Lie groups of motor integrals of generalized Kepler motion

The singularity of the Kepler motion can be eliminated by means of the spinor regularization. The extensive integrals of the Kepler motion form a Lie algebra with respect to the Poisson bracket operation. Mayer-Gürr has shown that in the caseH>0 the corresponding Lie group is the multiplicative group of all real 4×4 unimodular matrices SL(4,R). Kustaanheimo has posed the problem of the identification of the corresponding Lie groups in the elliptic and parabolic cases. We solve this problem here and use the opportunity to introduce the concept of the Clifford algebra which is needed in our solution.     

On Spinor Equations of Motion and their “Possible Integrals”

The motion of one point mass of the classical mechanics is treated by means of the relativistic spinor regularization (KUSTAANHEIMO 1975). Most general spinor equations of motion (2.9)‒(2.10)and the differential equations(3.16)‒(3.25)for the “possible integrals” (3.2)‒(3.11)of these quations of motion are deduced. If the force is a superposition of a conservative central force and of another force perpendicular to radius vector and velocity (Chapter 4, Case D), then the theory yields scalar and spinor integrals (4.7), (4.10)‒(4.12), (4.14)‒(4.15), (4.17), (4.28) that enable a parametric representation of the orbit by quadratures, as soon as one solution of a RICCATI differential equation (4.33) has been found.     

The two-body problem with drag and radiation pressure

The Kepler problem including radiation pressure and drag is treated. The equation of the orbit is derived and the scalar and vector integrals of motion are obtained by direct operation on the vector form of the equation of motion.     

Celestial mechanics with geometric algebra

Geometric algebra is introduced as a general tool for Celestial Mechanics. A general method for handling finite rotations and rotational kinematics is presented. The constants of Kepler motion are derived and manipulated in a new way. A new spinor formulation of perturbation theory is developed.This work was partially supported by JPL under contract with the National Aeronautics and Space Administration.     

An improved algorithm due to laguerre for the solution of Kepler's equation

A root-finding method due to Laguerre (1834–1886) is applied to the solution of the Kepler problem. The speed of convergence of this method is compared with that of Newton's method and several higher-order Newton methods for the problem formulated in both conventional and universal variables and for both elliptic and hyperbolic orbits. In many thousands of trials the Laguerre method never failed to converge to the correct solution, even from exceptionally poor starting approximations. The non-local robustness and speed of convergence of the Laguerre method should make it the preferred method for the solution of Kepler's equation.     

Extension of the Kepler problem towards minimization of energy and gravity softening

The classical Kepler Problem consists in the determination of the relative orbital motion of a secondary body (planet) with respect to the primary body (Sun), for a given time. However, any natural system tends to have minimum energy and is subjected to differential gravitational or tidal forces (called into play mainly due to the finite size and deformability of the secondary body). We formulate the Kepler Problem taking into account the finite size of the secondary body and consider an approximation which tends towards minimum energy orbits, by increasing the dimensionality of the problem. This formulation leads to a conceivable natural explanation of the fact that the planetary orbits are characterized by small eccentricities.     

Satellite onboard orbit propagation using Deprit’s radial intermediary

Short-term satellite onboard orbit propagation is required when GPS position measurements are unavailable due to an obstruction or a malfunction. In this paper, it is shown that natural intermediary orbits of the main problem provide a useful alternative for the implementation of short-term onboard orbit propagators instead of direct numerical integration. Among these intermediaries, Deprit’s radial intermediary (DRI), obtained by the elimination of the parallax transformation, shows clear merits in terms of computational efficiency and accuracy. Indeed, this proposed analytical solution is free from elliptic integrals, as opposed to other intermediaries, thus speeding the evaluation of corresponding expressions. The only remaining equation to be solved by iterations is the Kepler equation, which in most of cases does not impact the total computation time. A comprehensive performance evaluation using Monte-Carlo simulations is performed for various orbital inclinations, showing that the analytical solution based on DRI outperforms a Dormand–Prince fixed-step Runge–Kutta integrator as the inclination grows.     

Stochastic diffusion in inverse square fields: General formulation for interplanetary gas

The Fokker-Planck equation for small stochastic changes to particles in Kepler orbits has to be formulated in terms of the integrals of motion. We generalize the modelling of proton and electron collisional perturbations to gas particles on trajectories through the solar system in order to include both spatial and velocity diffusion. The general solution is obtained in terms of a 4-dimensional normal distribution. Treatment of the singularity in the Fokker-Planck operator reduces the dimensionality by one. In addition to extending earlier results for anisotropic collisional heating in the thermal approximation, the present formulation gives the changes in density due to the mean repulsive force and to perturbations of trajectories (spatial diffusion). The net diffusion is almost everywhere towards the sun and the density increase is significant in the downstream hydrogen wake, particularly where destructive depletion is strong and gravitational focussing weak.     

