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.     

Biaxial Mode of Rotation of a Satellite in the Orbit Plane

A mode of motion of a satellite with respect to its center of mass is studied, which is called the biaxial rotation in the orbit plane. In this mode of rotation, an elongated and nearly dynamically symmetric satellite rotates around the longitudinal axis, which, in turn, rotates around the normal to the plane of an orbit; the angular velocity of rotation around the longitudinal axis is several times larger than the orbital angular velocity, deviations of this axis from the orbit plane are small. Such a rotation is convenient in the case when it is required to secure a sufficiently uniform illumination of the satellite's surface by the Sun at a comparatively small angular velocity of the satellite. The investigation consists of the numerical integration of equations of the satellite's motion, which take into account gravitational and restoring aerodynamic moments, as well as the evolution of the orbit. At high orbits, the mode of the biaxial rotation is conserved for an appreciable length of time, and at low orbits it is destroyed due to the impact of the aerodynamic moment. The orbit altitudes and the method of constructing the initial conditions of motion that guarantee a sufficiently prolonged period of existence of this mode are specified.     

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.     

The use of refined model of aerodynamic moment in the problem of reconstruction of the <Emphasis Type="Italic">Foton</Emphasis> satellite rotational motion

A new mathematical model of the uncontrolled rotational motion of the Foton satellite is presented. The model is based on the Euler dynamic equations of rigid body motion and takes into account the action upon the satellite of four external mechanical moments: gravitational, restoring aerodynamic, moment with constant components in the satellite-fixed coordinate system, and moment arising due to interaction of the Earth’s magnetic field with the satellite’s proper magnetic moment. To calculate the aerodynamic moment a special geometrical model of the outer satellite shell is used. Detailed form of the formulas giving above-mentioned moments in the equations of satellite motion is agreed with the form of the considered motion. Model testing is performed by determining with its help the rotational motion of the Foton M-2 satellite (it was in orbit from May 31, 2005 to June 16, 2005) using the data of the onboard measurements of the Earth’s magnetic field strength. The use of the new model has led to a relatively small improvement in the accuracy of the motion determination, but allowed us to obtain physically real estimates of some parameters.     

Realization of modes of satellite attitude motion with a small level of microaccelerations using electromechanical executive devices

Quasi-static microaccelerations are estimated for a satellite specially designed to perform space experiments in the field of microgravity. Three modes of attitude motion of the spacecraft are considered: passive gravitational orientation, orbital orientation, and semi-passive gravitational orientation. In these modes the lengthwise axis of the satellite is directed along the local vertical, while solar arrays lie in the orbit plane. The second and third modes are maintained using electromechanical executive devices: flywheel engines or gyrodynes. Estimations of residual microaccelerations are performed with the help of mathematical modeling of satellite’s attitude motion under the action of gravitational and aerodynamic moments, as well as the moment produced by the gyro system. It is demonstrated that all modes ensure rather low level of quasi-static microaccelerations on the satellite and provide for a fairly narrow region of variation for the vector of residual microacceleration. The semi-passive gravitational orientation ensures also a limited proper angular momentum of the gyro system.     

Uncontrollable Rotational Motion of the Mir Orbital Station

The results of determining the rotational motion of the Mir orbital station are presented for four long segments of its unmanned uncontrolled flight in 1999–2000. The determination was carried out using the data of onboard measurements of the Earth's magnetic field intensity. These data, taken for a time interval of several hours, were jointly processed by the least squares method with the help of integration of the equations of station motion relative to its center of mass. As a result of this processing, the initial conditions of motion and the parameters of the mathematical model used were evaluated. The technique of processing is verified using the telemetry data on angular velocity of the station and its attitude parameters. Two types of motion were applied on the investigated segments. One of them (three segments) presents a rotation around the axis of the minimum moment of inertia. This axis executes small oscillations with respect to a normal to the orbit plane. Such a motion was used for the first time on domestic manned orbital complexes. The second type of motion begins with a biaxial rotation which, in a few weeks, goes over into a motion very similar to the rotation around the normal to the orbit plane, but around the axis of the maximum moment of inertia.     

