Direct derivation of short-periodic $$J_2$$ corrections using Fourier expansions

Abstract This paper presents a new semianalytical orbit propagation method designed to account for the $$J_2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>J</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:math> oblateness perturbation. Unlike common existing methods, such as those developed by Brouwer and Kozai, which typically rely on averaging methods to separate satellite motion into secular, short-periodic, and long-periodic terms, the new approach is based on the development of the Fourier series expansion of the Lagrange planetary equations. It provides an approximate solution that directly yields the short-periodic corrections of the classical orbital elements under the $$J_2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>J</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:math> perturbation. A key contribution is the detailed, explicit derivation of the Fourier series form of the $$J_2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>J</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:math> perturbing potential. An analysis of simulation data highlights the new method’s particular suitability and advantage in accuracy for orbits with low eccentricities and low inclinations, especially in the in-plane directions, compared to Kozai’s short-periodic solution. On the other hand, at high eccentricities and inclinations, the proposed method does not have an advantage in terms of accuracy. The utility of this method is demonstrated by propagating a two-satellite relative motion in low Earth orbit, where it results in significantly smaller in-plane errors.