# How fast is heat transferred by radiation in vacuum?

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.

• "Do not consider the sphere surface" not sure what that means. Are you saying we should assume the water magically stays in a sphere without some material holding it that way? – DanielSank Jul 16 '17 at 0:55
• Asking for other users to do a calculation for you is generally discouraged and considered off topic. Have you done any research into how to solve this problem? It's pretty easy if you look up some basics about radiation. Perhaps you could do that and then ask for help if you get stuck. – DanielSank Jul 16 '17 at 0:57
• You could adapt this to find the radiated power. The rate of cooling will not be linear so there will not be a definite time. – Keith McClary Jul 16 '17 at 0:58
• 3K is the temperature of outer space. The rate of cooling depends on the temperature difference between the sphere and 3K. This means that the sphere will approach 3K as a limiting temperature, but it does so "logarithmically", meaning that as time approaches infinity, the sphere approaches a 3K asymptote. For a more reasonable answer, choose a final sphere temperature that is somewhat higher than 3K. – David White Jul 16 '17 at 4:18
• @DanielSank: 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. – Llorenc Sole Jul 16 '17 at 7:28

As you said your question is based on science fiction movies, I guess we can simplify things by considering the following question: How long will it take for a person (80kg @ 37°C) to reach -20°C due to the energy loss by black body radiation?

• Why -20°C? Well, because water freezes at 0°C and we can assume that @ -20°C the body will be done: If you take a movie of a freezing body, I assume the face will show some indications of coolness @ -20°C.
• The physics can be describes by Stefan-Boltzmann's law, which states that the emitted power is given by $$P_{emitted} = A \sigma \epsilon T^4$$ Since the temperature of the universe is 3K, we can omit the absorbed power. Furthermore, for a first approx. you could just use $T=const=275K$ in the upper formula. If you like to get a better result, you need to integrate the formula and take the limits $-20°C \approx 250K$ and $+37° \approx 300K$. However,
• Now assume that the body is made up by water, so that the 80kg are pure water. Take the heat capacity of water and calc the total heat to cool the body from 37°C to 0°C.
• Next calc the total heat to freeze the body by using the latente heat capacity of water.
• Next, calc the total heat needed to freeze the body to -20°C by using the heat capacity of ice.
• Finally, note that $Power = \frac{energy}{time}$. So by dividing the "heat due to the heat capacities" by $P_{emitted}$ you get the time.