轨道改进中计算状态转移矩阵的分析方法

对当今人卫轨道改进问题,由于力学模型的复杂,精密星历和状态转移矩阵的计算均采用数值方法,这就需要积分两组常微分方程．本文针对状态转移矩阵在定轨中的作用,对定轨弧段不太长的情况,给出了状态转移矩阵的一种分析算法,从而避免数值求解两组常微分方程的问题,并以实际算例证实了这种算法的有效性     

Lunar perturbations of artificial satellites of the earth

Lunisolar perturbations of an artificial satellite for general terms of the disturbing function were derived by Kaula (1962). However, his formulas use equatorial elements for the Moon and do not give a definite algorithm for computational procedures. As Kozai (1966, 1973) noted, both inclination and node of the Moon's orbit with respect to the equator of the Earth are not simple functions of time, while the same elements with respect to the ecliptic are well approximated by a constant and a linear function of time, respectively. In the present work, we obtain the disturbing function for the Lunar perturbations using ecliptic elements for the Moon and equatorial elements for the satellite. Secular, long-period, and short-period perturbations are then computed, with the expressions kept in closed form in both inclination and eccentricity of the satellite. Alternative expressions for short-period perturbations of high satellites are also given, assuming small values of the eccentricity. The Moon's position is specified by the inclination, node, argument of perigee, true (or mean) longitude, and its radius vector from the center of the Earth. We can then apply the results to numerical integration by using coordinates of the Moon from ephemeris tapes or to analytical representation by using results from lunar theory, with the Moon's motion represented by a precessing and rotating elliptical orbit.     

Review of planetary and satellite theories

Planetary and satellite theories have been historically and are presently intimately related to the available computing capabilities, the accuracy of observational data, and the requirements of the astronomical community. Thus, the development of computers made it possible to replace planetary and lunar general theories with numerical integrations, or special perturbation methods. In turn, the availability of inexpensive small computers and high-speed computers with inexpensive memory stimulated the requirement to change from numerical integration back to general theories, or representative ephemerides, where the ephemerides could be calculated for a given date rather than using a table look-up process. In parallel with this progression, the observational accuracy has improved such that general theories cannot presently achieve the accuracy of the observations, and, in turn, it appears that in some cases the models and methods of numerical integration also need to be improved for the accuracies of the observations. Planetary and lunar theories were originally developed to be able to predict phenomena, and provide what are now considered low accuracy ephemerides of the bodies. This proceeded to the requirement for high accuracy ephemerides, and the progression of accuracy improvement has led to the discoveries of the variable rotation of the Earth, several planets, and a satellite. By means of mapping techniques, it is now possible to integrate a model of the motion of the entire solar system back for the history of the solar system. The challenges for the future are: Can general planetary and lunar theories with an acceptable number of terms achieve the accuracies of observations? How can numerical integrations more accurately represent the true motions of the solar system? Can regularly available observations be improved in accuracy? What are the meanings and interpretations of stability and chaos with respect to the motions of the bodies of our solar system? There has been a parallel progress and development of problems in dealing with the motions of artificial satellites. The large number of bodies of various sizes in the limited space around the Earth, subject to the additional forces of drag, radiation pressure, and Earth zonal and tesseral forces, require more accurate theories, improved observational accuracies, and improved prediction capabilities, so that potential collisions may be avoided. This must be accomplished by efficient use of computer capabilities.     

A semi-analytical method to study perturbed rotational motion

A semi-analytical method is presented to study the system of differential equations governing the rotational motion of an artificial satellite. Gravity gradient and non gravitational torques are considered. Operations with trigonometric series were performed using an algebraic manipulator. Andoyer's variables are used to describe the rotational motion. The osculating elements are transformed analytically into a mean set of elements. As the differential equations in the mean elements are free of fast frequency terms, their numerical integration can be performed using a large step size.     

An analytical iterative algorithm for the prediction of special satellite orbit points with the Brouwer orbit theory

Employing a direct recursive algorithm in relation with analytical theories will yield a considerable saving in computer time, as opposed to simulating a point by point integration through repeated evaluations of the orbit theory. As a case in point, we shall compute the set of osculating orbiting elements corresponding to special events within the revolution of an artificial satellite.     

Partial derivatives used in trajectory estimation

The Peano-Baker method is applied to the integration of the variational equations to produce the partial derivatives used in satellite navigation. In this method the analytic form of the state transition partial derivatives can be factored so that numerical integration is applied only to the departures from a simplified analytical model.The advantage of using the Peano-Baker approach rather than direct integration of the variational equations is that with the Peano-Baker method numerical integration can be performed adequately with low order formulae and relatively large step sizes. Numerical results are indicated.     

