Skip

Assume A Cylindrical Cow
July 10, 2013 3:07 AM   Subscribe


 
"A major problem with mathematical modeling is knowing that the model is correct. "

This is indeed a problem. As a wise man once told me, "All models are wrong, but some models are useful." In the end, you want to somehow verify your model in the real world, something they obviously did in the Manhattan Project.

Some of the worst fights in science are over different mathematical models of some process that is effectively unverifiable. Individuals/groups can argue for decades over this, with no endgame at all.
posted by grajohnt at 3:27 AM on July 10, 2013 [3 favorites]


Its an interesting argument. The thesis, as I understand it, is that throwing resources and talented people at a problem will only solve it rapidly if the problem is easily solvable. In the case of the manhattan project, producing a nuclear bomb was not a terribly difficult problem, so, an an exception, was solved in a few short years.

I think it might be a stronger thesis if the article could point to some new manhattan projects that are useless. If I hire a ton of scientists and give them lots of money to produce fusion I might not solve fusion, but I might actually get a lot of good results out anyway, because of the concentration of mind and money.
posted by Cannon Fodder at 4:31 AM on July 10, 2013 [1 favorite]




Wow, I disagree entirely.

The reason most "manhattan project" styled projects fail is simple: They in no way, shape, or form have the resources available that the real Manhattan Project did. There's a simple test for this. I call it the Calutron test.

Can you get permission to borrow a significant fraction of the United States' silver reserve, or other capital outlay of a similar magnitude.

Because that's what the Manhattan Project did. They needed to build a large number of electromagents for the calutrons needed in the U235 separation process. Copper, however, was in short supply, and needed everywhere -- as itself for electrical wiring, with tin as brass for ammo, and with aluminum for structural alloys for aircraft.

They didn't need the alloying properties, they just needed the electrical capacity. Could anything else do? Yep. Silver. So they went to the US treasury and borrowed it. 14,000 tons of silver, all told. Or, in units more familiar to the precious metal industry, 408,333,333 troy ounces.

Today's value of that? $7 billion. In 1944, it was literally $408 million dollar, a dollar coin then containing one troy ounce of silver (and a quarter had a quarter ounce, and so forth)

So: That's the scale. Can you just borrow $8B from the US?

I can think of exactly one project in the US that matched the scale of the Manhattan Project, and one that's close. The one that matched, of course, was the Apollo Project. The one that was close was the National Highway Defense System, better know as the Interstates.

I think we can file both of those as successes.
posted by eriko at 5:19 AM on July 10, 2013 [105 favorites]


This article seems to have been written in ignorance of a lot of verified information that has been published about how atomic bombs actually work. In particular the application of "cylindrical geometry" to Little Boy totally misses the point of how the bomb works and why its designers considered a test unnecessary.
posted by localroger at 5:28 AM on July 10, 2013 [2 favorites]


And, game over. eriko wins the thread.
posted by IAmBroom at 5:28 AM on July 10, 2013 [1 favorite]


This article discusses the mathematics of the Manhattan Project

I'm going to disagree and say that this article does nothing more than claim "Because spherical symmetry, the bomb was easy".

I could try and write down the challenges related to making the bomb that were not at all easy, and point out that spherical symmetry really only helped in designing the implosion mechanism (for the bombs that worked that way), but that could go on for days, so I think I'll just say this article is stupid.

Anyone with so much as a passing interest in the Manhattan project should skip this entirely and go read The Making of the Atomic Bomb, by Richard Rhodes. It's a fantastic book.
posted by kiltedtaco at 5:44 AM on July 10, 2013 [8 favorites]


I can think of exactly one project in the US that matched the scale of the Manhattan Project, and one that's close.

