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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