Showing papers in "Celestial Mechanics and Dynamical Astronomy in 1975"
TL;DR: In this article, a method is developed to study the stability of periodic motions of the three-body problem in a rotating frame of reference, based on the notion of surface of section.
Abstract: A method is developed to study the stability of periodic motions of the three-body problem in a rotating frame of reference, based on the notion of surface of section. The method is linear and involves the computation of a 4×4 variational matrix by integrating numerically the differential equations for time intervals of the order of a period. Several properties of this matrix are proved and also it is shown that for a symmetric periodic motion it can be computed by integrating for half the period only.
111 citations
TL;DR: In this article, the existence and description of regions in physical space and configuration space where motion cannot occur in a three-body configuration space is discussed. But the description of these regions turns out to be most similar to the Hill regions of the circular restricted problem of three bodies.
Abstract: For the general three-body problem we show the existence and describe regions in physical space and configuration space where motion cannot occur. The description of these regions turns out to be most similar to the Hill regions of the circular restricted problem of three bodies. These regions depend upon the constants of energy and angular momentum.
89 citations
TL;DR: In this article, it was proved that a symmetric periodic orbit of the restricted three-body problem can be continued analytically when the mass of the third body is small but not negligible.
Abstract: It is proved that a symmetric periodic orbit of the circular planar restricted three-body problem can be continued analytically, when the mass of the third body is small but not negligible, to a periodic motion of the general three-body problem in a rotating frame of reference whose origin coincides with the center of mass of the two bodies with large masses and itsx axis always contains these bodies The two bodies with the large masses describe periodic motion on thex axis of the rotating frame while the third body, with the small mass, describes a symmetric periodic orbit in this frame The motion of the two bodies lying on thex axis is always stable, whereas the periodic orbit of the third body in the rotating frame is stable or unstable depending on whether or not the nonzero characteristic exponents of the original periodic orbit of the restricted problem are of the stable or unstable type, respectively It is also shown that for a fixed value of the small mass of the third body, a family of symmetric periodic orbits of the restricted problem can be continued analytically to a family of periodic motions of the general three-body problem
87 citations
86 citations
TL;DR: In this paper, the location of the collinear libration points is investigated numerically, by taking the oblateness of the primaries into consideration, for 19 systems of astronomical interest.
Abstract: In this paper location of the collinear libration points is investigated numerically, by taking the oblateness of the primaries into consideration, for 19 systems of astronomical interest. It is found that in some of the systems the shifts are significant. These equilibria are shown to be unstable in general, though the existence of conditional, infinitesimal (linearized) periodic orbits around them can be established, in the usual way. It is shown that the eccentricity and synodic period of these orbits are functions of oblateness too. Numerical study, in this connection, with the above systems, revealed that the orbits around the libration point, which is farthest from the primary whose oblateness effect is included, exhibit a different trend from those around the other two points.
82 citations
TL;DR: In this article, the elastic deformation of a flexible body is represented as a time-varying linear combination of given mode shapes, which can be used to simulate a spinning Skylab and an interplanetary Viking.
Abstract: Equations of motion are derived for systems of rotationally interconnected bodies in which the terminal bodies may be flexible and the remaining bodies are rigid. The bodies may have an arbitrary ‘topological tree’ arrangement; that is, there are no closed loops of bodies. This derivation extends earlier results for systems of interconnected rigid bodies only, and is much simpler than several other recent works on terminal flexible bodies. The model for a flexible body assumes that the elastic deformation is representable as a time-varying linear combination of given mode shapes. The paper also derives the appropriate form for gravitational terms, so that the equations can be used for flexible satellites. Also included are expressions for kinetic energy and angular momentum so that in case these are theoretically constant, they can be used to monitor the accuracy of the numerical integration. The paper concludes with a section showing how interbody constraint forces and torques (which do not appear in the equations of motion) can be recovered from quantities available in this formulation, and also how to treat state variables which are prescribed functions of time. A digital computer program based on the equations derived here has been used to simulate a spinning Skylab (with flexible booms) and also the interplanetary Viking (with flexible solar panels and thrust vector control).
