A Curious Case of Solar System Gravities Clustering

Eugene V. Bobukh
(Click here for a full PDF paper)

This is a chart of surface gravity for 14 largest Solar System bodies. It looks rather… peculiar:

It looks as if gravity values are clustered. Clustered into four lumps separated by equal width (2.5x) gaps. These lumps consist of:

  • Jupiter with it’s almost triple Earth’s gravity (25 m/s2).
  • Venus, Earth, Saturn, Uranus, and Neptune. Though quite different in size and appearance, these planets all have nearly normal Earth’s gravity of ~10 m/s2
  • Mercury and Mars. The latter is twice heavier than the former, but their surface gravities are neatly positioned at the same 3.7 m/s2, or ~40% of normal.
  • Moon, Io, Europa, Ganymede, Callisto, and Titan, all loosely centered around 1.5 m/s2 despite being significantly different by size, mass and composition.
  • Triton and continuum of smaller bodies below 0.75 m/s2 (not shown here).

Why does Mother Nature seem to prefer certain gravity values so strongly that this results in visible grouping? After all, gravity is a continuous parameter of complex origin. The natural expectation is that surface gravities of planetary bodies so diverse should be scattered quite uniformly across the range.

But they are not. This is strange, and it calls for some questions.

 

1. Is it really unusual, or am I just exaggerating my subjective impressions?

The general approach to answering this question is simple. Generate large number (e.g., 10,000) of sets of 14 random values out of log-uniform distribution. Measure clustering quality of each by some objective metric. Count how many times these random sets show clustering stronger than measured for the Solar System’s by the same metric.

If that figure is high — e.g., 30% of all cases — we likely have a subjective illusion at hands. After all, if one third of completely random data sets with no grouping introduced into them clusters better than what I picked as “interesting”, there is probably nothing special about it.

Running this test with two measures (see Appendix B), I arrived at the answer: 0.08%.

In other words, only 0.08% of random data sets demonstrate clustering better than Solar System’s gravities.

So, these groups are real with 99.92% confidence.

 

2. Do any other parameters demonstrate clustering of similar quality?

The short answer is No.

Obviously, I could’ve not tested all physical variables in the whole Universe. But I tested Solar System masses, radii, Solar distances, densities, extrasolar gravities for systems with 5+ planets, and all extrasolar gravities known with at least 25% precision. None of them have demonstrated clustering more unlikely to arise by a random chance than 3% — and in those cases when it did fall below ~10% there was a known physical cause for it.

Thus I conclude that this clustering is not simply real, but kind of unique to gravities only. That supports the legitimacy of the next question:

 

3. Are there physical reasons behind it?

Partially. And now this is getting puzzling, somewhat.

One cluster, consisting of Saturn, Uranus, Neptune, and (probably?) Earth could be ascribed to transition from rocky to gaseous planet composition happening through this range of masses. As we know ([20, 25, 80]), the more massive a body within this range is, the more hydrogen, helium, and other gases it contains and so the larger it is. Because of that “swelling”, gravity remains approximately constant 10 m/s2 for planets with a broad range of masses between (roughly) 1 to 100 Earth’s mass, at least within the Solar System conditions. Our analysis detected that and showed a cluster there as expected.

Yet that is not enough to explain all the pieces. Even if you remove giant planets from the analysis, three prominent clusters still remain. Look at these gaps between Mars and Venus, or Io and Mercury:

Applying the same tests, I found the probability of “accidentally” producing the same or better clustering of (0.5-0.6)%.

 

4. Are there speculative physical reasons behind this, maybe?

Being scientifically honest, I shall say that quite likely this all is just a play of random chance. The world is full of coincidences with 1:200 probability of happening, and this could be one of them. Meaning that Solar System gravities are unusual, but there is no unusual reason for that.

After stating that, lets consider other — less likely, but more intriguing — possibilities.

It could be that the laws of planetary and satellite formation prohibited — at least within the Solar System — formation of bodies with surface gravities occupying any of those gaps. While I’m not aware of such laws, and to the best of my limited knowledge nobody is, they still might exist. Planetary formation process is an area of an active research and many fine details of it are still poorly understood.

Finally, the most thrilling possibility is that this clustering is a sign of… observational gap! Meaning that there are bodies in the Solar System with gravity values filling some of those gaps, but we just have not discovered them yet.

