We consider a restricted four-body problem, with a precise hierarchy between the bodies: two larger bodies and a smaller one, all three of oblate shape, and a fourth, infinitesimal body, in the neighborhood of the smaller of the three bodies. The three heavy bodies are assumed to move in a plane under their mutual gravity, and the fourth body to move in the three-dimensional space under the gravitational influence of the three heavy bodies, but without affecting them. We first find that the triangular central configuration of the three heavy oblate bodies is a scalene triangle (rather than an equilateral triangle as in the point mass case). Then, assuming that these three bodies are in such a central configuration, we perform a Hill approximation of the equations of motion describing the dynamics of the infinitesimal body in a neighborhood of the smaller body. Through the use of Hill’s variables and a limiting procedure, this approximation amounts to sending the two larger bodies to infinity. Finally, for the Hill approximation, we find the equilibrium points for the motion of the infinitesimal body and determine their stability. As a motivating example, we identify the three heavy bodies with the Sun, Jupiter, and the Jupiter’s Trojan asteroid Hektor, which are assumed to move in a triangular central configuration. Then, we consider the dynamics of Hektor’s moonlet Skamandrios.

我们考虑一个受限制的四体问题，在两体之间具有精确的层次结构：两个较大的体和一个较小的体，所有三个均为扁圆形，而第四个无穷小体位于三个体中较小的一个附近。假定三个重物在它们的相互重力作用下在一个平面中移动，并且第四个物体在三个重物的引力影响下在三维空间中移动，但不影响它们。我们首先发现三个重的扁圆体的三角形中心构型是一个斜角三角形（而不是点质量情况下的等边三角形）。然后，假设这三个主体处于这样的中央配置中，我们对运动方程进行希尔近似，描述了较小物体附近的无穷小物体的动力学。通过使用希尔变量和限制程序，这种近似等于将两个较大的物体发送到无穷大。最后，对于希尔近似，我们找到了无穷小物体运动的平衡点，并确定了它们的稳定性。作为一个激励性的例子，我们确定了三个重物，分别是太阳，木星和木星的特洛伊木星小行星海克托，它们假定以三角形的中心配置运动。然后，我们考虑赫克托尔的小号Skamandrios的动力学。对于希尔近似，我们找到了无穷小物体运动的平衡点并确定了它们的稳定性。作为一个激励性的例子，我们确定了三个重物，分别是太阳，木星和木星的特洛伊木星小行星海克托，它们假定以三角形的中心配置运动。然后，我们考虑Hektor的卫星Skamandrios的动力学。对于希尔近似，我们找到了无穷小物体运动的平衡点并确定了它们的稳定性。作为一个激励性的例子，我们确定了三个重物，分别是太阳，木星和木星的特洛伊木星小行星海克托，它们假定以三角形的中心配置运动。然后，我们考虑Hektor的卫星Skamandrios的动力学。