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 specific case of stability of cylindrical precession of a satellite

In a central Newtonian gravitational field, the motion of a dynamically symmetrical satellite along an elliptical orbit of arbitrary eccentricity is considered. The particular motion of the satellite is known when its axis of symmetry is perpendicular to the orbit plane, and the satellite rotates about this axis with a constant angular velocity (cylindrical precession). A nonlinear analysis of stability of this motion has been performed under the assumption that the geometry of the satellite mass corresponds to a thin plate. At small values of orbit eccentricity e the analysis is analytical, while numerical analysis is used for arbitrary values of e.     

Decay of high eccentricity satellite orbits with air drag using uniformly regular KS canonical elements

A non-singular analytical theory for the motion of high eccentricity satellite orbits under the influence of air drag is developed in terms of the Uniformly Regular Kustaanheimo and Stiefel (URKS) canonical elements, by assuming the atmosphere to be oblate with variation of density scale height with altitude. The series expansions include up to fourth-order terms of an independent variable Δ=λ² (function of eccentric anomaly) and c (a small parameter dependent on the flattening of the atmosphere). Only two of the nine equations are solved analytically due to symmetry in the equations of motion. Comparison of the important orbital parameters semi-major axis and eccentricity up to 1000 revolutions, obtained with the present analytical solution and the KS theory, shows the superiority of the present solution over the KS elements analytical solution. The theory can be used effectively for the orbital decay of aero-assisted orbital transfer orbits during mission planning.     

Rapid rotations of a satellite with a cavity filled with viscous fluid under the action of moments of gravity and light pressure forces

Rapid rotational motion of a dynamically asymmetric satellite relative to the center of mass is studied. The satellite has a cavity filled with viscous fluid at low Reynolds numbers, and it moves under the action of moments of gravity and light pressure forces. Orbital motions with an arbitrary eccentricity are supposed to be specified. The system, obtained after averaging over the Euler-Poinsot motion and applying the modified averaging method, is analyzed. The numerical analysis in the general case is performed, and the analytical study in the axial rotation vicinity is carried out. The motion in the specific case of a dynamically symmetric satellite is considered.     

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.     

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.     

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.     

Bifurcation Regime of a Synchronous Resonance in the Translational–Rotational Motion of Nonspherical Natural Satellites of Planets

Period-doubling bifurcations of the synchronous spin-orbit resonance in the motion of a nonspherical natural planetary satellite along the elliptic orbit are studied. The satellite spin axis is assumed to coincide with the axis of its largest principal moment of inertia and is perpendicular to the orbital plane. The period-doubling bifurcations take place when the value of satellite's dynamical asymmetry parameter falls in the parametric resonance domain. Theoretical dependences of the amplitude of the bifurcation oscillations of a satellite at the pericenter of its orbit upon the eccentricity and dynamical asymmetry parameter are investigated. Three different methods of calculating the amplitude of bifurcation oscillations are presented and compared. These theoretical estimates can be used to predict the opportunity to observe the bifurcation regime. The possibility of the occurrence of the bifurcation regime in the motion of natural planetary satellites is studied. It is concluded that the bifurcation regime is possible in the motion of Deimos, Epimetheus, Helen, Pandora, and Phobos. Phobos is the most probable candidate for finding the bifurcation regime of a synchronous rotation. The identification of such a regime would allow one to impose stringent constraints on the values of the inertial parameters of the satellite observed.     

Artificial satellites orbits in 2:1 resonance: GPS constellation

In this work, the resonance problem in the artificial satellites motion is studied. The development of the geopotential includes the zonal harmonics J₂₀ and J₄₀ and the tesseral harmonics J₂₂ and J₄₂. Through an averaging procedure and successive Mathieu transformations, the order of dynamical system is reduced and the final system is solved by numerical integration. In the simplified dynamical model, three critical angles are studied. The half-width of the separatrix is calculated through a linearized model which describes the behavior of the dynamical system in a neighborhood of each critical angle. Through the resonance overlap criterion the possible regular and irregular motions are investigated by the time behavior of the semi-major axis, argument of perigee and eccentricity. The largest Lyapunov exponent is used as tool to verify the chaotic motion.     