What would those speculative planets look like? The most likely candidates are:

a) A KBO object 6000-7000 km in size, with surface gravity of ~2.5 m/s2.

b) A hypothetical Planet X, intermediate in mass between Earth and Uranus, with surface gravity of ~6 m/s2. The latter figure at least does not contradict gravity ranges derived from mass and radius values given in [70] for putative Planet Nine.

Of course, I am far from seriously suggesting that based on the observation with just 200:1 statistical confidence, but… this is intriguing.

To summarize, I think there is some unsolved mystery here. And who and when will eventually solve it?

Want more detail? Here is a full PDF paper outlining this small research.

 

Appendix A. FAQ

Why gravity?
It may feel like a very artificial variable to study. But let’s remember that surface gravity is directly proportional to the product (ρ*R) of average planet density and its radius. So, when we study gravity, we effectively study that product — and it does have several interesting properties.

First, it is one of the simplest meaningful combinations capturing both planetary density and radius. If there is a non-trivial correlation between the two, it might potentially manifest itself via the peculiarities of the distribution of Log(g).

Next, both pressure at the center of the planetary body and its total gravitational energy are, to the first order of approximation, proportional to (ρ*R)2. Therefore, interesting phase transitions, or non-trivial interplay between the mechanical and gravitational energy of the planet, might surface through this variable.

That’s why it seems that Log(g) is a physically meaningful object of study.

Did you speak to any planetary scientists about this?
I did, with three actually. One of them had a chance to look at the full paper behind this article. They all pointed out — quite correctly — that without a good physical explanation this observation does not really qualify for a publishable scientific work, and I have to agree to that. Having no time to advance this research further, and with very little chance of completing it alone anyways, I resorted to publishing this for (and to the benefit of) general public.

 

Appendix B. Details on clustering quality measurements.

Measure 1: Gaps Area (GA).
This measure, which could be viewed as a variation of Davies–Bouldin index [140], is defined as the average size of inter-cluster areas relative to the total span of values considered:

Here c enumerates clusters in the order of increasing gravity, and K is the total number of them.

To apply it, one needs to cluster the objects using some of the standard methods first. For that, hierarchical agglomerative clustering with minimal (or “single-linkage”) merging [40, 50] criteria was chosen for its simplicity and ease of results interpretation, especially in one-dimensional case.

Here are the illustrative examples of artificial datasets with “good” and “poor” clustering quality.

“Good” clustering (large gray gaps, tight green groups):

“Poor” clustering quality: small gray areas, large and loose green groups:

Measure 2: Blur Tolerance (BT)
This approach builds on the ideas behind Kernel density estimation [50, 150] and assesses how discernable the clusters would be under poor observations/measurement conditions.

Starting with a set of gravity values gi, it converts them to a continuous function F(g) via blur transformation:

Intuitively, that transformation mimics image quality loss caused by imperfect observations. The tighter and more pronounced the clusters are, the more blurring their “image” can sustain while preserving enough contrast to see the clusters.

For illustration, four gravity clusters are detectable in the Solar System after blurring with radius r = 0.1:

If r is increased to 0.2, smoothing becomes stronger and only two clusters remain:

The blur radius r (relative to the total gravities span) that preserves at least 2:1 contrast between all K clusters is defined as Blur Tolerance clustering quality measure BT:

In most cases, these measures reasonably agree. However, Gap Area was found to be less robust in a face of degenerative clusters, containing only single elements, or when cluster size is smaller than the uncertainties of the measurements. It would produce elevated scores in such scenarios, as well as when there are multiple single-element “clusters” in the set. Blur Tolerance approach is more robust against those issues and its results should be favored in most conflicts, especially when the clusters are tight or singular.

 

Appendix C. References.

[10] Katharina Lodders, Bruce Fegley, Jr. The Planetary Scientist’s Companion. New York, Oxford, Oxford University Press, 1998.

[20] Diana Valencia, Dimitar D. Sasselov, Richard J. O’Connell. DETAILED MODELS OF SUPER-EARTHS: HOW WELL CAN WE INFER BULK PROPERTIES? The Astrophysical Journal, 665:1413–1420, 2007 August 20, http://iopscience.iop.org/article/10.1086/519554

[25] Leslie A. Rogers. MOST 1.6 EARTH-RADIUS PLANETS ARE NOT ROCKY. Accepted to ApJ, in press as of 05/12/14, http://arxiv.org/abs/1407.4457

[40] https://en.wikipedia.org/wiki/Hierarchical_clustering#Linkage_criteria

[50] Jiawei Han, Micheline Kamber, Jian Pei. Data Mining Concepts and Techniques, Third Edition. Morgan Kaufman Publishers, 2012.

