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Exercise_11 : Chaotic tumbling of Hyperion

Problem 4.19 & 4.20
2014301020065 熊毅恒

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1.Abstract

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As what we all know, the motion of asteriods located near the Kirkwood gaps is believed to be chaotic.Unfortunately, It is very hard to simulate these chaotic motions. 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 solve Problem 4.19 & 4.20 and further investigate the properities of the movement of Hyperion.And the behavior of two slightly different intial conditions.

2.Background

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PIA17193-SaturnMoon-Hyperion-20150531.jpg

2.1 Basic info

Hyperion is one of the largest bodies known to be highly irregularly
shaped (non-ellipsoidal, i.e. not in hydrostatic equilibrium) in the
Solar System.[c] The only larger moon known to be irregular in shape
is Neptune's moon Proteus. Hyperion has about 15% of the mass of
Mimas, the least massive known ellipsoidal body. The largest crater on
Hyperion is approximately 121.57 km (75.54 mi) in diameter and 10.2 km
(6.3 mi) deep. A possible explanation for the irregular shape is that

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Hyperion rotates chaotically, tumbling unpredictably through space as
it orbits Saturn. Hyperion orbits at a mean distance of 933,000 miles
(1,500,000 km) from Saturn in an eccentric orbit. This contributes to
variations in the spin or rotation of Hyperion. A stronger effect on
Hyperion's rotation is that it is in resonance with Saturn's largest
moon, Titan, which orbits at 759,200 miles (1,221,850 km). Thus, the
two objects speed up and slow down as they pass each other in a
complex set of variations. Because Hyperion is much smaller than
Titan, its rotation and orbit are affected vastly more than the larger
moon, and Titan apparently keeps the Hyperion orbit eccentric rather
than growing more circular over time.

2.2 Analysis part

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.With that goal in mind we consider the model with two bodies.We have two particles and ,connected by a massless rod in orbit around a massive object located at the origin.
There are two forces acting on each of the masses,the force of gravity from Saturn and the force from the rod.Since we are interested in the motion about the center of mass,the force from the rod does not contribute.

The Gravitational force on $m_{1}$ :\begin{eqnarray}
\vec F_1=-\frac{GM_sat m_1}{r_1^3}(x_1\vec i +y_1 \vec j)
\end{eqnarray}
where $M_{Sat}$ is the mass of Saturn, $r_{1}$ is the distance from Saturn $\vec i$ to $\vec j$ , and are unit vectors in the $x$ and $y$ directions. The coordinates of the center of mass are $(x_c,y_c)$ , so that $(x_1-x_c)\vec{i}+(y_1-y_c)\vec{j} $ is the vector from the center of mass to $m_{1}$. The torque on $m_{1}$ is then \begin{eqnarray}
\vec \tau_1=[(x_1-x_c)\vec i +(y_1-y_c)\vec j]\times \vec F_1
\end{eqnarray}
with a similar expression for $\vec \tau_2$. The total torque on the moon is just $\vec \tau_1+\vec \tau_2$and this is related to the time derivative of by: \begin{eqnarray}
\frac{d\vec w}{dt}=\frac{\vec \tau_1+\vec \tau_2}{I}
\end{eqnarray}
where $I=m_1|r_1|^2+m_2|r_2|2$is the moment of inertia. Putting this all together yields, after some algebra,
\begin{eqnarray}
\frac{d\vec w}{dt}\approx -\frac{3GM_{Sat}}{r_c^5} (x_csin\theta-y_ccos\theta)(x_c cos\theta+y_c sin\theta)
\end{eqnarray}
where $r_c$ is the distance from the center fo mass to Saturn.

3.Mainbody

• Code

→click here←

• result
  1.circular orbit
  1.1 single initial condition
  
  

  c1.png
  
  
  

  c2.png

As we can see, the motion is obviously not chaotic.

c3.png

Additionally,we also draw the Phase Plot.

1.2 varied initial condition

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Here we consider the divergence of two nearby trajecotries of the tumbling motion of Hyperion.We plot the differnece between two calculated results for $\Delta$ with $\Delta \theta=0.5 and 1.0$.Noting that it being a circular orbit,we are not able to observe the chaotic in the plot.However,we found some strange points that interfered with periodic.

1.Elliptical orbit
2.1 single initial condition

e1.png

e2.png

e3.png

It is easy to tell that it is chaotic.

2.2 varied initial condition

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13.png

14.png

Now let us investigate the chaotic situation.What is known to all is that the chaotic system is extrememly sensitive to the intial conditions.So we take a look what will happen with slight variation of $\theta$ .We now set $\Delta \theta=0.5and1.0$.It is obvious that the divergence could becomes so large with time passing by.

3. Reset and Non-Reset
   
   

   18.png
   
   In problem 4.20,we are asked to investigate the results between resetting the $\theta$ non-resetting $\theta$ the .With non-resetting program,the vertical lines in the plot vanish.It is reasonable due to the fact that the 'sudden turning points vanish'.And the becomes a continuous vaiable versus time

4.Influence of eccentricity

figure_1-10.png

figure_1-11.png

4.Conclusion

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. So from this passage we know that the motion of the Hyperion is not chaotic when we assume the circular orbit, while it becomes chaotic when the orbit is elliptical.

4.Acknowledgement

thank for

• Wikipedia
• zt