A Planet Effective Temperature Complete Formula Te = [ Φ (1-a) S (β*N*cp)¹∕ ⁴ /4σ ]¹∕ ⁴

Plus the introduction to the Reversed Milankovitch Cycle. Click on the box for more

A Planet Effective Temperature Complete Formula: Te = [ Φ (1-a) S (β*N*cp)¹∕ ⁴ /4σ ]¹∕ ⁴ (1)

My name is Christos J. Vournas, M.Sc. mechanical engineer.

I launched this site to have an opportunity to publish my scientific discoveries on the Climate Change.

I have been studying the Planet Earth’s Climate Change since November 2015; I have been studying it for four years now.

First I discovered the Reversed Milankovitch Cycle.

Then I found the faster a planet rotates (n2>n1) the higher is the planet’s average (mean) temperature T↑mean:

Tmin→ T↑mean ← Tmax

when n2>n1 (it happens because Tmin grows faster than Tmax goes down)

The further studies led me to discover the Rotating Planet Spherical Surface Solar Irradiation Absorbing-Emitting Universal Law and the Planet Effective Temperature Complete Formula.

 

The faster a planet rotates (n2>n1) the higher is the planet’s average temperature: Tmin↑→T↑mean←T↓max

My name is Christos J. Vournas, M.Sc. mechanical engineer.

I am 68 years old and I live in Athens Greece.

My e-mail address is: vournas.christos@yahoo.com

The date is October 11, 2019

 

All these discoveries are based on the Stefan-Boltzmann Law.

Rotating Planet Spherical Surface Solar Irradiation Absorbing-Emitting Universal Law: Jabs=Φ*S*(1-a)/4= Jemit=σΤe⁴/(β*N*cp)¹∕ ⁴ (W/m²)

A Rotating Planet Surface Solar Irradiation Absorbing-Emitting Universal Law

Planet Energy Budget:

Solar energy absorbed by a Hemisphere with radius "r" after reflection and dispersion:

Jabs = Φ*πr²S (1-a)  (W)

Total energy emitted to space from a whole planet:

Jemit = A*σΤe⁴ /(β*N*cp)¹∕ ⁴  (W)

Φ - is a dimensionless Solar Irradiation accepting factor

(1 - Φ) - is the reflected fraction of the incident on the planet solar flux

S  - is a Solar Flux at the top of atmosphere (W/m²)

Α - is the total planet surface (m²)

Te - is a Planet Effective Temperature (K)

(β*N*cp)¹∕ ⁴ - dimensionless, is a Rotating Planet Surface Solar Irradiation Warming Ability

A = 4πr² (m²), where r – is the planet's radius

Jemit = 4πr²σTe⁴ /(β*N*cp)¹∕ ⁴  (W)

global Jabs = global Jemit

Φ*πr²S (1-a) = 4πr²σTe⁴ /(β*N*cp)¹∕ ⁴

Or after eliminating πr²

Φ*S*(1-a) = 4σTe⁴ /(β*N*cp)¹∕ ⁴

The planet average Jabs = Jemit per m² planet surface:

Jabs = Jemit

Φ*S*(1-a) /4 = σTe⁴ /(β*N*cp)¹∕ ⁴     (W/m²)

Solving for Te we obtain the effective temperature:

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

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

N rotations/day, is planet’s sidereal rotation period

cp – is the planet surface specific heat

cp.earth = 1 cal/gr*oC, it is because Earth has a vast ocean. Generally speaking almost the whole Earth’s surface is wet. We can call Earth a Planet Ocean.

Here (β*N*cp)¹∕ ⁴ - is a dimensionless Rotating Planet Surface Solar Irradiation Warming Ability

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

The year-round averaged energy flux at the top of the Earth's atmosphere is Sο = 1.362 W/m².

With an albedo of a = 0,3 and a factor Φ = 0,47 we have Te = 288,36 K or 15°C.

This temperature is confirmed by the satellites measured Tmean.earth = 288 K.

A Planet Effective Temperature Complete Formula

The Effective Temperature Complete Formula has the wonderful ability to calculate Planets Surface Effective Temperatures (mean temperatures) getting almost the same results as the measured by satellites planet mean temperatures. This Complete Formula can be applied to all the without atmosphere planets and moons in a solar system.

A Planet Effective Temperature Complete Formula: Te = [ Φ (1-a) S (β*N*cp)¹∕ ⁴ /4σ ]¹∕ ⁴ (1)

We ended up to the following remarkable results:

 

Comparison of results the planet Te calculated by the Incomplete Formula:

Te = [ (1-a) S / 4 σ ]¹∕ ⁴

the planet Te calculated by the Complete Formula:

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

and the planet Te (Tsat.mean) measured by satellites:

 

Planet or             Te.incomplete       Te.complete             Te.satellites

  moon                     formula                formula                     measured

Mercury           437 K          346,11 K            340 K

Earth                255 K          288,36 K            288 K

Moon                271 Κ          221,74 Κ           220 Κ

Mars                211,52 K     215,23 K            210 K

To be honest with you, at the beginning, I got by surprise myself with these results.

