The eleventh homework of computational physics about Lyapunov exponent from calculations of $\Delta \theta$ of 4.19

defeng kong 2014301020081

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Abstract

I make some figures about $\theta$ versus time in circular and elliptical orbit of Hyperion and Lyapunov exponent from calculations of $\Delta \theta$ of 4.19.In the end ,I get the discipline that expoent varies as eccentricity of the orbit.

Background

1.Hyperion

Hyperion(/haɪˈpɪəriən/;Greek: Ὑπερίων), also known as Saturn VII , is a moon of Saturn discovered by William Cranch Bond, George Phillips Bond and William Lassell in 1848. It is distinguished by its irregular shape, its chaotic rotation, and its unexplained sponge-like appearance. It was the first non-round moon to be discovered.

The Voyager 2 images and subsequent ground-based photometry indicated that Hyperion's rotation is chaotic, that is, its axis of rotation wobbles so much that its orientation in space is unpredictable. Its Lyapunov time is around 30 days. Hyperion, together with Pluto's moons Nix and Hydra, is among only a few moons in the Solar System known to rotate chaotically, although it is expected to be common in binary asteroids. It is also the only regular planetary natural satellite in the Solar System known not to be tidally locked.

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.

2.problem 4.19

study the behavior of our model for Hyperion for diffierent initial conditions.Estimate the Lyapunov exponent from calculations of$\Delta \theta$,such as those shown in Figure 4.19.Examine how the exponent varies as a function of the eccentricity of the orbit.

3.equation and principle

Because Hyperion is unique among the large moons in that it is very irregularly shaped,we cannot regurd it as a big ball. so we can think it is m1 ,m2 and the rod that exists in the m1 and m2.There are two forces acting on each of the masses ,the force of gravity from Saturn and the force from the rod:

$$\vec{F}=-\frac{GM_{Sat}m_{1}}{r^{_{1}^{3}}}(x_{1}\vec{i}+y_{1}\vec{j})$$
$$\frac{d\omega }{dt}=-3\frac{GM_{Sat}}{r^{_{c}^{5}}}(x_{c}sin\theta -y_{c}cos\theta )(x_{c}cos\theta +y_{c}sin\theta )$$ .
$$\frac{d\theta }{dt}=\omega$$
for $x_{c},y_{c}$,we can get from:
$$\frac{dv_{x}}{dt}=-\frac{GM_{Sat}x}{r^{3}}$$
$$\frac{dv_{y}}{dt}=-\frac{GM_{Sat}y}{r^{3}}$$
$$\frac{dx}{dt}=v_{x}$$
$$\frac{dy}{dt}=v_{y}$$
(reference:problem 4.8)

The main body and conclusion

For circular orbit of Hyperion:

the code
we can find the tumbling is not chaotic.

For elliptical orbit of Hyperion:


the code
we can find when e=0.35 or 0.36 the tumbling is not chaotic,when e =0.38 or 0.40 the tumbling is chaotic.The boundary is about e=0.36.
For $\Delta \theta$ of circular orbit：

the code
The value of $\Delta \theta$ is not chaotic。
For $\Delta \theta$ of elliptical orbit:
Frist,we set $\Delta e$ is big($\Delta e$=0.1or0.05):


the code
when e=0.34~0.4（$\Delta e$=0.02） the change from that is not chaotic to that is chaotic is obvious.and we can find the boundary is about e=0.35 more precisly:

From we can find when e<0.35 the tumbling is not chaotic.when e>0.35,as the value of e increases,the value of expoent increases.in other word,the time when $\Delta \theta$ is about 2pi(6.28) is shorter.

Thanks

myself.
Wikipedia.