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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."
Many models of planet formation involve scenarios in which one or a few large protoplanets interact with a swarm of much smaller planetesimals. In such scenarios, three-body perturbations by the protoplanet as well as mutual collisions and gravitational interactions between the swarm bodies are important in determining the velocity distribution of the swarm. We are developing a model to examine the effects of these processes on the evolution of a planetesimal swarm. The model consists of a combination of numerical integrations of the gravitational influence of one (or a few) massive protoplanets on swarm bodies together with a statistical treatment of the interactions between the planetesimals. Integrating the planetesimal orbits allows us to take into account effects that are difficult to model analytically or statistically, such as three-body collision cross-sections and resonant perturbations by the protoplanet, while using a statistical treatment for the particle-particle interactions allows us to use a large enough sample to obtain meaningful results.
A simple resonance theory of three-body ion-atom association reactions is presented. The reaction is considered as proceeding through the formation of a long lived orbiting resonance complex between the atom and the ion. The population of these quasi-bound states is estimated assuming thermal equilibrium. A stable molecular ion may then be formed upon deactivation of the complex by collision with a third body. Various simplifying approximations to the potential curves and surfaces are employed. Furthermore, the deactivation cross sections for the relevant complexes are estimated from the corresponding atomic 'sizes.' A simple analytical formula for the three-body rate constant is thus derived. Reasonable agreement with experiment is obtained for He(+) in He and fair agreement for other light systems.
The sudden eccentricity increases discovered by Wisdom (1982) are reproduced in numerical integrations of the planar ecliptic restricted three-body problem, verifying that this phenomenon is real. Mapping derivations are qualitatively reviewed and the maximum Liapunov characteristic exponent and its importance for determining the character of a trajectory are explained. The results of a number of calculations of this exponent using the differential equations for the unaveraged three-body problem are shown and compared to equivalent calculations using a mapping. In all cases the two approaches agree whether the orbits are chaotic or quasiperiodic. The mappings are used to trace out the chaotic zone near the 3/1 commensurability, both in the planar-ecliptic problem and in the three-dimensional elliptic problem. The outer boundary of the chaotic zone coincides with the boundary of the 3/1 Kirkwood gap in the actual distribution of asteroids within the errors of the asteroid orbital elements.
This paper studies the possibility of lunar capture depending on variations of the solar mass under certain well specified conditions and assumptions regarding the behaviour of the three-body dynamical system formed by the Sun, Earth and Moon. It is found that a large amount of decrease in the solar mass (approximately 37%) would be required to allow capture if the model of the planar restricted problem of three bodies is assumed, if the masses of the Earth and Moon did not change and if the angular momentum of the Sun-Earth system did not change. Such large mass-changes of the Sun can not be associated with radiation mass losses only with catastrophic events, such as stellar close approaches.
We consider the effect of gravitational perturbations from Jupiter on the dynamics of asteroids, when Jupiter is itself perturbed by Saturn. The presence of Saturn introduces a number of additional frequencies into Jupiters orbit. These frequencies in turn produce chaos in narrow regions on either side of the chaotic zones associated with the mean motion resonances between the asteroids and Jupiter. The resonant arguments of these three-body resonances contain the longitudes of Jupiter and the asteroid together with either the secular frequency 9-6, or the longitude of Saturn. Resonances involving the longitude of Saturn are analogs of the Laplace resonance in the Jovian satellite system. We show that many three-body resonances involving the longitude of Saturn are chaotic. We give simple expressions for the width of the chaotic region and the associated Lyapunov time. In some cases the chaos can produce a diffusive growth in the 4 eccentricity of the asteroid that leads to ejection of the asteroid on times shorter than the age of the solar system. We give simple estimates for the diffusion time. Finally, we present the results of numerical integrations testing the theory.
