I was interested to see an article about nuclear bombs in the Daily Mail which mentioned the loss of a Mark 15 nuclear bomb by the USAF (Tybee Island B-47 crash). This was a hydrogen bomb which was intended to use a fission primary to trigger a fusion secondary. The neutrons from the fusion stage were intended to cause fission of high enriched uranium surrounding the secondary thus releasing the majority of the energy in this bomb design.
Now as the exact details of the bomb design are not in the public domain I will have to make some educated guesses. If we assume that the primary is a plutonium fission bomb then after having been at the bottom of the sea since 1958 then there are some things which will make a nuclear detonation less likely now.
If we assume that the bomb was of the same type as the plutonium bomb used in world war two, then the neutron trigger based on beryllium and polonium-210 will no longer be able to work. The bomb if triggers would not generate the neutron pulse at the right moment to power up the primary. Thus the bomb would not give a high fission yield, it might fizzle but it would never be able to detonate at full yield. With a defective primary I think that the secondary (fusion) stage will fail to go off.
If we were to be dealing with a fusion boosted primary stage then as 58 years have passed, as 4.7 half lives have passed then almost all the tritium in the bomb will have decayed. This would reduce the chance that the bomb would be able to function. Better still unless the bomb has been designed to make a helium / hydrogen separation in the final moments the helium-3 formed by the decay of the tritium will act as a neutron poison. This will have the opposite effect to the fusion boosting of the D-T reaction. So again this will make the bomb less able to function.
Also if we were dealing with a bomb which used an electrostatic D-T neutron generator tube, then the reduction in the amount of tritium in the tube would reduce its ability to generate neutrons. A bomb using this method would be a very advanced bomb design. I am not sure if a 1950s designer would have been able to use this option. My calculations suggest that about 4 % of the tritium would be left. This could have a large effect on the ability of the D-T fusion tube to generate a neutron pulse.
I would also like to point out that to cause a implosion a very well timed electrical command has to be delivered to the detonators. This is something which would be hard to do by chance, also in the decades since the bomb was lost the chemical explosives will have aged, this may alter their properties and thus reduce the chance of the implosion crushing the plutonium pit to take the fuel into a super prompt critical state.
I also reason that if tritium was added to the secondary fuel to help start up the secondary then the decay of the tritium which was present would also help render the bomb less dangerous.
Another reason why the bomb might fail to detonate with a nuclear detonation would be if the plutonium pit was missing or defective. According to the pilot the bomb he signed for the bomb did not have a vital plutonium part inside it. This lack of plutonium would be able to prevent the bomb operating.
While without seeing the bomb design and knowing the exact state of it right now, I think that the bomb is not very likely to explode without some form of human intervention. For the Daily Mail to print an article claiming that this bomb could create a vast tidal wave which could wipe out Savannah is in my view an unreasonable claim unless the Daily Mail can argue a good case that this bomb is still viable.
I think that the Daily Mail may well be publishing an alarmist article.
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