Probably been said before, but I can't help but write this response:
There are basically two totally different types of submarine: ballistic missile submarines, and hunter/killer submarines. Note that propulsion is totally arbitrary: a "nuclear" submarine normally means a submarine carrying a nuclear reactor as its main propulsion system, not a submarine armed with nuclear weapons.
Ballistic missile subs carry around 20 SLBMs - that's "Submarine Launched Ballistic Missiles", which in this time and age are meant to be used for waging unlimited nuclear war. The missiles mostly carry multiple, really big warheads. These are strategic weapons, used in deterrence, MAD and disarmament negotiations. The only thing these boats do is go out on patrol, cruise around in a big box of ocean and keep well away from everybody else in that box. If Armageddon isn't scheduled for this month, they go back to port, change crews, and go back out.
I choose to call all other submarines "hunter/killer", which is kind of an over-simplification. Bear with me. Maybe calling them "tactical" would be a better choice, since that is what mostly distinguishes them from the ballistic missile boats. These boats come in all sizes and kinds, nuclear-propelled and conventional, and all kinds of armament. Their missions can be diverse: surveillance, putting commando soldiers on a beach somewhere, escorting a surface battle group, denying sea room to somebody else, hunting merchant shipping, hunting surface warships, hunting submarines... Many of these missions are interrelated: if your sub escorts an aircraft carrier, it will most definitely keep an ear cocked for enemy submarines that are bent on killing the carrier, and do a bit of sub hunting themselves.
Looking at these missions, it's not hard to imagine that a lot of possible design solutions exist to fulfill them. The U.S. Navy, and indeed most western navies, seem to go the way of the multi-purpose boat: build a submarine that can handle submarine hunting - possibly the most demanding mission - and you get a submarine that should do all other things reasonably well, too.
Now for the Russians. Think back a dozen years or so, and remember that in those cold war days, U.S. Navy carrier battle groups roamed the high seas pretty much unchallenged by anybody else. Especially the Soviets (the Russians still were Soviets back then). And from what I remember, the Soviets had a healthy fear of those carriers, in the case of a larger conflict between the two power blocs, driving up the coast of Norway, right into the Barents Sea, back yard of the Soviets, and start pummeling the submarine and air bases located there with air strikes. Since the Soviet Navy didn't have proper carriers with which to go after the U.S. ones, they had to find something else. A relatively cheap (compared to carriers of their own) way was with submarines - a sub and its upkeep costs only fractions of a carrier battle group. Unfortunately, one main purpose of the "group" part of the carrier battle group was to keep enemy submarines from launching torpedoes at the carrier... so again, a solution was sought and found: The Soviet Navy built a number of classes of submarine dubbed, in the western naval lingo, SSG - "guided missile submarine". This indicates that these submarine's main mission was to shoot guided missiles - mostly of the cruise missile kind - at enemy ships. These missiles were built with both nuclear and conventional warheads; this is not surprising, since the Russians put both kinds of warhead on pretty much everything they built.
Back to the subject at hand: The Kursk was just such a boat - an SSGN of the Oscar II class, launched in 1994 and therefore one of the newer units in the Russian Navy. It carries as its main armament 24 SS-N-19 missiles, with a range of around 500km and either a 750 kg HE or 500 kT nuclear warhead (source: Jane's Naval Forces). These missiles, despite wearing a nuclear warhead, are classed as tactical weapons.
So the Kursk might have been nuclear -armed, but since agreements have been reached about removing tactical nuclear weapons from ships, and it is more complicated and more expensive to carry nuclear weapons, I find it highly unlikely that she was.
Having said this much, I notice that it really doesn't matter that much - unless some Baltic terrorists drive up there in a Zodiac and try to get their hands on a warhead... The only real problem in environmental terms will probably be the reactor and propulsion system - if any pipes are damaged, or corrode to an extent that they start to leak, evil things may happen. It would, unfortunately, not be the first nuclear reactor ending its life on the seafloor.
The Kursk is being called "the best ship in the fleet" and all kinds of other names after the fact that she sunk. I would regard that as a bit of hero talk for the rest of the world. She wasn't the newest boat, neither in her class nor absolutely.
The "flagships" don't always get the "cool new stuff" "to play with". Often, new stuff is put on older units which aren't that useful in their primary role anymore, tested and tweaked, and then put on the next class of new units to be built - but in a finished state.
The USS Enterprise (assuming the aircraft carrier CVN-65 is meant here) isn't just the first in its class, but the only one! She was the first nuclear-powered carrier, and there are others newer than her, but they aren't considered to be in her class. As for the "flagship": she's probably a flagship for the admiral commanding the battle group, and maybe for a fleet, but I've never heard Enterprise being referred to as anything more than that in this context.
The reports of other subs in the area by the Russian Navy are, in my opinion, more talk than based on proper facts or pieces of information in the Russian Navy. They could be pretty sure that there would be foreign boats there to monitor their exercise - these things aren't that secret - so saying "They hit us! They hit us!" comes in pretty handy in case of an accident. Not that there never were accidents - go read "Blind Man's Bluff" for some exciting stories.
