Keplerian Dynamics Research Articles

Spacecraft to Spacecraft Absolute Tracking for Autonomous Navigation of a Distributed Space System from Relative Sensors

Navigation for a distributed space system can be fully and autonomously provided by relative measurements between vehicles under non-linear dynamics, thereby removing any need for external observation. This paper derives the observability of a generic distributed system using relative measurements through an investigation of non-linear dynamical moments. The absolute state becomes observable due to second and higher dynamical moments that produce differential motion between vehicles, while the relative state observability is a function of the full system motion as traditionally expected. The results from observability analysis are leveraged to instantiate simulations which demonstrate observability for a distributed space system with relative optical measurements. The simulations assess uncertainty via a linearized covariance analysis in various settings including Keplerian, perturbed Keplerian, and cislunar dynamics. This navigation method excels when targets have highly differential motion and falters as the difference subsides. Therefore, highly perturbed environments with multiple targets generate the best results.

Orbit determination from one position vector and a very short arc of optical observations

In this paper, we address the problem of computing a preliminary orbit of a celestial body from one topocentric position vector P1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal{P}_1$$\\end{document} and a very short arc (VSA) of optical observations A2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal{A}_2$$\\end{document}. Using the conservation laws of the two-body dynamics, we write the problem as a system of 8 polynomial equations in 6 unknowns. We prove that this system is generically consistent, namely, for a generic choice of the data P1,A2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal{P}_1, \\mathcal{A}_2$$\\end{document}, it always admits solutions in the complex field, even when P1,A2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal{P}_1, \\mathcal{A}_2$$\\end{document} do not correspond to the same celestial body. The consistency of the system is shown by deriving a univariate polynomial v\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathfrak {v}$$\\end{document} of degree 8 in the unknown topocentric distance at the mean epoch of the observations of the VSA. Through Gröbner bases theory, we also show that the degree of v\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathfrak {v}$$\\end{document} is minimum among the degrees of all the univariate polynomials solving this problem. Even though we can find solutions to our problem for a generic choice of P1,A2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal{P}_1, \\mathcal{A}_2$$\\end{document}, most of these solutions are meaningless. In fact, acceptable solutions must be real and have to fulfill other constraints, including compatibility with Keplerian dynamics. We also propose a way to select or discard solutions taking into account the uncertainty in the data, if present. The proposed orbit determination method is relevant for different purposes, e.g., the computation of a preliminary orbit of an Earth satellite with radar and optical observations, the detection of maneuvres of an Earth satellite, and the recovery of asteroids which are lost due to a planetary close encounter. We conclude by showing some numerical tests in the case of asteroids undergoing a close encounter with the Earth.

On mean elements in artificial-satellite theory

The merits of a perturbation theory based on a mean-to-osculating transformation that is purely periodic in the fast angle are investigated. The exact separation of the perturbed Keplerian dynamics into purely short-period effects and long-period mean frequencies is achieved by a non-canonical transformation, which, therefore, cannot be obtained by Hamiltonian methods. For this case, the evolution of the mean elements strictly adheres to the average behavior of the osculating orbit. However, due to the unavoidable truncation of perturbation solutions, the fact that this kind of theory confines in the mean variations the long-period terms of the semimajor axis, how tiny they may be, can have adverse effects in the accuracy of long-term semi-analytic propagations based on it.

Optimal Floquet Stationkeeping under the Relative Dynamics of the Three-Body Problem

Deep space missions, and particularly cislunar endeavors, are becoming a major field of interest for the space industry, including for the astrodynamics research community. While near-Earth missions may be completely covered by perturbed Keplerian dynamics, deep space missions require a different modeling approach, where multi-body gravitational interactions play a major role. To this end, the Restricted Three-Body Problem stands out as an insightful first modeling strategy for early mission design purposes, retaining major dynamical transport structures while still being relatively simple. Dynamical Systems Theory and classical Hamiltonian Mechanics have proven themselves as remarkable tools to analyze deep-space missions within this context, with applications ranging from ballistic capture trajectory design to stationkeeping. In this work, based on this premise, a Hamiltonian derivation of the Restricted Three-Body Problem co-orbital dynamics between two spacecraft is introduced in detail. Thanks to the analytical and numerical models derived, connections between the relative and classical Keplerian and CR3BP problems are shown to exist, including first-order linear solutions and an inherited Hamiltonian normal form. The analytical linear and higher-order models derived allow the theoretical finding and unveiling of natural co-orbital phase space structures, including relative periodic and quasi-periodic orbital families, which are further exploited for general proximity operation applications. In particular, a novel reduced-order, optimal low-thrust stationkeeping controller is derived in the relative Floquet phase space, hybridizing the classical State Dependent Ricatti Equation (SDRE) with Koopman control techniques for efficient unstable manifold regulation. The proposed algorithm is demonstrated and validated within several end-to-end low-cost stationkeeping missions, and comparison against classical continuous stationkeeping algorithms presented in the literature is also addressed to reveal its enhanced performance. Finally, conclusions and open lines of research are discussed.

