Astronomical seeing is the limiting factor for the resolution of all but the smallest Earthbound telescopes.

Astronomical Seeing Source

Stunning advances in adaptive optics (along with it's predecessor speckle interferometry and it's budget-minded cousin lucky imaging) get around this but only with substantial compromises in (some combination of) throughput, cost, complexity and wavelength range (see answers to Why aren't ground-based observatories using adaptive optics for visible wavelengths?)

If identical telescopes sat on the surface of Earth and Mars and looked at a distant and equi-distant body, how would the following differ between the Mars telescope and the Earth telescope?:

  • Angular scale for the blurring, or $\mathbf{r_0}$
  • Time scale for the blurring, or $\mathbf{t_0}$
  • Aperture diameter where seeing starts to dominate over diffraction

caveats: The answer wouldn't have to use $\mathbf{r_0}$ and $\mathbf{t_0}$ if there is another way to address the angular and temporal differences between the astronomical seeing two observing sites. Both Mars and Earth have significant topography, but both have a "sea level" or reference 0 altitude, let's go with that.

edit: I've added a bounty, and if you would like to quantitatively compare to Maunakea or Atacama at 4.2 or 2.6 km to similar and similarly difficult-to-reach elevations on Mars, that's fine. But at an astounding 25 km, Olympus Mons on Mars has no remotely comparable site on Earth.

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    $\begingroup$ Lucky imaging is also mentioned in this recent answer and many others: astronomy.stackexchange.com/search?q=lucky+imaging $\endgroup$ – uhoh Oct 25 '18 at 23:45
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    $\begingroup$ FWIW, the atmospheric pressure on top of Olympus Mons is about 12% of the mean Martian atmospheric pressure; OTOH, Olympus Mons creates orographic clouds. $\endgroup$ – PM 2Ring Oct 26 '18 at 4:21
  • $\begingroup$ @PM2Ring here's some closer to home, for some of us at least ;-) earthscience.stackexchange.com/q/10478/6031 $\endgroup$ – uhoh Oct 26 '18 at 5:15
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    $\begingroup$ Sure, the atmospheric pressure at the Mars datum level is already much lower than at Earth's sea level, and it's much more homogeneous, due to the low water content, but why not put your telescope as high as you can? The only problem is those pesky orographic clouds. $\endgroup$ – PM 2Ring Oct 26 '18 at 14:09
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    $\begingroup$ The Moon is nearer and maybe better: no atmosphere and the far side is well shielded from Earth interference. $\endgroup$ – badjohn Jul 30 at 13:52

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