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In the preliminary design of space missions it can be useful to identify regions of dynamics that drive the system's behaviour or separate qualitatively different dynamics. The Lagrangian Coherent Structure (LCS) has been widely used in the analysis of dynamical systems, and generalises the concept of the stable and unstable manifolds to systems with arbitrary time-dependence. However, the use of three-dimensional LCS in astrodynamics has thus far been limited. This paper presents the application of a new numerical method introduced by the authors, DA-LCS, to astrodynamics systems using the Elliptic-Restricted Three-body Problem (ER3BP) as a test case. We are able to construct the full, three-dimensional LCS associated with the Sun-Mars ER3BP directly from the variational theory of LCS even for numerically challenging initial conditions. The LCS is analysed in detail, showing how it in this case separates regions of qualitatively different behaviour without any a priori knowledge. The paper then studies the effect of integration time and the parameterisation of the initial condition on the LCS found. We highlight how round-off errors arise from limits of floating-point arithmetic in the most challenging test cases and provide mitigating strategies for avoiding these errors practically., Comment: 26 pages; submitted for publication at the journal of Celestial Mechanics and Dynamical Astronomy in September 2022

In dynamical systems, it is advantageous to identify regions of flow which can exhibit maximal influence on nearby behaviour. Hyperbolic Lagrangian Coherent Structures have been introduced to obtain two-dimensional surfaces which maximise repulsion or attraction in three-dimensional dynamical systems with arbitrary time-dependence. However, the numerical method to compute them requires obtaining derivatives associated with the system, often performed through the approximation of divided differences, which can lead to significant numerical error and numerical noise. In this paper, we introduce a novel method for the numerical calculation of hyperbolic Lagrangian Coherent Structures using Differential Algebra called DA-LCS. As a form of automatic forward differentiation, it allows direct computation of the Taylor expansion of the flow, its derivatives, and the eigenvectors of the associated strain tensor, with all derivatives obtained algebraically and to machine precision. It does so without a priori information about the system, such as variational equations or explicit derivatives. We demonstrate that this can provide significant improvements in the accuracy of the Lagrangian Coherent Structures identified compared to finite-differencing methods in a series of test cases drawn from the literature. We also show how DA-LCS uncovers additional dynamical behaviour in a real-world example drawn from astrodynamics., Comment: Updates preprint to incorporate reviewer feedback. Main argument and results unchanged. Submitted to Journal of Computational Sciences. LaTeX, 45 pages

Recent observational missions have uncovered a significant number of compact multi-exoplanet systems. The tight orbital spacing of these systems has led to much effort being applied to the understanding of their stability; however, a key limitation of the majority of these studies is the termination of simulations as soon as the orbits of two planets cross. In this work we explore the stability of compact, three-planet systems and continue our simulations all the way to the first collision of planets to yield a better understanding of the lifetime of these systems. We perform over $25,000$ integrations of a Sun-like star orbited by three Earth-like secondaries for up to a billion orbits to explore a wide parameter space of initial conditions in both the co-planar and inclined cases, with a focus on the initial orbital spacing. We calculate the probability of collision over time and determine the probability of collision between specific pairs of planets. We find systems that persist for over $10^8$ orbits after an orbital crossing and show how the post-instability survival time of systems depends upon the initial orbital separation, mutual inclination, planetary radius, and the closest encounter experienced. Additionally, we examine the effects of very small changes in the initial positions of the planets upon the time to collision and show the effect that the choice of integrator can have upon simulation results. We generalise our results throughout to show both the behaviour of systems with an inner planet initially located at $1$ AU and $0.25$ AU.

We present TES, a new n-body integration code for the accurate and rapid propagation of planetary systems in the presence of close encounters. TES builds upon the classic Encke method and integrates only the perturbations to Keplerian trajectories to reduce both the error and runtime of simulations. Variable step size is used throughout to enable close encounters to be precisely handled. A suite of numerical improvements are presented that together makes TES optimal in terms of energy error. Lower runtimes are found in all test problems considered when compared to direct integration using ias15. TES is freely available.