As demonstrated by ongoing concept designs and the recent ARTEMIS mission, there is, currently, significant interest in exploiting three-body dynamics in the design of trajectories for both robotic and human missions within the Earth-Moon system. The concept of an interactive and 'dynamic' catalog of potential solutions in the Earth-Moon system is explored within this paper and analyzed as a framework to guide trajectory design. Characterizing and compiling periodic and quasi-periodic solutions that exist in the circular restricted three-body problem may offer faster and more efficient strategies for orbit design, while also delivering innovative mission design parameters for further examination.
Bond-orientational order in liquid Si via Monte Carlo simulation in conjuncation with empirical two- and three-body potentials of the form proposed by Stillinger and Weber are studied. Bond-orientational order (BOO) is described in terms of combinations of spherical harmonic functions. Liquid Si is found to have pronounced short-range BOO corresponding to l = 3, as expected for a structure with local tetrahedral order. No long-range BOO is found either in the equilibrium or the supercooled liquid. When the three-body potential is artificially removed, the tetrahedral bond-orientation order disappears and the liquid assumes a close-packed structure.
An experimental method was developed to determine the droplet impingement characteristics on 2-D and 3-D bodies. The experimental results provide the essential droplet impingement data required to validate water droplet trajectory codes, which are used in the analysis of aircraft icing. A body, whose water droplet impingement characteristics are required, is covered at strategic locations by thin strips of moisture absorbing (blotter) paper, and is exposed to an air stream containing a water dye solution spray cloud. Water droplet impingement data are extracted from the dyed blotter strips by measuring the optical reflectance of the dye deposit on the strips, using an automated reflectometer. Models tested include a 4-inch diameter cylinder, a NACA 652015 airfoil section, a MS(1)-0317 supercritical airfoil section, three simulated ice shapes, an axisymmetric inlet and a Boeing 737-300 inlet model. Detailed descriptions of the dye tracer technique, instrumentation, data reduction method and the results obtained are presented. Analytical predictions of collection efficiency characteristics for most test configurations are included for comparison.
Kellogg (1961), suggested that transport of atomic oxygen from the summer into the winter hemisphere and subsequent release of energy by three body recombination, O+O+N2 yields O2+N2+E, may contribute significantly to the so-called mesopause temperature anomaly. Earlier model calculations have shown that Kellogg's mechanism produces about a 10-percent increase in the temperature from summer to winter at 90 km. This process, however, is partly compensated by differential heating from absorption of UV radiation associated with dissociation of O2. In the auroral region of the thermosphere, there is a steady energy dissipation by Joule heating causing a redistribution and depletion of atomic oxygen due to wind-induced diffusion. With the removal of O, latent chemical energy normally released by three body recombination is also removed, and the result is that the temperature decreases by almost 2 percent near 90 km. Through dynamic feedback, this process reduces the depletion of atomic oxygen by about 25 percent and the temperature perturbation in the exosphere from 10 to 7 percent at polar latitudes. Under the influence of the internal dynamo interaction, the prevailing zonal circulation in the upper thermosphere changes direction when the redistribution of recombination energy is considered.
