Element sets for high-order Poincar\'e mapping of perturbed Keplerian motion

The propagation and Poincar\'e mapping of perturbed Keplerian motion is a key topic in celestial mechanics and astrodynamics, e.g. to study the stability of orbits or design bounded relative trajectories. The high-order transfer map (HOTM) method enables efficient mapping of perturbed Keplerian orbits over many revolutions. For this, the method uses the high-order Taylor expansion of a Poincar\'e or stroboscopic map, which is accurate close to the expansion point. In this paper, we investigate the performance of the HOTM method using different element sets for building the high-order map. The element sets investigated are the classical orbital elements, modified equinoctial elements, Hill variables, cylindrical coordinates and Deprit's ideal elements. The performances of the different coordinate sets are tested by comparing the accuracy and efficiency of mapping low-Earth and highly-elliptical orbits perturbed by $J_2$ with numerical propagation. The accuracy of HOTM depends strongly on the choice of elements and type of orbit. A new set of elements is introduced that enables extremely accurate mapping of the state, even for high eccentricities and higher-order zonal perturbations. Finally, the high-order map is shown to be very useful for the determination and study of fixed points and centre manifolds of Poincar\'e maps.Comment: Pre-print of journal articl

High-Fidelity Semianalytical Theory for a Low Lunar Orbit

We have developed a semi-analytical theory for low-altitude lunar orbits with the aim of verifying what the minimum order of the gravitational model to be considered should be in order to produce realistic results that can be applied to the analysis and design of real missions. With that purpose, we have considered a perturbation model that comprises a 50x50 gravitational field and the third-body attraction from the Earth. Initially, the process of developing the theory is briefly described. Then, the discussion is focused on the search for frozen orbits, for which the effect of each harmonic term of the gravitational model is analyzed separately. As higher-order zonal harmonics are included, new families of frozen orbits can appear. In addition, the eccentricity and inclination values for which frozen orbits can exist change. This effect is very important and needs to be taken into consideration, because ignoring high-order harmonics can lead to predict the existence of frozen orbits at certain inclinations at which the frozen-orbit eccentricity actually falls beyond the impact limit. Consequently, it has been verified that, in agreement with other authors, a 50x50 gravitational model should be the minimum to be considered for real applications.Comment: 8 pages, 5 figure

Absolute and Relative Motion Satellite Theories for Zonal and Tesseral Gravitational Harmonics

In 1959, Dirk Brouwer pioneered the use of the Hamiltonian perturbation methods for constructing artificial satellite theories with effects due to nonspherical gravitational perturbations included. His solution specifically accounted for the effects of the first few zonal spherical harmonics. However, the development of a closed-form (in the eccentricity) satellite theory that accounts for any arbitrary spherical harmonic perturbation remains a challenge to this day. In the present work, the author has obtained novel solutions for the absolute and relative motion of artificial satellites (absolute motion in this work refers to the motion relative to the central gravitational body) for an arbitrary zonal or tesseral spherical harmonic by using Hamiltonian perturbation methods, without resorting to expansions in either the eccentricity or the small ratio of the satellite’s mean motion and the angular velocity of the central body. First, generalized closed-form expressions for the secular, long-period, and short-period variations of the equinoctial orbital elements due to an arbitrary zonal harmonic are derived, along with the explicit expressions for the first six zonal harmonics. Next, similar closed-form expressions are obtained for the sectorial and tesseral (collectively referred to as tesserals henceforth) harmonics by using a new approach for the exact Delaunay normalization of the perturbed Keplerian Hamiltonian. This approach reduces the solution for the tesseral periodic perturbations to quadratures. It is shown that the existing approximate approaches for the normalization of the tesseral problem, such as the method of relegation, can be derived from the proposed exact solution. Moreover, the exact solution for the periodic variations due to the tesseral harmonics produces a unified artificial satellite theory for the sub-synchronous and super-synchronous orbit regimes without any singularities for the resonant orbits. The closedform theories developed for the absolute motion are then used to develop analytic solutions in the form of state transition matrices for the satellite relative motion near a perturbed elliptic reference orbit. The expressions for differential equinoctial orbital elements for establishing a general circular orbit type satellite formation are also derived to avoid singularities for the equatorial and circular reference orbits. In order to negate the along-track drifts in satellite formations, an ana- lytic expression for the differential semimajor axis is derived by taking into account the secular effects due to all the zonal harmonics. The potential applications of the proposed satellite theories range from fuel-efficient guidance and control algorithms, formation design, faster trade and parametric studies to catalog maintenance, conjunction analysis, and covariance propagation for space situational awareness. Two specific applications, one for solving a perturbed multiple revolution Lambert’s problem and the other for rapid nonlinear propagation of orbit uncertainties using point clouds, are also given. The theories presented in this work are implemented for computer simulations in a software tool. The simulation results validated the accuracy of these theories and demonstrated their effectiveness for various space situational awareness applications

