I've seen often in movies an astronaut exposed to vacuum (i.e. his helmet broken) and the depiction is him freezing nearly instantly. Considering that in vacuum radiation is the only way a body may transfer heat outside it may seem a bit surprising. I suppose this is what it happens but I wonder how fast this really occurs.
To put a simpler case, let's imagine a 1 liter sphere filled with water and uniformly heated at 323.15 K (50 C) left in outer space at 3 K. Do not consider the sphere surface. How many time will it take to cool down to 3 K?
Edit: This problem is well beyond my knowledge of physics and maths, I'm a bit old but still curious. I said not to consider surface just to make it simpler but if necessary then do it. My goal is to understand if radiation can cool down a body so fast as it is depicted un movies.
Edit: As per comment received let's consider a final temperature of 5 K so it doesn't take infinite time to reach it.