On saddle-node bifurcation and chaos of satellites (original) (raw)
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Three-Dimensional Subsatellite Motion under Air Drag and Oblateness Perturbations
Celestial Mechanics and Dynamical Astronomy
AbstraeL The three-dimensional relative motion of a subsatellite with respect to a reference station in an elliptical orbit is studied. A general theory based on the variation of the relative elements, i.e. the instantaneous differences between the orbital parameters of the subsatellite and those of the station, is formulated in order to incorporate arbitrary perturbing forces acting on both satellites. The loss of precision inherent in the subtraction of almost identical quantities is avoided by the consistent use of difference variables. In the absence of perturbations exact analytical representations can be obtained for the relative state parameters. The influences of air drag and Earth's oblateness on the relative motion trajectories are investigated and illustrated graphically for a number of cases.
Knowledge of orbital motion is essential for a full understanding of space operations. Motion through space can be visualized using the laws described by Johannes Kepler and understood using the laws described by Sir Isaac Newton. Thus, the objectives of this chapter are to provide a conceptual understanding of orbital motion and discuss common terms describing that motion. The chapter is divided into three sections. The first part focuses on the important information regarding satellite orbit types to provide an understanding of the capabilities and limitations of the spaceborne assets supporting the war fighter. The second part covers a brief history of orbital mechanics, providing a detailed description of the Keplerian and Newtonian laws. The third section discusses the application of those laws to determining orbit motion, orbit geometry, and orbital elements. This section has many facts, figures, and equations that may seem overwhelming at times. However, this information is essential to understanding the fundamental concepts of orbital mechanics and provides the necessary foundation to enable war fighters to better appreciate the challenges of operating in the space domain.
A Theory of Low Eccentricity Earth Satellite Motion
Journal of The Astronautical Sciences, 2012
Earth satellite motion is considered from the point of view of periodic orbits and Floquet theory in the Earth's zonal potential field. Periodic orbits in the zonal potential are nearly circular, except near the critical inclination. The local linear solution near the periodic orbit includes two degenerate modes that locally mirror the global invariance to time and nodal rotation, at least in the zonal potential. Since the Earth's oblateness is included in the periodic orbit, perturbations generally begin at one part in 10 5 , not one part in 10 3. Perturbations to the periodic orbit are calculated for sectoral and tesseral potential terms, for air drag, and for third body effects. The one free oscillatory mode of the periodic orbit is the eccentricity / argument of perigee analogues, and this can be extended past the first order in small quantities. There results a compact, purely numerical set of algorithms that may rival numerical integration in their accuracy, but have the usual "general perturbations" advantage of calculation directly at the time of interest, without having to perform a long propagation.
Attitude Dynamics of a Satellite on a Circular and Elliptic Low Earth Orbit
1992
The paper deals with the effect of the length on the type and stability of the inplane attitude motion of a dumbbell satellite moving on circular and elliptic equatorial Low Earth Orbits (LEO) by which the air drag also has a weak influence. For a circular LEO, a saddle-node bifurcation is found at some critical value of the length. This investigation can be performed analytically using imperfect bifurcation theory. When the orbit is elliptic using the results from the circular case, numerical simulation is used to approach the phase trajectories for sub-and supercritical values of the length. Depending on the order of the orbit eccentricity, three kinds of behaviour seem to be possible.
Analytical and Semi-Analytical Treatment of the Satellite Motion in a Resisting Medium
Journal of Applied Mathematics, 2012
The orbital dynamics of an artificial satellite in the Earth's atmosphere is considered. An analytic first-order atmospheric drag theory is developed using Lagrange's planetary equations. The short periodic perturbations due to the geopotential of all orbital elements are evaluated. And to construct a second-order analytical theory, the equations of motion become very complicated to be integrated analytically; thus we are forced to integrate them numerically using the method of Runge-Kutta of fourth order. The validity of the theory is checked on the already decayed Indian satellite ROHINI where its data are available.
Special Inclinations Allowing Minimal Drift Orbits for Formation Flying Satellites
Journal of Guidance, Control, and Dynamics, 2008
The possibility of obtaining a natural periodic relative motion of formation flying Earth satellites is investigated both numerically and analytically. The numerical algorithm is based on a genetic strategy, refined by means of nonlinear programming, that rewards periodic relative trajectories. First, we test our algorithm using a point mass gravitational model. In this case the period matching between the considered orbits is a necessary and sufficient condition to obtain invariant relative trajectories. Then, the J 2 perturbed case is considered. For this case, the conditions to obtain an invariant relative motion are known only in approximated closed forms which guarantee a minimal orbit drift, not a motion periodicity. Using the proposed numerical approach, we improved those results and found two couples of inclinations (63.4 and 116.6 deg, the critical inclinations, and 49 and 131 deg, two new "special" inclinations) that seemed to be favored by the dynamic system for obtaining periodic relative motion at small eccentricities.
Symbolic Calculations in Studying the Stability of Dynamically Symmetric Satellite Motion
The stability of cylindrical precession of the dynamically symmetric satellite in the Newtonian gravitational field is studied. We consider the case when a center of mass of the satellite moves in an elliptic orbit, while the satellite rotates uniformly about the axis of its dynamical symmetry that is perpendicular to the orbit plane. In the case of the resonance 3:2 (Mercury type resonance) we have found the domains of instability of cylindrical precession of the satellite in the Liapunov sense and domains of its linear stability in the parameter space. Using the infinite determinant method we have calculated analytically the boundaries of the domains of instability as power series in the eccentricity of the orbit. All the calculations have been done with the computer algebra system Mathematica.
A Dynamical Systems Theory Solution to an Orbiting Earth-Satellite Model
Several models of earth satellites that exist in the literature are not verified for real satellite configurations. An analytic procedure is applied to examine the equations representing the pitch librations of a satellite. The veracity of the model and the method of analysis are confirmed for a typical earth-satellite measurements.
The motion of artificial satellites in the set of Eulerian redundant parameters
Earth Moon and Planets, 1991
In this paper, the connections between orbit dynamics and rigid body dynamics are established throughout the Eulerian redundant parameters, the perturbation equations for any conic motion of artificial satellites are derived in terms of these parameters. A general recursive and stable computational algorithm is also established for the initial-value problem of the Eulerian parameters for satellites prediction in the Earth's gravitational field with axial symmetry. Applications of the algorithm are considered for the two cases of short and long term predictions. For the short-term prediction, we consider the problem of the final state prediction of some typical ballistic missiles in the geopotential model with zonal harmonic terms up to J 36, while for the long-term prediction, we consider the perturbed J 2 motion of Explorer 28 over 100 revolutions.
Journal of scientific research, 2021
We derive a set of non-linear, non-homogeneous and non-autonomous differential equations for the motion of a system of two inelastic cable-connected artificial satellites under the influence of shadow of the earth, solar radiation pressure, oblateness of the earth, air resistance and earth's magnetic field. The motion of the system is studied relative to its center of mass which has been assumed to move along a Keplerian elliptical orbit. The equation of relative motion of the system has been obtained. Equations of motion have been obtained in Rotating frame of reference and thereafter in Nechvile's Coordinate System. Applying simulations, the equations of motion lead to Jacobian integral of motion of the system. Further simulations and the equations of motion give rise to one equilibrium position of motion of the system concerned under the above mentioned perturbative forces.