A new analytic approach to the Sitnikov problem

A new analytic approach to the solution of the Sitnikov Problem is introduced. It is valid for bounded small amplitude solutions (z _max = 0.20) (in dimensionless variables) and eccentricities of the primary bodies in the interval (–0.4 < e < 0.4). First solutions are searched for the limiting case of very small amplitudes for which it is possible to linearize the problem. The solution for this linear equation with a time dependent periodic coefficient is written up to the third order in the primaries eccentricity. After that the lowest order nonlinear amplitude contribution (being of order z ³) is dealt with as perturbation to the linear solution. We first introduce a transformation which reduces the linear part to a harmonic oscillator type equation. Then two near integrals for the nonlinear problem are derived in action angle notation and an analytic expression for the solution z(t) is derived from them. The so found analytic solution is compared to results obtained from numeric integration of the exact equation of motion and is found to be in very good agreement. CERN SL/AP     

Geometrical and physical outlook on the cross product of two quaternions

As an outcome of our previous notes [13, 14] on the quaternion regularization of the classical Kepler problem and pre-quantization of the Kepler manifold we show, first, that both the cross product of two quaternions and the cross product of their anti-involutes are susceptible of a simple geometrical representation in the ordinary 3-dimensional euclidean spaceR ³ and, secondly, that they satisfy anSO(4)-invariant relation that implies projection of curves from the quaternion space onto the spaceR ³. ThisSO(4)-invariance allows—in the particular case of orthogonal quaternions of equal norm—a straight derivation: (i) of the correspondence between the free motion on the surface of a sphereS ³ and the physical elliptical Kepler motion (collisions included) on a plane denoted by _w ; (ii) of the celebrated Kepler equation and (iii) of the Levi-Civita regularizing time transformation. With (i) and (ii) we recover some of Györgyi's [3] results. The aforesaid orbital plane _w and the orbital plane _*, arrived at independently by exploiting the Kustaanheimo-Stiefel regularizing transformation, are shown to be inclined exactly at an angle characterizing the ratio of the semi-axes of the elliptical orbits and intimately related to the cross product representation. Thus the eventual superimposition of the two planes confirms the intimate connection between the various regularization procedures—transforming the classical Kepler problem into the geodesic flow onS ³—and the Fock's procedure for the quantum theoretical Kepler problem of the hydrogen atom (accidental degeneracy).This research was supported by the Consiglio Nazionale delle Ricerche of Italy (C.N.R.-G.N.F.M.).     

New absolute measurements of the solar spectrum 310–685 nm

During 1986–1989 at the high-altitude station on the Peak Terskol, Caucasus (h = 3000 m) absolute measurements of the solar disk-centre intensity were performed. The observations were carried out with the specialized solar telescope (D = 23 cm,F = 3 m) and grating spectrometer (F = 2 m, grating 140 × 150 mm, 600 grooves mm^–1). The ribbon tungsten lamps used for absolute calibration were calibrated to the USSR standard of spectral intensity and were also compared with the irradiance standard of the PMO/WRC (Davos, Switzerland), with the lamps used in the Alma-Ata Observatory (Kazakhstan) and in Simferopol University for absolute measurements of stellar spectra. Methods and apparatus were improving step by step during 1985–1988. Special care was paid to the study of all possible sources of errors, in particular to the method of correction for atmospheric extinction, to polarization properties of optical elements of the apparatus, and to establishing the most reliable absolute calibration system. Finally, the observations performed during 1989 utilized only the refined methods and apparatus. As a result, the absolute integrals of the solar disk-centre intensity for 1-nm wide spectral bands in the range 310–685 nm are available. We estimate the total error is 2.5% at 310 nm and 2.1% at 680 nm. The absolute irradiance for 5-nm wide spectral bands is also obtained. We compare our results with results by Neckel and Labs (1984), with the irradiance filter measurements performed in PMO/WRC and calibration of the Sun's spectral irradiance to the stellar irradiance standard Vega by Lockwood (1992). Our results show a systematic difference with data by Neckel and Labs in the near-ultraviolet. The results by Neckel and Labs are probably underestimated in this spectral range by 8%.Deceased 20 January 1994.     

