Problem Of Satellite Theory Research Articles

Evolution of the Motion Around a Slowly Rotating Body

The motion of a satellite about a rotating triaxial body will be investigated, stressing on the case of slow rotation. The Hamiltonian of the problem will be formed including the zonal harmonic J2 and the leading tesseral harmonics C22 and S22. The small parameter of the problem is the spin rate (� ) of the primary. The solution proceeds through three canonical transformations to eliminate in succession; the short, intermediate and long-period terms. Thus secular and periodic terms are to be retained up to orders four and two respectively. To understand the dynamics of a spacecraft or a natural particle around a celestial body (a planet, an asteroid or even a comet) it's convenient to take into account the spin rate (� ) of the primary, since it is important for several applications especially for geodetic satellites and when dealing with communication satellites where there's commensur-ability between the satellite period and � , and a case of resonance arises (1-3). Considering a slowly rotating earth like planet, the addition of tesseral harmonics is necessary (4, 5) ana- lyzed the orbital dynamics about an asteroid and established that the major perturbations acting on the orbiter are due to the leading harmonics of the geopotential. In a subsequent work (6) he tacked the problem of secular motion in a 2 nd degree and order-gravity field with no rotation qualitatively. In this respect the main problem of artificial satellite theory is very useful; the major contributors to this subject were (7- 10) formed the Hamiltonian of the motion of an A.S. about a planet with an inhomogeneous gravitational field including the leading zonal and tesseral harmonics. He used the Whit- taker variables and then he normalized the Hamiltonian us- ing the method of elimination of the parallax developed by (11). Instead of obtaining an explicit solution of the problem he performed an exhaustion analysis of the problem. The method adopted by Palacian, through elegant, but is very difficult to include higher order terms and higher order grav- ity coefficients. In this paper the gravitational force exerted by an earth like planet on an artificial satellite will be considered, the Hamiltonian of the problem will be formed, in terms of the Delaunay variables, with the earth's spin ratetaken as a small parameter of O(1).The planet's potential will be considered up to the leading zonal and leading tesseral harmonics, an outline of the perturbation technique is given which is based on the Lie- Deprit - Kamel transform. The Hamiltonian is then normalized through three suc- cessive canonical transformations to eliminating succession of the short, intermediate, and long period terms. The proce- dure followed facilitates such including higher order terms and higher coefficients of the geopotential. 2. THE GEOPOTENTIAL

A note on lower bounds for relative equilibria in the main problem of artificial satellite theory

In the analytical approach to the main problem in satellite theory, the consideration of the physical parameters imposes a lower bound for normalized Hamiltonian. We show that there is no elliptic frozen orbits, at critical inclination, when we consider small values of H, the third component of the angular momentum. The argument used suggests that it might be applied also to more realistic zonal and tesseral models. Moreover, for almost polar orbits, when H may be taken as another small parameter, a different approach that will simplify the ephemerides generators is proposed.

Bifurcation of Relative Equilibria in the Main Problem of Artificial Satellite Theory for a Prolate Body

In this paper, we study circular orbits of the J2 problem that are confined to constant-z planes. They correspond to fixed points of the dynamics in a meridian plane. It turns out that, in the case of a prolate body, such orbits can exist that are not equatorial and branch from the equatorial one through a saddle-center bifurcation. A closed-form parametrization of these branching solutions is given and the bifurcation is studied in detail. We show both theoretically and numerically that, close to the bifurcation point, quasi-periodic orbits are created, along with two families of reversible orbits that are homoclinic to each one of them.

The Main Problem in Satellite Theory Revisited

Using the elimination of the parallax followed by the Delaunay normalization, we present a procedure for calculating a normal form of the main problem (J2 perturbation only) in satellite theory. This procedure is outlined in such a way that an object-oriented automatic symbolic manip- ulator based on a hierarchy of algebras can perform this computation. The Hamiltonian after the Delaunay normalization is presented to order six explicitly in closed form, that is, in which there is no expansion in the eccentricity. The corresponding generating function and transformation of coordinates, too lengthy to present here to the same order; the generator is given through order four.

