TUNGSTEN VALLEY - Last month, we received a riddle posed by a beautiful young woman nearly half a century ago. At what intervals do a clock’s hour and minute hands align? It’s the same problem as how often planets align with each other, called conjunctions. What’s the formula for how often clock hands align and planets conjunct?

The way I thought about it back then, was to imagine Planet A swinging a full revolution around the sun, only to find on its arrival that Planet B had, in the mean-time, moved forward by some smaller angle of its own. As A then moves forward through that new, additional angle, B moves yet again a little further ahead.

And so forth, forever. Gradually A catches up to B, in an infinite series of ever-smaller increments. If we can sum that infinite series of ever-smaller fractions, as in this month’s drawing, then we’ll know the time between conjunctions.

The ancient Greeks considered this kind of summing in Zeno’s Paradox describing an arrow speeding to a target, and of Achilles chasing Mr. Tortoise. Philosophically, they were puzzled because they couldn’t see how an infinitely-long series of numbers could have a finite sum.

We, however, with the gift of algebra from our Arab forbears, know how. (Math students: Ask your teacher to walk you through this.) The answer turns out to be the product of the two planets’ periods, divided by the difference.

With this formula, we have a super-power. We can instantly predict all sorts of planetary positions with a calculator. Here are some examples:

Jupiter takes twelve of our years to orbit the sun; Saturn needs thirty. The product divided by the difference is 360 divided by 18, which means these two planets come together in our sky every 20 years. They last joined in 2020; they will do so again in 2040.

For Mercury’s best viewing in the evening or morning sky, we use its period of 88 days and ours, of 365 days, to discover that we get good evenings every 116 days. The same interval (on a shifted cycle) goes for mornings.

The formula tells us Venus repeats as an Evening Star every 584 days, a count that ancient Mayans used in their war calendar. The same cycle, date-shifted, applies for Venus as a Morning Star.

Mars and Jupiter reach their oppositions (when we see them best) every 779 days (Mars) and every 13 months (Jupiter). Rocket-launch opportunities to these planets occur at the same intervals. The formula further tells us they conjunct together, side-by-side in our sky, about every two and a quarter years.

And oh, from my (eventual) girlfriend’s original challenge: A clock’s hour and minute hands align (or have any other given angle between them) at intervals of twelve times one divided by twelve minus one, or 12/11 hours, which is once every hour, five minutes, and 27.3 seconds.

Thank you, J–. You were worth the wait.

In December Skies:

The sun begins the month in Scorpius, entering Sagittarius on December 17. At the winter solstice (the shortest day and the sun’s lowest noon-point of the year) on December 21, Nederland days and nights are 9.3 and 14.7 hours long, respectively.

The moon’s dates are: New December 1; First Quarter December 8; Full (Cold Moon) December 15; Last Quarter December 23; Second New Moon (called Black) December 30.

December Meteors: The Geminids (parent body asteroid 3200 Phaethon) run December 4-17, peaking on the 14th. Visible from 7 p.m. until dawn, but looking best around 3 a.m.

Best Sky Viewing Nights (Minimal Moon): December 1-8 and 23-31.

Sunset (Mid-Month): The Pleiades are rising with red Aldebaran just below. Brilliant Capella is in the northeast. The Andromeda Galaxy is high in the northeast, presenting a nice binocular view. Saturn, Fomalhaut, and Diphda form a striking triangle in the SSW. Saturn is above, Fomalhaut below, and Diphda to the left. Deneb, Vega, and Altair form the Navigator’s Triangle high in the northwest to west. 

Midnight (Mid-Month): The Gemini Twins, Castor above and Pollux below, are nearly overhead in the northeast above Mars. Orion is due south, with Sirius. The Great Nebula in Orion presents a terrific view in binoculars and low-power telescopes. Taurus with the Pleiades is nearly overhead.

Sunrise (Mid-Month): Vega is low in the northeast. Reddish Arcturus is high in the east. Orangish Regulus in Leo is almost overhead, slightly offset toward the southwest. The Gemini Twins are close to setting in the west. Capella sets in the northwest.

Mercury, in Scorpius, is highest in the southeast at dawn on December 15.

Venus, in Capricornus, is the Evening Star in the southwest. It reaches maximum elongation from the sun at mid-month.

Mars, in Cancer, rises at 7:30 p.m. High in the southeast, below the Gemini Twins, at midnight. It is low in the west at sunrise. 

Jupiter, in Taurus, is at opposition on December 5. Rising at sunset and setting at sunrise, it gives wonderful viewing of itself and its moons in telescopes and binoculars.

Saturn, in Aquarius, is high in the southwest at sunset, setting by 11 p.m. It forms a striking triangle with Fomalhaut below and Diphda to the left.

Notable Space Missions: NASA’s deep-space Lucy spacecraft does a gravity-assist earth flyby on December 12. Lucy has already made asteroid belt discoveries; she’s heading back out to visit two more main-belt asteroids, eventually visiting exotic asteroids near Jupiter.

Frank Sanders, a spectrum scientist, takes astronomy-related inquiries at backyardastronomy1@gmail.com.