[70] Jonathan J. Fortney, Mark S. Marley, Gregory Laughlin, Nadine Nettelmann, Caroline V. Morley, Roxana E. Lupus, Channon Visscher, Pavle Jeremic, Wade G. Khadder, Mason Hargrave. THE HUNT FOR PLANET NINE: ATMOSPHERE, SPECTRA, EVOLUTION, AND DETECTABILITY, https://arxiv.org/abs/1604.07424

[80] Fernando J. Ballesteros, B. Luque Walking on exoplanets: Is Star Wars right? https://arxiv.org/abs/1604.07725

[110] The Extrasolar Planets Encyclopaedia, http://exoplanet.eu/

[140] Some well-known measures of clustering quality on Wikipedia:
https://en.wikipedia.org/wiki/Cluster_analysis#Evaluation_and_assessment

[150] Kernel density estimation overview:
https://en.wikipedia.org/wiki/Kernel_density_estimation

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LPSC 2017 Trip Report, 4/4

[To the previous chapter][To the beginning]

CLIMATE OPTIMUM ON MARS INITIATED BY ATMOSPHERIC COLLAPSE

Cold, dry and sterilized are three words that best describe Mars today. Yet we know that there were better times there. Warmer times with denser atmosphere and liquid water that left riverbeds clearly visible even today. How long ago was that? How long did that period last, what have started and stopped it? That’s where the certainty steps away to be replaced with various hypotheses.

One of them is proposed in this paper. At the first glance, it looks awkward. The concept behind it is akin to a fire being accidentally started by a firefighter machine arriving on a false call.

It states that warmer climate was a response to the very first atmosphere collapse on Mars. That collapse triggered release of large quantities of methane clathrates that have accumulated under martian soil earlier. Photolysis and oxidation products of that methane increased greenhouse warming to the point of climate change, enabling liquid water flows on the surface.

But did it all happen exactly like that? I guess nobody knows for sure yet.

===

CAN MARS BE TERRAFORMED?

The short answer is No.

Of course, there are details. In theory, many approaches to that problem are imaginable. But at least the most straightforward of them isn’t likely to work. It suggests that if enough CO2 from the Martian polar cups is artificially evaporated, the greenhouse effect on Mars could be boosted to the point where the planet’s climate would become warm and self-sustaining.

However, after accounting for CO2 supply based on the most recent data, the authors concluded that there is not enough of that gas there for this project to work. And even complete evaporation of the Martian polar cups would warm the planet by ~10 C (~20 F) only. That would not be enough to shift the atmosphere into a self-sustaining warm mode.

===

EFFICIENCY OF ATMOSPHERIC EROSION BY IMPACTS: ENERGY CONSIDERATIONS AND APPLICATIONS

Let’s start with Shuvalov parameters. They are formulas that relate efficiency of planetary atmosphere erosion by impacting asteroids to parameters of those asteroids, mainly mass and velocity:

That efficiency has a maximum. A meteorite (or an asteroid) that is too large or too fast is less efficient in “splashing out” an atmosphere than a smaller or a slower one:


[Efficiency of atmosphere erosion by an asteroid as a function of its Shuvalov parameter ξ. Image Credit: Shuvalov, V., 2009. Meteorit. Planet. Sci. 44, 1095-1105.]

By choosing asteroids of the maximum efficiency, one can calculate how many of them would be required to completely strip off the atmosphere of Venus. The answer is “a lot”. Unrealistically many. We are talking about (0.5-9)% of the mass of the Moon. The upper limit of that estimate exceeds Solar System asteroids’ mass combined. Using that figure, I can crudely estimate that the energy needed to deflect that enormous fleet of asteroids is no less than ~1027 Joules. That is a million times more than the current yearly energy “budget” of the whole human race.

Alas, the plan of terraforming Venus by removing its atmosphere with asteroid impacts remains a sci fi – at best.

===

A GEOPHYSICAL PLANET DEFINITION

From the standpoint of geophysics, Pluto and all sufficiently large bodies like Titan, Ganymede or Europa are genuine planets irrespective of their orbits.

Their descriptions usually mention most features absent in small bodies but common in “real” planets, such as spherical equilibrium shape, differentiation, tectonic activity (current or past), atmosphere, certain resurfacing processes, formation and thermal history. In planetary science textbooks Pluto or Ganymede are more naturally presented in the same chapter with Mars or Venus rather than with small asteroids or comets. Language used when writing about Pluto is closer to that of Mars rather than of comets. Finally, the authors mention over 40 peer-reviewed publications referring to Titan and Europa as planets, “both pre- & post-IAU planet decision”.

