Restricted quantum-mechanical three-body problems

Eigenvalues and normalized wave functions of an electron are derived in the field of a generalized dipole with chargesZ ₁ andZ ₂ (Z ₁+Z ₂ 0) in the asymptotic region up to the third order (inr ₂) where the distance,r ₂, between the two charges is small. These asymptotic wave functions render it possible to calculate the asymptotic expansion for the coefficients of a coupled infinite system of second order differential equations arising from a perturbative (analytic) solution to the Schrödinger equation of helium-like ions if the nuclear charge is not less than 2.     

Restricted quantum-mechanical three-body problems

A new method is presented in a general form to solve the Schrödinger equation of helium-like ions. The wave function is expanded in terms of the eigenfunctions of a moving electron in the field of two Coulombic ions which are fixed in space. This makes the method similar to the Dirac perturbation theory (perturbation theory for time-dependent problems). In the present method an infinitely coupled system of infinitely many second-order ordinary differential equations must be solved instead of one second-order partial differential equation of three variables. The nature of the singular points and boundary conditions are discussed and some general relations are given which are useful for the numerical treatment.     

Restricted quatum-mechanical three-body problems

On the basis of a perturbative procedure in which the eigenfunctions of a helium-like ion are expanded in the Hilbert space built up from the eigenfunctions of an electron in two fixed Coulomb charges,all asymptotic eigenfunctions are constructed for the Schrödinger equation of helium-like ions. If the nuclear chargeZ ₁ is not less than 2, then our asymptotic considerations clarify the singularities of the Schrödinger equation of a helium-like ion (atr ₁=0, ,r ₂=0, ,r ₁₂=0, ), while in the case ofZ ₁=1 (negative hydrogen ion)r ₂=0 will not be treated in this paper. The established order (inr ₂) of asymptotics at 0 or in an exceptional case the zeroth-order term of the functions (actually the coefficients of an expansion of the desired eigenfunctions) as one of the electron coordinates (r ₂, the distance of the two fixed Coulomb charges, a parameter of the set of the basic functions) enables us to classify the eigenfunctions of a helium-like ion. This classification resembles the classification scheme for one-electron configurations. The asymptotics forr ₂ indicate bounded, pseudobounded (auto-ionizing) and free states. (Doubly ionized continuum states are not discussed here.) The use of the asymptotic solutions is indicated for the complete solution of the problem which may be either numerical integration or a variational procedure.Neutral muonic helium is included in the discussion.     

Restricted three-body problem when one of the primaries is an ellipsoid

This paper deals with the existence and the stability of the libration points in the restricted three-body problem when the smaller primary is an ellipsoid. We have determined the equations of motion of the infinitesimal mass which involves elliptic integrals and then we have investigated the collinear and non collinear libration points and their stability. This is observed that there exist five collinear libration points and the non collinear libration points are lying on the arc of the unit circle whose centre is the bigger primary. Further observed that the libration points either collinear or non-collinear all are unstable.     

A note on independent variables for restricted three-body problems

In studies of the elliptic restricted three-body problem, the true anomaly of the motion of the primaries is often used as the independent variable. The equations of motion then show invariancy in form from the circular case. It is of interest whether other independent variables exist, such that the invariant form of the equations is maintained. It is found that true anomaly is the only such variable.     

Dynamical model of binary asteroid systems through patched three-body problems

The paper presents a strategy for trajectory design in the proximity of a binary asteroid pair. A novel patched approach has been used to design trajectories in the binary system, which is modeled by means of two different three-body systems. The model introduces some degrees of freedom with respect to a classical two-body approach and it is intended to model to higher accuracy the peculiar dynamical properties of such irregular and low gravity field bodies, while keeping the advantages of having a full analytical formulation and low computational cost required. The neighborhood of the asteroid couple is split into two regions of influence where two different three-body problems describe the dynamics of the spacecraft. These regions have been identified by introducing the concept of surface of equivalence (SOE), a three-dimensional surface that serves as boundary between the regions of influence of each dynamical model. A case of study is presented, in terms of potential scenario that may benefit of such an approach in solving its mission analysis. Cost-effective solutions to land a vehicle on the surface of a low gravity body are selected by generating Poincaré maps on the SOE, seeking intersections between stable and unstable manifolds of the two patched three-body systems.     

