### 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.**

** .........................................**

** 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**

** ...................**

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

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### 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 Absorbing-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.**

** .......................................... **

**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**

** ...........................................**

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

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