The Sky’s The Limit

Another meeting of the Acme Pizza and Science Society, at our usual big round table in Pizza Eddie’s place on the Acme Building’s second floor. (The table’s also used for after‑hours practical studies of applied statistics, “only don’t tell nobody, okay?“) It’s Eddie’s turn to announce the topic for the evening. “This one’s from my nephew, guys. How high up is the sky on Mars?”

General silence ensues, then Al throws in a chip. “Well, how high up is the sky on Earth?”

Being a pilot, Vinnie’s our aviation expert. “Depends on who’s defining ‘sky‘ and why they did that. I’m thinking ‘the sky’s the limit‘ and for me that’s the highest altitude I can get up to legal‑like. Private prop planes generally stay below 10,000 feet, commercial jets aren’t certified above 43,000 feet, private jets aren’t supposed to go above 51,000 feet.”

Eddie counters. “How about the Concorde? And those military high-flyers?”

“They’re special. The SST has, um, had unique engineering to let it go up to 60,000 feet ’cause they didn’t want sonic boom complaints from ground level. But it don’t fly no more anyhow. I’ve heard that the Air Force’s SR-71 could hit 85,000 feet but it got retired, too.”

Al’s not impressed. “All that’s legal stuff. There’s a helicopter flying on Mars but the FAA don’t make the rules there. What else we got?”

Geologist Kareem swallows his last bite of cheese melt. “How about the top of the troposphere? That’s the lowest layer of our atmosphere, the one where most of our weather and sunset colors happen. If you look at clouds in the sky, they’re inside the troposphere.”

“How high is that?”

“It expands with heating, so the top depends where you’re measuring. At the Equator it can be as high as 18½ kilometers; near a pole in local winter the top squeezes down to 6 kilometers or so. And to your next question — above the troposphere we’ve got the stratosphere that goes up to 50 kilometers. What’s that in feet, Sy?”

<drawing Old Reliable and screen-tapping…> “Says about 31.2 miles or 165,000 feet. Let’s keep things in kilometers from here on, okay?”

“Then you’ve got the mesosphere and the exosphere but the light scattering that gives us a blue sky happens below them so I’d say the sky stops at 50 kilometers.”

Al’s been rummaging through his astronomy magazines. “I read somewhere here that you’re not an astronaut unless you’ve gone past either 80 or 100 kilometers, which is weird with two cut‑offs. Who came up with those?”

Vinnie’s back in. “Who came up with the idea was a guy named von Kármán. One of the many Hungarians who came to the US in the 30s to get away from the Nazis. He did a bunch of advanced aircraft design work, helped found Aerojet and JPL. Anyway, he said the boundary between aeronautics and astronautics is how high you are when the atmosphere gets too thin for wings to keep you up with aerodynamic lift. Beyond that you need rockets or you’re in orbit or you fall down. He had equations and everything. For the Bell X‑2 he figured the threshold was around 52 miles up. What’s that in kilometers, Sy?”

“About 84.”

“So that’s where the 80 comes from. NASA liked that number for their astronauts but the Europeans rounded it up to 100. Politics, I suppose. Do von Kármán’s equations apply to Mars as well as Earth?”

“Now we’re getting somewhere, Vinnie. They do, sort of. It’s complicated, because there’s a four‑way tug‑of‑war going on. Your aircraft has gravity pulling you down, lift and centrifugal force pulling you up. Lift depends on the atmosphere’s density and your vehicle’s configuration. The fourth player is the kicker — frictional heat ruining the craft. Lift, centrifugal force and heating all get stronger with speed. Von Kármán based his calculations on the Bell X‑2’s configuration and heat‑management capabilities. Problem is, we’re not sending an X‑2 to Mars.”

“Can you re‑calibrate his equation to put a virtual X‑2 up there?”

“Hey, guys, I think someone did that. This magazine says the Karman line on Mars is 88 kilometers up.”

“Go tell your nephew, Eddie.”