It is most likely correct that the foreign boats monitoring the exercise got the best picture of what happend right away, and that they also concluded that the Kursk suffered a catastrophic accident. The fact that no owner of those boats stepped forward to say anything isn't surprising, since it would have meant to disclose information about their own capabilities.
I have never heard of the U.S. government being involved in the rescue effort. The rescue submarine responsible for this piece of the globe is the British LR5, which was brought to the scene of the accident as "soon as possible" - this time frame being determined by Russian authorities.
This probably isn't specific to submariners, but to soldiers and sailors, and groups of people in similar situations, everywhere.
Probably been said before, but I can't help but write this response:
Ballistic missile subs carry around 20 SLBMs - that's "Submarine Launched Ballistic Missiles", which in this time and age are meant to be used for waging unlimited nuclear war. The missiles mostly carry multiple, really big warheads. These are strategic weapons, used in deterrence, MAD and disarmament negotiations. The only thing these boats do is go out on patrol, cruise around in a big box of ocean and keep well away from everybody else in that box. If Armageddon isn't scheduled for this month, they go back to port, change crews, and go back out.
I choose to call all other submarines "hunter/killer", which is kind of an over-simplification. Bear with me. Maybe calling them "tactical" would be a better choice, since that is what mostly distinguishes them from the ballistic missile boats. These boats come in all sizes and kinds, nuclear-propelled and conventional, and all kinds of armament. Their missions can be diverse: surveillance, putting commando soldiers on a beach somewhere, escorting a surface battle group, denying sea room to somebody else, hunting merchant shipping, hunting surface warships, hunting submarines... Many of these missions are interrelated: if your sub escorts an aircraft carrier, it will most definitely keep an ear cocked for enemy submarines that are bent on killing the carrier, and do a bit of sub hunting themselves.
Looking at these missions, it's not hard to imagine that a lot of possible design solutions exist to fulfill them. The U.S. Navy, and indeed most western navies, seem to go the way of the multi-purpose boat: build a submarine that can handle submarine hunting - possibly the most demanding mission - and you get a submarine that should do all other things reasonably well, too.
Now for the Russians. Think back a dozen years or so, and remember that in those cold war days, U.S. Navy carrier battle groups roamed the high seas pretty much unchallenged by anybody else. Especially the Soviets (the Russians still were Soviets back then). And from what I remember, the Soviets had a healthy fear of those carriers, in the case of a larger conflict between the two power blocs, driving up the coast of Norway, right into the Barents Sea, back yard of the Soviets, and start pummeling the submarine and air bases located there with air strikes. Since the Soviet Navy didn't have proper carriers with which to go after the U.S. ones, they had to find something else. A relatively cheap (compared to carriers of their own) way was with submarines - a sub and its upkeep costs only fractions of a carrier battle group. Unfortunately, one main purpose of the "group" part of the carrier battle group was to keep enemy submarines from launching torpedoes at the carrier... so again, a solution was sought and found: The Soviet Navy built a number of classes of submarine dubbed, in the western naval lingo, SSG - "guided missile submarine". This indicates that these submarine's main mission was to shoot guided missiles - mostly of the cruise missile kind - at enemy ships. These missiles were built with both nuclear and conventional warheads; this is not surprising, since the Russians put both kinds of warhead on pretty much everything they built.
Back to the subject at hand: The Kursk was just such a boat - an SSGN of the Oscar II class, launched in 1994 and therefore one of the newer units in the Russian Navy. It carries as its main armament 24 SS-N-19 missiles, with a range of around 500km and either a 750 kg HE or 500 kT nuclear warhead (source: Jane's Naval Forces). These missiles, despite wearing a nuclear warhead, are classed as tactical weapons.
So the Kursk might have been nuclear -armed, but since agreements have been reached about removing tactical nuclear weapons from ships, and it is more complicated and more expensive to carry nuclear weapons, I find it highly unlikely that she was.
Having said this much, I notice that it really doesn't matter that much - unless some Baltic terrorists drive up there in a Zodiac and try to get their hands on a warhead... The only real problem in environmental terms will probably be the reactor and propulsion system - if any pipes are damaged, or corrode to an extent that they start to leak, evil things may happen. It would, unfortunately, not be the first nuclear reactor ending its life on the seafloor.
The "flagships" don't always get the "cool new stuff" "to play with". Often, new stuff is put on older units which aren't that useful in their primary role anymore, tested and tweaked, and then put on the next class of new units to be built - but in a finished state.
The USS Enterprise (assuming the aircraft carrier CVN-65 is meant here) isn't just the first in its class, but the only one! She was the first nuclear-powered carrier, and there are others newer than her, but they aren't considered to be in her class. As for the "flagship": she's probably a flagship for the admiral commanding the battle group, and maybe for a fleet, but I've never heard Enterprise being referred to as anything more than that in this context.
I have never heard of the U.S. government being involved in the rescue effort. The rescue submarine responsible for this piece of the globe is the British LR5, which was brought to the scene of the accident as "soon as possible" - this time frame being determined by Russian authorities.