Thank you for clarification of your position.
I think that you need not to move the bomb to stratospere. Smith in Doomsday men gave estimate that doomsday cobalt bomb should weight near the weight of lincor Missuri that is 70 000 tonn. So you could detonate it on spot - and the enegry of explosion will bring isotopes to upper atmosphere.
Also, if we go in technical details about global radiological contamonation, I think it would be better to use not only cobalt but other isotopes. Gold was discussed as another one. But the best could be some kind of heavy gas like radon because it is does not (as I think) solve in the see but tend to stay in lower atmosphere. It is not a fact but just my opinion about making nuclear doomsday divice more efective, and while I think this partilur opinion is wrong, some one who really wants to make such device could find the ways to make it much more effective , taking different isotopes as blanket of the bomb.
The FHI's mini advent calendar: counting down through the big five existential risks. The first one is an old favourite, forgotten but not gone: nuclear war.
Nuclear War
Current understanding: medium-high
Most worrying aspect: the missiles and bombs are already out there
It was a great fear during the fifties and sixties; but the weapons that could destroy our species lie dormant, not destroyed.
But nuclear weapons still remain the easiest method for our species to destroy itself. Recent modelling have confirmed the old idea of nuclear winter: soot rising from burning human cities destroyed by nuclear weapons could envelop the world in a dark cloud, disrupting agriculture and the food supplies, and causing mass starvation and death far beyond the areas directly hit. And a creeping proliferation has spread these weapons to smaller states in unstable areas of the world, increasing the probability that nuclear weapons could get used, leading to potential escalation. The risks are not new, and several times (the Cuban missile crisis, the Petrov incident) our species has been saved from annihilation by the slimmest of margins. And yet the risk seems to have slipped off the radar for many governments: emergency food and fuel reserves are diminishing, and we have few “refuges” designed to ensure that the human species could endure a major nuclear conflict.