Third-body perturbations have been extensively studied in recent years. Almost all previous works, however, assumed that the perturbations are caused by a third body that orbits the primary on a radius larger than the semimajor axis of the perturbed object. This assumption is justified as long as the primary is not accompanied by a third body in close orbit. In this work, we present an analytic model for the dynamics of a perturbed object that orbits the primary on an orbit with a semimajor axis larger than the semimajor axis of the third body. Such a third body is referred to as an inner third body. An analysis of the long-term evolution of the orbital elements is presented, followed by simulation results, which demonstrate the validity of the model. A more generalized model is then developed, which includes a nonzero eccentricity for the orbit of the third body. An analogy between the $$J_2$$ problem and the inner third-body perturbation is indicated as well.

中文翻译：

---

内部第三体扰动

近年来，第三体扰动得到了广泛的研究。然而，几乎所有以前的工作都假设扰动是由第三个物体引起的，该物体绕主轨道的半径大于被扰动物体的半长轴。只要主星体没有在近轨道上伴随第三体，这个假设是合理的。在这项工作中，我们提出了一个扰动物体的动力学分析模型，该物体绕主轨道运行的轨道的长半轴大于第三体的长半轴。这种第三体被称为内部第三体。对轨道元素的长期演化进行了分析，随后给出了模拟结果，证明了模型的有效性。然后开发了一个更通用的模型，其中包括第三个天体轨道的非零偏心率。还指出了 $$J_2$$ 问题和内部第三体扰动之间的类比。