Periodic orbits in the restricted three body problem with earth-moon masses
Classification scheme for symmetric periodic orbits in restricted three body problem with two dimensions and earth-moon mass ratio
Broucke, R. A.↗
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Application of local Lyapunov exponents to maneuver design and navigation in the three-body problem
Dynamical systems theory has recently been employed to design trajectories within the three-body problem for several missions. This research has applied one stability technique, the calculation of local Lyapunov exponents, to such trajectories. Local Lyapunov exponents give an indication of the effects that perturbations or maneuvers will have on trajectories over a specified time. A numerical comparison of local Lyapunov exponents was first made with the distance random perturbations traveled from a nominal trajectory, and the local Lyapunov exponents were found to correspond well with the perturbations that caused the greatest deviation from the nominal. This would allow them to be used as an indicator of the points where it would be important to reduce navigation uncertainties.
dynamical systems theory↗
New periodic solutions of the plane three-body problem.
Poincare method generalized to nonparametric case to obtain existence of periodic solutions of restricted three body problem
Arenstorf, R. F.↗
On relative periodic solutions of the planar general three-body problem
We describe two relatively simple reductions to order 6 for the planar general three-body problem. We also show that this reduction leads to the distinction between two types of periodic solutions: absolute or relative periodic solutions. An algorithm for obtaining relative periodic solutions using heliocentric coordinates is then described. It is concluded from the periodicity conditions that relative periodic solutions must form families with a single parameter. Finally, two such families have been obtained numerically and are described in some detail.
Broucke, R.↗
The restricted three-body problem including radiation pressure
The force of radiation pressure is included in the restricted three-body problem by considering the major radial components of the pressure force for the case of a particle in the vicinity of two luminous massive bodies, as well as by introducing the Poynting-Robertson effect for the case of one luminous body. The positions of the Lagrangian points L4 and L5 are found as functions of the ratio of radiation to gravitational forces. The Poynting-Robertson effect renders the L4 and L5 points unstable on a time scale that is long compared to the period of rotation of the two massive bodies. Implications for space colonization and a mechanism for producing azimuthal asymmetries in the interplanetary dust complex are discussed.
Schuerman, D. W.↗
Minimum impulse three-body trajectories.
A rapid and accurate method of calculating optimal impulsive transfers in the restricted problem of three bodies has been developed. The technique combines a multi-conic method of trajectory integration with primer vector theory and an accelerated gradient method of trajectory optimization. A unique feature is that the state transition matrix and the primer vector are found analytical without additional integrations or differentiations. The method has been applied to the determination of optimal two and three impulse transfers between the L2 libration point and circular orbits about both the earth and the moon.
D'Amario, L.↗
Further studies on criteria for the onset of dynamical instability in general three-body systems
Numerical experiments designed for the elucidation of the conditions under which self-gravitating, three-body systems become dynamically unstable are examined of the cases of four orbital configuration types: circular, prograde, and coplanar; circular, retrograde, and coplanar; circular, direct, and inclined; and eccentric, direct, and coplanar. Results indicate that orbital inclination does not significantly affect stability in 'outer planet' configurations, while the stability of 'inner planet' configurations, where the tertiary is in close orbit about one member of the binary, is markedly less affected, once the relative orbital inclination is greater than 50 deg. It is found that the onset of dynamical instability is only weakly dependent on the eccentricity of either the binary or tertiary orbit, as long as the mass of the tertiary is comparable to the reduced mass of the binary.
Pendleton, Y. J.↗
Calculation of three-body collisional recombination coefficients for cesium and argon atomic ions with an assessment of the Gryzinski cross sections.
Three-body collisional recombination coefficients calculated for cesium and argon atomic ions, assessing Gryzinski cross sections
ARGON↗
Numerical studies of transitions between the restricted problem of three bodies and the problem of two fixed centers and the Kepler problem
Numerical analyses of transitions between restricted problem of three bodies and problem of two fixed centers and Kepler problem
Hoelker, R. F.↗
The resonance overlap criterion and the onset of stochastic behavior in the restricted three-body problem
The resonance overlap criterion for the onset of stochastic behavior is applied to the planar circular-restricted three-body problem with small mass ratio (mu). Its predictions for mu = 0.001, 0.0001, and 0.00001 are compared to the transitions observed in the numerically determined Kolmogorov-Sinai entropy and found to be in remarkably good agreement. In addition, an approximate scaling law for the onset of stochastic behavior is derived.
Wisdom, J.↗
Motion about the stable libration points in the linearized, restricted three-body problem
The motion of a point particle in the neighborhood of a triangular libration point (L sub 4 or L sub 5) in the linearized, restricted problem of three bodies in the plane is described. The derivation of the equations of motion is standard. From these equations, three invariants of the motion are obtained; the Jacobi integral is expressed linearly in terms of two of these. The trajectories for varied initial conditions are drawn, and a complete geometric description of the particle motion is given in elementary terms. Each trajectory has an exterior boundary curve; its equation is found. An approximation to this boundary curve was known; the two curves are compared graphically. For certain initial conditions, there is an interior region from which the trajectory is excluded; the equation of the boundary of this region is given.