The hyperbolic Kepler equation (and the elliptic equation revisited)

A procedure is developed that, in two iterations, solves the hyperbolic Kepler's equation in a very efficient manner, and to an accuracy that proves to be always better than 10^–20 (relative truncation error). Earlier work on the elliptic equation has been extended by the development of a new procedure that solves to a maximum relative error of 10^–14.     

The importance of the Lie algebra so (4, 2) for the Kepler problem

It is shown that the Lie algebra so(4, 2) is characteristic for the three dimensional Keplerian motion provided the eccentric anomaly is used as the independent variable. This algebra generates all the integrals of motion and yields the guiding principle for reformulating all the Keplerian formulas.Proceedings of the Sixth Conference on Mathematical Methods in Celestial Mechanics held at Oberwolfach (West Germany) from 14 to 19 August, 1978.     

Estimation of long-term density variations in the upper atmosphere of the earth at minimums of solar activity from evolution of the orbital parameters of the earth’s artificial satellites

Long-term data on the evolution of the parameters of motion of 15 artificial satellites of the Earth in orbits with minimal heights of 400–1100 km were used to study the density variations in the upper atmosphere at minimums of four cycles of solar activity. It was found that the density at these heights considered increased by about 7% at the minimum of solar cycle 20 as compared to solar cycle 19. Later, the density fell rather linearly at the minimums of cycles 21 and 22. The statistical processing of the data for solar cycles 20–22 demonstrated that the density decreased by 4.6% over ten years and by 9.9% over 20 years. Analyzing the density variations during the four cycles of solar activity, we found that the long-term decrease in density observed at the minimums of cycles 20–22 is caused mainly by specific variations of the solar activity parameters (namely, the solar radio flux and the level of geomagnetic disturbance).__________Translated from Astronomicheskii Vestnik, Vol. 39, No. 2, 2005, pp. 177–183.Original Russian Text Copyright © 2005 by Volkov, Suevalov.     

Kepler discretization in regular celestial mechanics

In this paper, a special extrapolation method for the numerical integration of perturbed Kepler problems (given in KS-formulation) is worked out and analyzed in detail. The underlying so-called Kepler discretization isexact for the pure (elliptic) Kepler motion. A numerically stable realization is presented together with a backward error analysis: this analysis shows that the effect of the arising rounding errors can be regarded as a small perturbation inferior to the physical perturbation. For test purposes, a well-known example describing the motion of an artificial Earth satellite in an equator plane subject to the oblateness perturbation is used to demonstrate the efficiency of the new extrapolation method.Proceedings of the Sixth Conference on Mathematical Methods in Celestial Mechanics held at Oberwolfach (West Germany) from 14 to 19 August, 1978.     

Geometry of spinor regularization

The Kustaanheimo theory of spinor regularization is given a new formulation in terms of geometric algebra. The Kustaanheimo-Stiefel matrix and its subsidiary condition are put in a spinor form directly related to the geometry of the orbit in physical space. A physically significant alternative to the KS subsidiary condition is discussed. Derivations are carried out without using coordinates.     

Orbit Propagation with Lie Transfer Maps in the Perturbed Kepler Problem

The Lie transfer map method may be applied to orbit propagation problems in celestial mechanics. This method, described in another paper, is a perturbation method applicable to Hamiltonian systems. In this paper, it is used to calculate orbits for zonal perturbations to the Kepler (two-body) problem, in both expansion in the eccentricity and closed form. In contrast with a normal form method like that of Deprit, the Lie transformations here are used to effect a propagation of phase space in time, and not to transform one Hamiltonian into another.     

An analytical theory for the orbit of nereid

A complete analytical dynamic theory for the motion of Nereid has been constructed, accurate to approximately 0.01 arc second over several hundred years. The solution uses the Lie transform approach advanced by Deprit and is consistent with respect to the magnitudes of the disturbing functions, including all perturbations to an accuracy of 10^–8 relative to the two-body potential (oblateness and third-body). Multiple short-period variables in the third-body perturbations are related via the ratio of their mean motions, reducing the number of independent variables. Extensive use is made of expansions giving trigonometric functions of the true anomaly as analytical Fourier series in the mean anomaly. Initial constants and mass parameters come from the data obtained during the Voyager II encounter with Neptune in 1989.     

Relativistic effects in the motion of artificial satellites: The oblateness of the central body II

The motion of artificial satellites in the gravitational field of an oblate body is discussed in the post — Newtonian framework using the technique of canonical Lie transformations. Two Lie transformations are used to derive explicit results for the longperiodic and secular perturbations for satellite orbits in the Einstein case.