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 did 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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Why Are Some Still Using Methylene Chloride to Extract Caffeine?

Methylene chloride, also known as dichloromethane, is a solvent that, compared to water, can do a better job at dissolving certain substances such as grease, paint and caffeine. The so-called non polar solvent is mass-produced. It’s used by the fracking and plastic industry and in the production of HFC-32, a widely used replacement for CFCs that wreck the ozone, But there’s a catch. Once considered too short-lived as a molecule to act as a catalyst to break down ozone, dichloromethane, according to atmospheric model simulations, is causing damage that is not only negligible but becoming increasingly significant.

Then there’s the fact that IARC also considers it a probable human carcinogen, a classification based on several cohort studies of occupational exposure to the solvent’s vapors.

In the 1970s, dichloromethane was detected as a residual compound in decaffeinated coffee and tea. The levels ranged from < 0.05 to 4.04 mg/kg(ppm) in coffee, and < 0.05 to 15.9 mg/kg in tea. Because of concern over residues, most producers no longer use dichloromethane. Some people argue that no evidence links low levels of consumption to the kind of elevated risk for cancer experienced by workers who are exposed to much higher concentrations. But without implementing precautions, dichloromethane remains part of a much larger soup of invasive and potentially dangerous compounds.

Besides, there is a much more clever and innocuous method of extracting caffeine known as the Swiss Water Method.

First green (unroasted) coffee beans are soaked in hot water. This not only extracts a lot of caffeine but also aromatic and flavorful oils, and carbohydrates. But the Swiss method uses the problem as part of the solution, pun-intended! After soaking, the aqueous-solution from the first extraction is passed through a charcoal filter to remove caffeine. The charcoal is made in some countries from the discarded shells of coconuts. The caffeine molecules get trapped in the filter while its more desirable compounds, for the most part, pass through and stay in solution, a mixture known as “green coffee extract”. All other green beans that have to be decaffeinated are then soaked, not in hot water again, but in the green coffee extract. Because the solution is already saturated with the compounds that should remain in the coffee beans, for every non-caffeine molecule that leaves the beans, there is one that returns at the same rate, with the net result that the beans keep their flavor. But since the charcoal had removed the caffeine, the solution is not saturated with caffeine and these new beans still lose their caffeine to the extract. The extract is then passed through the filter again to remove the newly dissolved caffeine, and the process is repeated with yet another batch of beans to be decaffeinated.

From the Swiss Water Decaffeinated Coffee Company

Given all this, I was surprised to see that Costco’s Kirkland brand of decaffeinated coffee still uses methylene chloride. (see update below)I promptly wrote to Costco. Oddly, they passed the complaint to the specific warehouse we shop at, which of course is not responsible for selecting the method of extraction. Perhaps coincidentally, and ever since my letter, I have not been able to see the product on their shelves. Another coincidence is that the product is currently unavailable at Amazon. Furthermore, the Costco website itself only offers an organic brand that uses the Swiss Water Method!

At the Canadian grocery chain, Loblaws, their name brand of decaffeinated coffee advertises the Swiss Water Method on its label, but their no-name brand does not specify. I called them yesterday, and I am happy to report that it too does not rely on methylene chloride. Its caffeine is extracted exactly the way its more expensive counterpart does it.

A friend asked me, “Please forgive me for asking but…what’s the point of decaf coffee? 😁“. My response: I mix it with real coffee so I could drink 6 cups a day instead of 3!

April 27, 2021 Update: Encouraging news! After months of absence on Costco shelves, decaffeinated coffee is back, and surely enough the caffeine has been removed using the Swiss water method! Coincidence or not, the important thing is that the giant retailer has gotten its act together.

Until last year, Kirkland used methylene chloride, as discussed earlier. But they’ve mended their ways and now extract it sensibly using water that’s saturated with flavours so as to remove only caffeine.