And none of that depends on the body’s orbital parameters.

Therefore, the proposal is to stop messing around with “dwarf planets” and, without waiting for IAC approval (which is not needed anyways), just start calling all these bodies planets.

===

Planetary Topography from Laser Altimetry

A great lecture on laser altimeters in service around other worlds. The video opens with honors and awards. If you are eager to see the technical part, jump to the 16th minute.

===

NASA Headquarters Briefing

In this video, NASA reports on past progress, budget and plans. Q&A start at the 51st minute and is worth attention in its own.

LPSC 2017 Trip Report, 3/4

[To the previous chapter][To the beginning]
BISMUTH TELLURIDES AND SULFIDE MIXTURES AND THEIR RELATION TO METAL FROST ON VENUS

In some radar images of Venusian highlands, there is something that looks almost like a snow. Of course, at 500 degrees Centigrade, that must be something else. Something with high reflectivity and high dielectric constant. Probably of an almost metallic nature. What could that be?


[“Snow” in Maxwell Montes on Venus. Image Credit: NASA/Magellan]

The authors hypothesized that this could be tellurides or sulfides-tellurides of Bismuth (Bi2Te3, Bi2Te2S).  They recreated Venusian conditions in the lab to test how these compounds sublimate and interact with CO2-rich Venusian atmosphere, and whether they would, like our snow, make “frost” in Venusian mountains at 300-degree “coldness” out there.

Their conclusions did not look very solid to me, and I’m not sure why only these two compounds were tested with quite many alternatives potentially imaginable. But the notion of “semiconductor snow” has certainly resonated within my mind, so I’ll keep that publication in the list.


[Sulfide-telluride of Bismuth Bi2Te2S, also known as telluric bismuth. Image Credit: Wikipedia]

===

THE BIOPAUSE PROJECT: BALLOON EXPERIMENTS FOR SAMPLING STRATOSPHERIC BIOAEROSOL

Whoever have read Michael Chriton’s The Andromeda Strain would immediately appreciate this work. The idea is the same: go as close to the outer space as possible, scoop life samples from out there and bring them back to study. Why? To see how far the Earth’s live continues into space, and what it is like there.

Of course, many countries and organizations have conducted these studies over the past 70 years. USSR, for example, have brought microorganisms from altitudes of 48-77 kilometers back in 1976. But Japan Aerospace Exploration Agency (JAXA) is notable for coming up with new interesting projects for rather modest money.

Typically, such research relies upon cultivation of specimens for analysis. Obtain, seed, grow, study what has germinated. But, according to the author, “more than 99% of the microbes in nature are thought to be uncultivated species” (++another link). So cultivation-based analysis is bound to miss 99% of the catch’s biological diversity – including perhaps the most bizarre and unusual microbes.

To work around that problem, JAXA decided to not cultivate. Instead, they simply studied all collected samples with a fluorescence microscope and a scanning electron microscope. And even though their balloon brought the microbes from a relatively “modest” altitudes of 13-27 kilometers, they (according to the abstract) “estimated the number density of stratospheric microbes including those that cannot be cultivated for the first time in the world.”

Unfortunately, during the mission return they’ve lost the negative test chamber. So the next flight, tentatively scheduled for June 2017, should help with verifying the results.

===

SURVIVABILITY OF RNA AND PROTEIN MONOMERS AGAINST EFFECTS OF SHOCK PRESSURES

Suppose a meteorite contaminated with terrestrial DNA hits another planet. Explosion, pressure spike, instantaneous heating — would organic matter survive such an ordeal?

Sometimes there the best way to find out is to run an experiment. That’s what the authors did. They shot artificial “meteorites” with proteins and RNA fragments added against solid targets and measured how much of the organic matter survived impacts. It turns out, rather little:

Shock stress – (approximate impact velocity) — % of organics survived

10.5 GPa — (~2.2 km/c) — 4.3%
28 GPa — (~4 km/c) — 0.7%
40 GPa — (~6 km/c) — 0%

Does that mean that panspermia does not work? Not at all. It’s possible to imagine numerous less stressful ways of organics delivery by meteorites. However, a direct impact against a body the size of Mars without significant slowdown by atmosphere is apparently fatal even for relatively simple organic molecules.

