RETROGRESSION OF THE PLANETS 765

A DEVICE TO SHOW THE RETROGRESSIONOF THE PLANETS

BY LUISE LANGECrane Junior College, Chicago

In the following I am going to describe a little self-madeastronomical model which I have found very useful and effec-tive to demontrate one of the central points of the Copernicandoctrine: how the combined motion of the earth and the planetsaround the sun brings about the observed periodic retrogres-sions of the planets. The idea of the model in itself is in no wayoriginal, it is indeed almost obvious. But the great simplicityof its execution and the universal and immediate availabilityof the material used justify, I believe, its being presented at thisplace. Moreover the new interest in the motions of the heavenlybodies which the Adier planetarium in Chicago is arousing inthousands of people may be said to impart a new actuality tothis old problem.

In any clock the two hands travel about the center withtheir angular velocities in the ratio 12:1. This suggests thatthey might be used to represent the respective motions of theearth and the planet Jupiter, the former traveling in one year,the latter in only a little less than twelve years around the sun.However, in the case of the clock it is the large hand whichtravels twelve times faster whereas in the case of the planets theouter one, Jupiter, is the one to travel twelve times slower thanthe inner one. Accordingly the small, slow hand of the clock hasto be made the longer of the two, indeed about five times aslong as the fast hand, Jupiter being a little over five times as farfrom the sun as is the earth.With this idea in mind we took an old alarm clock and re-

moved it from its metal case. Next we shortened the long handby turning up its end, and extended the small hand by meansof a piece of thick copper wire likewise with its sharpened endturned up. On these turned-up ends we placed two beads, alittle one for the earth and a big one for Jupiter. Next we re-moved the regulator so that the clockwork, if released, will justrun off, driving the fast hand around in about one second. How-ever the speed of the motion can easily be regulated by placinga finger lightly against any conveniently located gear. On re-leasing the clockwork we see the two planets traveling around

766 SCHOOL SCIENCE AND MATHEMATICS

with approximately correct relative velocities and distances,alternately drawing closer together and farther apart as theyget on the same or on opposite sides of the sun. So far themodel illustrates what we call the ^true motion" of the twoplanets.

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FIG. 1. "True Motion57 of Earth and Jupiter

But now comes the more interesting point: the apparentmotion of Jupiter as observed from the earth. The place in theheavens at which Jupiter is seen from the earth lies at everymoment in the direction of the straight line between the twoplanets. The motion of this line as the two planets travel in theirrespective orbits constitutes Jupiter’s apparent motion. To dem-onstrate this we made a pointer, about twelve inches long.(cut from the flattened out discarded case of the clock) andplaced it over the two turned up ends of the hands, fitting itwith a hole over the Jupiter-end, and with a long enough slot(about two and a half inches) over the earth-end to allow fortheir varying distance apart. Finally the bead planets wereput back in their places.

FIG. 2. Apparent Motion of Jupiter

Now we again set the universe in motion. As before the twoplanets travel uniformly arround the sun. But the pointer con-necting them is now seen to move alternately onward and back-ward just as Jupiter appears to do in the heavens. Whenever

RETROGRESSION OF THE PLANETS 767

the earth comes near passing between the sun and Jupiter thepointer begins to gradually move backward, then on again,reaching its greasest forward speed when the two planets areon opposite sides of the sun. Because of the nearly correctratios in the model this apparent motion of Jupiter is shownvery approximately correct also in its quantitative aspects. Tofollow it up in this respect the dial of the clock may be usedto great advantage. On it one reads not only the year (one fullrevolution of the earth) but by means of the division intotwelve parts also the months, down indeed to a fifth of amonth.* Writing the names of the months along the succes-sive hours (e.g., beginning at 12:00 with March, the month ofthe spring equinox) any position of the earth now signifies adefinite time of the year. To add further astronomical mean-ing one may also write the constellations of the zodiac dia-metrically opposite the respective months in which the sun isseen to pass through them, Virgo at 12:00 Libra at 1:00, etc.Watching now on the dial how often and where Jupiter ap-pears to retrograde we easily read off: it retrogrades aboutevery thirteen months; the retrograde arcs lie a little over 30°apart; e.g., if at one time Jupiter is in the middle of its retro-gression during February in the constellation of Leo (as it wasthis year) the next year this happens in March in the constella-tion of Virgo, etc. One further reads that the retrogression frombeginning to end lasts about four months, and that the retro-grade arc is about 10° long. The model makes it clear that thetime and arc distance between two successive retrogressionsdepends solely on the angular velocities of the two planets,whereas the duration and arc of any one retrogression dependsin addition on their relative distances from the sun. This latterratio our model would easily allow to vary, not however therelative velocities. Therefore it cannot quantitatively demon-strate the behavior of the other planets.

But with its moving line of sight between earth and Jupiterthe model demonstrates just as well the motion of the earthas seen from Jupiter, which means the motion of an innerplanet seen from an outer one. That is the kind of motion ex-hibited to us by Venus and Mercury. Watching the end of the

* The clock itself adds another little astronomical feature: the occasional rattlingof the rudimentary alarm announces each time when Jupiter (the original small handof the clock) has made a complete revolution, that is, it rings Jupiter’s sidereal period.

76S SCHOOL SCIENCE AND MATHEMATICS

pointer beyond the "earth^ we see it moving periodically on-ward and backward, but always within a limited arc distancefrom the sun.

Finally one may use the model to demonstrate�though notin the best possible way�still another interesting point: thatif two planets move concentrically around the sun either onemoves relative to the other in an epicyclic curve, as had beenmaintained by the Ptolemaic theory. To show this we havesimply to regard Jupiter as stationary, that is, refer the posi-tion of the sun and the earth to it as a fixed point. This can

FIG. 3. Epicyclic Path of the Earth with Jupiter Stationary

be accomplished by concentrating the attention on Jupiter aspoint of reference. This effort can be effectively aided by guid-ing the model, while in motion, in such a way as to keep Jupiterin a fairly stationary position relative to the table beneath.With Jupiter^ position fixed the sun is seen to move aroundit in a circle, and the earth, traveling in a circle around the sun,is seen to describe relative to Jupiter an epicyclic path. Jupiter,vice versa moves in that very same curve relative to the earth,namely in an epicyclic curve in which the center of the epicycletravels around the deferent in twelve years while Jupiter trav-els in the epicycle in one year. This was the way in which thePtolemaic theory had described the motion of Jupiter. This lastdemonstration is therefore useful in bringing home the com-plete equivalence of the Ptolemaic and Copernican theory fromthe merely mathematical point of view.