The celestial bodies motion theories: on the required accuracy of the motion theories of the perturbing bodies

The equation for calculation of the required accuracy of the perturbing bodies motion theories is obtained. The equation relates the accuracy required to take into account perturbing acceleration, acting on the perturbed body, with the accuracy of the motion theory of the perturbing body. The solutions for estimation of the required accuracy both for the inner and the external cases in the spherical coordinates are coincided. The solution for the calculation of the required accuracy for the general case (combining the inner and the external cases) in Cartesian coordinates is obtained. The special cases for the solution in Cartesian coordinates are studied. As an example, the estimations of the required accuracy of the motion theories of the solar system planets for some perturbed bodies (the near-Earth asteroid 4179 Toutatis, the main belt asteroid 208 Larcimosa, the trojan asteroid 588 Achilles, the centaur asteroid 5145 Pholus, the Kuiper belt asteroid 1995 QZ9, the comet Halley) are obtained. The conditions of the use of the obtained results are discussed.     

Stability of numerical method as applied to motion in a circular orbit

In this paper we discuss the numerical stability of Cowell's method when applied to the Keplerian circular motion. The critical stepsize h_m is given for the PE, PECE and CE algorithms of various orders. A comparison of our results with others' shows our method to be more precise.     

Integration rules from the equivalent differential equation method applied to equations of motion

The numerical integration of equations of motion necessarily implies the presence of errors that depend on initial conditions as well as the different physical parameters under consideration. More particularly, dumping or dissipative terms can appear and it is especially interesting to determine its causes. The equivalent differential equation method may allow the errors from a certain numerical scheme to be analyzed and, together with other considerations, can help us to eliminate or reduce them.     

On the motion of three rigid bodies: Central configurations

In the present paper, the motion of three rigid bodies is considered. With a set of new variables, and the 10 first integrals of the motion, the problem is reduced to a system of order 25 and one quadrature. The plane motions are characterized, and finally, an equation for the existence of central configurations (in particular, Lagrangian and Eulerian solutions) has been found. Besides, the case of three axisymmetric ellipsoids is studied.     

Virial oscillations of celestial bodies IV: The Lyapunov stability of motion

It is shown that with a virial approach to the solution of the many-body problem the integral characteristics of a system (Jacobi's function and total energy), being present in Jacobi's equation, are immanent to its own integrals. Estimating the Lyapunov stability of motion of a system they play the role of Lyapunov functions.Studying Lyapunov stability of the virial oscillations of celestial bodies we used the Duboshin criterion applicable when permanent perturbations are present. In the case of conservative systems the potential energy of the system plays the role of such a perturbation. Thus, the nature of the virial oscillations can be understood as an effect of non-linear resonance between the kinetic and the potential energies.It is shown that the stability of virial oscillations of conservative systems relative to variations of the form-factors product is only a necessary condition in the proof of the hypothesis that =const. for celestial bodies. The sufficient condition for the proof of this equality consists of the given direct derivation of the equation of virial oscillations of celestial bodies from Einstein's equation, as well as of the equivalence of Schwarzschild's solution and the solution of Jacobi's equation at .The stability of virial oscillations for dissipative systems is studied. It is shown that the stability is limited by the period of time of its bifurcation.     

Post-Newtonian limitations on measurement of the PPN parameters caused by motion of gravitating bodies

We derive an explicit Lorentz-invariant solution of the Einstein and null geodesic equations for data processing of the time delay and ranging experiments in the gravitational field of moving gravitating bodies of the Solar system – the Sun and major planets. We discuss the general-relativistic interpretation of these experiments and the limitations imposed by motion of the massive bodies on measurement of the parameters  γ_PPN, β_PPN  and  δ_PPN  of the parametrized post-Newtonian (PPN) formalism.     

Coupled motion of rigid bodies about their center of mass

A study is made of the motion of a system consisting of two rigid bodies coupled by a massless rigid boom. Relative translational and rotational motions are examined with the assumption that no external forces are acting on the system. For specific sets of initial conditions and assumptions on the symmetries of the two bodies, nontrivial analytic solutions are observed. The stability and the internal torques are also examined for a few selected cases.This research was conducted while the author was a senior research associate of the National Research Council at the National Aeronautics and Space Administration (NASA) Lyndon B. Johnson Space Center.     

Equations of motion for interconnected rigid and elastic bodies: A derivation independent of angular momentum

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).We announce with regret that Bill Hooker died in an avalanche while on a mountain-climbing expedition in Peru, July 1974.     

The general motion of three mutually-interacting,axisymmetric spherical rigid bodies

In this paper, the translational-rotational motions of an axisymmetric rigid body and two spherical rigid bodies under the influence of their mutual gravitational attraction are considered. The equations of motion in the canonical elements of Delaunay-Andoyer are obtained. The elements of motion in the zero and first approximations can be determined.     