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.     

Determination of attitude motion of the Foton M-2 satellite using the data of onboard measurements of its angular velocity

We present the resutls of a prompt determination of the uncontrolled attitude motion of the Foton M-2 satellite, which was in orbit from May 31 to June 16, 2005. The data of onboard measurements of the angular velocity vector were used for this determination. The measurement sessions were carried out once a day, each lasting 83 min. Upon terminating a session, the data were transmitted to the ground to be processed using the least squares method and integrating the equations of motion of the satellite with respect to its center of mass. As a result of processing, the initial conditions of motion during a session were estimated, as well as parameters of the mathematical model used. The satellite’s actual motion is determined for 12 such sessions. The results obtained in flight completely described the satellite’s motion. This motion, having begun with a small angular velocity, gradually became faster, and in two days became close to the regular Euler precession of an axisymmetric solid body. On June 14, 2005 the angular velocity of the satellite with respect to its longitudinal axis was approximately 1.3 degrees per second, and the angular velocity projection onto a plane perpendicular to this axis had a magnitude of about 0.11 degrees per second. The results obtained are consistent with more precise results obtained later by processing the data on the Earth’s magnetic field measured on the same satellite, and they complement the latter in determination of the motion in the concluding segment of the flight, when no magnetic measurements were performed.     

Uncontrolled Attitude Motion of the Foton-12 Satellite and Quasi-Steady Microaccelerations onboard It

The results of determination of the uncontrolled attitude motion of the Foton-12 satellite (placed in orbit on September 9, 1999, terminated its flight on September 24, 1999) are presented. The determination was carried out by the onboard measurement data of the Earth's magnetic field strength vector. Intervals with a duration of several hours were selected from data covering almost the entire flight. On each such interval the data were processed simultaneously using the least squares method by integrating the satellite's equations of motion with respect to the center of mass. The initial conditions of motion and the parameters of the mathematical model employed were estimated in processing. The results obtained provided for a complete representation of the satellite's motion during the flight. This motion, beginning with a small angular velocity, gradually sped up. The growth of the component of the angular velocity with respect to the longitudinal axis of the satellite was particularly strong. During the first several days of the flight this component increased virtually after every passage through the orbit's perigee. As the satellite's angular velocity increased, its motion became more and more similar to the regular Euler precession of an axisymmetric rigid body. In the last several days of flight the satellite's angular velocity with respect to its longitudinal axis was about 1 deg/s and the projection of the angular velocity onto the plane perpendicular to this axis had a magnitude of approximately 0.15 deg/s. The deviation of the longitudinal axis from the normal to the orbit plane did not exceed 60°. The knowledge of the attitude motion of the satellite allowed us to determine the quasi-steady microacceleration component onboard it at the locations of the technological and scientific equipment.     

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.     

Uncontrolled motion of the <Emphasis Type="Italic">Foton M-2</Emphasis> satellite and quasistatic microaccelerations on its board

The results of determining the uncontrolled rotational motion of the Foton M-2 satellite (in orbit from May 31 to June 16, 2005) are presented. The determination was made using the data of onboard measurements of the Earth’s magnetic field strength. Segments 270 min long (three orbits) were selected from these data covering the first two thirds of the flight. On each such segment the data were processed jointly by the least squares method using integration of the equations of attitude motion of the satellite. In processing, the initial conditions of motion and parameters of the used mathematical model were estimated. The thus obtained results gave a complete overview of the satellite motion. This motion, having started with a small angular velocity, gradually accelerated, and in two days became close to the regular Euler precession of an axisymmetric solid body. On June 09, 2005 (the last day of measurements) the angular velocity of the satellite relative to its lengthwise axis was about 1.1 deg/s, while the projection of the angular velocity onto a plane perpendicular to this axis had an absolute value of about 0.11 deg/s. Deviations of the lengthwise axis from a normal to the orbit plane did not exceed 60°. Based on the results of determination of the rotational motion of the satellite, calculations of quasi-static microaccelerations on its board are performed.     