The three-body problem

Book review

The three-body problemby Christian Marchal, Elsevier, Amsterdam, 1990, (ISBN 0444874402)     

Cas special du probleme restreint des trois corps (special cases of restricted three-body problems)

Resume Nous envisageons ici le problème, posée par l'académicien A. J. Ichlinsky (Académie des Sciences, U.R.S.S.). Deux masses actifs égales tournent par une orbite circulaire kepleriènne. Deux autres masses égales, simétriques par rapport à, le centre de cette orbite, se trouvent sous l'attraction des masses actifs et s'attirent l'un l'autre, mais n'agissent sur les masses actifs. Il est montré, que le mouvement de chaque masses passives est suffisament proche au mouvement d'une masse passive dans le problème de Kopenhague. En particulier problème d'Ichlinsky admet les solutions particulières, voisines aux solutions lagrangiènnes du problème de Kopenhague. Toutes ces solutions particulières sont instables mais admettent les solutions périodiques, voisines aux solutions lagrangiènnes. Le problème posée peut être applicable aux quelques problèmes de la mécanique celeste stellaire.

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In this paper we consider the some special cases of restricted three-body problems, proposed by academician A. J. Ichlinsky.

     

The stellar three-body problem

Two applications of von Zeipel's method to the stellar three-body problem eliminate the short period terms and establish two new integrals of the motion beyond the classical integrals. The remaining time averaged problem with only the second order Hamiltonian has one additional integral and can be solved. The motion with the third order averaged Hamiltonian included is more complex, in that there may be additional resonances, and the additional integral does not exist in all cases.     

The small-energy three-body systems

We study near-Lagrange collinear configurations in the three-dimensional three-body problem, with arbitrary gravitating masses. A general method, previously developed for Coulombian systems, which provides a unique formalism for treating few-body systems close to the breakup threshold, has been employed to study the motion of three bodies both for bounded and unbounded configurations. The dependence of the triple-escape function on the small total energy E has been evaluated, as well as rovibronic configurations for bounded motion. An approximate characteristic constant is found for symmetrical systems, /, where and are the vertical libration mode and rotational mode angular frequencies, respectively, and is the threshold exponent. For equal-mass systems this constant acquires the value 1.05288.     

The rectilinear three-body problem

The rectilinear equal-mass and unequal-mass three-body problems are considered. The first part of the paper is a review that covers the following items: regularization of the equations of motion, integrable cases, triple collisions and their vicinities, escapes, periodic orbits and their stability, chaos and regularity of motions. The second part contains the results of our numerical simulations in this problem. A classification of orbits in correspondence with the following evolution scenarios is suggested: ejections, escapes, conditional escapes (long ejections), periodic orbits, quasi-stable long-lived systems in the vicinity of stable periodic orbits, and triple collisions. Homothetic solutions ending by triple collisions and their dependence on initial parameters are found. We study how the ejection length changes in response to the variation of the triple approach parameters. Regions of initial conditions are outlined in which escapes occur after a definite number of triple approaches or a definite time. In the vicinity of a stable Schubart periodic orbit, we reveal a region of initial parameters that corresponds to trajectories with finite motions. The regular and chaotic structure of the manifold of orbits is mostly defined by this periodic orbit. We have studied the phase space structure via Poincaré sections. Using these sections and symbolic dynamics, we study the fine structure of the region of initial conditions, in particular the chaotic scattering region.     

