PREFACE TO THE ARTICLE
The following article is a translation of a technical paper published in Vol. 68, No. 31 of the Journal of the Vereines Deutscher Ingenieure and is descriptive of an instrument known as the Planetarium, which projects images depicting the relative movements of the sun, of the moon, of the planets, and of the fixed stars on the inner surface of a hemispherical dome; in other words, it simulates, or rather demonstrates, the relative movements of the heavenly bodies and presents a vivid, realistic picture of all changes in the relative positions and relative movements of these bodies. The translation is the work of G. N. Saegmuller, vice president of the Bausch and Lomb Optical Company, and it is through his courtesy that I have been successful in obtaining the article for publication in the U. S. Naval Institute Proceedings. My interest was first aroused while in Europe during the summer of 1924, when on a brief visit to the Carl Zeiss Works at Jena, Germany, the builders of the instrument. While there I was fortunate in finding the Planetarium, described herein, set up and in operation preparatory to being disassembled and shipped to Munich. The instrument, in external appearance, reminded me very much of a gun director and its operation showed a beautiful mechanical solution of the problem of demonstrating simply, readily and in a very realistic manner, the relative movements of the heavenly bodies.
In view of the weight given to the study of navigation and of astronomy at the U. S. Naval Academy, it occurred to me that the translation of the original description of this instrument would be of interest to the naval service, and it is, therefore, my desire to express appreciation to Mr. Saegmuller.
It is further of interest to know that the mounting of one of these instruments in Rochester, New York, is under contemplation. The writer is informed that the following instruments are building in Germany or under consideration:
- Munich—The dome has an internal diameter of thirty-one feet approximately.
- Jena—The dome has a diameter of fifty feet.
- Berlin—The dome will have a diameter of sixty feet; it is understood that estimates for the building complete with the instrument approximate $125,000.
- Dresden—The dome under consideration will have a diameter of eighty feet.
In permitting me the use of this article, the translator prefaced his work with the following remarks, which are of interest:
Planetariums or astrolabiums showing the relative motions of sun, moon and planets have been constructed at various times and are nothing new, but all were built on a small scale and the observer had to watch the various movements of the heavenly bodies from the outside of the apparatus. The new Zeiss Planetarium is of such construction that the observer is located on the inside of the apparatus and observes the motions of the heavenly bodies as they actually occur.
The ingenious idea of employing projection apparatus, and the fine solutions of the many difficult mechanical problems are due to Dr. Bauersfeld. He was, indeed, fortunate to have had the vast resources of the Carl Zeiss firm at his command; they again have demonstrated that they are leaders in scientific development, either in optical or mechanical directions.
Haller Belt,
Formerly Lieutenant-Commander, U. S. Navy.
SOME time before the outbreak of the war Dr. Oskar von Miller, the director of the German National Museum at Munich, approached the Zeiss firm regarding the construction of a planetarium which would show the movement of the heavenly bodies according to the Ptolemaic system on the interior of a semicircle dome in the same manner as they appear in the heavens. The first idea which suggested itself was to represent the stars by small electric bulbs attached to the hemispherical dome which would have to be rotated around an axis parallel to the earth’s axis. Sun, moon and planets were to be represented by illuminated discs driven by suitable gearing in such a way that the epicycle orbits of the heavenly bodies would be truly represented. The movements were to be so rapid that the happenings of one year could, at will, be reduced to a few minutes. It soon became evident that it was impossible to solve the problem in this manner, and the outbreak of the war, of course, put a stop to the work. It was taken up again after its close, but the problem was attacked from an entirely different point of view and the solution is such that it will also interest those who are not interested in mechanical problems.
The basic idea of the solution was to have the hemispherical dome fixed and to throw the images of the heavenly bodies on the dome by means of a system of projection apparatus which naturally had to be placed as near as possible to the center of the dome.
The available space in the Munich Museum allowed only ten meters for the inside diameter of the dome, with the horizontal line (Fig. 1) two meters above the floor. The system of the projection apparatus is placed at the same height and moves with a velocity corresponding to the movement of the heavenly bodies.
It is comparatively easy to project the system of the fixed stars. In the center of a hollow brass ball of one half meter diameter placed a Nitra lamp of 200 watts which furnishes illumination for thirty-one projectors which are attached to the outside of this hollow ball and which are so placed that each one projects a part of the starry heavens (Fig. 2).
