Motor integrals of a generalized Kepler motion

The ordinary spinor differential Equation (20) of the unperturbed Kepler motion is obtained from the classical equation of motion (19) if one uses the spinor regularization (9) and postulates an essential subsidiary condition (10). A natural generalization for the Kepler motion follows by dropping this subsidiary conditions; it is the 8-parameter set of solutions of the spinor equation of motion (20). The sixteen natural extensive integrals (30)–(35) for this generalized Kepler motion are here deduced by means of the relativistic motors (2), (7) of the Spinor Ring Algebra. These integrals form, with respect to the Poisson bracket operation, a 15-dimensional Lie algebra (40)–(44), closely related to the Lie algebras in quantum mechanics.Dedicated to Professor G. Järnefelt on his 70th anniversary.     

FRW and Bianchi type I cosmology of f-essence

F-essence is a generalization of the usual Dirac model with the nonstandard kinetic term. In this paper, we introduce a new model of spinor cosmology containing both Ricci scalar and the non minimally coupled spinor fields in its action. We have investigated the cosmology with both isotropy and anisotropy, where the equations of motion of FRW and Bianchi type-I spacetimes have been derived and solved numerically. Finally the quantization of these models through Wheeler-De Witt (WD) wave function has been discussed.     

Regularization and the computation of optimal trajectories

A completely regular form for the differential equations governing the three-dimensional motion of a continuously thrusting space vehicle is obtained by using the Kustaanheimo-Stiefel regularization. The differential equations for the thrusting rocket are transformed using the K-S transformation and an optimal trajectory problem is posed in the transformed space. The canonical equations for the optimal motion in the transformed space are regularized by a suitable change of the independent variable. The transformed equations are regular in the sense that the differential equations do not possess terms with zero divisors when the motion encounters a gravitational force center. The resulting equations possess symmetry in form and the coefficients of the dependent variables are slowly varying quantities for a low-thrust space vehicle.Presented at the Conference on Celestial Mechanics, Oberwolfach, Germany, August 17–23, 1969.     

Studies in the application of recurrence relations to special perturbation methods

Using the rectangular equations of motion for the restricted three-body problem a comparison is made of the Encke and Cowell methods of integration. Each set of differential equations is integrated using Taylor series expansions where the coefficients of the powers of time are determined by recurrence relationships. It is shown that for fairly highly eccentric orbits in which the perturbing force is less than one thousandth of the two-body force the Encke method achieves a considerable saving in machine time. This is also true for almost circular orbits when low or moderate accuracy is required. When very high accuracy is required, however, the Cowell method is faster unless the perturbing force is less than 10^–6 of the two-body force. There is little difference in the accuracy of the two methods, the Cowell method being slightly more accurate when a low or moderate accuracy criterion is imposed.     

The relative motion of two spheroidal rigid bodies

We consider two spheroidal rigid bodies of comparable size constituting the components of an isolated binary system. We assume that (1) the bodies are homogeneous oblate ellipsoids of revolution, and (2) the meridional eccentricities of both components are small parameters.We obtain seven nonlinear differential equations governing simultaneously the relative motion of the two centroids and the rotational motion of each set of body axes. We seek solutions to these equations in the form of infinite series in the two meridional eccentricities.In the zero-order approximation (i. e., when the meridional eccentricities are neglected), the equations of motion separate into two independent subsystems. In this instance, the relative motion of the centroids is taken as a Kepler elliptic orbit of small eccentricity, whereas for each set of body axes we choose a composite motion consisting of a regular precession about an inertial axis and a uniform rotation about a body axis.The first part of the paper deals with the representation of the total potential energy of the binary system as an infinite series of the meridional eccentricities. For this purpose, we had to (1) derive a formula for representing the directional derivative of a solid harmonic as a combination of lower order harmonics, and (2) obtain the general term of a biaxial harmonic as a polynomial in the angular variables.In the second part, we expound a recurrent procedure whereby the approximations of various orders can be determined in terms of lower-order approximations. The rotational motion gives rise to linear differential equations with constant coefficients. In dealing with the translational motion, differential equations of the Hill type are encountered and are solved by means of power series in the orbital eccentricity. We give explicit solutions for the first-order approximation alone and identify critical values of the parameters which cause the motion to become unstable.The generality of the approach is tantamount to studying the evolution and asymptotic stability of the motion.Research performed under NASA Contract NAS5-11123.     

The precession and nutation of deformable bodies

The aim of the present paper will be to derive from the fundamental equations of hydrodynamics the explicit form of the Eulerian equations which govern the motion about the centre of gravity of self-gravitating bodies, consisting of compressible fluid of arbitrary viscosity, in an arbitrary external field of force. If the problem is particularized so that the external field of force represents the attaction of the sun and the moon, this motion would represent the luni-solar precession and nutation of a fluid viscous earth; if, on the other hand, the external field of force were governed by the earth (and the sun), the motion would define the physical librations of the moon regarded as a deformable body. The same equations are, moreover, equally applicable to the phenomena of precession and nutation of rotating fluid components in close binary systems, distorted by mutual tidal action; and the present paper contains the first formulation of the effects of viscosity on such phenomena.Investigation supported in part by the U.S. National Aeronautics and Space Administration under Contract No. NASW-1470.     

