扁圓土星的重力力矩：土衛二物理天平動之研究

Abstract

土星施加在其衛星上的重力力矩對於他們各自的轉動力學扮演著重要的角色，其中土衛二在經度上的物理天平動已經被精確的測量，並且被用來約束其內部的結構。目前相信土衛二的內部是由三層主要的結構組成：岩石內核、液態海洋、冰殼層。然而，儘管土星是太陽系中最扁的行星之一，並且土星和土衛二之間的距離相對於土星表面半徑的差距不大，重力力矩的公式中排除了非球對稱土星形狀的效應。本研究中，我們利用多極展開的方法推導一般性的力矩公式，該公式能同時考慮任意形狀的土星的影響。對於扁圓土星來說，常規的力矩公式需要乘上一個修正係數，該係數正比於土星的扁率以及土星土衛二之間距離平方的倒數。帶入數值後得到該修正係數大約為千分之二，而相對應土衛二冰層厚度的估算被修正的幅度大約小於百分之一。

The gravitational torque of Saturn exerted on its moons plays an important role in their rotational dynamics, one of which, known as the physical libration in longitude, has been accurately measured for Enceladus (Thomas et al., 2016). This observation has been used to constrain the interior structure of Enceladus, which is believed to consist of three layers: rock core, water ocean, and ice shell. However, the formulation of the gravitational torque excludes the non-spherical shape of Saturn itself, even though Saturn is one of the most oblate planets in the Solar System and its distance to Enceladus is comparable to its mean radius. In this study, we derive the general torque equation using multipole expansion to include the effect due to the arbitrary shape of Saturn. For an oblate Saturn, the conventional torque is multiplied by a correction factor, which depends on Saturn’s oblateness and its distance to Enceladus. Given the physical parameters, the correction to the torque is about a few thousandth and the resultant correction to the estimate of Enceladus’ shell thickness is less than one percent.