Second-Order Solution for the Zonal Problem of Satellite Theory

The analytical solution for the perturbations of an artificial satellite due to the zonal part of the geopotential is presented. The Hamiltonian is fully normalized up to the second order by a single averaging transformation and the generating function is given explicitly. The formulas allow an arbitrarily high degree of geopotential harmonics to be included. The transformation from mean to osculating variables or vice versa is performed by means of a numerical method proposed by the author in a previous paper (Breiter,1997): periodic perturbations are computed by means of a Runge-Kutta method of order 2 instead of being explicitly derived from a generator.

Numerical continuation of families of frozen orbits in the zonal problem of artificial satellite theory

In the zonal problem of a satellite around the Earth, we continue numerically natural families of periodic orbits with the polar component of the angular momentum as the parameter. We found three families; two of them are made of orbits with linear stability while the third one is made of unstable orbits. Except in a neighborhood of the critical inclination, the stable periodic (or frozen) orbits have very small eccentricities even for large inclinations.

Numerical integration of periodic orbits in the main problem of artificial satellite theory

We describe a collection of results obtained by numerical integration of orbits in the main problem of artificial satellite theory (theJ2 problem). The periodic orbits have been classified according to their stability and the Poincare surfaces of section computed for different values ofJ2 andH (whereH is thez-component of angular momentum). The problem was scaled down to a fixed value (−1/2) of the energy constant. It is found that the pseudo-circular periodic solution plays a fundamental role. They are the equivalent of the Poincare first-kind solutions in the three-body problem. The integration of the variational equations shows that these pseudo-circular solutions are stable, except in a very narrow band near the critical inclincation. This results in a sequence of bifurcations near the critical inclination, refining therefore some known results on the critical inclination, for instance by Izsak (1963), Jupp (1975, 1980) and Cushman (1983). We also verify that the double pitchfork bifurcation around the critical inclination exists for large values ofJ2, as large as |J2|=0.2. Other secondary (higher-order) bifurcations are also described. The equations of motion were integrated in rotating meridian coordinates.

New Intermediaries for the Main Problem in Satellite Theory

AbstractAfter reviewing the original approach leading to the introduction of intermediaries in Satellite Theory, a general procedure to define intermediaries for the Main Problem in this Theory is proposed. This procedure is susceptible to an intuitive interpretation analogous to solving a simple puzzle. The application of this method to the Main Problem allows us not only to recover the well known classical intermediaries but also to obtain several completely new ones, all admitting simple solutions.

Increased accuracy of computations in the main satellite problem through linearization methods

The set of canonical redundant variables previously introduced by the first author is derived from Cartesian coordinates in a simplified form which allows the reduction of the Kepler problem to four harmonic oscillators with unit frequency. The coordinates are defined to be the direction cosines of the position of the particle along with the inverse of its distance. True anomaly is the new independent variable. The behavior of this new transformation is studied when applied to the numerical integrations of the main problem in satellite theory. In particular, computation time and accuracy of orbits in the new variables are compared with those in K-S and Cartesian variables. It is noteworthy that for high eccentricities the new variables require the least computation time for comparable accuracy, regardless of the integration scheme.

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.

Note on Cid's radial intermediary and the method of averaging

In the main problem of artificial satellite theory, the difference between the Hamiltonian and Cid's radial intermediary is a function of the argument of latitude whose average over the mean anomaly is zero.

The critical inclination in artificial satellite theory

Certain it is that the critical inclination in the main problem of artificial satellite theory is an intrinsic singularity. Its significance stems from two geometric events in the reduced phase space on the manifolds of constant polar angular momentum and constant Delaunay action. In the neighborhood of the critical inclination, along the family of circular orbits, there appear two Hopf bifurcations, to each of which there converge two families of orbits with stationary perigees. On the stretch between the bifurcations, the circular orbits in the planes at critical inclinmation are unstable. A global analysis of the double forking is made possible by the realization that the reduced phase space consists of bundles of two-dimensional spheres. Extensive numerical integrations illustrate the transitions in the phase flow on the spheres as the system passes through the bifurcations.

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.

