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<blockquote data-quote="Agemegos" data-source="post: 2810793" data-attributes="member: 18377">Oops. Escape velocity is actually proportional to the square root of the product of surface gravity and radius. Surface gravity is proportional to density times radius. Which is to say that escape velocity is proportional to radius times the square root of density. A habitable planet or moon will be about as dense as Earth or a little less.The velocity of a molecule in a gas is proportional to the square root of its molecular weight. Water has a molecular weight at little over four times that of helium. So the fact that helium has almost completely escaped from Earth and that water has not establishes that a planet or moon requires an escape velocity at least about half that of Earth. That means a radius about half that of Earth (or more, if density is lower). And that means a surface gravity at least about half that of Earth.So I ended up at the right place because my two errors cancelled out. For a naturally habitable world in a mature solar system there is a lower bound on surface gravity somewhere above half a gee. But if an ocean's worth of water is delivered to a planet or moon with lighter gravity if can hang on to it for a long time by human standards.</blockquote>

[QUOTE="Agemegos, post: 2810793, member: 18377"] Oops. Escape velocity is actually proportional to the [i]square root[/i] of the product of surface gravity and radius. Surface gravity is proportional to density times radius. Which is to say that escape velocity is proportional to radius times the square root of density. A habitable planet or moon will be about as dense as Earth or a little less. The velocity of a molecule in a gas is proportional to the square root of its molecular weight. Water has a molecular weight at little over four times that of helium. So the fact that helium has almost completely escaped from Earth and that water has not establishes that a planet or moon requires an escape velocity at least about half that of Earth. That means a radius about half that of Earth (or more, if density is lower). And that means a surface gravity at least about half that of Earth. So I ended up at the right place because my two errors cancelled out. For a naturally habitable world in a mature solar system there is a lower bound on surface gravity somewhere above half a gee. But if an ocean's worth of water is delivered to a planet or moon with lighter gravity if can hang on to it for a long time by human standards. [/QUOTE]

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