I’ll try here in few simple sentences explain the very essence of how the planet rotational warming Phenomenon occurs.

Lets consider two identical planets F and S at the same distance from the sun.

Let’s assume the planet F spins on its axis Faster, and the planet S spins on its axis Slower.

Both planets F and S get the same intensity solar flux on their sunlit hemispheres. Consequently both planets receive the same exactly amount of solar radiative energy.

The slower rotating planet’s S sunlit hemisphere surface gets warmed at higher temperatures than the faster rotating planet’s F sunlit hemisphere.

The surfaces emit at σT⁴ intensity – it is the Stefan-Boltzmann emission law.

Thus the planet S emits more intensively from the sunlit side than the planet F.

So there is more energy left for the planet F to accumulate then. That is what makes the faster rotating planet F on the average a warmer planet. That is how the Planet Surface Rotational Warming Phenomenon occurs.

It is the Planet Surface Rotational Warming Phenomenon we observe here. It states that

Planets' Mean Surface Temperatures (everything else equals) relate as their (N*cp) products' sixteenth root.

"A planet from rotating faster is getting energy from nowhere."

Yes, the faster rotation does not provide any additional incident SW radiative solar energy. No matter how slow or fast a planet rotates the incident on the planet's surface solar energy is always the same.

What changes with the faster rotation is the planet surface sunlit hemisphere IR emission ratio.

There is the Planet Surface Rotational Warming Phenomenon.

Lets consider two identical planets F and S at the same distance from the sun. Let’s assume the planet F spins on its axis Faster, and the planet S spins on its axis Slower. Both planets F and S get the same intensity solar flux on their sunlit hemispheres. Consequently both planets receive the same exactly amount of solar radiative energy.

The slower rotating planet’s S sunlit hemisphere surface gets warmed at higher temperatures than the faster rotating planet’s F sunlit hemisphere.

The surfaces emit at σT⁴ intensity.

Thus at every given moment the planet S sunlit surface emits IR outgoing radiative energy more intensively from the sunlit side than the planet F.

So there is more energy every given moment left for the planet F to accumulate for the night then. That is what makes the faster rotating planet F on the average a warmer planet. That is how the Planet Surface Rotational Warming Phenomenon occurs.

So we have under consideration four Moon mean surface temperatures:

193 K,

https://springerplus.springeropen.com/articles/10.1186/2193-1801-3-723

220 K ( the 220 K is what I support ),

https://www.sciencedirect.com/science/article/pii/S0019103516304869

https://simple.wikipedia.org/wiki/Moon

243 K,

http://www.digipac.ca/chemical/mtom/contents/chapter1/marsfacts.htm

250 K,

https://nssdc.gsfc.nasa.gov/planetary/factsheet/

https://en.wikipedia.org/wiki/Moon

Also we know the Moon's effective temperature

Te.moon = 270,4 K

Notice: All the above four Moon mean surface temperatures are below the Te.moon = 270,4 K.

So Moon's without atmosphere mean surface temperature (no matter which one) is below the Te.moon = 270,4 K.

On the other hand, Earth's mean surface temperature is

Tmean.earth = 288 K

And Earth's effective temperature

Te.earth = 255 K.

We have a very interesting paradox here. Earth with atmosphere has a higher mean surface temperature, than the Earth's effective temperature

Tmean.earth > Te.earth

288 K > 255 K

Moon has a lower mean surface temperature, than the Moon's effective temperature

Tmean.moon < te.moon="">

193 or 220 or 243 or 250 K < 270,4="">

This very interesting paradox

Tmean.earth > Te.earth

288 K > 255 K

and

Tmean.moon <>

193 or 220 or 243 or 250 K < 270,4="">

cannot be explained otherwise but because of a major difference Earth and Moon have. And this isn't the existence of atmosphere on the Earth's surface.

What it is then?

The major factor which determines the Earth's mean surface temperature Tmean.earth being HIGHER than Earth's effective temperature Te.earth,

and Moon's mean surface temperature Tmean.moon being LOWER than Moon's effective temperature Te.moon

is the Earth's and Moon's VERY DIFFERENT AXIAL SPIN.

Earth performs N.earth =1 rotation /day

Moon performs N.moon = 1 /29,531 rotation /day

or N.moon = 0,033863 rotation /day

Consequently, Earth's surface gets warmer than Moon's, because of the PLANET SURFACE ROTATIONAL WARMING PHENOMENON.