One could add the Human Genome project to that list. (And it doesn't involve incinerating millions of people, which has its own cost!)
posted by Blazecock Pileon at 5:47 AM on July 10, 2013 [3 favorites]


You can also add the current big data domestic and international spying programs run by the NSA, which is probably grabbing this sentence as soon as I hit the Post button. It certainly matches the Manhattan project in scope, cost, numbers of people involved, although its success or failure is still to be determined.
posted by beagle at 6:00 AM on July 10, 2013 [2 favorites]


BP, actually I don't think the Human Genome Project meets Eriko's criteria -- but it's nevertheless a fantastic example, because of that.

The HGP was expensive, but it wasn't nearly that expensive, and the reason was that the technology required got really cheap at a fortuitous time (and probably in no small part because people suddenly cared so much about genetics, for reasons only tangentially related to the HGP).

So I think you've both exposed some really interesting dimensions to the problem.
posted by lodurr at 6:11 AM on July 10, 2013


I agree with eriko that most "manhattan projects" aren't given the resources that the actual one received -- roughly, call it one percent of a year's GDP, or about $150 billion in today's environment. Or call it $30 billion a year for five years. But on the other hand, this kind of scale isn't laughably huge. We're planning to spend something like $400 billion on F-35s over several years, and the US Department of Justice is running pretty close to Manhattan Project funding levels (ie about $36 billion/year). Likewise, before the Great Recession threw so many people into poverty, food stamps ran at about that level and are currently a bit over double Manhattan Project funding levels (or, if you want, a full Manhattan Project every other year).

The Manhattan Project did have some very real advantages over other potential Manhattan Projects. If you have enough U-235, building an atomic bomb is indeed pretty simple: take two subcritical masses and whack them together really fucking hard. Simple enough not to bother to test. The implosion technique is obviously harder, hence the testing, but again relatively simple if you have the plutonium. And of course it seems to get much much harder once you're trying to use multistage designs that use their fissile material efficiently.

And the core problems, obtaining enough U-235 or plutonium, are eminently achievable simply by throwing money at the problems. Need to skim U-235 from natural uranium? If you're willing to say "Fuck it, we'll just use all the available techniques simultaneously at a rate which will more or less guarantee the production of enough by 1945 and we don't care if we use more electricity than New York," you can do this. Want plutonium? BUILD MOAR REACTORZ; it also helps if you don't need to give a shit about the people in Washington or South Carolina.

Not being an engineer or physical scientist, I'm not sure how many other potential Manhattan Projects have the same mix of relatively simple raw research and engineering challenges that mostly boil down to just spending money to increase scale. Some, for sure -- the Human Genome Project comes to mind. But stuff about fusion power generation always seems to include "If we knew how to X, which we have no idea how to do," or "If we had the following kind of unobtainium," which doesn't strike me as the same sort of problems the Manhattan Project faced.
posted by ROU_Xenophobe at 6:13 AM on July 10, 2013 [5 favorites]


I'm going to disagree and say that this article does nothing more than claim "Because spherical symmetry, the bomb was easy".

It may be a simple thesis, but it's still

© 2013 John F. McGowan

So don't use it without attribution!
posted by RonButNotStupid at 6:14 AM on July 10, 2013 [1 favorite]


The implosion technique is obviously harder, hence the testing, but again relatively simple if you have the plutonium.

Sorry; I meant to say that we know this is relatively simple because hardly anyone fucks up this stage of the problem.
posted by ROU_Xenophobe at 6:15 AM on July 10, 2013 [1 favorite]


Agree with the critiques above. This is not a very good blog post. For example, I'm kind of amazed that it doesn't mention Stanislaw Ulam and the Monte Carlo method.
posted by Ivan Fyodorovich at 6:16 AM on July 10, 2013


In addition to some of the other links to further reading on the topic, I want to throw out my introduction to MeFi favorite John McPhee:The Curve of Binding Energy, an engaging look at the design of nuclear weapons. It certainly gives the impression that making atomic bombs is not as simple as the article implies.
posted by TedW at 6:30 AM on July 10, 2013 [2 favorites]


If you have enough U-235, building an atomic bomb is indeed pretty simple: take two subcritical masses and whack them together really fucking hard.

