The energy coming from the Sun looks approximately like this:
The energy of the Sun becomes less intense with increased distance. The Earth is far away from the Sun.
The Earth's atmosphere is nearly transparent to visible light, but is opaque to light with higher and lower frequencies.
The spectrum of light hitting the Earth:
When this light hits the Earth some of it is absorbed, warming the atmosphere and surface. A warm object radiates energy. The Earth radiates a spectrum similar to the Sun, but since it is much cooler than the Sun the peak of the spectrum is at much lower frequencies than that of the Sun.
This lower frequency energy can't escape as easily through the atmosphere as the visible light peak of the Sun can because most of the energy is at frequency where the atmosphere is opaque.
As a result the Earth will continue to heat up.
To simplify things, I'm going to consider only the energy escaping through the transparent region, and not the energy absorbed and radiated out to space by the atmosphere. In this simplification, we can imagine that we are building a 'perfect' greenhouse that will maximize the greenhouse effect. In our perfect greenhouse we are using a material with a transparent region like the atmosphere, but one which is totally reflecting at all other frequencies. In this case our perfect greenhouse will heat up until all of the energy can escape from the transparent frequency region of our, otherwise, reflecting material. We imagine we might get something like this where the Earth has to get very hot in order for the incoming and outgoing flows to balance.
However the above image is not correct. The above diagram could only be correct if energies did not constantly redistribute to flow into the visible region. If energy could not easily redistribute, then when light leaves the transparent region, but can't leave the other areas, it would leave a hole in the spectrum.
In reality no hole will form, as energy is constantly redistributing itself through photon/molecule interactions in the environment. Initially light frequencies might be in bands and not have this spectrum shape at all. The shape of the spectrum is the most probable shape when examining the random distributions of energy at a quantum level. The shape of the spectrum is statistical in nature.
The final spectrum intensity will be much lower as the energy constantly flows into the transparent visible region. Our greenhouse is a sphere that has a 'hole' in the frequency spectrum that allows light to enter and be absorbed and which, to some extent, 'traps' this light. Similarly we could have a sphere with a small physical hole in it to let in light and with which to 'trap' it. The two are equivalent. The second case of the sphere with a small physical hole is often used in Physics explanations of the concept of a blackbody. A blackbody is an object that absorbs all the light that hits it. Even though this blackbody absorbs all the light that hits it, it does not keep all this energy, but will heat up and radiate energy outward again until the outward energy balances the absorbed energy. The temperature of a blackbody can be calculated using a formula known as the Stephan-Boltzmann equation:
S=σT⁴ ,
where S is the incoming flux, σ is the Stephan-Boltzmann constant, and T is the internal temperature of the sphere in Kelvin.
The energy flux (S) from the Sun is reported to be around 1366 W/m². If we plug this into the Stephan-Boltzmann equation we get an internal temperature for our perfect greenhouse of 394°K (or about 120°C). This is the maximum temperature the greenhouse effect can achieve.
Now we return to the Earth and its atmosphere, which is not a mirror. Instead for a more realistic Earth maximum greenhouse effect temperature we assume that there instead of mirror surrounding the Earth we have a perfect absorber (except at the window frequency as before). The Earth will absorb sunlight over its cross-section, which is a circle, but will radiate from the area of a sphere. The ratio of the the radiating area to the absorbing area is 4. This means that there is 4 times as much area radiating energy as there is 'equivalent' area receiving energy from the Sun. Our absorbing layer will radiate 1/4 of the Sun incoming flux, or 341.5 W/m². The absorbing layer will also radiate this same flux towards the Earth's surface. The total flux at the Earth's surface would then be double this, or 683 W/m². Plugging this value into the Stephan-Boltzmann equation we get a black body temperature of 331°K (58°C). This is a much more realistic theoretical maximum greenhouse temperature for the Earth and is pretty close to the maximum temperatures occasionally achieved at the equator.
There are many factors that will lower this maximum. Some of the energy from the sun in the transparent region is reflected by the surface and will not heat the Earth. The flux is not distributed evenly but is concentrated at the equator. This uneven distribution of flux will also lower the average temperature. There is also the extreme complexity of heat conduction and convection in the Earth's atmosphere and ocean's which our simple model ignores. Once this extra factors are taken into account, I hypothesize that the Earth is at or near its maximum temperature. There is strong evidence for this from the temperature history and general behavior during and in-between ice-ages
The Earth is not always nice and warm, but spends most of its time being rather cold with ice covering a significant fraction of the surface. Below is a graph of estimated temperatures from Vostok, Antarctica over the last 450,000 years, reconstructed using proxy temperature data from ice cores:
To me, this graph is peculiar. Why? Well the fact the temperatures coming out of an ice age rise at a linear rate and then suddenly stop. Here is what I see when I look at this data:
Why do temperatures stop rising when coming out of an ice age? Why do they stop rising so abruptly? Why do they rise to approximately the same stopping point temperature each time? This is all easily accounted for with a model that assumes the Earth has a maximum temperature (a saturation temperature). The abrupt stopping, the stopping at the same approximate value, are both exactly what we would expect if the temperature reaches a maximum. In fact, it would be very difficult to come up with a system that shows this behaviour without saturation. Systems not driven to saturation do not typically behave like this. In order for the system to stop, there must be something stopping it, a negative feedback. Systems with negative feedback do not rise linearly and then stop. They curve, they overshoot and they oscillate.
When negative feedback is introduced with a delay, the oscillations become larger. If the delay is large enough the system might even become chaotic. The delay in Earth negative feedback is likely very large, since it take a long time for things to heat up, for glaciers to recede, and so on. Yet we don't see any of this oscillation behaviour. Looking at these graphs it seems that the simplest explanation is that the system is being driven to saturation.
Through the above explanation and diagrams I hope I was able to give you:
Argument: "All of your arguments are about blackbodies. Grey-bodies have emissivity and absorptivity coefficients and these vary with wavelength and don't have to be equal. An emissivity less than absorptivity allows the temperature to be higher than a blackbody."
Reply: At equilibrium absorptivity and emissivity must be equal at all wavelengths and angles. This is Kirchoff's law.
Argument: "You misunderstand blackbodies. It is only what the temperature looks like from looking at the outgoing radiation. It is like the equivalent temperature of the outside layer only. The inside can still be hotter."
Reply: The outer layer must balance with the inside. If we imagine our system as a smaller sphere excluding the outer layer, then we can treat the outer layer now as the source. The new smaller outer layer of our smaller sphere must balance flux with the source layer. We can see that our new smaller outer layer must have a temperature less than or equal to a blackbody.
Argument: "Venus is an example that clearly shows the greenhouse effect can achieve temperatures higher than that of a blackbody."
Reply: Venus temperatures can not be explained using the greenhouse effect. The atmosphere of Venus is so thick that sunlight does not reach the ground. The only energy that can heat the ground comes from the sunlight absorbed in the atmosphere, yet the ground is much hotter than the atmosphere (ref.). How can the atmosphere heat the ground to be hotter than itself?
Or consider that Venus day is 243 days long, yet the darkside of Venus is nearly the same temperature as the dayside. How is this possible?
Both of these features of Venus climate can only be possible if the source of Venus high temperatures is due to something else other than the greenhouse effect. One possible explaination is that Venus was hit by a large meteor, or equivalent, which penetrated the crust and caused magma inside Venus to come to come to the surface. The thick atmosphere then creates a good insulator that keeps this heat from quickly escaping. This would explain the high temperatures, hot dark surface and the small difference between day and night temperature with such an extremely long day.