Notes on Hori Method for Non-Canonical Systems

In this paper the new approach for the integration theory of the canonical version of Hori method recently proposed is extended to the non-canonical one. It will be shown that the non-homogeneous ordinary differential equation with an auxiliary parameter t* associated with the mth order equation of the algorithm can also be replaced by a non-homogeneous partial differential equation in the time t. Using a generalized canonical approach, the general algorithm proposed by Sessin is then revised; as well as the Lagrange variational equations for the non-canonical version of Hori method. A simplified algorithm derived from Sessin's algorithm is presented for non-linear oscillations problem.     

Canonical Elements for Öpik Theory

The purpose of this paper is to find a set of canonical elements to use within the framework of Öpik theory of close encounters of a small body with a planet (Öpik, Interplanetary Encounters, 1976). Since the small body travels along a planetocentric hyperbola during the close approach and Öpik formulas are valid, without approximations, only at collision, we derive a set of canonical elements for hyperbolic collision orbits (eccentricity e → 1⁺, semi-major axis a fixed) and then we introduce the unperturbed velocity of the small body and the distance covered along the asymptote as a new canonically conjugate pair of orbital elements. An interesting result would be to get a canonical set containing the coordinates in the Target Plane (TP), useful for the analysis of the future encounters: in the last part we prove that this is not possible.     

Canonical theory of perturbations using eccentric anomaly as independent variable

It is shown in this paper how to build a canonical transformation of variables, so that the eccentric anomaly becomes the new independent variable. In the case of eccentric elliptical orbits it changes the equations of motion so, that they can be integrated analytically to any order of approximation comparatively easy.     

Perturbed Gylden Systems and Time-Dependent Delaunay-Like Transformations

We study the possibilities and limitations of the application of generalized Delaunay-like transformations (in the 6-dimensional phase space) and TR-like mappings (in the 8-dimensional, extended phase space) to perturbed two-body problems with a time-varying Keplerian parameter μ(t), that is, to Gylden-type systems. For the sake of theoretical completeness, both negative- and positive-energy motion (with nonstationary coupling parameter) are, in principle, considered. Our developments are intended to introduce canonical variables parallelling the classical ones of Delaunay and the Delaunay-Similar variables of Scheifele. This revised version was published online in August 2006 with corrections to the Cover Date.     

The General Method for Fixing the Gauges of Relativistic Astronomical Reference Systems

This article applies a new scheme of the first post-Newtonian theory (Damour et al., 1991–1994) to the problem of gauge in relativistic reference systems. Choosing and fixing gauge are necessary when the precision of time measurement and application needs to reach the 2PN level (10⁻¹⁶ or better). We present a general method for fixing the gauges of both the global and local coordinate systems, and for determining the expressions of gravitational potentials and coordinate transformations. The results relevant are consistent with the newest IAU resolutions, therefore they can be applied to astronomical practice.     

A Method for locating Singularities in the Complex Time Domain for Integrable Dynamical Systems

A simple method for the determination of the position of singularities in the complex time domain for dynamical systems which are described by ordinary differential equations is presented. The method is designed for integrable separable systems whose solutions are not expressible in closed form. A direct consequence of this method is that it ‘closes’ the phase space. Simple physical meaning is given to the singularity position. This revised version was published online in July 2006 with corrections to the Cover Date.     

Canonical planetary perturbation equations for velocity-dependent forces,and the Lense-Thirring precession

A form of planetary perturbation theory based on canonical equations of motion, rather than on the use of osculating orbital elements, is developed and applied to several problems of interest. It is proved that, with appropriately selected initial conditions on the orbital elements, the two forms of perturbation theory give rise to identical predictions for the observable coordinates and velocities, while the orbital elements themselves may be strikingly different. Differences between the canonical form of perturbation theory and the classical Lagrange planetary perturbation equations are discussed. The canonical form of perturbation theory in some cases has advantages when the perturbing forces are velocity-dependent, but the two forms of perturbation theory are equivalent if the perturbing forces are dependent only on position and not on velocity. The canonical form of the planetary perturbation equations are derived and applied to the Lense Thirring precession of a test body in a Keplerian orbit around a rotating mass source.     

