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This article Mars atmosphere states that

Today Mars atmosphere is very thin, but it was not always like this. Around 3.5 billion years ago, the red planet had a thick enough atmosphere for liquid water to run smoothly on its surface

Suppose you could increase Mars atmosphere volume by melting frozen CO2 or whatever. How much volume of gases could be added to Mars atmosphere that the planet gravity can retain?

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    $\begingroup$ It's not just a question of gravity. As mentioned in en.wikipedia.org/wiki/Terraforming_of_Mars Mars does not have an intrinsic global magnetic field, so compared to Earth, its atmosphere is more vulnerable to the solar wind. $\endgroup$ – PM 2Ring Sep 16 '19 at 6:22
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    $\begingroup$ It's not really about the volume, it's about time. However much you add, it will be more or less all be lost over many millions or a few billion years, as happened before, $\endgroup$ – Steve Linton Sep 16 '19 at 7:03
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    $\begingroup$ @PM2Ring - The notion that a magnetic field protects a planet from atmosphere loss has become increasingly suspect over the last decade. For example, see the A2A question at the Earth science SE, Does the magnetic field really protect Earth from anything? $\endgroup$ – David Hammen Sep 16 '19 at 9:45
  • $\begingroup$ @David Interesting! Thanks for that info. $\endgroup$ – PM 2Ring Sep 16 '19 at 9:57
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    $\begingroup$ Paradoxically, the more gas you add (by mass), the more can be retained (or, at least, for longer time periods) due to the increased density at the solid surface. The problem is that, given enough time, even a solid rock will evaporate into a vacuum. $\endgroup$ – Carl Witthoft Sep 16 '19 at 18:22
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There are hard limits on what types of gas that Mars can retain based on its temperature and mass (Graph of what gasses an astronomical body can retain).

Volume wise, it's not clear. Currently Mars is still losing its atmosphere, so it can't even retain that amount of atmosphere. But if you continually added gas to Mars, there isn't an end point where the planet could no longer hold the atmosphere being added. It just increases the pressure of the atmosphere at the surface and only slightly extends the height of the atmosphere. And after it stopped being added, Mars would slowly lose the atmosphere again.

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Terraforming Mars by melting the frozen CO2 we know is stored at the poles, mainly the South Pole, is a much more difficult enterprise than some people imagine and I doubt it will ever be done.

If by miraculous means you could restore the Martian atmosphere to what it was 4 billion years ago, it might last a couple of billion years.

You don't stipulate any particular length of time you want it to last, and a billion years is beyond most people's comprehension. They talk earnestly about how humans could avoid destruction when the Sun expands to its red giant phase in about five billion years time, not realising that man will have gone to join the dinosaurs in far less than one billion, so two billon is an almost unimaginably long time.

But the fact of the matter is that Mars is a very hostile place, and we will have to put up with all the difficulty and inconvenience of its thin atmosphere and bitter temperature for as long as we are there.

To look on the bright side, things won't be so bad once we have established a large heated and pressurised base there, but to go outside the base you will always need a pressure suit and space helmet.

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    $\begingroup$ -1 Whithout supporting citations or links, there's no way to verify if these statements are factual or fiction. This isn't how SE works. See comments below your other heavily down-voted naswer for recommendataions. $\endgroup$ – uhoh Sep 16 '19 at 8:33
  • $\begingroup$ I've read a similar number, but this exactly the sort of thing that's impossible to find on internet. Similarly, a 1000 hectopascal luna would have an an atmospheric lifetime of some 100,000 years or so. The numbers themselves are likely lost in an old article by Jerry Pournelle, back in the 1980's. Pournelle usually did the math. $\endgroup$ – Wayfaring Stranger Dec 6 '19 at 16:33

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