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Evolution of some quasicircular orbits in the planetary problem

机译：行星问题中某些准圆轨道的演化

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We derive in the plane problem a new closed solution of the Lagrangian equations for resonant motion, concomitantly including zeroth order, and approximate first order and secular perturbations. A major aim is the determination of simple lower limits for the maximum eccentricities and variations of semimajor axes (intrinsic values). Applications of the general solution are made for each perturbation separately, (i) Zeroth and first order perturbations: a new closed solution for the principal zeroth order variation of semimajor axes is obtained. The maximum eccentricity and relative change of semimajor axis of any lunar orbit cannot be lower than 0.019 and 0.018, respectively, (ii) Secular perturbations: with the angular momentum integral our secular perturbations can be easily extended to the spatial problem. The planetary Lidov-Kozai problem is extended to retrograde orbits, showing that large variations of eccentricity and inclination occur for initially circular orbits, if initial mutual inclinations are between about 40° and 150°. (iii) Resonant perturbations: for first and second order resonances, and initially circular orbits our formulas generally approximate just the calculated orbital elements during the whole motion. As a new unexpected result, the numerical exploration of the asteroid belt reproduces most of its overall characteristics up to third order resonances within the restricted three-body problem and modest initial eccentricities ≤0.05.
机译：我们在平面问题中导出了一个用于共振运动的拉格朗日方程的新封闭解，同时包括零阶，近似一阶和长期扰动。一个主要目的是确定最大偏心率和半长轴变化（固有值）的简单下限。通用解的应用分别针对每个扰动，（i）零阶和一阶扰动：获得了半长轴的主要零阶变化的新的闭解。任何月球轨道的最大偏心率和半长轴的相对变化分别不得低于0.019和0.018。（ii）长期扰动：利用角动量积分，我们的长期扰动可以轻松地扩展到空间问题。 Lidov-Kozai行星问题扩展到了逆行轨道，这表明，如果初始相互倾角在40°至150°之间，则最初的圆形轨道会发生大的偏心率和倾斜度变化。 （iii）共振扰动：对于一阶和二阶共振，以及最初的圆形轨道，我们的公式通常在整个运动过程中仅近似计算出的轨道元素。作为一个新的出乎意料的结果，对小行星带的数值研究再现了其大部分总体特征，直至在受限的三体问题和适度的初始偏心率≤0.05之内的三阶共振。

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