Eclipse Chasing in Cyberspace

Once upon a time (before there were digital computers), scientists, engineers, and afficionados of science engaged in Back Of The Envelope calculations (BOTEs). They knew how to produce subtle results using concise mathematics. Ideally, BOTE calculations and notes should fit on the back of a handy envelope, hence the name. The exercise honed intuitions and provided an excellent feel for subjects, a visceral familiarity with problems that can be lacking in modern computer models. In particular, developing BOTE computations forces the computer (you!) to continuously assess the quality of your approximations and data and to revise your expectations accordingly. Unlike state of the art numeric models, the results of BOTEs never depend on the nth decimal place.

When one of the regulars on a CompuServe forum recently told a visitor that there was nowhere else in the solar system from which the Sun was as neatly eclipsed as from Earth, it struck a false note with me. It's an oft-repeated observation, but it's just not so! I remembered having stumbled across such a place while learning to put a Hewlett-Packard 35 pocket calculator (remember those?) through its paces. Just to make the exercise sporting, and to keep to the spirit of BOTEs, let's rediscover this other eclipse-chasers' paradise using a slide rule instead of a calculator or a computer.

Where else in the solar system can you see a total solar eclipse that looks like those we so enjoy on Earth? One crucial issue is that the Sun and the eclipsing body should appear of almost identical angular size, so that during totality the sun's prominences and corona are neatly visible all the way around. The other is that the Sun and the eclipsing body must actually come into alignment as seen from the chosen vantage.

Consider Jupiter's swarm of moons.

From Jupiter's distance (call it J) of 486,000,000 miles, the Sun's diameter (D) of 865,000 miles has an apparent size of arctan(D/J). Instead of converting this to degrees or arc minutes for easy visualizaton (OK, it's 0.10 degrees, or six arc minutes, about a fifth the diameter of the sun and moon in earthly skies), we should recall one of the key codas of BOTE computation: At small angles, the angle in radians is equal almost to the sine and the sine is equal almost to the tangent. If we are willing to leave angles in the slightly arcane system of radians instead of converting to the more familiar degrees, minutes, and seconds, then we can use the proportion 865,000/486,000,000 = 0.00185 for both the angle and its tangent (an HP-42s calculator shows the error in this particular case is less than one part in a million!). Invert this fraction for a more convenient factor: at Jupiter's distance from the sun any ball held 541 times its own diameter away will just block the Sun. Try saying that another way: a ball 1 meter in diameter will exactly block out the Sun if the ball is 541 meters away. Likewise, a satellite 1,000 miles in diameter will exactly hide the sun at a distance of 541,000 miles.

That's all we need to go prospecting for solar eclipses.

This table shows the orbital radii and approximate diameters of Jupiter's four large moons (in miles, to "slide rule accuracy"). It also shows how far away from each moon you must be for it to just eclipse the sun:
  Moon      Dia    Orb Rad.      "Eclipse Distance"
Io       : 2,000   260,000    2,000 * 541 = 1,082,000
Europa   : 1,800   420,000    1,800 * 541 =   974,000
Ganymede : 3,100   670,000    3,100 * 541 = 1,680,000
Callisto : 2,800 1,200,000    2,800 * 541 = 1,515,000

As seen from the cloudtops of Jupiter, none of these distances and diameters works out quite right. Callisto almost makes it, but all the Galilean moons are too large to neatly but barely block out the sun. (Tiny Amalthea circles Jupiter inside Io's orbit, and it's about the right size, but the little moon is so badly out of round that I decline to book passage to Jupiter in order to tread ammonia while awaiting its fleeting shadow.)

There are better and more hospitable viewing sites than Jupiter's cloudtops. Check out Europa as seen from Ganymede, and Callisto as seen from Europa. When these moons are on the same side of Jupiter, their distances are even less friendly to eclipse watchers, but when they are on nearly opposite sides of their orbits, things are much more favorable:
Ganymede to Europa:  1,190,000 miles.
Europa to Callisto:  1,620,000 miles.

During mutual occultation seasons on Jupiter (1996 - 1998, for instance), solar eclipses very similar to those visible from earth are visible from at least two of Jupiter's moons. If you find yourself on Europa in such a season, watch for Callisto to produce familiar-looking solar eclipses. And if you find yourself on Ganymede, watch for Europa to do the honors.

Jupiter itself can get in the way, of course, but if an eclipse occurs just after or just before the observer's moon slides in or out of Jupiter's shadow, the distances are essentially unchanged (but they are changed and in the case of Europa's eclipses as seen from Ganymede, the position of Europa in its orbit makes the difference between annular and total eclipses -- just as the position of Earth's Moon does for us. Fortunately, eclipses that take place far enough from Jupiter in the sky of Ganymede can be total.

Since working this out, I've found a reference in Isaac Asimov's collection of science essays The Solar System and Back which provides a blow-by-blow description of similar events. In "The Dance of the Satellites," Asimov displays some BOTE celestial mechanics including eclipses as seen from Jupiter's inner moon Amalthea. It's just possible that the Great One overlooked the better eclipses visible across the satellites' orbits. In turn, he points out a phenomenon I overlooked: at the moment of totality, the "dark side" of the eclipsing satellite is bathed in the light of a nearly full Jupiter. At the moment of totality, features of the satellite hidden in the Sun's glare should stand out vividly in an ashen light far brighter than any which graces earth's crescent moon.

Holstering the slide rule and taking advantage of 25 years of computing progress, we can use a desktop PC to check out the exact circumstances of specific events using (for example) Bill Gray's excellent Guide software. (I suppose using a computer makes this a "power BOTE.") We can then render them with startling clarity using Adobe Photoshop.
(Close-up and details of the July 16, 1997, event.)

Furthermore, it's a simple matter to confirm the times of these occurances by searching the WWW for a list of upcoming mutual satellite phenomena (hint: start with "International Jupiter Watch" if you're paying by the minute). The event depicted in the accompanying illustrations is a solar eclipse by Europa for an observer on Ganymede on July 16, 1997.

All that remains is for Scientific Expeditions (no doubt in cahoots with Sky and Telescope) to book an eclipse cruise to Jupiter's moons. Make mine a cabin with a view, please.

Are there places other than the Earth and Jupiter for eclipse hounds? I leave Mars, Saturn, Uranus, Neptune, and Pluto to be sniffed out by others.

In the process of chasing down these circumstances, two things became perfectly plain. First: the sky is always clear in cyberspace. And second: Water Rat had it right in The Wind in the Willows when he told Mole that nothing, absolutely nothing is half so much worth doing as simply messing about in BOTEs.

Take me home!