The nebula around the binary post-AGB star 89 Herculis

Context. There is a class of binary post-asymptotic giant branch (post-AGB) stars that exhibit remarkable near-infrared (NIR) excess. These stars are surrounded by disks with Keplerian or quasi-Keplerian dynamics and outflows composed of gas escaping from the rotating disk. Depending on the dominance of these components, there are two subclasses of binary post-AGB stars: disk-dominated and outflow-dominated. Aims. We aim to properly study the hourglass-like structure that surrounds the Keplerian disk around 89 Her. Methods. We present total-power on-the-fly maps of 12CO and 13CO J = 2 − 1 emission lines in 89 Her. Previous studies are known to suffer from flux losses in the most extended components. We merge these total-power maps with previous NOEMA maps. The resulting combined maps are expected to detect the whole nebula extent of the source. Results. Our new combined maps contain the entirety of the detectable flux of the source and at the same time are of high spatial resolution thanks to the interferometric observations. We find that the hourglass-like extended outflow around the rotating disk is larger and more massive than suggested by previous works. The total nebular mass of this very extended nebula is 1.8 × 10−2 M⊙, of which ∼65% comes from the outflow. The observational data and model results lead us to classify the envelope around 89 Her as an outflow-dominated nebula, together with R Sct and IRAS 19125+0343 (and very probably AI CMi, IRAS 20056+1834, and IRAS 18123+0511). The updated statistics on the masses of the two post-AGB main components reveal that there are two distinct subclasses of nebulae around binary post-AGB stars depending on which component is the dominant one. We speculate that the absence of an intermediate subclass of sources is due to the different initial conditions of the stellar system and not because both subclasses are in different stages of the post-AGB evolution.

Rotating and Expanding Gas in Binary Post-AGB Stars

There is a class of binary post-AGB stars (binary system including a post-AGB star) that are surrounded by Keplerian disks and outflows resulting from gas escaping from the disk. To date, there are seven sources that have been studied in detail through interferometric millimeter-wave maps of CO lines (ALMA/NOEMA). For the cases of the Red Rectangle, IW Carinae, IRAS 08544-4431, and AC Herculis, it is found that around ≥85% of the total nebular mass is located in the disk with Keplerian dynamics. The remainder of the nebular mass is located in an expanding component. This outflow is probably a disk wind consisting of material escaping from the rotating disk. These sources are the disk-dominated nebulae. On the contrary, our maps and modeling of 89 Herculis, IRAS 19125+0343, and R Scuti, which allowed us to study their morphology, kinematics, and mass distribution, suggest that, in these sources, the outflow clearly is the dominant component of the nebula (∼75% of the total nebular mass), resulting in a new subclass of nebulae around binary post-AGB stars: the outflow-dominated sources.Besides CO, the chemistry of this type of source has been practically unknown thus far. We also present a very deep single-dish radio molecular survey in the 1.3, 2, 3, 7, and 13 mm bands (∼600 h of telescope time). Our results and detections allow us to classify our sources as O- or /C-rich. We also conclude that the calculated abundances of the detected molecular species other than CO are particularly low, compared with AGB stars. This fact is very significant in those sources where the rotating disk is the dominant component of the nebula.

