On the application of spinors to the problems of celestial mechanics
Vectorial concept for solving three body problem in celestial mechanics
Stumpff, K.↗
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Solution of the N-body problem with recurrent power series.
Adaptation of Steffensen's method of solution of both the restricted and general three-body problems in terms of power series in time to the solution of the equations of motion of N planets around the sun. The key feature of this adaptation lies in the introduction of N new variables and in the use of new relations which contain the time derivatives and can thus be treated as differential equations.
Broucke, R.↗
Dynamic Optimization of Multi-Spacecraft Relative Navigation Configurations in the Earth-Moon System
In this paper, the notion of relative navigation introduced by Hill, Lo and Born is analyzed for a large class of periodic orbits in the Earth-Moon three-body problem, due to its potential in supporting Moon exploration efforts. In particular, a navigation metric is introduced and used as a cost function to optimize over a class of periodic orbits. While the problem could be solve locally as an optimal control problem, a dynamical based approach that allows for a global/systematic view of the problem is proposed. First, the simpler problem of multiple spacecraft placement on a given periodic orbit is solved before the notion of continuation and bifurcation analysis is used to expand the range of solutions thus obtained.
Three Body Problem↗
Wide-Field Infrared Survey Telescope and Starshade Formation Flying Dynamics at Sun-Earth L2
The formation flying of an occulter with a telescope at the Sun-Earth L2 (SEL2) Libration Point can be a challenging problem. A good knowledge of the Restricted Three Body Problem dynamics is required to understand how these two spacecraft interact with each other in the SEL2 unstable environment, and how other perturbations such as Solar Radiation Pressure (SRP) affect their mutual trajectories. This paper focuses on the transfer trajectories to achieve specific relative positions between two spacecraft as they fly in formation at SEL2, andanalyzes the relevance of SRP in this formation, using the Wide-Field Infrared Survey Telescope (WFIRST) and the Starshade occulter as an example. Given that WFIRST and Starshade have very different area-to-mass ratios, SRP will affect their motion in different ways, and their relative position can be key to reduce the V cost. In this paper we intend on providing an explanation on how the relative position between both spacecrafts affects the transfer V from one observation to the other using dynamical system theory and Floquet modes.
Pressure↗
Abort Trajectory Design Strategies for the Artemis Missions
NASA's Artemis campaign plans to send astronauts to the lunar surface for the first time since 1972. The campaign relies on the Orion crew capsule to ferry the crew from Earth to an L2 9:2 lunar synodic resonant Near-Rectlinear Halo Orbit (NRHO) before descent to the lunar surface. This investigation examines a process to construct various abort families during transit from Earth to the NRHO using the Earth-Moon Circular Restricted Three-Body Problem (CR3BP). The initial trajectories are constructed using the CR3BP and categorized in families. A process is then summarized to transcribe and converge trajectories from the CR3BP into a higher-fidelity model. Ultimately, an effective methodology to develop, classify and converge these aborts in higher-fidelity is critical to the Artemis program in the event of an anomaly during the crew transit.
Trajectory Design↗
Lunar transfer trajectory design and the four-body problem
The existence of a ballistic trajectory from the Earth to orbit about the Moon was long considered to be impossible based on analysis of the three-body problem. In 1990 a ballistic trajectory from the Earth to lunar orbit was discovered while analyzing a plan to salvage the Muses A (Hiten) spacecraft. This trajectory utilized the Sun's gravity in conjunction with the Earth and Moon's gravity and was thus the first example of a practical four-body trajectory design. This paper presents a review of lunar transfer trajectories that go beyond three-body theory and the Jacobi integral. These include Hiten, Lunar A and the Genesis return trajectory from the vicinity of the Moon to Earth.It is shown that these trajectories may be analyzed by piecing together segments where three-body motion dominates.
Muses↗
Trajectories leaving a sphere in the restricted 3-body problem
The set of trajectories leaving/impacting the surface of the Galilean satellites of Jupiter is analyzed from theoretical and computational veiwpoints in the ciruclular restricted three body problem in order to characterize the sensitive impact regions for spacecraft trajectory applications as well as the main dynmical structures influencing this set of trajecotries.
