Analytical Orbit Predictions with Oblate Atmosphere using K-S Uniformly Regular Canonical Elements
- Paper number
IAC-05-C1.6.04
- Author
Dr. Ram Krishan Sharma, Indian Space Research Organisation (ISRO), India
- Coauthor
Mr. Xavier James Raj, Indian Space Research Organisation (ISRO), India
- Year
2005
- Abstract
Accurate orbit prediction of the Earth’s satellites is an important requirement for mission planning, satellite geodesy, spacecraft navigation, re-entry and orbital lifetime estimates. For this purpose, it has become necessary to use extremely complex force models to match with the present operational requirements and observational techniques. The problem becomes all the more complicated in the near-Earth environment due to the fact that the satellite is influenced by the non-spherical effects of the Earth’s gravitational field as well as the dissipative effects of the Earth’s atmosphere. The effects of the atmosphere are difficult to determine since the atmospheric density, and hence the drag, undergoes large modelled fluctuations. Though the accurate ephemeris of a near-Earth satellite can be generated by the numerical integration methods with respect to a complex force model, the analytical solutions, though difficult to obtain for complex force models and limited to relatively simple models, represent a manifold of solutions for a large domain of initial conditions and find indispensable application to mission planning and qualitative analysis. The method of the K-S total-energy element equations (Stiefel and Scheifele, 1971) is a powerful method for numerical solution with respect to any type of perturbing forces, as the equations are less sensitive to round-off and truncation errors in the numerical algorithm. The equations are everywhere regular in contrast with the classical Newtonian equations, which are singular at the collision of the two bodies. The equations are smoothed for eccentric orbits because eccentric anomaly is the independent variable. These equations have been used effectively to generate accurate analytical solution with respect to Earth=s oblateness for short-term motion (Sharma, 1997) and air drag force for long-term motion (Sharma, 1991, 1992, 1997; Nair and Sharma, 2003). A particular canonical form of the K-S differential equations, known as K-S uniform regular canonical equations, where all the ten elements are constant for unperturbed motion and the equations permit the uniform formulation of the basic laws of elliptic, parabolic and hyperbolic motion (Stiefel and Scheifele, 1971, p250) are found to provide accurate short- and long- term orbit predictions numerically, with Earth\'s zonal and tesseral harmonic terms (Sharma and James Raj, 1988;1999). Recently these equations were utilized to generate accurate analytical solutions for short-term orbit predictions with respect to Earth=s zonal harmonic terms J2, J3, J4 (James Raj and Sharma, 2003). In this paper we have extended the K-S uniform regular canonical equations of motion for inclusion of the canonical forces and developed a new analytical theory by analytically integrating the resulting equations of motion by a series expansion method with air drag force.The atmosphere is assumed to be oblate with constant density scale height. A non-singular solution up to third-order terms in eccentricity is obtained. 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. For comparison purpose, the KS element equations are integrated numerically with the same force model with a fixed step size 4th order Runge-Kutta-Gill method using a small step size of half degree in eccentric anomaly. Numerical experimentation with the analytical solution over a wide range of perigee altitude, eccentricity and orbital inclination has been carried out up to 1000 revolutions. The results obtained from the analytical expressions match quite well with the numerically integrated values. We have also compared the numerical results obtained from our theory with the third-order KS theory (Sharma, 1992) and the third-order theory of Swinerd and Boulton (1982). It is noticed that the present analytical solution provides better estimates of semi-major axis and eccentricity than the other two theories over a wide range of eccentricity, perigee height and orbital inclination. Refrences Nair, L.S. and Sharma, R.K. Decay of satellite orbits using K-S elements in an oblate diurnally varying atmosphere with scale height depending on altitude, Adv. Space Res., 31, 2011-2017, 2003. Sharma,R.K., Analytical approach using KS elements to near-Earth orbit predictions including drag, Proc. R. Soc. London, A 433, 121-130, 1991 Sharma, R.K., A third-order theory for the effect of drag on Earth satellite orbits, Proc.R. Soc. London, A 438, 467-475, 1992. Sharma, R.K., Analytical Integration of K-S element equations with J2 for short-term orbit predictions, Planetary and space Science, 45, 1481-1486, 1997. Sharma, R.K., Contraction of satellite orbits using KS elements in an oblate diurnally varying atmosphere, Proc. Royal Soc. Lond., A 453, 2353-2368, 1997. Sharma, R.K. and Xavier James Raj, M., Long-term orbit computations with K-S uniformly regular canonical elements with oblateness, Earth, Moon and Planets, 32, 63-178, 1988. Stiefel, E.L. and G.Scheifele, Linear and Regular Celestial Mechanics, Springer, Berlin, 1971. Swinerd, G. G. and Boulton, W. J., Contraction of satellites orbits in an oblate atmosphere with a diurnal density variation, Proc. Royal Soc. Lond., A 383, 12--145, 1982. Xavier James Raj, M. and Sharma, R.K., Orbit prediction with Earth’s flattening: KS canonical elements, proceedings of 44th ISTAM Congress, Regional Engineering College, Warangal, Dec.22-25, 1999. Xavier James Raj, M. and. Sharma, R.K, Analytical short-term orbit prediction with J2, J3, J4 in terms of K-S uniformly regular canonical equations, Adv. Space Res., 31, 2019-2025,2003.
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