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Cell[CellGroupData[{
Cell["Precession of Planetary Orbits", "Title"],

Cell["Student Projects in AP1603 investigating classical motion.", \
"Subsubtitle"],

Cell["\<\
All projects must have\[Ellipsis]
Name:____________
Email:_____@columbia.edu\
\>", "Subsubtitle"],

Cell[CellGroupData[{

Cell["Introduction", "Section"],

Cell[TextData[{
 "During our classroom discussions, we've investigated the motion of planets \
orbiting a massive star. In particular, we've shown how ",
 StyleBox["Mathematica",
  FontSlant->"Italic"],
 " can be used to visulaize elliptic orbits, to demnstrate the conservation \
of angular momentum, and to prove Kepler's \"harmonic law\" of planets. \
Kepler's laws result from the inverse-square dependence of the gravitational \
force on the separation between massive bodies. What happens when the force \
does not have a perfect inverse square dependence?\n\nIn this research \
project, you are to investigate the stability of plaentary motion when the \
radial-dependence of the gravitational force  changes from the ideal ",
 Cell[BoxData[
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 ". Nature produces these perturbations in two ways. First, the each planet \
accellerates according to a total force: the vector-sum of the forces from \
the Sun (very large) and the other plaents (much smaller, but still \
important). The motion of three or more interacting objects can not be \
integrated analytically, and computer simulation of the dynamics of three (or \
more) objects shows the possibility of orbit instability and chaos. Secondly, \
near extremely massive objects, Newton's law of gravity becomes inaccurate, \
and we need to include corrections due to Einstein's theory of general \
relativity. When the gravitational force deviates from a perfect \
inverse-square dependence, elliptical orbits precess.\n\nThe following  two \
sections present related suggestions for your study and research using ",
 StyleBox["Mathematica",
  FontSlant->"Italic"],
 ". Take one of these as a starting-point, and conduct \"computational \
experiments\" and analysis to explore how eliptical orbits precess. \
Re-examine the notebooks discussed in class, copy relevant expressions, and \
modify them during your investigations. \n\nYour goal is to produce a \
complete ",
 StyleBox["Mathematica",
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 " notebook containing a summary of your investigations. This will most \
likely be an answer to one or more questions concerning planetary orbits. It \
should contain ",
 StyleBox["Mathematica",
  FontSlant->"Italic"],
 " expressions and graphics which illustrate your solution. Your notebook \
should be concise and easy to read. Try not to include a large number of \
repeated expressions. Instead, generate a table or graphic of your results. \
In all cases, format your notebook and include texual comments and \
descriptions. Your notebook need not be long. It should be interesting and \
arrive at a definite conclusion."
}], "Text"]
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Cell[CellGroupData[{

Cell["Modification of the Inverse-Square Law", "Section"],

Cell[TextData[{
 "The inverse-square dependence of the gravitational force (and the Coulomb \
force between two charged objects) is related to  the ",
 Cell[BoxData[
  FormBox[
   SuperscriptBox["r", "2"], TraditionalForm]]],
 " radial dependence of the area of a surface enclosing one object. The force \
of gravity emanates from an object, and the force is directed towards that \
object. The total gravitational \"field lines\" or \"lines of force\" are \
conserved as the enclosing surface moves to different radii. Since the \
surface area scales as ",
 Cell[BoxData[
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 " the gravitational force must vary as ",
 Cell[BoxData[
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 ". In this way, the product of the inward gravitational force and the \
surface area of a sphere at radius ",
 StyleBox["r",
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 " is independent of ",
 StyleBox["r",
  FontSlant->"Italic"],
 ". (At least this is true in the classical picture of a Euclidean universe.)\
\n\nAn easy way to explore the consequences of the inverse square dependence \
of gravity is to exmine what happens when gravity behaves differently. For \
example, conside the orbit of a planet when the force of gravity is given by \
the equation\n\t",
 StyleBox["F",
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 "= ",
 Cell[BoxData[
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 StyleBox["m / ",
  FontSlant->"Italic"],
 Cell[BoxData[
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 "\nwhere \[Beta] is not precisely equal to 2. (In this equation, time is \
measured in years and radius in AU, the average distance between the Sun and \
the Earth.)\n\nExplore what happens to the planet's orbit as \[Beta] changes. \
When \[Beta] = 3, planets are eventually ejected from the solar system. For \
\[Beta] = 2.1, elliptic orbits precess. (What about circular ones?) Is there \
a simple relationship between orbit precession and the value of \[Beta]?"
}], "Text"]
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Cell[CellGroupData[{