Perturbation theory of two-fixed-center problem leading to approximation of three-body problem
Reduced three-body and four-body problems in connection with cislunar trajectories
Recent observations of this famous ring galaxy, including optical and near-infrared CCD surface photometry, and VLA radio continuum and 21 cm line mapping (Higdon 1992b, in prep.), have inspired a renewed modeling effort. Toomre's (1978, in The Large-scale Structure of the Universe, eds. Longair and Einasto) series of restricted three-body simulations demonstrated how the multiple rings could be produced in a nearly head-on galaxy collision. New models with a halo-dominated potential based on the 21 cm rotation curve are able to reproduce such details as the spacing between rings, ring widths, offset of the nucleus, and several kinematical features, thus providing strong support for the collisional theory. The new observations have shown there are little or no old stars in Cartwheel; it may consist almost entirely of gas and stars produced as a result of compression in the ring wave. To model this process Smooth Particle Hydrodynamics (SPH) simulations of the Cartwheel disk have been performed. Fixed gravitational potentials were used to represent the Cartwheel and a roughly 30 percent mass collision partner. The interaction dynamics was treated as in the usual restricted three-body approximation, and the effects of local self-gravity between disk particles were calculated. We are particularly interested in testing the theory that enhanced star formation in waves is the result of gravitational instability in the compressed region (see e.g. Kennicutt 1989, ApJ 344, 685). The gas surface density in a number of simulations was initialized to a value slightly below the threshold for local gravitational instability throughout most of the disk. The first ring wave produces relatively modest compressions (a factor of order a few), triggering instability in a narrow range of wavelengths. Self-gravity in the disk is calculated over a comparable range of scales. Simulations were run with isothermal, adiabatic, and adiabatic with radiative cooling characterized by a relatively short timescale. The isothermal approximation is good except in the vicinity of the strong second (inner) ring, and several snapshots from one case are shown in the figure below. Flocculent spiral segments are present before the collision, and these are compressed into dense knots in the ring wave. These knots are likely to be sites of vigorous star formation. In the strong rarefaction behind the outer ring most of the knots are radially stretched and sheared, giving rise to spoke-like features. A few dense knots are evidently very tightly bound, because they retain their coherence and are stretched relatively little through the rarefaction. This is in accord with evidence for continuing star formation in some spokes (Marcum, Appleton and Higdon 1992). The number and spacing of spokes is a direct function of the scale of the gravitational instability in the disk. Thus, the gravitational instability theory, together with the hypothesis that massive stars are only formed in dense knots of gas, can account for most of the distinct morphology of the Cartwheel.
The wave-driven fluctuations in the O2(0-1) atmospheric nightglow is modeled and the parameter (eta) is calculated using a model that accounts for either three-body recombination of atomic oxygen atoms alone to form the O2(b exp 1 Sigma(g)(+)) state directly, or by the further inclusion of the process that allows the formation of the O2(c exp 1 Sigma(u)(-)) intermediate state. The calculations are performed for a latitude of 18 deg N and for the months of March and June. The general results, which display how (eta) varies with wave period, horizontal wavelength, season, and chemical scheme, show that for given values of wave period and horizontal wavelength it is not possible to discriminate between seasonal effects and between the effects of different chemical schemes at evanescent and short gravity wave periods. It is shown that, when quenching by atomic oxygen is ignored, the resulting values of (eta) calculated with the complete chemistry are similar to those obtained from the three-body recombination scheme alone.
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.
The landing contact conditions for the HL-10, M2-F2/F3, and the X-24A lifting body vehicles are analyzed statistically for 81 landings. The landing contact parameters analyzed are true airspeed, peak normal acceleration at the center of gravity, roll angle, and roll velocity. Ground measurement parameters analyzed are lateral and longitudinal distance from intended touchdown, lateral distance from touchdown to full stop, and rollout distance. The results are presented in the form of histograms for frequency distributions and cumulative frequency distribution probability curves with a Pearson Type 3 curve fit for extrapolation purposes.
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Calculations can be performed for any atmospheric conditions and for all water drop sizes, from the smallest cloud droplet to large raindrops. Any subsonic, external, non-lifting flow can be accommodated; flow into, but not through, inlets also can be simulated. Experimental water drop drag relations are used in the water drop equations of motion and effects of gravity settling are included. Seven codes are described: (1) a code used to debug and plot body surface description data; (2) a code that processes the body surface data to yield the potential flow field; (3) a code that computes flow velocities at arrays of points in space; (4) a code that computes water drop trajectories from an array of points in space; (5) a code that computes water drop trajectories and fluxes to arbitrary target points; (6) a code that computes water drop trajectories tangent to the body; and (7) a code that produces stereo pair plots which include both the body and trajectories. Code descriptions include operating instructions, card inputs and printouts for example problems, and listing of the FORTRAN codes. Accuracy of the calculations is discussed, and trajectory calculation results are compared with prior calculations and with experimental data.