Towards a sustainable exploitation of the geosynchronous orbital region

In this work the orbital dynamics of Earth satellites about the geosynchronous altitude are explored, with primary goal to assess current mitigation guidelines as well as to discuss the future exploitation of the region. A thorough dynamical mapping was conducted in a high-definition grid of orbital elements, enabled by a fast and accurate semi-analytical propagator, which considers all the relevant perturbations. The results are presented in appropriately selected stability maps to highlight the underlying mechanisms and their interplay, that can lead to stable graveyard orbits or fast re-entry pathways. The natural separation of the long-term evolution between equatorial and inclined satellites is discussed in terms of post-mission disposal strategies. Moreover, we confirm the existence of an effective cleansing mechanism for inclined geosynchronous satellites and discuss its implications in terms of current guidelines as well as alternative mission designs that could lead to a sustainable use of the geosynchronous orbital region.Comment: Accepted for publication in Celestial Mechanics and Dynamical Astronom

Station-keeping for lattice-preserving Flower Constellations

2D-Lattice Flower Constellations present interesting dynamical features that al- low us to explore a wide range of potential applications. Their particular initial distribution (lattice) and their symmetries disappear when some perturbations are considered, such as the J2 effect. The new lattice-preserving Flower Constella- tions maintain over long periods of time the initial distribution and its symmetries under the J2 perturbation, which is known as relative station-keeping. This paper deals with the study of the required velocity change that must be applied to the satellites of the constellation to have an absolute station-keeping

On the Periodic Orbits of the Perturbed Two- and Three-Body Problems

In this work, a perturbed system of the restricted three-body problem is derived when the perturbation forces are conservative alongside the corresponding mean motion of two primaries bodies. Thus, we have proved that the first and second types of periodic orbits of the rotating Kepler problem can persist for all perturbed two-body and circular restricted three-body problems when the perturbation forces are conservative or the perturbed motion has its own extended Jacobian integral

Towards an analytical theory of the third-body problem for highly elliptical orbits

When dealing with satellites orbiting a central body on a highly elliptical orbit, it is necessary to consider the effect of gravitational perturbations due to external bodies. Indeed, these perturbations can become very important as soon as the altitude of the satellite becomes high, which is the case around the apocentre of this type of orbit. For several reasons, the traditional tools of celestial mechanics are not well adapted to the particular dynamic of highly elliptical orbits. On the one hand, analytical solutions are quite generally expanded into power series of the eccentricity and therefore limited to quasi-circular orbits [17, 25]. On the other hand, the time-dependency due to the motion of the third-body is often neglected. We propose several tools to overcome these limitations. Firstly, we have expanded the disturbing function into a finite polynomial using Fourier expansions of elliptic motion functions in multiple of the satellite's eccentric anomaly (instead of the mean anomaly) and involving Hansen-like coefficients. Next, we show how to perform a normalization of the expanded Hamiltonian by means of a time-dependent Lie transformation which aims to eliminate periodic terms. The difficulty lies in the fact that the generator of the transformation must be computed by solving a partial differential equation involving variables which are linear with time and the eccentric anomaly which is not time linear. We propose to solve this equation by means of an iterative process.Comment: Proceedings of the International Symposium on Orbit Propagation and Determination - Challenges for Orbit Determination and the Dynamics of Artificial Celestial Bodies and Space Debris, Lille, France, 201

Revisiting Vinti theory : generalized equinoctial elements and applications to spacecraft relative motion

