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Sunday, January 9, 2011

159 Degrees of Separation: One of the greatest escapes in history.

I'm a huge fan of Paul Nahin. A true polymath, his areas of expertise range far and wide. Paul has written a series of books targeted at the lay reader (perhaps better said as advanced lay reader) on a wide range of subjects, from electronics (The Science of Radio), probability (Duelling Idiots and Other Probability Puzzles), to general mathematics and physics (several).

I have them all, and enjoy them immensely. One of my favorites is Chases and Escapes: The Mathematics of Pursuit and Evasion, a fantastic overview of the history and mathematics of modern pursuit theory. Pursuit theory in this context includes the analysis of optimal strategies for both pursuing (catching) a target, and evasion (losing or avoiding the chaser) by the target.

I thought of Paul recently when I caught the excellent BBC documentary Hiroshima, part of the BBC History of World War II series. If you have any interest in the history changing atomic bombing of Japan, I can highly recommend this documentary / docudrama. The film intersperses interviews with members of the allied forces, and those of survivors from Hiroshima, with superb reenactments and CGI of the events leading to, and including, the bombing.

In the documentary, it is mentioned that after the release of the bomb, the pilot of the B-29 Enola Gay, Colonel Paul Tibbets, made a "159 degree diving turn" to maximize the distance from the bomber to the detonation point of the Little Boy Uranium gun-type bomb. It had been determined that the plane could be damaged and possibly crippled if the distance was less that eight miles. As it turns out, Tibbets put 10.5 miles between the detonation and the plane (according to science officers analyzing data after the bombing.)

Why 159 degrees?

Tibbets had said to Oppenheimer (the father, in essence, of Little Boy and the subsequent Fat Man Plutonium implosion bomb): "I told him that when we had dropped bombs in Europe and North Africa, we'd flown straight ahead after dropping them - which is also the trajectory of the bomb. But what should we do this time?"

According to Tibbets, Oppenheimer replied "You can't fly straight ahead because you'd be right over the top when it blows up and nobody would ever know you were there."

Tibbets continues "He said I had to turn tangent to the expanding shock wave. I said, 'Well, I've had some trigonometry, some physics. What is tangency in this case?' He said it was 159 degrees in either direction. 'Turn 159 degrees as fast as you can and you'll be able to put yourself the greatest distance from where the bomb exploded.'"

I'm quite sure that something got lost in the translation here, either by the interviewer, or in Tibbets' memory of what was said (I'm quite sure Oppenheimer would not have said such a thing!): As we'll see, the statement "I had to turn tangent to the expanding shock wave" doesn't make much sense, and I'm sure what was meant is more along the lines of "Turn until the line from the detonation to the plane is tangent to the turning radius..."

We can do a simplified version of the problem with simple trigonometry. We want to maximize our speed and distance away from the detonation point, within the reasonable limits of the aircraft (i.e., no Split-S maneuvers, though I think the plane could tolerate one, and it might have been even more effective at getting maximal distance.)

Viewed from above the bomb run, we can diagram the problem as below (most certainly not to scale):

D represents the distance from the release of the bomb to the detonation point.

R represents the turning radius of the B-29 Superfortress (I am not aware of any actual data on Tibbets' aircraft: having been lightened by removal of non-essential components, I would venture a slightly smaller number than a standard B-29.)

Plugging in some rough numbers garnered from various sources, we find R/D ≅ 0.19

Since the tangent of angle a is defined as that value, we find that a = Tan-1(0.19), or a = 10.8 degrees.

a is half of angle b, making b = 21.6 degrees.

We know from basic trigonometry that angle c is 180 - b, or 158.4 degrees. Close enough for government work!

Hence, a turn of this angle will point the Enola Gay directly away from the detonation point, maximizing the distance over time from  that moment of flight.

In reality, the problem is a bit more complex, since it really is a problem in three dimensions (the drop height, actual detonation height of ~ 1900 feet, the time from drop to detonation, and the speed of the aircraft and the effects of the dive and turn must all be taken into account.)

In a personal communication, I asked Paul Nahin if he was aware of any such analysis. His reply was negative. Perhaps in the next printing of the book...