Chemistry of nebulae around binary post-AGB stars: A molecular survey of mm-wave lines

Context. There is a class of binary post-asymptotic giant branch (post-AGB) stars that exhibit remarkable near-infrared excess. Such stars are surrounded by Keplerian or quasi-Keplerian disks, as well as extended outflows composed of gas escaping from the disk. This class can be subdivided into disk- and outflow-dominated sources, depending on whether it is the disk or the outflow that represents most of the nebular mass, respectively. The chemistry of this type of source has been practically unknown thus far. Aims. Our objective is to study the molecular content of nebulae around binary post-AGB stars that show disks with Keplerian dynamics, including molecular line intensities, chemistry, and abundances. Methods. We focused our observations on the 1.3, 2, 3 mm bands of the 30mIRAM telescope and on the 7 and 13 mm bands of the 40 m Yebes telescope. Our observations add up ~600 h of telescope time. We investigated the integrated intensities of pairs of molecular transitions for CO, other molecular species, and IRAS fluxes at 12, 25, and 60 μm. Additionally, we studied isotopic ratios, in particular 17O/18O, to analyze the initial stellar mass, as well as 12CO/13CO, to study the line and abundance ratios. Results. We present the first single-dish molecular survey of mm-wave lines in nebulae around binary post-AGB stars. We conclude that the molecular content is relatively low in nebulae around binary post-AGB stars, as their molecular lines and abundances are especially weaker compared with AGB stars. This fact is very significant in those sources where the Keplerian disk is the dominant component of the nebula. The study of their chemistry allows us to classify nebulae around AC Her, the Red Rectangle, AI CMi, R Sct, and IRAS 20056+1834 as O-rich, while that of 89 Her is probably C-rich. The calculated abundances of the detected species other than CO are particularly low compared with AGB stars. The initial stellar mass derived from the 17O/18O ratio for the Red Rectangle and 89 Her is compatible with the central total stellar mass derived from previous mm-wave interferometric maps. The very low 12CO/13CO ratios found in binary post-AGB stars reveal a high 13CO abundance compared to AGB and other post-AGB stars.

Optimal Q-laws via reinforcement learning with guaranteed stability

Closed-loop feedback-driven control laws can be used to solve low-thrust many-revolution trajectory design and guidance problems with minimal computational cost. Lyapunov-based control laws offer the benefits of increased stability whilst their optimality can be increased by tuning their parameters. In this paper, a reinforcement learning framework is used to make the parameters of the Lyapunov-based Q-law state-dependent, increasing its optimality. The Jacobian of these state-dependent parameters is available analytically and, unlike in other optimisation approaches, can be used to enforce stability throughout the transfer. The results focus on GTO–GEO and LEO–GEO transfers in Keplerian dynamics, including the effects of eclipses. The impact of the network architecture on the behaviour is investigated for both time- and mass-optimal transfers. Robustness to navigation errors and thruster misalignment is demonstrated using Monte Carlo analyses. The resulting approach offers potential for on-board autonomous transfers and orbit reconfiguration.

Keplerian disks and outflows in post-AGB stars: AC Herculis, 89 Herculis, IRAS 19125+0343, and R Scuti

Context. There is a class of binary post-AGB stars with a remarkable near-infrared excess that are surrounded by Keplerian or quasi-Keplerian disks and extended outflows composed of gas escaping from the disk. The Keplerian dynamics had been well identified in four cases, namely the Red Rectangle, AC Her, IW Car, and IRAS 08544−4431. In these objects, the mass of the outflow represents ~10% of the nebular mass, the disk being the dominant component of the nebula. Aims. We aim to study the presence of rotating disks in sources of the same class in which the outflow seems to be the dominant component. Methods. We present interferometric NOEMA maps of 12CO and 13CO J = 2–1 in 89 Her and 12CO J = 2–1 in AC Her, IRAS 19125+0343, and R Sct. Several properties of the nebula are obtained from the data and model fitting, including the structure, density, and temperature distributions, as well as the dynamics. We also discuss the uncertainties on the derived values. Results. The presence of an expanding component in AC Her is doubtful, but thanks to new maps and models, we estimate an upper limit to the mass of this outflow of ≲3 × 10−5 M⊙, that is, the mass of the outflow is ≲5% of the total nebular mass. For 89 Her, we find a total nebular mass of 1.4 × 10−2 M⊙, of which ~50% comes from an hourglass-shaped extended outflow. In the case of IRAS 19125+0343, the nebular mass is 1.1 × 10−2 M⊙, where the outflow contributes ~70% of the total mass. The nebular mass of R Sct is 3.2 × 10−2 M⊙, of which ~75% corresponds to a very extended outflow that surrounds the disk. Conclusions. Our results for IRAS 19125+0343 and R Sct lead us to introduce a new subclass of binary post-AGB stars, for which the outflow is the dominant component of the nebula. Moreover, the outflow mass fraction found in AC Her is smaller than those found in other disk-dominated binary post-AGB stars. 89 Her would represent an intermediate case between both subclasses.