Early phases of complex astrodynamics applications often require broad searches of large solution spaces. For these studies, mission complexity generally motivates the use of the coarsest dynamical models with analytical solutions because of the implied lightening of the computational load. In this context, two-body dynamics are typically employed in practice, but higher-fidelity models with analytical solutions exist, an attractive prospect for modern applications that may require or benefit from greater accuracy. Vinti theory, which prescribes one of the many alternative described models known as intermediaries, is revisited because it leads to a direct generalization of two-body dynamics, naturally incorporating the dominant effect of oblateness and optionally the top/bottom-heavy characteristic of a celestial body without recourse to perturbation methods. Prior to the innovations introduced in this dissertation, Vinti theory and associated solutions possessed many singularities in popular orbital regimes. The theory has received limited use. The goals of this dissertation are to assess Vinti theory's effectiveness in a modern application and remove its long-standing disincentives. These objectives inform the two main contributions, respectively: 1) Vinti theory is applied to the relative motion problem through the development of a state transition matrix (STM), enabled by improvements to the existing theory; 2) a new nonsingular element set is introduced. The relative motion application leverages Vinti's approximate analytical solution with J₃. An analytical relative motion model is derived and subsequently reformulated so that Vinti's solution is piecewise differentiable, developed alongside boosts in accuracy and removal of singularities in polar and nearly circular or equatorial orbits. Some of these singularities reside in the solution, others in the partials. Solving the problem in oblate spheroidal elements leads to large linear regions of validity. The new STM is compared with side-by-side simulations of a benchmark STM obtained from perturbation methods and is shown to offer improved accuracy over a broad design space. To defray the costs of software development, robust code is provided online. The second major thrust area is the introduction of a nonsingular element set that is at once novel and familiar. Vinti theory suffers from other well-known singularities, strictly artifacts of classical elements that are detrimental to many applications. To mitigate these singularities, the standard (spherical) equinoctial elements are chosen to inform in a natural way their generalization to a new nonsingular element set: the oblate spheroidal equinoctial orbital elements. The new elements are derived without J₃ and concise algorithms presented for common coordinate transformations. The transformations are valid away from the nearly rectilinear orbital regime and are exact except near the poles. When near the poles, the transformations match the accuracy of the approximate analytical solution. As a result, the singularity on the poles is completely eliminated for the first time. Analytical state propagation of the new elements in time for bounded orbits completes their formal introduction. Benefits of the new elements are identified. The dissertation is organized as follows. To convey Vinti theory's broader context, extensive background on intermediaries and related topics is provided in Chapter 1. General enhancements that grew out of the main efforts, including the removal of some singularities, are consolidated in Chapter 2 along with mathematical preliminaries. Relative motion is explored as the selected application in Chapter 3 and the major deficiencies of Vinti theory are removed in Chapter 4 with the introduction of the new element set. Analytical orbit propagation in the new set is developed in Chapter 5.Aerospace Engineerin

Measuring the Lense-Thirring precession using a second Lageos satellite

A complete numerical simulation and error analysis was performed for the proposed experiment with the objective of establishing an accurate assessment of the feasibility and the potential accuracy of the measurement of the Lense-Thirring precession. Consideration was given to identifying the error sources which limit the accuracy of the experiment and proposing procedures for eliminating or reducing the effect of these errors. Analytic investigations were conducted to study the effects of major error sources with the objective of providing error bounds on the experiment. The analysis of realistic simulated data is used to demonstrate that satellite laser ranging of two Lageos satellites, orbiting with supplemental inclinations, collected for a period of 3 years or more, can be used to verify the Lense-Thirring precession. A comprehensive covariance analysis for the solution was also developed

Orbit Determination Using Vinti\u27s Solution

Orbital altitudes congested with spacecraft and debris combined with recent collisions have all but negated the Big Sky Theory. As the sheer number of orbital objects to track grows unbounded so does interest in prediction methods that are rapid and minimally computational. Claimed as the \other solvable solution, the recently completed solution too orbital motion about the earth, based on Vinti\u27s method and including the major effects of the equatorial bulge, opens up the prospect of much more accurate analytical models for space situational awareness. A preliminary examination of this solution is presented. A numerical state transition matrix is found using Lagrange partial derivatives to implement a nonlinear least squares fitting routine. Orbit fits using only the solvable solution for non-circular, non-equatorial trajectories less than 60 degrees inclination are on the order of a few hundred meters with projected, average error growth of less than a kilometer per day which is similar to the expected performance of the Air Force\u27s method. Also, a classical perturbations approach to incorporate the dissipative effects of air drag using Hamiltonian action and angle formulation is developed. Predicted drag effects re 97.5 correct after one day and 87 correct after five days when compared to an integrated truth. Results are validated by performing a similar method on the two body problem