The diapositives for these star projectors were made photographically from drawings executed on a large scale which showed every star up to the sixth magnitude. Four thousand five hundred stars were thus located, which is about the number visible to the naked eye. The different magnitudes were represented by discs of different size; thus a star of the first magnitude presents a disc of twenty-three milimeters diameter on the projection surface. This system of different sized discs has proven perfectly satisfactorily in practice and to the observer they appear as stars of different magnitude. To project the milky way it was necessary to provide a number of small projectors with the diapositives more or less diffused; it was also necessary to have additional small projectors showing the names of the different constellations.
This hollow ball carrying all these projectors revolves around an axis parallel to the earth’s axis, which, for the latitude of Munich, makes an angle of about 42° with the vertical.
It was much more difficult to produce the movement of the sun, moon and planets. To obtain the loop shaped orbits of the planets mechanically, even approximately, proved to be impracticable and resource was had to the simple Copernican theory of their movements to obtain their correct orbits. This solution is easily understood when we consider the fixed stars immovably projected on a sphere in whose center the sun is also fixed. The earth and planets move around the sun in elliptical orbits according to well known law's. The spot on which a planet is to appear in the heavens is the prolongation of the line from the earth to the planet to the sphere.
It is easy to represent the motion of the earth and a planet mechanically on a small scale. If we imagine the point which represents the earth connected with a planet by means of a telescopic tube which can be lengthened or shortened, corresponding to the varying distance between earth and planet, and attach a small projecting apparatus to this tube, which will throw the image of the planet in the direction of the tube’s axis on the starry sphere, the problem is solved and the image of the planet must describe its path with great precision.
In order to prevent these projection apparatus from interfering with each other, it was necessary to arrange them serially, one behind the other. This necessitated the reproduction of the earth’s orbit for each planet, also that the various mechanisms should be driven by individual axes of small diameter, in order not to cast too much of a shadow, as they could not be driven by a central axis for the reason that the above mentioned system of telescopic tubes had to move freely over the center of the orbits.
Fig. 3 represents the movement for Mercury. Two round metal discs “a” and “b” are held together at their circumference by the studs “c” “c.” The upper disc carries an axis which is inclined y°, corresponding to the inclination of Mercury’s orbit to the ecliptic (earth’s orbit) and to this is attached a lever carrying a small steel ball representing Mercury. In a corresponding manner a small steel ball representing the earth is attached in correct radial proportion to a lever rotating on an axis secured to the lower disc.
These two balls, representing Mercury and the earth, are not connected by telescopic tubes mentioned above, but by a diagonal lever system (or movable parallelograms) which allows a greater range in the lengthening or shortening distance between the balls representing Mercury and the earth. At the end of this system the small projection apparatus is attached and its optical axis is always in line with the centers of earth and Mercury. This, however, is not strictly true as the planetary movement had to be moved slightly away from the center, due to the fact that the fixed star apparatus already occupied this position as shown in Fig. 2. On this account, the projectors are inclined a certain amount from the true position.
The planetary orbit as shown in Fig. 3 describes a circle with the sun in its center. This is not strictly correct as these orbits are really ellipses, but the differences in planetary orbits between circles and ellipses are very slight; the greatest difference is found in the orbit of Mercury where the small axis of the ellipse is two per cent smaller than the great one. This difference would not amount to very much, but the error would become far greater if the variations in the velocity in different parts of the orbit were not taken into account. To diminish this error, a crank movement is being made use of based on the following considerations:
Fig. 4 depicts the ellipse of the orbit of a planet “P” with “S” as the sun in one of its foci. If we commence with the point “A” which corresponds to the shortest distance from the sun (perihel) and calculate the angles ? which the radius vector describes in the time “t,” the relation between “t” and ? can be expressed by the formula
“T” represents the full revolution in time, “e” the numerical eccentricity of the ellipse as given by the proportions of the distances “MS” to “MA.”
A very similar formula is obtained if we consider a point “P1” (Fig. 5) to move with perfectly uniform motion over a circle. The radius vector of an eccentrically located point “S” gives the angular valve 9 dependent on the time “t” very closely according to the formula
?1 expresses the proportion of the distances “M1 S1” to M1 A1. It is very easy to effect this mechanically if we substitute ?1 = 2 ? (as evidenced by comparing the two formulae) and we receive approximately the movement of the radius vector of the elliptic orbit. Without great error the elliptic motion of the planet may be assumed to be circular, taking care, however, that the position of the sun conforms to the numerical eccentricity of the real elliptic orbit.
