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Cell[TextData[{
 StyleBox["Retrograde Motion",
  FontWeight->"Bold"],
 "\nRetrograde motion is the phenomenon where over the course of time, on the \
scale of years, a planet will move in your sky in generally an arc but \
sometimes move backwards for a few days and then start moving in the normal \
direction again. This is called retrograde motion and is utterly \
unexplainable by a planetary model where Earth is at the center of the Solar \
System and everything revolves around it.\n",
 StyleBox["Circular Planetary Orbits\n",
  FontWeight->"Bold"],
 "The left model shows the Sun, Earth, and some other Planet all rotating in \
the same plane. The orbit of the planet can be controlled by the Orbital \
Radius slider and the Initial Position slider, which places the planet at an \
initial angle around the sun from 0 to 2\[Pi]. By Newton\[CloseCurlyQuote]s \
Third Law, the force of gravity between the Sun and a planet is in magnitude \
the centripetal force for circular motion, so",
 StyleBox[" ",
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     SuperscriptBox["r", "2"]], "="}], TraditionalForm]],
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 " ",
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     SuperscriptBox["v", "2"]}], "r"], TraditionalForm]],ExpressionUUID->
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 " \[Implies] v = ",
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     SubscriptBox["Gm", "2"], "/", "r"}]], TraditionalForm]],ExpressionUUID->
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 ", that is, the tangential velocity of the planet and therefore the orbital \
period of the planet is completely determined by the radius at which the \
circular orbit is occurring, meaning that the Orbital Radius slider is the \
only one we need to talk about all circular orbits, both closer to the Sun \
than Earth and farther.\n",
 StyleBox["The Celestial Sphere",
  FontWeight->"Bold"],
 "\nThe Celestial Sphere, as can be turned on with the check box, is the sky \
of the planet it\[CloseCurlyQuote]s surrounding. The lines connecting every \
body in the sky to the center of the Earth intersects the Celestial Sphere at \
one point, which is the point in the sky you would see if you looked up. The \
right model shows the projection of the objects in space onto the celestial \
sphere. Here the sizes of the objects don\[CloseCurlyQuote]t matter, the \
planet and the Sun show up as the same sized dots on the celestial sphere; \
back when retrograde motion was first noticed it is definitely plausible that \
their equipment was nowhere near sophisticated enough to measure the changes \
in sizes of celestial bodies over the course of a few years, meaning that a \
point in the sky is all they have to go off of anyway.\n",
 StyleBox["Not to Scale\n",
  FontWeight->"Bold"],
 "Making the sizes of bodies relative to each other to scale doesn\
\[CloseCurlyQuote]t work for this model, as the Sun would be one million \
times bigger than the Earth, which, when scaled to fit the model, would make \
the Earth invisible. Scaling for distance also doesn\[CloseCurlyQuote]t make \
sense, as even for the Sun, a very large object, it might only barely appear \
in the model if the distances were accurate, that is, the diameter of the Sun \
is less than 900,000 miles, 1/100th the 93 million miles from the Earth to \
the Sun. Finally the Moon\[CloseCurlyQuote]s orbital period cannot be \
accurate, as the Moon going around the Earth 13.4 times a year would be way \
too sensitive, the time slider would have to be slid extremely carefully to \
be able to see the Moon moving as opposed to jumping around the Earth.\n",
 StyleBox["Explanation of Retrograde Motion\n",
  FontWeight->"Bold"],
 "When we move the time slider, or alternatively hit play on the time \
animation (slowing it down is advised), we can see night to night the red \
planet moving along Earth\[CloseCurlyQuote]s sky counter-clockwise. The Sun \
is also moving counter-clockwise, as is the moon. The apparent similarity in \
behaviour could plausibly lead one to believe that Earth is at the center of \
the solar system. However, we see that at some point in the year, the red \
planet moves backwards in our sky for a while, and the continues on its \
counterclockwise trajectory. This retrograde motion cannot be explained by \
models putting Earth as the center of the solar system. By inspecting the \
left model in real time as the retrograde motion unfolds on the right model, \
we see that retrograde motion is really the effect of the Earth ",
 StyleBox["passing",
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 " the red planet in its orbit, this makes it clear how why a parallax effect \
is coming in to play. Parallax is the phenomenon by which an object appears \
to move with respect to the background. As we are catching up to the red \
planet in the orbit, it moves somewhat circularly (this, too, is not obvious \
- the model shows us though how early astronomers made the mistake, as it \
looks awfully similar to how the moon goes around us), but when we pass it \
the parallax effect takes over and the projection of the planet onto the \
celestial sphere moves backwards for a while. This model also allows us to \
visualize why retrograde motion occurs when the planet is both farther away \
and closer to the Sun than Earth is.",
 StyleBox["\nFixing at Midnight\n",
  FontWeight->"Bold"],
 "The horizontal blue line is the horizon. Time of day is really a measure of \
where on the planet you are standing, so by midnight I really mean opposite \
the Sun, and by horizon I mean the diameter of Earth perpendicular to the \
vector connecting the Sun and Earth, which can be thought of as the line \
connecting the positions of 6pm and 6am. When we fix at midnight and place \
the planet farther from the Sun than the Earth is, we do not see the planet \
move backwards on the projection - this is because when we don\
\[CloseCurlyQuote]t fix the horizon, it\[CloseCurlyQuote]s moving at a speed \
faster than the planet\[CloseCurlyQuote]s negative movement speed when it \
goes backward; fixing the horizon is equivalent to rotating everything by the \
opposite of the inclination of the horizon, so the planet\[CloseCurlyQuote]s \
motion will always be in the same direction. The way to interpret retrograde \
motion here is by the slowing down of the red dot on the projection, which is \
hard to see on this model. The inner planet still does though show the \
retrograde motion on the projection when we fix at midnight. We could get \
this more obvious effect for the outer planet if we increased the speed \
(reduced the period) of its orbit.",
 StyleBox["\nA Brief History of Heliocentrism\n",
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 "In short, Copernicus designed a model where the Earth is not the center of \
the solar system but rather the Sun is. His measurements allowed for \
predictions that matched the precision of the previously established \
geocentric Ptolemaic system, but additionally gave an explanation for \
retrograde motion.\n\n",
 StyleBox["A note on the code\n",
  FontWeight->"Bold"],
 "This notebook was created for expository purposes, to explain retrograde \
motion, rather than to showcase the code - consequently the code is rather \
dense and not particularly friendly to read. The variable decelerations at \
the top before the Manipulate function tell most of the story. The left model \
uses Graphics3D where the bodies are Spheres with a predetermined radius and \
a center that is a vector determined by a function that takes as argument the \
manipulated variable T. The rotation of these vectors is done with a 2x2 \
rotation matrix - the 3D model is inherently 2D mathematically speaking as \
all of the orbits lie in one plane. The right model is literally a projection \
of the left model, using the projection function CSProj defined in the top. \
Everything inside the Manipulate is to draw everything and allow for options \
- all of the options are If statements for turning things on and off (like \
the Celestial Sphere checkbox) and in the case of fixing at midnight a \
rotation matrix inside of an If statement, that is applied only if the Fix at \
Midnight checkbox is checked."
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