Why aren't there nuclear powered aircraft? I know this might sound like more of an engineering question that about physics, and it probably is, but bear with me: I'm still not sure if the answer to my question lies in the physics or in the engineering, or even in the politics.
Most if not all of modern submarines are powered by a nuclear reactor. At sea, only the most heavyweight (like air-carriers) use a nuclear reactor. But I've never heard of nuclear-powered aircraft.
Is there a weight-to-power density relation making it difficult or impossible to have nuclear aircraft? or is something related to perceived safety/politics?
 A: Nuclear power is the only way to make submarines work underneath the water for long distances without coming up because of the oxygen that all other types require. 
And batteries for electric engines are too heavy.
Historically a submarine had to go up to just below sea level to get air through the snorkel for the engines that charge the batteries. 
Then it could go down and drive only as far as the batteries could get and then repeat the process. But because the batteries are huge and heavy, this is only a short distance.
Under water the diesel engines do not work, they need the oxygen out of the air.
This was a huge disadvantage at war times.
Nuclear power does not need the fresh air (the oxygen part of it) like all other types of generators that work with combustion engines, which need oxygen. So someone built a nuclear submarine and this was so superior to the old ones that they chose to do it. They can travel and hide for months without coming up.
Some military ships are also nuclear powered:
http://www.radiationworks.com/nuclearships.htm
It allows them to travel longer distances without refueling. 
For civilian ships it is probably too expensive. 
For air planes it seems easier and safer and cheaper, compared to nuclear power, to just power them with kerosene. A reactor is heavy and has a certain size. Large ships can carry them. 
Submarines and ships have unlimited water cooling available. 
A: Needless to say, there is no idea too crazy not to have tried in the 
1950's, see for example  http://en.wikipedia.org/wiki/Project_Pluto 
and http://en.wikipedia.org/wiki/Aircraft_Nuclear_Propulsion .
A: This is a cost to benefit question and can only be answered by a guess in a physics board. There is a new generation of small compact reactors  that could be used for powering apartment buildings 

The new reactor, which is only 20 feet by 6 feet,

Seems compact enough, so it is not size but weight that is important, since this weight has to be lifted. Due to radiation concerns a lot of lead will be involved and I suspect that the advantage in energy costs of 

5 cents per kilowatt hour, about half the cost of grid energy.

will be lost. 
In addition there are the concerns about accidents and possible radiation released. Look at the aviation accident which released depleted uranium used as counterweights in some airplanes, I suppose in order to reduce volume. Think of the problems if it there were a reactor aboard. Submarines etc do not have the same risk factors.
A: To elaborate on Thomas's answer, there was serious research put into it, before the advent of good ballistic missiles: https://en.wikipedia.org/wiki/Aircraft_Nuclear_Propulsion
The idea was to build B-52 type planes that could just loiter in the air in an undisclosed location for a month, which would eliminate the risk of a Soviet first strike taking out the planes on the ground. Basically the same tactic as a ballistic missile submarine: Hide With Pride.
The basic principle of operation was the same as a jet engine: pull in air with a compressor fan, heat it, expel the now-much-higher-pressure air through a turbine which powers that compressor fan. But instead of heating air by burning jet fuel in it, they would pass the air through the core of a nuclear reactor.
It theoretically would have worked, but the project was killed because ICBMs mostly solved the same problem better. And also a crash would have smeared nuclear fission products across a wide area, which would have been expensive to clean up. It's a very interesting project though. To keep the vehicle light enough to fly, the radiation shielding was only between the crew compartment and the reactor, so the plan was to keep the plane in a radiation-shielded hangar when not flying. The plane never flew though.
