Lets say there's a super earth orbiting the sun at 3.00 AU, how thick would its atmosphere have to be to trap enough heat to make the planet warm enough for liquid water to flow on its surface?
imo the question is overly-simplified. Whether surface water can be liquid depends on the following factors:
- Intensity of heat from the Sun
- Distance from the Sun
- Thickness of atmosphere and ability to absorb and retain heat from sunlight (a major factor, heavily influenced by the presence and composition of greenhouse gases). It is important to keeping the atmosphere warm enough both in the day and night.
- Surface air pressure, which influences the boiling and melting points, and is influenced by both the thickness and composition of the atmosphere.
The above factors are related, but there's no "master control". That said, assuming all circumstances are equivalent to our own Earth, and you increase the size of the Earth dramatically into that of a super-earth, then the following needs to happen in order to stay habitable:
- Since the Earth is 3 AU from the Sun, a super-earth at this same distance will be able to ensure liquid surface water if the same atmospheric temperatures at the surface can be achieved.
- A super earth needs to retain more heat from the sun to counter the heat emitted. Assuming the sun is the same, then the atmosphere must be better at heat retention. This means a more effective composition of greenhouse gases and/or a thicker atmosphere.
- Balance must be struck, however, as a significant change can also change the air pressure.