As I understand it, the average atmospheric pressure at the surface of Mars is close to the triple point of water. Is this a coincidence or does water have some sort of controlling effect on Mars' atmosphere?
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Mars' surface atmosphere ranges widely in both temperature and pressure. This range occasionally includes the triple point of water. I think it's fairly safe to say that currently the triple point of water is not having much of a direct, controlling effect on the atmosphere, except an influence on what small amount of water might be found in the atmosphere. However, it certainly may have a large influence on the atmosphere's behavior over billions of years, as it would have the surface water and ice.
A triple point is a point in a 2D plot phase plot with temperature and pressure as axes. Below is a phase plot for water which is from this amazing site about water. Note - this site is extensively hyperlinked. Click on a feature inside the phase plot (on the web page) for example, and the reference PDF will appear!
To the left of the triple point - where temperature is below exactly +0.01C, liquid can not exist very long. Water will tend to change directly between solid and gas.
To the right of the triple point, up to some high temperature - water can exist as a stable liquid over a range of pressures that depends on temperature. At higher temperatures water can become supercritical and outside the scope of this answer (whew, dodged that one).
I've added a rounded rectangle to indicate roughly the range of temperatures and pressures that may occur on the Martian surface. I used ballpark values of 150K to 293K as guidelines for atmospheric temperature extremes (looking at numbers from here, and I found a table with approximate pressures for extremes of elevations on Mars.
note: "Armstrong Limit" does not refer to a location on mars in that table, although I think it should, and some locations are actually on Earth, not Mars.
The triple point of water falls in the upper right corner of this area. So at very low altitudes, like at the bottom of Hellas Planitia, on a very hot day, liquid water could rest comfortably exposed to atmosphere until it starts cooling down in the afternoon.
above phase diagram of water from here, with annotations about Mars added.
above surface elevation of Mars from here (larger size available). The giant depression in deep blue centered around 40S, 70W is Hellas Planitia - the lowest area on the surface. The highest point is the summit of Olympus Mons, the farthest mountain on the left at 19N, 134W.
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2$\begingroup$ Note: The graphic spells "Hellas Planitia" incorrectly. "Hellas" means "Greek", and "Planitia" is "Plain". $\endgroup$– PhiterosCommented Jul 11, 2016 at 1:53
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$\begingroup$ A comment of @Uwe on a recent question from me, space.stackexchange.com/questions/39650/… , learned me that only the partial pressure of water vapour prevents water to evaporate, not the pressure of the atmosphere of Mars. So can the liquid water at the bottom of Hellas Planitia indeed rest comfortably ? $\endgroup$– CornelisCommented Oct 31, 2019 at 15:25
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2$\begingroup$ @Conelisinspace evaporation and boiling are two different processes. It's the total pressure that can keep water from boiling, and the partial pressure that can keep it from evaporating. If the pressure is too low, then bubbles of steam can appear anywhere inside the liquid. It's only total pressure that water molecules in the bulk can sense. But evaporation only happens at the surface, where the partial pressure of water vapor has an effect. $\endgroup$– uhohCommented Oct 31, 2019 at 15:58
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$\begingroup$ O.k., but depending on the timescale evaporation can be quick ! [Wiki] [en.wikipedia.org/wiki/… tells us that rate measurements range from 75-300 cm in a year . $\endgroup$– CornelisCommented Nov 1, 2019 at 9:31
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2$\begingroup$ From sol 150 to 200 the atmospheric pressure at Gale Crater was > 9 mbar. At that same period the max. temp. was just above 0$⁰$ C, so taking the vapour pressure of 6.5 mbar at 5$⁰$ C, even there liquid water could rest comfortably for a few hours ! $\endgroup$– CornelisCommented Nov 1, 2019 at 9:46