A hypothetical four-body problem and its applications - An estimate of the effect of the moon and the sun on the syncom orbit
Hypothetical four-body problem applied to estimate effects of moon and sun on Syncom orbit
SYNCOM SATELLITE↗
SEARCH · Search NASA
Results for “FOUR-BODY METHOD”
Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.
Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.
Some periodic solutions of elliptical and restricted four-body problems.
Periodic solutions of elliptical and restricted four-body problems about libration points of restricted three-body problem
FOUR-BODY METHOD↗
Periodic motion around the triangular libration point in the restricted problem of four bodies.
Periodic motion around triangular libration point in restricted four-body problem of earth-moon-sun system
EARTH-MOON SYSTEM↗
Atomic r sub ij integrals for many-electron wavefunctions.
Integration scheme for variational r sub ij wave functions containing unlinked four-electron correlated terms for atoms up to neon
FOUR-BODY METHOD↗
Regularization of the restricted problem of four bodies.
Regularization of restricted three-body problem extended to case where three primaries of any mass revolve in circular orbits around common center of mass and fourth body of infinitesimal mass moves in their field
CIRCULAR ORBIT↗
A harmonic L4 orbit for the very restricted four-body problem
Harmonic L-4 orbit for very restricted four-body problem determined by method of general perturbations using Chebyshev series
EARTH-MOON SYSTEM↗
Four-body trajectory optimization
The two methods which are suitable for use in a 4-body trajectory optimization program are both multiconic methods. They include an approach due to Wilson (1970) and to Byrnes and Hooper (1970) and a procedure developed by Stumpff and Weiss (1968). The various steps in a trajectory optimization program are discussed, giving attention to variable step integration, the correction of errors by quadrature formulas, questions of two-impulse transfer, three-impulse transfer, and two examples which illustrate the implementation of the computational approaches.
Pu, C. L.↗
Four-body trajectory optimization
A collection of typical three-body trajectories from the L1 libration point on the sun-earth line to the earth is presented. These trajectories in the sun-earth system are grouped into four distinct families which differ in transfer time and delta V requirements. Curves showing the variations of delta V with respect to transfer time, and typical two and three-impulse primer vector histories, are included. The development of a four-body trajectory optimization program to compute fuel optimal trajectories between the earth and a point in the sun-earth-moon system are also discussed. Methods for generating fuel optimal two-impulse trajectories which originate at the earth or a point in space, and fuel optimal three-impulse trajectories between two points in space, are presented. A brief qualitative comparison of these methods is given. An example of a four-body two-impulse transfer from the Li libration point to the earth is included.
Pu, C. L.↗
Four-body trajectory optimization
A comprehensive optimization program has been developed for computing fuel-optimal trajectories between the earth and a point in the sun-earth-moon system. It presents methods for generating fuel optimal two-impulse trajectories which may originate at the earth or a point in space and fuel optimal three-impulse trajectories between two points in space. The extrapolation of the state vector and the computation of the state transition matrix are accomplished by the Stumpff-Weiss method. The cost and constraint gradients are computed analytically in terms of the terminal state and the state transition matrix. The 4-body Lambert problem is solved by using the Newton-Raphson method. An accelerated gradient projection method is used to optimize a 2-impulse trajectory with terminal constraint. The Davidon's Variance Method is used both in the accelerated gradient projection method and the outer loop of a 3-impulse trajectory optimization problem.