On the solution to every Lambert problem

Lambert’s problem is the two-point boundary value problem for Keplerian dynamics. The parameter and solution space is surveyed for both the zero- and multiple-revolution problems, including a detailed look at the stress cases that typically plague Lambert solvers. The problem domain, independent of formulation, is shown to be rectangular for each revolution case, making the elusive initial guess and the solution itself amenable for interpolation. Biquintic splines are implemented to achieve continuous derivatives and quick evaluation. Resulting functions may be used directly as low-fidelity solutions or used with a single update iteration without safeguards. A concise, improved vercosine formulation of the Lambert problem is presented, including new singularity-free and precision-saving equations. The interpolation scheme is applied for up to 100 revolutions. The domain considered includes all practically conceivable flight times, and every possible geometry except a small region near the only physical singularity of the problem: the equal terminal vector case. The solutions are archived and benchmarked for accuracy, memory footprint, and speed. For typical scenarios, users can expect $$\sim 6$$ or more digits of velocity vector accuracy using an interpolated solution without iteration. Using a single, unguarded iteration leads to solutions with near machine precision accuracy over the full domain, including the most extreme scenarios. Depending on desired resolution, coefficient files vary in size from $$\sim 3$$ to 65 MB for each revolution case. Evaluation runtimes vary from $$\sim 2$$ to 5 times faster than the industry benchmark Gooding algorithm. The coefficient files and driver routines are provided online. While the method is currently demonstrated on the vercosine formulation, the 2D interpolation scheme stands to benefit all Lambert problem formulations.

Geometric Solutions for Problems in Velocity-Based Orbit Determination

Classical techniques for initial orbit determination (IOD) require the analyst to find a body’s orbit given only observations such as bearings, range, and/or position. In these cases, one of the goals is often to solve for the unknown velocity vector at one or more of the observation times to fully define the orbit. Recently, however, a new class of IOD problems has been proposed that switches the knowns and unknowns in these classic IOD problems. Specifically, the objective is to find the unknown position vectors given only velocity measurements. This paper presents a detailed assessment of the geometric properties of this new family of velocity-only IOD problems. The primary tool for this geometric analysis is Hamilton’s orbital hodograph, which is known to be a perfect circle for all orbits obeying Keplerian dynamics. This framework is used to produce intuitive and efficient algorithms for IOD from three velocity vectors (similar structure to Gibbs problem) and for IOD from two velocity vectors and time-of-flight (similar structure to Lambert’s problem). Performance of these algorithms is demonstrated through numerical results.

High-order state transition polynomial with time expansion based on differential algebra

In this study, a high-order state transition polynomial with time expansion (STP-T) method is developed to propagate an initial orbital state around its reference value to a variable final time based on the differential algebra (DA) technique. STP-T is a high-order Taylor polynomial of the final orbital state expanded around the reference initial state and reference propagation time. Since the final state usually shows different nonlinearity with respect to different components of initial state and propagation time, a weighted-order scheme is combined with STP-T, which enables the STP-T to have higher orders on the components with higher nonlinearity and lower orders on the components with lower nonlinearity. Then, an error estimation method is presented, which can a priori provide the error profiles of a STP-T and is useful for selecting a proper order and determining the corresponding valid ranges of displacements. Finally, the STP-T method is tested for orbit propagation under three typical orbital dynamics: the unperturbed Keplerian dynamics, the J2 perturbed two-body dynamics, and the nonlinear relative dynamics. The numerical simulation results indicate that the STP-T supplies a good approximation of the final state within certain valid ranges of initial state and propagation time, the a priori estimated error is close to the exact error in sense of trend and magnitude, and the computational cost can be significantly saved by the weighted-order scheme without loss of accuracy.

