From a Math Puzzle to an Odd Moon

Asked for an odd moon, my friend Bob thought of Keith Moon, a witty but incorrect answer.

If you hated age-problems in high school algebra, you have plenty of company. I’ve even heard teachers complain that they are contrived instruments of torture. (I think they are merely tired of teaching a curriculum over which they have no control. 🙂 )Here’s one that I devised, not sadistically, but to lead to an interesting discussion of a rather peculiar moon.

It is 2021. I have a chemical element in mind discovered long ago. In a different year, an unusual moon was discovered. The sum of their current anniversary dates is 595 years.

In still another year, the anniversary of that moon’s discovery was 135 years less than the current anniversary of the element. The sum of the anniversaries in that same year was the product of 5 and of a number that is also prime in the National Basketball Association.

Identify the moon and the element.

The equations leading to the solution are

x + y = 595

y – a = x – 135

5(23) = x – a + y – a , where x = current anniversary year of the element; y is that of the moon and a is the difference between 2021 and the so-called “another year”. 23 was Michael Jordan’s number and also worn by Lebron James. Solving for y, we get 350 for the anniversary of the moon, so we are looking for a moon discovered in 1671, and there’s only one: Saturn’s Iapetus.

If you are a fan of Arthur C. Clarke’s science fiction, you will be happy to learn that at the end of his 1968 novel, 2001: A Space Odyssey, the monolith at the end of the book is located on Japetus(Iapetus). There’s a free app “Moons of Saturn” which, as its name suggests, is designed strictly to reveal the positions of the major moons of Saturn at half hour or longer intervals, and on that app you will notice that Iapetus has a slightly eccentric orbit. But compared to its sister moons, the orbit is highly inclined at 15.47 degrees. Its density is only 1.08 g/cm^3, but three other of Saturn’s moons have densities close to that of water. What makes Iapetus unusual is that half the planet is very reflective, while the other half is dark with an albedo of only 0.03 to 0.05 meaning that it absorbs 95 to 97 % of the light that strikes it. Giovanni Cassini, who discovered Iapetus, had observed the yin-yang characteristic of the moon’s surface. He only saw Iapetus on Saturn’s west side. Based on this observation, he surmised that one side of the moon was much darker than the other, and that it was tidally locked with its mother planet.

The part of 2001: A Space Odyssey which mentions Iapetus
Iapetus. A composite image taken by NASA’s Cassini’s mission

How this this happen? One hypothesis is that it’s linked to the presence of another Saturnian moon, Phoebe. This small body with an average radius of only about 100 km has a low albedo of 0.06. Its orbit is retrograde, which means it goes around Saturn in the opposite direction than most of its other satellites. The combination of those observations suggests that Phoebe is a Centaur, an object that escaped from an area filled with dark objects, the Kuiper Belt, and which, in this case, was captured by the giant Saturn. Since Phoebe is so small and dark a guess is that the dark material can easily escape from its surface and that some of it was picked up by nearby Iapetus. Another hypothesis is that the dark material consists of hydrocarbons emitted by Iapetus’ volcanoes.

An area on Iapetus with an original mix of dark and ” native” white, icy material would have absorbed heat and have its ice sublimate, leaving more dark material behind. Meanwhile the whitish material that would land in another location would not absorb as much radiation, not sublimate and would more likely remain there, progressively getting lighter as the cycle repeated itself. This mechanism seems likely given that Iapetus rotates very slowly with a day as long as 79 Earth days, giving ample time for the dark material to absorb solar radiation and cause icy volatiles to migrate.

Another weird Iapetian characteristic is the equatorial bulge that gives it some of the tallest mountains in the solar system: over 20 km high! Some guess that the bulge is from the collapse of a Saturnian ring. Others think it’s from the result of having slowed down, with conservation of momentum causing a shift in the center of mass.

Oh, and what was the element in the puzzle? Since x = 245, the one discovered that many years ago in 1776 is the most common atom in the universe and part of the ice in Iapetus—-hydrogen.

Sources:

Iapetus https://solarsystem.nasa.gov/moons/saturn-moons/iapetus/in-depth/

Phoebe https://solarsystem.nasa.gov/moons/saturn-moons/phoebe/in-depth/

Images from https://en.wikipedia.org/wiki/Iapetus_(moon)

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