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.     

Regularization of Equations of Motion of Celestial Bodies. 2. Correction of the KS-Coordinate System in Numerical Integration

The idea of using various L-matrices in numerical integration of the regular equations, which describe the motion of small bodies of the Solar System, is developed. The problem of the optimal position of the radius vector and velocity at numerical integration in the KS-coordinate system is posed. The solution of this problem, which reduces the number of calculations of the vector of perturbing accelerations, is given. The transformation providing this optimal solution is suggested, and the results of numerical integration are given.     

The uniform,regular differential equations of the KS transformed perturbed two-body problem

The Newtonian differential equations of motion for the two-body problem can be transformed into four, linear, harmonic oscillator equations by simultaneously applying the regularizing time transformation dt/ds=r and the Kustaanheimo-Stiefel (KS) coordinate transformation. The time transformation changes the independent variable from time to a new variables, and the KS transformation transforms the position and velocity vectors from Cartesian space into a four-dimensional space. This paper presents the derivation of uniform, regular equations for the perturbed twobody problem in the four-dimensional space. The variation of parameters technique is used to develop expressions for the derivatives of ten elements (which are constants in the unperturbed motion) for the general case that includes both perturbations which can arise from a potential and perturbations which cannot be derived from a potential. These element differential equations are slightly modified by introducing two additional elements for the time to further improve long term stability of numerical integration.Originally presented at the AAS/AIAA Astrodynamics Specialists Conference, Vail, Colorado, July 1973     

数值历表对动力系统参数的偏导数

本文推导了天体运动方程的数值解对积分初始条件、天体质量等动力系统参数的偏导数所满足的微分方程和初始条件。     

Computation of Partial Derivatives of Perturbed Motion Using the Arcs of Osculating and Superosculating Orbits

The methods for analytical determination of partial derivatives of the current parameters of motion with respect to their initial values are described. The methods take into account principal perturbations and are based on the use of the osculating and superosculating intermediate orbits constructed earlier by the author. These orbits ensure the first-, second-, and third-order contact to the real trajectory at the initial time. The solution for parameters of the intermediate motion and partial derivatives of these parameters is given in a universal closed form. The partial derivatives on long time intervals are computed using a step-by-step procedure combined with the Encke method of special perturbations, in which the intermediate orbits are used as the reference. The numerical results show that the new approach can be efficiently used for solving the problem of differential correction of orbits of asteroids and comets on the basis of observational data.     

The orbital evolution of NEA 30825 1900 TG1

The orbital evolution of the near-Earth asteroid (NEA) 30825 1990 TG1 has been studied by numerical integration of the equations of its motion over the 100 000-year time interval with allowance for perturbations from eight major planets and Pluto, and the variations in its osculating orbit over this time interval were determined. The numerical integrations were performed using two methods: the Bulirsch-Stoer method and the Everhart method. The comparative analysis of the two resulting orbital evolutions of motion is presented for the time interval examined. The evolution of the asteroid motion is qualitatively the same for both variants, but the rate of evolution of the orbital elements is different. Our research confirms the known fact that the application of different integrators to the study of the long-term evolution of the NEA orbit may lead to different evolution tracks.     

Numerical integration of the variational equations of satellite orbits

A recurrent power series (RPS) method is constructed for the numerical integration of the equations of motion together with the variational equations of N point masses orbiting around an oblate spheroid. By the term “variational equations” we mean the equations of the partial derivatives of the bodies’ position and velocity components with respect to the initial conditions, the relative masses and the spheroid's oblateness coefficients J₂ and J₄. The construction of recursive relations for the partial derivatives involved in the variational equations is based on partial differentiation of the corresponding recursive relations for the integration of the equations of motion. Since the number of the auxiliary variables needed for this complex system becomes tremendously large when N>1, special care must be taken during computer implementation, so as to minimize the amount of computer memory needed as well as the cost in CPU time. The RPS method constructed in this way is tested for N=1,…,4 using real initial conditions of the Saturnian satellite system. For various sets of satellites, we monitor the behaviour of all the corresponding partial derivatives. The results show a prominent difference in the behaviour of the partial derivatives between resonant and non-resonant orbital systems.     