65 citations
TL;DR: In this article, the authors describe two relatively simple reductions to order 6 for the planar general three-body problem, and show that this reduction leads to the distinction between two types of periodic solutions: absolute or relative periodic solutions.
Abstract: 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.
46 citations
TL;DR: In this article, a periodic orbit of the restricted circular three body problem, selected arbitrarily, is used to generate a family of periodic motions in the general three-body problem in a rotating frame of reference, by varying the massmcffff3 of the third body.
Abstract: A periodic orbit of the restricted circular three-body problem, selected arbitrarily, is used to generate a family of periodic motions in the general three-body problem in a rotating frame of reference, by varying the massm
3 of the third body. This family is continued numerically up to a maximum value of the mass of the originally small body, which corresponds to a mass ratiom
1:m
2:m
3≃5:5:3. From that point on the family continues for decreasing massesm
3 until this mass becomes again equal to zero. It turns out that this final orbit of the family is a periodic orbit of the elliptic restricted three body problem. These results indicate clearly that families of periodic motions of the three-body problem exist for fixed values of the three masses, since this continuation can be applied to all members of a family of periodic orbits of the restricted three-body problem. It is also indicated that the periodic orbits of the circular restricted problem can be linked with the periodic orbits of the elliptic three-body problem through periodic orbits of the general three-body problem.
45 citations
TL;DR: In this paper, the Lagrangian function is used to stabilize and regularize the equations of motion in the case of momenta in addition to the coordinates of the coordinates, and a modification of Ebert's theorem is proposed to the relation between the original and new independent variables.
Abstract: Time transformations involving momenta in addition to the coordinates are studied from the points of view of stabilization and regularization of the equations of motion. The generalization of Sundman's transformation by using the potential function to transform the time is further generalized by using the Lagrangian function for the same purpose. The possibility of the stabilization of the equations of motion is investigated similarly to Stiefel's and Baumgarte's recent results but instead of a factorial, an additive control function is introduced in all equations of motion. The relation between the original and new independent variables is integrated by a modification of Ebert's theorem and it is shown that the new independent variable is Hamilton's principal function. Numerical examples illustrate the method and seem to indicate that the computation of close approach trajectories benefit especially by the transformations discussed. The Appendix offers an analytic treatment regarding the stabilization of the constant of energy.
42 citations
TL;DR: In this article, the family of simple-periodic retrograde satellites of Jupiter is used as the generating family in order to find periodic orbits of the second generation of up to order eight.
Abstract: The familyf of simple-periodic retrograde satellites of Jupiter is used as the generating family in order to find periodic orbits of the second generation of up to order eight. Thirteen new families of long-periodic orbits of the type of retrograde satellites are found and the significane of their characteristic family curves for the determination of regions of stability is investigated. The use of these, family curves is suggested for the direct discovery of such regions in a two-dimensional space of parameters.
37 citations
TL;DR: In this article, a numerical study of periodic solutions of the Planar General Three-Body Problem with variable masses has been presented, extending all the way from the elliptic restricted problem to the general problem with three equal masses.
Abstract: The article contains a numerical study of periodic solutions of the Planar General Three-Body Problem. Several new periodic solutions have been discovered and are described. In particular, there is a continuous family with variable masses, extending all the way from the elliptic restricted problem to the general problem with three equal masses. All our examples have special symmetry properties which are described in detail. Finally we also suggest some important applications to the natural satellites of the solar system.
TL;DR: An extensive analysis of the motion of Phobos and Deimos from 1877 to 1973 has been fulfilled and the new values of the parameters of the orbital model first developed by Struve have been determined for both satellites.