Lo, Martin↗
Optimal Low Thrust Orbit Transfers for Space Telescope Refueling at SEL2
The James Webb Space Telescope (JWST), a ten billion-dollar infrared telescope with a 6.5m primary mirror to be launched in 2021, is designed to operate in a Halo orbit around the second Sun-Earth Lagrange point (SEL2) for five to ten years. At that point fuel for station keeping and attitude maneuvers will run out. Refueling missions to JWST, as well as to similar space telescope missions proposed for SEL2, could greatly enhance the “science-per-dollar” value and promote a more sustainable use of space assets. In this paper, we present a novel approach to designing fuel optimal trajectories that will allow the refueling spacecraft to arrive at the SEL2 Halo orbit with maximum final mass (i.e. fuel payload). The low thrust optimal control problem is formulated using an indirect optimization method, leading to a two-point boundary value problem with a bang-bang control structure. We make use of a hyperbolic tangent smoothing technique for performing continuation on the thrust magnitude to reduce the sharpness of the control switches in early iterations and, thus, promote convergence. The problem is posed and solved in the circular restricted three-body problem. In this dynamical system, invariant manifolds exist that can be utilized to reduce fuel consumption. The here presented methodology to this challenging and important problem in astrodynamics demonstrates a significant potential for low-cost refueling mission design.
Woollands, Robyn↗
Approximation of the restricted problem by the two-fixed-center problem second semi- annual report, part i
Perturbation theory of two-fixed-center problem leading to restricted three-body approximation
PERTURBATION THEORY↗
New contributions to the problem of capture
Libration-point capture is analyzed in the framework of the restricted three-body problem in order to obtain general results capable of predicting the major features of an arbitrary postcapture orbit, with specific reference to capture as the presumed mode of origin of Jupiter's outer satellites. Some examples of libration-point capture are computed, the time a body remains captured is estimated, and a long-term numerical integration of orbits is carried out which shows that the equation used to estimate capture lifetimes is quite conservative. An extended capture criterion is proposed and used to demonstrate that direct postcapture orbits lie outside retrograde ones. Mass change involving one or both primaries is evaluated as a means of producing permanent capture, it is found that mass-change effects can produce 'pull-down capture', and consideration is given to the orbital evolution of Jupiter's outer satellites. The effect of drag in the solar nebula or in a nebula around proto-Jupiter on the capture process is briefly investigated. It is concluded that a capture origin for the outer satellites is improbable within the last 4.6 billion years but that either mass change or nebula drag could have been effective in producing capture.
Heppenheimer, T. A.↗
Heteroclinic Connections Between Periodic Orbits and Resonance Transitions in Celestial Mechanics
This paper applies dynamical systems techniques to the problem of heteroclinic connections and resonance transitions in the planar circular restricted three-body problem.
heteroclinic Orbits genesis missions resonance tra↗
Direct numerical simulation of sheared turbulent flow
The summer assignment to study sheared turbulent flow was divided into three phases which were: (1) literature survey, (2) computational familiarization, and (3) pilot computational studies. The governing equations of fluid dynamics or Navier-Stokes equations describe the velocity, pressure, and density as functions of position and time. In principle, when combined with conservation equations for mass, energy, and thermodynamic state of the fluid a determinate system could be obtained. In practice the Navier-Stokes equations have not been solved due to the nonlinear nature and complexity of these equations. Consequently, the importance of experiments in gaining insight for understanding the physics of the problem has been an ongoing process. Reasonable computer simulations of the problem have occured as the computational speed and storage of computers has evolved. The importance of the microstructure of the turbulence dictates the need for high resolution grids in extracting solutions which contain the physical mechanisms which are essential to a successful simulation. The recognized breakthrough occurred as a result of the pioneering work of Orzag and Patterson in which the Navier-Stokes equations were solved numerically utilizing a time saving toggling technique between physical and wave space, known as a spectral method. An equally analytically unsolvable problem, containing the same quasi-chaotic nature as turbulence, is known as the three body problem which was studied computationally as a first step this summer. This study was followed by computations of a two dimensional (2D) free shear layer.
