Contraction of near-Earth satellite orbits using uniformly regular KS canonical elements in an oblate atmosphere with density scale height variation with altitude

A new non-singular analytical theory for the contraction of near-Earth satellite orbits under the influence of air drag is developed in terms of uniformly regular Kustaanheimo and Stiefel (KS) canonical elements using an oblate atmosphere with variation of density scale height with altitude. The series expansions include up to fourth power in terms of eccentricity and c (a small parameter dependent on the flattening of the atmosphere). 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. It is observed that the analytically computed values of the semi-major axis and eccentricity are consistent with the numerically integrated values up to 500 revolutions over a wide range of the drag-perturbed orbital parameters. The theory can be effectively used for re-entry of near-Earth objects.     

Prediction of satellite orbits contraction due to diurnally varying oblate atmosphere and altitude-dependent scale height using KS canonical elements

A new non-singular analytical theory for the motion of near-Earth satellite orbits with the air drag effect is developed in terms of uniformly regular KS canonical elements. Diurnally varying oblate atmosphere is considered with variation in density scale height dependent on altitude. The series expansion method is utilized to generate the analytical solutions and terms up to fourth-order terms in eccentricity and c (a small parameter dependent on the flattening of the atmosphere) are retained. 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 important drag perturbed orbital parameters: semi-major axis and eccentricity are obtained up to 500 revolutions, with the present analytical theory and by numerical integration over a wide range of perigee height, eccentricity and inclination. The differences between the two are found to be very less. A comparison between the theories generated with terms up to third- and fourth-order terms in c and e shows an improvement in the computation of the orbital parameters semi-major axis and eccentricity, up to 9%. The theory can be effectively used for the re-entry of the near-Earth objects, which mainly decay due to atmospheric drag.     

Analytical orbit predictions with air drag using KS uniformly regular canonical elements

A new nonsingular analytical theory for the motion of near Earth satellite orbits with the air drag effect is developed for long term motion in terms of the KS uniformly regular canonical elements by a series expansion method, by assuming the atmosphere to be symmetrically spherical with constant density scale height. The series expansions include up to third order terms in eccentricity. 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. Numerical comparisons of the important orbital parameters semi major axis and eccentricity up to 1000 revolutions, obtained with the present solution, with KS elements analytical solution and Cook, King-Hele and Walker's theory with respect to the numerically integrated values, show the superiority of the present solution over the other two theories over a wide range of eccentricity, perigee height and inclination.     

Long-term orbit computations with KS uniformly regular canonical elements with oblateness

This paper concerns with the study of KS uniformly regular canonical elements with Earth's oblateness. These elements, ten in number, are all constant in the unperturbed motion and even in the perturbed motion, the substitution is straightforward and elementary due to the transformation laws being explicit and closed expression. By utilizing the recursion formulas of Legendre's polynomials, we are able to include any number of Earth's zonal harmonics J _n in the package and also economize the computations. A fixed step-size fourth-order Runge-Kutta-Gill method is employed for numerical integration of the canonical equations.Utilizing 5 test cases covering a large range of semimajor axis and eccentricity, we have carried out computations to study the effects of Earth's zonal harmonics (up to J ₃₆) and integration step-size variation. Bilinear relations and energy equation are used for checking the accuracies of numerical integration. From the application point of view, the package is utilized to study the behaviour of 900 km height near-circular sun-synchronous satellite orbit over a longer duration of 220 days time (nearly 3078 revolutions) and the necessity of including more number of Earth's zonal harmonic terms is noticed. The package is also used to study the effect of higher zonal harmonics on three 900 km height near-circular orbits with inclinations of 60, 63.2, and 65 degrees, by including Earth's zonal harmonics up to J ₂₄. The mean eccentricity (e _m) is found to have long-periods of 459.6, 6925.1 and 1077.6 days, respectively. Sharp changes in the variation of _m near the minima to e_m are noticed. The values of _m are found to be very near to +-90 degrees at the extrema of e_m. The same orbit is employed to study the effect of variation of inclination from 0 to 180 degrees on long-period (T) of eccentricity with J ₂ to J ₂₄ terms. T is found to increase rapidly as we proceed towards the critical inclinations.     

Canonical elements and Keplerian-like solutions for intermediary orbits of satellites of an oblate planet

Within the framework of the Canonical Formalism in the extended phase space,a general Hamiltonian is investigated that covers a wide class of radial intermediaries accounting for themajor secular effects due to a planet's oblateness perturbations.An analytical, closed-form solution for this generic Hamiltonian is developed in terms of elementary functions via the corresponding Hamilton-Jacobi equation. The analytical solution so obtained can be contemplated according to a simple geometrical and dynamical interpretation in Keplerian language by means of the usual relations characterizing elliptic elements along ahypothetic Keplerian motion.Appropriate choices for the terms appearing in the proposed Hamiltonian lead to recovering the analogues of some well-known, classical radial intermediaries (those introduced by Deprit and the one built by Alfriend and Coffey), but also certain new ones derived by Ferrándiz for the Main Problem in the Theory of Artificial Satellites of the Earth. In any case, the results are also applicable to problems dealing with orbital motion of other planetary satellites.The generality of this pattern leads to asystematic obtaining of solutions to the considered intermediaries: special choices of the Hamiltonian yield the correspondinganalytical solution to the respective intermediary problem.     

