### We are able to Theoretically calculate the planet mean surface temperature Tmean

The Planet Surface Rotational Warming Phenomenon

The method we use in this research is the "Planet Surface Temperatures Comparison Method".

The data available are from observatories and satellite measurements.

The data:

1). The solar flux's intensity upon the planet surface "S".

2). The planet surface average Albedo "a".

3). Planet surface temperatures "T" K.

4). Planet rotational spin value "N" rotations/day.

5). Planet surface composition (planet average surface specific heat "cp" cal/gr.oC).

6). Planet surface Φ-factor - the planet surface Solar Irradiation Accepting Factor (the planet surface shape and roughness coefficient).

We have resulted to an important discovery:

The planet mean surface temperatures relate (everything else equals) according to their (N*cp) products’ sixteenth root.

The consequence of this discovery is the realization that a planet with a higher (N*cp) product (everything else equals) appears to be a warmer planet.

We call it the Planet Surface Rotational Warming Phenomenon.

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We are able to Theoretically calculate for the planet without-atmosphere the mean surface temperature.

For every planet without atmosphere there is the theoretical uniform surface effective temperature Te.

Te = [ Φ (1-a) S /4σ ]¹∕ ⁴

And for every planet without atmosphere there is the average surface temperature (the mean surface temperature) Tmean.

Thus we can write

Tmean = Te * X

where X is a coefficient which calculates the planet Tmean from the planet known Te.

The X is a different and very distinguished for every different planet number.

Notice:

The planet Te is theoretically calculated by the Stefan-Boltzmann emission law, when the planet average surface Albedo, and the solar flux upon the planet surface are known.

Te = [ Φ (1-a) S /4σ ]¹∕ ⁴

Now, we can accept that for every planet (ι) there is a Te.ι and there is a Tmean.ι

We can accept that for every planet (ι) there is a Xι, there is a Te.ι and there is a

Tmean.ι = Te.ι* Xι

So we have here

Tmean.ι = Te.ι * Χ.ι

or

Tmean.ι = [ Φ.ι (1 - a.ι) S.ι (X.ι)⁴ /4σ ]¹∕ ⁴

Conclusion:

We have admitted that for every planet (ι) there is a different for each planet (ι) a factor [(X.ι)⁴ ], which relates for the purpose to theoretically calculate for the planet (ι) the average (mean) surface temperature Tmean.ι

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by simply multiplying the X.ι with planet (ι) theoretical uniform surface effective temperature Te.ι

### A Simple Theorem, but a very important Theorem.

Also for every planet (ι) without atmosphere we have the planet (N.ι*cp.ι) product.

I have demonstrated in my website that planet mean surface temperatures relate (everything else equals) according to their (N*cp) products’ sixteenth root.

Thus we can in the equation

Tmean.ι = [ Φ (1-a) S (X.ι)⁴ /4σ ]¹∕ ⁴

the (X.ι)⁴ term to replace with the (β *N.ι *cp.ι) ¹∕ ⁴ term

where

a.ι – is the planet (ι) the average surface Albedo

Φ.ι – is the solar irradiation accepting factor (for smooth surface planets Φ = 0,47 and for rough surface planets Φ = 1)

N.ι – is planet (ι) rotational spin (rot/day)

cp.ι – is the planet average surface specific heat (cal/gr.oC)

β = 150 days*gr*oC/rotation*cal is a Rotating Planet Surface Solar Irradiation INTERACTING-Emitting Universal Law constant

Consequently, for every without-atmosphere planet (ι) we have:

Tmean.ι = [ Φ.ι (1 - a.ι) S.ι (β *N.ι *cp.ι)¹∕ ⁴ /4σ ]¹∕ ⁴

Conclusion:

The above formula theoretically calculates the planets without atmosphere mean surface temperatures with very closely matching to the satellite measured temperatures results.

Planet….Te……Tmean….Tsat.mean

Mercury..439,6 K..325,83 K…340 K

Earth…..255 K….287,74 K….288 K

Moon…..270,4 Κ…223,35 Κ…220 Κ

Mars….209,91 K…213,21 K….210 K

Notice:

The planet mean surface temperatures Tmean are very much precisely being measured by satellites.

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A Simple Theorem, but a very important Theorem.

From the above...

for every without-atmosphere planet (ι) we have:

Tmean.ι = [ Φ.ι (1 - a.ι) S.ι (β *N.ι *cp.ι)¹∕ ⁴ /4σ ]¹∕ ⁴

or

Tmean = [ Φ (1 - a) S (β*N*cp.)¹∕ ⁴ /4σ ]¹∕ ⁴

or it can be re-written as

Tmean = Te * [(β*N*cp.)¹∕ ⁴]¹∕ ⁴

The Theorem:

The planet mean surface temperature Tmean numerical value will be equal to the planet effective temperature Te numerical value Tmean = Te only when the term

(β*N*cp) = 1

and, since the

β = 150 days*gr*oC/rotation*cal

the planet N*cp product should be then

N*cp = 1 /β

or the numerical value of the product

N*cp = 1 /150

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The Theorem leads to the following very important conclusions:

1). In general, the planet effective temperature numerical value Te is not numerically equal to the planet without-atmosphere mean surface temperature Tmean.

2). For the planet without-atmosphere mean surface temperature numerical value Tmean to be equal to the planet effective temperature numerical value Te the condition from the above Theorem the (N*cp = 1 /150) should be necessarily met.

3). For the Planet Earth without-atmosphere the (N*cp) product is (N*cp = 1) and it is 150 times higher than the necessary condition of (N*cp = 1/150) .

Consequently, Earth's effective temperature Te the numerical value  cannot be equal to Earth's without-atmosphere mean surface temperature... not even close.

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