Regularization of the Limit Problem of Satellite Librations in a Keplerian Orbit Taking Light Pressure into Account

The planar librations of a satellite whose center of mass moves along an elliptic orbit are considered. It is assumed that not only the gravitational moment but also the forces of light pressure act upon the satellite. Account is taken of the fact that the right-hand sides of the differential equations are nonanalytic functions of the phase variables. When e 1, e being the orbit's eccentricity, the deformations of solutions are considered for the case of a satellite moving along a highly elongated orbit. Such transformation of the initial system of differential equations is carried out so that the new system becomes regular up to the value e = 1. A limit problem corresponding to the case e = 1 is considered. When the azimuth angle of the light source coincides with the direction to the pericenter, the dynamical system is reversible. In this case, the known families of the periodic solutions to the problem can be continued up to the limit case.     

Planar oscillations of a vibrating dumbbell-like body in a central field of forces

The problem of planar motions of a dumbbell-like body of variable length in a central field of Newtonian attraction is considered both in the exact formulation and in satellite approximation. In the satellite approximation the true anomaly of the center of mass is used as an independent variable, which has allowed us to represent the equation of planar oscillations of the dumbbell in the form similar to the Beletskii equation. Some exact solutions to the inverse problem are given both in the complete formulation and in satellite approximation. Under an assumption of small orbit eccentricity and amplitude of the dumbbell vibrations the conditions of existence are found for families of almost periodic motions and splitting separatrices. The phenomena of alternation of regular and chaotic motions are established numerically, as well as chaos suppression with increasing frequency of vibrations. Using the method of averaging the stabilization of tangent equilibria is proved to be impossible.     

A study of evolution of spin-up mode of a satellite in the plane of its orbit

We investigate the mode of spinning up a low-orbit satellite in the plane of its orbit. In this mode the satellite rotates around its principal central axis of the minimum moment of inertia which executes small oscillations with respect to the normal to the orbit plane; the angular velocity of the rotation around this axis several times exceeds the mean orbital motion. Gravitational and restoring aerodynamic moments are taken into account in the satellite’s equations of motion. A small parameter characterizing deviation of the satellite from a dynamically symmetric shape is introduced into the equations. A two-dimensional integral surface of the equations of motion, describing quasi-steady-state rotations of the satellite close to cylindrical precession of the corresponding symmetrical satellite in a gravitational field, has been studied by the method of small parameter and numerically. Such quasi-steady-state rotations are suggested to be considered as unperturbed motions of the satellite in the spin-up mode. Investigation of the integral surface is reduced to numerical solution of a periodic boundary value problem of a certain auxiliary system of differential equations and to calculation of quasi-steady-state rotations by the two-cycle method. A possibility is demonstrated to construct quasi-steady rotations by way of minimization of a special quadratic functional.     

Families of periodic solutions to Hamiltonian systems: Nonsymmetrical periodic solutions for a planar restricted three-body problem

A monodromy matrix calculated at a single arbitrary point of the periodic solution to a Hamiltonian system allows one to obtain both the direction of continuation for the family of solutions of the first (in Poincarés sense) kind and the multiplicity and direction of branching for periodic solutions of the second kind. In case of resonances 1 : 1 and 1 : 2 one needs to take into account the structure of elementary divisors of the monodromy matrix. Using the planar circular restricted three-body problem as an example, the infiniteness of the process of branching for a nonintegrable system and its finiteness for an integrable system are demonstrated. It is proved that periodic solutions of both first and second kinds which are obtained by continuation of symmetric periodic solutions of a restricted problem are also symmetric. The only exception is the case of resonance 1 : 1 and two second-order cells of the monodromy matrix in the Jordanian form. In this case, all periodic solutions of the second kind turned out to be nonsymmetrical. Examples of the families of nonsymmetrical periodic solutions are given.__________Translated from Kosmicheskie Issledovaniya, Vol. 43, No. 2, 2005, pp. 88–110.Original Russian Text Copyright © 2005 by Kreisman.     