Rendezvous Strategies in the Vicinity of Earth-Moon Lagrangian Points

In the context of Human Spaceflight exploration mission scenario, with the Lunar Orbital Platform- Gateway (LOP-G) orbiting about Earth-Moon Lagrangian Point (EML), Rendezvous and Docking (RVD) operational activities are mandatory and critical for the deployment and utilization of the LOP-G (station assembly, crew rotations, cargo delivery, lunar sample return). There is extensive experience with RVD in the two-body problem: in Low Earth Orbit (LEO) to various space stations, or around quasi-circular Low Lunar Orbits (LLO), the latter by Apollo by means of manual RVD. However, the RVD problem in non-Keplerian environments has rarely been addressed and no RVD has been performed to this date in the vicinity of Lagrangian points (LP) where Keplerian dynamics are no longer applicable. Dynamics in such regions are more complex, but multi-body dynamics also come with strong advantages that need to be further researched by the work proposed here. The aim of this paper is to present methods and results of investigations conducted to first set up strategies for far and close rendezvous between a target (the LOP-G, for example) and a chaser (cargo, crew vehicle, ascent and descent vehicle, station modules, etc.) depending on target and chaser orbit. Semi-analytical tools have been developed to compute and model families of orbits about the Lagrangian points in the Circular Restricted Three Body Problem (CR3BP) like NRHO, DRO, Lyapunov, Halo and Lissajous orbits. As far as close rendezvous is concerned, implementation of different linear and non-linear models used to describe cis-lunar relative motion will be discussed and compared, in particular for NRHO and DRO.

Preliminary orbits with line-of-sight correction for LEO satellites observed with radar

We propose a method to account for the Earth oblateness effect in preliminary orbit determination of satellites in low orbits with radar observations. This method is an improvement of the one described in (Gronchi et al 2015), which uses a pure Keplerian dynamical model. Since the effect of the Earth oblateness is strong at low altitudes, its inclusion in the model can sensibly improve the initial orbit, giving a better starting guess for differential corrections and increasing the chances to obtain their convergence. The input set consists of two tracks of radar observations, each one composed of at least 4 observations taken during the same pass of the satellite. A single observation gives the topocentric position of the satellite, where the range is very accurate, while the line of sight direction is poorly determined. From these data we can compute by a polynomial fit the values of the range and range rate at the mean epochs of the two tracks. In order to obtain a preliminary orbit we wish to compute the angular velocities, that is the rate of change of the line of sight. In the same spirit of (Gronchi et al 2015), we also wish to correct the values of the angular measurements, so that they fit the selected dynamical model if the same holds for the radial distance and velocity. The selected model is a perturbed Keplerian dynamics, where the only perturbation included is the secular effect of the $J_2$ term of the geopotential. The proposed algorithm models this problem with 8 equations in 8 unknowns.

Secular Evolution Driven by Massive Eccentric Disks/Rings: An Apsidally Aligned Case

Abstract Massive eccentric disks (gaseous or particulate) orbiting a dominant central mass appear in many astrophysical systems, including planetary rings, protoplanetary and accretion disks in binaries, and nuclear stellar disks around supermassive black holes in galactic centers. We present an analytical framework for treating the nearly Keplerian secular dynamics of test particles driven by the gravity of an eccentric, apsidally aligned, zero-thickness disk with arbitrary surface density and eccentricity profiles. We derive a disturbing function describing the secular evolution of coplanar objects, which is explicitly related (via one-dimensional, convergent integrals) to the disk surface density and eccentricity profiles without using any ad hoc softening of the potential. Our analytical framework is verified via direct orbit integrations, which show it to be accurate in the low-eccentricity limit for a variety of disk models (for disk eccentricity ≲0.1–0.2). We find that free precession in the potential of a disk with a smooth surface density distribution can naturally change from prograde to retrograde within the disk. Sharp disk features—edges and gaps—are the locations where this tendency is naturally enhanced, while the precession becomes very fast. Radii where free precession changes sign are the locations where substantial (formally singular) growth of the forced eccentricity of the orbiting objects occurs. Based on our results, we formulate a self-consistent analytical framework for computing an eccentricity profile for an aligned, eccentric disk (with a prescribed surface density profile) capable of precessing as a solid body under its own self-gravity.

Shadow Trajectory Model for Fast Low-Thrust Indirect Optimization

Preliminary design of low-thrust trajectories generally benefits from broad searches over the feasible space. Despite convergence issues, indirect methods are generally faster than direct methods and are therefore well-suited for such searches. Nonetheless, indirect solutions typically require expensive numerical integration of at least the state and costate equations. Here, based on the physical interpretation of the primer vector, a fast model that approximates solutions to all the dynamics is introduced. If the ballistic dynamics have a closed-form solution, then the costate equations can be approximated without resorting to numerical integration. Analogous to the Sims–Flanagan model for direct optimization, the new model approximates thrust arcs with a series of ballistic arcs and impulsive maneuvers. The closed-form solution is obtained using a Sundman-transformed independent variable, which also provides an efficient discretization. Furthermore, a low-order series solution is used for the ballistic propagation. The model is introduced and evaluated for speed and accuracy using examples with Keplerian dynamics. Speedups vary according to thrust and discretization levels. For accuracies relevant to preliminary design, order-of-magnitude speedups are achieved.

