Simulation of precise orbit determination of lunar orbiters

Based on the ongoing Chinese lunar exploration mission, i.e. the “Chang'e 1” project, precise orbit determination of lunar orbiters is analyzed for the actual geographical distribution and observational accuracy of the Chinese united S-band (USB) observation and control network as well as the very long baseline interferometry (VLBI) tracking network. The observed data are first simulated, then solutions are found after including the effects of various error sources and finally compared. We use the space data analysis software package, GEODYN, developed at Goddard Space Flight Center, NASA, USA. The primary error source of the flight orbiting the moon is the lunar gravity field. Therefore, the (formal) error of JGL165P1, i.e. the model of the lunar gravity field with the highest accuracy at present, is first discussed. After simulating the data of ranging and velocity measurement as well as the VLBI data of the time delay and time delay rate, precise orbit determination is carried out when the error of the lunar gravity field is added in. When the orbit is determined, the method of reduced dynamics is adopted with the selection of appropriate empirical acceleration parameters to absorb the effect of errors in the lunar gravity field on the orbit determination. The results show that for lunar missions like the “Chang'e 1” project, that do not take the lunar gravity field as their main scientific objective, the method of reduced dynamics is a simple and effective means of improving the accuracy of the orbit determination of the lunar orbiters.     

A robust estimation method in orbit improvement

Orbit improvement is studied via processing simulated optical data of 20 satellites. Using our criterion the fraction of outliers falls to less than 10% in most cases. Our method, which ensures the accuracy of orbit improvement, consists in using the Huber estimator twice, followed by iterating the Hampel estimator until convergence. The parameters C₀ and C₁ in the Hampel estimation may be given the fixed values of 2.2 and 3.6, while the value of C₂ varies in a definite way with the size of the outlier considered.     

The region of stable motions around a periodic eight-like orbit in the general three-body problem

We investigate the neighborhood of the periodic eight-like orbit found by Moore (1993) and Chenciner and Montgomery (2000). One-, two-, and three-dimensional scans in body coordinates, velocities, and masses were constructed. We found the regions of initial conditions in which the maximum mutual separation did not exceed 5 distance units during 2000 time units (about 300 periods of the initial solution). Larger deviations from the periodic solution lead to distant body ejections and escapes. The identified regions of finite motions are complex in structure. In some sections, these are simple-connected manifolds, while in other sections, stability zones alternate with escape zones. We estimated the fractal dimensions of the stability regions in three-dimensional scans: it typically ranges from 2 to 3. In some cases, we found transitions between motions along the figure of eight in its neighborhood and motions in the vicinity of a periodic Broucke orbit in the isosceles three-body problem.     

Application of two special orbits in the orbit determination of lunar satellites

Using inter-satellite range data,the combined autonomous orbit determination problem of a lunar satellite and a probe on some special orbits is studied in this paper.The problem is firstly studied in the circular restricted three-body problem,and then generalized to the real force model of the Earth-Moon system.Two kinds of special orbits are discussed:collinear libration point orbits and distant retrograde orbits.Studies show that the orbit determination accuracy in both cases can reach that of the observations.Some important properties of the system are carefully studied.These findings should be useful in the future engineering implementation of this conceptual study.     

The assessment of the semi-analytical method in the long-term orbit prediction of Earth satellites

To understand the long-term evolution and distribution of the space objects, it is necessary to predict their orbits. Compared with the short-term prediction of a few days, the priority concerns of the long-term orbit prediction are the calculation speed, and the accuracies of major orbital elements, including the semi-major axis and eccentricity which define the shape of the orbit, as well as the orbital inclination and the right ascension of ascending node which define the orientation of the orbit. Given these requirements, it is preferable to adopt the semi-analytical method, which averages the system over the orbital period, and integrates the averaged system using the numerical method. It is not new, however, in the available literature, we can hardly find a quantitative assessment regarding its accuracy and speed when it is applied to various types of orbits. In this paper, we would like to report our implementation and assessment of the semi-analytical method, expecting that it would help to estimate its feasibility in the long-term orbit prediction. The quantitative assessment covers the commonly used orbits for the Earth satellites. In some rare and special cases where the performance of our method appears abnormal, we discuss the reasons and possible solutions. We hope our results can provide some useful reference for the similar applications of the semi-analytical method since our method is a relatively common approach in terms of both accuracy and implementation.     

