Ex_11: The exponent varies as a function of the eccentricity of the orbit(4.19)

Author: 刘永杰 2014301020094

题目：

$\qquad$Study the behavior of our model for Hyperion different innitial conditions.Examine how this Lyapunov exponent varies as a function of the eccentricity of the orbit

背景：

$\qquad$Hyperion is unique among the large moons in that it is very irregularly shaped, has a fairly eccentric orbit, and is near a much larger moon, Titan. These factors combine to restrict the set of conditions under which a stable rotation is possible. The 3:4 orbital resonance between Titan and Hyperion may also make a chaotic rotation more likely. The fact that its rotation is not locked probably accounts for the relative uniformity of Hyperion's surface, in contrast to many of Saturn's other moons, which have contrasting trailing and leading hemispheres.---Wikipedia

计算过程：

$\qquad$Heyperion实际上是双星系统，则有：他们各自和土星之间靠万有引力束缚分别为$F_1,F_2$，以土星为坐标原点建立坐标系，在此坐标系中两质点坐标为$(x_1,y_1),(x_2,y_2)$,质心坐标为$(x_c,y_c)$,将卫星的运动看作质心绕着土星的运动和两个质点绕着质心的运动。
从课本Figure4.16的分析来看，我们可以得到：

$$\frac{dw}{dt}=-\frac{3GM_{Sat}}{{r_c}^5}·(x_csinθ-y_ccosθ)(x_ccosθ+y_csinθ)$$
$\qquad$再通过Euler-Coromer方法计算就可以得到θ与ω之间的关系

θ与ω关系代码1

圆轨道

椭圆轨道

轨道的偏心率对角度差的影响：代码2

取不同的偏心率，得到拉普洛夫指数如下：

结论

$\qquad$通过以上作图观察，可以看到随着轨道的偏心率的增大，拉普洛夫指数改变，表示当初始的角度相差很小时，圆轨道的情况下，在t>0时刻两个角度相差不大，而对于椭圆轨道，在t>0时刻两个角度相差增大，对应着卫星运动变得混沌。

致谢

感谢郭忠智同学的尽心帮助