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Endless sky.

I’ve been a nerd pretty much my whole life, so as a kid, I wondered nerdy things. I wondered, for instance, if it was possible to have an endless sky: clouds below you and clouds above. As a slightly older kid, I realized that skies don’t work that way. A sky is just an atmosphere, just a layer of gas wrapped around a solid (or liquid) planet.

Now, though, with a fair bit of obscure physics knowledge under my belt, I can finally decide not only whether or not an all-sky planet is possible, but if it is, what that planet would be like. I knew there had to be some good things about being a grownup.

Because I want my planet to be all sky, I’m going to build it from dry air and water vapor (the water vapor is so we can have clouds). Can you build a planet out of nothing but air? Well-informed intuition suggests you can: after all, Jupiter and the Sun are mostly hydrogen, and hydrogen is less dense and therefore harder to squeeze together than air. But as it turns out, we can get more precise. We can ask how large a cloud of air would have to be to collapse into a planet. This number is called the “Jeans length” or “Jeans radius” (and thanks to XKCD for making me aware of this formula). For air at Earth surface density (1.2 kilograms per cubic meter) and room temperature (68 Fahrenheit, 20 Celsius, or 293.15 Kelvin), the Jeans length is 35,400 kilometers, which is just over half the equatorial radius of Jupiter. You might think that would mean that the silly planet known as Endless Sky would be quite massive. In fact, it’s only got three times the mass of the moon.

Unfortunately, this manageably-small gas planet wouldn’t collapse very much under its own gravity. Its gravity would be weak, and as it collapsed, compression would heat it up and probably evaporate most or all of it. It’s important to note that the Jeans equation was intended to be used on nebulae, not inexplicable pockets of air floating in space for no reason. I guess Sir James Jeans just wasn’t thinking ahead. (Incidentally, Sir James Jeans would be a really pretentious name for a line of denim pants).

But no matter the details, what would Endless Sky actually be like? This is where the fun begins.

As anybody who’s ever looked into hydrodynamics knows, fluids are an enormous pain in the ass to deal with. They’re always swirling around and compressing and carrying pressure waves and expanding and contracting. You can simulate fluid behavior using the Navier-Stokes equations, which are frightening:

Navier-Stokes

(From the Wikipedia article.)

We’ve got partial derivatives and dot products and divergence operators all over the fucking place. And to actually turn these equations into a computer program, you need a whole set of conservation equations, as well as initial conditions and boundary values. These equations are complicated because things like air and water are swirly and their mass moves around all the time. So you’d think the equation for the pressure in an atmosphere would be horrifying. Actually, it’s not. It’s

(pressure at the surface) * exp(-1 * (altitude) / (scale height))

The scale height is the altitude at which the pressure is e times smaller than it is at the surface (where e is about 2.71). This means that pressure corresponds quite closely to altitude. The scale height on Earth is (roughly) 8500 meters, so at an altitude of 8.5 kilometers, the pressure is 1/e atmospheres  (0.37 atmospheres or 37 kilopascals). Because this dependency is pretty stable, you can also say that, if you’re measuring a pressure of 37 kilopascals, you’re at an altitude of 8.5 kilometers.

We can use this nifty formula to figure out what our Endless Sky will be like. Before I get to the amusing pictures, here’s a caveat: I’m assuming constant Earth surface gravity throughout the atmosphere, which is inaccurate. That’s why this is a thought experiment, and why I majored in English instead of physics.

But here’s roughly, what the atmosphere of Endless Sky would look like:

 An Endless Atmosphere

You can learn two things from this picture right away: First, how thin tropospheres are, which tend to be where human beings and similar organisms hang out. Second, don’t buy cheap graph-paper notebooks, because the paper can apparently detect when you’re trying to tear it out gently and violently self-destructs.

