TimG wrote:You cannot count on radiative cooling of electronics in a hard vacuum.
(I'm assuming an emissivity close to 1. The real emissivity will be slightly lower.)
I don't think this is true (the basic assumption that you cannot count on radiation to cool electronics). While in this case I underestimated the Pis power consumption in my mental back of envelope, the math still shows radiation sufficient. A more realistic consumption figure is closer to 1W and a good SOC max target temp is 85C, since it will begin to take cooling measures at this point.
I think the issue with your approximation is probably that you are assuming an emission area that is only the SOC -- that it is floating. In reality, the inefficiencies that consume power are spread across the board, and hence the area is larger. More important, the surrounding PCB will conduct heat away and also radiate. You'll find that as you make successively larger areas radiate at elevated temperatures, you will very rapidly eclipse 1W at well below 85.
Now, if we're actually in vacuum, there is zero conduction away from the board, and the entire thing will be isothermal, except for minor differences in emissivity and the gradient across which the conduction takes place. Sure, the thermal conductivity of FR4 is bad, and you would need to take into account the traces, so it gets complicated. But at first approximation, the entire board is emitting at an elevated temperature, whose temperatures depend on the distribution of consumption and how it is conducted across it. Under these conditions (radiation from surroundings notwithstanding), you will find that the board doesn't have to go far from room temperature to emit 1W. Now to bring our Pi back from a black hole and assume some background radiation (from your vacuum chamber, the sky, whatever), it needs to be some degrees over the ambient temperature of the surroundings to effect any radiation. I find that if the whole board is 45C it will radiate about 1.5W. Obviously the SOC and other components will need to be hotter to accomplish this, but then they'll radiate more .. you get the picture. It's a complicated problem that would need a numerically integrated solution.
As for conduction, unless there is a path off the board, the heat has nowhere to go. A good first-order approximation is that it doesn't.