The problem of the critical inclination of the third body

Abstract

Abstract The so-called critical inclination problem of the third body appears in various contexts of celestial mechanics, especially in studies of orbital stability, secular perturbations, frozen orbits, resonances, and the restricted three-body problem. In general terms, the critical inclination is a value from which the orbital dynamics change qualitatively, for example, large eccentricity oscillations, instabilities, or Lidov–Kozai-type resonances arise. In this work, we investigate the critical inclination problem using a more accurate model of the disturbing potential due to the third body than the classical model, where additional terms were taken into account by Lei and Grishin (2025a, 2025b). Considering the double-averaged problem, we show a comparison between the modified double-averaged model von Zeipel–Lidov–Kozai and the classical double-averaged model Lidov–Kozai. We consider both analytical and numerical methods. We show a formula to calculate the critical inclination considering the modified potential model. Note that all applications presented in this work are for the case of a high-altitude orbit around the Moon perturbed by the Earth’s gravitational attraction. From the results, we reproduced the classic case showing the critical inclination in a graph of eccentricity versus inclination and then plotted the same graph with the modified potential model. We also present $$e\sin i$$ e sin i versus $$e\cos i$$ e cos i diagrams to show the regions of libration (librating around the equilibrium point), bifurcation and circulation. In the particular case, when we consider $$e = 0$$ e = 0 , we obtain the classic critical inclinations of $$i_{c} = \mathbf{39.23}$$ i c = 39.23 degrees and $$i_{c} = \mathbf{140.77}$$ i c = 140.77 degrees, while in the modified model we find $$i_{c} = \mathbf{42.65}$$ i c = 42.65 degrees and $$i_{c} = \mathbf{144.13}$$ i c = 144.13 degrees, in this case, $$a = \textbf{13000}$$ a = 13000 km. In the modified model, the calculation of the critical inclination depends on some orbital and physical parameters of the disturbing body.