Other Sources:

The increasing threat to stratospheric ozone from dichloromethane. Nature CommunicationsDOI: 10.1038/ncomms15962

S Saloko et al 2020 IOP Conf. Ser.: Earth Environ. Sci. 443 012067

IARC Monograph on Dichloromethane (methylene chloride)

Caffeine in Coffee: Its Removal, Why and How? July 1999Volume39(Issue5) Pages, p.441- 456 – Critical Reviews in Food Science and Nutrition

Why are the orbits of our planets not equally eccentric?

How fitting that the 2 planets with the most circular orbits (lowest eccentricities) are named after Venus, the goddess of love and fertility, and Neptune, the god of fresh water and sea, while those with the least circular orbits honor the gods of commerce (Mercury) and the underworld (Pluto)!

There is a reason behind the differences in eccentricities, but the connection to names is a coincidence. The Romans had no notion of elliptical orbits or eccentricity when they named Mercury and Venus, and although I am not sure if the eccentricities were measured before Neptune and Pluto were named in 1846 and 1930, respectively, I am willing to bet that their elliptical nature did not influence the choice of names. Besides, Venus, at least from up close, does not inspire love. Without water to wash away any carbon dioxide from its atmosphere, Venus has a runaway greenhouse effect with an average surface temperature of 464 o C and to boot, clouds of sulfuric acid hover over its surface. Although Neptune is as blue as our oceans, its color comes from the methane in its atmosphere. There is water in its -200 o C icy surface, but it’s mixed with methane and ammonia.

On the left are the two planets with the greatest eccentricities, Mercury and Pluto. The least eccentric orbit is that of Venus, top right, followed by Neptune. Art works are by (clockwise from top left) Jean-Baptiste Pigalle, Alexandre Cabanel, Michel Anguier and Giovanni Battista di Jacopo.

How is eccentricity measured and why does it exist for planets?

An ellipse is a flattened circle with less symmetry than its counterpart. You can draw a circle with one taut string and one nail to keep it focused at the the center. But to draw an ellipse we have to attach a loose string to a pair of nails and then it pull it taut throughout its 360 degree journey.

The nails’ positions are the two focal points of the ellipse. If you put the shape of an ellipse on an xy plane and place the origin at the center, each focal point is said to be “c” units way from the origin. The horizontal distance from the center to the widest point of the ellipse is often denoted as “a” ; and the equivalent in the vertical direction is called “b”.

Drawing an ellipse is made possible with the above nail-set up because the total length of the string is constant. In fact, if you think about it, the length is equivalent to the entire horizontal ” diameter, equal to 2a. That in turn implies that a, b and c are related by the Pythagorean theorem, although the hypotenuse in this case is a and not the conventional c .

By using the relation a2 = b2 + c2 and applying the distance formula from the point, P = (x,y), to the focus (0, c), with the knowledge that half the string’s length is equal to “a”, we could derive, after some algebraic mud-wrestling,

x2 /a2 + y2/b2 = 1.

If we shift the graph so that one of the foci is at the origin, then we get (x – c)2 /a2 + y2/b2 = 1, which looks like this:

But how do you measure eccentricity(e) from all of this? The simplest way is to express it as a fraction of the focal distance, c, over the distance a, so e= c/a. (Incidentally, the purpose of shifting the focus to the origin is to make it more similar to the polar coordinate-system used in astronomy, given that the sun is at one of the foci. The planet’s closest approach(perihelion) and farthest (aphelion) become the distances to F1. The ratio of their differences to that of their sum ends up being equivalent to the ratio of a/c.) So in the case to the left, the eccentricity for the ellipse would be quite high at 4/5 = 0.8. To reduce it, we would have to move the middle of the ellipse closer to its foci, which would round it out. Of course, if we want to keep the focus’s value fixed we would have to change b’s value accordingly since it is bound by the relationship a2 = b2 + c2 . Here’s what an eccentricity or e value of 0.2 looks like—it’s far more circular.

