行星環動態模型之研究

Abstract

行星環是一個十分迷人的力學系統, 他有着很多有趣的特性。馬克士威（Maxwell）曾經對土星環的穩定性作了一個廣為人知的研究，他是第一位推導出土星環保持在線性穩定的條件的人。但是，我們現在知道土星環不只是一個美麗的環，環的旁邊還有許多月亮。因此我們跟随馬克士威的腳步，同時利用其他研究員的技巧及想法，加上各種不同的迫近手法去研宄行星環上的粒子的運動，最後利用電腦模擬去驗證我們的結果。我們分析的結果表示如果粒子的數目少於一個上限，那麼這個環系統是線性穩定的。另外，我們也研究了在行星環的其中當兩顆粒子互相靠近的時候他們之間的非線線交互作用。我們發現如果粒子的徑向速度比切線速度小很多的時候，兩顆粒子就會有互相排斥的現象出現。我們也根據這個現象推導出一些量化的結果。

Planetary rings are fascinating systems with interesting behavior. A very comprehensive analytical study of the stability of planetary rings goes back to Maxwell when he derived for the first time a criterion for the rings of Saturn to remain linearly stable. However, we now know that Saturn does not just possess a beautiful ring system but also has several moons. Following Maxwell’s footstep, and borrowing techniques and ideas developed by other researchers, we used several approaches to investigate the motion of the particles in the ring, and combined numerical methods to check against our analytical results. Our analysis indicates that the ring system is linearly stable if the number of particles in the ring does not exceed a certain limit. We also investigated the interaction of two particles in the ring when they get closer as a result of nonlinear interactions. We found that if the motion is more or less azimuthal (that is, the radial velocity is small), then the particles effectively repel each other. Certain quantitative results were derived concerning this behavior.