Isolated generating periodic solutions to the Beletsky equation

We study degeneracies in families of periodic solutions to the Beletsky equation which correspond to intersections of three manifolds of these solutions: the symmetric, the asymmetric, and the manifold belonging to one of the integrable cases, i.e., e = 0 or μ = 0. We obtain equations for these isolated solutions, which allow us to compute them with an arbitrary precision. It is shown that additional degeneracies take place at some of these solutions. The method we use is applicable to a wide class of nonlinear ordinary differential equations (ODEs) depending on parameters.     

Gravitation Maneuver Using the Families of Super-Unstable Orbits around the Libration Points

The families of periodic solutions to an autonomous Hamiltonian system in that part where the solutions are unstable have their specific field of influence. Under strong instability, the orbits that have fallen in such a field of influence are drawn into the family as in a whirlpool and then are thrown away from it. In the particular case of the restricted three-body problem, the orbits around the libration points L ₁ and L ₂ are super-unstable and the Keplerian elements in motion over these orbits change drastically. When the orbits fall into such a domain, for some time the motion is close to the motion around the libration point, and after being thrown out of this domain, the Keplerian orbital elements also change essentially.     

Periodic motions of a nearly dynamically symmetric satellite in the neighborhood of hyperboloidal precession

The motion of a satellite close to a dynamically symmetric solid body in a Newtonian gravitational field over a circular orbit is studied. The system of differential equations describing the body’s motion is close to a system with cyclic coordinate. New classes of periodic motions are constructed in the neighborhood of a known partial solution to an unperturbed problem, hyperboloidal precession of a dynamically symmetric satellite. In the resonance case, when the ratio of one frequency of small oscillations of a reduced system with two degrees of freedom in the neighborhood of a stable equilibrium position to the frequency of cyclic coordinate variation is close to an integer number, there exist one or three families of periodic motions that are analytic in terms of fractional powers of a small parameter. A study of stability of these motions was performed with the help of KAM (Kolmogorov-Arnold-Moser) theoty. Faling the described resonance there exists a unique family of periodic motions that is analytic in terms of integer powers of a small parameter. The check-up of stability of these motrons was carried out. We distinguished the cases of parametric resonance, resonances of the third and fourth orders, and a non-resonant case. In the resonance cases our study relies on well-known results on stability of Hamiltonian systems during resonances [1]. In the non-resonant case we use the KAM theory [2].     

Families of periodic solutions in three-dimensional restricted three-body problem

Planar orbits of three-dimensional restricted circular three-body problem are considered as a special case of three-dimensional orbits, and the second-order monodromy matrices M (in coordinate z and velocity v _z ) are calculated for them. Semi-trace s of matrix M determines vertical stability of an orbit. If |s| ≤ 1, then transformation of the subspace (z, v _z ) in the neighborhood of solution for the period is reduced to deformation and a rotation through angle φ, cosφ = s. If the angle ? can be rationally expressed through 2π,φ = 2π·p/q, where p and q are integer, then a planar orbit generates the families of three-dimensional periodic solutions that have a period larger by a factor of q (second kind Poincareé periodic solutions). Directions of continuation in the subspace (z, v _z ) are determined by matrix M. If |s| < 1, we have two new families, while only one exists at resonances 1: 1 (s = 1) and 2: 1 (s = ?1). In the course of motion along the family of three-dimensional periodic solutions, a transition is possible from one family of planar solutions to another one, sometimes previously unknown family of planar solutions.     

