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Cell[CellGroupData[{
Cell["Chaotic Tumbling of Hyperion", "Title"],

Cell["APAM1601", "Subsubtitle"],

Cell[CellGroupData[{

Cell["Hyperion (A tumbling moon of Saturn)", "Section"],

Cell[TextData[{
 StyleBox["Voyager 1",
  FontSlant->"Italic"],
 " took this photo of Saturn on Nov. 16, 1980 from a distance of 5.3 million \
km."
}], "Text"],

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Cell[TextData[{
 "Saturn and its ring system is one of the most unique objects in the sky. \
When viewed through a modern telescope, the planet appears as a light yellow \
and gray banded oblate spheroid. Like the other giant planets--Jupiter, \
Uranus, and Neptune--the visible planet is the cloud top of an extensive \
gaseous atmosphere.\nSaturn orbits the Sun every 29.4577 years; it's orbit \
has an eccentricity of 0.0556; Saturn's mass is 95.147 times the Earth's; and \
it rotates rapidly, once every 10.657 hours.\nSaturn's white rings were first \
seen by Galileo Galilei in 1610; his small, imperfect telescope showed the \
planetary disk flanked by what he first interpreted as being two smaller \
bodies. Christiaan Huygens correctly theorized (late 1650s) the ring nature \
of these alleged \"companions.\" James Clerk Maxwell mathematically \
demonstrated (1857) that the rings were composed of many small, unconnected \
particles, each orbiting near Saturn's equatorial plane.\nThe classical \
designations for the rings are based on the gross ring components identified \
from the ground, but the ",
 StyleBox["Voyager",
  FontSlant->"Italic"],
 " spacecraft have shown the ring system to be highly structured. The radial \
particle-density distribution changes over distances of hundreds of meters, \
but individual particles, whose estimated sizes range from tens to hundreds \
of centimeters, have not been resolved. The ring plane has a maximum \
thickness of 1 to 2 km (0.6 to 1.2 mi). Spectroscopy shows the presence of \
water ice, which probably covers rocky silicate cores. The dynamics of the \
rings are not presently well understood. The theory of satellite resonances \
predicts that particles whose orbital periods are integral fractions (such as \
1/2 or 2/3) of the periods of the satellites become either locked into or \
perturbed out of a particular orbit, but only a few of the observed gaps can \
be explained in this manner."
}], "Text"],

Cell[TextData[{
 "Voyager 2",
 StyleBox[" obtained this picture of Hyperion from a distance of 500,000 km \
on Aug. 24, 1981.",
  FontSlant->"Plain"]
}], "Text",
 FontSlant->"Italic"],

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Cell["\<\
Saturn has the most extensive satellite system in the solar system. Not \
counting the myriad ring particles, more than 20 bodies orbiting around \
Saturn have so far been identified. Six can be easily seen through the \
telescope.
Titan is the largest Saturnian satellite and, among all solar-system \
satellites, is second in size only to the Jovian satellite Ganymede. It is \
the only satellite with a substantial atmosphere. (Neptune's Triton has a \
much thinner one.)  With the exception of Phoebe, Iapetus, and Hyperion, they \
are in nearly circular orbits. You will see, Hyperion's elliptical orbit has \
a profound effect on its rotational dynamics. All of Saturn's satellites, \
with the exception of Enceladus, have highly cratered, old surfaces.
Hyperion is relatively small, irregular in shape, and heavily cratered. The \
long axis is not oriented toward the planet, and this indicates the Hyperion \
is dynamically unstable. Indeed, Hyperion is one of the finest examples of \
orbital chaos in the solar system.
Hyperion has a mass which is 20 billion times smaller than Saturn's and about \
52,000 times smaller than Earth. Its eccentricity is 0.11; it rotates once \
around Saturn in 21.3 days; and it is 0.01 A.U. (about 3.8 times farther than \
the distance between the Earth's moon and Earth).\
\>", "Text"]
}, Closed]],

Cell[CellGroupData[{

Cell["Introduction", "Section"],

Cell["\<\
We must describe (1) the equations of motion for the center of mass of \
Hyperion, and (2) the dynamical equation for the internal rotation and \
orientation of Hyperion. The center-of-mass motion is the familar Newtonian \
planetary motion. The rotational motion is determined by the torque applied \
by Saturn's gravitational field to the asymmertic moon, and the moon's moment \
of inerta.\
\>", "Text"],

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For our \"dumbell\" model of Hyperion, the torque is given by the vector \
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Since the separation between the two dumbell masses is small, the equations \
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Hyperion's rate of change of angular rotation is determined by the torque \
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The orientation of the long axis of Hyperion is an important quantity. When \
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A plot of the angle of the dumbell with respect to the radial.\
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In the same way, we'll compute the trajectory difference for elliptical \
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The exponetial growth of the separation of the orbits is proportional to the \
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We have studied the tumbling orbit of Hyperon. This is an example of coupled \
oscillators. 
For non-circular orbits, the rocking motion of the moon is coupled to the \
slowly changing gravitational force as the object moves closer and then \
farther away from the planet. 
Coupled oscillators can become chaotic. Notice that chaotic motion is not \
\"random\". Each solution to the differential equations is \"precise\" and \
determined. When a system is chaotic, this means that two solutions that \
differ by a very slight change of initial conditions have an exponentially \
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Deterministic chaos is found in many systems of equations and in many \
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