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Jason Kottke and Theodore von Kármán

March 12, 2012 at 11:05 PM by Dr. Drang

I’ve never subscribed to kottke.org’s RSS feed. I guess I figured that most of what Kottke links to gets relinked by others, so why bother. But today I followed a link there and poked around a bit, finally landing on this post about the edge of space. It struck me as being a perfect little writeup on an interesting topic; pitched just right to get me to follow up with more reading. I changed my mind about subscribing.

I don’t know when I first read that the “official” edge of space was at an altitude of 62 miles, but I know I immediately took it to be an arbitrary definition. Sixty-two miles is 100 kilometers, and that’s just too round of a number to be based on some real physical phenomenon.

Wrong. It turns out to have a real meaning as a dividing line between aeronautics and astronautics. Back in the 50s, Theodore von Kármán reasoned that the fundamental difference between aircraft and spacecraft was that aircraft fly and spacecraft orbit. (It’s possible that he came to this conclusion before there were, in fact, any spacecraft.) And there’s a way to distinguish between the two.

An aircraft flies because its weight, WW, is equalled or exceeded by its lift, FLF_L, which can be calculated as

FL=12ρACLv2F_L = \frac{1}{2} \rho A C_L v^2

where ρ\rho is the air density, AA the planform area, CLC_L the lift coefficient, and vv the airspeed.

The minimum airspeed to stay aloft, then, is

vmin=2WρACLv_\mathrm{min} = \sqrt{\frac{2 W}{\rho A C_L}}

An airplane’s design determines the W/ACLW/A C_L ratio, and the altitude determines ρ\rho. The higher the altitude, the lower the density. All other things being equal, then, the minimum airspeed increases with altitude.

Since the airspeed necessary to fly increases with altitude, there must be a point at which that airspeed equals the orbital velocity,

vo=GMR+av_o = \sqrt{\frac{G M}{R+a}}

where GG is the universal gravitational constant, MM is the mass of the Earth, RR is the radius of the Earth, and aa is the altitude.

If a plane has to travel at the orbital velocity to remain airborne, it might just as well be considered a spacecraft. The altitude at which this occurs was the boundary von Kármán was looking for.

Clearly, the design of the plane has some effect on where this boundary lies, but apparently enough designs put the boundary near 100 km that von Kármán chose that as the best reasonable approximation. The imaginary line he drew in the sky at 100 km is known as the Kármán line.

Theodore von Kármán

(Image from NASA via Wikipedia.)

Von Kármán was one of the giants of 20th century mechanics. You can’t study mechanics without running into him repeatedly. His greatest contributions were to fluid mechanics and aerodynamics, but his name is also associated with solid mechanics in the analysis of buckling and the bending of thin plates. He was one of the founders of the Jet Propulsion Laboratory, and there are prizes given in his name by SIAM and ASCE. He was continual source of analytical brilliance, still producing single-author papers1 until his death at 82.

I have a small but fun von Kármán story.

At the beginning of my career, I taught mechanical engineering at a small Big Ten university. Some of the senior guys in the department had been there for decades and were around when a very old Theodore von Kármán spent some time there as a Distinguished Visiting Scholar (or some such title). To make him more comfortable, the department bought a plush recliner chair for his office. When the great man’s visit was over, the recliner was snagged by one of the fluid mechanics professors, and it had, over the following quarter-century, been moved from office to office, always staying with the fluid mechanics guys.

The recliner became a revered object in the department and was known as The Von Kármán Chair in Mechanical Engineering.

(Jason Kottke is not to be blamed for the puns of academic engineers.)

Since von Kármán was one of our original rocket scientists (I’m pretty sure the blackboard behind him is filled with equations for rocket fuel—although I could be wrong because I’m not a rocket scientist), I feel obligated to include this Mitchell & Webb masterpiece.

  1. If you’re a venerable researcher, it’s easy to be listed as one of a paper’s coauthors, either through courtesy or by directing the work of others. But a single-author paper means you did the work yourself. ↩

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