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Hmm, good question. That's really high up, far higher than aircraft travel. I would guess that there's an interaction with solar radiation of some kind. Is it where the magnetic field deflects the "solar wind" perhaps? With the atmosphere that thin and pressure so low, I do think most heat energy absorbed near the Earth's surface would be dissipated, continuing the pattern seen on mountaintops.
Thinking about this abstractly, I realized I didn't know why the Earth's core is hot. If it was formed by a lot of space dust accreting, it would seem more likely to be very cold. And if the sun is the source of the heat, it should heat the surface more than the core. So I have to guess that the same mechanism is at work: the intense pressure due to gravity is enough to heat the core beyond iron's melting point.
- January 17, 2019
The real power here is held by government employees, especially those in critical jobs. Let’s say that more TSA screeners decided to walk off the job. It’s already the case that the TSA absentee rate has gone up to 7.6 percent, from 3.2 percent a year ago. It is possible to imagine screeners staying home in much greater numbers, thus crippling the entire nation. That could either force President Donald Trump’s hand or lead to a congressional override of a potential presidential veto.
(You might think the nation can do without TSA workers, but I doubt that is true. Even if they don’t make air travel safer, it is hard to imagine airports and airlines going about their business, in a litigious society, without TSA assistance.)
- Potassium–argon dating, abbreviated K–Ar dating, is a radiometric dating method used in geochronology and archaeology. It is based on measurement of the product of the radioactive decay of an isotope of potassium (K) into argon (Ar). Potassium is a common element found in many materials, such as micas, clay minerals, tephra, and evaporites. In these materials, the decay product 40Ar is able to escape the liquid (molten) rock, but starts to accumulate when the rock solidifies (recrystallizes).
I left out the cold mountain tops. This seems like an easy one. Hot air rises because it is less dense than the surrounding air, and the density is related to the ambient pressure. As the hot air rises, the pressure decreases, so the temperature of the rising air will decrease according to Boyle's or Charles' law.
That explains why a column of hot air would cool off as it rises, but not why the atmosphere is typically colder at elevation. I suppose what heat there is in the atmosphere comes from the sun warming the Earth's surface, heating the air by conduction, so temperatures drop as you get farther from the surface. If that's true, then we would expect high-elevation places like Denver to have similar temperatures to low-altitude places at a similar latitude, but higher places appear to have lower temperatures even when they are not mountain tops.
Maybe this one is not so easy. Ideas?
It's big numbers all the way down. I would like to understand this stuff better than I do. But I enjoyed the author's comments:
- I’m sometimes intimidated by young mathematicians who know (∞,1)-categories and the like better than I do… until I say something about other branches of math or physics and discover they are completely clueless about many basic facts. Then I’m reassured that my life hasn’t actually been wasted.
For example, I remember explaining to some mathematicians why the Moon rises a bit later each day. I can easily imagine not remembering whether it’s earlier or later. But this is something one can work out from first principles if one knows the Moon orbits the same way the Earth turns. Given that fact, a good mathematician should be able to figure out pretty quickly about much later the Moon rises each day. If they can’t do that, no amount of (∞,1)-categorical expertise will impress me.
I also remember stumping people with the question “if a solar eclipse happens when the Moon comes between the Sun and the Earth, why isn’t there one every month?”
A more significant challenge: “Since all rocks come from material that formed the Earth about 3 or 4 billion years ago, how can we use radioactive dating to measure the age of rocks and get different answers for different rocks?”
And: “If hot air rises, why is it colder on mountain tops?”
Confusion over the movement of the Moon caused me to miss an occultation of the Pleiades. Experience tells me that the Moon rises about an hour later each day. But my first intuition on seeing this question is that the Moon's orbit should put it ahead, not late, when my position on the Earth's surface repeats after 24 hours. Of course, that's right, and that's why I need some more time to catch up with the Moon: the observer is late. How much more time? Well, after about 28 days the catching-up periods should add up to a full revolution, so I suppose it's about 1/28 of a day, 0.86 hours, or 51 minutes. I'll make some observations if my memory and the weather hold up (both poor prospects).
