Problem Of Critical Inclination Research Articles

Analysis on the long term orbital evolution of Molniya satellites

Long term evolution of the Molniya satellites are investigated by means of historical data analysis, theoretical analysis and numerical integration. Both the mean motion resonance problem and the critical inclination problem are studied. The period and the amplitude of the semi-major axis for each satellite are obtained analytically and compared with the observational data. In addition, the reason of the observed sudden changes in the center and the amplitude of the oscillating semi-major axes is determined as the effect of the atmosphere drag. For the long period perigee motion, the dominant perturbations come from the luni-solar gravity. A two-degree-of freedom system is established by adding the two periodic terms of the neighbor resonances to the Hamiltonian of the classical single resonance model. In theory, the resulting resonance overlap model is responsible for the chaotic layer between the libration region and the circulation region. In practice, it is applied to explain the quick decay of the earliest Molniya satellites and to study the satellites that still orbiting the Earth at present.

Long term evolution of Molniya orbit under the effect of Earth’s non-spherical gravitational perturbation

A double resonance model is applied to study the long term evolution of a Molniya orbit, which is highly elliptical (e⩾0.7), critically inclined (i≈63.4°), and in the state of the 2:1 mean motion resonance with the Earth rotation. The dynamics of a Molniya orbit can be divided into three kinds: short (12h), intermediate (several years) and long (several centuries) period motions, with the latter two studied in this paper. The J2 and Jl2 (l=2,3,…,8) harmonics are modelled, based on a careful selection. The analytic solution for the intermediate period motion is obtained, a first integral, I¯3, for the long period motion is derived analytically, and the phase structures are obtained by the level curves of I¯3. Three types of the phase structures, depending on the equilibria and stabilities, are observed when the Hamiltonian constant varies. Compared with the near circular 12-h satellite orbits and with the critically inclined orbits without mean motion resonance with the Earth rotation, the features of the Molniya orbits are discussed in detail. It is pointed out that (1) unlike the case of near circular orbits, the J32 term does not dominate the 2:1 mean motion resonance problem (intermediate period motion), and that (2) instead of the J22 terms, the resonant tesseral harmonics dominate the critical inclination problem (long period motion).

The post-Newtonian effects in the critical inclination problem in artificial satellite theory

The critical inclination problem in artificial satellite theory and the post-Newtonian (PN) effects on it are discussed. The Hamiltonian is constructed and is expressed in terms of the Delaunay variables containing the zonal harmonics up to J 4 and the PN contribution up to order J 2/ c l 2. The Hamiltonian is ordered out such that the PN terms are of order J 2. Two successive canonical transformations based on Lie series and transform are carried out to remove short and long period terms from the Hamiltonian, respectively. Terms are retained up to the third order in the secular perturbations and to the second order in the periodic perturbations, where J 2 is considered of first order.

Canonical solution of the critical inclination problem taking into account PN-corrections

The critical inclination problem in artificial satellite theory and the post-Newtonian (PN) effects on it are discussed. The Hamiltonian is constructed and is expressed in terms of the Delaunay variables containing the zonal harmonics up to J 4 and the PN contributions up to order J 2/ c l 2. The Hamiltonian is ordered out such that the PN terms are of order J 2. Two successive canonical transformations based on Lie series and transform are carried out to remove short- and long-period terms from the Hamiltonian, respectively. Terms are retained up to third order in the secular and to the second order in the periodic terms, where J 2 is considered of first order.

The problem of critical inclination combined with a resonance in mean motion in artificial satellite theory

In this paper the two-degree of freedom problem of a geosynchronous artificial satellite orbiting near the critical inclination is studied. First a local approach of this problem is considered. A semi-numerical method, well suited to describe the perturbations of a non-trivial separable system, is then applied such that surfaces of section illustrating the global secular dynamics are obtained. The results are confirmed by numerical integrations of the full Hamiltonian.

The problem of critical inclination combined with a resonance in mean motion in artificial satellite theory

The problem of critical inclination combined with a resonance in mean motion in artificial satellite theory

Relativistic effects in the critical inclination problem in artificial satellite theory

It is well known that in artifical satellite theory special techniques must be employed to construct a formal solution whenever the orbital inclination is sufficiently close to the critical value cos−1 (1/√5). In this article the authors investigate the consequences of introducing certain relativistic effects into the motion of a satellite about an oblate primary. Particular attention is paid to the critical inclination(s), and for such critical motions an appropriate method of solution is formulated.

The critical inclination problem with small eccentricity

The general theory described in an earlier paper is used to investigate the long-term behavior of the eccentricity and argument of the perigee for Earth satellites which have nearly circular orbits and orbital inclinations in the neighbourhood of the critical value 63°.4. Taking into account the effect of all the zonal harmonics, it is shown that there are just twoTypes of behaviour possible, compared with the six described in the earlier general theory and the four found by Aoki (1963).

The critical inclination problem with small eccentricity

The critical inclination problem with small eccentricity

Formal solution in the problem of small divisors

view Abstract Citations (87) References (6) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS Formal solution in the problem of small divisors Garfinkel, Boris Abstract Small divisors in mechanics arise from resonance, i.e., the near-commensurability of two of the natural frequencies of the motion. Under very general assumptions, the problem can be reduced to a canonical form of one degree of freedom, with the Hamiltonian F defined by F=Ao(y)+Ao(y) cos2z. Here z is the coordinate, y the momentum, and the coefficients A0 and A0 satisfy the relation A1/A0=O(k), where k is a small parameter. Resonance is associated with the vanishing of the derivative A 0' for some value Y=Y* In the neighborhood of Y*, the resonance problem is solved to O(k-") by the Bohlin-von Zeipel technique, and the solution is expressed in terms of elliptic functions. It is shown that the resonance solution reduces to the classical solution with a singularity at Y=Y*, when the divisor * is sufficiently large. Applications to celestial mechanics are illustrated by the problem of critical inclination and by the 24-h satellite in the artificial satellite theory. Publication: The Astronomical Journal Pub Date: October 1966 DOI: 10.1086/110171 Bibcode: 1966AJ.....71..657G full text sources ADS |