The adjacent figure is a close representation of Mercury’s orbit (e=0.204). But it’s the second most elliptical orbit after that of the dwarf planet Pluto. Scaled down to the scale of a page, the orbits of Venus (e = 0.007), Neptune(0.009) and even that of Earth(0.017) would look like perfect circles.

What then is behind the discrepancies? It’s the particular details of each planet’s history of formation, very loosely akin to the way the micro-environment of each snowflake affects its shape and the way its coming to existence will in turn affect the conditions of another flake.

An artist’s conception of the solar nebula.

Our planets’ family history began with a dense cloud of interstellar dust particles and gas molecules. A nearby supernova may have compressed the gas-dust cloud into a spinning, swirling disk of material known as a solar nebula. Gravity acted from the nebula’s center of mass, pulling in an increasing number of particles. At one point the most abundant particles, hydrogen atoms, fused at the core leading to the birth of our sun, which used up most of the nebula’s mass.

Farther out in the disk the remaining matter also clumped together. The clumps violently collided with one another, growing in size the way the mass of dead insects got larger on the radiators of old 1970s cars as they sped on highways through forests. The collisions increased the eccentricity of the bigger clumps’ orbits around the newborn sun. But as the surrounding bodies and debris stuck to the growing planets, they worked to have the opposite effect and decreased the eccentricity.

Just as the way that the order and arrangement of the planets and other bodies in our solar system is rooted in its history, so are the eccentricities of the planets. None of them experienced the same collisions or scooped up the same amount of mass. The four terrestrial planets planets, Mercury, Venus, Earth and Mars, the ones with rocky material that could withstand the heat from the young solar system close to the sun, each had different neighborhoods that influenced their orbits and continue to do so. The changes are just smaller than those they experienced in their youth.

Meanwhile, materials such as ice, liquid or gas could only survive the heat at a greater distance from the sun. They too were pulled together by gravity. In these outer regions of the solar system, Jupiter, Saturn, Uranus and Neptune formed. At the beginning, these planets also amassed more matter from the swirling disk. This made them migrate because of the angular momentum that was exchanged. Simulations reveal that Jupiter was forced to move inward, while Saturn, Uranus and Neptune drifted further away from the Sun. Before they moved, their eccentricities were as low as those of Venus and Earth, but when Jupiter and Saturn became locked into a 1:2 orbital resonance, the eccentricities of the giants and Uranus quickly escalated to the ones we observe today. That of Neptune, which was further away, remained low.

Having said all that, we should not go away without reiterating that eccentricities are still in flux. They constantly change, albeit very slowly and not too wildly. Here’s that of Mars as an example, which changes mostly due to the influence of Jupiter. According to the Solex program, it takes a couple of million years for Mars to go through a full cycle from its minimum (an eccentricity as low as Venus’s current one) to a maximum ( 30% more than its present one, but still much lower than Mercury’s.)

From the Solex program https://web.archive.org/web/20080218133456/http://chemistry.unina.it/~alvitagl/solex/index.html

Earth’s variations in eccentricity are more minor and have very few consequences, unlike those of other Milankovitch cycles. Hence we have another way of rationalizing our feelings for the most startling of planets: its eccentricity is, and will forever be, low, sometimes even lower than that Venus, and unlike the latter, our planet is fertile and covered with clouds that help sustain it.

From Nature: https://www.nature.com/scitable/knowledge/library/milankovitch-cycles-paleoclimatic-change-and-hominin-evolution-68244581/

Other Sources:

NASA: https://solarsystem.nasa.gov/solar-system/our-solar-system/in-depth

Origin of the orbital architecture of the giant planets of the Solar System. K. Tsiganis1, R. Gomes and al. Vol 435|26 May 2005|doi:10.1038/nature03539

Formation and Accretion History of Terrestrial Planets From Runaway Growth Through to Late Time: Implications for Orbital Eccentricity. Ryuji Morishima and Max W. Schmidt. The Astrophysical Journal, 685:1247Y1261, 2008 October 1