Comparison of Numerical Methods for the Integration of Natural Satellite Systems

We present a new implementation of the recurrent power series (RPS) method which we have developed for the integration of the system of N satellites orbiting a point-mass planet. This implementation is proved to be more efficient than previously developed implementations of the same method. Furthermore, its comparison with two of the most popular numerical integration methods: the 10th-order Gauss–Jackson backward difference method and the Runge–Kutta–NystrRKN12(10)17M shows that the RPS method is more than one order of magnitude better in accuracy than the other two. Various test problems with one up to four satellites are used, with initial conditions obtained from ephemerides of the saturnian satellite system. For each of the three methods we find the values of the user-specified parameters (such as the method's step-size (h or tolerance (TOL)) that minimize the global error in the satellites' coordinates while keeping the computer time within reasonable limits. While the optimal values of the step-sizes for the methods GJ and RKN are all very small (less than T/100, the ones that are suitable for the RPS method are within the range: T/13<h<T/6 (T being the period of the innermost satellite of the problem). Comparing the results obtained by the three methods for these step-sizes and for the various test problems we observe the superiority of the RPS method over GJ in terms of accuracy and over RKN both in accuracy and in speed. This revised version was published online in July 2006 with corrections to the Cover Date.     

Processing Symplectic Methods for Near-Integrable Hamiltonian Systems

Processing techniques are used to approximate the exact flow of near-integrable Hamiltonian systems depending on a small perturbation parameter. We study the reduction of the number of conditions for the kernel for this type of Hamiltonians and we build third, fourth and fifth order methods which are shown to be more efficient than previous algorithms for the same class of problems. This revised version was published online in July 2006 with corrections to the Cover Date.     

The Extended Delaunay Method Applied to First Order Resonance

This paper is concerned with the extended Delaunay method as well as the method of integration of the equations, applied to first order resonance. The equations of the transformation of the extended Delaunay method are analyzed in the (p + 1)/p type resonance in order to build formal, analytical solutions for the resonant problem with more than one degree of freedom. With this it is possible to gain a better insight into the method, opening the possibility for more generalized applications. A first order resonance in the first approximation is carried out, giving a better comprehension of the method, including showing how to eliminate the ‘Poincaré singularity’ in the higher orders. This revised version was published online in July 2006 with corrections to the Cover Date.     

Efficient Method for Computing with Certainty Periodic Orbits on a Surface of Section

We present an improved method for locating periodic orbits of a dynamical system of arbitrary dimension. The method first employs the characteristic bisection method (CBM) to roughly locate a periodic orbit, followed by the quadratically convergent Newton method to rapidly refine its position. The method is applied to the physically interesting example of the two degrees of freedom photogravitational problem, and shown to surpass the CBM algorithm and Newton's method alone.     

Symplectic Tangent map for Planetary Motions

Equations are presented for the computation of tangent maps for use in nearly Keplerian motion, approximated by use of a symplectic leapfrog map. The resulting algorithms constitute more accurate and efficient methods to obtain the Liapunov exponents and the state transition matrix, and can be used to study chaos in planetary motions, as well as in orbit determination procedures from observations. Applications include planetary systems, satellite motions and hierarchical, nearly Keplerian systems in general. This revised version was published online in July 2006 with corrections to the Cover Date.     

Optimal Compensation and Implementation for Adaptive Optics Systems

This paper develops a compensation algorithm based on Linear–Quadratic–Gaussian (LQG) control system design whose parameters are determined (in part) by a model of the atmosphere. The model for the atmosphere is based on the open-loop statistics of the atmosphere as observed by the wavefront sensor, and is identified from these using an auto-regressive, moving average (ARMA) model. The (LQG) control design is compared with an existing compensation algorithm for a simulation developed at ESO that represents the operation of MACAO adaptive optics system on the 8.2 m telescopes at Paranal, Chile. This revised version was published online in July 2006 with corrections to the Cover Date.     

An Implementation of the Logarithmic Hamiltonian Method for Artificial Satellite Orbit Determination

We discuss the use of a recently discovered exact two-body leapfrog for accurate symplectic integration of perturbed two-body motion and for the computation of the state-transition matrix. We pay special attention to artificial satellite orbit determination and describe in detail the evaluation of the perturbing acceleration. Inclusion of air drag and other non-canonical forces are also discussed. The main advantage of this new formulation is conceptual simplicity, for easy programming and high accuracy for orbits with large eccentricity. The method has been evaluated in real artificial satellite orbit determinations.This revised version was published online in October 2005 with corrections to the Cover Date.     