Artificial satellites orbits in 2:1 resonance: GPS constellation

In this work, the resonance problem in the artificial satellites motion is studied. The development of the geopotential includes the zonal harmonics J₂₀ and J₄₀ and the tesseral harmonics J₂₂ and J₄₂. Through an averaging procedure and successive Mathieu transformations, the order of dynamical system is reduced and the final system is solved by numerical integration. In the simplified dynamical model, three critical angles are studied. The half-width of the separatrix is calculated through a linearized model which describes the behavior of the dynamical system in a neighborhood of each critical angle. Through the resonance overlap criterion the possible regular and irregular motions are investigated by the time behavior of the semi-major axis, argument of perigee and eccentricity. The largest Lyapunov exponent is used as tool to verify the chaotic motion.     

Single-impulse transfers from orbits of artificial satellites to orbits around <Emphasis Type="Italic">L</Emphasis><Subscript>1</Subscript> or <Emphasis Type="Italic">L</Emphasis><Subscript>2</Subscript> libration points

In the context of the restricted circular three-body problem a method for constructing families of periodic orbits is described. Each orbit contains a segment of transfer from artificial satellite orbit of a smaller body to an orbit around L ₁ or L ₂ points of the Sun-Earth and Earth-Moon systems, a segment of multiple flyby of this libration point, and a segment of return to the artificial satellite orbit. Dependences of velocities at the pericenter on the pericenter radius are given.     

Orbit prediction for IRAS using vector and analytic techniques

This paper describes the techniques of a vector approach to the solution of the differential equations of motion of a near-Earth satellite. The method provides a good stable foundation for developing the orbital elements, thus allowing an analytic approach to be used in subsidiary algorithms. The mathematical concepts used in these algorithms are explained, and equations are developed for calculating Earth and Moon eclipses, radiation zone crossings, atmospheric density effects, solar cell decay, look angles and a geographical ephemeris. Results are presented for the IRAS satellite, and show that prediction errors of less than $\text{1}\text{1}\text{2}$ sec over one week or errors of less than 15 sec over $\text{3}\text{1}\text{2}$ months are possible.     

全球星摄动运动及摄动补偿运控策略研究

绕地卫星摄动运动是导致星座设计构型发散的主要因素,摄动补偿轨道控制是维持全     

Analysis of Low-Frequency Microaccelerations onboard the Foton-11Satellite

The method and the results of investigating the low-frequency component of microaccelerations onboard the Foton-11satellite are presented. The investigation was based on the processing of data of the angular velocity measurements made by the German system QSAM, as well as the data of measurements of microaccelerations performed by the QSAM system and by the French accelerometer BETA. The processing was carried out in the following manner. A low-frequency (frequencies less than 0.01 Hz) component was selected from the data of measurements of each component of the angular velocity vector or of the microacceleration, and an approximation was constructed of the obtained vector function by a similar function that was calculated along the solutions to the differential equations of motion of the satellite with respect to its center of mass. The construction was carried out by the least squares method. The initial conditions of the satellite motion, its aerodynamic parameters, and constant biases in the measurement data were used as fitting parameters. The time intervals on which the approximation was constructed were from one to five hours long. The processing of the measurements performed with three different instruments produced sufficiently close results. It turned out to be that the rotational motion of the satellite during nearly the entire flight was close to the regular Eulerian precession of the axially symmetric rigid body. The angular velocity of the satellite with respect to its longitudinal axis was about 1 deg/s, while the projection of the angular velocity onto the plane perpendicular to this axis had an absolute value of about 0.2 deg/s. The magnitude of the quasistatic component of microaccelerations in the locations of the accelerometers QSAM and BETA did not exceed 5 × 10^–5–10^–4m/s²for the considered motion of the satellite.     