Nonlinear analytic solution for perturbed relative motion using differential equinoctial elements

This paper develops a nonlinear analytic solution for satellite relative motion in J₂-perturbed elliptic orbits by using the geometric method that can avoid directly solving the complex differential equations. The differential equinoctial elements (DEEs) are used to remove any singularities for zero-eccentricity or zero-inclination orbits. Based on the relationship between the relative states and the DEEs, state transition tensors (STTs) for transforming the osculating DEEs and propagating the mean DEEs have been derived. The formulation of these STTs has been split into a set of vector and matrix operations, which avoids directly expanding the complex second-order terms, and thus, the obtained STTs could be easy-to-understand and easy-to-code. Numerical results show that the proposed nonlinear solution is valid for zero-eccentricity and zero-inclination reference orbit and is more accurate than the previous linear or nonlinear methods for the long-term prediction of satellite relative motion.     

Studies in the application of recurrence relations to special perturbation methods

The procedure of numerical integration of the elliptic three dimensional restricted threebody problem by the use of recurrence relations to evaluate successively higher derivatives of the relative position and velocity vectors of the bodies and of the variational matrix is investigated. A set of recurrence relations is developed which involves the introduction of fewer auxiliary variables than in previous papers of this series, while the recurrence relations themselves are of a simpler form than those in other treatments involving the same number of such auxiliary variables. A technique for automatic adjustment of the integration step-length at each point in the orbit, such that the local truncation error remains close to, but always less than, some specified amount, is incorporated. This technique involves the comparison of pre-integration values with those obtained after consecutive forward and reverse integration steps, and has decided advantages over step-adjustment methods currently in use.Both these modifications to previous techniques are shown, by presentation of sample computational results, to represent considerable savings in machine time for a given calculation and desired accuracy; these savings are generally around a factor of two and become greater as the desired accuracy in the computations increases.     

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.     

A note on relative motion in the general three-body problem

It is shown that the equations of the general three-body problem take on a very symmetric form when one considers only their relative positions, rather than position vectors relative to some given coordinate system. From these equations one quickly surmises some well known classical properties of the three-body problem such as the first integrals and the equilateral triangle solutions. Some new Lagrangians with relative coordinates are also obtained. Numerical integration of the new equations of motion is about 10 percent faster than with barycentric or heliocentric coordinates.     

Poisson equations of rotational motion for a rigid triaxial body with application to a tumbling artificial satellite

Differential equations are derived for studying the effects of either conservative or nonconservative torques on the attitude motion of a tumbling triaxial rigid satellite. These equations, which are analogous to the Lagrange planetary equations for osculating elements, are then used to study the attitude motions of a rapidly spinning, triaxial, rigid satellite about its center of mass, which, in turn, is constrained to move in an elliptic orbit about an attracting point mass. The only torques considered are the gravity-gradient torques associated with an inverse-square field. The effects of oblateness of the central body on the orbit are included, in that, the apsidal line of the orbit is permitted to rotate at a constant rate while the orbital plane is permitted to precess (either posigrade or retrograde) at a constant rate with constant inclination.A method of averaging is used to obtain an intermediate set of averaged differential equations for the nonresonant, secular behavior of the osculating elements which describe the complete rotational motions of the body about its center of mass. The averaged differential equations are then integrated to obtain long-term secular solutions for the osculating elements. These solutions may be used to predict both the orientation of the body with respect to a nonrotating coordinate system and the motion of the rotational angular momentum about the center of mass. The complete development is valid to first order in (n/w ₀)², wheren is the satellite's orbital mean motion andw ₀ its initial rotational angular speed.     

On error bounds and initialization in satellite orbit theories

The order of magnitude of the error is investigated for a first-order von Zeipel theory of satellite orbits in an axisymmetric force field, i.e., first-order long period and short-period effects are included along with second order secular rates. The treatment is valid for zero eccentricity and/or inclination. In the case where initial position and velocity vectors are known, the in-track position error over time intervals of order 1/J ₂ is kept at 0(J ₂ ²), like the other position errors and velocity errors, by calibration of the mean motion with the aid of the energy integral. The results are specifically applicable to accuracy comparisons of the Brouwer orbit prediction method with numerical integration. A modified calibration is presented for the general asymmetric force field which includes tesseral harmonics.     

