Long-term orbit computations with KS uniformly regular canonical elements with oblateness
TL;DR: In this paper, a fixed step-size fourth-order Runge-Kutta-Gill method is employed for numerical integration of the canonical equations with Earth's oblateness.
Abstract: 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 Ωm near the minima to em are noticed. The values of Ω
m are found to be very near to +-90 degrees at the extrema of em. 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.
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01 Feb 1992
TL;DR: In this paper, the classical and generalized Sundman time transformations are used to establish new generating set of differential equations of motion in terms of the Eulerian redundant parameters for the Earth's gravitational field with axial symmetry with zonal harmonic terms up to J36.
Abstract: In this paper, the classical and generalized Sundman time transformations are used to establish new generating set of differential equations of motion in terms of the Eulerian redundant parameters. The implementation of this set on digital computers for the commonly used independent variables is developed once and for all. Motion prediction algorithms based on these equations are developed in a recursive manner for the motions in the Earth's gravitational field with axial symmetry whatever the number of the zonal harmonic terms may be. Applications for the two types of short and long term predictions are considered for the perturbed motion in the Earth's gravitational field with axial symmetry with zonal harmonic terms up to J36. Numerical results proved the very high efficiency and flexibility of the developed equations.
15 citations
TL;DR: The full recurrent power series solution is established for J2-gravity perturbed motion in terms of the Eulerian redundant parameters, and a final state of very high accuracy is obtained for each case study.
Abstract: In this paper the full recurrent power series solution is established for J2-gravity perturbed motion in terms of the Eulerian redundant parameters. Applications of the method for the problem of the final state prediction are illustrated by numerical examples of some typical ballistic missiles, a final state of very high accuracy is obtained for each case study.
14 citations
TL;DR: In this paper, the Earth's zonal harmonic J2 perturbation is considered, and analytical solutions using KS elements are derived for short-term orbit computations, where only two of the nine KS element equations are integrated analytically due to the reasons of symmetry.
Abstract: Analytical solutions using KS elements are derived. The perturbation considered is the Earth's zonal harmonic J
2. The series expansions include terms of fourth power in the eccentricity. Only two of the nine KS element equations are integrated analytically due to the reasons of symmetry. The analytical solution is suitable for short-term orbit computations. Numerical studies show that reasonably good estimates of the orbital elements can be obtained in one step of 10 to 30 degrees of eccentric anomaly for near-Earth orbits of moderate eccentricity. For application purposes, the analytical solution can be effectively used for onboard computation in the navigation and guidance packages, where the modelling of J
2 effect becomes necessary.
10 citations
01 Jan 2004
TL;DR: In this article, a 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.
Abstract: 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. r 2006 Elsevier Ltd. All rights reserved.
7 citations
TL;DR: In this article, the perturbed motion of the artificial satellites under the effects of the earth's oblateness and atmospheric drag was established in terms of the Eulerian redundant parameters.
Abstract: In this paper, the perturbed motion of the artificial satellites under the effects of the earth's oblateness and atmospheric drag will be established. These equations will be expressed in terms of the Eulerian redundant parameters. Applications of the method for the perturbed motion are illustrated by numerical examples for some test cases of the orbits.
7 citations
References
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TL;DR: In this article, Kustaanheimo et al. developed a regularization of Kepler motion using a simple mapping of a four-dimensional space R* onto a 3D space Ä, where the equations of any undisturbed Kepler motion are linear differential equations.