This circular orbit is produced by means of a crank, and the crank pin, which represents the planet, receives its motion from the crank arm “S1 P1” (Fig. 5). This is embodied in the mechanism sketched in Fig. 6. _
Fig. 6. Mechanism to Produce Planetary Orbit
The pin “p” at the end of the crank “h,” represents the planet and the motion is produced by means of a slotted lever “h2” which itself is moved by pin “p1” attached to the wheel “Z." This wheel “Z” receives a uniform motion by means of gears whose axes are placed outside the discs “a” and “b” (Fig. 3) along the various planetary movements. The lever “h2” represents the radius vector “S1 P1” in Fig. 5. The intersection “S1” of its axis with the plane of the circle produced by the planet ball on lever “h1” represents the sun’s position. The drive pin of the three members “h1 h2” and “Z” in Fig. 6 are, as a result of the reasons given above, placed excentrically in relation to each other.
In spite of the introduction of this movement considerable errors still remain in the orbit of Mercury on account of its great excentricity of 0.2. The angular errors ? of the radius vector (Fig. 4) vary from —3.9° up to +3.9° which in the most unfavorable position of Mercury, at its nearest approach to the earth, amount to about 7°. Without the use of the above described correction mechanism the errors would amount to ±19° in the projected position of Mercury on the sphere. The position errors of the other planets are much smaller on account of the smaller ex- centricity of their orbits. Mars, which has, next to Mercury, the greatest excentricity, shows errors of only about one fifth of those of Mercury.
The orientation of the various planetary movements and the moon’s path had to be considered in relation to the ecliptic, the position of which, in regard to the fixed stars, could be considered as fixed. This resulted in the arrangement of these various mechanisms around an axis which represents the axis of the ecliptic and which has a fixed relation to the fixed star sphere (Fig. 2). The axis of the ecliptic forms an angle of ±23.5° with the polar axis and the small changes due to the changes of this inclination, together with the periodic effects of nutation could without appreciable error be neglected. It seemed desirable, however, to take into consideration the influence of precession which is caused by the earth’s axis describing a small circle around the axis of the ecliptic, the period being about 26,000 years. The effect of this precession is easily accounted for by giving the axis of the ecliptic, carrying the projector and the movements for sun, moon, and planets, the necessary additional movement, see Fig. 2.
According to the scheme shown in Fig. 3, which represents Mercury, similar mechanisms were designed for each planet and encased in circular boxes all of them connected and arranged as shown in Fig. 2.
Close to the lower bearing of the axis of the ecliptic the sun’s projector is placed whose movement is readily effected as the sun is represented by a fixed central stud and only the earth’s motion had to be taken into account.
The movement next to the sun represents the moon and here the earth can be represented by a fixed stud. The movement of the moon presents many difficulties. Whereas the orbits of the planets can be represented without appreciable error as fixed in relation to the fixed stars, the moon’s orbit changes rapidly. Its inclination to the ecliptic is a little more than 50, which can easily be represented by inclining the central axis of the moon’s orbit to that extent just as shown in Fig. 3 for Mercury. The small changes in the inclination of the moon’s orbit can be overlooked as they occur periodically.
But the line of intersection of the moon’s orbit with the ecliptic changes so rapidly that a full revolution is completed in 18.6 years. This change has been taken care of by giving the inclined central axis an additional amount of motion around the axis of the ecliptic. A further complication arises from the fact that the direction of the great axis of the moon’s orbit changes every year about 40°. To represent this change mechanically would have complicated the mechanism and the necessary gearings. For this reason it was deemed best to assume the moon’s orbit to be circular instead of being elliptic since the small amount of eccentricity of 0.055 causes an error of only about 6° in the projected moon’s position.
To represent the phases of the moon the simplest solution seemed to be to employ a small electric bulb with one-half of its surface blackened, and to throw its image directly on the sphere.
It is evident that the turning of the lamp would show the moon’s phases, but it was found that the bulbs could not be made small enough and that the required brightness could not be obtained.
FIGS. 7 TO 9
The problem was solved by placing back of the diaphragm in the projector which represents the full moon, an additional diaphragm having the forms shown in Figs. 7-9. This works very well from one full moon’s phase to the next, but by continuing rotation the real phases would not appear but their reflected images. To obtain the correct phases it would have been necessary to advance the diaphragm 180° at once at full moon time, and this not being feasible it was thought best to employ a second projector rigidly connected with the first one which alternately comes into action. By this means it was possible to obtain uniform motion for the diaphragms producing the phases.
Next to the apparatus representing the motion of the moon are placed the projection mechanisms for the planets Mercury, Venus, Mars, Jupiter and Saturn, which are all constructed on the same general plan as already described for Mercury. The phases of Venus were not taken into consideration as they are not visible to the naked eye.
