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comment by wasoxygen
wasoxygen  ·  324 days ago  ·  link  ·    ·  parent  ·  post: Patterns That Eventually Fail

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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.

Devac  ·  324 days ago  ·  link  ·

I'll guess that something about hot lava prevents some kind of radioactive decay from occurring, but I can't imagine how.

That's because it doesn't influence decay rates. The age of stones is measured by effects of decaying nuclei or other forms of radiation on them, from within (Uranium-Lead or other forms of isotope ratio dating) and without ((thermo)luminescence). Once the rock is formed, it stops releasing (some) products of decay or replenishing less stable isotopes, which is not unlike removing a living organism from C14 cycle after it dies.

EDIT: Do keep in mind that the above explanation is somewhat idealised. Also, wikipedia has a nice overview of chronological dating methods if you are interested.

wasoxygen  ·  323 days ago  ·  link  ·

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).

Crazy.

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?

Devac  ·  323 days ago  ·  link  ·

Then how would you explain thermosphere with only the facts you mention? It starts just below the Kármán line at 90km above the sea level, where the pressure is much lower than even the peak of Mt. Everest, and yet the temperature range within is between 500 and 2500 °C.

You aren't saying anything that's wrong on its own, but it's not the full picture of the problem. Think of what happens in the atmosphere at different heights, its composition and how it affects the flow of energy.

wasoxygen  ·  322 days ago  ·  link  ·

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.

Devac  ·  319 days ago  ·  link  ·

Let's focus on one problem at a time. Also, let's not forget that every step was simplified.

You can simplify the initial problem to the following: you have a source of energy (Sun), a black body screen (Earth's surface) and a transfer medium between the two that can act as a buffer.

Since both of them radiate, there should be a point where energy going in, is equal to the energy going out. If you'd have two bodies at T1 and T2 (T1 < T2) and connect them with a strip of copper, you could reasonably expect there's going to be a point on the strip where T = (T1 + T2) / 2. That's your basic concept of heat transfer.

Now, time for some facts regarding our buffer:

The density and pressure of the atmosphere change with the distance from the Earth's surface.

The amount of energy per volume that air can retain is proportional to the number of particles enclosed in said volume (and their energies).

The average velocity of those particles is proportional to temperature and kinetic energy. Mass is another important factor, pertinent to the original question.

Heavier gasses stay close to the surface while lighter ones go up. It affects the distribution of gasses throughout the atmosphere. All gasses in the atmosphere (at least up to at least 80-90 km) are composed of more than one atom. N2, O2, CO, CO2, H2O, O3, Ar, CH4, NOx. Those are the most important ones. I'm going to use the word 'particle' when referring to them.

The amount of energy a particle can retain, and how hard or easy it is to change its temperature, is related to its heat capacity and specific heat. This has three components, each related to the type of motion a particle can make. First is your basic translational motion, which is to say "how a particle travels in 3D space". Up, down, left, right, forwards, backwards and their combinations. Second is the way a particle can rotate. Because we have only three axes (and three moments of inertia associated with them), it can only spin around X, Y or Z axis (or a combination of them). The third way a particle can move is the way it can vibrate. Imagine balls connected by a spring, and you have the basic idea. For an n-atom particle, there are (n - 1) ways it can vibrate.

The more ways a particle can move, the number of degrees of freedom, gives it a broader range of energy it can absorb. Each of those plays their part due to the equipartition of energy. Broadly speaking, each degree of freedom gets proportional for its type share of energy. Meaning: they have more ways to absorb and emit radiation.

FYI: Wikipedia has a good article on rotational-vibrational spectroscopy.

I told you to consider what happens at various heights of the atmosphere. I did it to point you toward things like clouds and water vapour in general. They scatter light and absorb tremendous amounts of energy on their own. Combine it with all the greenhouse gasses, some mentioned above, that are most excited in the infrared spectrum, and you get even more buffer properties. They act like a one-way mirror, absorbing energy radiated from the ground and radiate almost all of it it back towards the ground. That's part of the reason (if not the sole reason, I'm not certain about the specifics) why satelite images of clouds are done in IR spectrum.

Now, let's bring those facts together. Close to Earth's surface, you have more of those massive, multi-atom particles. This skews the point of energy transfer equilibrium closer to the surface.

Wait, so where's the role of pressure? It's talked about indirectly. Pressure and temperature are related to a collective force given by all the particles hitting surfaces (translational motion). The higher you go and the fewer of those particles are present, the less energy they carry on average.

There's a second part to this post that I'm still writing. It's more about convective processes and why/how they don't matter nearly as much as you might be thinking. My goal is to write it from the first principles approach like this one, but it's going to take me a while.

flagamuffin  ·  315 days ago  ·  link  ·

Let's focus on one problem at a time.

nah i think the other problem is more interesting than gases. one at a time is so '90s. witness:

-it seems intuitive that random space dust which became the core is cold because space is cold... but remember space is cold because it's empty-ish and the core is very much not empty, so lots of heat got trapped in the core. there was, at the time, lots of jostling. jostling is heat.

-as mr. devac pointed out, we can figure out how old rocks'n'stuff is sometimes by measuring radiation, which is just more heat

-also the layers of the earth trap heat pretty well right, i mean there's a ton of magma down there and not many access points on the surface

and the sheer magnitude of energy is unnerving--

temp of sun at surface 6000 kelvin. temp of earth, at same, a few hundred. massive difference in energy, so a sort of "arbitrage" occurs (obviously not really, and not forever, pace second law) and order springs from disorder. because order is just spending energy to create a temporary equilibrium. anyway, i don't know a damn thing about the thermosphere but the point is we don't really blast that much heat out. we trap it, and have since day 1, and we also accidentally used it to create a bunch of neat shit. https://en.wikipedia.org/wiki/Autocatalysis#Creation_of_order

borrowed time, but then we're in good company -- the whole universe is borrowing too

flagamuffin  ·  315 days ago  ·  link  ·

hubski paragraph spacing still sucks