Pu, C. L.↗
Energy-Screened Many-Body Expansion for Protein–Ligand Interactions: Examining Convergence for Metalloenzymes Through Seven–Body Interactions
Fragment-based quantum chemistry is a powerful strategy for calculating protein−ligand interaction energies using quantum chemistry methods. Rigorous convergence often requires hundreds of atoms in the protein binding-site model, especially if that model is constructed using distance-based criteria to select amino acid residues, while three- and four-body calculations exhibit instability related to combinatorial proliferation in the number of subsystem calculations. Here, we report an energy-based screening protocol for the many-body expansion applied to protein−ligand interactions, implemented in the open-source FRAGME∩T code. Using a combination of aggressive screening based on semiempirical quantum chemistry, with an improved graph-theoretical algorithm to eliminate unimportant subsystems, we are able to perform n-body calculations up to n = 7 using density functional theory in triple-ζ basis sets. Distance cutoffs further reduce the cost without compromising accuracy. Rapid and stable convergence of the many-body expansion is obtained by n = 4, for a pair of metalloenzymes in which a divalent ion coordinates directly to the ligand. As compared to previous results that relied solely on distance cutoffs, oscillations in the n-body corrections are reduced or eliminated, although residual errors remain in one case. This work demonstrates that benchmark-quality protein−ligand interaction energies can be systematically converged using a method with excellent parallel efficiency and scalability.
37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
A uniformly valid solution for motion about the interior libration point of the perturbed elliptic-restricted problem
Bounded motion about Lagrange collinear libration points is considered for a perturbed elliptic-restricted problem. A practical application is the motion of a satellite near a libration point collinear with the sun and the earth-moon barycenter. Such a system is treated here as an earth-sun-satellite elliptic restricted problem with lunar perturbations. The method of dual time scales is utilized to develop a uniformly valid three-dimensional analytical solution to the satellite's equations of motion. The analytical development applies somewhat generally to that class of four-body problems where the second primary mass is much greater than the first, and the third primary mass is much greater than the second.
Richardson, D. L.↗
An investigation into critical aspects of a new form of low energy lunar transfer, the Belbruno-Miller trajectories
This paper documents an integrated Belbruno-Miller (B-M) trajectory and its corresponding injection period. The B-M trajectories use Weak Stability Boundaries (WSB) resulting from four-body perturbative dynamics between the earth, moon, sun, and spacecraft to significantly reduce maneuver requirements for lunar transfer. It is determined that the presented nominal trajectory has a viable 4-day injection period considering maneuver constraints. The nominal B-M trajectory produces saving varying from 150 m/s to 222 m/s over traditional Hohmann means thus pointing to the practical utility of the WSB method. Energy savings are extracted at the price of extended flight time of approximately 6-months. Usefulness of this procedure was recently dramatized by the Japanese spacecraft Hiten when it arrived a the moon on October 2, 1991 following its entry into B-M trajectory on April 25, 1991.
Krish, V.↗
Five-body systems with Bethe-Salpeter equations
We extend the Bethe-Salpeter formalism to systems made of five valence particles. Restricting ourselves to two-body interactions, we derive the subtraction terms necessary to prevent overcounting. We solve the five-body Bethe-Salpeter equation numerically for a system of five scalar particles interacting by a scalar exchange boson. To make the calculations tractable, we implement properties of the permutation group S 5 and construct an approximation based on intermediate two- and three-body poles. We extract the five-body ground and excited states along with the spectra obtained from the two-, three-, and four-body equations. In the limit of a massless exchange particle, the two-, three, four- and five-body states coexist within a certain range of the coupling strength, whereas for heavier exchange particles the five-body system becomes Borromean. Our study serves as a building block for the calculation of pentaquark properties using functional methods.
Eichmann, Gernot [Univ. of Graz (Austria)] (ORCID:↗
Space shuttle with common fuel tank for liquid rocket booster and main engines (supertanker space shuttle)
An operation and schedule enhancement is shown that replaces the four-body cluster (Space Shuttle Orbiter (SSO), external tank, and two solid rocket boosters) with a simpler two-body cluster (SSO and liquid rocket booster/external tank). At staging velocity, the booster unit (liquid-fueled booster engines and vehicle support structure) is jettisoned while the remaining SSO and supertank continues on to orbit. The simpler two-bodied cluster reduces the processing and stack time until SSO mate from 57 days (for the solid rocket booster) to 20 days (for the liquid rocket booster). The areas in which liquid booster systems are superior to solid rocket boosters are discussed. Alternative and future generation vehicles are reviewed to reveal greater performance and operations enhancements with more modifications to the current methods of propulsion design philosophy, e.g., combined cycle engines, and concentric propellant tanks.
Thorpe, Douglas G.↗