Singularity-Free Formulations for Drag Temperature Model and Its Analytical Gradient

Determining orbits for low-Earth-orbit objects involves computing atmospheric density and its gradient. Existing atmospheric density models, which typically use spherical coordinates to represent the position of a space object, present challenges in computing atmospheric density and deriving its analytical gradient. These challenges stem from two main issues: a singularity issue arising from undefined longitudes over the Earth’s poles and a mathematical complexity issue associated with using spherical coordinates to derive the analytical gradient of atmospheric density. This paper proposes a singularity-free formulation for the Drag Temperature Model (DTM), a widely applied and well-structured model, and develops its analytical gradient formulation. The method is applicable across the entire DTM series, from the earliest version to the latest DTM2020 release. Experimental results, validated using the publicly available algorithm of DTM94, demonstrate the high fidelity of the singularity-free formulations compared to the original model. Specifically, the relative deviation of atmospheric density is on the order of 1E-3, and that of gradient deviation is less than 1E-5 at nearly all spatial locations. The method is particularly well suited for sequential orbit determinations that require precise state transition matrices.