Sorry, I want to argue about this a little bit. Yes, figuring out the mechanics of the gun-type design are pretty simple, if you already know that it's the answer. So much of the work of the Manhattan project was figuring out the properties of uranium and plutonium, which were simply not known when the project started. If you go look at the early estimates for the critical mass for uranium they vary by orders of magnitude, even when calculated by really preeminent scientists like Heisenberg. Figuring this out is not easy, especially because the fission cross section for uranium-235 had to be inferred from the cross section of unenriched uranium (enriched 235 obviously wasn't available at the time), which contained several different isotopes all of which could have wildly different cross sections. Figuring this out was not Miracle From Above level of genius, but took serious effort and resources. Understanding the properties of Plutonium was substantially harder still. Neither of these types of bombs were anything close to a sure thing until a significant way into the project.
posted by kiltedtaco at 6:37 AM on July 10, 2013 [10 favorites]


"Sorry; I meant to say that we know this is relatively simple because hardly anyone fucks up this stage of the problem."

Is this true? I don't really know anything about the history of other weapons programs.

It is suggestive that "the gadget", used at the Trinity test, was an implosion bomb and worked, but they went to great lengths on it because the implosion design was the one they were worried about and they didn't want to waste any pu-239. They knew the gun-type u-235 bomb was going to work and were ready to deploy it, which they did a few weeks later at Hiroshima. But that was all the u-235 that they had! So if they needed to use a second bomb on Japan, which is what happened, it would be a pu-239 implosion device so they absolutely had to test the design. So, as I said, they were extremely careful in its design and engineering because they needed the test to work and the subsequent deployed bomb to work.

In the 1940s, the implosion design certainly was not trivial. Contrary to this blog post, the mathematics of the actual design were difficult — using a combination of slow and fast high-explosive to shape the shock waves. The calculations were especially difficult, calculating that propagation through those materials, the damper, and the plutonium.

But that's not all — the engineering had to be incredibly precise for this to actually work. All these parts, including the high-explosives, had to be machined to very, very tight tolerances.

I don't know if this level of engineering expertise is (relatively) trivial these days — I imagine that it is. But I doubt that it was in the 50s when the other nuclear powers were building their weapons. The gun-type bomb is much, much easier. Dead simple. Except for the production of uranium-235, which is even now a huge industrial undertaking.
posted by Ivan Fyodorovich at 6:38 AM on July 10, 2013 [6 favorites]


@ Ivan Fyodorovich - they were so concerned about losing pu-239 that they designed a container in case it didn't go nuclear.

Meet Jumbo. A 214 ton cylinder, walls 14 inches thick.

http://www.brookings.edu/about/projects/archive/nucweapons/jumbo

http://www.atomicarchive.com/History/trinity/jumbo.shtml
posted by JB71 at 6:45 AM on July 10, 2013 [2 favorites]


eriko: "In 1944, it was literally $408 million dollar, a dollar coin then containing one troy ounce of silver (and a quarter had a quarter ounce, and so forth)"

This a sidetrack, but this is not accurate. The last silver dollar coin minted (the Peace Dollar) had last been struck in 1935. Not sure how many were still in circulation in 1944, but in any case, they were 90% silver, 10% copper. They contained 0.77344 troy oz of silver (about .85 avoirdupois ounces).

The silver Washington quarters then in circulation were also 90% silver, and contained 0.18084 troy oz (about 0.2 avoirdupois).
posted by Chrysostom at 6:51 AM on July 10, 2013 [1 favorite]


I hate to keep posting, but honestly this article is not just wrong, it's super insulting to those who worked on the Manhattan project. Just surveying the mathematics of the project off the top of my head, Los Alamos deserves credit for driving the first computers, inventing Monte Carlo (Ulam/Metropolis/von Neumann), and really building the field of radiation hydrodynamics, amongst many other subjects, and to ignore all that is inexcusable. It's like writing an article called "Contributions of Einstein to Physics" and then describing how he approved patents at the patent office.
posted by kiltedtaco at 6:53 AM on July 10, 2013 [9 favorites]


Is this true? I don't really know anything about the history of other weapons programs.

