Sunday, 28 July 2019

Saturn’s Peculiar Flying Saucer Shaped Moons

Saturn’s Peculiar Flying Saucer Shaped Moons

Saturn has some pretty weird looking moons, something I discovered a while back while reading a paper about crater counts and how these related to the age of these moons (link below).  Here are a few key paragraphs from the paper, describing these unusual moons, and the processes that may have produced their current appearance: 

“Saturn’s small icy satellites present a broad variety of shapes and have radii between 300 m and 135 km. They have high porosities and very low densities, about half the density of water  ice… Except for Hyperion, they are strongly related to the ring system, being some of them embedded in them, others producing a gap or just orbiting in the border of a ring. Others are related even to the mid-sized satellites as trojans of Tethys or Dione.”
Observations for Pan and Atlas show that these objects have a distinct equatorial ridge, which may be present on Daphnis as well (Charnoz et al. 2007; Porco et al. 2007). These features can be appreciated in the striking images obtained by the Cassini mission (Fig. 1). The origin of these formations is unclear. On one hand, the rotational periods of these satellites are much too long (approximately 14 hours) for centrifugal force to compensate the gravitational force. On the other hand, tidal force generated by Saturn would deform bodies in the radial direction generating an ellipsoidal object and not a "flying saucer" object (Charnoz et al. 2007). According to Porco et al. (2007) one possible explanation is that all three satellites likely grew from cores that were one third to one half their present sizes by accumulation of A ring material during an initial formation stage that took place inside a more vertically extended ring. Once they had filled their Roche lobe, a secondary stage of accretion formed their equatorial ridges in a disk that was already 20 meter thick (as the current rings), which would explain the accumulation of particles along the equator. More recently, Leleu et al. (2018) found an alternative explanation considering head-on merging collisions between similar-sized bodies (also known as the pyramidal regime) as they migrated away from the rings.

Basically, the paper took advantage of Cassini’s high resolution photographs of Saturn’s rings, to do crater counts and relate those to theory regarding collision rates and erosion rates.  The crater count evidence seems to indicate that some of these moons (Pan, Daphnis, and Atlas among them) suffered from large scale collisions within the past hundred million years or so.  The cratering also indicates that the surfaces undergo ongoing resurfacing processes (i.e. erosion, sculpting, matter accumulation, etc.).  Besides interacting with the ring system, these moons probably were hit by objects originating in the trans-Neptunian region (also known as Centaurs).

You can download the paper from the arXiv site for further details.  For my own part, I will focus on these moons as interesting objects in their own right, as I like moons.  And judging from blog counts, lots of other people do, too.

Here are some pictures and descriptions of a few of these moons, focussing on those known as “ring shepherds”.  Most of these photos were taken by NASA’s Cassini spacecraft, though some of the discoveries were made much earlier by the Voyager spacecraft.

Saturn’s Moon Pan

This one has been variously described as a flying saucer, a walnut, a ravioli or a poached egg.  I think it looks a bit like someone wearing a sunhat at a jaunty angle in this photo – but lots of imagery could be applied to Pan.  By the way, Pan was a Greek messenger god, associated with shepherds.  Pan, the moon, is considered a “shepherd moon”, that herds particles in the ring system (e.g. clears a gap).

Pan is the innermost moon of Saturn that has been given a name.  It orbits at about 134,000 km, with a very small eccentricity (a very circular orbit).  It’s about 35 km across and 23 km wide.  So, obviously not big enough to become spherical due to its own gravity.

It was predicted to exist from some disturbances noted in the Encke Gap, which is an opens space in the ring system of about 325 km wide.  The predicted size and position of Pan turned out to be very accurate.

The funny shape is due to Pan sweeping up material from the ring system, in which it is embedded.  That material accreted preferentially in a plane, as a sort of ring itself, around Pan.  By now it is part of Pan, but it is hard to say how solid and tightly bound it is to the surface.

Saturn’s Moon Atlas

Atlas is another small moon, at about 40 km by 35 km by 19 km.  It is also somewhat poached egg shaped, like Pan.  At about 138,000 km, it is somewhat farther out than Pan.  The orbit may be somewhat chaotic, though, as it is influenced by some of the other inner moons. It is also considered to be a ring shepherd moon.

The moon was named Atlas, from Greek mythology, as it “held up the rings, as the titan Atlas held up the Earth”.   As the titan Atlas was supposed to be standing on a giant Turtle, this moons somewhat turtle-like appearance may be appropriate.

Atlas’s shape is due to picking up matter from the ring system of Saturn, same as Pan.  It has been calculated that Atlas probably has picked up about all it can this way, as it has about filled its Roche Lobe, meaning that any new matter would tend to be flung off by the moon’s centrifugal force overcoming the rather small gravity that it possesses. 

Atlas’s “pancake” component, if we may call it that, appears to be very smooth, but that may just be an artifact of the photography (i.e. somewhat out of focus).

