Planets. Restless wanderers of the sky. Worshipped as Gods.
The motion of the planet Mars, also known as the “Red Planet”(due to its distinctive orange-red colour on the night sky), has been studied by ancient and more modern astronomers for several centuries. Having been named after Mars, the planet always had a special place in my heart. Uncovering the particulars of its peculiar motion proved to be one of the most difficult problems for mathematicians and astronomers during the 14th, 15th and 16th centuries A.D. The person responsible for solving the problem, cracked the centuries-old mystery behind the laws of planetary motion. That man was no other than Johannes Kepler, who published his discovery in his groundbreaking opus, called Astronomia Nova (or the The New Astronomy) in 1609 A.D. His monumental work paved the way for the genius of Sir Isaac Newton and the formulation of the law of Universal Attraction.
Planets tend to do interesting things in the sky. Sometimes they move relatively quickly with respect to the background stars, while at others they speed up. What was even more baffling, was that occasionally, a planet will appear to slow down, stop, move backwards, stop again, and then move forwards again. This is called planetary prograde and retrograde motion. Astrologers make a meal out of this, suggesting that retrograde motion is a portent of evil influences. Rubbish.
The ancient Babylonians had a branch of their civil service dedicated to observing the night sky. They would take careful notes of the movements of celestial objects, and note them down in clay tablets that served as a record for centuries. It is believed that they were already keeping meticulous astronomical records as early as the 7th century B.C. Their treatment of the data was entirely numerical, not geometrical. They were very capable in computing planetary ephemerides and coming up with numerical methods to predict celestial events, but no geometric model of the universe seems to have featured in their line of work.
For the ancient Greeks, however, geometry became a partner to astronomy, as the ancient philosophers, astronomers and mathematicians attempted to make a working model of the universe. This auspicious marriage between mathematics and astronomy, would serve as one of the major driving forces in the progress of Science for centuries to come. Apollonius of Perga (2nd century B.C.) came up with a clever geometrical device to explain planetary motion. Planets, he claimed, moved on circles, whose centre was itself moving around the Earth in a circular orbit. These were called epicycles [epi Gr; ‘on’ + cycle, i.e. circles on circles]. Apollonius’ model was refined to perfection a few centuries later by another Greek astronomer, Claudius Ptolemy, who lived in Alexandria in the 2nd century A.D. You can interact with an excellent simulation here. To simulate the motion of the planets, Ptolemy had to evict the Earth just ever so slightly from the centre of their orbit, in order to make the model compatible with the observations.
Today, we know that planets move in elliptical orbits, with the Sun on one of the foci of those ellipses. How does this modern model explain retrograde motion? It is simply a matter of perspective; an appearance rather than a true backwards and forwards motion. Chasing the Red Planet is an activity that I made for students that were studying the laws of uniform circular motion, and that were also attending the Astronomy Club I was running at the time.
The activity uses a simple tool made with Geogebra to demonstrate prograde and retrograde motion. You can find the material here. The idea was to compare the predictions of a simple model of the Solar System with real measurements of planetary motions. The “measurements” would come from the Microsoft World Wide Telescope.
The following question can now be posed to the students: what is the exact geometrical arrangement of the Earth and Mars, during the beginning (and the end) of each retrograde motion of the latter? In other words, what is the angle between the radii of the planets when the phenomenon begins (or ends)?
The orbital periods of most of the planets around the Sun have been known since antiquity, and were certainly known to the astronomers of the 16th century. Playing the part of such an astronomer, one can attempt to calculate the angular distance between the two planets, with respect to the Sun. Starting with the orbital periods of the planets, one can easily calculate their respective –approximate– angular velocities (assuming uniform circular motion). By applying the laws of uniform circular motion, one only need have the period of the phenomenon (either the beginning or the end of the retrograde motion) in order to calculate the angular distance of the two planets at the start or the end of the phenomenon, with respect to the Sun.
Instead of relying on age-old astronomical tables and the tedious calculations which go along with those, or even actual observations which would take the better part of a decade, the students were set with the task to chase the red planet as it moves on the celestial sphere, using the WorldWideTelescope. By accelerating time by x1.000.000 times, they kept track of its planetary motion, and marked the dates at which the retrograde motion either begins or ends (either halting point). This was done for about 6 occurrences, and a mean period can be determined using an Excel spreadsheet, in order to demonstrate the results. A useful online tool for speedily calculating the days between two dates can be found here.
The students discovered that the period is not fixed, and fluctuates around a mean value of roughly 790 days. They were then instructed to enter a similar value in their physics formulas, along with the angular velocities of each planet that they had already calculated. Following a few simple calculations using their formulas, a value for the angular distance can be derived, which can then be compared with the value which is predicted by the Geogebra tool.
The activity ended with a discussion on how science works. One creates a model of a phenomenon by postulating a number of hypotheses. That model then makes certain predictions with regard to the phenomenon it purports to describe. By comparing the predictions of the model with the actual phenomenon, one can then proceed to find ways to spot weaknesses of the model and improve them – or, sometimes, even discard the model completely. Kepler had the courage to scrap seven years of hard work and start anew, discarding old hypotheses which mankind had held as dogma for centuries. His quest for truth led him to the discovery of the three Laws of planetary motion, that now bear his name.
“I demonstrate by means of philosophy that the earth is round, and is inhabited on all sides; that it is insignificantly small, and is borne through the stars.”
— Johannes Kepler