===

PULMONARY INFLAMMATORY RESPONSES TO ACUTE METEORITE DUST EXPOSURES – IMPLICATIONS FOR HUMAN SPACE EXPLORATION

Again, this area of research isn’t new. I vaguely remember some papers from 1990s concluding that lunar dust is harmful for lungs and causes strong silicosis in rats.

In this study, inflammatory stress response of human lung tissue to lunar, Martian, Vestian and terrestrial dust was measured. The first three were obtained from corresponding meteorites. The last one was made from terrestrial basalts.

The observations are rather gloomy. All dust caused significant negative effect on lungs. But the worst of four types studied was the dust from Mars. Its effect is comparable to that of terrestrial mine tailings, notable for causing severe health problems in mine workers. Lunar dust is the next, followed by least harmful terrestrial basalts (although they weren’t completely benign, to be clear).


[Inflammatory Stress Response to various types of dust. Image Credit: A.D. Harrington, F.M. McCubbin, J. Kaur, A.Smirnov, K. Galdanes, M.A.A. Schoonen, L.C. Chen, S.E. Tsirka, T. Gordon / NASA Johnson Space Center; Dept. of Environmental Medicine, New York University School of Medicine; Dept. of Geosciences, Stony Brook University; Geology Dept., Lone Star College; Environmental Sciences Dept., Brookhaven National Laboratory; Pharmacological Sciences, Stony Brook University]

I can see several implications here. First, if humans would ever walk on Mars, they would have to invest considerably into protecting against dust there. Second, my pile of paper sheets with imprecisions noted in “The Martian” just grew one item larger :) Third, we often underestimate seriousness of dust effects on other planets. For example, Lunar dust is so strongly abrasive that it destroys moving parts and surfaces exposed to it an order of magnitude faster than what’s expected on Earth. If interested, take a look at the last passage on the 5th page of this document.

To part 4/4.

LPSC 2017 Trip Report, 2/4

[Back to the first part]

HOW DIELECTRIC BREAKDOWN MAY WEATHER THE LUNAR REGOLITH

Not only meteorite impacts reshuffle lunar surface. Solar wind, especially during long lunar nights, probably makes a comparable contribution to that process. How? Via a flow of energetic electrons. Which can reach the night side of the Moon and hit and charge regolith particles up to electric field strengths of ~106 V/m. Which would cause electric breakdowns within particles, resulting in their destruction or partial re-melting. Which amounts to slow mixing and resurfacing of the lunar soil. The darker and the colder it is, the better this process presumably works (in “warmer” locations, electric charges dissipate via normal conductivity).

We don’t know for sure whether this actually happens as described. But it would be worth figuring that out. Who knows – maybe in countless millions of years the astronaut’s footprints would disappear not under the dust of nearby meteorite impacts, but via trillions of microscopic discharges?


[This is an electric discharge, too. Just a larger one. Image Copyright: Eugene V. Bobukh]

===

SPACECRAFT IMPACTS ON THE MOON: CHANG’E 1, APOLLO LM ASCENT STAGES
Electronic poster

Over the past 60 years, Solar System exploration has gotten its own history… and archaeology. Yes, people search for landers of 1970s in contemporary satellite imagery from other planets, and sometimes find them, as that wonderfully happened with the mysteriously silenced Mars-3 Soviet probe.

The author of this presentation specializes on lunar impacts and has built a whole collection of them. This year, four more were added to it: ascent stages of Apollo-12 and -14, Chinese Chang’E 1 and (very recently) European SMART-1.


[Impact traces from ascent stages of Apollo-12. Image Credit: NASA / Philip J. Stooke]

I found it surprising that many impacts leave behind not craters, but streaks. Apparently, grazing collisions at velocities of ~1 km/s just result in violent rotation and breakup of the impactor, scattering the debris over long distances along the orbital path.

===

SIZE AND SOLAR INCIDENCE DISTRIBUTION OF SHADOWS ON THE MOON

It may be hard to accept that after studying the Moon for hundreds years, walking on it, making 12-inch resolution photos of it, and dumping 180 tons of trash on its surface we can still have something unclear about it. Especially about such a seemingly trivial thing as shadows.

Yet it appears to be the case.

The authors conducted a research that may look like a simple and boring activity. They studied the distribution of lunar shadows by size. This is pure statistics – but with a twist. Shadows sizes and shapes could be rather complex and depend on lunarscape characteristics in non-obvious ways. That is particularly pronounced at lunar “mornings” or “evenings”, when the sunlight is parallel to the surface and minute roughness variations cause great impact on the projected shadows.

