It would be far less than 38 days until the surface of the planet is frozen; likely the span of just a few days, and once the water precipitates eliminating any cloud layer and attendant greenhouse effects, the oxygen and nitrogen in the atmosphere will chill and freeze quickly. The oceans would certainly freeze over and remain liquid in the benthic regions for a long period of time, but without substantial tectonic activity, they too would freeze eventually except right around seismic zones and hotspots. Thermochemical life may still exist in such areas, but with limited resources and ability to sustain habitable conditions away from vents it probably won’t evolve into much.
Previously threads on the topic:
[THREAD=611731]How long could humanity survive if Earth were flung out of its orbit?[/THREAD]
[THREAD=411531]What Would Happen If The Sun Burned Out?[/THREAD]
[THREAD=427700]How long do humans survive if sun stops producing light?[/THREAD]
[THREAD=394257]The Sun turns off! How long do we have?[/THREAD]
[THREAD=544823]Poof goes the sun. Can humans survive?[/THREAD]
Since the basis for the calculations isn’t really discussed in the article I can only guess at the assumptions and models used, but I’ve run my own back-of-envelope calculations (as well as having developed heat transfer models for space habitats of various sizes) and I think 38 days to a fully frozen surface is wildly optimistic. While it is true that we don’t freeze overnight while the Sun is blocked, the flow of radiant enegy to the Sunward side of the Earth keeps the climate in net equilibrium, and flow of moist air from the oceans provides sustaining warmth during the nocturnal portion of the cycle. Without regular diurnal warming, the atmosphere will continue to cool, and as it does so the moisture in the air that provides most of the heat capacity will precipitate out, resulting in rapid heat loss and increase emissivity of the Earth as a whole. Hell, just during a full solar eclipse you can sense a temperature drop of around 5 °C over the span of just a few minutes, and that is just local cooling.
Trivial nitpick - because the Earth and the Moon are non-uniform and quite close to each other, there could be observable differences between the effect of the Moon on the Earth and the effect of a Moon-massed black hole in the Moon’s orbit on the Earth.
I tried to look up what the vapor pressure of dry ice is at different temperatures and relate that to the partial pressure of CO2 in the atmosphere but got bogged down on terminology and units. But my point is that the standard freezing/sublimation point of CO2 is the temperature at which its vapor pressure is equal to standard atmospheric pressure, which would have to be much greater than the partial pressure of carbon dioxide in the Earth’s atmosphere. In fact, I’m not sure you could condense CO2 from the air before the air would liquify anyway.