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It has been observed since antiquity that at certain times some of the planets appear to reverse their direction in the 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 within 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.

PLANETS.ps

Maple's computing abilities can be used to predict when retrograde motion will occur. 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. Mars revolves around its axis once every 1.88 years and the radius of its orbit is kms. Set up functions XMars and YMars which yield the x and y coordinates of Mars as a function of time.

Now define a function which yields the angle between the line connecting Mars and the Earth and the x-axis. Assuming that measuremanets are made at a fixed time of night and disregarding questions of latitude, this angle will correspond to the angle one would have to look up at to see Mars in the night sky. Using the unapply function, define a function which describes this angle. Since we are interested in the motion, rather than the position, of Mars the next step is to calculate the derivative of this function. The expression Maple yields will be quite complicated.

Since it is of interest to know when Mars reverses its motion a rewsonable course of action at this point is to set the expression obtained by differentiating equal to 0 and then solving 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. Plot the derivative to determine where it is negative.

Try plotting the angular velocity for the other planets given the following data:

The assumption of a circular orbit is not justified for Mercury and Pluto.

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Juris Steprans
Mon Apr 21 13:11:32 EDT 1997