半人马座和海王星外天体的扰动函数
The disturbing function for polar Centaurs and transneptunian objects
摘要：
三体问题的经典扰动函数是基于近共面轨道附近引力相互作用的展开。因此，它不适合于确定和研究在近极轨道上的半人马座和海王星外天体与太阳系行星的共振。本文给出了极轨附近引力相互作用的级数展开算法，并明确地给出了偏心率和倾角余弦的四阶扰动函数。极级数的性质与经典扰动函数的性质有很大不同：极级数可以模拟任何共振，因为展开阶与共振阶无关。$p$：$q$共振的力振幅的偏心率和倾斜率的幂不取决于共振阶数$|p-q|$的值，而仅取决于其宇称。因此，所有偶数谐振阶偏心率振幅为$\propto e^2$，奇数谐振阶偏心率振幅为$\propto e$，直到偏心率最低阶$e$。通过对极扰动函数结构和可能的共振临界参数的新发现，我们说明了跨海王星天体471325目前可能被锁定的极共振1：3，3：1，2：9和7：9的动力学。
英文摘要：
The classical disturbing function of the three-body problem is based on an expansion of the gravitational interaction in the vicinity of nearly coplanar orbits. Consequently, it is not suitable for the identification and study of resonances of the Centaurs and transneptunian objects on nearly polar orbits with the solar system planets. Here, we provide a series expansion algorithm of the gravitational interaction in the vicinity of polar orbits and produce explicitly the disturbing function to fourth order in eccentricity and inclination cosine. The properties of the polar series differ significantly from those of the classical disturbing function: the polar series can model any resonance as the expansion order is not related to the resonance order. The powers of eccentricity and inclination of the force amplitude of a $p$:$q$ resonance do not depend on the value of the resonance order $|p-q|$ but only on its parity. Thus all even resonance order eccentricity amplitudes are $\propto e^2$ and odd ones $\propto e$ to lowest order in eccentricity $e$. With the new findings on the structure of the polar disturbing function and the possible resonant critical arguments, we illustrate the dynamics of the polar resonances 1:3, 3:1, 2:9 and 7:9 where transneptunian object 471325 could currently be locked.
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