On linear equations of variation in dynamical problems

The linear equations of variation, associated with a motion of a particle moving in a plane under a field of force which admits a first integral of the motion of any form, are drawn up in terms of the tangential and normal displacements. The existence of the first integral implies that the normal displacement satisfies a single second-order differential equation, the tangential displacement being given from the solution of this by a single quadrature. The special cases are examined in which the integral is one of energy, and in which it is one of angular momentum. The extension is made to the motion of two particles moving in a plane under a conservative force-field depending on their positions, which admits also an integral of angular momentum. (The study of the relative motion in the gravitational problem of three bodies in the plane may be put into this form by Jacobi's formulation). An equation is given for finding the non-zero characteristic exponents of a periodic solution of this second problem.Presented at the Conference on Celestial Mechanics, Oberwolfach, Germany, August 17–23, 1969.     

Enhancement of on-board forecasting accuracy of GEO SC COG motion using the compensative transversal acceleration

A method is suggested for enhancing the on-board forecasting accuracy of the COG motion of a GEO SC with a long time of independent operation. The suggested method consists of introducing so-called compensative transversal acceleration (CTA), along with zonal harmonics into the right sides of the differential equations of SC motion among other disturbances due to the Earth’s gravitational field eccentricity. The CTA compensates the integral effect of the sectoral and tesseral harmonics; its value is constant for a specified point of GEO SC location (standing point) and is calculated on the Earth from numerical integration of differential equations of motion taking into account the complete set of gravitational field harmonics. The CTA value is transmitted on-board of an SC as program command data. The method is implemented in algorithms of on-board forecasting of Electro-L SC motion and can be used to enhance the on-board forecasting accuracy of the COG motion of GEO SCs with a long time of independent operation.     

A perturbation method and some applications

For differential equations with one fast variable, a perturbation method is introduced that transforms a solution valid over only a short time interval to a new solution composed of averaged variables plus a periodic function of the averaged variables. The averaged variables are governed by a set of differential equations where the fast variable has been removed and thus can be numerically integrated quickly or solved directly. This method is applied to a perturbed harmonic oscillator with a cubic perturbation, van der Pol's equation, coorbital motion in the restricted three-body problem, and to nearly circular motion of a particle near one of the primaries in the restricted three-body problem.     

Efficient numerical routines to find the averaged motion of small dust particles under the Poynting-Robertson force

The orbit-averaged differential equations of motion of dust particles under gravity, radiation pressure and Poynting-Robertson drag were given by Wyatt and Whipple (1950). An integral of motion enables the system of two equations in semi-major axis a and eccentricity e to be reduced to one equation, the solution of which is presented here in terms of analytical formulae. An efficient numerical algorithm to compute the solution is given. Listings of two FORTRAN routines are included.     

Quasiintegrals of the photogravitational eccentric restricted three-body problem with Poynting–Robertson drag

A restricted three-body problem for a dust particle, in presence of a spherical cometary nucleus in an eccentric (elliptic, parabolic or hyperbolic) orbit about the Sun, is considered. The force of radiation pressure and the Poynting– Robertson effect are taken into account. The differential equations of the particle’s non-inertial spatial motion are investigated both analytically and numerically. With the help of a complex representation, a new single equation of the motion is obtained. Conversion of the equations of motion system into a single equation allows the derivation of simple expressions similar to the integral of energy and integrals of areas. The derived expressions are named quasiintegrals. Relative values of terms of the energy quasiintegral for a smallest, largest, and a mean comet are calculated. We have found that in a number of cases the quasiintegrals are related to the regular integrals of motion, and discuss how the quasiintegrals may be applied to find some significant constraints on the motion of a body of infinitesimal mass.     

Numerical stabilization of the differential equations of Keplerian motion

A stabilization of the classical equations of two-body motion is offered. It is characterized by the use of the regularizing independent variable (eccentric anomaly) and by the addition of a control-term to the differential equations. This method is related to the KS-theory (Stiefel, 1970) which performed for the first time a stabilization of the Kepler motion. But in contrast to the KS-theory our method does not transform the coordinates of the particle. As far as the theory of stability and the numerical experiments are concerned we restrict ourselves to thepure Kepler motion. But, of course, the stabilizing devices will also improve the accuracy of the computation of perturbed orbits. We list, therefore, also the equations of the perturbed motion.     

Dynamics of multibody chains in circular orbit: non-integrability of equations of motion

This paper discusses the dynamics of systems of point masses joined by massless rigid rods in the field of a potential force. The general form of equations of motion for such systems is obtained. The dynamics of a linear chain of mass points moving around a central body in an orbit is analysed. The non-integrability of the chain of three masses moving in a circular Kepler orbit around a central body is proven. This was achieved thanks to an analysis of variational equations along two particular solutions and an investigation of their differential Galois groups.     