One method of gravitational attitude control of a rotating satellite

The mode of spinning up a low-orbit satellite in the plane of its orbit is studied. In this mode, the satellite rotates around its longitudinal axis (principal central axis of the minimum moment of inertia), which executes small oscillations with respect to the normal to the orbit plane; the angular velocity of the rotation around the longitudinal axis is several tenths of a degree per second. Gravitational and restoring aerodynamic moments were taken into account in the equations of satellite’s motion, as well as a dissipative moment from eddy currents induced in the shell of the satellite by the Earth’s magnetic field. A small parameter characterizing deviation of the satellite from a dynamically symmetric shape and nongravitational external moments are introduced into the equations. A two-dimensional integral surface of the equations of motion, describing quasistationary rotations of the satellite close to cylindrical precession of the corresponding symmetrical satellite in a gravitational field, has been studied by the method of small parameter and numerically. We propose to consider such quasistationary rotations as unperturbed motions of the satellite in the spin-up mode.     

基于T-H方程的卫星轨迹模型参考输出跟踪控制方法

当目标卫星沿椭圆轨道运行时,描述追踪星与目标星相对运动的线性化方程为T-H方程。将描述航天器相对运动的T-H方程变换为周期系统的状态空间形式,并给出卫星轨迹跟踪控制问题的数学描述。基于周期系统的参量Lyapunov方法和模型参考跟踪控制理论,提出了卫星轨迹跟踪控制器的设计方法。利用该方法设计了带有收敛速率保障的反馈镇定控制器和具有自由参数的前馈控制器。对追踪星相对目标星悬停任务进行了数值仿真,仿真结果表明提出的控制方案是有效的。     

Periodic motions of a satellite-gyrostat relative to its center of mass under the action of gravitational torque

We investigated periodic motions of the axis of symmetry of a model satellite of the Earth, which are similar to the motions of the longitudinal axes of the Mir orbital station in 1999–2001 and the Foton-M3 satellite in 2007. The motions of these spacecraft represented weakly disturbed regular Euler precession with the angular momentum vector of motion relative to the center of mass close to the orbital plane. The direction of this vector during the motion was not practically changed. The model satellite represents an axisymmetric gyrostat with gyrostatic moment directed along the axis of symmetry. The satellite moves in a circular orbit and undergoes the action of the gravitational torque. The motion of the axis of symmetry of this satellite relative to the absolute space is described by fourth-order differential equations with periodic coefficients. The periodic solutions to this system with special symmetry properties are constructed using analytical and numerical methods.     

Evolution of Rotational Motion of a Symmetrical Satellite with Viscoelastic Rods

We study the motion of a symmetrical satellite with a pair of flexible viscoelastic rods in a central Newtonian gravitational field. A restricted problem formulation is considered, when the satellite's center of mass moves along a fixed circular orbit. A small parameter is introduced which is inversely proportional to the stiffness of flexible elements. Another small parameter is equal to the ratio of the squared orbital angular velocity and the squared magnitude of the initial angular velocity of the satellite. In order to describe the satellite rotational motion relative to the center of mass, we use the canonical Andoyer variables. In the undisturbed formulation of the problem, i.e., at = 0 and = 0, these variables are the action–angle variables. Equations describing the evolution of motion are derived by an asymptotic method which combines the method of separating motions for systems with an infinite number of degrees of freedom and the Krylov–Bogolyubov method for systems with fast and slow variables. The manifolds of stationary motions are found, and their stability is investigated on the basis of equations in variations. Phase portraits are constructed which describe the rotational motion of a satellite at the stage of slow dissipative evolution.     

Hori method for generalized canonical systems

In this paper, some special features on the canonical version of Hori method, when it is applied to generalized canonical systems (systems of differential equations described by a Hamiltonian function linear in the momenta), are presented. Two different procedures, based on a new approach for the integration theory recently presented for the canonical version, are proposed for determining the new Hamiltonian and the generating function for systems whose differential equations for the coordinates describe a periodic system with one fast phase. These procedures are equivalent and they are directly related to the canonical transformations defined by the general solution of the integrable kernel of the Hamiltonian. They provide the same near-identity transformation for the coordinates obtained through the non-canonical version of Hori method. It is also shown that these procedures are connected to the classic averaging principle through a canonical transformation. As examples, asymptotic solutions of a non-linear oscillations problem and of the elliptic perturbed problem are discussed.     