Constrained normalization of Hamiltonian systems and perturbed Keplerian motion

Consider a Hamiltonian system (H, ℝ2n ,ω). LetM be a symplectic submanifold of (ℝ2n ,ω). The system (H, ℝ2n ,ω) constrained toM is (H∣M, M, ω∣M). In this paper we give an algorithm which normalizes the system on ℝ2n in such a way that restricted toM we have normalized the constrained system. This procedure is then applied to perturbed Kepler systems such as the lunar problem and the main problem of artificial satellite theory.

Third-Order Solution to the Main Problem in Satellite Theory

In the present paper we announce a completely analytic closed-form third-order solution to the main problem in the theory of an artificial satellite. This is the first time an analytic solution of the main problem has been produced to order 3 which is valid for satellites with any eccentricity 0<Kl. The solution is accomplished by constructing a progression of three canonical transformations from the state variables to a set of action-angle variables in which the Hamiltonian for the problem is a function of the action variables only. The transformed Hamiltonian is developed, without omitting terms, to order 4 in the small parameter €= -J2; by way of verification it was found to agree through order 3 with the theory of Brouwer as extended by Kozai. The algebraic expressions for these transformations were produced by computer in a very compact form; conciseness is achieved in two ways, by eliminating the parallax and by controlling the computer automated calculations so as to avoid infinite series expansions in the eccentricity. Our programs prove that the main problem in satellite theory can be solved in closed form to order 3.

A note on ?The main problem of satellite theory for small eccentricities, by A. Deprit and A. Rom, 1970?

Criticism of Deprit and Rom's (1970) extension of Brouwer's (1959) first-order satellite theory to the third order in J sub 2. It is noted that Deprit and Rom's theory suffers from the drawback that the perturbations of all orders are expressed as infinite power series in the eccentricity e. It is demonstrated that Deprit and Rom's reported failure to extend Brouwer's theory to the second order in a closed form by means of Lie transforms is due to an oversight and is not caused by difficulties inherent in the problem or in the particular perturbation method used.

The main problem of artificial satellite theory for small and moderate eccentricities

Perturbation techniques based on Lie transforms as suggested by Deprit were used as the theoretical foundation for programming the analytical solution of the Main Problem in Satellite Theory (all gravitational harmonics being zero exceptJ 2). The collection of formulas necessary and sufficient to construct an ephemeris is given in the exposition. Short and long period displacements, as well as the secular terms, have been obtained up to the third order inJ 2 as power series of the eccentricity. They result from two successive completely canonical transformations which it has been found convenient not to compose into a unique transformation. Division by the eccentricity appears nowhere in the developments-neither explicitly nor implicitly. The determination of the constants of motion from the initial conditions has been given an elementary solution that is both complete and explicit without being iterative. The program was developed by Rom from MAO's package of subroutines forMechanizedAlgebraicOperations. Reliability tests have been run in two instances: in-track errors for ANNA 1B are only 20 cm after 210 days in orbit, while for RELAY II, they are 2.4 m, even after 350 days in orbit.

Solution of the problem of artificial satellite theory without drag

view Abstract Citations (476) References (2) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS Solution of the problem of artificial satellite theory without drag Brouwer, Dirk Abstract . Sections i-6 give the solution of the main problem for a spheroidal earth with the potential limited to the principal term and the second harmonic which contains the small factor k2. The solution is developed in powers of k2 in canonical variables by a method which is basically the same as that used in treating a different problem by von Zeipel (1916). The periodic terms are divided in two classes: the short-period terms contain the mean anomaly in their argu- ments; the arguments of the long-period terms are multiples of the mean argument of the perigee. The periodic terms, both of short and long period, are developed to 0(k2); the secular motions are obtained to 0(k22). The results are obtained in closed form; no series developments in eccentriciLy or inclination arise. The solution does not apply to orbits near the critical inclination, 63?4, but is otherwise valid for any eccentricity I and any inclination. Section 7 gives the long-period terms and the additions to the secular motions caused by the fourth harmonic in the potential; section 8 gives the contt-ibutions by the third and fifth harmonics; section 9 contains formulas for computation. Publication: The Astronomical Journal Pub Date: November 1959 DOI: 10.1086/107958 Bibcode: 1959AJ.....64..378B full text sources ADS |