It has been observed since antiquity that, at certain times, some of the planets appear to reverse their direction in the night sky; in other words, their motion appears to be retrograde. This is difficult to explain within the Ptolemian model of the solar system which places the Earth at the centre of the orbits of the other planets. However it is easily explained using the model of Copernicus which places the Sun at the centre of the orbits of all the planets, including the Earth. The retrograde motion results from the relative motion of Earth with respect to the other planets. Maple's computing abilities can be used to predict not only retrograde motion, but, other behaviour as well.

The orbits of most of the planets are quite well approximated by circles
centred at the sun and, moreover, all of these circles can be assumed to lie
in the same plane; let us choose this plane to be the *xy*-plane. Letting
one AU (Astronomical Unit) denote the radius of the orbit of the earth (
kms), it is possible to set up functions which yield the *x* and *
y* coordinates of the Earth as a function of time, *t*, which will
be measured in earth years:

To keep things simple, assume that the *x*-axis has been chosen so
that at time *t*=0 both the earth and planet Mars lie on the positive
*x*-axis. Using available
data on Mars
, set up functions `XMars` and `YMars` which yield the *x*
and *y* coordinates of Mars as a function of time.

At this point one must deal with the question of how to determine the position of an object in the sky. Perhaps the first answer to spring to mind would be to use the horizon and the directions of the compass as reference points. One could, for example, describe the position of an object in the sky by measuring a given angle from the eastern horizon and another angle from the southern horizon. The difficulty one encounters is that the earth spins in its axis, or, from the point of view of pre-Copernican astronmers, the heavens rotate about the earth along an axis containg the north star. This means that, over the course of an evening, the stars change their position. Consequently, in order to compare the positions of the planets one must make all measurements at the same time of night.

Therefore, define a function which yields the angle between the line connecting Mars and the Earth and the horizon as seen at midnight. (Keep in mind that at midnight an astronomer observing the planets would be directly opposite the sun. Hence, the horizon, as seen by the astronomer would be perpendicular to the imaginary line connecting the sun and the earth.) Assuming that all measuremanets are made at a midnight and, disregarding questions of latitude, this angle will correspond to the angle at which one would have to look up to see Mars in the night sky. Define a function which describes this angle. A good idea is to think of the positions of the planets as well as the horizon as vectors since the angle between vectors is readily obtained using dot products and the arccos function. When using the arccos function you must remember to be careful when dealing with angles between 180 and 360 degrees since this is outside the range of the arccos function. You should also keep in mind that the dot product of the horizon vectore and the vector from earth to a planet will not tell you when the angle between horizon and planet has esceeded 180 degrees. For this you might want to consider the dot product of the planet vector and the line connecting the sun and earth. What value, positive or negative, will this dot product have if the angle between planet and is greater than 180 degrees?

Explain how this function can be used to locate Mars in the night sky. What additional information about Mars is required to actually be able to make use of this function?

If you have done your calculations correctly you will observe that there is no retograde motion in this scheme! It turns out that retrograde motion is only observed with respect to an imaginary, fixed coordinate system (which it is convenient to have centered at the Sun). However, why would astronomers who believed the Earth was the centre of the universe, use such a coordinate system when it would be much more natural for them to use the frameweork of the Earth's horizons? Indeed, how would they have conceived of such a coordinate system? The answer is to be found in the fact that the planets move across the sky against the background of the fixed stars. Recall however that the fixed stars are not fixed, but rotate about the poles once every 24 hours. Nevertheless, they provide a natural coordinates system. To observe retrograde motion, you must determine the positions of the planets with respect to this system

To determine when retrograde motion occurs, you must focus on the motion, rather than the position, of Mars. Hence, the next step is to calculate the derivative of this function. However, the expression Maple yields will be quite complicated. A reasonable course of action at this point is to set the expression obtained by differentiating equal to 0 and then solve to find the points at which motion is reversed. Try this. Depending on which version of Maple you are using you will either fail or come up with a very complicated expression. There are now two ways to appraoch the problem One is to give Maple a hand by suggesting some trigonometric substitutions in order to simplify the derivative and then solve. The other is to try plotting the derivative to determine where it is negative.

You should be aware that the assumption of circular orbits is not really justified for Mercury and Pluto . and, even for Mars the assumption is lead to discrepancies with the observations made by Copernicus' opponent Tycho Brahe. Indeed it was in resolving these discrepancies that Kepler hypothesized elliptic orbits for the planets. In order to fully and accurately track the position of all the planets in the night sky one would also need information on the angles of the major axes of the elliptical paths of the planets. The point of perihelion, or closest approach to the Sun, occurs along the major axis of the elliptic orbit of a planet. Therefore knowing the angle at perihelion yields the required information. This can be found at NASA's educational site . Of course, the determination of the position of the planets discussed so far has ignored the latitude of the observer. Using the eccentricty given in the data on Mars , but still assuming a circular orbit for the earth, plot the position of Mars in the night sky. Compare this plot with the one you would obtain assuming a circular orbit. Comment on the accuracy of Tycho Brahe's observations.

Modify your equation that you developed for Mars to also work for Uranus, Saturn and Neptune using data available on the internet. Plotting this information should allow you to answer questions such as the following: How often will Uranus and Saturn be within 10° of each other in the night sky?

The other assumption made was that the orbits of all the planets ar in the same plane as the orbit of the earth. This is a reasonably good approximation for most of the planets. For example, checking the data on Saturn reveals that its orbit inclination is only 2.49°; in other words, the angle between the plane of the orbit of Saturn and that of Earth is 2.49 °. Modify the preceding work to take into account the inclination of the orbit of Neptune to define a function which determines its position in the sky more accurately. Compare this function with the one you obtained without taking into account the angle of inclination of Neptune's orbit.