Abstract: A regularization of Kepler motion in the three-dimensional space Ä is developed using a simple mapping of a four-dimensional space R* onto Ä. In Ä* the equations of any undisturbed Kepler motion are linear differential equations with constant coefficients thus remaining completely regulär at the center of attraction. This agreeable property of the equations makes them well suited for computation of perturbations. Many classical theories of perturbations in rectangular coordinates begin with a linearization of the equations of motion. In our theory such a preconditioning of the problem is avoided since the equations in jß are already linear. From a practical point of view it may be a disadvantage that the regularized methods require more integrations than the classical procedures. The paper was written during P. Kustaanheimo's stay at the Zürich in summer 1964. No knowledge of spinors is needed for understanding the paper. The authors are indebted to IBM for Sponsoring this work. 1. Levi Civita's regularization o! the plane motion Let z = xl + ix2 be the plane of the orbit of a moving body. For regularization purposes Levi-Civita proposed in [1] to introduce a parameter-plane w = u^ + iu2 mapped onto the physical z-plane conformally by the transformation / j x o #1 = \\ — U\\ (1) z = w, or l l 2 3/2 — 2iU^U^· Accordingly parabolic coordinates are introduced in the physical plane. A conical section centered at the origin of the w-plane is transformed into a conical section of the z-plane having one focus at the origin; therefore the transformation is a very practical method for the discussion of a Kepler motion. In order to get an idea about a generalization of such transformations in higherdimensional spaces, we observe the transformation of the differentials (2) dx1 = 2(w1dii1 — u2du2), dx2 = 2(ztgdzt1 + Wjdwg) attached to the transformation of the coordinates. In matrix notation this can be written äs ( _ (u, —»t\\/<*»i\\ (3) W U «JW The matrix of this linear transformation has the following basic properties. Kustaanheimo and S t i e f e l , Perturbation theory of Kepler motion 205 1. Each element is a homogeneous linear function of the parameters ? w 2 . In our case the elements are very trivial functions of this type because only one parameter appears in each element. However this specialisation is not essential. As a matter of fact it disappears if other cartesian coordinate-systems are used in the two planes. 2. The matrix is orthogonal. The scalar product of the two lines vanishes and each line has the norm (u\\ + Mg). From this it follows (4) dx\\ + dx\\ = 4(»J + u\\) (du\\ + du\\) . Property 2 is essentially equivalent to the fact that the mapping is conformal. The local ratio of magnification is äs follows from (4). 2. Oeneralizations We want to investigate whether an analogous Situation is possible in the /i-dimensional space R. Let w1? w 2 ? · · · » n he rectangular coordinates in R. A matrix A with n rows and columns is desired with properties analogous to l, 2 that is 1. The elements are linear homogeneous functions of the ut. 2. The matrix is orthogonal in the following sense: a) The scalar product of two different rows vanishes. b) Each row has the norm (u\\ + u\\ + · · · + un). A famous result of A. Hurwitz [2] states that such a matrix can only be produced if n — l, 2, 4 or 8. This problem was later studied and generalized by many authors [3 — 6]. The case n = l is trivial, the case n = 2 was discussed above. Unfortunately the case n = 3 is out of question; therefore there is no easy generalization of Levi-Civita's transformation in Ä. Hence we proceed to the case n = 4. A matrix satisfying our desires is for example
709 citations
TL;DR: In this paper, economical and stable recurrence formulae for the Earth's zonal potential and its gradient for the KS regularized theory were established for any number of n of the zonal harmonic coefficients.
Abstract: In this paper, economical and stable recurrence formulae for the Earth's zonal potential and its gradient for the KS regularized theory will be established for any numberN of the zonal harmonic coefficient A general recursive computational algorithm based on these formulae is also established for the initial value problem of the KS theory for the prediction of artificial satellites in the Earth's gravitational field with axial symmetry Applications of the algorithm for the problem of the final state prediction are illustrated by numerical examples of three test orbits each for two geopotential models corresponding toN=2 andN=36 A final state of any desired accuracy is obtained for each case study, a result which shows the flexibility of the algorithm
11 citations
TL;DR: In this article, the first order perturbations of the elements of near-circular earth satellite orbits are calculated to an accuracy of the order of and including the first power of eccentricity.
Abstract: The subject of this paper is the calculation of perturbations of the elements of near-circular earth satellite orbits Literal expressions are found for calculating first-order perturbations to an accuracy of the order of and including the first power of eccentricity The improvement of the elements of near-circular orbits is also discussed One section deals with the determination of constants of integration
3 citations