A numerical-analytical method for studying the orbital evolution of distant planetary satellites

We describe an approximate numerical-analytical method for calculating the perturbations of the elements of distant satellite orbits. The model for the motion of a distant satellite includes the solar attraction and the eccentricity and ecliptic inclination of the orbit of the central planet. In addition, we take into account the variations in planetary orbital elements with time due to secular perturbations. Our work is based on Zeipel’s method for constructing the canonical transformations that relate osculating satellite orbital elements to the mean ones. The corresponding transformation of the Hamiltonian is used to construct an evolution system of equations for mean elements. The numerical solution of this system free from rapidly oscillating functions and the inverse transformation from the mean to osculating elements allows the evolution of distant satellite orbits to be studied on long time scales on the order of several hundred or thousand satellite orbital periods.     

On the transfer of radiation at asteroidal surfaces in relation to their orbit deflection — II

The efficiency of absorption of X-rays generated by a nuclear explosion at the surface of an asteroid, estimated earlier, is used to calculate the explosion yield needed to deflect the orbit of an asteroid. Following the work of Ahrens &38; Harris, it is shown that a recoil velocity of 1 cm s⁻¹ is required to deflect an asteroid from a collision course with the Earth, and the necessary yield of explosion energy is estimated. If it is assumed that the scaling law between the energy and the diameter of the resulting crater, obtained from experiments carried out on the Earth, remains valid on the asteroid surface, where gravity is much weaker, an explosion energy of 8 and 800 megaton (Mton) equivalent of TNT would be required for asteroids of diameter 1 and 10 km respectively. If, on the other hand, the crater diameter is proportional to a certain power of the gravity g , the power being determined from a dimension analysis, 130 kton and 12 Mton would be required to endow asteroids of diameters 1 and 10 km with the required velocity, respectively. The result indicates that in order to estimate the required explosion energy, a better understanding of cratering under gravity much weaker than on the Earth would be required.     

Astronomical data reduction with total least squares

Although astronomers have been involved with the development and use of least squares, and alternatives, they have made insufficient use of total least squares: least squares that allow for error in the equations of condition as well as the observations. There exist, however, problems of astronomical data reduction for which total least squares represents the ideal mathematical tool. Among these problems are the differential correction of an orbit and the determination of parameters of Galactic kinematics. Total least squares, although more computationally demanding than ordinary least squares, can be used in any situation where the latter is applicable. However, care must be paid to the proper scaling of the data matrix. The method merits greater attention by the astronomical community.     

On the effect of the eccentricity of a planetary orbit on the stability of satellite orbits

The effect of the eccentricity of a planet’s orbit on the stability of the orbits of its satellites is studied. The model used is the elliptic Hill case of the planar restricted three-body problem. The linear stability of all the known families of periodic orbits of the problem is computed. No stable orbits are found, the majority of them possessing one or two pairs of real eigenvalues of the monodromy matrix, while a part of a family with complex instability is found. Two families of periodic orbits, bifurcating from the Lagrangian points L₁, L₂ of the corresponding circular case are found analytically. These orbits are very unstable and the determination of their stability coefficients is not accurate, so we compute the largest Liapunov exponent in their vicinity. In all cases these exponents are positive, indicating the existence of chaotic motions     

Numerical investigation of the orbit of Interball‐1

We investigated the motion of the Earth's artificial satellite Interball‐1 by using a method suitable for the computation of large eccentricity orbits. Though the measured and the computed orbital elements differ from each other within the measured error bound, we found a slight tendency for secular decreasing in the semi‐major axis, caused probably by electromagnetic drag. We analysed the dominant role of the Moon in the variations of the orbital eccentricity, leading to zero perigee height and the end of the lifetime of the satellite. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A simple algorithm for orbit classification

We describe a simple algorithm for classifying orbits into orbit families. This algorithm works by finding patterns in the sign changes of the principal coordinates. Orbits in the logarithmic potential are studied as an application; we classify orbits into boxlet families and examine the influence of the core radius on the set of stable orbit families.     

