Can Superman walk on the sun?
However, let me give you the answer then you can listen to see the work / proofs / logic.
MOS Kryptonian super powers are powered entirely by Earth's Yellow Sun (this includes sensory abilities). That is the only mandate on having their powers. Thus their powers can persist in a vacuum, underwater, in an oil rig fire (which isn't normal Earth atmosphere), or in all the environments in between Earth and space.
Jor-El describes this using the metaphor of "drinking" and like water, you aren't only hydrated while you are drinking the water. You can also store the drink in your body to keep you hydrated while not drinking. This means Superman can exhibit powers even when not in direct sunlight.
However, a more apt metaphor is of a laptop with a removable battery and a power plug. The laptop will run on either battery, or plug, or both... but not neither.
The Kryptonian Atmosphere acts to unplug (but not drain!) the battery. Additionally, space ships are shielded against radiation. Therefore, being exposed to Kryptonian Atmosphere disconnected his battery and being on the Black Zero unplugged his power cord. So Superman lost his strength. After the Kryptonian Atmosphere was replaced (either by Earth Atmospherics OR vacuum), Superman's battery was reconnected and thus his strength immediately returned. He was NOT empowered by Earth's Atmosphere (otherwise vacuum would have weakened him). As for what Zod and company were breathing on Earth, we can't be certain, but even if they were breathing Kryptonian Atmosphere, they would still have power in the form of direct sunlight or a power cord (irrespective of a disconnected battery).
This theory is fully consistent with the entire film and the Podcast reconciles every instance (including the helmets and sensory "weakness"; and Superman briefly losing flight while fighting the World Engine) except the Prequel comic, but also explains why that example can be excluded.
As for why, mechanically, Kryptonian Atmosphere weakens Superman, it's a combination of two general factors: 1) He has organs and they work; 2) It's a biochemical reaction.
We know Superman tends to follow biological rules for a number of reasons, not the least of all is how Zod is stopped. Additionally, there's a strong suggestion that even with all his powers, Clark still biochemically reacts with the world... he drinks beer while watching football, implying that it is reacting with his body so that he can taste it. It is unlikely that Kryptonian Atmosphere is radiological simply because of how much was spewed into the air and Lois's exposure to it.
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