Estimation of residual microaccelerations on board an artificial earth satellite in the monoaxial solar orientation mode

The mode of monoaxial solar orientation of a designed artificial Earth satellite (AES), intended for microgravitational investigations, is studied. In this mode the normal line to the plane of satellite’s solar batteries is permanently directed at the Sun, the absolute angular velocity of a satellite is virtually equal to zero. The mode is implemented by means of an electromechanical system of powered flywheels or gyrodynes. The calculation of the level of microaccelerations arising on board in such a mode, was carried out by mathematical modeling of satellite motion with respect to the center of masses under an effect of gravitational and restoring aerodynamic moments, as well as of the moment produced by the gyrosystem. Two versions of a law for controlling the characteristic angular momentum of a gyrosystem are considered. The first version provides only attenuation of satellite’s perturbed motion in the vicinity of the position of rest with the required velocity. The second version restricts, in addition, the increase in the accumulated angular momentum of a gyrosystem by controlling the angle of rotation of the satellite around the normal to the light-sensitive side of the solar batteries. Both control law versions are shown to maintain the monoaxial orientation mode to a required accuracy and provide a very low level of quasistatic microaccelerations on board the satellite.     

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.     

Modes of uncontrolled rotational motion of the <Emphasis Type="Italic">Progress M-29M</Emphasis> spacecraft

We have reconstructed the uncontrolled rotational motion of the Progress M-29M transport cargo spacecraft in the single-axis solar orientation mode (the so-called sunward spin) and in the mode of the gravitational orientation of a rotating satellite. The modes were implemented on April 3–7, 2016 as a part of preparation for experiments with the DAKON convection sensor onboard the Progress spacecraft. The reconstruction was performed by integral statistical techniques using the measurements of the spacecraft’s angular velocity and electric current from its solar arrays. The measurement data obtained in a certain time interval have been jointly processed using the least-squares method by integrating the equations of the spacecraft’s motion relative to the center of mass. As a result of processing, the initial conditions of motion and parameters of the mathematical model have been estimated. The motion in the sunward spin mode is the rotation of the spacecraft with an angular velocity of 2.2 deg/s about the normal to the plane of solar arrays; the normal is oriented toward the Sun or forms a small angle with this direction. The duration of the mode is several orbit passes. The reconstruction has been performed over time intervals of up to 1 h. As a result, the actual rotational motion of the spacecraft relative to the Earth–Sun direction was obtained. In the gravitational orientation mode, the spacecraft was rotated about its longitudinal axis with an angular velocity of 0.1–0.2 deg/s; the longitudinal axis executed small oscillated relative to the local vertical. The reconstruction of motion relative to the orbital coordinate system was performed in time intervals of up to 7 h using only the angularvelocity measurements. The measurements of the electric current from solar arrays were used for verification.     

Single-Axis Solar Orientation of a Satellite of the Earth

The possibility of using the mode of single-axis solar orientation is considered for a satellite placed into a nearly circular orbit with an altitude of 900 km and bearing a solar sail. The satellite (together with the sail) has an axisymmetric structure, its symmetry axis being the principal central axis of the maximum moment of inertia. The center of the sail pressure lies on this axis and is displaced with respect to the satellite's center of mass. The symmetry axis of the satellite is set to the Sun so that its center of mass would be located between the Sun and the pressure center and would rotate around this axis with an angular velocity of a few degrees per second. The satellite's axis of symmetry makes a slow precession under the action of the gravitational moment and the moment of light pressure forces. Though the maximum magnitudes of these moments are comparable, the moment of the light pressure forces dominates and controls the precession in such a way that the symmetry axis orientation to the Sun remains unchanged.     

