On the Evolution of Orbits in a Photo-Gravitational Circular Three-Body Problem: The Inner Problem

Abstract

Abstract We consider the spatial restricted circular three-body problem in the nonresonant case. The massless body (satellite) is assumed to have a large sail area and, therefore, the light pressure is taken into account. We study the evolution of the satellite orbit based on Gauss’s scheme: the averaged equations of motion are investigated in Keplerian phase space, when a Keplerian ellipse with its focus in the main body (Sun) is taken as an unperturbed orbit located inside a sphere whose radius is equal to the orbital radius of the outer planet (inner problem). An investigation of the averaged model in the classical case, where the light pressure is neglected, is known to run into considerable difficulties both in calculating the averaged force function and in analyzing the evolving orbits. We have shown for the first time that the twice-averaged force function admits of an explicit analytical representation via hypergeometric (generalized hypergeometric) functions expandable into convergent power series based on the application of Parseval’s formula. We have also shown that the averaged equations of motion including the additional influence of light pressure are Liouville-integrated (we have three independent first integrals in involution). We have investigated, at fixed values of the Lidov–Kozai integral, the stationary regimes of oscillations in the case of low values of the satellite’s unperturbed semimajor axis (Hill’s case), their bifurcation as a function of the light pressure coefficient $$\\delta$$ . In the plane of Keplerian elements $$e$$ and $$\\omega$$ we have constructed the phase portraits of the oscillations at various values of the light pressure coefficient. The portrait rearrangement due to both equilibrium position bifurcations and separatrix splitting is described. The separatrix splitting is shown to reverse the direction of evolution of the argument of pericenter $$\\omega$$ in the case of rotational motions.