They found that the number of shadows with sizes between 3 and 100 meters (10 to 300 feet) does not depend on their size. More importantly, they saw no clear good explanation for that. So perhaps there is something in the distribution of lunar bumps and ditches that we don’t understand well? This conclusion caused some serious discussion at the conference, with people was trying to interpret it via crater size statistics, but I did not quite capture all the details so can’t convey them here.

So yes, this is a seemingly trivial statistic of black and white patches. But done right, it reveals something enigmatic about our well-studies Moon – something hidden within its shadows…


[Shadows area distribution from the discussed work. Image Credit: Oded Aharonson, Paul O. Hayne, Norbert Schorghofer]

===

PENITENTES AT TARTARUS DORSA, PLUTO

Penitentes are bizarre needle-like snow formations known so far only on Earth:


[Image Credit: Wikipedia]


[Image Credit: Wikipedia]

But, according to the presentation, they may also exist on… Pluto! While we haven’t seen them close enough to be 100% confident, these longs stripes “are well described by the theoretical penitente models of [4] with spacing, orientation and growth rates matching well with observations for the methane ices observed by New Horizons (NH) [5] on Pluto”. And yes, they most likely are made of methane (CH4) ice.

[Tartarus Dorsa are on Pluto. Image Credit: NASA/JHUAPL/SWRI]

To part 3/4

Lunar and Planetary Science Conference 2017 Trip Report, 1/3

Couple months ago I attended Lunar and Planetary Science Conference (LPSC) 2017 in Houston, Texas. Of roughly 3000 talks and posters presented there, I’m sharing summaries on twenty that I personally found most inspiring or thought-provoking.

Disclaimer. I am a physicist by education. I generally understand the language in use at LPSC. But I am not a planetary scientist. My selection of articles is almost certainly subjective and biased. My interpretations may be incorrect. If anything serious (such as your job, finances, or professional reputation) depends upon the accuracy of my narrative, I’d suggest verifying it with the authors. Their contact info is available within the referenced abstracts.

===

TRANSIENT BROAD SPECULAR REFLECTIONS FROM TITAN’S NORTH POLE

Titan, Saturn’s moon, features extensive set of lakes and seas (called lacus and mare) filled with liquid hydrocarbons. Sunlight reflecting off them has been photographed numerous times over the past ten years:


[Image Credit: NASA/JPL/University of Arizona/DLR]

However, sometimes similar bright spots are seen at locations with no lakes. Moreover, the effect is transient. On one Cassini flyby it glitters, but not on the next one with very similar geometry. What’s going on?

As an explanation, a “wet asphalt” hypothesis is proposed.

Suppose a methane rain pours over a rough surface similar to asphalt in texture. “Asphalt” gets wet and starts reflecting the sunlight, just like it happens on Earth after rain:


[Image Copyright: Eugene V. Bobukh]

By the next Cassini flyby, it all dries out — and we see no bright spots anymore.

===

METHANE, ETHANE AND NITROGEN LIQUID STABILITY ON TITAN

This talk is about Titan’s hydrosphere, too. Except that the “hydro” part is not really appropriate, since there is no water in Titan’s lakes and seas. Instead, they are filled with liquid methane (CH4), ethane (C2H6), and nitrogen (N2). But we poorly know their proportions and don’t know well how they interact under Titan’s conditions.

After cooling those three gases to cryogenic (-300F…-350F) temperatures, the authors studied how they mix and freeze under various temperatures and pressures. What they found makes Titan’s hydrosphere a far more complex system than Earth’s oceans. Well, that is somewhat expected if three, not just one type of “water” and “ice” play together.

First, methane-ethane mixture freezes under lower temperatures than either of those gases. Thus, they act as mutual “antifreeze”.

When they solidify, methane ice floats to the surface, while ethane ice sinks to the bottom. The initial concentration defines which ice forms first. This is more complex than what we see on Earth, where ice always stays at the top:


[An photo of the experimental camera from the abstract. On the left, ethane ice is on the bottom of the liquid mixture. On the right, methane ice is at the surface. Image Credit: J. Hanley, L. Pearce, G.Thompson, W. Grundy, H. Roe, G. Lindberg, S. Dustrud, D. Trilling, S. Tegler / Lowell Observatory, Flagstaff, AZ; Northern Arizona University, Flagstaff, AZ; University of Texas, Austin, TX.]

