The theory of a GHE in a planetary atmosphere is rather well known. Many claim it is no longer a theory but actually proven as a fact.
However I have been reading a lot of things Richard Feynman said about science and have decided to apply some of them to the GHE. I am going to start from an assumption this is something new I have just come across and after listening to the explanations ask: "How do we know it is like that, what can we measure or test to confirm it?"
Now Richard Feynman had a very useful piece of lab equipment which he used quite a lot. I've got one of them too, albeit a version with a lower spec and with a rusty outer casing that shows its age. However I have tested it recently and it is up to spec and passes its boot up self-test. You might have a higher spec one.
So please apply your brain to this:
Planetary Atmosphere Thought Experiment.
You have 2 identical planets, same mass, same rotational speed, same orbit around a sun. They have gaseous atmospheres of identical heat capacity and of the same height. The orbits are perfectly circular and regular. The planets have reached a thermal equilibrium. The ONLY difference between the 2 planets is that one has greenhouse gases in its atmosphere and the other does not.
What differences would the GHE make to the conditions on the 2 planets?
Well fairly obviously if there is a GHE on one its surface temperature will be higher. We will call the planet with the GHE planet G, and the one without planet N.
G will have a higher surface temperature. Now suppose we are looking from outside, from space, at G and N. Will they look different to any measuring apparatus? We know the solar input to each is the same (orbits are the same) so, as we know G and N are in thermal equilibrium it MUST be the case that quantity of radiation leaving the planets is the same. So if we measure it like we do for any celestial body we will measure the same "effective temperature" looking at G and N.
G and N will have the same effective temperature. The surface of planet G will be hotter than N. The height of the atmospheres is the same so the temperature gradient in the atmosphere will be higher on G. Its lapse rate is higher. For any given height in the atmosphere the temperature of the atmosphere on G is higher at that height, than the temperature on N.
So G has a higher surface temperature and the temperature of its atmosphere is higher. Yet, as both planets are radiating into space at the same rate their measured effective temperatures are identical.
This then leads to my main question; What physical parameter can we measure if given ONE planet, without any knowledge of what the conditions are on the other, in order to determine whether we have been given planet G or N?
EDIT 22/4/17. I am adding this edit as the answers attempted so far have all really been an explanation of the details of how the GHE works on Earth and the specifics of CO2. I would like the question properly addressed as it is in connection with the thought experiment proposed. The answer I am looking for will likely be given in general terms, applicable to the GHE from any greenhouse gas or a combination of 2 or more and also be one generally applicable to any planet you might find anywhere. Multiple parameters may be measured. You do have access to the planet's surface or any chosen point in the atmosphere. To clarify the gaseous atmospher may be any combination of gases or a single gas, the specific heat capacity of the atmosphere is the same on each planet, the height of the atmospheres is defined as the height of the troposphere so the tropospheres are the same height. Layers of atmosphere above the troposphere can vary between the 2 planets as I believe there is no claim or evidence that the presence or absence of a greenhouse gas in these upper layers is significant (if you have knowledge otherwise please state it).
EDIT 26/4/17 . This edit follows the question being put "on hold" as being too broad. The actual question posed is quite specific and narrow and is also brief and to the point. It is the "What physical parameter(s) can we measure ?" type of question. That is not broad at all, it is specific and it follows the format of what Richard Feynman suggests. The answer is expected to be in the form of "We would measure X" or "We would measure X and Y" or "We would measure X, Y and Z". Obviously the answer could be expanded via something like "We would measure X and if it is below n then that shows...". So based on a simple and specific questions, and the expected simple answers, the claim of those who put this on hold for being too broad looks erroneous and I suggest they revise their view as it does not accord with a logical examination of the situation as I have explained.