Groups of symmetries in the two‐body problem associated to Einstein's PN field

The two‐body problem associated to a spherical post‐Newtonian (PN) field with Einsteinian parameterization is revisited from the single standpoint of symmetries. The corresponding vector fields, in Hamiltonian and standard polar coordinates, or in collision‐blow‐up and infinity‐blow‐upMcGehee‐type coordinates, present symmetries that form diffeomorphic commutative groups endowed with a Boolean structure. The existence of such symmetries is of much help in understanding characteristics of the global flow, or in finding symmetric periodic orbits in more complex problems depending on a small parameter.     

An exact solution to determination of an open orbit

We present an exact solution of the equations for orbit determination of a two body system in a hyperbolic or parabolic motion. In solving this problem, we extend the method employed by Asada, Akasaka and Kasai (AAK) for a binary system in an elliptic orbit. The solutions applicable to each of elliptic, hyperbolic and parabolic orbits are obtained by the new approach, and they are all expressed in an explicit form, remarkably, only in terms of elementary functions. We show also that the solutions for an open orbit are recovered by making a suitable transformation of the AAK solution for an elliptic case.     

Determination of an intermediate perturbed orbit from three position vectors

We suggest a new approach to solving the problem of finding the orbit of a celestial body from its three spatial position vectors and the corresponding times. It allows most of the perturbations in the motion of a celestial body to be taken into account. The approach is based on the theory of intermediate orbits that we developed previously. We construct the orbit the motion along which is a combination of two motions: the motion of a fictitious attracting center whose mass varies according to Mestschersky’s first law and the motion relative to the fictitious center. The first motion is generally parabolic, while the second motion is described by the equations of the Gylden-Mestschersky problem. The constructed orbit has such parameters that their limiting values at any reference epoch define a superosculating intermediate orbit with a fourth-order tangency. We have performed a numerical analysis to estimate the accuracy of approximating the perturbed motion of two minor planets, 145 Adeona and 4179 Toutatis, by the orbits computed using two-position procedures (the classical Gauss method and the method that we suggested previously), a three-position procedure based on the Herrick-Gibbs equation, and the new method. Comparison of the results obtained suggests that the latter method has an advantage.     

Orbit determination by four observations

An elliptic orbit is determined from two short-arc pairs of observations at different oppositions by the angular momentum integral. Other methods for initial orbit determination than the classical do exist.     

Analysis of the neighborhood of the 2: 1 resonance in the equal-mass three-body problem

We consider the trajectories in the neighborhood of a 2: 1 resonance (in periods of osculating motions of the outer and inner binaries) in the plane equal-mass three-body problem. We identified the zones of motions that are stable on limited time intervals. All of them correspond to the retrograde motions of the outer and inner subsystems. The prograde motions are unstable: the triple system breaks up into a final binary and an escaping component. In the barycentric nonrotating coordinate system, the trajectories occasionally form symmetric structures composed of several leaves. These structures persist for a long time, and, subsequently, the trajectories of the bodies fill compact regions in coordinate space.     

Effects of the physical libration of the moon on lunar orbiters

Lunar physical libration, which is true oscillation of lunar equator in the space, alters the lunar gravitational field in the space coordinate system and affects the orbiting motion of lunar orbiters (hereafter called as lunar satellites) correspondingly. The effect is very similar to that of the precession and nutation on the earth satellites, and a similar treatment can be used. The variations in the gravitational force and in the orbit perturbation solution are clearly given in this paper together with numerical illustrations.     

On critical values concerning the evolution of the long period families

In a previous paper, we proposed another special critical value concerning the evolution of the long period family around the equilateral equilibrium points, besides the two values given by Henrard. Are there any other special critical values? After studying the stability curves of the long period family carefully, we gave a negative answer. During the study, we found an interesting family of periodic orbits which we called the homo family. We studied the evolution of this family following the increase of μ. With these findings, we were able to explain the origin of the four branches of periodic families emanating from L4 and the stability results of the equilateral equilibrium points.     

Periodic solutions near a central libration point in the problem of the motion of a star inside an elliptical galaxy

We consider the three-dimensional problem of the motion of a star inside an inhomogeneous rotating elliptical galaxy with a homothetic density distribution. We construct and analyze the periodic solutions near a central libration point by using Lyapunov’s method.