Saturn’s Moon Daphnis

Daphnis is a very small moon, only 8 km by 8 km by 6 km.  It too resides in one of the gaps of the ring system, at about 136,000 km.  Like Pan, it is a shepherd moon – in the photo above, you can see it peeking out, from the gap that it produces in the ring system.  It’s name also relates to a shepherd in Greek mythology.

A closeup of Daphnis reveals that it is not all that weirdly shaped, compared to Pan and Atlas, though it does seem to have a bit of the flange like structure (equatorial ridge seems to be the accepted term), similar to the other two moons, though much reduced, probably because of its small gravitational field.  It’s hard to say what it looks like – in my mind its shape is a bit like a Star Trek shuttlecraft, or perhaps an elongated walnut.


Saturn’s Moon Prometheus

As the photos above show, the appearance of Prometheus depends a lot on the perspective from which the picture was taken, as Prometheus has an irregular shape of about 135 km by 80 km by 60 km.  So, it can have a sort of flying saucer look, as in the photo on the left, or an elongated potato shape, as on the right.  It orbits at about 139,000 km, and is also a ring shepherd.

It interacts gravitationally with some other nearby moons and the ring system, so its orbit seems to be a bit chaotic.  Here is a still photo from a time lapse video taken by Cassini, showing Prometheus going through the F Ring, and dragging matter with it as it does so.

Prometheus is named after the Titan that stole fire from the gods in Greek mythology.  As a punishment, he suffered from severe liver troubles.

Though it looks pretty elongated in the photo, it is still nowhere nearly as stretched out as the interstellar body Oumuamua (shown below), which recently visited the solar system.  Oumuamua’s extreme elongation is very unusual – there is speculation that it may have been “sculpted” by collisions with interstellar dust during its long journey between the stars, which is reminiscent of the way Saturn’s moons have had their shapes altered by their long travel among the dust of Saturn’s ring system.


1 - Cratering and age of Saturn’s small satellites.  N.L. Rossignoli, R. P. Di Sisto, M. Zanardi and A. Dugaro1, arXiv:1904.13011 [astro-ph.EP]
2 Wiki, Saturn’s moons
3- Oumuamua’s Elongated Shape:

Now that you have read some real science (astronomy and astrophysics), you should read some science fiction.  Either of the Kati of Terra series or the Witch’s Stones series would be excellent choices.  Alternatively, you could try the short story “The Magnetic Anomaly”, which has lots of physics, and plenty about magnetic fields, perhaps affecting the brain’s hippocampus.  J

Kati of Terra

How about trying Kati of Terra, the 3-novel story of a feisty young Earth woman, making her way in that big, bad, beautiful universe out there. 


The Witches’ Stones

Or, you might prefer, the trilogy of the Witches’ Stones (they’re psychic aliens, not actual witches), which follows the interactions of a future Earth confederation, an opposing galactic power, and the Witches of Kordea.  It features Sarah Mackenzie, another feisty young Earth woman (they’re the most interesting type – the novelist who wrote the books is pretty feisty, too).

The Magnetic Anomaly: A Science Fiction Story

“A geophysical crew went into the Canadian north. There were some regrettable accidents among a few ex-military who had become geophysical contractors after their service in the forces. A young man and young woman went temporarily mad from the stress of seeing that. They imagined things, terrible things. But both are known to have vivid imaginations; we have childhood records to verify that. It was all very sad. That’s the official story.”

Monday, 15 July 2019

Breaking the Bank in Roulette, with Statistics (or not)

Breaking the Bank in Roulette, with Statistics

Gambling is an interesting intersection of statistics, probability, real life and greed.  That generated the following question on Quora:

As the spindle on a roulette wheel becomes worn over time, does the probability of certain numbers showing up as winners increase?


I can recall a story about a gambler in the 19th century that noticed that the roulette wheels at Monte Carlo were becoming biased by overuse, and supposedly broke the bank by betting on numbers that had become favored in this way.  Eventually the casino figured out that there was a problem, and responded by moving the wheels around the casino overnight, so nobody could be sure about which wheel was which, from day to day.  There is some speculation that the casino colluded in this gambling coup, as the stunt was good publicity.  This was supposed to be the basis of the song “The Man Who Broke the Bank of Monte Carlo”.

There is a related 20th century version of this tale, whereby some physicists, mathematicians and computer scientists were supposed to have secretly tracked roulette wheels in Vegas in the 1980’s, and used equations of motion from classical physics, in an attempt to “beat the house”.  This is described in the book “The Pleasures of Probability” (Richard Isaac, page 72-73). They claimed to have computers in their shoes (shades of Maxwell Smart) that performed the calculations.  Supposedly, their method worked (at least in theory), but was plagued by technical problems, so they didn’t upset Vegas too badly.

Both of these stories seem rather far-fetched, as it would take a lot of observations to be sure that a number on a roulette wheel was truly coming up much more frequently than chance would predict, especially given the shift in probability needed to break even, let alone make money.

