Long-period Perturbations Research Articles

Perturbation solutions to the nonlinear motion of displaced orbits

This paper analytically describes the nonlinear motion of displaced orbits obtained by low-thrust propulsion, and focuses on the first-order analytical derivation of perturbation solutions of osculating Keplerian element (OKE) time histories. A virtual Earth (VE) model is developed to transform non-Keplerian orbits above the Earth into Keplerian orbits around the VE by treating the thrust and coordinate transformation effects as perturbations. The numerical OKEs computed from the Cartesian position and velocity exhibit secular and periodic variations. The short-period perturbations of the orbital elements are derived by the conventional quasi-mean element method. The differential equations of the secular and long-period perturbations are linearized for an analytical solution. A comparison between the analytical OKEs and numerical OKEs of a specific displaced orbit validates the correctness and good accuracy of the analytical derivation. One contribution of this paper is that the analytical OKEs can be applied for a fast calculation of a displaced orbit in the preliminary mission design phase, which has more significance than accurate but complicated time-consuming calculations by complete variational equations. Another contribution is revealing the connections between the amplitude as well as frequency of perturbation solutions and displaced orbits from a physical perspective, from which the topology and geometrical size of a displaced orbit can be easily inferred from the time history of OKEs.

New Closed-Form Solutions for Optimal Impulsive Control of Spacecraft Relative Motion

This paper addresses energy-optimal guidance and control of satellite relative motion for formation flying and rendezvous using impulsive maneuvers. To meet the requirements of future multisatellite missions, closed-form solutions of the inverse relative dynamics are sought in arbitrary orbits. Time constraints dictated by mission operations and relevant perturbations acting on the formation are accounted for by splitting the optimal reconfiguration into guidance (long-term) and control (short-term) layers. Both problems are cast in relative orbit element space, which allows the simple inclusion of secular and long-periodic perturbations through a state-transition matrix and the translation of the energy-optimal optimization into a minimum-length path-planning problem. From the choice of state variables, both guidance and control problems can be solved in closed form, leading to optimal, predictable maneuvering schemes for simple onboard implementation. Besides generalizing previous work, this paper finds five new in-plane and out-of-plane closed-form solutions to the relative motion control problem in unperturbed eccentric and perturbed near-circular orbits. A delta-v lower bound is formulated, which provides insight into the optimality of the control solutions. Finally, the functionality and performance of the new maneuvering schemes are rigorously assessed through numerical integration of the equations of motion and systematic comparison with primer vector optimal control.

Constructive analytical solution of the evolution hill problem

A new analytical solution of the system of differential equations describing secular perturbations and long-period solar perturbations of mean orbits of outer satellites of giant planets was obtained. As distinct from other solutions, the solution constructed using von Zeipel’s method approximately takes into account, in the secular part of the perturbing function, the totality of fourth order with respect to the small parameter m of the ratio of the mean motions of the primary planet and the satellite. This enables us to describe more accurately the evolution of satellite orbits with large apocentric distances, which in the course of evolution may exceed the halved radius of the Hill sphere of the planet with respect to the Sun. Among these are the orbits of the two outermost Neptunian satellites N10 (Psamathe) and N13 (Neso). For these satellites, the parameter m amounts to 0.152 and 0.165, respectively. Different from a purely analytical solution, the proposed solution requires preliminary calculations for each satellite. More precisely, in doing so, we need to construct some simple functions to approximate more complex ones. This is why we use the phrase “constructive analytical.” To illustrate the solution, we compare it with the results of the numerical integration of the strict motion equations of the satellites N10 and N13 over time intervals 5–15 thousand years.

Semi-analytical investigations of high area-to-mass ratio geosynchronous space debris including Earth’s shadowing effects

This paper investigates the long-term perturbations of the orbits of geosynchronous space debris influenced by direct radiation pressure including the Earth’s shadowing effects. For this purpose, we propose an extension of our homemade semi-analytical theory [Valk, S., Lemaître, A., Deleflie, F. Semi-analytical theory of mean orbital motion for geosynchronous space debris under gravitational influence. Adv. Space Res., submitted for publication], based on the method developed by Aksnes [Aksnes, K. Short-period and long-period perturbations of a spherical satellite due to direct solar radiation. Celest. Mech. Dyn. Astron. 13, 89–104, 1976] and generalized into a more convenient non-singular formalism. The perturbations accounting for the direct radiation pressure with the Earth’s shadow are computed on a revolution-by-revolution basis, retaining the original osculating Hamiltonian disturbing function. In this framework, we compute the non-singular mean longitude at shadow entry and shadow exit at every orbital revolution in opposition to classical approaches where the singular eccentric anomalies at shadow entry and shadow exit are computed. This new algorithm is developed using non-singular variables. Consequently, it is particularly suitable for both near-circular and near-equatorial orbits as well as orbits which transit periodically around null eccentricities and null inclinations. The algorithm is tested by means of numerical integrations of the equations, averaged over the short periods, including radiation pressure, J 2, the combined Moon and Sun third body attraction as well as the long-term effects of the 1:1 resonance occurring for geosynchronous objects. As an extension of [Valk, S., Lemaître, A., Anselmo, L. Analytical and semi-analytical investigations of geosynchronous space debris with high area-to-mass ratios influenced by solar radiation pressure. Adv. Space Res., doi:10.1016/j.asr.2007.10.025, 2007b], we especially apply our analysis to space debris with area-to-mass as high as 20 m 2/kg. This paper provides numerical and semi-analytical investigations leading to a deep understanding of the long-term evolution of the semi-major axis. Finally, these semi-analytical investigations are compared with accurate numerical integrations of the osculating equations of motion over time scales as high as 25 years.