Semi-analytic solution of a two-body problem with drag

An approximate semi-analytic solution of a two-body problem with drag is presented. The solution describesnon-lifting orbital motion in a central, inverse-square gravitational field. Drag deceleration is a non-linear function of velocity relative to a rotating atmosphere due to dynamic pressure and velocity-dependent drag coefficient. Neglected are aerodynamic lift, gravitational perturbations of the inverse-square field, and kinematic accelerations due to coordinate frame rotation at earth angular rate. With these simplifications, it is shown that (i) orbital motion occurs in an earth-fixed invariable plane defined by the radius and relative velocity vectors, and (ii) the simplified equations of motion are autonomous and independent of central angle measured in the invariable plane. Consequently, reduction of the differential equations from sixth to second-order is possible. Solutions for the radial and circumferential components of relative velocity are reduced to quadratures with respect to radial distance. Since the independent variable is radial distance, the solutions are singular at zero radial velocity (e. g., for circular orbits). General atmospheric density and drag coefficient models may be used to evaluate the velocity quadratures. The central angle and time variables are recovered from two additional quadratures involving the velocity quadratures. Theoretical results are compared with numerical simulation results.Presently affiliated with AVCO Systems Division, Wilmington, MA 01887, U.S.A.     

A Simplified Kinetic Element Formulation for the Rotation of a Perturbed Mass-Asymmetric Rigid Body

Euler's equations, describing the rotation of an arbitrarily torqued mass asymmetric rigid body, are scaled using linear transformations that lead to a simplified set of first order ordinary differential equations without the explicit appearance of the principal moments of inertia. These scaled differential equations provide trivial access to an analytical solution and two constants of integration for the case of torque-free motion. Two additional representations for the third constant of integration are chosen to complete two new kinetic element sets that describe an osculating solution using the variation of parameters. The elements' physical representations are amplitudes and either angular displacement or initial time constant in the torque-free solution. These new kinetic elements lead to a considerably simplified variation of parameters solution to Euler's equations. The resulting variational equations are quite compact. To investigate error propagation behaviour of these new variational formulations in computer simulations, they are compared to the unmodified equations without kinematic coupling but under the influence of simulated gravity-gradient torques.     

Numerical Determination of Homoclinic and Heteroclinic Orbits at Collinear Equilibria in the Restricted Three-Body Problem with Oblateness

Asymptotic motion to collinear equilibrium points of the restricted three-body problem with oblateness is considered. In particular, homoclinic and heteroclinic solutions to these points are computed. These solutions depart asymptotically from an equilibrium point and arrive asymptotically at the same or another equilibrium point and are important reference solutions. To compute an asymptotic orbit, we use a fourth order local analysis, numerical integration and standard differential corrections.     

Generalized Lagrangian orbital elements for central force problems

We consider the definitions and resulting equations of motion for the Lagrangian orbital elements associated with conventional osculating orbit theory for central forces. The analysis indicates that the definitions themselves lead to difficulties which are most apparent in the circular limit. An alternate set of defining relations is presented which eliminates the problems associated with osculating elements. The remaining equation of motion based on these new definitions is reduced to quadratures. This solution completely expresses the orbits for central force problems with no restriction on the eccentricity. Both bounded and open orbits are considered. A generalized Laplace-Runge-Lenz vector is developed and a number of example solutions are presented.     

Almost rectilinear halo orbits

Numerical studies over the entire range of mass-ratios in the circular restricted 3-body problem have revealed the existence of families of three-dimensional halo periodic orbits emanating from the general vicinity of any of the 3 collinear Lagrangian libration points. Following a family towards the nearer primary leads, in 2 different cases, to thin, almost rectilinear, orbits aligned essentially perpendicular to the plane of motion of the primaries. (i) If the nearer primary is much more massive than the further, these thin L₃-family halo orbits are analyzed by looking at the in-plane components of the small osculating angular momentum relative to the larger primary and at the small in-plane components of the osculating Laplace eccentricity vector. The analysis is carried either to 1st or 2nd order in these 4 small quantities, and the resulting orbits and their stability are compared with those obtained by a regularized numerical integration. (ii) If the nearer primary is much less massive than the further, the thin L₁-family and L₂-family halo orbits are analyzed to 1st order in these same 4 small quantities with an independent variable related to the one-dimensional approximate motion. The resulting orbits and their stability are again compared with those obtained by numerical integration.     

The gravity-perturbed Lambert problem: A KS variation of parameters approach

A new formulation is presented for the perturbed Lambert problem. The formulation employs the variation-of-parameters method in the KS transformed state space to determine perturbations of a Keplerian Lambert solution. The approach is universal (in that its validity is not restricted to a particular energy domain). For the case of the second zonal harmonic (oblateness) perturbation, first order perturbations are carried out entirely analytically; non-iterative corrections are determined through solution of a pair of algebraic equations. For more general perturbations, numerical quadratures are required.