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An exactly Earth-sized tidally-locked planet orbits its host star, which is functionally identical to our Sun, at an average distance of 1 AU, the same distance that the Earth orbits the Sun. The atmosphere of this Earth-sized planet is identical to ours in every way.

Carbon dioxide freezes at -109 degrees Fahrenheit/-78 degrees Celsius/195 degrees Kelvin. Does the side of this planet that faces away from its star reach temperatures low enough to form carbon dioxide ice caps similar to those on Mars?

Note that, since this planet has an atmosphere, atmospheric convection will draw heat from the star-facing side to the opposite side; this might make it too warm there for the formation of dry ice.

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OK, let's do the math...

We have a temperature of -89.2 C for Antarctica = 184 K vs. -78.5 C for dry ice sublimation point = 194.7 K. (The first is fairly irrelevant because summer will come to Antarctica) Dry ice has a heat of sublimation at 180 K of 26 kJ/mol.

The big deal is the beautifully named "Clausius-Clapeyron" relationship. The temperature that will freeze a terrarium filled with CO2 gas is different from the temperature that will freeze a trace of CO2 at very low pressure. This is probably more easily calculated with a formula in the article linked, but it's best remembered and understood by considering a basic thermodynamic equilibrium (Boltzmann distribution). To escape from dry ice, CO2 molecules have to have 26 kJ/mol of energy, and their actual energy is RT = 1.987 cal/(K mol) * T = 8.314 J/(K mol) * T. If we can treat the gas as the whole volume of the system, we can say the pressure is proportional to c exp (-H/RT) = c exp ((-26000J/mol)/(8.314 J/(K mol) T) = c exp (-3127K / T). Now we know the pressure is 1 atm at T = 194.7 K, so we have 1 atm = c exp (-16.07) and c = 9.530E+6. That means -3127K/T for your circumstance is ln(0.000412/9.530E+6)= -23.86. [by the time you read this that 0.0412% figure someone gave above is probably gone with the wind) T = 3127/23.86 = 131.1 K = -142.1 C. That's still almost double the temperature of liquid nitrogen, say, so we're not talking drastic cold temperature on an astronomical scale. But we still have to get 53 C below the coldest temperature on Earth!

The good news is that the water will freeze out on the far side, and the atmosphere should get drier and less capable of carrying heat. The bad news is that all that accumulated water might raise the night side atmosphere atop a huge mass of ice, reducing the atmospheric pressure and with it ... the partial pressure of CO2.

All told, I'm getting to what the others said -- this is "unlikely" given the circumstances. Unlikely doesn't mean impossible - looking at a paper like this, simulating an Earthlike planet with a 150 K nightside seems doable by having a very low albedo on the nightside. The role of convection is also ... well, complicated. As in I-ought-to-read-that-paper-but-it-bites-back complicated. Is anything involving turbulence simple?

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  • $\begingroup$ Would a temperature of -142 degrees Celcius not be reached ? Back side of Mercury is -300F which is about -149 degrees Celcius. Guess dark side of Earth would be colder.. nasa.gov/audience/forstudents/5-8/features/nasa-knows/… $\endgroup$
    – Goodies
    Mar 6 at 22:52
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    $\begingroup$ Mercury doesn't have an atmosphere, but this planet has an Earth-like atmosphere by definition. I added two Arxiv links in there, but I'm not going to pretend to have really figured them out. $\endgroup$ Mar 6 at 23:09
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You want to consult Simulations of the Atmospheres of Synchronously Rotating Terrestrial Planets Orbiting M Dwarfs: Conditions for Atmospheric Collapse and the Implications for Habitability (and related papers by Joshi & Haberle).

Ultimately, the answer is probably not--but there is some wiggle room, because your hypothetical synchronously-rotating Earth rotates much more slowly than any of the planets actually simulated in studies of M-dwarf planet atmospheres. Precise circulation patterns depend quite a lot on rotation rate, and this slow-rotating world might, as a consequence, end up completely lacking upper atmosphere super-rotation, or having much more symmetrical ground-level prevailing wind flows, which might drastically change the atmosphere's heat transfer properties. But, given an atmospheric composition and pressure equal to Earth's, it seems most likely that you'll be fine, and the atmosphere will not collapse.

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Percentage of Carbon dioxide in atmosphere

Carbon dioxide in Mars atmosphere = 95% (9532000ppm)

Carbon dioxide in Earth atmosphere = 0.04% (412 ppm)

On your planet, the atmosphere is earth-like i.e.

pressure = 1 atm

Carbon dioxide in atmosphere = 0.04%

A small amount of dry ice may form

On earth, inspite of heat convection by air or water waves or ocean currents, the average minimum temperature of North Pole is −40° F (−40° C) and on South Pole is −76° F (−60° C).

As told by Robert Williscroft here,

Regarding dry ice at the poles, during my tenure at the South Pole, we registered the lowest temperature ever recorded there — -117.8° F. The temperature at which carbon dioxide will fall out of the air as “dry snow” is about -109° F, so we were able to measure a small amount of carbon dioxide snow on the surface of the 10,000 feet thick snow/ice pack at the pole when the temperature dropped below about -110° F.

It is possible that on your planet, the minimum temperature reaches below -109 degrees Fahrenheit/-78 degrees Celsius/195 degrees Kelvin and a small amount of dry ice may be seen.

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    $\begingroup$ The lowest temperatures recorded on Earth are not necessarily a good proxy for the lowest temperatures on an otherwise-identical planet with a permanent darkside and correspondingly less efficient heat distribution. $\endgroup$ Mar 6 at 14:37
  • $\begingroup$ @Logan R. Kearsley You are right. But still this lowest on earth is below freezing point of CO2 and we don't see dry ice on earth. $\endgroup$
    – imtaar
    Mar 6 at 14:44
  • $\begingroup$ So what? Your answer would be enormously improved if you could explain why you think a tidally-locked world wouldn't experience colder temperatures. As is, it does not seem to address the core feature of the question. $\endgroup$ Mar 6 at 19:18
  • $\begingroup$ Thanks @Logan R. Kearsley, I have improved the answer. $\endgroup$
    – imtaar
    Mar 7 at 8:04

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