Can you build a nuke plane?
I can't build anything. Also I think they are generally a stupid idea.
Yes, there are a couple of possibilities. Can you build one that is "reasonably economical" and not a disaster waiting to happen? No.
How do you build a nuke plane. 2 options, basically, you can either do a nuclear jet (e.g.
http://en.wikipedia.org/wiki/Project_Pluto) where you use a VERY, VERY hot reactor to heat air and push it out the back. Doing that without spilling radioactivity into the air being pushed through means shields between the radioactive fuel and the air, which slows heat transfer, which...
Or you can go with a prop plane with electric engines and a nuclear generating station on board. There you have even more fun. Work out the mass/power ratio of even the lightest reactors, calculate how much power e.g. a Boeing Pelican would need and figure out the weight of the reactor. My back-of-the envelop calculations done a while back suggested the plane wouldn't even get off the ground (by about an order of magnitude if my memory is correct, which it may not be). You MIGHT get around this by using really highly enriched fuel, but then you'd better hope the plane never crashes...
There was this 1970s RAND study book, on possible alternative fuel approaches to really big airplanes, that doted on a third nuclear power scheme you haven't acknowledged as possible--using a relatively compact liquid-metal cooled reactor to transfer heat, through a secondary coolant loop, directly to radiators corresponding to the combustion chamber of turbofan engines. For purposes of keeping the arrangement compact, he had the reactor occupying the region of the fuselage right at the joint of the wings to the main body, and the engines all clustered on top right there, to minimize the distance the liquid metal (some sodium/potassium mix) had to travel).
So I suspect that might work; another approach would be to have an advanced gas-cooled reactor that does not use air but say helium as the working gas, which is run directly through sealed power turbines and then cooled and recycled (one might compactly cool it by using the still-hot turbine exhaust to boil water and run a secondary steam engine for auxiliary power, for instance), then take the power from the main turbines, via mechanical shafts or electricity, to drive fans that perform aerodynamically more like turbofans than turboprops. Or for that matter, props can be very effective, as per say the Soviet/Russian Tupolev "Bear" bomber for instance!
Would it be smart? I doubt it. The sheer mass breakdown of the system becomes more competitive with conventionally combusting systems at larger sizes, especially when one factors in minimum necessary shielding mass--this becomes proportionately smaller at bigger sizes. Go somewhat bigger and it breaks even economically even with cheap fuel rivalling it, somewhat sooner if avgas and other combustible alternates are very expensive. But it's a big cost upfront; it takes a lot of financing and a lot of government bureaucracy giving it a thumbs-up to be done, no easy starts with small scale versions to prove the concept.
Then too safety is a huge and uncertain cost. What does it take to guarantee a crashing plane won't shatter or splash its reactor core all over the landscape and inject it as fine poisonous dust into the air, or leak all over the seafloor? (Not to mention the risks of rouge nations and money-grubbing opportunists catering to them, or various non-governmental fringe groups, scavenging the stuff for their own fell purposes!) The RAND guy made all sorts of promises of the safe integrity of his conceptual liquid-metal cooled reactor, asserting that it could be sealed off and hardened to survive a crash intact. Should we believe him?
And as I think I sketched out already here, it isn't clear to me what mission the dang things should have. The RAND study wound up mostly favoring simply synthesizing conventional hydrocarbon fuel as least risk and competitive overall cost, with the prospect of a hydrogen-fueled version (bigger fuselage than any other for the target, but smaller take-off weight and hence smaller wings) but just couldn't let the nuke go. Certainly for a narrow niche of missions where the plane would ideally be kept aloft for many days or more, ie very long loiter time, he made a case. To be sure we hardly see the need for that OTL! For long range I personally like hydrogen; for wacky stuff like supersonic airborne rocket launch I think Nukes would be a profoundly stupid approach.
I actually think airborne nukes have their best niche in really big airships, but since those are relegated to ASB by the establishment as it is, I don't think it's any loss we haven't developed airborne nukes to speak of.
OTL that is. ITTL, the Air Force snookered itself into splurging on one* and now they are stuck with a military white elephant. They'd do well to make some lemonade out of this lemon, so surely studies are well underway for applications where the damn thing makes at least marginal sense. Like, airborne command posts much bigger than the 747 based ones of OTL, or perhaps subsonic airborne ballistic missile launchers suitable for something like the MX, or a whole quiver of Minutemen. The thing about such admittedly marginal missions is that they'd benefit from staying airborne for a good long time without refueling, so the nuke comes in handy there.
And for some of these missions they can promise to orbit around far from inhabited land, so if something goes wrong and it crashes, at least it isn't crashing on Sheboygan!
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*A lot of the OTL inflation of Soviet capabilities and subsequent 'Gap' panics was the Air Force cherry-picking the evidence for a worst-case scenario that would get them maximum funding, I suppose the same thing happened here--building on more objectively alarming real Soviet evidence, they made an even bigger scarecrow for Congress and Presidential candidates to gibber about, and hand their favorite contractors yet another big blank check.
In general I guess ITTL, especially if it turns out that the POD is basically the Russians doing better, Eisenhower still gives his warnings about the overweening Military Industrial Complex and all that, but his personal tipping point is on a greater scale, given objective evidence of clearly better Soviet capability than OTL. And the tax burden on the USA is somewhat higher, which might not necessarily translate into a worse standard of living as a certain amount of Keynesian recycling worked well in the '50s and early '60s. The big taxpayers, the big corporations and their stockholders, were also likely to benefit directly from the pork. Perhaps the Devil's due bill comes in the form of a sooner or sharper slowdown into "stagflation" in the late '60s; perhaps it never comes due at all and there is just a proportionately bigger downturn from a proportionately bigger maximum economic performance.
So in that latter optimistic case, the money for extra frilly high-tech wankery a la Gerry Anderson's Thunderbirds comes from simply "making the pie higher" as George W Bush so masterfully put it. Or if one believes a price must be paid somewhere--I leave it up to the timeline author to tell us where the axe falls in his view. I'm willing to believe that there was a certain margin to be expanded in that generation and dynamics go much as OTL.
If someone wants to suggest that this timeline actually works out better economically, in the sense of totally avoiding stagflation eventually happening or something like that, I'll tsk at them. Just arguing that we could probably have inflated the 50s-60s booms some more without changing the basic dynamic strikes me as Santa Clausish enough for the season!