I did my own Monte Carlo simulation (the technique is named after the gambling locale) to simulate games of roulette, just looking at single number bets for simplicity.  A couple of relevant points about roulette are:

  • There are 38 numbers on the wheel (1-36, 0, and 00).  So, the probability of the wheel landing on any given number, over the long run, is 1 in 38, which is about 2.63%.
  • The return on a correct bet is 36 to 1, which would require a probability of about 2.78% for the bet to be an even money proposition.  In other words, the “bad wheel” would have to move the probability of some particular number winning from 2.63% to 2.78%.

Having a number go from 2.63% hits to 2.78% might not seem all that noticeable to the house (and thus might be ignored), but the player would have to be sure that the wheel truly had become biased by overuse, before it would be a reliable bet.

So, for this technique to be profitable, the gambler would have to watch the wheel, note the numbers that come up, and compute those probabilities.  Once it was clear that a particular number really was coming up more than 2.78% of the time, the gambler could hit that number, confident that the game had turned in the gambler’s favor.

So, how long would a gambler have to watch and record a wheel to be sure that it was a wheel that had “gone bad”, and not a fair wheel showing random variation?  Here are results from some simulations that I did in Excel using the random number function (note that different runs of the simulations could produce different results, but the general trend would remain):

  • After 100 spins, the gambler would barely know anything reliable about the wheel.  A perfectly fair wheel could have any particular number getting anywhere from 0% to 7% hits, in 100 spins, just by random chance.
  • After 1000 spins, he/she would still be in the dark.  A perfectly fair wheel could have a particular number getting anywhere from 1.5% to 3.6% hits, in 1000 spins. 
  • After 10,000 spins, the situation would be little better.  A perfectly fair wheel could have a particular number getting anywhere from 2.2% to 3.0% hits, in 1000 spins.
  • At 100,000 spins, the gambler could finally be reasonably sure that the wheel had turned, and the game could be trusted to be break-even.  With that many simulations, any particular number was hit from 2.53% to 2.78% of the time, in my simulation.


  • With one million spins, the fair wheel will show all 38 numbers coming up almost the same proportion of times, from 2.61% to 2.67%.


So, it looks to me like a gambler would have a difficult time being sure that an apparent “hot number” wasn’t just random variation in a fair wheel.  A bias in an actual bad wheel might become obvious to the gambler much quicker than this, but it would probably be just as noticeable to the house, so it would quickly be shut down.  In fact, I would assume that in this day and age, roulette wheels probably have some kind of counter device installed, to ensure that they don’t develop a noticeable bias.

So, to answer the original question, the probability of certain numbers coming up as winners could increase, via a worn spindle (or other mechanical problem), but it probably wouldn’t be useful for a given gambler to help him/her beat the house.  And, as long as gamblers were ignorant of the bias, it probably wouldn’t affect the casino’s profits, as the gamblers would win a bit more when they happed to bet on the hot number, but lose a bit more when they happened to bet on the others.


The Man Who Broke the Bank of Monte Carlo

Physics, Roulette Gambling Coup Attempt:
The Pleasures of Probability, Richard Isaac (Page 72-73, and lots more about roulette and other gambling games)

And here are some other gambling related posts (horseracing).  It’s about my attempt, in younger years, to beat the horses via computer-based statistical analysis:

Now that you have read about gambling systems, here’s a short story about a system that worked, but far too well:

A Dark Horse

Just what might a gambler give up, to go on the winning streak of his life? Even he can't know for sure. Christopher Marlowe's Doctor Faustus legend is given a Damon Runyon spin, in this short story.  For those who aren’t familiar with it, the Faustus legend is about someone who sells his soul to you know who, for fame and fortune.  Things are not nearly so simple for the character in the story, though.

This is a short story of about 6500 words, or about 35 to 45 minutes reading time, for typical readers.

Or how about a nice travel story.

A Drive Across Newfoundland

Newfoundland, Canada’s most easterly province, is a region that is both fascinating in its unique culture and amazing in its vistas of stark beauty. The weather is often wild, with coastal regions known for steep cliffs and crashing waves (though tranquil beaches exist too). The inland areas are primarily Precambrian shield, dominated by forests, rivers, rock formations, and abundant wildlife. The province also features some of the Earth’s most remarkable geology, notably The Tablelands, where the mantle rocks of the Earth’s interior have been exposed at the surface, permitting one to explore an almost alien landscape, an opportunity available on only a few scattered regions of the planet.

The city of St. John’s is one of Canada’s most unique urban areas, with a population that maintains many old traditions and cultural aspects of the British Isles. That’s true of the rest of the province, as well, where the people are friendly and inclined to chat amiably with visitors. Plus, they talk with amusing accents and party hard, so what’s not to like?

This account focusses on a two-week road trip in October 2007, from St. John’s in the southeast, to L’Anse aux Meadows in the far northwest, the only known Viking settlement in North America. It also features a day hike visit to The Tablelands, a remarkable and majestic geological feature. Even those who don’t normally consider themselves very interested in geology will find themselves awe-struck by these other-worldly landscapes.