Evolution of the orbital-plane orientations in the two-protoplanet three-body problem with variable masses

The three body problem with variable masses with two of the bodies being protoplanets is analyzed. The protoplanetary masses are assumed to be much less than the protosolar mass: m ₁(t) ? m ₀(t), m ₂(t) ? m ₀(t). The variations of the body masses over time are assumed to be known. The masses vary isotropically with different rates: ? ₀/m ₀≠? ₁/m ₁, ? ₀/m ₀≠? ₂/m ₂, ? ₁/m ₁≠? ₂/m ₂. The bodies are treated like material points. The problem is described by analogy to the second system of Poincaré elements, based on the equations of motion in a Jacobian coordinate system. Individual aperiodic motions in a quasi-conical cross section are used as the initial, unperturbed, intermediate motions. The expression for the perturbing function does not include terms proportional to third and higher powers of the small masses m ₁ and m ₂. A new analytical expression for the perturbing function analogous to the second system of the Poincaré variables is obtained in the formulation considered using a classical scheme. The analogs of the eccentricities e ₁ and e ₂ and the orbital inclinations i ₁ and i ₂ are considered to be small. The perturbing function accurate to within terms of second order in the small quantities e ₁, e ₂, i ₁, and i ₂ is calculated in a symbolic form using Mathematica package. The equations for the secular perturbations in this protoplanetary three-body problem, with the bodies treated as points with masses varying isotropically with different rates, are obtained. General rigorous analytical solutions to these equations for the secular perturbations describing the evolution of the orbital planes are derived for oblique elements, for arbitrary mass-variation laws. An analog of the Laplace theorem is proved for the orbital inclinations. Analytical formulas are obtained for the temporal variation of the longitudes of the ascending nodes and the inclinations for arbitrary mass variations with different rates. It is shown that the Jacobian node theorem, which is valid in the classical three-body problem with constantmasses, is violated in this problem, unless special initial conditions apply.