AFAIK, the only people who didn't get the proverbial earth-shattering kaboom on their first try were the North Koreans.

Yes, figuring out the mechanics of the gun-type design are pretty simple, if you already know that it's the answer.

Fair enough.
posted by ROU_Xenophobe at 7:02 AM on July 10, 2013


Well thank goodness the Germans didn't have good old American ingenuity, or we'd all be driving our BMWs through a post-apocalyptic wasteland!
posted by blue_beetle at 7:10 AM on July 10, 2013 [1 favorite]


+1 for The Curve of Binding Energy. Here's a Google books link with a taste. I'm looking at the three-page section that begins "A small group has not had the opportunity before to rearrange people and buildings this way," ends "So far as we know, everybody in the world who has tried to make a nuclear explosion since 1945 has succeeded on the first try," and in the middle lists eight research requirements for success of the Manhattan Project, none of which would be obstacles to building a bomb now that we know the answers.
posted by jhc at 7:12 AM on July 10, 2013 [1 favorite]


Carey Sublette nuclear weapons faq is still online! (Don't download too much of it if you want to stay off the terror watch list.)
posted by bukvich at 7:22 AM on July 10, 2013


I agree with eriko's statement, and the article surely overstates the mathematical dimension of the problem (you know--actually making stuff is hard, and most mathematical models suck, pretty much by their nature). But it's not about engineering, is it? It's at a math blog, after all...

All that said, I didn't interpret the article as insulting. It came across as a palliative for the common tendency to overly aggrandize (often, ironically, to the point of mysticism) the role of mathematics in science and engineering. For educated lay people this seems like a useful point to make. I usually tell my students (I teach a few applied math and classical physics courses at the university level--mostly to engineers) that, though it's counter-intuitive, they need to realize they spend most their time studying math that's relatively easy, precisely because that's the stuff that people have been able to say the most about.

For example, linear algebra and linear system theory are hugely important in applied science. Even most of our methods for tackling nonlinear problems essentially involve converting them into a collection of "sequential" or "localized" linear problems. Is linear analysis important because nature is linear? I'd argue, no, that's bass-ackwards: it's important because after a few hundred years we understand the stuff very, very well, because it's relatively simple. So that's a big hammer we have in our toolbox, and we have learned to see many things as a linear nails.
posted by mondo dentro at 7:35 AM on July 10, 2013 [2 favorites]


The physics of the implosion device are far from simple, you have a sphere, covered with a series of very carefully designed and shaped layers of high explosives. There are a limited number of detonating caps which fire everything else off, timed to exceedingly tight tolerances.

Assuming the device works at this point you have a number of shockwaves at a cluster of points around your core, you then need to shape all those wavefronts in such a way as to result in a very strong "tube" of toothpaste forcing inward uniformly from all directions in towards the center, even where those wave fronts meet and pass through each other.

There have got to be insane amounts of work in terms of trial and error to get that kind of stuff into the ballpark that makes it even possible to achieve the type of compression necessary to raise the density of the core by a factor of 2 or more (I have no idea what the number really is) for the microseconds it takes to have the neutron pulse generator then kick off the real action.

It sounds simple until you really step back and consider what you're really doing. Compressing a metal to less than 1/2 of it's volume from all sides, then hitting it with a neutron pulse, all within timing tolerances that were probably 1000 times faster than any previous industrial timing requirements. Too soon, and you roast part of the core and shoot the thing asymmetrically apart before the big boom... too late and it's already heating itself up and pushing back apart before the right big boom.
posted by MikeWarot at 7:42 AM on July 10, 2013 [2 favorites]


About models and such, I've always wanted to get a personalized t-shirt reading "Project failure likelihood is proportional to the distance between your homomorphism and reality." but we have so many nitpicking career mathematicians around campus that I'd be promptly hanged from the balconies.
posted by Iosephus at 8:22 AM on July 10, 2013


Yeah, this article is just wrong on a ton of different levels. Just because the bombs were cylindrical or spherical doesn't make them easy to solve -- as MikeWarot pointed out, the actual implosion charges could not be spherical as there were finitely many of them and they were made of flat plates. The calculations needed to figure out this exact geometry were exceedingly difficult and, as the author points out, high-powered computers were not available so enormous amounts of work had to be done by hand.