Abstract: An extensive analysis of the motion of Phobos and Deimos from 1877 to 1973 has been fulfilled. The new values of the parameters of the orbital model first developed by Struve have been determined for both satellites. The new sets of the orbital parameters compete with the solutions of similar accuracy found by Wilkins and Sinclair. A secular acceleration in longitude of Phobos is found to be equal to +(0.107±0.011)×10−7 deg day−2. The value of the acceleration is little affected when one or another group of oppositions is omitted. The acceleration of Deimos is determined with great uncertainty: +(0.06±0.34)×10−9 deg day−2. Values found for the orbital parameters seem to be in good agreement since the mass, oblateness and coordinates of the pole of Mars inferred from the motion of each satellite have similar values in both cases.
TL;DR: In this paper, the authors determined the positions of the two Martian satellites Phobos and Deimos from 80 television photographs of the satellites taken by the imaging system of the Mariner 9 spacecraft.
Abstract: Orbit elements for the two Martian satellites Phobos and Deimos have been determined from 80 television photographs of the satellites taken by the imaging system of the Mariner 9 spacecraft. Phobos was found to be within 60 km of its positions predicted by recently published ephemeris theories which include a secular acceleration term in the longitude. This tends to corroborate the existence of a secular acceleration in the longitude of Phobos. Deimos was found to be within 100 km of its position predicted from Earth-based observations.
TL;DR: In this paper, the authors present a formulaire sous a forme compacte for the construction des perturbations planetaires d'ordres eleves par rapport aux masses perturbatrices.
Abstract: Les auteurs presentent un formulaire sous une forme compacte pour la construction des perturbations planetaires d'ordres eleves par rapport aux masses perturbatrices. Elles ont ete construites par un processus iteratif et donnent les variations des elements osculateurs. Il n'y a pas de singularites pour les excentricites et inclinaisons nulles dans les equations differentielles. Toutes les operations elementaires sont des manipulations de series de Fourier a coefficients numeriques, et un grand soin a ete apporte pour economiser les operations algebriques. Le formulaire est presente sous trois formes:
(a)
Une forme vectorielle, a composantes reelles qui peut etre utilisee pour l'integration numerique.
(b)
Une forme complexe, afin de mettre en evidence les symetries du systeme de variables.
(c)
Une forme scalaire qui est la plus elaboree. Cette derniere forme a ete utilisee pour la construction des perturbations du premier ordre, pour tous les couples de planetes. Deux illustrations sont donnees (Jupiter et Saturne, Venus et Terre). Par ailleurs, des remarques sont faites sur l'utilisation pratique des series de Fourier, la resolution de l'equation de Kepler sous forme complexe et la construction par iteration de l'inverse de la distance entre deux corps.
TL;DR: In this article, the applicability of several methods of analytical mechanics to the modern problem of formulating generic equations of motion of a point-connected set of rigid bodies in a topological tree was explored.
Abstract: The purpose of the present paper is to explore the applicability of several methods of analytical mechanics to the modern problem of formulating generic equations of motion of a point-connected set of rigid bodies in a topological tree, in order to compare the results of the previously published Hooker-Margulies/Hooker equations. The unexpected result of the inquiry is the discovery that with the substitution of a key kinematical identity from the Hooker and Margulies vector-dyadic equations for the multiple-rigid-body tree, identical equations emerge from each of four quite different derivation procedures.
TL;DR: In this article, modified multirevolution methods are derived that will integrate exactly products of linear and periodic functions, and information obtained from an approximate analytical solution of the satellite equations of motion, may be used to increase the accuracy and/or efficiency of the multire-volution integration.
Abstract: Multirevolution methods allow for the computation of satellite orbits in steps spanning many revolutions. The methods previously discussed in the literature are based on polynomial approximations, and as a result they will integrate exactly (excluding round-off errors) polynomial functions of a discrete independent variable. Modified methods are derived that will integrate exactly products of linear and periodic functions. Numerical examples are given that show that these new methods provide better accuracy for certain satellite problems. It is also shown that information obtained from an approximate analytical solution of the satellite equations of motion, may be used to increase the accuracy and/or efficiency of the multirevolution integration.
TL;DR: In this article, the main theorems of the theory of averaging are formulated for slowly varying standard systems and it is shown that it is possible to extend the class of perturbation problems where averaging might be used.
