### The planet blackbody effective temperature formula is not capable to provide any realistic planet average temperature approach…

**The planet blackbody effective temperature formula**

** Te = [ (1 – a)S /4σ ]¹∕ ⁴ **

**is not capable to provide any realistic planet average temperature approach… **

**Let’s see why,**

** Moon’s average distance from the sun**

** R = 150.000.000 km**

** or R = 1 AU**

** (AU is Astronomical Unit, 1AU = 150.000.000 km which is the Earth’s distance from the sun)**

** In the solar system, for convenience reasons, astronomers use for distances comparison the AU instead of the kilometers.**

** Moon’s satellite measured average surface temperature (the mean surface temperature)**

** Tmoon = 220 K**

** Mars’ distance from the sun**

** R = 1,524 AU**

** Tmars = 210 K**

**Let's continue...**

** There is the planet blackbody temperature formula, which calculates the planet uniform effective temperature… **

**It is a theoretical approach to the planet mean surface temperature estimation. It is defined as the temperature planet without atmosphere would have, if planet is considered as a uniformly irradiated blackbody surface.**

** And therefore it is initially assumed a blackbody planet effective temperature being a uniform surface temperature. **

**The planet blackbody effective temperature formula:**

**Te = [ (1 – a)S /4σ ]¹∕ ⁴ **

**a ****– is the planet average Albedo (dimensionless) **

**S – is the solar flux on the planet surface W/m²**

** So – is the solar flux on Earth. (since Earth has atmosphere with clouds, the So is measured above the clouds at the Top of the Atmosphere, or TOA)**

** So = 1.361 W/m² **

**S = So*(1/R² ) **

**it is the from the sun distance the square inverse law. **

**The formula can be written also as**

** Te = [ (1 – a) So*(1/R² ) /4σ ]¹∕ ⁴ **

**Now, since the formula is a fundamental physics the planet surface average temperature approach, the planets’ effective temperatures should relay accordingly.**

** So we can write the planet average surface temperature comparison coefficient:**

** [(1 – a) So*(1/R² ) /4σ ]¹∕ ⁴**

** Let’s assume comparing the planet’s 1 and the planet’s 2 effective temperatures Te1 and Te2. **

**Then we shall have: **

**Te1 /Te2 = [(1 – a1) So*(1/R1² ) /4σ ]¹∕ ⁴ / [(1 – a2) So*(1/R2² ) /4σ ]¹∕ ⁴**

** (Te1 /Te2 )⁴ = [(1 – a1) /(1 – a2) ]* [(1/R1² ) /(1/R2² )] **

**Let’s compare Moon’s and Mars’ satellite measured temperatures**

** Tmoon = 220 K**

** Tmars = 210 K**

** (Tmoon /Tmars)⁴ = (220 /210)⁴ = 1,0476⁴ = 1,2045**

** Let’s compare Moon’s and Mars’ comparison coefficients **

**[ (1 – a.moon) /(1 – a.mars) ]* [(1/Rmoon² ) / (1/Rmars² ) ]**

** [ (1 – 0,11) /(1 – 0,25) ]* [(1/1² ) /(1/1,524² ) ]**

** ( 0,89 /0,75)* (1,524² ) = (0,89 /0,75) * 2,32 = 2,75**

** Conclusion: We obtained on the left side of the comparison equation the **

*1,2045* number

** (for satellite measured planet average surface temperatures comparison) and on the right side the**

* 2,75* number

**(for planets’ coefficients comparison) **

**Consequently we may conclude here, that the planet blackbody effective temperature formula is not capable to provide any realistic planet average temperature approach…**

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### The Planet Corrected Effective Temperature : Te.correct = [ Φ (1-a) S /4σ ]¹∕ ⁴

**Te - planet effective temperature **

**Te = [ (1-a) S /4σ ]¹∕ ⁴**

**Te.correct - the planet corrected effective temperature **

**Te.correct = [ Φ (1-a) S /4σ ]¹∕ ⁴ **

**Φ - is the solar irradiation accepting factor (it is the planet surface spherical shape, and planet surface roughness coefficient)**

**Φ = 0,47 - for smooth surface planets without atmosphere**

** Φ = 1 - for heavy cratered without atmosphere planets**

**Φ = 1 - for gases planets **

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

**Te.correct = [ Φ (1-a) S /4σ ]¹∕ ⁴ **

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

**Or **

**Tmean = Te.correct * [ (β*N*cp)¹∕ ⁴ ]¹∕ ⁴**

**Table 1. Comparison of Predicted (Tmean) vs. Measured (Tsat) Temperature for All Rock-type Planets**

** ......................Φ..Te.correct .. [(β*N*cp)¹∕ ⁴]¹∕ ⁴..Tmean ...Tsat**

** ................................°K ......................................°K ........°K**

**Mercury .....0,47....364,0 ........ 0,8953............. 325,83 ...340 **

**Earth ..........0,47....211 ........... 1,368................287,74 ....288. **

**Moon ..........0,47....224 ........... 0.9978.............223,35 .....220 **

**Mars ...........0,47....174 ........... 1,227..............213,11 .....210 **

**Io ..................1.......95,16 ........ 1,169..............111,55 .....110 **

**Europa ........0,47....78,83 ........ 1,2636.............99,56 .....102**

** Ganymede...0,47....88,59 ........ 1,209.............107,14 ....110 **

**Calisto ..........1.....114,66 ........ 1,1471...........131,52 ....134 ±11**

** Enceladus .... 1 ......55,97 ........ 1,3411............75,06 .......75 **

**Tethys ..........1.......66,55 ......... 1,3145 ...........87,48 .......86 ± 1 **

**Titan .............1.......84,52 ......... 1,1015 ...........96,03 .......93,7 **

**Pluto .............1.......37 .............. 1,1164 ...........41,6 .........44 **

**Charon .........1......41,90 ........... 1,2181 ...........51,04 .......53 **

**Conclusion: We can calculate planet mean surface temperature obtaining very close to the satellite measured results. **

**Tmean = Te.correct * [ (β*N*cp)¹∕ ⁴ ]¹∕ ⁴**

**where**

** [ (β*N*cp)¹∕ ⁴ ]¹∕ ⁴ - is the planet surface warming factor**

**Warming Factor = (β*N*cp)¹∕₁₆**

### The Planet's Without-Atmosphere Mean Surface Temperature Equation

** from the incomplete effective temperature equation**

** Te = [ (1-a) S / 4 σ ]¹∕ ⁴**

** which is in common use right now, but actually it is an incomplete planet's Te equation and that is why it gives us very confusing results.**

** to the Planet's Without-Atmosphere Mean Surface Temperature Equation **

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

**The Planet's Without-Atmosphere Mean Surface Temperature Equation is also based on the radiative equilibrium and on the Stefan-Boltzmann Law. **

**The Equation is being completed by adding to the incomplete Te equation the new parameters Φ, N, cp and the constant β. **

**Φ - is the dimensionless Solar Irradiation accepting factor **

**N - rotations /day, is the planet’s axial spin**

** cp – is the planet's surface specific heat**

** β = 150 days*gr*oC/rotation*cal – is the Rotating Planet Surface Solar Irradiation Absorbing-Emitting Universal Law constant.**

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