Navigation Solution to Maneuver a Spacecraft Relative to a Sphere Centered on a Cooperative Satellite

A differential correction algorithm is presented to deliver an impulsive maneuver to a satellite to place it within a sphere, with a user defined radius, centered around a non-maneuvering satellite within a constrained time. The differential correction algorithm develops and utilizes the State Transition Matrix along with the Equations of Motion and multiple satellite?s state information to determine the optimum trajectory to achieve the desired results. The results from the differential correction algorithm are very accurate for prograde orbits, as presented. The results allow for orbit design trade-offs, including satellites? initial inclinations, semi-major axes, as well as the ballistic coefficients. The results also provide an empirical method to determine the optimum ΔV$\mathrm{\Delta }V$ solution for the provided problem. Understanding that the minimum fuel solution lies with a semi-major axis ratio of 1, a very accurate empirical approximation is presented for semi-major axis ratio values less than and greater than 1. This work ultimately provides the generalized framework for applying the algorithm to a unique user defined maneuvering spacecraft scenario.     

A series of small scientific satellite with flexible standard bus

Japan Aerospace Exploration Agency has a plan to develop the small satellite standard bus for various scientific missions and disaster monitoring missions. The satellite bus is a class of 250–400 kg mass with three-axis control capability of 0.02 accuracy. The science missions include X-ray astronomy missions, planetary telescope missions, and magnetosphere atmosphere missions. In order to adapt the wide range of mission requirements, the satellite bus has to be provided with flexibility. The concepts of modularization, reusability, and product line are applied to the standard bus system. This paper describes the characteristics of the small satellite standard bus which will be firstly launched in 2011.     

Determining Spacecraft Motion from Four Star Sensor Measurements

The BOKZ-M60 star sensor (a module that measures the coordinates of stars) has been designed for determining the parameters of the orientation of the intrinsic coordinate system relative to the inertial system from observations of stellar sky sections. The methods and results of processing of measurements by a set of four BOKZ-M60 sensors on the Resurs-P satellite no. 2 have been described. The time interval at which the satellite was in orbital orientation exceeds three orbital revolutions (19003 s). The joint processing of measurements by the four sensors conducted at the same time instants allowed the sensors to be associated with the universal coordinate system. With a root-mean-square error of less than 0.4′′ for each angle of rotation around its axes, this system is consistent with the model of the satellite’s rotational motion. The position of the universal system with respect to the instrumental coordinate system of the satellite was determined from the angular velocity measurements. Here, the root-mean-square errors for the values determined by the angles of rotation of the universal system around its axes were 0.044°, 0.051°, and 0.18°. The low-frequency (with frequencies less than 0.05 Hz) variations in the positions of intrinsic sensor coordinate systems relative to the universal system do not exceed 10′′. These are periodic variations with a fundamental frequency equal to the orbital frequency. The root-mean-square values of high-frequency components of these variations do not exceed 18′′.     

Periodic solutions to a Hamiltonian system

This paper presents a sufficiently complete and relatively elementary account (and justification) of a method of investigation of solutions to a Hamiltonian system with two degrees of freedom by calculating the families of periodic solutions. The goal of this paper is to present a theoretical basis for studies of the families of symmetrical periodic solutions to the restricted three-body problem by their intersections with the plane of symmetry.     

