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In order to save development costs, Mars2020 will basically be the same rover as MSL Curiosity, but with another set of instruments. NASA's next big Mars ambition seems to be sample return. But after that, are there plans to use a third copy of the same rover?

Could the same basic rover work on the Moon? Or do maybe temperature variations and lunar dust require another basic design? And could it be run in real time on the Moon? Is it the time lag to Mars, or its basic design, that makes it use only one batch of commands per day?

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But after that, are there plans to use a third copy of the same rover?

After that is way, way in the future. So any "plans" are more like notions. They wouldn't mean anything. It is not even clear what kind of rover would be used as part of a return of Martian samples after 2020. If that happens at all. The approval of such an effort is well into the future.

Could the same basic rover work on the Moon?

The fundamental issue is thermal control. The rover is designed to work in the Martian atmosphere, with a Martian diurnal cycle of close to an Earth day. No atmosphere and a 29-day diurnal cycle is very different. The thermal control approach would have to be completely reconsidered and redesigned from the ground up. Note that thermal control is not just about the electronics inside, but also all the actuators and other devices hanging out in the breeze, so to speak. Wheel actuators, arm actuators, instruments, etc.

The communications system would also need to be redesigned for more, higher-rate direct-to-Earth links. Or you could send a relay orbiter along with the thing.

It wouldn't be the same instruments. An entirely different set of instruments would be selected for a lunar investigation.

And could it be run in real time on the Moon? Is it the time lag to Mars, or its basic design, that makes it use only one batch of commands per day?

It is some combination of the time lag, the opportunities for communication through a relay satellite, and the pace of activities on the rover along with energy limitations that lead to a daily (solly) command cycle for Mars rovers. In principle joy-sticking a rover on the Moon should be possible, but it will still move slowly, and instrument activities will still take many hours or days. I'd expect some benefit to traverse speeds from more frequent interactions with a rover on the Moon. However you would be energy limited in how far you can go.

In short, no, an exact copy would neither work, nor be very scientifically appropriate.

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From what I gather, the main problems with putting MSL on the moon would be:

  • Lunar regolith is much coarser than Mars' dust.
  • There is no atmosphere so the EDL mechanism would have to be completely redesigned.
  • The rover is designed to perform science in the daylight, and moon nights are two weeks long.
  • No more available plutonium for an RTG to power the rover.

Some minor problems would be:

  • The heat-sensitive components would have to be adapted.
  • MSL is designed to communicate with a (relatively) low-altitude orbiter, and lunar orbits are unstable.

In conclusion, I believe that the project would work. However, the goals of the MSL are not compatible with the moon. We do not need to search for life or water on the moon. Therefore the science payload itself would probably have to be almost entirely scrapped and a moon-specific payload would have to be adapted. But the MSL as a platform for delivering a science payload seems feasible.

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