~~ Rich Olcott

The Titanic Winds of Titan (And Venus)

Last week we saw that the atmosphere of Saturn’s moon Titan wasn’t quite as weird as we thought.  But there another way it’s really weird, completely unlike Earth but yet very much like Venus.  Titan’s a superrotater, a world whose atmosphere circles the planet much faster than its surface does.

Let’s start with a relatively simple Earthside phenomenon, a hurricane.  Warm air rises, right?  When the warmth comes from bathtub-temperature sea-water, it’s wet warm air.  As the air rises it cools and releases the moisture as rain.  But the air can’t just keep rising forever or we’d squirt out all our atmosphere. So where does it go?

From a physicist’s perspective, that’s the key question.   If we can track/predict the path of a small parcel of air molecules through a weather system, then we’ve got at least a rough understanding of how that system works.

For the past half-century, atmosphere physicists have been engaged on a project grandly entitled the General Circulation Model (GCM), a software mash-up of the Ideal Gas Law, Newton’s Laws of motion, thermodynamic data for solid/liquid/gas transformations, the notoriously difficult Navier-Stokes equations for viscous fluids, and careful data management for input streams from thousands of disparate sources.  Oh, and it’s important that the Earth is a rotating spheroid rather than a flat plane.

How a tropical cyclone works
Illustration by Kevin Song, from Wikimedia Commons

Kevin Song’s diagram summarizes much of what we know about hurricanes.  An air packet rises until it hits the tropopause (the top edge of the troposphere), then expands horizontally.  While the packet’s spreading out, the planet’s rotation generates Coriolis “forces” that bend straight-line radial paths into the spirals we’ve seen so often in satellite photos.

A hurricane may look big on your weathercaster’s screen, but it’s less than 0.1% of Earth’s surface area.  Nonetheless, many of the same principles that drive a hurricane underlie global weather patterns.

wind-cells-and-jets-2Air warmed by the equatorial Sun rises, only to sink as it heads poleward.  Our packet loops between the Equator and about 30ºN (see the diagram).

Actually that loop is a slice through a big doughnut that stretches all the way around the Earth.  Another doughnut lies southward just below the Equator.  Two more pairs of doughnuts reside polewards of those as indicated by the other arrows in the diagram.  The doughnuts act like a set of interlocking gears, each reinforcing and moderating the motion of its neighbors.

Thanks to the same geometric phenomenon that spins a hurricane, air packets in these doughnuts don’t loop back to the points they started from.  The Earth turns under the packets as they journey, so each packet takes a spiraling tour around the planet.

Because of all those doughnuts, on average Earth wears a set of cloud-top necklaces.  Regions within 15º of the Equator are rain-forested, as are the Canadian and Siberian forested belts near 60ºN.  The world’s most prominent deserts cluster beneath the dry downdrafts near the 30º latitudes.  Jupiter, “the Easter egg planet,” gets its pink and blue bands from similar doughnuts except that Jupiter has room for many more of them.

Those green circles in the diagram are important, too.  They also represent Earth-circling doughnuts, but ones whose winds flow parallel to Earth’s surface rather than perpendicular to it.  The ones close to the surface give rise to the trade winds.  The high-altitude ones are the jet streams that steer storm systems and give the weathercasters something to talk about, especially in the wintertime.

Jet streams flow briskly — 60 to 200 mph, on a par with a middling hurricane.  Here’s a benchmark: Earth’s equatorial circumference is 25,000 miles, so Ecuadorian palm trees circle the planet at (25000 miles/24 hours)=1041 mph.  Our jet streams go about 15% of that.  Theory and GCM agree that the jets are powered by the Coriolis effect — spiraling air packets in the primary donuts cooperate to push jet stream air packets like oars on a galley ship.  That adds up.

Titan and Venus can’t possibly work that way.  Both of them rotate much more slowly than Earth (Titan about 30 mph, Venus only 4), so Coriolis forces are negligible.  But Titan’s jet streams do 75 mph and Venus’ race at 185.  What powers them?  The physicists are still arguing.

~~ Rich Olcott