### Earth’s Without-Atmosphere Corrected Effective Temperature calculation Te.correct.earth = 210 Κ

Earth's Corrected Effective Temperature is Te.correct.earth = 210 Κ

To calculate Earth's Corrected Effective Temperature we should use the following data values

σ = 5,67*10⁻⁸ W/m²K⁴, the Stefan-Boltzmann constant

Φ = 0,47 solar irradiation accepting factor (dimensionless)

a = 0,306 Earth's average albedo

So = 1.361 W/m², solar flux on the top of the Earth's atmosphere

Earth’s Without-Atmosphere Corrected Effective Temperature Equation  Te.correct.earth is:

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

Te.correct.earth = [ 0,47 (1-0,306) 1.361 W/m² /4*5,67*10⁻⁸ W/m²K⁴ ]¹∕ ⁴ =

Te.correct.earth = [ 0,47 (0,694) 1.361 W/m² /4*5,67*10⁻⁸ W/m²K⁴ ]¹∕ ⁴ =

Te.correct.earth = ( 1,957.367.636,68 )¹∕ ⁴ = 210,34 K

Te.correct.earth = 210,34 K or Te.correct.earth = 210 K

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

Te - planet effective temperature Te = [ (1-a) S /4σ ]¹∕

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....210 ...........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

### Why is (for Earth Te =255K) NASA calculation so inaccurate – too high?

Earth without-atmosphere and higher than Moon Albedo (a=0,306), when measured by NASA the Earthen equilibrium temperature should be even less than 210K.

Why is (for Earth Te =255K) NASA calculation so inaccurate – too high?

And Tse – Te = 288K - 210K = 78C the measured GHE then?

### The planet effective temperature Te is not the limit to the avg. temperature rise.

There is a deeply established concept that "The avg. planetary temperature changes with rotation speed rising to equilibrium temperature as the spin rate increases.”

This concept determines the planet effective (equilibrium) temperature Te as a kind of cut-off point. This concept states, planet avg. temperature (the avg. surface without-atmosphere temperature) cannot exceed the planet effective (equilibrium) temperature Te, no matter how fast the planet rotational spin.

What we actually observe is the following:

The avg. planetary temperature changes with rotation speed rising to equilibrium temperature and overgoing it as the spin rate increases...

Notice, there is a limit to the avg. planetary temperature rise, but it is not the Te or the Te.corrected.

Also the calculated Te and Te.corrected assume planet having reached uniform surface temperature, which is impossible, because planets always are solar irradiated by one side, and, no matter how fast they rotate, the solar lit side is always warmer…

And there are not measured data for planets' blackbody temperatures, because planet blackbody temperatures, (either the not corrected Te and the corrected Te.corrected) are only mathematical abstractions.

### When rotating faster – more areas get exposed to solar flux in unit of time.

(Orphan planet is a planet not having a mother star to orbit).

Orphan planet is not solar energy irradiated, therefore it has a surface temperature because of its own internal heat sources.

Two orphan planets may have the same average surface temperature, but the more differentiated surface temperatures orphan planet has the greater amount of IR outgoing radiative energy the orphan planet emits. (It is in accordance with Stephan-Boltzmann emission law nonlinearity.)

Let’s consider two orphan planets emitting the same amount of IR outgoing radiative energy. The more surface temperatures differentiated orphan planet – the colder on average surface temperature planet.

An orphan planet with uniform surface temperature would have approached the planet effective radiative temperature Te. Te is the highest possible average surface temperature for an orphan planet.

When rotating the planet surface has larger surface areas get exposed to solar flux in unit of time. When rotating faster – more areas get exposed.

Since surface's the slower ability to accumulate HEAT than emit IR, the faster rotating planet is capable to TRANSFORM larger amounts of SW EM radiative solar energy into HEAT.

Thus the faster rotating planet (everything else equals) is capable to accumulate larger amounts of transformed into HEAT solar EM energy.

That is what makes a faster rotating planet on average surface a warmer planet.

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The planet specular reflection was neglected

For planets and moons with smooth surface, the surface’s specular reflection is not negligible.

The smooth surface planets and moons have a very strong the surface’s specular reflection.

The specular reflection is not included in albedo.

So we had (for those planets and moons with smooth surface, and, therefore, with surface’s strong specular reflection), we had to correct their respective the planet effective temperature Te.

Correcting the Effective temperature (Te) formula:

Te = [(1-a) S /4σ ]¹∕ ⁴ (K) (1)

We insert the

Φ – the solar irradiation accepting factor (the planet spherical shape and planet surface roughness coeficient)

Φ =0,47 for smooth surface planets and moons

Φ =1 for heavy cratered (rough surface) planets and moons

Te.correct = [Φ(1-a) S /4σ ]¹∕ ⁴ (K) (2)

Te.correct, for the smooth surface planets and moons, has a much lower, than Te, numerical values.

Thus, for Earth, the Te =255K, when corrected,
became Te.correct =210K.

But, notice, it is very important:

The planet effective temperature, even when it is corrected, the planet effective temperature does not exist, the planet effective temperature is a mathematical abstraction.

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Opponent:

“The specular reflection is not included in albedo.”

Evidence?

“The specular reflection is not included in albedo.”

“Evidence?”

-

Let's demonstrate

“The specular reflection is not included in albedo.”

on Moon's Te example:

Moon’s Te = 270,4 K

Moon’s Te.correct = 224 K (corrected for smooth surface planets and moons with

Φ = 0,47)

Moon’s satellite measured average surface temperature Tsat = 220 K.

When comparing those three temperatures

Te = 270,4 K (calculated with moon’s Albedo

a =0,11)
Te.correct = 224 K (calculated with moon’s Albedo

a =0,11
and Φ =0,47 )

And the measured Tsat =220 K,

The theoretical Te.correct = 224 K is very much close to the satellite measured

Tsat = 220K.

On the opposite, the Te = 270,4 K is very much higher, than the satellite measured

Tsat =220 K.

And this is an undeniable evidence of the fact that

The specular reflection is not included in albedo.

********

Another undeniable evidence of the fact that the

specular reflection is not included in albedo.

The Planet Mars’ Te =210K
and Planet Mars’ Tsat =210K

COINCIDENCE!

Planet Mars’ Te =210K is calculated by the use
of Mars’ Albedo a =0,25

Planet Mars’ Tsat =210K is the Mars’ satellite measured average surface temperature.

Those two temperatures, the theoretically calculated 210K and the measured one 210K -there is not any physical explanation of them to coincide, except of the Mars’ specular reflection being ignored.

Why the specular reflection is ignored – because it was considered too small.

And yes, there are planets and moons where the specular reflection is too small to take in consideration. For them Φ =1.

But there also are the smooth surface planets and moons with very strong specular reflection.

Those planets and moons are:

Mercury
Earth
Moon
Mars
Europa
Ganymede

And for those planets and moons the Φ =0,47

Thus for the planet Mars, instead of Te =210K
it is Te.correct =174K