Dynamics of an axisymmetric satellite under the action of gravitational and aerodynamic torques

Dynamics of attitude motion of an axisymmetric satellite moving in a circular orbit under the action of gravitational and aerodynamic torques is investigated. All equilibrium positions of the satellite in the orbital coordinate system are determined numerically, and sufficient conditions of stability of the equilibrium positions are derived.     

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.     

On the possibility of long-time existence of submicron particles injected in elongated elliptical orbits with low perigee in the near-Earth space

On the basis of numerical experiments the theoretical possibility of long-time (longer than 1 month) and superlong-time (longer than 1 year) existence in orbit of technogenic microparticles (MPs) with radii of a few hundredths of a micrometer is demonstrated. MPs are injected into the near-Earth space (NES) in elongated elliptical low-perigee orbits with parameters, corresponding to Molniya satellite’s orbital parameters. Calculations were carried out taking into account disturbing effects on the MP orbital motion in NES of the following factors: the gravitational disturbance caused by polar oblateness of the Earth, the solar pressure force (calculated with using the techniques of the Mie theory), the drag force of a neutral component of background gas, as well as the electrodynamic forces caused by interaction of electric charge, induced on MPs, with the magnetic and electric fields of the NES.     

A small parameter method in problems of maneuvering space vehicles with aerodynamic efficiency

The motion of a satellite with aerodynamic efficiency along a low near-circular orbit is considered in the paper. The controls of bank angle γ and lift coefficient C_y are used as control functions. The introduction of a small parameter ($\text{? = (}\text{?}{}_0\text{· S · g}{}_0\text{2G}\text{)}$) makes it possible to integrate an adjoint system of equations and to obtain an approximate solution to the complete problem in the class of piecewise-constant control functions. Maximum values for the coordinates of heading angle η and lateral derivation from the plane of a reference orbit ?, which are connected with orbit plane angle by the relation cos i = cos ? · cos h, are used as criteria of maneuvering capability for a satellite with aerodynamic efficiency. Optimal programs for bank angle and incidence variation are derived and the influence of lift-to-drag ratio on the vehicle maneuvering capabilities has been estimated.It is shown that the process of the optimal motion is a special kind of gravitational skipping similar to the Keplerian motion but with continuous descent.     

Modeling of the aerodynamic moment acting upon a satellite

We consider the issues of modeling the moments of aerodynamic forces acting upon a satellite with gravitational system of stabilization. It is assumed that satellite orbits are almost circular with heights in the range 550–750 km. Simplified analytical expressions are suggested for the aerodynamic moment in the case when a satellite moves in the regime of gravitational orientation. Accuracy of the obtained expressions is estimated to be compared with that of expressions derived under the assumption of constant coefficient of frontal resistance. An analysis is made of short-periodic variations of the atmosphere density occurring due to orbital motion of a satellite. It is demonstrated that these variations can result in a substantial change of the aerodynamic moment, and their approximation by a truncated Fourier series is suggested. Estimates of the accuracy of the suggested approximation are given.     

Gravity-oriented satellite dynamics subject to gravitational and active damping torques

The dynamics of the rotational motion of a satellite moving in the central Newtonian field of force over a circular orbit under the effect of gravitational and active damping torques, which depend on the satellite angular velocity projections, has been investigated. The paper proposes a method of determining all equilibrium positions (equilibrium orientations) of a satellite in the orbital coordinate system for specified values of damping coefficients and principal central moments of inertia. The conditions of their existence have been obtained. For a zero equilibrium position where the axes of the satellite-centered coordinate system coincide with the axes of the orbital coordinate system, the necessary and sufficient conditions for asymptotic stability are obtained using the Routh–Hurwitz criterion. A detailed analysis of the regions where the conditions of the asymptotic stability of a zero equilibrium position are fulfilled have been obtained depending on three dimensionless parameters of the problem, and the numerical study of the process of attenuation of satellite’s spatial oscillations for various damping coefficients has been carried out. It has been shown that there is a wide range of damping parameters from which, by choosing the necessary values, one can provide the asymptotic stability of satellite’s zero equilibrium position in the orbital coordinate system.     

Investigation of the Time of Ballistic Existence for Elliptic Orbits Evolving under the Influence of Gravitational Perturbations of External Bodies

A comparative analysis of the time of ballistic existence for different elliptic orbits evolving under the influence of gravitational perturbations of external bodies is performed. For this purpose, the solution of a satellite variant of the restricted circular double-averaged three-body problem obtained by M.L. Lidov [1] is used. The use of a geometrical interpretation of Lidov's solutions has allowed us to create a visualized tool for a choice of evolving satellite orbits with a long time of ballistic existence.     