Orbit period modulation for relative motion using continuous low thrust in the two-body and restricted three-body problems

This paper presents rich new families of relative orbits for spacecraft formation flight generated through the application of continuous thrust with only minimal intervention into the dynamics of the problem. Such simplicity facilitates implementation for small, low-cost spacecraft with only position state feedback, and yet permits interesting and novel relative orbits in both two- and three-body systems with potential future applications in space-based interferometry, hyperspectral sensing, and on-orbit inspection. Position feedback is used to modify the natural frequencies of the linearised relative dynamics through direct manipulation of the system eigenvalues, producing new families of stable relative orbits. Specifically, in the Hill–Clohessy–Wiltshire frame, simple adaptations of the linearised dynamics are used to produce a circular relative orbit, frequency-modulated out-of-plane motion, and a novel doubly periodic cylindrical relative trajectory for the purposes of on-orbit inspection. Within the circular restricted three-body problem, a similar minimal approach with position feedback is used to generate new families of stable, frequency-modulated relative orbits in the vicinity of a Lagrange point, culminating in the derivation of the gain requirements for synchronisation of the in-plane and out-of-plane frequencies to yield a singly periodic tilted elliptical relative orbit with potential use as a Lunar far-side communications relay. The \(\Delta v\) requirements for the cylindrical relative orbit and singly periodic Lagrange point orbit are analysed, and it is shown that these requirements are modest and feasible for existing low-thrust propulsion technology.     

Non-integrability of the three-body problem

We consider the planar problem of three bodies which attract mutually with the force proportional to a certain negative integer power of the distance between the bodies. We show that such generalisation of the gravitational three-body problem is not integrable in the Liouville sense.     

Robe's restricted three-body problem revisited

Robe's restricted three-body problem is reanalyzed with a view to incorporate a new assumption, namely that the configuration of the fluid body is that described by an hydrostatic equilibrium figure (Roche's ellipsoid). In the concomitant gravitational field a full treatment of the buoyancy force is given. The pertinent equations of motion are derived, the linear stability of the equilibrium solution is studied and the connection between the effect of the buoyancy forces and a perturbation of the Coriolis force is pointed out.     

The photogravitational restricted three-body problem

The photogravitational restricted three-body problem is reviewed and the case of the out-of-plane equilibrium points is analysed. It is found that, when the motion of an infinitesimal body is determined only by the gravitational forces and effects of the radiation pressure, there are no out-of-plane stable equilibrium points.     

Plasma polarization shift: An improved quantum-mechanical treatment

Using an improved quantum mechanical treatment of the plasma polarization shift theory, calculations have been made for plasma conditions corresponding to available measurements of the shift of Heii resonance lines. Agreement with experimental results at low density is found but large discrepancies still exist at high density. Comparison with other theories leads to agreement with a corrected version of the semi-classical treatment of the effect although discrepancies of one order of magnitude are found with the only quantum treatment presently available.     

On the double averaged three-body problem

The Hamiltonian of three point masses is averaged over fast variablel and ll (mean anomalies) The problem is non-planar and it is assumed that two of the bodies form a close pair (stellar three-body problem). Only terms up to the order of (a/á)⁴ are taken into account in the Hamiltonian, wherea andá are the corresponding semi-major axes. Employing the method of elimination of the nodes, the problem may be reduced to one degree of freedom. Assuming in addition that the angular momentum of the close binary is much smaller than the angular momentum of the motion of the binary around a third body, we were able to solve the equation for the eccentricity changes in terms of the Jacobian elliptic functions.     

On the elliptic restricted three-body problem

The main goal of this paper is to show that the elliptic restricted three-body problem has ejection-collision orbits when the mass parameter µ is small enough. We make use of the blow up techniques. Moreover, we describe the global flow of the elliptic problem when µ = 0 taking into account the singularities due to collision and to infinity.     

Robe's restricted three-body problem with drag

Robe's restricted three-body problem is investigated with regards to the effects of a linear drag force. In particular. the stability of the model's equilibrium points is studied in this respect. Two scenarios are envisaged: the one originally discussed by Robe himself and the one suggested by him and recently analyzed by the present authors, that assumes for the fluid body the structure of a Roche's ellipsoid.     

Stability of the planetary three-body problem

We present a direct method for the expansion of the planetary Hamiltonian in Poincaré canonical elliptic variables with its effective implementation in computer algebra. This method allows us to demonstrate the existence of simplifications occurring in the analytical expression of the Hamiltonian coefficients. All the coefficients depending on the ratio of the semi major axis can thus be expressed in a concise and canonical form.