R-Toroid as a Three-Dimensional Generalization of a Gaussian Ring and Its Application in Astronomy

Abstract

A new analytical model (R-toroid), representing a 3D generalization of the precessing Gaussian ring, is constructed for the study of secular perturbations in celestial mechanics. Our approach is based on triple averaging of the motion of a material point and is reduced to a chain of transformations: 1D Gaussian ring–2D R-ring–3D R-toroid. The figure, structure and gravitational potential of the R-toroid are studied. We obtained the expression for the mutual energy of the R-toroid and the outer Gaussian ring to study the motion of bodies in the gravitational field of the model in two forms (in the integral and in the form of a power-law series). Two equation systems of the secular evolution of osculating orbits (Gaussian rings), in the gravitational field of an R-toroid and in the field of a central precessing star, are derived using the mutual energy. The periods of nodal TΩ and apsidal Tω orbital precession were obtained. Examples of three hot Jupiters with a known period of nodal precession are considered. For the exoplanet Kepler-413b, the R-toroid describes the evolution of any orbit with a ≥ 5.48 AU, and for the exoplanet PTFO 8-8695b, the critical value of the semi-major axis turned out to be only amin ≈ 0.2 AU. The frequency profile of the precession of the test orbit in the field of the star and planet PTFO 8-8695b has been calculated. The minimum value of the period of nodal precession was TΩ ≈ (26.1 ± 3.0) × 103 years.