Orbital stability of planar oscillations of a satellite in a circular orbit

We consider the attitude motion of a satellite with a circular orbit in a central Newtonian gravitational field. The satellite is a solid body whose mass geometry is that of a plate. A nonlinear analysis is made of orbital stability of planar oscillations of the satellite at which its middle or major axis of inertia is perpendicular to the orbit plane. At small amplitudes of oscillations the analysis of stability was made analytically, while for arbitrary amplitudes the numerical analysis was performed.     

Experimental investigation of the modes of operation of uncontrolled attitude motion of the Progress spacecraft

Results of in-flight tests of three modes of uncontrolled attitude motion of the Progress spacecraft are described. These proposed modes of experiments related to microgravity are as follows: (1) triaxial gravitational orientation, (2) gravitational orientation of the rotating satellite, and (3) spin-up in the plane of the orbit around the axis of the maximum moment of inertia. The tests were carried out from May 24 to June 1, 2004 onboard the spacecraft Progress M1-11. The actual motion of this spacecraft with respect to its center of mass, in the above-mentioned modes, was determined by telemetric information about an electric current tapped off from solar batteries. The values of the current obtained during a time interval of several hours were processed jointly using the least squares method by integration of the equations of the spacecraft’s attitude motion. The processing resulted in estimation of the initial conditions of motion and of the parameters of mathematical models used. For the obtained motions the quasi-static component of microaccelerations was computed at a point onboard, where installation of experimental equipment is possible.     

The Mode of Gravitational Orientation for the International Space Station

The results of the determination of the uncontrolled attitude motion of the International Space Station during its unmanned flight in 1999 are presented. The data of onboard measurements of three components of the angular velocity are used for this determination. These data covering an interval of slightly less than one orbit were jointly processed by the least squares method, by integrating the equations of motion of the station relative to its center of mass. As a result of this processing, the initial conditions of the motion and the parameters of the mathematical model used were estimated. The actual motion of the station has been determined for 20 such intervals during April–November. Throughout these intervals, the station rotated about the axis of the minimum moment of inertia, the latter executing small oscillations relative to the local vertical. Such a mode, known as the mode of gravitational orientation of a rotating satellite or the mode of generalized gravitational orientation, was planned before the flight. The measurements were made to verify it. The quasistatic component of the microaccelerations aboard the station is estimated for this mode.     

Orbit design for the Spektr-R spacecraft of the ground-space interferometer

In order to carry out tasks of the RadioAstron mission, a high-apogee orbit was designed. On average, the period of its satellite’s orbit around the Earth is 8.5 days with evolution due to gravitational perturbations produced by the Moon and the Sun. The perigee and apogee of this orbit vary within the limits 7500–70000 km and 270000–333000 km, respectively. The basic evolution of the orbit represents a rotation of its plane around the line of apsides. Over 3 years, the plane normal to the orbit draws on the celestial sphere an oval with a semi-major axis of about 150° and semi-minor axis of about 45°.     

Secular evolution of rotary motion of a charged satellite in a decaying orbit

An electrostatically charged Earth satellite whose orbit is decaying due to the Earths oblateness is considered. Secular perturbations of the orbit are taken into account: they are caused by the second zonal harmonic of the geopotential. These perturbations represent deviations of the longitude of the ascending node and perigee argument, the orbit form being invariable and the orbit inclination to the equatorial plane being constant. The attitude rotary motion of the satellite under the action of perturbing moments of the gravitational and Lorentz forces is studied. The magnetic field of the Earth is taken in a quadrupole approximation. The evolution of the satellites rotary motion is investigated on the basis of new differential equations in s-parameters specially constructed for this purpose. Using the method of averaging, basic regularities of the secular evolution of rotary motion of a screened satellite are revealed. It is found that the rotary motion of a charged satellite essentially depends on the quadrupole component of the geomagnetic potential.__________Translated from Kosmicheskie Issledovaniya, Vol. 43, No. 2, 2005, pp. 111–125.Original Russian Text Copyright © 2005 by Tikhonov.