For solar eclipses, I'm not sure it is so far off to say there is about one (at least partial) eclipse every month, somewhere on Earth. The tilt of the Earth's axis should not be a factor in eclipse frequency, but if the Moon's orbit is tilted with respect to the Earth's orbit around the Sun, that would make alignments less frequent.
For the rocks I have no idea. I think with once-living material, the different carbon isotopes are kept in balance by metabolism, and when the creature dies there is a predictable shift as the less-stable isotope decays. Rocks should be pretty uniform to start with. Maybe it's a clue in the question that the rock-forming "material" is uniform, but different rocks were formed at different times. If they are sedimentary, like limestone, they might contain material from living creatures. I can't remember the difference between igneous and metamorphic, but one of them is something about lava cooling down. I'll guess that something about hot lava prevents some kind of radioactive decay from occurring, but I can't imagine how.
And in 2019, I just entered the phrase "is 4294967297 prime" into a search engine. That level of effort might explain why I only got two out of three of the examples above correct (no better than chance!) before looking at the solutions.
But #1 stumped Fermat too! I didn't know that this problem sparked Euler's interest in number theory, according to "How Euler Did It" (4-page PDF). It was one of the many problems left over from the famous Fermat-Descartes correspondence.
- Fermat and Descartes did not like each other very much. In fact, some people describe their relationship as a “feud,” but it seems that Descartes resented Fermat more than Fermat disliked Descartes. They probably never met.
I figured Euler must have scribbled out a lot of long division problems to crack the Fermat number conjecture. But apparently he found a shortcut.
- Euler’s mentor in St. Petersburg, Christian Goldbach, alerted Euler to the conjecture in 1729. Euler responded almost immediately that he could make no progress on the problem, but by 1732, close to a hundred years after Fermat had originally made the conjecture, Euler had a solution: Fermat was wrong. In Euler’s first paper on number theory [E26] Euler announced that 641 divides 4,294,967,297.... What Euler did not tell us in E26 was how he thought to try to divide 4,294,967,297 by 641. He hadn’t simply been dividing by prime numbers until he got to 641. He had a much better way, but he waited about fifteen years, until E134, to reveal that secret.
With time to spare, I am in sight of ten thousand pages for the year, a goal I don't often achieve. E-books made it possible, contributing about two-thirds of the total. A lot of these were free downloads, starting with a bootleg copy of Orwell's amazing The Road to Wigan Pier, which paired well with The People of the Abyss, but mostly Gutenberg classics.
The Adventures of Tom Sawyer and Huckleberry Finn were good reading; the first made me wonder what all the fuss over content was about, the second relieved my confusion. Additional hits were The Autobiography of Benjamin Franklin and The Adventures of Sherlock Holmes.
- Sherlock Holmes sat moodily at one side of the fireplace cross-indexing his records of crime, while I at the other was deep in one of Clark Russell's fine sea-stories...
William Clark Russell was a great discovery. I much enjoyed The Wreck of the Grosvenor, The Death Ship, and The Frozen Pirate (though this last title had some passages that made Mark Twain's racial insensitivity seem mild).
Most of my more serious reading is still on paper, where I can scribble notes. The News: A User’s Manual was a reminder to focus on things of lasting importance, mostly unheeded, with some memorable exceptions.
And one more Alain de Botton title, thanks to a Hubski tip.
Vaclav Smil's Making the Modern World was relatively small and densely-packed, like the landfills he says are the endpoint for much of Europe's recycling.
- Collection rates of post-consumer plastic waste are impressive, more than 90% in eight EU countries and in Switzerland, and an average of 43% (25.1 Mt) in 2011 for the EU-27; but more than 40% (10.3 Mt) of collected plastics were then landfilled and of the 14.9 Mt (60%) that were recovered nearly 9 Mt were incinerated (energy recovery) and only about 6 Mt (or close to 10% of annual production) were recycled (Plastics Europe, 2019).