Prediction of trajectories in the Earth's gravitational field with axial symmetry using KS uniform regular elements

In this paper, a motion prediction algorithm based on the KS regular elements is developed for the motion in the Earth's gravitational field with axial symmetry. The algorithm is of recursive nature and general in the sense that it could be applied for any conic motion whatever the number of zonal harmonic coefficientsN 2 may be. Applications of the algorithm for the problem of the final state prediction are illustrated by numerical examples of eight typical ballistic missiles for geopotential model with zonal harmonic terms up toJ ₃₆. A final state of any desired accuracy is obtained for each case study, a result which shows the efficiency and the flexibility of the algorithm.     

Drag perturbations of artificial satellite orbits in a spherically symmetrical atmosphere

The equations characterizing the motion of an artificial satellite in a non-rotating spherically symmetrical atmosphere are integrated in the assumption of a linear variation of the density scale height with height, and using a new variable instead of the true anomaly. The secular perturbations in the semi-major axis and eccentricity are deduced.     

The possible motions of a satellite about an oblate planet

The regions of motions of a satellite for given values of energy and angular momentum about polar axes are shown. Special attention is paid to the circular equatorial orbits which have been shown to be Hill stable. The anomalistic and the nodal period for the motions near to the circular equatorial orbits have been found.     

First-order analytic propagation of satellites in the exponential atmosphere of an oblate planet

The paper offers the fully analytic solution to the motion of a satellite orbiting under the influence of the two major perturbations, due to the oblateness and the atmospheric drag. The solution is presented in a time-explicit form, and takes into account an exponential distribution of the atmospheric density, an assumption that is reasonably close to reality. The approach involves two essential steps. The first one concerns a new approximate mathematical model that admits a closed-form solution with respect to a set of new variables. The second step is the determination of an infinitesimal contact transformation that allows to navigate between the new and the original variables. This contact transformation is obtained in exact form, and afterwards a Taylor series approximation is proposed in order to make all the computations explicit. The aforementioned transformation accommodates both perturbations, improving the accuracy of the orbit predictions by one order of magnitude with respect to the case when the atmospheric drag is absent from the transformation. Numerical simulations are performed for a low Earth orbit starting at an altitude of 350 km, and they show that the incorporation of drag terms into the contact transformation generates an error reduction by a factor of 7 in the position vector. The proposed method aims at improving the accuracy of analytic orbit propagation and transforming it into a viable alternative to the computationally intensive numerical methods.     

On the roto-translatory motion of a satellite of an oblate primary

We deal with the problem of the motion of a triaxial satellite of an oblate primary of larger mass. We show that the treatment is simplified by using a canonical set of variables previously introduced by the authors, that allows a drastic reduction in the expansions of the potential. A general method to avoid the appearance of virtual singularities when the angles between certain planes are small is designed. Our approach is applicable either to natural or artificial satellites.     

Influence of a second satellite on the rotational dynamics of an oblate moon

The gravitational influence of a second satellite on the rotation of an oblate moon is numerically examined. A simplified model, assuming the axis of rotation perpendicular to the (Keplerian) orbit plane, is derived. The differences between the two models, i.e. in the absence and presence of the second satellite, are investigated via bifurcation diagrams and by evolving compact sets of initial conditions in the phase space. It turns out that the presence of another satellite causes some trajectories, that were regular in its absence, to become chaotic. Moreover, the highly structured picture revealed by the bifurcation diagrams in dependence on the eccentricity of the oblate body’s orbit is destroyed when the gravitational influence is included, and the periodicities and critical curves are destroyed as well. For demonstrative purposes, focus is laid on parameters of the Saturn–Titan–Hyperion system, and on oblate satellites on low-eccentric orbits, i.e. \(e\approx 0.005\).     

Periodic orbits in the Chermnykh-like restricted problem of oblate bodies with radiation

In this paper, the periodic orbits around triangular points in the range of linear stability of the restricted three body problem, when the smaller primary and the test particle have the shape of an oblate spheroid and the larger primary is a radiation emitter with the allowance for the gravitational potential from the belt, is studied. It is observed that the orbits around these points are elliptical and have long and short periodic orbits. The period, orientation, eccentricities, the semi-major and semi-minor axis of the elliptic orbits are found. The study includes some numerical examples in the case of the Sun-Earth and Sun-Jupiter systems.     

