Exercise_11 : Chaotic Tumbling of Hyperion

Problem 4.19 & 4.20

冉峰 弘毅班 2014301020064

Abstract

• It is known that the 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 assignment, the motion of Hyperion will be discussed according to the instruction of Problem 4.19 and 4.20.

Background

The Basic Information of Hyperion:

• Hyperion, also known as Saturn VII (7), 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 moon is named after Hyperion, the Titan god of watchfulness and observation – the elder brother of Cronus, the Greek equivalent of Saturn – in Greek mythology. It is also designated Saturn VII. The adjectival form of the name is Hyperionian.
  Hyperion's discovery came shortly after John Herschel had suggested names for the seven previously-known satellites of Saturn in his 1847 publication Results of Astronomical Observations made at the Cape of Good Hope. William Lassell, who saw Hyperion two days after William Bond, had already endorsed Herschel's naming scheme and suggested the name Hyperion in accordance with it. He also beat Bond to publication.
  

Rotation of Hyperion:

• 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. 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.
  

Analysis of The Motion of Hyperion:

• There are two forces acting on each of the masses, the force of the 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 m1 can be written as:
  $$\begin{eqnarray*} \vec F_1=-\frac{GM_{Sat} m_1}{r_1^3}(x_1\vec i +y_1 \vec j) \end{eqnarray*}$$
  where $M_{Sat}$ is the mass of Saturn, $r_1$ is the distance from Saturn to $m_1$, $\vec i$ and $\vec j$ 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.
  

The Main Body

• We study the behavior of our modle for Hyperion for different initial conditions. For problem 4.19 and 4.20, we need to study the behavior that the divergence of two trajectories $\theta_1(t)$ and $\theta_2(t)$ in the chaotic regime. Also we have two types of modles. One is restricted $\theta$ to the range $[-\pi,\pi]$, the other do not restrict the value of $\theta$
  

Code

@author: RanFeng
import math
import matplotlib.pyplot as plt
GM=4*(math.pi**2)
class precession:
    def __init__(self,e=0.5,time=9,dt=0.0001,vcoefficient=1,beta=2,alpha=0,intial_w=0,intial_sita=0):
        self.intial_sita=intial_sita
        self.alpha=alpha
        self.beta=beta
        self.vcoefficient=vcoefficient
        self.e=e
        self.a=1/(1+e)
        self.x0=self.a*(1+e)
        self.y0=0
        self.vx0=0
        self.vy0=self.vcoefficient*math.sqrt((GM*(1-e))/(self.a*(1+e)))
        self.X=[self.x0]
        self.Y=[self.y0]
        self.Vx=[self.vx0]
        self.Vy=[self.vy0]
        self.w=[intial_w]
        self.sita=[intial_sita]
        self.T=[0]
        self.dt=dt
        self.time=time
        self.ThetaPrecession=[]
        self.TimePrecession=[]
    def calculate_reset(self):
        while self.T[-1]<self.time:
            r=math.sqrt(self.X[-1]**2+self.Y[-1]**2)
            newVx=self.Vx[-1]-(GM*(1+self.alpha/r**2)*self.X[-1]/r**(1+self.beta))*self.dt
            newX=self.X[-1]+newVx*self.dt
            newVy=self.Vy[-1]-(GM*(1+self.alpha/r**2)*self.Y[-1]/r**(1+self.beta))*self.dt
            newY=self.Y[-1]+newVy*self.dt
            newW=self.w[-1]-3*GM/(r**5)*(self.X[-1]*math.sin(self.sita[-1])-self.Y[-1]*math.cos(self.sita[-1]))*(self.X[-1]*math.cos(self.sita[-1])+self.Y[-1]*math.sin(self.sita[-1]))*self.dt
            newSita=self.dt*self.w[-1]+self.sita[-1]         
            self.Vx.append(newVx)
            self.Vy.append(newVy)
            self.X.append(newX)
            self.Y.append(newY)
            self.w.append(newW)
            self.sita.append(newSita)
            self.T.append(self.T[-1]+self.dt)            
            while self.sita[-1]>=1*math.pi:
                self.sita[-1]=self.sita[-1]-2*math.pi
            while  self.sita[-1]<=-1*math.pi:
                self.sita[-1]=self.sita[-1]+2*math.pi       
        def plot(self,slogan=''):
            plt.scatter(self.sita,self.w,label=slogan,s=0.1)
            plt.legend(loc='upper right',frameon=False,fontsize='small')
            plt.grid(True)
            plt.title('Hyperion(Elliptical orbit) $\\omega$ versus $\\theta$')
            plt.ylabel('$\\omega(radians/yr)$')        
            plt.xlabel('$\\theta(radians)$')

Result

First we study the modle with the $\theta$ restricted.