To revolve the entire planetarium two possibilities had to be considered. To represent the daily motions the entire system has to revolve around the polar axis, which is effected by means of an electric motor attached to the side of the main column of the planetarium. By means of change gears different velocities can be obtained and a day can be represented in four and one half minutes, two minutes, or fifty seconds. All other movements are rigidly connected with the polar axis by gearing. The movements of sun, moon or planets relative to the fixed stars are, of course, very slow. To make these movements more perceptible it is necessary to throw the polar axis out of gear which makes it possible to show the movements of the planets relatively to the fixed stars, which are now at a standstill, by imparting more rapid motion to their mechanism by means of an auxiliary motor, which by means of gears is directly connected with the main axis of the planetary movements. Three velocities are provided and the happenings of a year can be shown in four and one half minutes, fifty or seven seconds. This last speed is primarily for the purpose of being able to set the position of the stars for a particular time. Reverse motion is also provided.
By using this yearly motion the loop shaped orbits of the planets are beautifully represented. As a matter of course the fixed stars do not strictly stand still, as on account of precession a small movement is given the axis of the ecliptic.
All mechanisms are rigidly connected by means of gears (excepting the daily motion which can be disconnected). A revolution counter on the planet axis permits setting it to any particular time. The accuracy of these movements over long periods of time depends, of course, on the proper selection of gears which are necessary to produce the motions of sun, moon and planets. It would have been very easy to produce the utmost accuracy if the size of gears did not have to be considered. As, however, a limit to these dimensions had to be set, it was merely a question of patience to select such numbers for the teeth of the various gears as finally gave the nearest result to absolute accuracy.
For example, the data for Mercury may be given. For the main driving axis common to all planets ten revolutions were taken for one sidereal year; thus for the movement of the sun a proportion of 1:10 resulted. The astronomical data for Mercury gave the proportion 0.415209106, which can very well
error of 1:107. This error is so small that for Mercury an error in position of 1° would not occur before a lapse of 5,000 years.
Regarding the other planets and the moon the errors for the same space of time are considerably less without making use of additional gears.
Although a great number of gears are necessary to drive all these mechanisms the friction has been reduced to a minimum by the use of ball bearings wherever it was possible to employ them.
Figures 10 and 11 show the planetarium in high and low position of ecliptic.
On the side of the main column the switchboard is seen by means of which all projectors and motors are manipulated.
In the background the Munich horizon is shown as it appears from the platform of the National Museum.
In demonstrating the daily motions it would have been necessary to produce such a bright illumination at the rising of the sun that the stars would disappear. This was not done as it seemed more important to show the movement of the sun through the various constellations and the movements of the planets in its proximity.
A very effective demonstration of the movement of the heavenly bodies is obtained by throwing in gear both the daily and yearly motions in such a ratio that the sun retains its position on the meridian. The effect is the same which an observer on the earth would obtain if the earth always presented the same side to the sun, as is the case of the moon in regard to the earth. In this case the orbits of the inner planets (Mercury and Venus) appear clearly as ellipses while the outer planets move through the entire Zodiac without forming loops. The sun does not remain fixed but performs a periodic movement up and down on account of its varying declination throughout the year.
The reproduction of the firmament by means of the described projectors is very effective and charming from an aesthetic point of view as the smooth spherelike projection surface destroys the faculty of differentiating distances and produces in the observer the impression of infinity. Another illusion is produced when the fixed starry heaven is moved around the polar axis; the observer is apt to imagine that the floor is moving around and that the heavens remain firm.
The planetarium was first shown in Munich at the annual meeting of the National Museum in the fall of 1923, the apparatus being far enough advanced to produce the projections.
The numerous exhibitions proved that the complicated mechanism stood the trial exceedingly well although it revolved apparently several thousand years.
With this apparatus the firm of Carl Zeiss has produced a very effective means of furthering the knowledge of the movement of heavenly bodies and awakening an interest in astronomy.
The cost of constructing such a planetarium is necessarily high, but not so prohibitive that they could not be erected in large cities, especially if a small entrance fee were to be collected. The usefulness of such an apparatus for instruction in schools can not be overestimated.
A second planetarium with a dome of sixteen meters inside diameter is being built, to be erected on the grounds of the Zeiss factory. This new apparatus will have some additional features not possessed by the one erected in Munich.
For instance, the polar axis may be made adjustable in latitude, which would make it possible to produce the heavenly motions as they appear to an observer at the north pole or the equator. The phenomena of the midnight sun can also be produced very effectively.