Inclination
Longitude of
Ascending Node
Argument of
Perihelion
Arm
♄
☉
and
Gegenschein
☽
Arm
☿
♀
♂
♃
|
Abweichungen der Planetenpositionen in Folge begrenzter Zahnradübersetzungen gehören der Vergangenheit an. Errors in planetary locations due to gear transmissions with their periodic inaccuracies are a matter of the past. |
||
|
—Zeiss web site for current models |
Yonkers’s answer to the Antikythera mechanism is here at the Hudson River Museum. The planets are listed from north to south. We use the term planets in the ancient sense, including the Sun and Moon but excluding Uranus, Neptune, and Pluto.
Each photo is 640 × 425 pixels. They were taken on Sunday, December 31, 2006, and downloaded from Kodak. Frank Wortner took the photos; Ron Walker explained what they showed. Thanks, guys!
The Zeiss Model II had the seven planets in the same order.
| Arm | Planet | i
Inclination |
Ω
Longitude of Ascending Node | ω
Argument of Perihelion |
Zeiss M101-5 | Zeiss Model II | |
|---|---|---|---|---|---|---|---|
| Ceiling | Floor | ||||||
| North Arm |
Saturn
♄ |
2.48446° | 113.7153281104° | 33.71690° | |
|
|
| Sun
☉ and Gegenschein |
0° | 114.20783° | |
||||
| Moon
☽ |
5.145° | |
|
||||
| South Arm |
Mercury
☿ |
7.00487° | |
|
|
||
| Venus
♀ |
3.39471° | 76.68069° | 54.85229° | |
|||
| Mars
♂ |
1.85061° | 49.57854° | 286.46230° | |
|
||
| Jupiter
♃ |
1.30530° | 100.55615° | 274.19770° | |
|||
The Zeiss is a hardware implementation of the Copernican solar system. To keep the clockwork simple, all gears are circular and move at a constant speed. There are no Keplerian ellipses or equal area laws.
The planetary cages form a seven-storey apartment house. Saturn lives on the top (northernmost) floor; the Sun is one level below. A shared bulkhead serves as Saturn’s floor and the Sun’s ceiling, making it easy for the Sun to transmit its once-per-year rotation to Saturn’s clockwork. The same transmission takes place from Mercury’s floor to Venus’s ceiling, and from Mars’s floor to Jupiter’s ceiling.
All the gears (except those on the Moon’s ceiling) are powered by a common driveshaft. It turns 10 times per year at a constant rate, in the direction in which the sun seems to travel around the ecliptic. When looking down from the northern hemisphere (our usual orientation), we refer to this direction as “right to left” or “counterclockwise”; astronomy books call it “west to east”. The gears on the driveshaft are tightened with the 5/64″ “Planet Projectors” allen wrench.
The Earth goes around the sun at a varying speed, fastest each year at aphelion on January 3 and slowest at perihelion on July 4th. From our point of view, however, it looks like the Sun goes around the Earth at a varying speed, once again fastest on January 3 and slowest on July 4th. It is this point of view that the Zeiss projects. When we want the true picture, we use the orrery. Similarly, the above table gives the inclination of each planet’s orbit to the plane of the Earth’s orbit. When we study the Solar System itself, rather than its appearance from Earth, we use the inclination of each planet’s orbit to the plane of the Sun’s equator.
The gear on the driveshaft that moves the sun has 32 teeth; the big gear has 320 teeth. This is how I discovered that the driveshaft turns 10 times per year. Since there is only one gear between them (the one with the four screws), they turn in the same direction, west to east.
Each planet is projected by a cylindrical “flashlight” connected to the Zeiss by two bearings or pins. One end of the sun’s flashlight (the “business end”—the end that the light comes out of) is attached to a point on the circumference of the 320-tooth gear. Let’s assume for the moment that the other end (the “butt end”) is attached to the hub of the gear. Then the sun would swing around the dome at a constant speed as the gear turns ata constant speed.
The butt end is actually attached to a pin near, but not at, the center of the 320-tooth gear. This pin is fixed to the floor through a wide hole in the center of the gear, like the hole in a 45 RPM phonograph record. As the 320-tooth gear turns, the distance between the two pins changes. The flashlight has sliders to accomodate this.
Since the butt-end pin in not in the center, the flashlight swings at a varying speed: fastest when the pins are closest together, slowest when they are farthest apart. The butt-end pin is only about an eighth of an inch from the center of the 320-tooth gear, which makes it hard to determine the moment when the sun moves fastest. To determine when this happens, there is a dimple near the circumference of the 320-tooth gear. It’s visible in this photo alongside the four-screw gear, at the same longitude on the 320-tooth gear as the blue vertical pin that holds the flashlight. On January 3, the dimple is adjacent to the out-of-focus and otherwise purposeless bolt near the top of the 320-tooth gear in this photo. The butt-end pin lies along the line segment connecting this bolt to the center of the 320-tooth gear.