The LAGEOS satellites orbital residuals determination and the Lense–Thirring effect measurement

The method applied since 1996 for the analysis of the orbital residuals of the LAGEOS satellites in order to measure the Lense–Thirring effect has been the subject of the present work. This method, based on the difference between the orbital elements of consecutive arcs, is explained and analysed also from the analytical point of view. It is proved that this “difference method” works well for the determination of the secular effects, as in the case of the relativistic precession induced by the Earth's angular momentum, but also very useful for the determination and study of the long-term periodic effects. Indeed, the only limitation in the determination of the periodic effects is the possibility of the reduction of their amplitude by a factor which depends from the periodicity of the given perturbation and from the orbital arc length chosen for the satellite during the data analysis. In the case of the Yarkovsky–Schach effect, the main non-gravitational perturbation seen in the LAGEOS satellites orbital residuals, in particular in its perigee rate and eccentricity vector excitation residuals, we show that the “difference method” is quite good also for the determination of the long-period perturbations induced by this subtle non-conservative force.

The impact of the new Earth gravity models on the measurement of the Lense–Thirring effect with a new satellite

In this paper we investigate the opportunities offered by the new Earth gravity models from the dedicated CHAMP and, especially, GRACE missions to the project of measuring the general relativistic Lense–Thirring effect with a new Earth’s artificial satellite. It turns out that it would be possible to abandon the stringent, and expensive, requirements on the orbital geometry of the originally prosed LARES mission (same semimajor axis a = 12,270 km of the existing LAGEOS and inclination i = 70°) by inserting the new spacecraft in a relatively low, and cheaper, orbit ( a = 7500–8000 km, i ∼ 70°) and suitably combining its node Ω with those of LAGEOS and LAGEOS II in order to cancel out the first two even zonal harmonic coefficients of the multipolar expansion of the terrestrial gravitational potential J 2, J 4 along with their temporal variations J ˙ 2 , J ˙ 4 . The total systematic error due to the mismodelling in the remaining even zonal harmonics would amount to ∼1% and would be insensitive to departures of the inclination from the originally proposed value of many degrees. No semisecular long-period perturbations would be introduced because the period of the node, which is also the period of the solar K 1 tidal perturbation, would amount to ∼10 2 days. Since the coefficient of the node of the new satellite would be smaller than 0.1 for such low altitudes, the impact of the non-gravitational perturbations of it on the proposed combination would be negligible. Then, a particular financial and technological effort for suitably building the satellite in order to minimize the non-conservative accelerations would be unnecessary.

Prediction of artificial satellite positions on the basis of the second order theory of motion

Some contemporary solutions of an artificial satellite orbital motion need analytical theories on a high level of accuracy. For example, the application of the analytical methods (general perturbations) becomes advantageous when a large number of satellite orbits have to be predicted simultaneously. Since the general perturbations method usually uses mean orbital elements as arguments, while the osculating elements play the role of initial conditions for predictions, the relation between the osculating and mean elements has to be determined on the same level of accuracy as that of the theory of motion applied. The paper presents a new relation between osculating and mean orbital elements based on the Mersman (1970) algorithm for the Hori-Lie perturbation method. The inverse transformation from the osculating into mean elements has the same general form as the direct transformation from the mean into osculating elements. Different are the signs of the generator in the transformations. The two kinds of transformations are realized with the same accuracy. The theory of motion includes all terms of the second order of the short- and the long-period perturbations due to zonal and tesseral harmonics of an arbitrary degree and order. Secular perturbations are included on an appropriate level of accuracy. The formulas for perturbations are based on the generalized lumped coefficients and have relatively compact forms.

