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Maybe I'm stupid, but if the moon can orbit Earth without fuel, why can't we put something human-made in a similar but different orbit?

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There are geo stationary orbits that don’t degrade. If you’re orbiting earth or moon they will always be tugging at it Edit: as mentioned above getting it to geo stationary would be costly. It’d also be in a place hard for missions to reach. Also meant Lagrange point not geostationary

You're not stupid. Atmospheric drag drops off rapidly as you go higher. The ISS would deorbit within a few years due to atmospheric drag at it's current altitude, but if you got up to ~1000km it would take thousands of years to decay.

That could work & would be substantially less crazy like the suggestions to put it to GEO and Lagrange points.

Still, unless actually maintained in some form, there will be a risk of the ex-ISS complex shedding stuff over time - either due to collisions or possibly material degradation, which could still be an issue for anything below or crossing the new higher orbit.

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