天文动力学方程数值积分中的一种有效变步法

利用积分曲线的曲率控制步长的技巧，使天文动力学方程数值解法的精度和速度有较大提高，这种方法适用于天体精密定轨以及一些精度要求高的常微分方程初值问题的数值积分。     

薄施密特校正板的加工方法研究

针对薄板玻璃的刚度差，在研磨中容易产生变形这一特点，提出了一种新的加工方法。以加工薄施密特校正板（厚径比小于1／40）为例，用传统的上盘粘接固定工件进行加工，工件在下盘前的面形精度较好，但下盘后产生的变形使其面形精度下降，不能满足要求。采用真空模具法，吸附工件从而使工件固定产生预期的变形，面形精度易于控制，使加工非球面转变为加工平面，降低了加工难度，消除了工件在粘接过程中温度所引起的变形，降低了加工时的不均匀压应力，从而减少了变形，使得加工后的面形与加工过程中的面形接近，最终保证了工件的面形精度。     

The influence of shapes of the bodies on the dynamics of planetary and star systems

The dynamics of planetary and star systems including perturbing forces due to the flattering and quadrupole distortion of the bodies is studied. The analytical model describing the perturbations which influence on the orbital motion of extrasolar planetary systems is presented. The calculations of the secular evolution of the mean orbital elements have shown that the effects related to the shape of the body are more important than the ones due to the quadrupole distortion.     

A New Method to Determine the Derivatives of the Laplace Coefficients

The determination of the secular variations of the orbital elements of objects in N-body systems is based on the literal development of the perturbing function. The development makes use of the Laplace coefficients and their derivatives. In this paper a new method is described for the analytical computation of the derivatives of the Laplace coefficients. It is an explicit formula in the sense that it only contains the Laplace coefficients and the parameter on which the Laplace coefficients depend. The advantage of this method is that it is unnecessary to calculate all the derivatives up to the desired order. It is enough to calculate the Laplace coefficients. Easy coding is a further benefit of the method and it provides more accurate numerical results. The paper describes in detail the application of the method through an example and gives comparison with former methods.This revised version was published online in October 2005 with corrections to the Cover Date.     

基于差拍的频标比对系统的性能比较

针对差拍计数器测量系统和差拍数字频标比对系统各自的频率测量特性、精度表现等方面的差异,建立了基于差拍技术的频标比对平台以便进行测试比较。分析了这两套系统的技术要点及实现流程,详细阐述了差拍数字频标比对系统性能较高的原因,并对这两套系统的综合性能进行了测试和比较分析。差拍计数器测量系统所得测量值的稳定度（A11an方差,τ=1S）为4．7×10^-12,差拍数字频标比对系统所得测量值的稳定度为3×10^-14,后者比前者提高了近两个数量级。     

Frozen orbits for satellites close to an Earth-like planet

We say that a planet is Earth-like if the coefficient of the second order zonal harmonic dominates all other coefficients in the gravity field. This paper concerns the zonal problem for satellites around an Earth-like planet, all other perturbations excluded. The potential contains all zonal coefficientsJ ₂ throughJ ₉. The model problem is averaged over the mean anomaly by a Lie transformation to the second order; we produce the resulting Hamiltonian as a Fourier series in the argument of perigee whose coefficients are algebraic functions of the eccentricity — not truncated power series. We then proceed to a global exploration of the equilibria in the averaged problem. These singularities which aerospace engineers know by the name of frozen orbits are located by solving the equilibria equations in two ways, (1) analytically in the neighborhood of either the zero eccentricity or the critical inclination, and (2) numerically by a Newton-Raphson iteration applied to an approximate position read from the color map of the phase flow. The analytical solutions we supply in full to assist space engineers in designing survey missions. We pay special attention to the manner in which additional zonal coefficients affect the evolution of bifurcations we had traced earlier in the main problem (J ₂ only). In particular, we examine the manner in which the odd zonalJ ₃ breaks the discrete symmetry inherent to the even zonal problem. In the even case, we find that Vinti's problem (J ₄+J ₂ ² =0) presents a degeneracy in the form of non-isolated equilibria; we surmise that the degeneracy is a reflection of the fact that Vinti's problem is separable. By numerical continuation we have discovered three families of frozen orbits in the full zonal problem under consideration; (1) a family of stable equilibria starting from the equatorial plane and tending to the critical inclination; (2) an unstable family arising from the bifurcation at the critical inclination; (3) a stable family also arising from that bifurcation and terminating with a polar orbit. Except in the neighborhood of the critical inclination, orbits in the stable families have very small eccentricities, and are thus well suited for survey missions.