On Construction of a Long-Term Moon’s Theory

An analytical long-term theory of the motion of the Moon is constructed within the framework of the general planetary theory (Brumberg, 1995). A method, different from the one of (Ivanova, 1997) designated below as (*), for the determination of the perturbations depending on the eccentricities and inclinations of lunar and planetary orbits is used which allows to obtain the solution of the problem in the purely trigonometric form up to any order with respect to the small parameters.The aim of this paper is to construct the long-term Lunar theory in the form consistent with the general planetary theory (Brumberg, 1995). For this purpose the Moon is considered as an additional planet in the field of eight major planets (Pluto being excluded). In the result the coordinates of the Moon may be represented by means of the power series in the evolutionary eccentric and oblique variables with trigonometric coefficients in mean longitudes of the Moon and the planets. The long-period perturbations are determined by solving a secular system in Laplace-type variables describing the secular motions of the lunar perigee and node and taking into account the secular planetary inequalities.

The Use of Lie Series in The Construction of a Perturbation Theory and some Recent Results in The Theory of The Motion ofd Hyperion

AbstractThis 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.

Long-periodic and secular perturbations to the orbit of a spherical satellite due to direct solar radiation pressure

A theory is developed for the perturbations to the orbit of a spherically symmetric satellite which accounts for the changes in the perigee and nodal positions and the variations of the Sun-Earth distance and direction over an orbital revolution. The theory is semi-analytical, the equations of motion being integrated with respect to time over the sunlit period of each orbital revolution. Long-periodic and short-periodic perturbations may be treated separately, and this is important for long-term analyses in terms of mean elements where short-period terms are averaged or omitted.

The fifth-order analytical solution of the equations of motion of a satellite in orbit around a non-spherical planet

A high-precise analytical theory of a satellite in orbit around a non-spherical planet has been developed. The Poisson's small parameter method has been used. All secular and short-periodic perturbations proportional up to and including a product of five arbitrary harmonic coefficients of the planetary potential expansion are calculated. Long-periodic perturbations are derived with the accuracy of up to the fourth-order, inclusive. The influence of the high-order perturbations on the motion of ETALON-1 satellite has been investigated. The results of comparison of the numerical and analytical integration of the equations of its motion over a five year interval are as follows: - the r.m.s. difference between the positions is 1.1 cm; - the r.m.s. difference between the ranges is 0.5 cm.

Evaluation of JGM 2 geopotential errors from Geosat, TOPEX/Poseidon and ERS-1 crossover altimetry

World-ocean distribution of the crossover altimetry data from Geosat, TOPEX/Poseidon (T/P) and the ERS 1 missions have provided strong independent evidence that NASA's/CSR's JGM 2 geopotential model (70 × 70 in spherical harmonics) yields accurate radial ephemerides for these satellites. In testing the sea height crossover differences found from altimetry and JGM 2 orbits for these satellites, we have used the sea height differences themselves (of ascending minus descending passes averaged at each location over many exact repeat cycles) and the Lumped Latitude Coefficients (LLC) derived from them. For Geosat we find the geopotential-induced LLC errors (exclusive of non-gravitational and initial state discrepancies) mostly below 6 cm, for TOPEX the corresponding errors are usually below 2 cm, and for ERS 1 (35-day cycle) they are generally below 5 cm. In addition, we have found that these observations agree well overall with predictions of accuracy derived from the JGM 2 variance-covariance matrix; the corresponding projected LLC errors for Geosat, T/P, and ERS 1 are usually between 1 and 4 cm, 1 – 2 cm, and 1 – 4 cm, respectively (they depend on the filtering of long-periodic perturbations and on the order of the LLC). This agreement is especially impressive for ERS 1 since no data of any kind from this mission was used in forming JGM 2. The observed crossover differences for Geosat, T/P and ERS 1 are 8, 3, and 11 cm ( rms), respectively. These observations also agree well with prediction of accuracy derived from the JGM 2 variance-covariance matrix; the corresponding projected crossover errors for Geosat and T/P are 8 cm and 2.3 cm, respectively. The precision of our mean difference observations is about 3 cm for Geosat (approx. 24,000 observations), 1.5 cm for T/P (approx. 6,000 observations) and 5 cm for ERS 1 (approx. 44,000 observations). Thus, these “global” independent data should provide a valuable new source for improving geopotential models. Our results show the need for further correction of the low order JGM 2 geopotential as well as certain resonant orders for all 3 satellites.

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.

Asteroid long-periodic perturbations: generic term representation of the determining function

A simple scheme is introduced for representing, in a computer readable form, the determining function of the canonical transformation needed to compute the asteroid long-periodic perturbations. The theory and the procedures involved in this computation are described in detail. The literal development of the determining function is coded by means of a “generic term” technique, and the resulting lists of integers are compressed in a single flat file. Examples of the lists are given and some important applications are explained.