Abstract: The main theorems of the theory of averaging are formulated for slowly varying standard systems and we show that it is possible to extend the class of perturbation problems where averaging might be used.
TL;DR: In this paper, the behavior of the argument of the pericentre is investigated for the orbit of an artificial satellite which is moving under the potential of double libration when the inclination of the orbit is close to the critical value tan−1 2.
Abstract: The behaviour of the argument of the pericentre is investigated for the orbit of an artificial satellite which is moving under the potential
when the inclination of the orbit is close to thecritical value tan−1 2. The theory is developed to first order and it is applicable to all values of the eccentricity, with the exception of those in the neighbourhood of zero and unity. Four principal types of behaviour are noted and these are illustrated in appropriate phase-plane diagrams. It is shown that the two types which exhibit double libration in the argument of the pericentre are restricted to a relatively small domain in the (a, e)-plane of possible motions. Moreover, it is demonstrated that for double libration to occur it is necessary, but not sufficient, that\(e > \sqrt 6/13\). The ranges of values of the inclination for which libration of the pericentre is a possibility are given for the more important cases.
TL;DR: In this paper, it is shown that it is possible to make a change of variables in a Lagrangian in such a way that the number of variables is increased without the use of Lagrange multipliers.
Abstract: It is shown that it is possible to make a change of variables in a Lagrangian in such a way that the number of variables is increased. The Euler-Lagrange equations in the redundant variables are obtained in the standard way (without the use of Lagrange Multipliers!). These equations are not independent but they are all valid and consistent. In some cases they are simpler than if the minimum number of variables are used. The redundant variables are supposed to be related to each other by several constraints (not necessarily holonomic), but these constraints are not used in the derivation of the equations of motion. The method is illustrated with the well known Kustaanheimo-Stiefel Regularization. Some interesting applications to perturbation theory are also described.
TL;DR: In this paper, the attitude motion of spinning axisymmetric satellites in the presence of gravity-gradient and solar radiation pressure torques is studied analytically, and an approximate closed-form solution for the nonlinear, nonautonomous, coupled fourth-order system proves to be an excellent tool in locating periodic solutions of the system in both circular and noncircular orbits.
Abstract: Attitude motion of spinning axisymmetric satellites in presence of gravity-gradient and solar radiation pressure torques is studied analytically. The approximate closed-form solution developed for the nonlinear, nonautonomous, coupled fourth-order system proves to be an excellent tool in locating periodic solutions of the system in both circular and noncircular orbits. The variational stability of the periodic motion is examined using the Floquet theory. The resonance analysis suggests the existence of critical combinations of system parameters leading to large amplitude oscillations.
TL;DR: In this paper, a new method was described for representing the motion in the planar problem of three bodies when all three point masses simultaneously come close to each other, and it was shown that the motion during the critical phase of closest approach is intimately connected with triple parabolic escape.
Abstract: A new method is described for representing the motion in the planar problem of three bodies when all three point masses simultaneously come close to each other. The main results are (1) that the motion during the critical phase of closest approach is intimately connected with triple parabolic escape and (2) that a sufficiently close triple approach generally leads to the escape of one body witharbitrarily high asymptotic velocity.
TL;DR: In this paper, an algorithm is proposed for generating sequences of Chebyshev series which are the best approximations of an astronomical ephemeris in the sense of Chebyhev over large intervals of time.
Abstract: An algorithm is proposed for generating sequences of Chebyshev series which are the best approximations of an astronomical ephemeris in the sense of Chebyshev over large intervals of time. The criterion for a polynomial approximation of a function to be the best polynomial approximation of the function is that the error function present certain rippling characteristics as described by Remez (1957). General features of the program in PL/1 are described.
TL;DR: In this article, a procedure for sequentially estimating the state of a lunar orbiting space vehicle acted upon by unmodeled terms in the lunar potential has been developed, which provides more precise estimates of the vehicle state than conventional orbit determination procedures and, hence, provides an accurate input for navigation purposes.