Periodic orbits of the second kind in the restricted three-body problem when the more massive primary is an oblate spheroid

Utilizing secular perturbing potential due to oblateness, the existence of periodic orbits of the second kind is established through analytic continuation using Delaunay's canonical variables in the planar restricted three-body problem when the more massive primary is an oblate spheroid with its equatorial plane coincident with the plane of motion.     

The regular motion of an axisymmetrical satellite in the field of a triaxial body

This paper investigates the regular motions of an axisymmetrical satellite in the field of Newton's attraction of a triaxial body. Both the orbital and the self rotational motions of the two bodies are taken into consideration. The exact solutions are discussed using Poincaré's method of small parameter. In the decomposition of the force function all the harmonic terms up to the third order are taken into account.The results show the existence of eight solutions. The stability of the new group of solutions is discussed using two methods to get the necessary and sufficient conditions required for the stability of these motions.     

Classes of orbits in the main problem of satellite theory

We consider the main problem in satellite theory restricted to the polar plane. For suitable values of the energy the system has two unstable periodic orbits. We classify the trajectories in terms of their ultimate behavior with respect these periodic orbits in: oscillating, asymptotic and capture orbits. We study the energy level set and the existence and properties of the mentioned types of motion.     

Exact linearization of non-planar intermediary orbits in the satellite theory

In this paper we consider the reduction of the equations of motion for non-planar perturbed two body problems into linear form. It is seen that this can be easily accomplished for any element of the class of radial intermediaries to the satellite problem proposed by Deprit in 1981, since they have a functional dependence suitable for linearization. The transformation is worked out by using an adequate set of redundant variables. Four harmonic oscillators are obtained, of which two are coupled through gyroscopic terms. Their constant frequencies contain the secular contribution of the main problem of artificial satellite theory up to the order of the considered intermediary. Therefore, this result may well be interesting in relation to the study and prediction of accurate long-term solutions to satellite problems.     

Analytical short-term orbit predictions withJ 3 andJ 4 in terms of KS elements

Analytical theory for short-term orbit motion of satellite orbits with Earth's zonal harmonicsJ ₃ andJ ₄ is developed in terms of KS elements. Due to symmetry in KS element equations, only two of the nine equations are integrated analytically. The series expansions include terms of third power in the eccentricity. Numerical studies with two test cases reveal that orbital elements obtained from the analytical expressions match quite well with numerically integrated values during a revolution. Typically for an orbit with perigee height, eccentricity and inclination of 421.9 km, 0.17524 and 30 degrees, respectively, maximum differences of 27 and 25 cm in semimajor axis computation are noted withJ ₃ andJ ₄ term during a revolution. For application purposes, the analytical solutions can be used for accurate onboard computation of state vector in navigation and guidance packages.     

Approximate analysis for relative motion of satellite formation flying in elliptical orbits

This paper studies the relative motion of satellite formation flying in arbitrary elliptical orbits with no perturbation. The trajectories of the leader and follower satellites are projected onto the celestial sphere. These two projections and celestial equator intersect each other to form a spherical triangle, in which the vertex angles and arc-distances are used to describe the relative motion equations. This method is entitled the reference orbital element approach. Here the dimensionless distance is defined as the ratio of the maximal distance between the leader and follower satellites to the semi-major axis of the leader satellite. In close formations, this dimensionless distance, as well as some vertex angles and arc-distances of this spherical triangle, and the orbital element differences are small quantities. A series of order-of-magnitude analyses about these quantities are conducted. Consequently, the relative motion equations are approximated by expansions truncated to the second order, i.e. square of the dimensionless distance. In order to study the problem of periodicity of relative motion, the semi-major axis of the follower is expanded as Taylor series around that of the leader, by regarding relative position and velocity as small quantities. Using this expansion, it is proved that the periodicity condition derived from Lawden’s equations is equivalent to the condition that the Taylor series of order one is zero. The first-order relative motion equations, simplified from the second-order ones, possess the same forms as the periodic solutions of Lawden’s equations. It is presented that the latter are further first-order approximations to the former; and moreover, compared with the latter more suitable to research spacecraft rendezvous and docking, the former are more suitable to research relative orbit configurations. The first-order relative motion equations are expanded as trigonometric series with eccentric anomaly as the angle variable. Except the terms of order one, the trigonometric series’ amplitudes are geometric series, and corresponding phases are constant both in the radial and in-track directions. When the trajectory of the in-plane relative motion is similar to an ellipse, a method to seek this ellipse is presented. The advantage of this method is shown by an example.     

Concerning the formation of giant regular structures in the solar atmosphere

Using the Mt. Wilson magnetic synoptic charts from the recent solar activity cycles the dynamics of the formation of giant regular structures formed in the plus (leading) polarity of older more extended magnetic fields are studied. Although their diameters are about one order greater than those of supergranules, the processes of their development go analogously to those of supergranulation and granulation.The close relation of the minus (following) polarity distribution to the new formed active regions is demonstrated and the possible different mode of distribution and different role of both opposite polarities on activity processes is discussed.