Antenna radiation effects on the orbits of GPS and INTELSAT satellites

A large number of disturbances add to the main force exerted by Earth's gravitational field and affect the actual orbital trajectory of artificial satellites. They possess antennas with specific purposes, such as telecommunication systems operating at specific ranges of frequencies and radiated power. For instance, the antennas used in the GPS and INTELSAT satellites are quadrifilar helix and parabolic reflectors respectively. The radiation emitted by the antennas produces a radiation reaction force on the satellite making its orbital elements deviate from their expected values. Using a mathematical model for the radiation reaction force caused by the antenna, derived from the electromagnetic theory and the energy-momentum conservation law, the perturbation effects on the orbits of the GPS and INTELSAT satellites were studied. The numerical integrator used to solve the satellite equations of motion is based on the Runge–Kutta method of fourth and fifth orders. The theoretical model of antenna radiation reaction takes into account the satellite mass, antenna radiated power and maximum gain of the antenna.     

Translational–Rotational Motion of a Viscoelastic Sphere in Gravitational Field of an Attracting Center and a Satellite

We study the translational–rotational motion of a planet modeled by a viscoelastic sphere in the gravitational fields of an immovable attracting center and a satellite modeled as material points. The satellite and the planet move with respect to their common center of mass that, in turn, moves with respect to the attracting center. The exact system of equations of motion of the considered mechanical system is deduced from the D'Alembert–Lagrange variational principle. The method of separation of motions is applied to the obtained system of equations and an approximate system of ordinary differential equations is deduced which describes the translational–rotational motion of the planet and its satellite, taking into account the perturbations caused by elasticity and dissipation. An analysis of the deformed state of the viscoelastic planet under the action of gravitational forces and forces of inertia is carried out. It is demonstrated that in the steady-state motion, when energy dissipation vanishes, the planet's center of mass and the satellite move along circular orbits with respect to the attracting center, being located on a single line with it. The viscoelastic planet in its steady-state motion is immovable in the orbital frame of reference. It is demonstrated that this steady-state motion is unstable.     

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.     

Orbit lifetime for microparticles in highly elliptical orbits with low perigee

On the basis of numerical experiments, we have shown the principal possibility of long (more than 1 month) and extremely long (more than 1 year) orbit lifetime of technogenic microparticles with radii from 1 to 100 μm injected into the near-Earth space in highly elliptical orbits with low perigee, including the case of an orbit with parameters corresponding to the orbital parameters of the Molniya satellite. Calculations are performed taking into account the perturbing effect on the orbital microparticle motion in the near-Earth space of gravitational perturbation caused by the Earth’s polar oblateness, the solar pressure force (calculated using methods of the Mie theory), and the drag force of neutral component of the background gas under conditions of low, medium, and high levels of solar and geomagnetic activities.     

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.     

Modifying the atlas of low lunar orbits using inert tethers

For long enough tethers, the coupling of the attitude and orbital dynamics may show non-negligible effects in the orbital motion of a tethered satellite about a central body. In the case of fast rotating tethers the attitude remains constant, on average, up to second order effects. Besides, for a tether rotating in a plane parallel to the equatorial plane of the central body, the attitude–orbit coupling effect is formally equal to the perturbation of the Keplerian motion produced by the oblateness of the central body and, therefore, may have a stabilizing effect in the orbital dynamics. In the case of a tethered satellite in a low lunar orbit, it is demonstrated that feasible tether lengths can help in modifying the actual map of lunar frozen orbits.     

A Retrospective Geometric Analysis of the Long-Term Evolution of Orbits and the Ballistic Lifetime for Satellites of the Prognoz Series

We consider the applied aspects of the geometrical analysis of solutions in a restricted circular double-averaged three-body problem that are concerned with the design of high-apogee satellite orbits. Based on the analysis of the long-term evolution and the ballistic lifetime for orbit families of the Prognoz satellites launched into their orbits in the period 1972–1995, we suggest some practical suggestions for choosing long-lived high-apogee orbits with taking into account various requirements for the domain of evolution of the orbital elements.     

超低轨卫星气动力辅助轨道控制研究

针对超低轨卫星所受气动力显著的特点,提出一种利用气动舵的气动力辅助轨道控制方法。通过分析大气旋转、卫星所处空间位置以及气动舵偏转角度对气动力的影响,对提出的轨道控制方法进行了优化。该方法通过调整气动舵产生连续微小的气动力对卫星轨道进行控制,使各轨道要素均保持在误差容限范围内。将其应用于太阳同步轨道上的对地观测卫星,仿真结果表明,该方法可以在卫星姿态保持三轴对地稳定的前提下,实现轨道保持控制,保证任意纬度下卫星实际位置与标称轨道位置偏差在给定范围内。     

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.     

