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 T_Ω and apsidal T_ω 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 a_min ≈ 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 T_Ω ≈ (26.1 ± 3.0) × 10³ years.

中文翻译：

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R-环形作为高斯环的三维推广及其在天文学中的应用

摘要

为了研究天体的长期扰动，构建了一个新的分析模型（R形环），该模型表示进动的高斯环的3D泛化。我们的方法基于物质点运动的三次平均，并简化为一系列转换：1D高斯环– 2D R环– 3D R环形。研究了R形环的图形，结构和引力。我们获得了R形环与外高斯环的互能表达式，以两种形式（以幂律级数的形式和整数形式）研究模型在重力场中的运动。R形环的引力场和中心进动恒星的场中，两个闭合轨道（高斯环）的长期演化的方程组，是利用共同的能量得出的。节点时期Ť _Ω和拱线Ť _ω获得轨道的进动。考虑三个已知节点进动时期的热木星的例子。对于外行星开普勒-413B中，R-环状描述了具有任何轨道的进化一个≥5.48 AU，和用于外行星PTFO 8-8695b，半长轴的临界值被证明是仅一个_分≈0.2 AU。已经计算出了PTFO 8-8695b恒星和行星测试场进动的频率曲线。节点进动的周期的最小值为Ť _Ω听，说：（26.1±3.0）×10 ³年。