Addition of nitrogen (simply from the atmosphere) increases solidification temperature of the mixture and thus can cause sudden freezing of the liquid. And another LPSC publication suggests that nitrogen dissolves very well in liquid methane – but rather poorly in liquid ethane. As a result, evaporation of methane from the mixture could trigger bubbling nitrogen release from a lake. As if it was filled with sparkling wine, but at -300F and with nitrogen bubbles instead of carbon dioxide.

And here is the most interesting observation. At pressures exceeding 2.5 bars (corresponding to depths over ~300 feet on Titan) methane-ethane-nitrogen mixture separates into two liquid phases. There was an amazing video in the presentation on that, but here we’ll have to resort to a single picture:

[Image Credit: J. Hanley, L. Pearce, G.Thompson, W. Grundy, H. Roe, G. Lindberg, S. Dustrud, D. Trilling, S. Tegler / Lowell Observatory, Flagstaff, AZ; Northern Arizona University, Flagstaff, AZ; University of Texas, Austin, TX.]

Gas is at the top of the experimental cell. The next layer is ethane-rich liquid. Underneath it is the second liquid phase rich with nitrogen. And at the “ceiling” of the first liquid phase you can see a tiny hint of a droplet. That is the 2nd phase condensed. Insoluble in the first phase, it flows and drops through it just like water through oil.

When I saw that, I immediately thought of layered seas possibility on Titan. Did you think of the same, too?

===

CLASSIFICATION OF LABYRINTH TERRAINS ON TITAN

Titan’s labyrinth terrains are highlands carved with complex system of ridges and valleys, visually resembling tree bark:


[Image Credit: NASA/JPL]

Their nature and origin and aren’t completely clear. At the first glance, they look like riverbeds – and we know there are rivers on Titan. But if you examine them carefully, you might notice that some of those labyrinths are… closed. They don’t “flow” anywhere!

Does it mean at least some labyrinths are not riverbeds? The authors conducted morphological analysis of labyrinth terrains and concluded that at least some of them could be… karsts!

Even if that is the case, we still don’t know whether they are produced via material evaporation, wash-away, or dissolution. But maybe some of them are similar in appearance to this terrestrial karst carved by an ancient waterfall?


[Image Copyright: Eugene V. Bobukh]

===

TOPOGRAPHIC ASSESSMENT OF HOLLOWS ON MERCURY: DISTINGUISHING AMONG FORMATION HYPOTHESES

Now let’s transition to Mercury, while staying on the same subject of karsts. Or hollows, actually.

Hollows are curious depression spots on the surface of Mercury. Their origin isn’t fully clear yet. One of the explanations suggests that they are… karsts, too. But karsts formed through evaporation of some volatile material from beneath the dusty surface.


[Mercurian hollows. Yes, you should try to see depressions, not protrusions in these spots. And the “bumps” are craters. Sometimes rotating an image can help, so I added a rotated version. Another trick to assist with “flipping” the picture inside out is stepping back and temporarily defocusing one’s eyes. Image Credit: NASA]

We don’t know what that material is, although most likely it is not water. And I don’t know whether the hypothesis is correct. I just enjoyed some pictures of prominent hollows presented at the talk. I recorded their NASA numbers, I found one of them in the Messenger catalog and posted here after some contrast enhancement. To my defense I shall mention that author’s photos were processed, too — and some looked much better than my version:


[Image Credit: NASA/JPL]

===

OXYGEN DEPLETION ON THE SURFACE OF MERCURY: EVIDENCE OF SILICON SMELTING?

After spending four years in orbit around Mercury MESSENGER, the space probe, has measured the abundances of primary rock-building chemical elements (O, Si, Ti, Al, Cr, Fe, Mn, Mg, Ca, Na, K, S, Cl) in Mercury’s surface.

Yet MESSENGER’s instruments see only abundances of those atoms – not the chemical bonds between them or mineralogical composition. To obtain the latter, usually it is assumed that all metallic elements are fully oxidized. For instance, that Silicon is present as SiO2, Titanium as TiO2, Aluminium as Al2O3, and so on. Then, the minerals are “constructed” out of these oxides (e.g., Andalusite Al2SiO5 is represented as Al2O3 + SiO2).

That usually works well – but not for Mercury. There seems to be not enough Oxygen to bind all metallic elements while maintaining the measured Oxygen to Silicon ratio O/Si = 1.4 ± 0.03 (and even less, as the abstract suggests).

