Non-integrability of the problem of motion around an oblate planet

Celletti, Alessandra; Negrini, Piero

doi:10.1007/bf00051896

Cited by 35 publications

(28 citation statements)

References 8 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In spite of the known general nonintegrability of the main problem (Irigoyen and Simo 1993;Celletti and Negrini 1995), approximate solutions to the dynamics determined by the Hamiltonian in Eq. (1) may be quite useful in practical applications.…”

Section: The Main Problem Hamiltonianmentioning

confidence: 98%

“…The problem of the motion in this potential field is known as the main problem in artificial satellites theory, and was the subject of many studies over the years. Because the dynamics are nonintegrable (except for equatorial motion), and might even exhibit chaotic behaviour (Irigoyen and Simo 1993;Celletti and Negrini 1995), the only possible closed-form solutions may be obtained by approximating and/or averaging the full J 2 gravitational potential.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Satellite onboard orbit propagation using Deprit’s radial intermediary

Gurfil

Lara²

2014

Celest Mech Dyn Astr

View full text Add to dashboard Cite

Short-term satellite onboard orbit propagation is required when GPS position measurements are unavailable due to an obstruction or a malfunction. In this paper, it is shown that natural intermediary orbits of the main problem provide a useful alternative for the implementation of short-term onboard orbit propagators instead of direct numerical integration. Among these intermediaries, Deprit's radial intermediary (DRI), obtained by the elimination of the parallax transformation, shows clear merits in terms of computational efficiency and accuracy. Indeed, this proposed analytical solution is free from elliptic integrals, as opposed to other intermediaries, thus speeding the evaluation of corresponding expressions. The only remaining equation to be solved by iterations is the Kepler equation, which in most of cases does not impact the total computation time. A comprehensive performance evaluation using Monte-Carlo simulations is performed for various orbital inclinations, showing that the analytical solution based on DRI outperforms a Dormand-Prince fixed-step Runge-Kutta integrator as the inclination grows.

show abstract

Section: The Main Problem Hamiltonianmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

Satellite onboard orbit propagation using Deprit’s radial intermediary

Gurfil

Lara²

2014

Celest Mech Dyn Astr

View full text Add to dashboard Cite

show abstract

“…The integrability of the IVP (18) with the potential having the form (19) is still an open problem. It was proven that when only the J 2 term is retained, the problem is generally non-integrable (Celletti and Negrini 1995;Irigoyen and Simo 1993).…”

Section: Absolute Motion In An Axially-symmetric Gravitational Fieldmentioning

confidence: 99%

Solutions and periodicity of satellite relative motion under even zonal harmonics perturbations

Martinusi

Gurfil

2011

Celest Mech Dyn Astr

View full text Add to dashboard Cite

Finding relative satellite orbits that guarantee long-term bounded relative motion is important for cluster flight, wherein a group of satellites remain within bounded distances while applying very few formationkeeping maneuvers. However, most existing astrodynamical approaches utilize mean orbital elements for detecting bounded relative orbits, and therefore cannot guarantee long-term boundedness under realistic gravitational models. The main purpose of the present paper is to develop analytical methods for designing long-term bounded relative orbits under high-order gravitational perturbations. The key underlying observation is that in the presence of arbitrarily high-order even zonal harmonics perturbations, the dynamics are superintegrable for equatorial orbits. When only J 2 is considered, the current paper offers a closed-form solution for the relative motion in the equatorial plane using elliptic integrals. Moreover, necessary and sufficient periodicity conditions for the relative motion are determined. The proposed methodology for the J 2 -perturbed relative motion is then extended to non-equatorial orbits and to the case of any high-order even zonal harmonics (J 2n , n ≥ 1). Numerical simulations show how the suggested methodology can be implemented for designing bounded relative quasiperiodic orbits in the presence of the complete zonal part of the gravitational potential.

show abstract

“…the second zonal harmonic J2, draws great attention due to the nonintegrability of the main problem in the artificial satellite theory and it may also induce chaotic motion under certain conditions [4]. Brouwer [3] and Kozai [9] made significant contributions on the study of J2 perturbed satellite motion, among several theories that emerged in the past.…”

Section: Introductionmentioning

confidence: 99%

On J2 short-term orbit predictions in terms of KS elements

Saji

Sellamuthu

Sharma

2017

IJAA

View full text Add to dashboard Cite

Sharma's singularity-free analytical theory for the short-term orbital motion of satellites in terms of KS elements in closed form in eccentricity with Earth's zonal harmonic term J2, is improved by using King-Hele's expression for the radial distance 'r' which includes the effect of J2, and is suitable for low eccentricity orbits. Numerical experimentation with four test cases with perigee altitude of 200 km and eccentricity varying from 0.01 to 0.3 for different inclinations is carried out. It is found that the orbital elements computed with the analytical expressions in a single step during half a revolution match very well with numerically integrated values and show significant improvement over the earlier theory. The solution can be effectively used for computation of mean elements for near-Earth orbits, where the short-term orbit perturbations due to J2 play most important role. The theory will be very useful in computing the state vectors during the coast phase of rocket trajectories and flight algorithms for on-board implementation.

show abstract

Non-integrability of the problem of motion around an oblate planet

Abstract: We provide a result of non-analytic integrability of the so-called J 2-problem. Precisely by using the Lerman theorem we are able to prove the existence of a region of the phase space, where the dynamical system exhibits chaotic motions

Cited by 35 publications

References 8 publications

Satellite onboard orbit propagation using Deprit’s radial intermediary

Satellite onboard orbit propagation using Deprit’s radial intermediary

Solutions and periodicity of satellite relative motion under even zonal harmonics perturbations

On J2 short-term orbit predictions in terms of KS elements

Contact Info

Product

Resources

About