Mittleman, D.↗
Stability regions for quasiperiodic motion in the restricted problem of three bodies
Surfaces of section, plotted in configuration space, have been computed for the motion of the massless particle in the restricted problem of three bodies. Nine mass ratios and a wide variety of Jacobi constants were investigated; over four thousand orbits were computed, about half of which were finally used. The plots of surface of section have been reduced to plots of stability regions, following a method due to Henon (1965a, 1965b, 1966b, 1969). Sample surfaces of section are also given. The complete set of 276 surfaces of section has been published as a report (Jefferys, 1971) and is available upon request from the author.
Jefferys, W. H.↗
Analytic Expressions for Derivatives from Series Solutions to the Three Body Problem
This paper presents a notation system to facilitate to solution of differential equations via Taylor series expansions and applies it to solve the circular restricted three body problem. Unlike previous Taylor series methods in the astrodynamics literature, computer algebra solvers are not used. Instead the notation system allows one to solve a system of differential equations analytically “by hand” without resorting to computer algebra software. This method produces recurrence relations explicitly in terms of a sequence of derivatives of the state with respect to time for the coefficients of Taylor Series solutions that can be evaluated numerically or manipulated further to investigate properties of the solution. For example, additional derivatives with respect to other parameters may also be found, including those that describe the dependence of the solution on initial conditions.
Strange, Nathan↗
Model Predictive Control in the Three-body Problem Using Invariant Funnels As Terminal Sets
This paper describes a method for augmenting Model Predictive Control techniques using invariant funnels computed in the Circular Restricted Three-Body Problem. We use ellipsoids that roughly approximate the boundary of the invariant funnel as convex terminal sets for a short look-ahead optimization problem at each time step. We apply this method to a hypothetical low-thrust mission to land on Jupiter’s mooon Europa and show that including the invariant funnels as terminal sets reduces the amount of control effort required by almost an order of magnitude.
Close, Sigrid↗
Effect of abrasive grit size on wear of manganese-zinc ferrite under three-body abrasion
Wear experiments were conducted using replication electron microscopy and reflection electron diffraction to study abrasion and deformed layers produced in single-crystal Mn-Zn ferrites under three-body abrasion. The abrasion mechanism of Mn-Zn ferrite changes drastically with the size of abrasive grits. With 15-micron (1000-mesh) SiC grits, abrasion of Mn-Zn ferrite is due principally to brittle fracture; while with 4- and 2-micron (4000- and 6000-mesh) SiC grits, abrasion is due to plastic deformation and fracture. Both microcracking and plastic flow produce polycrystalline states on the wear surfaces of single-crystal Mn-Zn ferrites. Coefficient of wear, total thickness of the deformed layers, and surface roughness of the wear surfaces increase markedly with an increase in abrasive grit size. The total thicknesses of the deformed layers are 3 microns for the ferrite abraded by 15-micron SiC, 0.9 microns for the ferrite abraded by 4-micron SiC, and 0.8 microns for the ferrite abraded by 1-micron SiC.
Miyoshi, Kazuhisa↗
Predictions of three-body-decays of mesons from pole-dominated dispersion relations
The investigation shows that the method of finite dispersion relations (FDR) as developed by Aviv and Nussinov (1970) is consistent with a wide range of three-body meson decays. The specific results vary with each reaction. For each possibility a definite form for the t-channel Regge-pole terms is selected. The results of the investigation indicate the basic reliability of an approach based on the FDR method and the finite-energy sum rules.
Thews, R. L.↗
Periodic orbits in the restricted three-body problem in the neighborhood of resonances of type /k plus 1//k /Ueber periodische Bahnen im eingeschraenkten Dreikoerperproblem und in der Naehe der Kommensurabilitaeten vom Typus /k plus 1//k/.
Numerical analysis of periodic solutions of restricted three body problem in sun-Jupiter system, noting genealogy of periodic orbits
Carpenter, L. H.↗
Calculations of rate constants for the three-body recombination of H2 in the presence of H2
A new global potential energy hypersurface for H2 + H2 is constructed and quasiclassical trajectory calculations performed using the resonance complex theory and energy transfer mechanism to estimate the rate of three body recombination over the temperature range 100 to 5000 K. The new potential is a faithful representation of ab initio electron structure calculations, is unchanged under the operation of exchanging H atoms, and reproduces the accurate H3 potential as one H atom is pulled away. Included in the fitting procedure are geometries expected to be important when one H2 is near or above the dissociation limit. The dynamics calculations explicitly include the motion of all four atoms and are performed efficiently using a vectorized variable-stepsize integrator. The predicted rate constants are approximately a factor of two smaller than experimental estimates over a broad temperature range.
Schwenke, David W.↗