Equations of motion of elliptic restricted problem of three bodies with variable mass

The differential equations of motion of the elliptic restricted problem of three bodies with decreasing mass are derived. The mass of the infinitesimal body varies with time. We have applied Jeans' law and the space-time transformation of Meshcherskii. In this problem the space-time transformation is applicable only in the special case whenn=1,k=0,q=1/2. We have applied Nechvile's transformation for the elliptic problem. We find that the equations of motion of our problem differ from that of constant mass only by a small perturbing force.     

Special perturbations employing osculating reference states

The concept of employing osculating reference position and velocity vectors in the numerical integration of the equations of motion of a satellite is examined. The choice of the reference point is shown to have a significant effect upon numerical efficiency and the class of trajectories described by the differential equations of motion. For example, when the position and velocity vectors on the osculating orbit at a fixed reference time are chosen, a universal formulation is yielded. For elliptical orbits, however, this formulation is unattractive for numerical integration purposes due to Poisson terms (mixed secular) appearing in the equations of motion. Other choices for the reference point eliminate this problem but usually at the expense of universality. A number of these formulations, including a universal one, are considered here. Comparisons of the numerical characteristics of these techniques with those of the Encke method are presented.     

A second-order solution to the two-point boundary value problem for rendezvous in eccentric orbits

A new second-order solution to the two-point boundary value problem for relative motion about orbital rendezvous in one orbit period is proposed. First, nonlinear differential equations to describe the relative motion between a chaser and a target are presented considering the second-order terms in the gravity. Then, by regarding the second-order terms as external accelerations, we establish second-order state transition equations. Moreover, the J2 perturbations effects can also be considered in the state transition equations. Last, the initial relative velocity to fulfill a rendezvous is determined by solving the state transition equations. Numerical simulations show that the new second-order state transition equations are accurate. The second-order solution to the two-point boundary value problem on eccentric orbits is valid even if the relative range is farther than 500 km.     

On the “rotational angular momentum” of the oceans and the corresponding polar motion

Periodic polar motions caused by ocean tides are predicted. In the Liouville equations for rotational motion the complete excitation functions for the ocean tides have to be used. This does not depend on the fact that hydrodynamical ocean tide models do not consider the centrifugal acceleration. The observable polar motion of the Celestial Ephemeris Pole CEP (more exactly: the terrestrial location of the CEP) is tabulated for the ten ocean tides M₂, S₂, N₂, K₁, O₁, P₁, M f, M f′, M m, Ssa. Typical amplitudes for the largest ocean tides are 0.4 milliarcseconds. This is within the reach of geodetic VLBI and SLR observations.     

A noncanonical analytic solution to theJ 2 perturbed two-body problem

The motion of a satellite subject to an inverse-square gravitational force of attraction and a perturbation due to the Earth's oblateness as theJ ₂ term is analyzed, and a uniform, analytic solution correct to first-order inJ ₂, is obtained using a noncanonical approach. The basis for the solution is the transformation and uncoupling of the differential equations for the model. The resulting solution is expressed in terms of elementary functions of the independent variable (the ‘true anomaly’), and is of a compact and simple form. Numerical results are comparable to existing solutions.     

Gravitational perturbations of equatorial orbits

Asymptotic solutions are developed for the motion of a geocentric satellite in the equatorial plane due to gravitational perturbations such as nonsphericity (especially oblateness) of the primary body. Axisymmetric potentials are considered. A class of transformations is developed and the equations of motion are solved by the method of generalized multiple scales. Further it is shown that the equations of motion can be transformed into the required form to within any specified degree of accuracy. The transformations form an Abelian group of infinite order which leaves the differential equations of motion invariant. Solutions are developed in terms of elementary functions instead of elliptic or other higher transcendental functions and are shown to agree with known results.This investigation was carried out under NASA Grant NGR-31-001-152 with the author as a consultant to Princeton University.     

>“Resolving” neutron stars physics and geometry

The unprecedented harvest of X‐ray photons detected from dozens of isolated neutron stars has made it possible to glimpse at their emission mechanisms as well as at their emission geometry. Rotating hot spot(s), superimposed to the global thermal emission from the neutron star surface, are seen from several objects, allowing to probe the stars' external heating sources. Non‐thermal emission is also seen to vary as the stars rotate. Moreover, absorption features have been detected in the spectra of several objects, allowing to probe (tentatively) the stars' magnetic fields. Spectacular tails, trailing the stars' supersonic motion, trace the boundaries of the relativist winds streaming from the star's magnetosphere. Apart from classical radio pulsar and certified radio‐quiet neutron stars, XMM‐Newton has devoted significant observation time to the enigmatic central compact objects, presumably isolated neutron stars shining at the center of their supernova remnants. Far from showing a unifying behaviour, XMM‐Newton data have unveiled a surprising diversity. Understanding the reason(s) behind such diversity is the challenge for the next decade of X‐ray observations. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)