This is just an educated guess, while Mars has a weaker atmosphere than ours which can cause trouble for sustaining life I do not believe this is the reason it does not trap oxygen. In fact Mars’ atmosphere is composed largely of Carbon dioxide this is a molecule which contains oxygen atoms which are also found in the Earths atmosphere. Mars however, unlike Earth’s 20% contains less than 0.5% O2, oxygen to oxygen molecules. On Earth the main thing that creates O2 molecules are plants, which are noticeably missing on Mars. We could in theory introduce oxygen gas to mars, but this is highly impractical. What could be better would be to take the CO2 already on Mars and split it (using plants or equipment) to turn it from CO2 to O2 in a chemical process. [Though you would also need that once created there is no process to remove O2 from the atmosphere, but I feel this is okay with Mars’ largely CO2 atmosphere].
The ability of a planet to hold onto it’s gases is to do with:
1. How much gravity there is
2. The molecular mass of the gas
3. The temperature of the gas near the top of it’s atmosphere
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The more gravity there is and the colder it is, the tighter the planet will hold onto its lighter gas molecules. Earth for example cannot hold onto its hydrogen or helium gas at all so it just leaves.
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Unfortunately, if you create Earth like conditions on Mars (so humans can live outside and wear t-shirts), the gravity on Mars isn’t enough to hold onto its oxygen molecules (or water molecules for that matter) for very long. So even if we take lots with us to make an atmosphere it’ll just float away eventually.
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There is a very thin atmosphere on Mars, but this is constantly being eroded because the molecules have enough thermal energy so that they can reach escape velocity and just say bye bye to Mars. It’s only the heavier molecules that will hang around.
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