Cell["The Precession of Mercury", "Section"],

Cell[TextData[{
 "Mercury (the inner most planet) and Pluto (the outermost) have orbits which \
deviate most from circular. Since Mercury is close to the Sun, the variations \
to the usual gravitational force caused by Einstein's general theory of \
relativity cause the elliptical orbit of Mercury to precess. The precession \
of Pluto's elliptic orbit is inperceptable since Pluto is much further from \
the Sun than Mercury.\n\nIn this problem, you are to investigate the \
precession of Mercury due to the first (and most significant) correction to \
Newton's law of gravity due to Einstein. Let the force of gravity be\n\t",
 StyleBox["F",
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 "= ",
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 " (1",
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  FontSlant->"Italic"],
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 " + ",
 Cell[BoxData[
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    SuperscriptBox["r", "4"]}], TraditionalForm]]],
 ")\nwhere \[Alpha] \[TildeTilde] 1.1 \[Times] ",
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 " when radius is measured in units of AU and time in units of years. The \
correction, ",
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   RowBox[{"\[Alpha]", "/", 
    SuperscriptBox["r", "4"]}], TraditionalForm]]],
 ", is very, very small. Mercury's orbit precesses once every 230,000 years! \
Therefore, in this problem you will need to examine larger values of \
\[Alpha]. The precessional rate will depend on the value of \[Alpha]. When \
\[Alpha] is large, the orbit precession rate will be larger than when \
\[Alpha] is small. \n\nIn order to calculate Mercury's orbit, you will have \
to initialize the orbit with realilstic values for its position and velocity. \
Mercury's semimajor radius (the longer radius of its ellipse, ",
 StyleBox["a",
  FontSlant->"Italic"],
 ") is about 0.39 AU, and the eccentricity is ",
 StyleBox["e ",
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 "= 0.206. (Remember, the eccentricity relates the semiminor radius, ",
 StyleBox["b, ",
  FontSlant->"Italic"],
 " to the semimajor radius, ",
 StyleBox["a",
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 ", by ",
 StyleBox["b = a ",
  FontSlant->"Italic"],
 Cell[BoxData[
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 ". The distance from the Sun to Mercury is (1 + ",
 StyleBox["e) a = ",
  FontSlant->"Italic"],
 "0.47 AU.)  The velocity of Mercury at its greatest distance from the Sun is \
about 8.2 AU/year. This value can actually be determined by consideration of \
conservation of total energy (kinetic plus potential) and conservation of \
angular momentum. \n\nOnce you've initialized Mercury's orbit, examine its \
orbit for various values of \[Alpha]. Carefully consider how to let ",
 StyleBox["Mathematica",
  FontSlant->"Italic"],
 " determine the changing angle of the elliptic axes of the orbit over time. \
See if you can develop a relationship between the value of \[Alpha] and the \
rate of precession of the ellipse. This relation is linear for small values \
of \[Alpha]. By calculating the precession rate for values of \[Alpha] \
between 0.001 and 0.01, you can perform a linear extrapolation to \[Alpha] \
\[TildeTilde] 1.1 \[Times] ",
 Cell[BoxData[
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 "and compare your estimate of Mercury's precession to observations. "
}], "Text"]
}, Open  ]],

Cell[CellGroupData[{

Cell["Other Ideas", "Section"],

Cell[TextData[{
 "Students are free to investigate other topics related to planetary \
dynamics. Other topics examine previously include: (i) stellar oscillations \
induced by the orbits of large planets and used to observe planets within \
other solar systems (see ",
 ButtonBox["http://exoplanets.org/",
  BaseStyle->"Hyperlink",
  ButtonData:>{
    URL["http://exoplanets.org/"], None}],
 "), (ii) destabilization of asteroids using known (and fixed) orbits of \
Mars, Jupiter, and Earth to consider the process of an asteroid impact  on \
Earth  (see ",
 ButtonBox["http://neo.jpl.nasa.gov/risk/",
  BaseStyle->"Hyperlink",
  ButtonData:>{
    URL["http://neo.jpl.nasa.gov/risk/"], None}],
 "), (iii) adding one-body at a time, and consider the Keplarian orbits of \
two, three, four, \[Ellipsis], planar (or non-planar) interacting bodies. How \
do you characterize your solutions/simulations?"
}], "Text"]
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