KS Elements Research Articles

Effectiveness of KS elements in satellite orbit prediction using earth's gravity, drag and solar radiation pressure

Satellite moving under the gravitational field of Earth deviates from its two-body elliptic orbit, due to the combined effects of the gravitational field of Earth, atmospheric drag, solar radiation pressure, third-body gravitational effects, etc. This paper utilizes the KS regular element equations to solve Newtonian equations of motion to obtain numerical solution with respect to perturbing forces, like, Earth's gravity (includes zonal, sectorial and tesseral harmonics terms), atmospheric drag and solar radiation pressure. Effectiveness of the theory is illustrated by comparing the results with some of the existing theories in literature.Â

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

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.

Decay of high eccentricity satellite orbits with air drag using uniformly regular KS canonical elements

A non-singular analytical theory for the motion of high eccentricity satellite orbits under the influence of air drag is developed in terms of the Uniformly Regular Kustaanheimo and Stiefel (URKS) canonical elements, by assuming the atmosphere to be oblate with variation of density scale height with altitude. The series expansions include up to fourth-order terms of an independent variable Δ= λ 2 (function of eccentric anomaly) and c (a small parameter dependent on the flattening of the atmosphere). Only two of the nine equations are solved analytically due to symmetry in the equations of motion. Comparison of the important orbital parameters semi-major axis and eccentricity up to 1000 revolutions, obtained with the present analytical solution and the KS theory, shows the superiority of the present solution over the KS elements analytical solution. The theory can be used effectively for the orbital decay of aero-assisted orbital transfer orbits during mission planning.

Decay of Satellite Orbits Due to the Drag of Rotating Oblate Atmosphere with Scale Height Dependent on Altitude

The motion of near-earth satellite orbits with the air drag effect is developed in terms of the KS elements utilizing the Earth’s zonal harmonic terms up to J 6 and an oblate atmospheric model with varying scale height dependent on altitude. The variation of the rotational velocity of the atmosphere is taken into account. Numerical studies are carried out (over a wide range of orbital parameters) with Matlab software (version7). The accuracies of the numerical computations are checked with the bilinear relations in KS elements, and are found to be satisfactory. We compared our results with those of Saad et al. (Planet Space Sci 56(3–4):537–541, 2008) and found more agreement with the observed decay of RS-1.

Analytical approach using KS elements to high eccentricity orbit predictions including drag

Abstract A new non-singular analytical theory for the contraction of high eccentricity satellite orbits under the influence of air drag is developed in terms of the KS elements using an oblate atmosphere with variation of density scale height with altitude. The series expansions include up to fourth power in terms of an independent variable Λ (function of the eccentric anomaly) 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 (a) and eccentricity (e) match very well with the numerically integrated values up to 1000 revolutions over a wide range of the drag perturbed orbital parameters. Inclusion of the density scale height variation with altitude is found to increase the decay of the high eccentricity orbits up to eight percent. The theory can be used effectively for the orbital decay of aero-assisted orbital transfer orbits during mission planning.

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.

ENCKE METHODS ADAPTED TO REGULARIZING VARIABLES

In this paper formulations of Encke's type corresponding to the best known sets of linearizing coordinates KS and BF are presented. They are easier to derive than the usual formulation in Cartesian variables, since no numerical difficulties appear in the equations of motion. The experiments performed show a remarkably good behavior of the new methods for the numerical integration of satellite orbits, the performance of the classic version being clearly improved, at least in the analyzed cases corresponding to nearly circular low Earth orbits. Some questions concerning the possibility of increasing the accuracy by means of rectifications are briefly treated. Moreover our methods are proved to be able to compete with the corresponding methods in linear KS or BF elements.

Contraction of high eccentricity satellite orbits using KS elements in an oblate atmosphere

A non-singular analytical theory for the contraction of high eccentricity orbits (eccentricity e > 0.5) under the influence of air drag is developed in terms of the KS elements, using an oblate exponential atmospheric model. With the help of MACSYMA software, the series expansions include up to the sixth power in terms of an independent variable λ, introduced by Sterne as cos E =1 − H λ 2 a e , where E and a are the eccentric anomaly and semi-major axis of the orbit and H, the density scale height, is assumed constant. The solution is tested numerically over a wide range of orbital parameters: perigee height (H p), e and inclination (i) up to 100 revolutions and is found to be quite accurate. The % error in the semi-major axis computation when compared with numerically integrated values, for test cases whose perigee heights vary from 150 to 300 km and eccentricities increase from 0.6 to 0.8, is found to be less than one. The decay rates of a and e are found to be lower than those obtained with spherically symmetrical atmospheric models and increasing with increase in inclination. The theory can be effectively used for the orbital decay of Molniya type of communication satellites, decay of geostationary transfer orbits and during mission planning of aeroassisted orbital transfer orbits.

Contraction of satellite orbits using KS elements in an oblate diurnally varying atmosphere

A non‐singular analytical theory for the motion of near‐Earth satellite orbits with air drag is developed in terms of KS elements, utilizing a model of the atmosphere that allows for oblateness and in which the density behaviour approximates to the observed diurnal variation. The series expansions include up to cubic terms in e (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 at the end of each revolution due to symmetry in the KS element equations. Numerical comparisons between the analytical and numerically integrated values of the drag perturbed orbital parameters (semi‐major axis, eccentricity and argument of perigee) are made up to 100 revolutions for the test cases with different inclinations, over a range of perigee and apogee heights, which experience significant diurnal bulge effects. The comparisons are found to be good. The accuracies of the numerical computations are checked with the bilinear relation in KS elements, and are found to be very satisfactory.

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.

A third-order theory for the effect of drag on Earth satellite orbits

Analytical theory for the motion of near-Earth satellite orbits with the air drag effect is developed in terms of the KS elements, utilizing an analytical oblate exponential atmospheric model. The series expansions include up to cubic terms in e (eccentricity) and c (a small parameter dependent on the flattening of the atmosphere). Due to the symmetry of the KS element equations, only one of the eight equations is integrated analytically to obtain the state vector at the end of each revolution. Numerical comparisons are made with nine test cases, selected to cover a wide range of eccentricity with perigee heights near to 300 km at three different inclinations. A comparison of three orbital parameters: semi-major axis, eccentricity and argument of perigee, perturbed by air drag with oblate atmosphere is made with the previously developed second-order theory. It is found that with the present theory with increase in eccentricity there is improvement in semi-major axis and eccentricity computations over the second-order theory.

Analytical approach using KS elements to short-term orbit predictions including J 2

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.