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The Anthropogenic Global Warming hypothesis is based upon flat earth mathematics.
It would appear the guys-n-gals at Quantumville recalled the high-school formula for determining the surface area of a disc consists of its radius squared times pi (i.e., r² × π) and how the greater surface area of a sphere consists of its radius squared times pi times four (i.e., r²π × 4).
Thus a disc with a radius of 1 will have a surface area of pi (i.e., 3.14) and the same disc converted to a sphere will therefore have a surface area four times more than the original disc (i.e., 12.57). Applying this mathematical thinking to the terraqueous planet we all live on their abstractive minds effectively reduced it to a flat disc which has been expanded four times.
Then they mentally painted this flat earth black.
Thereupon they reduced the (measured) insolation of the oblate spheroid known as Planet Earth to a quarter of its real-world strength – and directed it to impinge upon every square millimetre of their phantom planet twenty-four-seven forever and a day – thereafter applying a radiance versus temperature constant and thus they had their flat earth temperature of -18° Celsius.
As this is at odds with the +15° Celsius real-world temperature they confected the “greenhouse effect” to account for the missing 33° Celsius.
This is where the weirdity of a blackbody flat earth bathed twenty-four-seven with quadruply-weakened sunbeams turns into utter bizarrerie as they deemed some trace gases in the frigid upper troposphere, busily absorbing infrared light radiating from the blackbody surface and emitting it in all directions, to be thus radiating some of it back to the heat-source and thereby raising its temperature via this ‘back-radiation’ thaumaturgy the requisite thirty-three degrees (from -18°C to +15°C).
Moreover, this sci-fi scenario conveniently ignores how the other atmospheric gasses constituting some 99% of the air we all breathe – which are heated at ground-level by conduction and thence by convection as hot air rises and cold air sinks to such an extent as to dominate in determining the thermal structure of the lower atmosphere (troposphere) – are also emitting infrared light in all directions.
Furthermore, no externally heated substance – be it heated by conduction (direct transference from the heat source), by convection (heated gases rising and mingling and mixing with sinking cooler gases), or by radiation (via the heat source emitting infrared light) – can actually raise the temperature of its heat-source (let alone to such a precise degree as 33° Celsius, no more and no less, provided some specified trace gases remain at pre-industrial parts-per-million levels).
(This real-world matter-of-factness does not apply to all the phantom planets in the noncausative quantum solar system, of course, which are busily raising the surface temperature of its central star – the heat-source for all those blackbody flat planets – above 5778° K via a massive-scale variant version of this phantasmagorical ‘back-radiation’).
To summarise:
The physical earth is not flat.
The physical earth is not black.
The physical earth is not static (it is constantly rotating).
The physical earth is not bathed with quadruply-weakened sunbeams.
Sunlight does not impinge upon every square millimetre of the physical earth twenty-four-seven (only during daytime).
Sunlight does not impinge with equal intensity upon every square millimetre of the daytime hemisphere (most obliquely at polar latitudes and dawn-dusk regions).
All atmospheric gases are heatable (not just several trace gases).
All heated atmospheric gases radiate infrared light (not just several trace gases).
The physical atmosphere insulates the daytime hemisphere from heating-up to unliveable temperatures (unlike its nearest neighbour the airless moon).
The physical atmosphere insulates the nighttime hemisphere from freezing to unliveable temperatures (unlike its nearest neighbour the airless moon).
In the physical world no externally heated substance can raise the temperature of its heat-source.
In the physical world some specified trace gases can (as evidenced in notable past eras) exceed by several thousand parts-per-million those several hundred parts-per-million pre-industrial levels deemed sacrosanct by influential doomsayers and/or panicmongers.
tl;dr: there is no “greenhouse effect” in reality (nor “greenhouse gases” either).
...are you really complaining that they didn't write out and evaluate the surface integral over the daylit hemisphere ∫∫S Isuncos θincidencedS? I assure you that it simplifies to Isunπr²
I assume you are referring to the second law of thermodynamics? The greenhouse effect doesn't violate the second law. The net heat flow is still surface>atmosphere>space. The fact that infrared-absorbing gasses reflect some of this heat back to the surface would not cause the surface to "heat up" in the absence of external solar radiation. The net effect is to slow down the rate of energy emission from the surface out into space. Net heat flows from hot to cold in every step.
I appreciate putting someone in their place with the power of calculus, but a simpler demonstration would be a diagram of light hitting the earth. The portion of light captured by earth will be a circle, not the surface area of a half sphere, let alone the surface area of a full sphere.
The last (which is @Sky is assuming) would be attained if, with a paper earth, we cut earth open and lay it flat on the table to capture sunlight on all its surface.
Oh sure, that's the simplifying assumption which is presumed to be understood. Unfortunately, OP did not understand it. He may be confused by how the locally-valid rule I = Isuncos θincidence can give rise to the much simpler rule PEarth = Isunπr² when summed over all localities. The way to resolve such confusion is to evaluate the integral yourself and see that it all cancels out.
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