It is really interesting to read physics papers from before the computer era for exactly this reason -- so much work has to be done by hand that solutions have to be worked out in much greater detail in order to be useful. In a modern paper, one might derive the necessary differential equations, figure out the needed parameters, and then present a series of relevant numerical simulations. In older papers, though, typically half of the paper is spent taking the differential equation (or similar) and making relevant and useful approximations that render it more amenable to hand calculations. The process of doing this requires that you first understand all of the relevant approximations (which gives you a sense of which parameters are actually important and which are not), but the reward is that you end up with simpler results that can often hint at much more important physics than you may have initially suspected. Of course, you also get a lot of stuff that looks like nonsense nowadays -- I've seen huge, important papers that have two pages of groundbreaking work followed by ten pages of how to solve the equations using log tables instead of more expensive calculators.

The idea that because computers were not invented, therefore the Manhattan project was easy is almost unbelievably dumb. I guess building the pyramids must have been pretty easy, too, seeing as how the Egyptians didn't have modern earth movers. Also, just like Atomic bombs, the pyramids have simple geometric shapes and not overly complicated interiors. Maybe he can download the Mathworld graphic of a pyramid and write an article about that as well!
posted by artichoke_enthusiast at 8:40 AM on July 10, 2013 [6 favorites]


I'm just kind of perplexed by the conclusions here. If mathematical modeling sucks and full system tests rarely work, what about the Apollo Program, which saw the development of a giant rocket which succeeded on its 1st launch in its final configuration? What about the novel navigational systems and custom computers that guided a spacecraft to the lunar surface? What about ... every bridge ever built? Etc.

The author has written on the Manhattan Project before, and I kind of agree that it was unusually successful -- but maybe it was due to assembling the dream team of physicists in an environment with no politics, high budget, and an urgent mission, while uncovering low-hanging-fruit in a brand new field of study.

I agree that the term is overused though.
posted by RobotVoodooPower at 9:11 AM on July 10, 2013


So: That's the scale. Can you just borrow $8B from the US?

Well,
The US flew nearly $12bn in shrink-wrapped $100 bills into Iraq, then distributed the cash with no proper control over who was receiving it and how it was being spent.
posted by jamjam at 9:21 AM on July 10, 2013 [4 favorites]


RobotVoodooPower - The Apollo program came after many spectacular launch failures. It worked in the end because there was a ton of trial and error by a very large crew of very skilled people all working towards one simple mission statement, which had the popular support of the nation.

Jamjam - The Manhattan project borrowed 14,000 TONS of silver (and returned it)... the current war gave away a few truck loads of fiat paper in a land that doesn't enforce the law that gives it value. Big difference in scale here.
posted by MikeWarot at 9:59 AM on July 10, 2013


Not only did they borrow 14,700 tons of silver, they returned all but 1/3,600,000th of it!
posted by fings at 1:21 PM on July 10, 2013


Well, you know, someone probably just pocketed the missing 119 troy ounces.
posted by radwolf76 at 2:05 PM on July 10, 2013


blue_beetle: Well thank goodness the Germans didn't have good old American ingenuity, or we'd all be driving our BMWs through a post-apocalyptic wasteland!
Actually, Einstein is thought to have written letters to Heisenberg (I'm hoping I get the details right) which suggested obliquely it would be a terrible thing if the Nazis studied atomic power.

The implication is that Herr Heisenberg may actually have made a moral choice to avoid looking into that dark cauldron, and no one else in Germany had the intellect to lead them there.
posted by IAmBroom at 2:39 PM on July 10, 2013


Well thank goodness the Germans didn't have good old American ingenuity

They had bad luck and less determination.