• During this time, we can study the behavior of the Circular orbit at the begining. Just like the book, we chose $dt=0.0001yr$ and plot the figure of Hyperion $\theta$ versus $time$.
  
  Then we can plot the figure of Hyperion $\omega$ versus $time$ as the same initial conditions.
  
  Then we want to study the behavior of the influence of the initial conditions's influence. We plot the figure that $\Delta\theta$ versus $time$,with the initial conditions $\theta_1(0)=0$ and $\theta_2(0)=0.01$.
  
  Also we can chose different $\theta_2(0)=0.05,0.1,0.5$.
  $$\theta_2(0)=0.05$$
  
  $$\theta_2(0)=0.1$$
  
  $$\theta_2(0)=0.5$$
  
  Meanwhile we can plot phase figure.
  
• Also we need to study the behavior of the Elliptical orbit.
  First we chose $dt=0.0001yr$ and $e=0.365$ to plot the figure of Hyperion $\theta$ versus $time$.
  
  We can chose difference $e=0.4,0.5,0.206$ to plot the figure again.
  $$e=0.206$$
  
  $$e=0.4$$
  
  $$e=0.5$$
  
  We can see that when $e$ is small, the behavior is just like the circular.
  Then we can plot the figure of Hyperion $\omega$ versus $time$ as the same initial conditions.
  
  We chose difference $e=0.4,0.5,0.206$ to plot the figure again.
  $$e=0.206$$
  
  $$e=0.4$$
  
  $$e=0.5$$
  
  It's different with different $e$, even if $e$ is small.
  Then we want to study the behavior of the influence of the initial conditions's influence. We plot the figure that $\Delta\theta$ versus $time$ as the same initial conditions.
  First we can chose $e=0.365$, $\theta_1(0)=0$ and $\theta_2(0)=0.01$.
  
  We chose different $e=0.5$ to see the difference.
  $$e=0.206$$
  
  $$e=0.5$$
  
  Now, we want to change the initial $\theta_2(0)=0.05,0.1,0.5$
  to see the difference.
  For $e=0.365$
  $$\theta_2(0)=0.05$$
  
  $$\theta_2(0)=0.1$$
  
  $$\theta_2(0)=0.5$$
  
  For $e=0.206$
  $$\theta_2(0)=0.05$$
  
  $$\theta_2(0)=0.1$$
  
  $$\theta_2(0)=0.5$$
  
  For $e=0.5$
  $$\theta_2(0)=0.05$$
  
  $$\theta_2(0)=0.1$$
  
  $$\theta_2(0)=0.5$$
  
  Meanwhile we can plot phase figure.
  We chose $e=0.206,0.365,0.5$.
  $$e=0.206$$
  
  $$e=0.365$$
  
  $$e=0.5$$
  

Then we study the modle without the $\theta$ restricted.

• We can study the behavior of the Circular orbit at the begining. We want to study the behavior of the influence of the initial conditions's influence. We plot the figure that $\Delta\theta$ versus $time$,with the initial conditions $\theta_1(0)=0$ and $\theta_2(0)=0.01$.
  
  Chose different $\theta_2(0)=0.05,0.1,0.5$.
  $$\theta_2(0)=0.05$$
  
  $$\theta_2(0)=0.1$$
  
  $$\theta_2(0)=0.5$$
  
  It's just like the the modle with the $\theta$ restricted.
  Meanwhile we can plot phase figure.
  
• Then study the Elliptical orbit.
  First we can chose $e=0.365$, $\theta_1(0)=0$ and $\theta_2(0)=0.01$.
  
  Chose different $\theta_2(0)=0.05,0.1,0.5$.
  $$\theta_2(0)=0.05$$
  
  $$\theta_2(0)=0.1$$
  
  $$\theta_2(0)=0.5$$
  
  Also we can chose different $e=0.206,0.5$ when $\theta_1(0)=0$ and $\theta_2(0)=0.01$.
  $$e=0.206$$
  
  $$e=0.5$$
  
  Meanwhile we can plot phase figure.
  We chose $e=0.206,0.365,0.5$.
  $$e=0.206$$
  
  $$e=0.365$$
  
  $$e=0.5$$
  

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.

Thanks For

• The book of Computational Physics, chapter 4
• Wikipedia