Archie E. Roy

Studies in the application of recurrence relations to special perturbation methods

The integration by recurrent power series of certain differential equations occurring in celestial mechanics is shown to be very much more efficient and accurate than that produced by classical one step methods. It is shown that for any such system of differential equations the machine time taken to carry out an integration is a minimum for a certain choice of the number of terms taken in the recurrent power series. In the two-body orbits considered this number is about 15. For the same accuracy criterion the power series is faster than the Runge-Kutta method of the fourth order by a factor which varies between 6 and 15 depending on the eccentricity of the orbit.

The origin of the constellations

Abstract The problem of the origin of the stellar constellations familiar to western astronomers from ancient times is discussed in an attempt to answer the classical detective story questions: Who? When? Why? and Where? The available astronomical, literary and archaeological evidence is examined to suggest a possible solution.

Hill stability of satellite orbits

Szebehelys criterion for Hill stability of satellites is derived from Hills problem and a more exact result is obtained. Direct, Hill stable, circular satellites can exist almost twice as far from the planet as retrograde satellites. For direct satellites the new result agrees with Kuipers empirical estimate that such satellites are stable up to a distance of half the “radius of action” of the planet. Comparison with the results of numerical experiments shows that Hill stability is valid for direct satellites but meaningless for retrograde satellites. Further accuracy for the maximum distance of Hill stable orbits is obtained from the restricted problem formulation. This provides estimates for planetary distances in double star systems.

Significant high number commensurabilities in the main lunar problem. I : The Saros as a near-periodicity of the Moon's orbit

A description is given of certain historically known cycles associated with high-number near commensurabilities among the synodic, anomalistic and nodical lunar months and the anomalistic year. Using eclipse records, the JPL ephemeris and results from three-body numerical integrations, any dynamical configuration of the Earth-Moon-Sun system (within the framework of the main lunar problem) is shown to repeat itself closely after a period of time equal in length to the classical Saros cycle of 18 years and 10 or 11 days. The role played by mirror configurations in reversing solar perturbations on the lunar orbit is examined and it is shown that the Earth-Moon-Sun system moves in a nearly periodic orbit of period equivalent to the Saros. The Saros cycle is therefore the natural averaging period of time by which solar perturbations can be most effectively removed in any search into the long term evolution of the lunar orbit.

The effect of orbital eccentricities on the shape of the hill-type analytical stability surfaces in the general three-body problem

Hill-type stability surfaces are computed for the general hierarchical three-body problem for non-zero eccentricities of the initial osculating orbits. Significant differences are found between them and the one obtained for initial zero eccentricities. Application is made to the triple subgroups of the Solar System; in particular it is found that no analytical guarantee of Hill-type stability can be given to any of the satellites against solar perturbations.

Stability criteria in many-body systems: III. Empirical stability regions for corotational, coplanar, hierarchical three-body systems

The stability parameters developed and discussed in the first paper of this series (Walkeret al., 1980) are used to determine empirically, by means of numerical integration experiment, regions of stability for corotational, coplanar, hierarchical three-body systems. The initially circular case of these systems is studied: the components of the close binary are taken to move initially in circular orbits with respect to their common mass-centre, the third mass initially moving in a circular orbit with respect to the same mass-centre such that its orbit lies wholly outside those of the former two masses.The stability of these systems is then studied by reference to the empirical stability parameters and the initial ratio of the semi-major axes of the orbit of the close binary to that of the third mass about the binarys mass-centre, which is less than unity. For given values of the stability parameters it is determined how the stability of a system is affected by changes in the ratio of the semi-major axes. It is found that an upper limit to this ratio exists which determines the region of stability for such systems. It is also found possible, in the region of instability, to predict how unstable a system will be i.e. crudely speaking, the number of orbits it may be expected to execute before some gross instability sets in.The effect commensurabilities in mean motion have on the stability of these systems is also considered. It is generally found that these commensurabilities enhance the stability of these systems. The predictive powers of the method are then tested: using many test cases it is seen how accurately the stability or instability of a system may be predicted.

The Caledonian Symmetrical Double Binary Four-Body Problem I: Surfaces of Zero-Velocity Using the Energy Integral

The Caledonian four-body problem introduced in a recent paper by the authors is reduced to its simplest form, namely the symmetrical, four body double binary problem, by employing all possible symmetries. The problem is three-dimensional and involves initially two binaries, each binary having unequal masses but the same two masses as the other binary. It is shown that the simplicity of the model enables zero-velocity surfaces to be found from the energy integral and expressed in a three dimensional space in terms of three distances r 1, r 2, and r 12, where r 1 and r 2 are the distances of two bodies which form an initial binary from the four body system’s centre of mass and r 12 is the separation between the two bodies.

Stability criteria in many-body systems: V: On the totality of possible hierarchical general four-body systems

In previous papers of this series the stability of hierarchical many-body dynamical systems has been considered in terms of parameters which are a measure of the perturbation imposed on the disturbed Keplerian orbit of each member of a system by the other bodies present.If these parameters are small enought then the initial hierarchical order will be undisturbed for some time i.e. the status quo will be maintained and to that extent stability will exist. In the light of this criterion the appropriate parameters for the general four-body problem are considered. Two distinct hierarchical arrangments of four-body systems are possible; these are classifid and an examination of the relevant stability parameters is made in each case.It is shown how regions may be determined within which real four-body systems can exist and may be stable. It is also shown how the various types of possible systems, (e.g.,Star-Star-Star-Star, Star-Planet-Planet-Star, etc.) within these regions may be identified.

The Stability of N-Body Hierarchical Dynamical Systems

The equations of motion of n-body hierarchical dynamical system (HDS) in a generalized Jacobi coordinate system enable empirical stability parameters to be readily defined. The magnitudes of these parameters, together with the ratios of successive radius vectors in the HDS may make it possible to compute the stability of the HDS, so providing a measure of the time interval in which there is an even chance of the status quo of the HDS being altered by mutual perturbations.

Studies in the application of recurrence relations to special perturbation methods: IV: Comparison of the Encke method with the use of perturbational equations

Using the rectangular equations of motion for the restricted three-body problem a comparison is made of the integration of these equations by the Encke method and by a set of perturbational equations. Each set of differential equations is integrated using Taylor series expansions where the coefficients of the powers of time are determined by recurrence relations. It is shown that for very small perturbations the use of the perturbational equations is more efficient than the use of the Encke method. A discussion is also given of when Cowells method is more efficient than either of these techniques.