In recent years, the retrieval of entire asteroids has received significant attention, with many approaches leveraging the invariant manifolds of the Circular-Restricted Three-body Problem to capture an asteroid into a periodic orbit about the $L_1$ or $L_2$ points of the Sun-Earth system. Previous works defined an `Easily Retrievable Object' (ERO) as any Near-Earth Object (NEO) which is retrievable using these invariant manifolds with an impulsive $\Delta v$ of less than $500$ m/s. We extend the previous literature by analysing the Pareto fronts for the EROs discovered for the first time, using high-performance computing to lift optimisation constraints used in previous literature, and modifying the method used to filter unsuitable NEOs from the NEO catalogue. In doing so, we can demonstrate that EROs have approximately the same transfer cost for almost any possible transfer time, including single-impulse transfers, which could offer significant flexibility to mission designers. We also identify $44$ EROs, of which $27$ are new, and improve on previously-known transfer solutions by up to $443$ m/s, including $17$ new capture trajectories with $\Delta v$ costs of less than $100$ m/s., Comment: Updated to the accepted manuscript; to be published in Acta Astronautica (2021)

A hybrid orbit propagator based on the analytical integration of the Kepler problem is designed to determine the future position and velocity of any orbiter, usually an artificial satellite or space debris fragment, in two steps: an initial approximation generated by means of an integration method, followed by a forecast of its error, determined by a prediction technique that models and reproduces the missing dynamics. In this study we analyze the effect of slightly changing the initial conditions for which a hybrid propagator was developed. We explore the possibility of generating a new hybrid propagator from others previously developed for nearby initial conditions. We find that the interpolation of the parameters of the prediction technique, which in this case is an additive Holt-Winters method, yields similarly accurate results to a non-interpolated hybrid propagator when modeling the J2 effect in the main problem propagation., Comment: 12 pages, 3 figures

The end-of-life disposal of Galileo satellites is needed to avoid collisions with operational spacecraft and to prevent the generation of space debris. Either disposal in stable graveyard orbits or disposal into the atmosphere exploiting eccentricity growth caused by lunisolar resonances are possible. However, there is a concern about the predictability of MEO orbits because of possible chaotic behaviour caused by the overlap of resonances. In this work, we investigate if Galileo disposal orbits are predictable and if safe disposal is possible under initial uncertainties. For this, we employ finite-time Lyapunov exponents (FTLE) and sensitivity analysis and compare these two methods regarding their practicality for analysing the predictability of disposal orbits. The results show that FTLE are not suitable for determining if an orbit behaves chaotically or not on the time scale of interest. Sensitivity analysis, on the other hand, can be used to quantify the effect of uncertainties on the orbital evolution and to determine if safe disposal is possible. In addition, we show that reliable re-entry disposal is feasible with a limited \Delta V budget. However, safe disposal into a graveyard orbit is not always feasible considering uncertainties in the disposal manoeuvre and dynamical model when the available \Delta V for disposal is low., Comment: 29 pages, 20 figures

A differential algebra based importance sampling method for uncertainty propagation and impact probability computation on the first resonant returns of Near Earth Objects is presented in this paper. Starting from the results of an orbit determination process, we use a differential algebra based automatic domain pruning to estimate resonances and automatically propagate in time the regions of the initial uncertainty set that include the resonant return of interest. The result is a list of polynomial state vectors, each mapping specific regions of the uncertainty set from the observation epoch to the resonant return. Then, we employ a Monte Carlo importance sampling technique on the generated subsets for impact probability computation. We assess the performance of the proposed approach on the case of asteroid (99942) Apophis. A sensitivity analysis on the main parameters of the technique is carried out, providing guidelines for their selection. We finally compare the results of the proposed method to standard and advanced orbital sampling techniques.

We study the local shadow of the Schwarzschild black hole with a quadrupole distortion and the influence of the external gravitational field on the photon dynamics. The external matter sources modify the light ring structure and lead to the appearance of multiple shadow images. In the case of negative quadrupole moments we identify the most prominent mechanism causing multiple shadow formation. Furthermore, we obtain a condition under which this mechanism can be realized. This condition depends on the quadrupole moment, but also on the position of the observer and the celestial sphere., Comment: 18 pages, 14 figures

Utilizing concepts from dynamical systems theory, we demonstrate how the existence of light rings, or fixed points, in a spacetime will give rise to families of periodic orbits and invariant manifolds in phase space. It is shown that these structures define the shape of the black hole shadow as well as a number of salient features of the spacetime lensing. We illustrate this through the analysis of lensing by a hairy black hole., Comment: 6 pages