The above mechanism is how Ptolemaic astronomy modeled the varying speed of a planet in its orbit. The planet was attached to a point on the circumference of a circle (a “deferent”) that spins at a constant speed. But the planet was viewed from a point not exactly at the center of the circle (the deferent was “eccentric”). The direction and motion of the beam of light projected by the flashlight from the butt-end pin to the business-end pin (and beyond to the planetarium dome) corresponds to the direction and motion of our line of sight from the Ptolemaic earth to the planet.
The floors of Mercury and Mars have exactly the same pair of 32- and 320-tooth gears, turning in synch with the sun gear. It’s not so complicated.
Incidentally, the Gegenschein flashlight is mounted on the Sun flashlight, but pointing in the opposite direction. It’s below the octagon butt-end of the Sun is this photo, and in the lower right corner of this photo.
The small gear on Saturn’s floor turns once per Earth year. The big gear on Saturn’s ceiling turns once every 29.45 Earth years. The butt-end pin is on the circumference of the small gear; the business-end pin is on the circumference of the big gear. The beam of light from the butt-end to the business-end corresponds to our line of sight from the Earth to Saturn.
The small gear is easily turned by the Sun. After all, Saturn’s floor is the Sun’s ceiling.
The big gear on Mercury’s floor turns once per Earth year. It’s identical to the big gear on the Sun’s floor and their dimples should be in the same place. A smaller gear on Mercury’s ceiling (which is, none the less, the biggest gear on the ceiling) turns once per 88 Earth days. The butt-end pin is on the circumference of the big gear on the floor; the business-end pin is on the circumference of the smaller gear on the ceiling. The beam of light corresponds to our line of sight from the Earth to Mercury.
Future research. The Mars ceiling seems to have a pair of stacked gears that do not share the same center (not sure). Jupiter might have its “Earth pin” 180° from where the Earth is (not sure).
The yellow label on the Moon’s floor is the Präzession 1900 label. The big gear on the moon floor turns once every 6793.5 Earth days or 18.5996 Earth years. Since an even number (4) of gears separate it from the driveshaft, it turns “backwards”, from east to west. This implements the regression of the Moon’s nodes around the ecliptic. A dimple on the big gear marks the location of the descending node. The dimple is at 9’clock in this photo, next to a hole in the big gear.
The clockwork on the Moon’s ceiling has its own private driveshaft, barely visible along the lower edge of this photo. It turns approximately 2 and 3/4 times per sidereal month. It’s gear (32 teeth) is the only driveshaft gear that is brass, and the adjacent gear (66 teeth) is the only one that’s vinyl. The next pair of gears has 84 and 108 teeth. The driveshaft gear take a 1.5 mm allen wrench, which is smaller than the gears on the other driveshaft.
The big gear on the Moon’s ceiling, in the drum with the parallel copper contact strips, turns once every 27.321582 days. A smaller gear in the drum walks around a gear in the center of the drum. The latter gear is attached to the Sun, on the other side of the ceiling, synchronizing the Moon’s phases with the Sun. The curved cable (actually a spring) in the photo connects the walking gear with the gearbox in the flashlight that produces the phases.
Here are our first and second attempts to photograph the walking gear. The curved cables leads directly into it, but it’s hidden in the shadow between the big gear and the copper contact strips. To see it better, put the photo into Photoshop and turn up the brightness and contrast. An extreme closeup reveals that the allegedly German gearbox was actually made by Huco in England, a company that specializes in “innovative misalignment couplings” (sounds like a Shaw play). Here are our first and second views of the gears atop the gearbox.
There is no evidence (yet) that the Zeiss knows about the Moon’s apogee and perigee. Each 24 hours of forward diurnal motion moves the moon about 13.5 days along the ecliptic. Does it know about the progression of the perigee at the rate of 360° per 8.85 years? This would seem to require gearing comparable to that on the Moon’s floor.
The Zeiss moon is currently three days late.
In the above step 1, it is important to position the Moon east of the Sun. When the Moon passes the Sun from east to west, a tiny crescent persists. When passing the Sun from west to east, the Moon is completely invisible. Let’s exploit this asymmetry for the present, and fix it at some future time.
The Zeiss predicts a descending node for February 10, 2007, but the actual date of the node is February 5, 2007 at 21 UT.