An Analysis of the Low-Frequency Component in Measurements of Angular Velocity and Microacceleration onboard the Foton-12 Satellite

We compare the results of two methods used to determine the angular velocity of the Foton-12 satellite and the low-frequency component of microaccelerations onboard it. The first method is based on reconstruction of the satellite's rotational motion using the data of onboard measurements of the strength of the Earth's magnetic field. The motion (time dependence of the orientation parameters and angular velocity) was found from the condition of best approximation of the measurement data by the functions calculated along the solutions to equations of attitude motion of the satellite. The solutions found were used to calculate the quasistatic component of microaccelerations at certain points of the satellite, in particular, at the point of location of an accelerometer of the QSAM system. Filtration of the low-frequency component of the angular velocity and microacceleration from the data of measurements by a sensor of angular velocity and by the accelerometer of this system served as a second method. The filtration was made using the discrete Fourier series. A spectral analysis of the functions representing the results of determining the angular velocity and microacceleration by both methods is performed. Comparing the frequencies and amplitudes of the harmonic component of these functions allowed us to estimate the accuracy of measurements made by the QSAM system in the low-frequency range.     

Variable structure control for satellite attitude stabilization in elliptic orbits using solar radiation pressure

The paper proposes the use of solar radiation pressure for satellite attitude control in elliptic orbits based on variable structure control. The system comprises of a satellite with two-oppositely placed solar flaps. Sliding mode control and terminal sliding mode control techniques have been adopted to develop nonlinear control laws for suitably rotating the control solar flaps to neutralize the adverse effect of eccentricity normally responsible for a considerable deterioration in the attitude control performance. The detailed numerical simulation of the governing nonlinear equation of the motion including the effects of various system parameters on the controller performance, establishes the feasibility of the proposed control strategy. The proposed controller is found to be robust against parameter uncertainties and external disturbances and thereby, the control strategy presented in the paper may be applicable to future satellite missions.     

Fourth-order theories for orbit predictions for low- and high-eccentricity orbits in an oblate atmosphere with scale height dependent on altitude

Two new fourth-order non-singular analytical theories for the motion of near-Earth satellite orbits with air drag are developed for low- and high-eccentricity orbits in an oblate atmosphere with variation of density scale height with altitude. Uniformly regular Kustaanheimo–Stiefel (KS) canonical elements are utilized for low-eccentricity orbits and KS element equations are employed for high-eccentricity orbits. 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 analytical solutions are compared with the numerically integrated values up to 100 revolutions, and found to be quite accurate over a wide range of eccentricity, perigee height and inclination.     

Low-Frequency Microaccelerations onboard the Foton-11 Satellite

We analyze the microacceleration measurements carried out onboard the Foton-11 satellite with the three-component accelerometer BETA. The microaccelerations were recorded virtually throughout the entire orbital flight of the Foton-11 satellite. The data obtained were analyzed in the following way. First they were used to determine the actual rotational motion of the satellite for several arbitrarily selected time intervals 4 h long. This problem was solved by constructing the approximation of the microacceleretation low-frequency component (previously determined from the data) by its calculated analog computed along the solutions to differential equations of rotational motion of the satellite. The approximation was made by the least squares method. As a result, those mathematical model parameters and the solutions to equations of motion were found that gave the best consistency of the microacceleretation low-frequency component and its calculated analog. Then the spectral analysis of the low-frequency component and its calculated analog was made. It was shown that, although basic harmonics of these functions coincided sufficiently well, some harmonics of the low-frequency component failed to be interpreted in terms of the satellite's rotational motion.     

Detection and Prediction of Absorbed Radiation Doses from Solar Proton Fluxes onboard Orbital Stations

We consider cases of simultaneous detection of the absorbed doses produced by proton fluxes of powerful solar events onboard the Mir and ISS orbital stations and the Ekspress A3 geosynchronous satellite. Experimental data are analyzed using a software package that takes into account the energy spectra of protons at the Earth's orbit depending on the time of event evolution, as well as their penetration to near-earth orbits and through the protective shields of spacecraft. Based on a comparison of the experimental data of dosimeters with the calculation of absorbed doses under the action of solar proton events, we developed a method of estimating the effective thickness of the shielding of dosimeters and made some estimates. A possibility is considered for predicting the radiation hazard onboard orbital stations upon the appearance of solar proton events using dosimeter data from a geosynchronous orbit.