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If space is cold at 2.7 Kelvin and heat can't escape in a vacuum then why is the ISS wrapped in space blankets if the sun is hot?

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As noted in another question, the ISS faces some pretty hot temps. Remember, the Sun heats radiantly. When you're sitting in that much radiant heat, without an atmosphere to dissipate it, you're going to get hot really quick (emphasis mine)

Without thermal controls, the temperature of the orbiting Space Station's Sun-facing side would soar to 250 degrees F (121 C), while thermometers on the dark side would plunge to minus 250 degrees F (-157 C).

And

"In space there is no air for conduction or convection," he added. Space is a radiation-dominated environment. Objects heat up by absorbing sunlight and they cool off by emitting infrared energy, a form of radiation which is invisible to the human eye.

As a result, insulation for the International Space Station doesn't look like the fluffy mat of pink fibers you often find in Earth homes. The Station's insulation is instead a highly-reflective blanket called Multi-Layer Insulation (or MLI) made of Mylar and dacron.

"The Mylar is aluminized so that solar thermal radiation can't get through it," explains Hong.

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    $\begingroup$ In other words, the ISS needs space blankets because the sun is hot. $\endgroup$ – Jörg W Mittag Mar 30 '18 at 16:45
  • $\begingroup$ Would it be better to think of the mylar blanket as a flexible mirror to reflect the radiation? $\endgroup$ – RonJohn Mar 31 '18 at 4:55
  • $\begingroup$ @RonJohn That's literally what they describe it as $\endgroup$ – Machavity Mar 31 '18 at 17:51
  • $\begingroup$ Thanks. I didn't see that in your answer, so wasn't sure. $\endgroup$ – RonJohn Mar 31 '18 at 18:51
  • $\begingroup$ This also points out a problem that a lot of things in space have to take into account. One side can be hundreds of degrees, while the side facing away from the sun will be minus hundreds of degrees. That's a serious temperature gradient. $\endgroup$ – Wayne Werner Mar 31 '18 at 20:01
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Insulation can function in both ways, keeping heat on whichever side is desired. In space limiting the amount of thermal input from the sun is very valuable since that heat is easily acquired but hard to get rid of.

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'Heat can't escape in a vacuum' is false. While there's no air to absorb the surface heat and transport it elsewhere you're still losing heat through infrared radiation.
You're still losing significant amounts of heat while the ISS is in earth's shadow.

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    $\begingroup$ While true, ISS is concerned with shedding heat and reducing the gain of more heat, not with the retention of heat. $\endgroup$ – Saiboogu Mar 30 '18 at 19:12
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    $\begingroup$ They are concerned with both. Plenty of hardware has heaters on it too to keep things from getting too cold. $\endgroup$ – Tristan Mar 30 '18 at 20:45

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