Equations of motion for rotating finite bodies in the extended PPN formalism

The equations of motion for rotating finite bodies in the extended PPN formalism of Will and Nordtvedt are developed. These equations may be used in conjunction with precise radio and laser ranging data in solar system tests of viable metric gravitational theories.This paper presents the results of one phase of research carried out at the Jet Propulsion Laboratory, California Institute of Technology, under Contract No. NAS 7-100, sponsored by the National Aeronautics and Space Administration.     

Symplectic methods and their application to the motion of small bodies in the Solar System

Symplectic methods have been widely used in Solar System dynamics. This paper discusses both single step and multistep symplectic methods. For single step methods we point out that the modified algorithm (Wisdom et al., 1991, Kinoshita et. al., 1991) can be executed in the mass center coordinate system and in the Jacobian coordinate system. For multistep methods we describe the connections between symmetric and symplectic methods.     

Symplectic methods and their application to the motion of small bodies in the Solar System

Symplectic methods have been widely used in Solar System dynamics. This paper discusses both single step and multistep symplectic methods. For single step methods we point out that the modified algorithm (Wisdom et al., 1991, Kinoshita et. al., 1991) can be executed in the mass center coordinate system and in the Jacobian coordinate system. For multistep methods we describe the connections between symmetric and symplectic methods.     

Equations of motion of elliptically-restricted problem of a body with variable mass and two tri-axial bodies

The differential equations of motion of the elliptic restricted problem of three bodies, an infinitesimal spherical body with decreasing mass and two tri-axial bodies are derived. We have applied Jeans's law and the space-time of Meshcherskii in the special case whenn=1,k=0,q=1/2. Also Nechvíle's transformation for the elliptic problem be applied for this case.     

Surface exposure dating of non-terrestrial bodies using optically stimulated luminescence: A new method

We propose a new method for in situ surface exposure dating of non-terrestrial geomorphological features using optically stimulated luminescence (OSL); our approach is based on the progressive emptying of trapped charge with exposure to light at depth into a mineral surface. A complete model of the resetting of OSL with depth and time is presented for the first time; this model includes the competing effects of both optical resetting and irradiation. We consider two extreme conditions at the time the resetting is initiated: (a) a negligibly small trapped charge population and (b) a saturated trapped charge population. The potential dating applications for (a) include dust accumulation, volcanic rocks and impact-related sediments, and for (b) fault scarps, rock-falls, landslides and ice-scoured bedrock. Using assumptions based on terrestrial observations we expect that this approach will be applicable over the last 100 ka. The method is ideally suited to in situ measurement using existing technology developed for space applications, and so offers for the first time the realistic possibility of direct determination of exposure ages of young non-terrestrial surfaces.     

Equations of motion of the restricted problem of (2 + 2) bodies when primaries are magnetic dipoles and minor bodies are taken as electric dipoles

In this problem of the restricted (2 + 2) bodies we have considered two magnetic dipoles of masses M ₁ and M ₂(M ₁ > M ₂) moving in circular Keplarian orbit about their centre of mass. Two minor bodies of masses m ₁, m ₂(m _j< M ₂) are taken as electric dipoles in the field of rotating magnetic dipoles. These minor bodies interact with each other but do not perturb the primaries.We have found equations of motions which differ from that of Goudas and Petsagouraki's (1985).     

Multiplier hysteresis and image motion errors in limb darkening scans obtained in April 1981 (and later)

In response to the comment of P. Foukal (1989) we show that in our limb darkening observations multiplier hysteresis and image motion (seeing) can be ruled out as major sources of error.     

An averaging method to study the motion of lunar artificial satellites II: Averaging and Applications

On this, the second part of a two part study (Steichen, 1998) we further develop a semi-analytical theory for a lunar artificial satellite. This theory is obtained by averaging analytically the Hamiltonian function over period up to a month. The averaged equations are then numerically integrated. The solution is free from singularities at e = 0 and I = 0 and is not expanded in powers of these variables. In the last section, the analytic work is applied to characteristic examples to validate the method used. This revised version was published online in July 2006 with corrections to the Cover Date.     

Construction of confidence regions in problem on probabilistic study into motion of minor bodies of the solar system

Different approaches to constructing regions of possible motions of minor bodies are examined. An economical approach to the minimization of the number of points of the initial region by means of defining this region with its boundary surface is offered, and the estimations of its efficiency are given. The methods for estimating the admissibility of the linear approach are considered. For this purpose, simple methods for calculating the nonlinear factors are offered, which makes it possible to classify a problem to be solved as either strongly or weakly nonlinear. Recommendations are given on the possibility of reducing the concrete estimation problem to a weakly nonlinear one, where the more economical linear approach can be used. The combined method of mapping the initial region in time is also offered which unites the linear and nonlinear approaches. By example of two asteroids, the area of applicability of linear mappings is estimated.