## ORBIT PREDICTIONS OF A NEAR-EARTH SATELLITE UNDER THE COMBINED EFFECTS OF THE EARTH’S OBLATENESS AND AIR-DRAG IN TERMS OF KS ELEMENTS

ORBIT PREDICTIONS OF A NEAR-EARTH SATELLITE UNDER THE COMBINED EFFECTS OF THE EARTH’S OBLATENESS AND AIR-DRAG IN TERMS OF KS ELEMENTS

REPORT OF MAJOR RESEARCH PROJECT

Submitted to

University Grants Commission,

Bahadur Shah Zafar Marg,

New Delhi

Pin: 110 002

Principal Investigator

Dr LILA S NAIR

Associate Professor (Retd.),

H H M S P B NSS COLLEGE FOR WOMEN,

NEERAMANKARA,

THIRUVANANTHAPURAM. 695040

**SUMMARY**

1. The work is carried out in three stages. The KS element equations are integrated analytically by a series expansion method by assuming

(i) spherically exponential atmosphere including Earth’s oblateness which corresponds to zonal harmonic J2

(ii) an oblate diurnal atmosphere together with Earth’s zonal harmonics J3 and J4

(iii) an oblate diurnal atmosphere when density scale height varies with altitude and including the terms corresponds to Earth’s zonal harmonics J2, J3 and J4

First stage provides a non-singular analytical theory of long-term orbital motion of satellites under the effect of Earth’s oblateness and air drag due to spherically symmetrical exponential atmosphere, in terms of the KS elements. The series expansion method is used to compute terms, due to the Earth’s zonal harmonic J2 and analytically developed density model of a spherically symmetrical exponential atmosphere, in the generalized form of the KS element equations. The chapter ends with the comparison of the analytically integrated values of the changes in the orbital element, argument of perigee, caused by the perturbation due to the Earth’s zonal harmonics J2 and air drag of spherically symmetrical exponential atmosphere with the numerically integrated results.

The orbital theory in terms of KS elements to find the combined effect of the perturbation due to J3, J4 and the diurnally varying and oblate atmospheric density in satellite’s orbit prediction is explained in second stage. The analytical model for the density is developed up to third-order terms in e and c. The KS element equations are formulated for long-term orbit predictions with the terms J3, J4 and using the analytically developed atmospheric density model. Numerical experimentation with the test cases reveals that reasonably accurate values of the argument of perigee are obtained after 100 revolutions.

In third stage an analytical solution with J2, J3 and J4 terms for an oblate diurnally varying atmosphere with variation of the scale height depending on altitude for long-term motion is described. An analytical density model with fourth-order terms in c and e as well as the second order terms in μ, gradient of the scale height altitude is developed. The KS element equations are analytically integrated to obtain the changes in the KS elements due to the terms added to the equations which are contributed by the new density function. Comparison of the analytical solution is made with the numerically integrated values.

The analytical expressions are singularity free and can be used efficiently for near-Earth non-circular orbits. A wide range of eccentricity and inclination is considered for calculating the change in argument of perigee by present analytical theory and by numerical integration. The accuracies of the numerical computations examined with the help of the bilinear relation in the KS elements are found to be very satisfactory Comparison between analytically and numerically integrated values for 1 and 100 revolutions shows that the analytically integrated values are reasonably accurate and thus highlights the usefulness of the analytical expressions. Graphical representation as well as the tables presented here emphasizes the importance of developing the theory to find the decrease in argument of perigee and from the theory developed through chapters 2, 3 and 4, it is clear that terms corresponding to J2 contributes maximum change in argument of perigee.