As an attempt to explain that discrepancy a hypothesis is proposed. It states that deep within Mercurian interiors smelting of metallic Silicon is possibly happening. The ingredients for it are Silicon dioxide (quartz, SiO2) and graphite which, as we strongly suspect, is present in significant quantities within Mercury’s crust.

If that is true then 12.6–17.9% (by weight) of Mercurian northern hemisphere could be made by pure Silicon or Silicon-iron alloy.

Why is it interesting? First, you don’t often meet metallic Silicon in nature. More importantly, on all other rocky bodies (such as Moon, Earth, or Mars) oxygen is plentiful to have oxidized everything that could be oxidized. Why is it different on a well-fried-through Mercury? That remains an open question.



[Mercury and Silicon. Left Image Credit: NASA/JPL; Right Image Credit: Wikipedia]

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LUNAR CRUSTAL MAGNETIZATION INFERRED FROM CHARACTERISTICS OF LUNAR SWIRLS

I thought that I knew quite a lot about the Moon. Yet I had no idea of so-called lunar swirls.

They were discovered in 1960s, but their nature remains elusive. They are strange white markings on the surface of the Moon. Often bended, often curved, they frequently display somewhat periodic patterns, resembling those inflicted by a magnet to a CRT monitor. That lends credibility to interpreting lunar swirls as areas protected from the solar wind by natural magnets buried underneath the surface. Whether those magnets are solidified ancient lava tubes, or remains of iron meteorites, we don’t know. In fact, we still don’t know whether this explanation is correct — although swirls are associated with magnetic anomalies detected from the orbit.

The rest of the work was concerned with estimating the parameters of those putative magnets. And I enjoyed a very thought of a natural invisible shield protecting lunar surface from solar radiation…



[Lunar swirls. First Image Credit: Wikipedia; Second Image Credit: NASA]

Next part (2/4 — yes, I broke it into smaller chunks)

Make an identity?

There is some pleasure in finding a universal solution to a class of problems that people pass around. Here is one of them.

You have probably seen mathematical puzzles of this type before. Someone gives you a few numbers and asks to place arithmetic and “other mathematical signs” between or around them so that the resulting expression is evaluated as some target number.

For example, if the numbers are 2, 3, 4, and the the target is 10, one of the solutions might be:

10 = –(2-3*4)

Or this:

10 = 2*3 + 4

Here I would like to present a general solution that works for all problems of this class, provided that mathematical signs include elementary analytic functions and all numbers are integers.

Let’s start with the core relation between hyperbolic functions (https://en.wikipedia.org/wiki/Hyperbolic_function):

Cosh2(x) – Sinh2(x) = 1

From that, the relation between the inverse hyperbolic functions immediately follows:

Cosh(ArcSinh(x)) = √(1+x2)

Using x = √k, one arrives then to formula that converts k into k+1:

Cosh(Arcsinh(√k)) = √(1+k)    (1)

By applying that formula repeatedly as many times as needed, we can increment integers. For example:

Cosh(Arcsinh(Cosh(Arcsinh(Cosh(Arcsinh(Cosh(Arcsinh(Cosh(Arcsinh(2)))))))))) = 3

(you can check that with a calculator)

The rest is easy. We just apply that formula to one of the input numbers, and play it as many times as needed to produce the target.

What do you do with other numbers? You decrement them away to zeros with a reverse of formula (1):

Sinh(Arccosh((√k)) = √(k-1)   (2)

So, for instance, if you wanted to make 10 out of numbers 8, 3, 5, 2, you would’ve written:

10 = Cosh(Arcsinh(…(Cosh(Arcsinh(8)))…)) + Sinh(Arcсosh(…(Sinh(Arcсosh(3)))…)) + Sinh(Arccosh(…(Sinh(Arccosh(5)))…)) + Sinh(Arccosh(…(Sinh(Arccosh(2)))…))

— where the first pair of calls is repeated 36 times, the second 9 times, the third 25 times, and the last one 4 times.

Obviously, further optimizations are possible, but the solution is complete at this point.

Have a nice day,
Eugene

13 heaviest gases

Nothing predicted a destination of a seemingly benign Internet search. I just Googled “the heaviest gas”. I chuckled at the output:

searchresultsf2

For sure, the first result (Radon) was wrong by a great margin. The second (WF6) was much closer to truth, but I decided to spend 10 minutes to check if Mother Nature knows anything heavier than that.

Ten months later I said to myself: “you’ve got to stop”.

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