[modeless]

The temperature that you raise to the fourth power is not Celsius, it's Kelvin. Otherwise things at -200 C would radiate more heat than things at 100 C. Also the temperature of space is ~3 K (cosmic microwave background), not 10 C.

[ithkuil]

There is a large region of the upper atmosphere called the thermosphere where there is still a little bit of air. The pressure is extremely low but the few molecules that are there are bombarded by intense radiation and thus reach pretty high temperatures, even 2000 C!

But since there are so few such molecules in any cubic meter, there isn't much energy in them. So if you put an object in such a rarefied atmosphere. It wouldn't get heated up by it despite such a gas formally having such a temperature.

The gas would be cooled down upon contact with the body and the body would be heated up by a negligible amount

[modeless]

These satellites will certainly be above the themosphere. The temperature of the sparse molecules in space is not relevant for cooling because there are too few of them. We're talking about radiative cooling here.

[ithkuil]

indeed. talking about temperature is incomplete without other aspects such as pressure

[terminalshort]

Yeah, if you forget about the giant fucking star nearby

[modeless]

The Sun is also not 10 C. Luckily you have solar arrays which shade your radiators from it, so you can ignore the direct light from it when calculating radiator efficiency. The actual concern in LEO is radiation from the Earth itself.