Because I decided to make the parameters of my endless sky pretty much the same as Earth’s atmosphere, it all looks pretty familiar from the 1 atmosphere level to the 0.00001 atmosphere level (where we pass the Kármán line and go into space). Because Endless Sky has an oxygen atmosphere and (I’ve just decided) a sun, it’ll have an ozone layer much like Earth’s. I always wondered why thunderstorms grow to a certain height and then stop and spread out. It turns out to be because, below a certain level (called the tropopause), the higher you go, the colder the air gets. But above this level, and in fact throughout the stratosphere, the air actually gets warmer as you go higher. This is partly because of the ozone layer. Thunderstorms happen when hot, moist air rises into the lower-pressure air above, expands, and its water condenses. But this can only happen if it’s warmer than the surrounding air, and that pretty much becomes impossible at the tropopause, since the air above is already warmer.

Basically, if you were flying around Endless Sky in a hot-air balloon, and you sat so that the gondola obscured the horizon, it would be easy to think you were flying around Earth: bright blue skies, fluffy white clouds (no happy trees, though, unfortunately).

But if you looked down, you would see something horrifying. Namely, you would see no ground. Depending on atmospheric conditions below you, you might get different kinds of exotic clouds, but these would be smaller than the ones you see on Earth, because the higher pressure wouldn’t allow them to expand as much.

Looking into the sky below you would probably be quite a lot like looking into a deep, clear ocean: it would grow a deeper and darker blue. The blue is due to Rayleigh scattering, which (combined with the spectrum of light the Sun puts out) is why the sky above is blue. The darker is because less of the sunlight would make it through all that air.

You might think that the pressure would just keep going up and up, and the air would just get denser and denser as you got deeper. On most planets, solid, liquid, and gaseous alike, temperatures tend to go up as you go down. In the case of rocky planets, this is partly because of radioactive decay. But no matter what kind of planet, there’s internal heat left over from its formation. Therefore, it gets pretty hot down there. And when the temperature and pressure rise above the so-called critical point of the gases involved, something magical happens: the gases don’t quite liquefy, but they don’t remain gaseous, either. They sort of forget what they are, and become supercritical fluids. Supercritical fluids are amazing. They can be as dense as water (or denser), but they compress and expand like a gas and fill their containers. Here’s a video from awesome YouTuber Ben Krasnow showing you what supercritical carbon dioxide looks like:

Nitrogen’s critical point is lower than oxygen’s, so at a depth of about 128 kilometers (below the 1-atmosphere level), you would encounter a broiling opalescent sea of semi-liquid nitrogen containing a lot of dissolved oxygen. Looking down, you would see city-sized Bénard cells, which look approximately like this:

Benard Cells

(Image courtesy of NOAA, the US government atmospheric science people, who are pretty neat.)

They probably wouldn’t be quite that orderly, though. But there would be nothing but broiling opalescent clouds rising and falling as far as the eye could see, twisted into peculiar shapes or into alien jet streams by Coriolis forces.

Being denser, the supercritical ocean would attenuate light much faster than the gaseous part of the atmosphere. You would see deep, dark blue down to the opalescent layer, and then nothing. But, farther down still (about 200 kilometers deep), the sky would no longer be endless: the pressure would pass 88,000 atmospheres, which is the pressure at which oxygen solidifies into pretty blue crystals. This means there would be another layer of weather on Endless Sky: down there, the crust of oxygen and nitrogen “ice” would evaporate into the supercritical fluid above, rise, and cool, driving powerful convection currents and stirring the deep, and where it fell and cooled, the supercritical fluid would condense into oxygen and nitrogen snow. Perhaps it would form enormous dune fields like the ones we see on Earth and, amazingly, on Saturn’s moon Titan:

Titan_dunes_crop

(Earth dunes on top. Titan dunes (probably made of water ice) on the bottom. Image courtesy of NASA, via Wikipedia.)

I hope my eight-year-old self is vindicated. He’s probably dancing around like a lunatic, the hyperactive little freak…

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6 thoughts on “Endless sky.

  1. Pingback: The weather in hell. | Sublime Curiosity

  2. Pingback: A great big pile of money. | Sublime Curiosity

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