Harris, Vascar G.↗
Sun-earth libration point trajectories that avoid the solar exclusion zone
Three-dimensional orbits in the restricted three-body problem have been the subject of a number of recent studies. One type of three-dimensional, quasi-periodic orbit that emanates from the general vicinity of the collinear libration points is known as a 'Lissajous' trajectory. Future mission plans include trajectories near the interior libration point. (L1) in the sun-earth system and may consider Lissajous orbits as part of the trajectory design. Such orbits may be constrained, however, to remain beyond the region of the solar disk as viewed from earth. This effort involves the numerical determination of Lissajous trajectories of arbitrary, but predetermined, length that never violate such a constraint as a result of maneuvers directed, in general, perpendicular to the ecliptic plane.
Howell, K. C.↗
Low-Thrust Trajectory Optimization with Simplified SQP Algorithm
The problem of low-thrust trajectory optimization in highly perturbed dynamics is a stressing case for many optimization tools. Highly nonlinear dynamics and continuous thrust are each, separately, non-trivial problems in the field of optimal control, and when combined, the problem is even more difficult. This paper de-scribes a fast, robust method to design a trajectory in the CRTBP (circular restricted three body problem), beginning with no or very little knowledge of the system. The approach is inspired by the SQP (sequential quadratic programming) algorithm, in which a general nonlinear programming problem is solved via a sequence of quadratic problems. A few key simplifications make the algorithm presented fast and robust to initial guess: a quadratic cost function, neglecting the line search step when the solution is known to be far away, judicious use of end-point constraints, and mesh refinement on multiple shooting with fixed-step integration.In comparison to the traditional approach of plugging the problem into a “black-box” NLP solver, the methods shown converge even when given no knowledge of the solution at all. It was found that the only piece of information that the user needs to provide is a rough guess for the time of flight, as the transfer time guess will dictate which set of local solutions the algorithm could converge on. This robustness to initial guess is a compelling feature, as three-body orbit transfers are challenging to design with intuition alone. Of course, if a high-quality initial guess is available, the methods shown are still valid.We have shown that endpoints can be efficiently constrained to lie on 3-body repeating orbits, and that time of flight can be optimized as well. When optimizing the endpoints, we must make a trade between converging quickly on sub-optimal endpoints or converging more slowly on end-points that are arbitrarily close to optimal. It is easy for the mission design engineer to adjust this trade based on the problem at hand.The biggest limitation to the algorithm at this point is that multi-revolution transfers (greater than 2 revolutions) do not work nearly as well. This restriction comes in because the relationship between node 1 and node N becomes increasingly nonlinear as the angular distance grows. Trans-fers with more than about 1.5 complete revolutions generally require the line search to improve convergence. Future work includes: Comparison of this algorithm with other established tools; improvements to how multiple-revolution transfers are handled; parallelization of the Jacobian computation; in-creased efficiency for the line search; and optimization of many more trajectories between a variety of 3-body orbits.
Parrish, Nathan L.↗
Application of dynamical systems theory to a very low energy transfer
We use lobe dynamics in the restricted three-body problem to design orbits with prescribed itineraries with respect to the resonance regions within a Hill's region. the application we envision is the design of a low energy trajectory to orbit three of Jupiter's moons using the patched three-body approximation. We introduce the "switching region," the P2BA analogue to the "spere of influence."
Marsden, J. E.↗
Lectures on regularization
Regularization of relative many-body equations of celestial mechanics and relative three-body equations in case of binary collisions
MANY-BODY PROBLEM↗
Global search for planar and three-dimensional periodic orbits near Europa
A global grid search is performed to find simply- and doubly-symmetric periodic orbits in the Restricted Three Body Problem using the dimensioned parameters associated with the Jupiter-Europa system.
Europa↗
Nonperiodic variations in astrophysical systems: Investigating frequency evolution
We present a method related to the wavelet transform, the Gabor transform, for investigating astronomical time series containing nonconsistent frequencies. Instances in which such data sets may arise include variable star light curves, numerical studies of the gravitational three-body problem, X-ray binaries, and signals from more exotic objects such as planets around pulsars and mass infall from accretion structures onto compact objects. As an illustration of its power, we apply the technique to a numerical data set of a gravitational three-body interaction and to photometry of the rapidly oscillating peculiar A star HD 60435. In the three-body example, the method provides an insightful shorthand that allows for the determination of episodes where the system behaves as two nearly Keplerian orbits. For HD 60435, the power in the main frequency exhibits unusual evolution over the duration of the observation.
Boyd, Patricia T.↗