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Is there any particular deep area, trench, fissure, lava tube or otherwise on of Mars in which a person could survive with only an oxygen supply? How deep would plants or animals have to be in Mars not to be pressurized or insulated?

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To

Or how deep would one have to be in Mars not to need a pressurized suit?

and starting with @Rob's values and Planetery-Science.org's scale height of about 10.8 km to at least roughly ballpark an answer:

altitude (km)   pressure (kPa)
  -7.15             1.16
   0.               0.6
  25.               0.03

$$P(h) = P_0 \exp\left( -\frac{h-h_0}{h_{scale}} \right)$$

I get an altitude of -25 kilometers for the pressure to reach roughly the Armstrong Limit described in @Rob's excellent answer.

That doesn't mean that I would advocate doing so though!

edit: Based on @Uwe's comment I've extended the plot to -38 km altitude where the pressure reaches about 20 kPa, a slightly more people-friendly pressure than the absolute Armstrong limit.

note: I chose the two higher pressure points for the extrapolation, deviations from simple scale height behavior may be worse up there. Ideally one would estimate a temperature profile and use it to generate a temperature-dependent scale height as at least a step in the right direction. None the less, the answer will remain several tens of kilometers below the surface.

estimated pressure versus altitude on Mars

import numpy as np
import matplotlib.pyplot as plt

hscale    = 10.8  # km
kms, kPas = np.array([-7.15, 0.0, 25.  ]), np.array([ 1.16, 0.6,  0.03])
P0, h0    = kPas[0], kms[0]
alts      = np.arange(-38, 25.)
pressures = P0 * np.exp(-(alts - h0) / hscale)

if True:
    plt.figure()
    for i in range(2):
        plt.subplot(1, 2, i+1)
        plt.plot(kms, kPas, 'ok')
        plt.plot(alts, pressures)
        if i == 0:
            plt.yscale('log')
            plt.ylabel('pressure (kPa)', fontsize=16)
        else:
            plt.title('fitted region (linear)', fontsize=16)
            plt.xlim(-10, None)
            plt.ylim(None, 1.5)
        plt.xlabel('altitude (km)',  fontsize=16)
    plt.show()
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    $\begingroup$ What does the dots represent? $\endgroup$ – Muze the good Troll. Aug 7 '18 at 5:56
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    $\begingroup$ @Muze The three data points in the plot are the three data points in the table. The one at very high altitude doesn't fit as nicely because scale height is a simple approximation and doesn't work well for the outer atmosphere. $\endgroup$ – uhoh Aug 7 '18 at 5:57
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    $\begingroup$ Actually those values are from those links, subject to errors associated with those places. But hey, @GdD figured it would need to be "greater than greater than 10 miles" ... Luckily it didn't get it wronged. If they can't dig that far on Earth for gold they won't dig that far on Mars for "S03E01 061 Spock". -- Thanks for doing a Chart, I don't have an Equation Solver on my phone. Good job. $\endgroup$ – Rob Aug 7 '18 at 14:10
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    $\begingroup$ @Rob Indeed! That's one of my all time favorites. I remember even as a kid wondering what that "tick tick tick" sounds was. $\endgroup$ – uhoh Aug 7 '18 at 15:46
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    $\begingroup$ The question asked for survival when breathing oxygen without a suit, but survival at the Armstrong limit is a matter of some minutes only. The partial pressure of oxygen within a passenger airplane allows unlimited survival of healthy humans. That would require a pressure of about 0.16 bar or 16 kPa. Would you please extend your nice graph to a upper limit of 20 kPa? $\endgroup$ – Uwe Aug 22 '18 at 10:04
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Is there any particular deep areas of Mars in which a person could survive with only an oxygen supply without a pressurized suit?

No.

Hellas Planitia is the lowest point on Mars, the basin floor is about 7,152 m (23,465 ft) deep and the pressure is 1.16 kPa (0.168 psi). The average surface pressure of Mars is 0.6 kPa (0.087 psi). The highest point, Olympus Mons, has a height of nearly 25 km (13.6 mi or 72,000 ft) and a pressure of 0.03 kPa (0.0044 psi) - so you'd be digging over 10 miles - the deepest mine on Earth is 2.5 miles.

The Armstrong limit is 6.25 kPa (0.906 psi).

See Wikipedia's "Armstrong Limit":

"The Armstrong limit or Armstrong's line is a measure of altitude above which atmospheric pressure is sufficiently low that water boils at the normal temperature of the human body. Humans absolutely cannot survive above this limit in an unpressurized environment; above Earth, this begins 18-19 km (59,000-62,000 ft) above sea level. It is named after United States Air Force General Harry George Armstrong, who was the first to recognize this phenomenon.

At or above the Armstrong limit, exposed body fluids such as saliva, tears, urine, blood and the liquids wetting the alveoli within the lungs—but not vascular blood (blood within the circulatory system)—will boil away without a full-body pressure suit, and no amount of breathable oxygen delivered by any means will sustain life for more than a few minutes.".

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  • $\begingroup$ This is close @Rob, I'm not happy with it as an answer as people still cannot survive for long at the Armstrong limit as there's not enough oxygen. They'll die very quickly above it, and a bit less quickly below. Humans can survive on top of Everest - barely, and that's 30kPa. Your hole would need to be deeper. $\endgroup$ – GdD Aug 7 '18 at 7:49
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    $\begingroup$ @GdD: only healthy, trained and hight adapted humans may survive on top of Everest for a very limited time. A lot of climbers died there. $\endgroup$ – Uwe Aug 20 '18 at 20:09
  • $\begingroup$ Just to also mention, the deepest hole ever dug on earth is 7.5miles deep: Kola Superdeep Borehole in Russian 9 inches wide. Good luck digging deeper than than on Mars :P. Also, that Armstrong limit is horrifying... $\endgroup$ – Magic Octopus Urn Aug 20 '18 at 20:59
  • $\begingroup$ @GdD: There is no oxygen at all within the lungs at the Armstrong limit, the lungs are filled with pure water vapor. The vapor pressure of water at body temperature is equal to the ambient pressure. For survival an oxygen partial pressure in the lungs of about 0.16 bar is necessary. $\endgroup$ – Uwe Aug 21 '18 at 20:24
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    $\begingroup$ @GdD The question specifically says that it's asking only about pressure, and assumes the person has an oxygen supply. $\endgroup$ – Avi Cherry Oct 23 '18 at 0:30

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