A Synthetic Theory of Motion for Titan-Hyperion

We present an iterative method allowing to synthetize a semi-numerical solution for the equations of motion of the resonant Saturn's satellites Titan-Hyperion (limited now to the planar problem). The current theory of Hyperion by Taylor, Sinclair & Message (1987) gives the greatest terms of the long-period part of the solution (depending on two angles: the libration angle τ, and the angular distance of the pericenters ζ). Using it as a first approximation, this solution is substituted numerically in the exact Lagrange equations of motion for Titan and Hyperion, computed for many values of the three angles: τ, ζ and ϕ (the mean synodic longitude). Then, a multivariable Fourier transform allows to reconstruct the equations in these three angles, that is in same form as the initial one with, in addition, the short-period terms. Then, a solution may be obtained and used as a better approximation in an iterative process. Besides a complete determination of the short-period perturbations of Hyperion obtained here completely for the first time, some long-period perturbations of Titan by Hyperion are also found which would be non negligible at the 10 km level.

Asteroid long-periodic perturbations: The second order Hamiltonian

The Hamiltonian of the second order with respect to the disturbing mass, as defined in the higher order-higher degree theory of asteroid secular perturbations by Yuasa (1973), is expressed in the heliocentric, ecliptic coordinate system. Errors found in the original paper with terms coming from the principal part of the disturbing function are removed, and corrected values of the coefficients are computed. The importance of second-order perturbations and the improvement in the accuracy of proper element determination, achieved by using the newly-obtained coefficients, are demonstrated. Finally, a table of the secular frequencies as functions of the semimajor axis is given, and compared with the analogous one by Kozai (1979).

Simultaneous measurements of Antarctic mesospheric gravity waves by meteor radar and lidar.

Simultaneous measurements of high-latitude upper mesospheric gravity waves by a 50MHz meteor radar and a dye lidar were made at Syowa Station, Antarctica, in 1985. The radar measured nearly north-south components of neutral wind velocities at altitudes of 80-100km and at slant ranges of 110-600km from Syowa Station, while the lidar measured the sodium density profile and sodium abundance at the same altitudes at the zenith of Syowa Station. From measurements on six nights, it is found that short-period (1-2 hours) and long-period (≥2 hours) perturbations of the wind velocity due to gravity waves are well manifested in the sodium abundance and sodium density perturbations, thus suggesting that the response of the neutral winds to gravity waves is fundamentally consistent with the sodium layer response. From a case study, it is demonstrated that during a mediumscale auroral substorm, a strong equatorward wind attaining a maximum of 60m/s appeared in accord with a gradual decrease in sodium abundance. A mechanism explaining this causal relation is unknown.

The use of computer algorithms in the construction of a theory of the long-period perturbations of Saturn's satellite hyperion

This paper briefly indicates some computer-based techniques which have been developed in the construction of a theory of the long-period perturbations of the orbit of Saturn's satellite Hyperion, in the method used to expand the disturbing function and its derivatives, in the fitting into the theory of observationally-derived values of the co-efficients of the main long-period terms, and in the construction of a differential correction system to find a dynamically consistent set of values of these co-efficients.

Density scale height and zonal wind speed deduced from 296 orbits of Intercosmos 10 rocket (1973-82B)

Orbital parameters of the satellite Intercosmos 10 rocket (1973-82B) have been determined at 296 epochs using about 14,000 optical and radar observations during the 1439 d of its life. 1973–1982B was in an orbit inclined at about 74.0° to the Equator and it decayed on 8 October 1977. The orbital elements were found using the RAE orbit refinement program PROP, in the MACPROP version, giving average orbital accuracies in perigee height and inclination of 110 m and 0.002° respectively. The influence of aerodynamic drag on the motion of 1973-82B is considered. A model of the atmosphere is adopted that allows for oblateness and the diurnal variation. The observed values of eccentricity and inclination are modified by the removal of perturbations due to luni-solar attraction, solid Earth and ocean tides, solar radiation pressure, zonal and low-order long-periodic tesseral harmonic perturbations and changes due to precession. The resulting variations due to drag are analysed to give values of density scale height and three values of the zonal rotation rate of the upper atmosphere at heights between 200 and 300 km.

Earth albedo and the orbit of Lageos

The long-period perturbations in the orbit of Lageos satellite due to the earth's albedo have been found using a new analytical formalism. The earth is assumed to be a sphere whose surface diffusely reflects sunlight according to Lambert's law. Specular reflection is not considered. The formalism is based on spherical harmonics; it produces equations which hold regardless of whether the terminator is seen by the satellite or not. Specializing in the case of a realistic zonal albedo shows that Lageos' orbital semimajor axis changes periodically by only about a centimeter and the eccentricity by two parts in 105. The longitude of the node increases secularly by about 6×10−4 arc sec yr−1. The effect considered here can explain neither the secular decay of 1.1 mm day−1 in the semimajor axis nor the observed along-track variations in acceleration of order 2×10−12 ms−2.