Abstract: In previous investigations, a procedure for sequentially estimating the state of a lunar orbiting space vehicle acted upon by unmodeled terms in the lunar potential has been developed. Results obtained by processing tracking data from the Apollo 10 and 11 missions indicate that the algorithm provides more precise estimates of the vehicle state than conventional orbit determination procedures and, hence, provides an accurate input for navigation purposes. The question of the agreement of the estimates with the actual unmodeled accelerations has not been established.
TL;DR: In this paper, the authors deduce input-output equations from Breakwell-Perko's matching theory, which make it possible to give a linear description of an arc of a second species solution starting far from the Moon and also finishing far from a close approach to it.
Abstract: From Breakwell-Perko's matching theory, we deduce input-output equations which make it possible to give a linear description of an arc of a second species solution starting far from the Moon and also finishing far from the Moon after a close approach to it. This allows a global analysis of the second species solutions. Special attention is drawn to the periodic symmetric second species solutions.
TL;DR: The main features of this theory are presented with special emphasis on the most specific of them: choice of the parameters, separation of the problem into subproblems (main problem, generalized main problem, complete problem), special adaptation of the method required by the resonant situation of the Galilean system.
Abstract: The main features of this theory are presented with special emphasis on the most specific of them: choice of the parameters, separation of the problem into subproblems (main problem, generalized main problem, complete problem), special adaptation of the method required by the resonant situation of the Galilean system
TL;DR: In this article, the authors extended Breakwell and Perko's first-order matching theory to a more general matching theory which is applicable to a wider class of second species solutions.
Abstract: This paper extends Breakwell and Perko's ‘first order’ matching theory (1965, 1966) to a more general matching theory which is applicable to a wider class of second species solutions In a first stage, the matching theory is elaborated on the basis of new assumptions on the orders of magnitude of the small parameters In a second stage, we construct a matching theory which takes into account general assumptions which include our assumptions and Breakwell-Perko's
TL;DR: In this paper, the authors describe the resonance among three of the Galilean satellites in a way intended to demonstrate the similarities it has to the normal two-satellite resonance, and the hypothesis that the resonance was formed by the action of tidal forces is discussed.
Abstract: The resonance amongst three of the Galilean satellites is described in a way intended to demonstrate the similarities it has to the normal two-satellite resonance. The hypothesis that the resonance was formed by the action of tidal forces is discussed. The problem is too complicated to reach any firm conclusions, but the tidal hypothesis does not seem to be a satisfactory explanation.
TL;DR: In this article, the authors define the notion of system lie and define a system mecanique comportant des variables cycliques, on montre que l'ensemble des mouvements stationnaires du systeme lie coincide avec l''sensemble of stations lie, i.e., if le system lie est stable, il en est de meme for le systeme initial.
Abstract: Ayant defini la notion de ‘systeme lie’ associe a un systeme mecanique comportant des variables cycliques, on montre que l'ensemble des mouvements stationnaires du systeme coincide avec l'ensemble des mouvements stationnaires du systeme lie. L'etude de la stabilite de ces mouvements montre que si le systeme lie est stable, il en est de meme pour le systeme initial. La recherche des mouvements stationnaires des gyrostats fournit une application de cette etude.
TL;DR: In this article, a short derivation of the regularized equations of motion for the perturbed two-body problem is given, which is then applied to the slightly modified time transformation dt/ds=r/ω.
Abstract: A short derivation is given of the regularized equations of motion for the perturbed two-body problem. This method is then applied to the slightly modified time transformation dt/ds=r/ω.
TL;DR: In this article, the general-altitude transformation from geocentric to geodetic coordinates and vice versa is derived and the error incurred in these expansions is of the order of one part in 3×107.
Abstract: Formulas for the general-altitude (height above the ellipsoid) transformation from geocentric to geodetic coordinates and vice versa are derived. The set of four formulas is expressed in each of two useful forms: series expansions in powers of the Earth's flattening and series expansions in powers of the Earth's eccentricity. The error incurred in these expansions is of the order of one part in 3×107.