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Phase change of water at different pressures and temperatures tell us that with no atmospheric pressure 200 Kelvin (minus 73 Celcius) is enough to turn ice into vapor directly

Wouldn't that suggest that solar insolation at the distance of earth to the sun heating potential is enough to kickstart the hypothetical ice planet to a waterworld with an atmosphere? Even when averaging day/night?

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The answer is no.

Explanation:

No atmosphere is equivalent to all molecules being able to evade to space (This happens on planets with low gravitational force). Thus also $H_2O$ molecules will evade and I guess that many molecules already evaded during the frozen state of the planet due to sublimation. So this could be the end of this answer.

But let's suppose for a moment, even if it is not realistic, that an atmosphere could form. Top of atmosphere power received from the sun is around $S_0 = 1368 \, \frac{\text{W}}{\text{m}^2} $ ($S_0$ being the solar constant). Assuming earth is a blackbody we can equate power received from the sun and power radiated from the hypothetical earth (see also Stefan-Boltzmann law).

$S_0(1-\alpha) \pi r^2 = \sigma T^4 4 \pi r^2$

$\alpha = 0.6$ is the albedo, $\sigma = 5.67 \cdot 10^{-8}$ is the Stefan-Boltzmann constant, $\pi = 3.1415$, $T$ is earths surface temperature and $r = 6371 \, \text{km}$ is earths radius. Simplifying and rearranging yields

$T = \left(\frac{S_0(1-\alpha)}{4\sigma}\right)^{\frac{1}{4}} \approx 222 \, \text{K} = -51.15°\text{C}$.

Taking a look at the phase diagram you provided, we see that at $222 \, \text{K}$ and low pressure water naturally occurs as gas. Given the assumption that an atmosphere could indeed form (for some magical reason someone increases gravtiy) the answer is yes an atmosphere could form.

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