Plasma flow disturbances in the magnetospheric plasma sheet during substorm activations

We have considered variations in fields and particle fluxes in the near-Earth plasma sheet on the THEMIS-D satellite together with the auroral dynamics in the satellite-conjugate ionospheric part during two substorm activations on December 19, 2014 with K _p = 2. The satellite was at ~8.5R_E and MLT = 21.8 in the outer region of captured energetic particles with isotropic ion fluxes near the convection boundary of electrons with an energy of ~10 keV. During substorm activations, the satellite recorded energetic particle injections and magnetic field oscillations with a period of ~90 s. In the satellite-conjugate ionospheric part, the activations were preceded by wavelike disturbances of auroral brightness along the southern azimuthal arc. In the expansion phase of activations, large-scale vortex structures appeared in the structure of auroras. The sudden enhancements of auroral activity (brightening of arcs, auroral breakup, and appearance of NS forms) coincided with moments of local magnetic field dipolarization and an increase in the amplitude Pi2 of pulsations of the B_z component of the magnetic field on the satellite. Approximately 30–50 s before these moments, the magnetosphere was characterized by an increased rate of plasma flow in the radial direction, which initiated the formation of plasma vortices. The auroral activation delays relative to the times when plasma vortices appear in the magnetosphere decreased with decreasing latitude of the satellite projection. The plasma vortices in the magnetosphere are assumed to be responsible for the observed auroral vortex structures and the manifestation of the hybrid vortex instability (or shear flow ballooning instability) that develops in the equatorial magnetospheric plane in the presence of a shear plasma flow in the region of strong pressure gradients in the Earthward direction.     

Variation of the Satellite Orbit Altitude by the Light Pressure Force

The possibility of the uncontrolled increase of the altitude of an almost circular satellite orbit by the force of the light pressure is investigated. The satellite is equipped with a damper and a system of mirrors (solar batteries can serve as such a system). The flight of the satellite takes place in the mode of a single-axis gravitational orientation, the axis of its minimum principal central moment of inertia makes a small angle with the local vertical and the motion of the satellite around this axis constitutes forced oscillations under the impact of the moment of force of the light pressure. The form of the oscillations and the initial orbit are chosen so that the transverse component of the force of the light pressure acting upon the satellite be positive and the semimajor axis of the orbit would continuously increase. As this takes place, the orbit remains almost circular. We investigate the evolution of the orbit over an extended time interval by the method which employs separate integration of the equations of the orbital and rotational motions of the satellite. The method includes outer and inner cycles. The outer cycle involves the numerical integration of the averaged equations of motion of the satellite center of mass. The inner cycle serves to calculate the right-hand sides of these equations. It amounts to constructing an asymptotically stable periodic motion of the satellite in the mode of a single-axis gravitational orientation for current values of the orbit elements and to averaging the equations of the orbital motion along it. It is demonstrated that the monotone increase of the semimajor axis takes place during the first 15 years of motion. In actuality, the semimajor axis oscillates with a period of about 60 years. The eccentricity and inclination of the orbit remain close to their initial values.     

On Saturn’s rotation relative to a center of mass under the action of the gravitational moments of the Sun and Jupiter

Saturn’s rotation relative to a center of mass is considered within an elliptic restricted three-body problem. It is assumed that Saturn is a solid under the action of gravity of the Sun and Jupiter. The motions of Saturn and Jupiter are considered elliptic with small eccentricities e_S and e_J, respectively; the mean motion of Jupiter n_J is also small. We obtain the averaged Hamiltonian function for a small parameter of ε = n_J and integrals of evolution equations. The main effects of the influence of Jupiter on Saturn’s rotation are described: (α) the evolution of the constant parameters of regular precession for the angular momentum vector I₂; (β) the occurrence of new libration zones of oscillations I₂ near the plane of the celestial equator parallel to the plane of the Jupiter’s orbit; (γ) the occurrence of additional unstable equilibria of vector I₂ at the points of the north and south poles of the celestial sphere and, as a result, the existence of homoclinic trajectories; and (δ) the existence of periodic trajectories with arbitrarily large periods near the homoclinic trajectory. It is shown that the effects of (β), (γ), and (δ) are caused by the eccentricity e of the Jupiter’s orbit and are practically independent of Jupiter’s mass (within satellite approximation).