I am planning a series of science fiction novels that take place on an imaginary binary planet system. Both planets have a lower surface gravity than the Earth and one has slightly more mass than the other.
If average temperature is the same as earth or cooler and the radius of the larger planet is allowed to vary, what is the minimum amount of gravity required for the larger planet to sustain a 50:50 Nitrogen/Oxygen atmosphere with a total pressure of 100 mmHg at 2000 m above sea level?
What law or theorem would I need to solve this problem.
Extra stuff about the planets, my motivations and other things I'm trying to figure out that you don't have to read if you don't want to:
I want the larger planet to have an atmosphere that is thinner and whose pressure varies to a greater degree by altitude than the atmosphere of the Earth, but is still breathable by humans at sea level. Basically I want Denver to feel more like Mount Everest and Mount Everest should be pretty much vacuum. Mount Everest has an oxygen partial pressure of about 43 mmHg, and Denver is at an altitude of 1.6 km. Let's say we want 50 mmHg of oxygen at 2km. I figure if oxygen concentration is higher on the planet then on Earth, then atmosphere can be thinner and overall pressure can be lower. I picked a 50:50 mix not knowing whether it is really feasible, but if it is then pressure could be 100 mmHg at 2km. I don't know how to calculate how big the radius and mass of the planet has to be to satisfy these or what the pressure would be at sea level and at what altitude pressure will be 0.
The smaller planet should have an atmosphere that is too thin for humans but could conceivably host sentient life.
Both planets should have lower gravity than the earth. Each planets has a different species of sentient beings and I would like them to be able to lift stuff into space for much cheaper than here on earth. Basically by the time they have something like a Saturn V rocket, there should already be some trade in manufactured goods between the two.
Both planets have awesome magnetic fields that are way better than Earth's at doing the stuff that magnetic fields do. I also would like the binary planets to be in a binary star system and have one or two small moons each, but this is probably pushing it. Even one small ice moon would serve as a handy plot device but I don't want to make it too improbable and I do want to get as much of the Physics right as I can and figuring out the orbits, climate map and the seasons will be difficult enough as it is with two planets, especially considering how little I know.