     

Further results concerning exact analytical solutions basic to two-body orbits

Complex-variable techniques are used to establish exact analytical solutions to a class of two-body problems. In view of Lambert's theorem, two points on the conic, the chord-distance between the two points, and the time interval are considered given, and subsequently the solutions for the semi-major axis required to define the orbit are developed and expressed ultimately in terms of elementary quadratures.     

A satellite relative motion model including $$J_2$$ and $$J_3$$J3 via Vinti’s intermediary

Vinti’s potential is revisited for analytical propagation of the main satellite problem, this time in the context of relative motion. A particular version of Vinti’s spheroidal method is chosen that is valid for arbitrary elliptical orbits, encapsulating \(J_2\), \(J_3\), and generally a partial \(J_4\) in an orbit propagation theory without recourse to perturbation methods. As a child of Vinti’s solution, the proposed relative motion model inherits these properties. Furthermore, the problem is solved in oblate spheroidal elements, leading to large regions of validity for the linearization approximation. After offering several enhancements to Vinti’s solution, including boosts in accuracy and removal of some singularities, the proposed model is derived and subsequently reformulated so that Vinti’s solution is piecewise differentiable. While the model is valid for the critical inclination and nonsingular in the element space, singularities remain in the linear transformation from Earth-centered inertial coordinates to spheroidal elements when the eccentricity is zero or for nearly equatorial orbits. The new state transition matrix is evaluated against numerical solutions including the \(J_2\) through \(J_5\) terms for a wide range of chief orbits and separation distances. The solution is also compared with side-by-side simulations of the original Gim–Alfriend state transition matrix, which considers the \(J_2\) perturbation. Code for computing the resulting state transition matrix and associated reference frame and coordinate transformations is provided online as supplementary material.     

Averaged model to study long-term dynamics of a probe about Mercury

This paper provides a method for finding initial conditions of frozen orbits for a probe around Mercury. Frozen orbits are those whose orbital elements remain constant on average. Thus, at the same point in each orbit, the satellite always passes at the same altitude. This is very interesting for scientific missions that require close inspection of any celestial body. The orbital dynamics of an artificial satellite about Mercury is governed by the potential attraction of the main body. Besides the Keplerian attraction, we consider the inhomogeneities of the potential of the central body. We include secondary terms of Mercury gravity field from \(J_2\) up to \(J_6\), and the tesseral harmonics \(\overline{C}_{22}\) that is of the same magnitude than zonal \(J_2\). In the case of science missions about Mercury, it is also important to consider third-body perturbation (Sun). Circular restricted three body problem can not be applied to Mercury–Sun system due to its non-negligible orbital eccentricity. Besides the harmonics coefficients of Mercury’s gravitational potential, and the Sun gravitational perturbation, our average model also includes Solar acceleration pressure. This simplified model captures the majority of the dynamics of low and high orbits about Mercury. In order to capture the dominant characteristics of the dynamics, short-period terms of the system are removed applying a double-averaging technique. This algorithm is a two-fold process which firstly averages over the period of the satellite, and secondly averages with respect to the period of the third body. This simplified Hamiltonian model is introduced in the Lagrange Planetary equations. Thus, frozen orbits are characterized by a surface depending on three variables: the orbital semimajor axis, eccentricity and inclination. We find frozen orbits for an average altitude of 400 and 1000 km, which are the predicted values for the BepiColombo mission. Finally, the paper delves into the orbital stability of frozen orbits and the temporal evolution of the eccentricity of these orbits.