# Extension of the King-Hele orbit contraction method for accurate, semi-analytical propagation of non-circular orbits

## TL;DR

This paper extends the King-Hele semi-analytical orbit contraction method to better handle Earth's complex atmospheric density profiles, improving accuracy for non-circular orbit propagation while maintaining low computational costs.

## Contribution

The authors introduce a novel atmosphere model derivative and superimposed exponential atmospheres to enhance the KH method's accuracy for realistic density profiles.

## Key findings

- Extended KH method achieves accuracy comparable to numerical quadrature.
- Method effectively propagates non-circular orbits with low computational load.
- Refined series expansion improves the handling of complex atmospheric conditions.

## Abstract

Numerical integration of orbit trajectories for a large number of initial conditions and for long time spans is computationally expensive. Semi-analytical methods were developed to reduce the computational burden. An elegant and widely used method of semi-analytically integrating trajectories of objects subject to atmospheric drag was proposed by King-Hele (KH). However, the analytical KH contraction method relies on the assumption that the atmosphere density decays strictly exponentially with altitude. If the actual density profile does not satisfy the assumption of a fixed scale height, as is the case for Earth's atmosphere, the KH method introduces potentially large errors for non-circular orbit configurations. In this work, the KH method is extended to account for such errors by using a newly introduced atmosphere model derivative. By superimposing exponentially decaying partial atmospheres, the superimposed KH method can be applied accurately while considering more complex density profiles. The KH method is further refined by deriving higher order terms during the series expansion. A variable boundary condition to choose the appropriate eccentricity regime, based on the series truncation errors, is introduced. The accuracy of the extended analytical contraction method is shown to be comparable to numerical Gauss-Legendre quadrature. Propagation using the proposed method compares well against non-averaged integration of the dynamics, while the computational load remains very low.