Satellite dynamics under the influence of gravitational and aerodynamic torques. A study of stability of equilibrium positions

The dynamics of the rotational motion of a satellite, moving in the central Newtonian force field under the influence of gravitational and aerodynamic torques, is investigated. The paper proposes a method for determining all equilibrium positions (equilibrium orientations) of a satellite in the orbital coordinate system for specified values of aerodynamic torque and the major central moments of inertia; the sufficient conditions for their existence are obtained. For each equilibrium orientation the sufficient stability conditions are obtained using the generalized energy integral as the Lyapunov function. The detailed numerical analysis of the regions where the stability conditions of the equilibrium positions are satisfied is carried out depending on four dimensionless parameters of the problem. It is shown that, in the general case, the number of satellite’s equilibrium positions, for which the sufficient stability conditions are satisfied, varies from 4 to 2 with an increase in the value of the aerodynamic torque magnitude.     

Spacecraft attitude maneuver using two single-gimbal control moment gyros

In this paper, arbitrary rest-to-rest attitude maneuver problems for a satellite using two single-gimbal control moment gyros (2SGCMGs) are considered. Although single-gimbal control moment gyros are configured in the same manner as the traditional pyramid-array CMG, only two CMGs are assumed to be available. Attitude maneuver problems are similar to problems involving two reaction wheels (RWs) from the viewpoint of the number of actuators. In other words, the problem treated herein is a kind of underactuated problem. Although 2SGCMGs can generate torques around all axes, they cannot generate torques around each axis independently. Therefore, control methods designed for a satellite using two reaction wheels cannot be applied to three-axis attitude maneuver problems for a satellite using 2SGCMGs. In this paper, for simplicity, maneuvers around the x- and z-axes are first considered, and then a maneuver around the y-axis due to the corning effect resulting from the maneuver around the x- and z-axes is considered. Since maneuvers around each axis are established by the proposed method, arbitrary attitude maneuvers can be achieved using 2SGCMGs. In addition, the maneuvering angles around the z- and x-axes, which are required in order to maneuver around the y-axis, are analytically determined, and the total time required for maneuvering around the y-axis is then analyzed numerically.     

A practical approach to the disposal of highly toxic and long-lived spent nuclear fuel waste between Venus and Earth

Extraterrestrial disposal, while not the only alternative (there is at least one very safe terrestrial method), nevertheless assures definite and irreversible removal of the most toxic and long-lived waste from the biosphere. In the foreseeable future, there is far less need to retrieve nuclear waste for later use then to dispose of it in a publicly acceptable manner, beginning in the near future (middle 1980s). It is, therefore, more important to assure safety in the weakest link of this disposal process—transportation into near-Earth orbit—than to engage in a retrievable disposal beyond Earth. The disposal “site” should lie at minimum safe transfer energy level. Primary candidate is the space between Venus and Earth. The number of propulsion phases should be a minimum, preferably only one (out of near-Earth orbit). Lunar gravity assist can be helpful to achieve higher inclination of the heliocentric orbit relative to the ecliptic.Solidified spent fuel isotopes and actinides, sufficient to reduce the residual terrestrial waste to the radiation level of natural uranium deposits after 30–40 yr instead of 1000–1500 yr, is deposited into heliocentric orbits. Transportation systems, requirements, costs and the associated socio-economic benefit potentials of an environmentally more benign and a more vigorous nuclear power generation program are presented.Prior to solidification, an interim storage of 10 yr, following removal from the reactor, may be required. The Shuttle, with one Orbiter modified as Nuclear Waste Carrying Orbiter (NWCO) and an out of near-Earth orbit booster, provides a safe and economic transportation system at (1979$) disposal mission costs from surface to disposal orbit of less than $\text{0.5¢/}\text{kWhe or}\text{? 0.1¢/}\text{kWhe}$ (some 70% of overall exo-disposal cost), depending on level of orbital operations (this at basic Shuttle flight cost of $30M). The orbital operations mode can be selected according to lead time and costs, and can be advanced sequentially, lowering disposal costs while at the same time financing the development of orbital operations techniques needed also for other and larger-scale exo-industrial activities. An average of 10–15 disposal missions of the NWCO is required annually, at the given conditions, to service the spent fuel of 173 reactors 1978 in operation in W. Europe, Japan and North America.