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Supposing a roughly mars sized planet, how little surface water would need to exist to sustain a population of a size similar to pre-industrial earth (say 750 million?) The situation is somewhat unique in the sense that I'm hoping for a very significant portion of the planet to arid, Tatooine style. Is that actually feasible? Note, not all water needs to be on the surface. The population has the means to collect water from underground and atmospheric sources. But I imagine some significant body of water would need to exist.

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An arid planet without surface water would likely have very little plant and animal life. That in itself is no problem, but it makes the ecological balance extremely fragile.

Humans themselves could do fine on a very arid world without any surface water. Check this anecdote for the effects of working in very low humidity (<1%). With their water sources and by using specially designed clothing (like Dune Stillsuits) to conserve most water, it would not be too hard to maintain a steady supply.

Food crops would have to be grown in greenhouses, either special strains of the local flora, bred over thousands of years to grow faster than the native desert plants or imported species if there is contact with other worlds). I imagine that if underground water sources are not a problem, greenhouses wouldn't be either.

Trouble would rear its ugly head if the humans ever reached the industrial age and started burning fossil fuels. (There could be some if the planet used to have more water and plant life in its early life). The carbon dioxide emitted would not be absorbed by plants or dissolved in ocean water, leading to global warming and eventually a toxic atmosphere. Growing irrigated crops in the open would quickly drain underground water reserves, and it is doubtful that reclaiming water from the atmosphere would be productive enough to maintain the outdoor crops.

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