The 11th homework : Hyperion

Problen 4.19
Study the behavior of our model for Hyperion for different initial conditions.Estimate the Lyapunov exponent from calculations of $△θ$,such as those shown in Figure 4.19.Examine how this exponent varies as a function of the accentricity of the orbit

Abstract

$\qquad$ Three-body system is famous for its complexity and an analytical solution is known not to exist. This is where numerical solutions can show its values in scientific significance.It is known that te motion of asteriods located near the Kirkwood gaps is believed to be chaotic.And there is various chaotic motion of planet which will be a fairly difficult stimulation.However,there is one case of chaos in our solar system that is accessible to a fairly simple stimulation,that is the Hyperion.In this passage,we are going to investigate the properities of the movement of Hyperion.And the behavior of two slightly different intial conditions.
$\qquad$ This work utilizes the simplified model of the Titan Hyperion to complete the 4.19 observation Lyapunov exponent with the eccentricity of the orbital eccentricity relationship, by observing the elliptical orbit eccentricity increases, Lyapunov exponent increases, which can be Judging from the slope of the line in the figure.

Background

$\qquad$ 土卫七密度较低，表面犹如海绵，是太阳系已知天体中自转混沌的最大天体，每21.3天绕土星旋转一周，土卫七可能是一颗更大卫星遭到撞击后剩余的残骸。土卫七绕土星轨道的偏心率较大，且与土星最大卫星土卫六轨道接近，容易受到土星与土卫六引力的共同影响。这些因素集合在一起限制了它自转稳定的条件。

$\qquad$ 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

Calculation

$\qquad$ To simulate the motion of Hyperion we will first make few simplifying assumptions.Our goal will not be to perform a relastic simulations.Rather,our objective is simply to show that the motion of such an irregularly shaped moon can be chaotic.Think of the Heyperion as two particles with mass m1 and m2 linked together by a rod of unqualified mass,and each of them are constrainted by the gravitational bound $\vec{F_1} ,\vec{F_2}$ by Saturn.To coordinate the origin of Saturn coordinate system,The coordinates of the two particles are$(x_1,y_1),(x_2,y_2)$,and their centroid coordinates are $x_c,y_c$.The movement of the satellite as a center of mass around the movement of Saturn and two particles around the center of mass movement.

$$\vec{F_1}=-\frac{GM_{Sat}m_1}{r_1^3}(x_1\hat{i}+y_1\hat{j})$$
$$\vec{F_2}=-\frac{GM_{Sat}m_2}{r_1^3}(x_2\hat{i}+y_2\hat{j})$$

Torsional moment:

$$\vec{M_1}=[(x_1-x_c)\hat{i}+(y_1-y_c)\hat{j}]×\vec{F_1}$$
$$\vec{M_2}=[(x_2-x_c)\hat{i}+(y_2-y_c)\hat{j}]×\vec{F_2}$$
So rotation equation of the two particles around the mass center is:
$$\frac{d\vec{w}}{dt}=\frac{\vec{M_1}+\vec{M_2}}{I}$$

Where I is the moment of inertia,and $I=m_1|r_1|^2+m_2|r_2|^2$. So we get:

$$\frac{dw}{dt}=-\frac{3GM_{Sat}}{{r_c}^5}·(x_csinθ-y_ccosθ)(x_ccosθ+y_csinθ)$$

Main body

for circular orbit,reset θ:

for elliptical orbit circular orbit :

Effect of orbital eccentricity on angular difference:

change the initial conditions,$△θ=0.01$change the initial velocity

Taking the different eccentricity, the Laplov index is obtained as follows:

Conclusion

●From the above, we can see that with the increase of eccentricity of the orbit, the Laplov index changes, indicating that when the initial angle difference is very small, in the case of circular orbits, the two angles are different at t> 0 And for elliptical orbitals, the difference between the two angles increases at t> 0, corresponding to the chaotic motion of the satellite.
● This model is sensitive for initial condition.
● The origin of these dips is that the angle difference decreases as the angular velocity of the larger angle changes from positive to negative.

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Acknowledemet

Thanks to Guozgongzhi
Reference:
1. wikipedia.
2. Computational physics by Nicholas J.Giordano,Hisao Nakanishi.