Accurate analytical representation of Pluto modern ephemeris

An accurate development of the latest JPL’s numerical ephemeris of Pluto, DE421, to compact analytical series is done. Rectangular barycentric ICRF coordinates of Pluto from DE421 are approximated by compact Fourier series with a maximum error of 1.3 km over 1900–2050 (the entire time interval covered by the ephemeris). To calculate Pluto positions relative to the Sun, a development of rectangular heliocentric ICRF coordinates of the Solar System barycenter to Poisson series is additionally made. As a result, DE421 Pluto heliocentric positions by the new analytical series are represented to an accuracy of better than 5 km over 1900–2050.     

New estimation of usually neglected forces acting on Galilean system

We studied small perturbations acting on Galilean satellites. Most of them are still not computed in the analytical theories and could probably improve the ephemeris of these satellites which are outside the precision of the observations. We used a numerical method to test the effect of such perturbations. Here are reporting the main results we obtained.     

Evolution of orbits and encounters of distant planetary satellites. Study tools and examples

This study of the orbital evolution and encounters of distant satellites of planets is aimed at determining their origin. It is also important for understanding the distribution of matter in the early stages of evolution of the Solar System. The mutual encounter of satellites is very weak because of their small sizes and masses. However, at very large time intervals, mutual encounter can be quite close to significantly changing the orbits of satellites. In order to study these factors, we have developed a special method and computer programs. For 107 distant satellites of Jupiter, Saturn, Uranus, and Neptune, motion parameters have been determined using observational data. On the basis of these parameters, a numerical integration of the equations of motion of the satellites has been carried out in time intervals of several thousand years. Using the original method of frequency analysis, we found rather simple analytical functions that correspond to the results of the numerical integration and make it possible to calculate orbital parameters at any time during a long interval. These tools make it possible to conduct extensive studies of changes in the form and relative position in space of the orbits of distant satellites of Jupiter, Saturn, Uranus, and Neptune. Several examples illustrate the possibilities offered by these tools. The computer software in the form of a service ephemeris of satellite orbits over a long interval of time is available via the Internet (http://www.sai.msu.ru/neb/nss/evolu0e.htm) on the website of the State Astronomical Institute of the Moscow State University.     

An Encke-type special perturbation method

In this paper, a combination analytical-numerical integration method for solving the differential equations of a modified set of Lagrange's planetary equations is described. The integration method is an Encke-type method because it involves integrating the deviations between the actual trajectory and a reference trajectory. The reference trajectory is obtained from an analytical solution containing the dominant secular and periodic effects of the gravitational field of the primary body. A set of nonsingular elements is used so that the method will be valid for all circular and elliptical motions. It is shown that the method is an accurate and efficient means of satellite ephemeris generation.This paper was presented at the AIAA/AAS Meeting, Princeton University, August 1969.     

A new method of initial orbit determination

Up to now we have been dealing with the construction of entirely analytical planetary theories such as VSOP82 (Bretagnon, 1982) and TOP82 (Simon, 1983). These theories take into account the whole of the Newtonian perturbations of nine point masses: the Sun, the Earth-Moon barycentre, the planets Mercury, Venus, Mars, Jupiter, Saturn, Uranus and Neptune. They also take into account perturbations due to some minor planets, to the action of the Moon and the relativistic effects. The perturbations of these last three types are in a very simple way under analytical form but they considerably increase the computations when introduced in the numerical integration programs.In the present paper we thus study a solution in which the Newtonian perturbations for the ten point masses are treated through numerical integration, the other perturbations being analytically added.     

A numerical comparison with the Ceplecha analytical meteoroid orbit determination method

Abstract– Analytic methods by Ceplecha have long been used for the determination of meteoroid heliocentric orbits. These methods include both the derivation of an initial atmospheric contact position and velocity state, and the calculation of an orbit at infinity based on zenithal attraction assumptions. Herein, we describe a numerical integration‐based verification for a portion of the Ceplecha methods, a verification driven by the need for an accurate meteoroid ephemeris in the hours before atmospheric contact. We show a close correspondence in analytic and numerical results, with a previously undocumented minor correction to a meteoroid’s longitude of the ascending node.     

