Analytical theory in terms of J2, J3, J4 with eccentric anomaly for short-term orbit predictions using uniformly regular KS canonical elements

A new non-singular analytical theory with respect to the Earth’s zonal harmonic terms J2, J3, J4 has been developed for short-periodic motion, by analytically integrating the uniformly regular KS canonical equations of motion using generalized eccentric anomaly ‘E’ as the independent variable. Only one of the eight equations need to be integrated analytically to generate the state vector, due to the symmetry in the equations of motion, and the computation for the other equations is done by changing the initial conditions. King-Hele’s expression for radial distance ‘r’ with J2 is also considered in generating the solution. The results obtained from the analytical expressions in a single step during half a revolution match quite well with numerically integrated values. Numerical results also indicate that the solution is reasonably accurate for a wide range of orbital elements during half a revolution and is an improvement over Sharon et al. [17] theory, which is generated in terms of KS regular elements. It can be used for studying the short-term relative motion of two or more space objects and in collision avoidance studies of space objects. It can be also useful for onboard computation in the navigation and guidance packages.


Introduction
In the artificial satellite theory, the motion of a satellite under the effect of Earth's oblateness, namely the second zonal harmonic J2 in the gravitational potential field is known as the main problem. Any Earth satellite mission requires precise orbit computation under the influence of this dominating perturbation. The non-integrability dynamics of the J2 problem [2] allows the avenue for analytical theories to be developed. In the past, several authors had treated this problem to obtain closed form solutions using different methods. Several analytical theories for the motion of Earth's satellite under the effect of Earth's first few zonal harmonic terms are available in the literature. Some of the notable are by [10], [12], [6], [3], [1], [11] and [7]. The KS transformation regularizes the non-linear equations of motion and converts into linear differential equations of a harmonic oscillator. KS formulation was used by [5] and [9] for short-term orbit predictions with J2 effect. The KS uniform regular canonical equations of motion [19] are a particular canonical form where all the ten elements are constant for unperturbed two-body problem and are applicable to elliptic, parabolic and hyperbolic orbital motion. In [13] these equations were numerically integrated to obtain accurate orbits under the effect of Earth's oblateness with zonal harmonic terms up to J36. Analytical theory in terms of KS elements with J2 [14] and [16], and with J3 and J4 [15] was developed for short-term orbit predictions. [8] analytically integrated the uniformly regular KS canonical elements with Earth's zonal harmonics J2, J3 and J4. The independent variable, fictitious time 's' given by dt/ds = r with t and r being the physical time and radial distance, respectively, and used for analytical integration, resulted in complicated integrals. Because of the complexity of the integrals in evaluation for practical problems, the utility of the analytical solution was limited for operational purposes. {18] developed a new non-singular analytical solution with J2 in close form in eccentricity 'e' for short-term orbit predictions by analytically integrating the uniformly regular KS canonical equations of motion, using the generalized eccentric anomaly 'E' as the independent variable. The integrals are found to be much simpler than obtained in [8]. In this paper, the analytical solution of [18] is improved by using King-Hele's expression [10] for radial distance r as function of J2. Further, new non-singular analytical solutions with J3 and J4 in close form in eccentricity and inclination for short-term orbit predictions by analytically integrating the uniformly regular KS canonical equations of motion, using the generalized eccentric anomaly 'E' as the independent variable are developed. Numerical study has been carried out for a wide range of orbital parameters. The theory is found to provide reasonably accurate results over half a revolution.
The solutions can have number of applications. It can be used for studying the short-term relative motion of two or more space objects and in collision avoidance studies of space objects and generation of mean orbital elements. It can be also useful for onboard computation in the navigation and guidance packages, where the modeling of J2 effect becomes necessary.

Equations of motion
The K-S uniformly regular canonical equations of motion in terms of fictitious time s are [13], [18] The relation between s and E is given by Equation (1) in terms of E can be written as When the perturbation due to Earth's oblateness J2 is considered: Perturbing potential, ), x = (x 1 , x 2 , x 3 ) = L(u)u, r = √(x 1 2 + x 2 2 + x 3 2 ) = u 1 2 + u 2 2 + u 3 2 + u 4 2 , Where h, K 2 , R, E, r, Jn are total energy, gravitational parameter, Earth's equatorial radius, eccentric anomaly, radial distance and nth zonal harmonic term of Earth, respectively.

Expression for radial distance 'r' in terms of J2
Substituting the value of * , we get

Numerical results
For computing the results with J2, three test cases at an inclination of 85° for eccentricities 0.01, 0.1 and 0.2, having perigee height of 200 km are chosen to show the effectiveness of the present theory. The other initial conditions for the orbit are given as right ascension of ascending node (Ω) = 60° and argument of perigee (ω) = 0°. The results were generated to validate the improvement obtained with the addition of King Hele's [10] expression with J2. The difference between the numerically integrated and analytically computed values with the modified theory (ANAL1) and existing theory (ANAL2) by Sharon et al. [17] during half a revolution with a single analytical step size are given in Table 1. It may be noted that the modified theory provides more accurate values of the important orbital parameter 'semi-major axis' during half a revolution than Sharon et al. theory [17]. To generate the results with J3 and J4, three test cases A, B and C having eccentricities of 0.03791, 0.17524 and 0.53964 with inclination of 30° are chosen to show the effectiveness of the present theory for very small to very high eccentricity orbits. Details of the initial state vector x, ẋ along with the resulting orbital elements are provided in Table 2. As may be seen from Tables 3 and 4, the error is found to be less than 0.55 % with J3 and less than 0.85% with J4 during half a revolution in all the numerical simulations carried out.

Conclusion
K-S uniformly regular canonical equations of motion with generalized eccentric anomaly provide a very efficient and accurate analytical integration method for short-term orbit computation with Earth's oblateness for short-term motion. Only one of the eight equations need to be integrated analytically to generate the state vectors, because of symmetry in the equations of motion. Numerical results indicate that the solution is quite accurate for a wide range of eccentricity. The solution obtained using King-Hele's expression for radial distance as function of J2 is an improvement over the existing theory of Sharon et al. [17] which uses KS regular elements. The percentage error is found to be less than 0.55% for J3 and less than 0.85% for J4. The solutions can have number of applications. It can be used for studying the short-term relative motion of two or more space objects and in collision avoidance studies of space objects. It can be also useful for onboard computation in the navigation and guidance packages and generation of mean orbital elements.