Linearized Lambert’s Problem Solution

Lambert’s problem is the well-known problem to determine the orbit that allows an object to travel between two given position vectors in a given time of flight under Keplerian dynamics. One drawback of solving this problem is that it requires solving a transcendental equation, which involves commonly using an iterative method that can be undesirable for autonomous onboard applications. In this paper, we present a linearization to the solution of Lambert's problem using Lagrange parameters. It is shown that, around a wide variety of nominal transfer trajectories, high-accuracy solutions to the neighboring transfers can be rapidly determined using this linearized solution. Sensitivity studies are presented that show that errors in the terminal velocities are typically well below 1% for cases with 5% perturbations in the terminal positions and time of transfer. Unlike many similar solutions for the relative Lambert’s problem, the linearized results presented here do not suffer in general when the nominal transfer trajectory has high eccentricity. Examples are presented that demonstrate that this linearization can be used effectively for targeting and orbit determination applications. Important for possible onboard applications, it is shown that the cost of computing the partials needed to create a linearized solution is very cheap; in the case tested, computation of the partials takes only 0.1% of the time needed to compute a full Lambert’s solution.

Dealing with uncertainties in angles-only initial orbit determination

A method to deal with uncertainties in initial orbit determination (IOD) is presented. This is based on the use of Taylor differential algebra (DA) to nonlinearly map uncertainties from the observation space to the state space. When a minimum set of observations is available, DA is used to expand the solution of the IOD problem in Taylor series with respect to measurement errors. When more observations are available, high order inversion tools are exploited to obtain full state pseudo-observations at a common epoch. The mean and covariance of these pseudo-observations are nonlinearly computed by evaluating the expectation of high order Taylor polynomials. Finally, a linear scheme is employed to update the current knowledge of the orbit. Angles-only observations are considered and simplified Keplerian dynamics adopted to ease the explanation. Three test cases of orbit determination of artificial satellites in different orbital regimes are presented to discuss the feature and performances of the proposed methodology.

Uncertainty propagation in orbital mechanics via tensor decomposition

Uncertainty forecasting in orbital mechanics is an essential but difficult task, primarily because the underlying Fokker–Planck equation (FPE) is defined on a relatively high dimensional (6-D) state–space and is driven by the nonlinear perturbed Keplerian dynamics. In addition, an enormously large solution domain is required for numerical solution of this FPE (e.g. encompassing the entire orbit in the $$x-y-z$$ subspace), of which the state probability density function (pdf) occupies a tiny fraction at any given time. This coupling of large size, high dimensionality and nonlinearity makes for a formidable computational task, and has caused the FPE for orbital uncertainty propagation to remain an unsolved problem. To the best of the authors’ knowledge, this paper presents the first successful direct solution of the FPE for perturbed Keplerian mechanics. To tackle the dimensionality issue, the time-varying state pdf is approximated in the CANDECOMP/PARAFAC decomposition tensor form where all the six spatial dimensions as well as the time dimension are separated from one other. The pdf approximation for all times is obtained simultaneously via the alternating least squares algorithm. Chebyshev spectral differentiation is employed for discretization on account of its spectral (“super-fast”) convergence rate. To facilitate the tensor decomposition and control the solution domain size, system dynamics is expressed using spherical coordinates in a noninertial reference frame. Numerical results obtained on a regular personal computer are compared with Monte Carlo simulations.

Detection of Keplerian dynamics in a disk around the post-AGB star AC Herculis

So far, only one rotating disk has been clearly identified and studied in AGB or post-AGB objects (in the Red Rectangle), by means of observations with high spectral and spatial resolution. However, disks are thought to play a key role in the late stellar evolution and are suspected to surround many evolved stars. We aim to extend our knowledge on these structures. We present interferometric observations of CO J=2-1 emission from the nebula surrounding the post-AGB star AC Her, a source belonging to a class of objects that share properties with the Red Rectangle and show hints of Keplerian disks. We clearly detect the Keplerian dynamics of a second disk orbiting an evolved star. Its main properties (size, temperature, central mass) are derived from direct interpretation of the data and model fitting. With this we confirm that there are disks orbiting the stars of this relatively wide class of post-AGB objects