Computer-developed construction of analytic expressions for the coordinates and partial derivatives of Jupiter's Galilean satellites

In his effort to develop series expressions for the coordinates of the Galilean satellites accurate to one are second (Jovicentric), R. A. Sampson was forceda priori to adopt certain numerical values for several constants imbedded in his theory. His final numerical values for the series expressions are not amenable to adjustment of the constants of integration nor of physical constants which affect the motion of the satellites. A method which utilizes computer-based algebraic manipulation software has been developed to reconstruct Sampson's theory, to remove existing errors, to introduce neglected effects and to provide analytical expressions for the coordinates as well as for the partial derivatives with respect to orbital parameters, Jupiter and satellite masses, Jupiter's oblateness (J ₂,J ₄) and Jupiter's pole and period of rotation. The computer-based manipulations enable one to perform, for example, the approximately 10⁸ multiplications required in calculating some perturbations (and their partial derivatives) of Satellite II by Satellite III with ease, and provide algebraic expressions which can readily be adjusted to generate theories corresponding to revised constants of integration and physical parameters.     

Ephemeris of a highly eccentric orbit: Explorer 28

An ephemeris has been obtained for Explorer 28 (IMP 3) which agrees well with 2 years of radio observations and with SAO observations a year later. This ephemeris is generated over the 3 year lifetime by a numerical integration method utilizing a set of initial conditions, at launch and without requiring further differential correction. Because highly eccentric orbits are difficult to compute with acceptable accuracy and because a long continuous arc has been obtained which compares with actual data to a known precision, this ephemeris may be used as a standard for computing highly eccentric orbits in the Earth-Moon system.Orbit improvement was used to obtain the initial conditions which generated the ephemeris. This improvement was based on correcting the energy by adjusting the semimajor axis to match computed times of perigee passage with the observed. This procedure may generate errors in semimajor axis to compensate for model errors in the energy; however this compensation error is also implicit in orbit determination itself.     

建立同步卫星精密星历表的最佳数值方法

简要介绍当前天体力学中常用的各种数值计算方法,结合同步卫星运动方程的特点和轨道解的性质,分析各种数值计算方法在同步卫星情况下使用的优劣,确定一次和分形式的Cowell方法是建立同步卫星精密星历表的最佳方法,最后通过有效的数值实验,给出不同精度要求下Cowell方法的最佳阶和相应的最大步长.     

Prediction of satellite orbits contraction due to diurnally varying oblate atmosphere and altitude-dependent scale height using KS canonical elements

A new non-singular analytical theory for the motion of near-Earth satellite orbits with the air drag effect is developed in terms of uniformly regular KS canonical elements. Diurnally varying oblate atmosphere is considered with variation in density scale height dependent on altitude. The series expansion method is utilized to generate the analytical solutions and terms up to fourth-order terms in eccentricity and c (a small parameter dependent on the flattening of the atmosphere) are retained. Only two of the nine equations are solved analytically to compute the state vector and change in energy at the end of each revolution, due to symmetry in the equations of motion. The important drag perturbed orbital parameters: semi-major axis and eccentricity are obtained up to 500 revolutions, with the present analytical theory and by numerical integration over a wide range of perigee height, eccentricity and inclination. The differences between the two are found to be very less. A comparison between the theories generated with terms up to third- and fourth-order terms in c and e shows an improvement in the computation of the orbital parameters semi-major axis and eccentricity, up to 9%. The theory can be effectively used for the re-entry of the near-Earth objects, which mainly decay due to atmospheric drag.     

Analytical theories of satellite motion constructed with a new package of formula manipulation and compared with numerical integration

Specialized to the Lie series based perturbation method of Kirchgraber and Stiefel (1978) a new computer algebra package called ANALYTOS has been developed for constructing analytical orbital theories either in noncanonical or canonical form. We present results on the (extended) Main Problem of orbital theory of artificial earth satellites and related issues. The order of the solutions achieved is generally one order higher than those known from literature. Moreover, the analytical orbits have been checked succesfully against precise numerical ephemerides. This revised version was published online in July 2006 with corrections to the Cover Date.     

Enhancing the Accuracy of Numerical Integration of the Equations of Asteroid Motion with Perturbations from Major Planets and the Moon from the DE Ephemerides

The discontinuous behavior of coordinates of planets and the Moon and their derivatives, which are determined from their modern ephemerides, at the boundaries of adjacent interpolation intervals is illustrated using the example of the DE436 ephemerides. The numerical integration of the equations of motion of two asteroids demonstrates that the integration accuracy increases by several orders of magnitude if the step of numerical integration is matched to the boundaries of ephemeris interpolation intervals. In addition, an algorithm for ephemeris smoothing at the boundaries of interpolation intervals is developed and applied in order to eliminate the jumps of coordinates and their first-order derivatives emerging in extended- and quadprecision calculations. This algorithm allows one to remove the jumps of coordinates and their derivatives up to any given order. It is demonstrated that the use of ephemerides smoothed to the first-order derivatives in quad-precision calculations increases the accuracy of numerical integration by ~10 orders of magnitude.