They explored the istope-separation route toward a U235 bomb, but they had the resources to pursue only one avenue. They tried thermal diffusion, since its inventor Klaus Clusius was on their side. But it turned out that the Clusius method doesn't work for Uranium Hexaflouride. Our side went on to successfully explore gaseous barrier diffusion and electromagnetic separation, but the Germans abandoned the idea.

And Germany's path to a Plutonium bomb was blown by a single mistaken measurement. When Walther Bothe measured the neutron absorption cross-section of carbon in January 1940, he came up with a value high enough to rule it out as a moderator for the nuclear reactors which would be needed to breed Plutonium. Bothe didn't know that his graphite was contaminated with boron. On our side Enrico Fermi, used a different methodology to make the measurement which gave a much more promising number.

That mistake left the Germans with only heavy water as a potential reactor moderator, and when the British bombed the Norsk Hydro heavy water plant in Vermork and Norwegian commandoes sunk the ferry carrying the train with the last of its output, the German atomic program was made irrelevant. If they'd had a bit more time their "atomic machine" might have sustained criticality, but not at a scale useful for anything but pure research.
posted by localroger at 3:04 PM on July 10, 2013 [7 favorites]


Oh and all the difficulties faced on all sides are nicely portrayed at some detail in Richard Rhodes' The Making of the Atomic Bomb which richly deserves its Pulitzer Prize.
posted by localroger at 3:06 PM on July 10, 2013


The HGP was expensive, but it wasn't nearly that expensive, and the reason was that the technology required got really cheap at a fortuitous time (and probably in no small part because people suddenly cared so much about genetics, for reasons only tangentially related to the HGP).

More than the cost, which continues more or less in offshoot projects and is similar in scope to the Manhattan Project, the HGP has also brought together many of the best minds in biology, bioinformatics and computation, much as the MP did for physics and mathematics. Of course, it doesn't involve blowing up people and huge chunks of land, so it isn't as sexy, but we are already reaping returns from the massive amount of data collected and analysis performed. Unlike a war effort that has sunk billions into weaponry that has more or less only caused fear, sickness, death and suffering (and consequentially sucked the economic air out of the room for decades) this project will actually provide a net positive benefit to generations to come.
posted by Blazecock Pileon at 4:12 PM on July 10, 2013


It was Leo Szilard who grilled the representatives from the National Carbon Company about possible elements in commercial graphite which might absorb (instead of slowing) neutrons over lunch at Columbia Men's Faculty Club, when he found out that Boron (the worst possible contaminant) was used. Had he just used supplies procured from the National Bureau of Standards, the measurement errors the Germans had made would have been repeated here.

You can read all about it in the December 1992 Bulletin of the Atomic Scientists.
posted by MikeWarot at 4:24 PM on July 10, 2013 [2 favorites]


Ah, Richard Rhodes did not get Szilard's contribution to the carbon / boron fiasco into TMOTAB. Although it was rather brilliant that Rhodes thought Szilard, who let's face it nobody ever heard of IRL, was so central to the whole endeavour that it was his footstep Rhodes chose to start the story.
posted by localroger at 6:21 PM on July 10, 2013


Several other people have pointed out the huge flaws in this paper, so I won't repeat them other than to emphasise that no, the explosive lens layer of Gadget / Fat Man most emphatically was not spherically symmetrical, and that is what made developing it so damned difficult.

This is, however, a nice example of that special sort of academic arrogance I sometimes encounter that manifests itself as 'I am very smart in my own narrow field so obviously my initial appraisal of any problem that looks even remotely related must be right. Pah! Why does everyone else say that problem is hard? I don't even need to try to solve it to know it's easy.'
posted by Major Clanger at 4:40 AM on July 11, 2013 [1 favorite]


« Older D.O.G.   |   I am plant, and as such I... Newer »


This thread has been archived and is closed to new comments



Post