You see I was searching for a mathematical approach…

A Planet Universal Law Formula

As you know, to maintain a Planet Universal Law Formula one has to study all the planets' behavior. In that way only one may come to general conclusions. That is why I call our Earth as a Planet Earth. After all Earth is a Planet and as a Planet it behaves in accordance to the Universal Laws - as all Planets in the Universe do.

Φ - is the dimensionless solar irradiation spherical surface accepting factor, Φ=0,47

The 288 K - 255 K = 33°C difference does not exist

When I saw the Earth’s both measured by satellites and calculated with Formula temperatures being 288 K, I felt extremely well and satisfied. It was a nice feeling. It was a discovery, it worked and it was promising sigh, and the 33°C difference did not exist anymore.

The Tsat.earth - Te.incompl = 288 K - 255 K = 33°C difference does not exist.

The first thing I had to do was to check the Complete Formula on some other planets. And it didn’t take me too long to realize that a Planet-Without-Atmosphere Effective Temperature Complete Formula was working on all the planets and moons without atmosphere in the solar system.

I dare to assume now that this Formula works for all planets and moons without atmosphere in the whole universe…

A Planet Effective Temperature Complete Formula succesfully calculates planet's effective temperature.

A Planet Effective Temperature Complete Formula. Earth's and Moon's Effective Temperature Calculation.

A Planet-Without-Atmosphere Effective Temperature Complete Formula derives from the incomplete Te formula which is based on the radiative equilibrium and on the Stefan-Boltzmann Law.

from the    Te = [ (1-a) S / 4 σ ]¹∕ ⁴  

which is in common use right now, but actually it is an incomplete Te formula and that is why it gives us very confusing results.

A Planet-Without-Atmosphere Effective Temperature Complete Formula is also based on the radiative equilibrium and on the Stefan-Boltzmann Law.

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

to the complete Te = [ Φ (1-a) S (β*N*cp)¹∕ ⁴ /4σ ]¹∕ ⁴  (1)

(.......)¹∕ ⁴ is the fourth root 

S = So(1/R²), where R is the average distance from the sun in AU (astronomical units)

S - is the solar flux W/m² 

So = 1.362 W/m² (So is the Solar constant)

Planet’s albedo: a

Φ - is the dimensionless solar irradiation spherical surface accepting factor

Accepted by a Hemisphere with radius r sunlight is S*Φ*π*r²(1-a), where Φ = 0,47 for smooth surface planets, like Earth, Moon, Mercury and Mars…

(β*N*cp)¹∕ ⁴ is a dimensionless Rotating Planet Surface Solar Irradiation Warming Ability

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

N rotations/day, is planet’s sidereal rotation period

cp – is the planet surface specific heat

cp.earth = 1 cal/gr*oC, it is because Earth has a vast ocean. Generally speaking almost the whole Earth’s surface is wet. We can call Earth a Planet Ocean.

cp = 0,19 cal/gr*oC, for dry soil rocky planets, like Moon and Mercury.

Mars has an iron oxide F2O3 surface, cp.mars = 0,18 cal/gr*oC

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

This Universal Formula (1) is the instrument for calculating a Planet-Without-Atmosphere Effective Temperature. The results we get from these calculations are almost identical with those measured by satellites.

 

1. Earth’s-Without-Atmosphere Effective Temperature Calculation:

So = 1.362 W/m² (So is the Solar constant)

Earth’s albedo: aearth = 0,30

Earth is a rocky planet, Earth’s surface solar irradiation accepting factor Φearth = 0,47

(Accepted by a Smooth Hemisphere with radius r sunlight is S*Φ*π*r²(1-a), where Φ = 0,47)

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

N = 1 rotation per day, is Earth’s sidereal rotation period

cp.earth = 1 cal/gr*oC, it is because Earth has a vast ocean. Generally speaking almost the whole Earth’s surface is wet. We can call Earth a Planet Ocean.

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

Earth’s-Without-Atmosphere Effective Temperature Complete Formula Te.earth is

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

Τe.earth = [ 0,47(1-0,30)1.362 W/m²(150 days*gr*oC/rotation*cal *1rotations/day*1 cal/gr*oC)¹∕ ⁴ /4*5,67*10⁻⁸ W/m²K⁴ ]¹∕ ⁴ =

Τe.earth = [ 0,47(1-0,30)1.362 W/m²(150*1*1)¹∕ ⁴ /4*5,67*10⁻⁸ W/m²K⁴ ]¹∕ ⁴ =

Te.earth = 288,36 Κ

And we compare it with the

Tsat.mean.earth = 288 K, measured by satellites.

These two temperatures, the calculated one, and the measured by satellites are almost identical.

 

2. Moon’s Effective Temperature Calculation:

So = 1.362 W/m² (So is the Solar constant)

Moon’s albedo: amoon = 0,136

Moon’s sidereal rotation period is 27,3216 days. But Moon is Earth’s satellite, so the lunar day is 29,5 days

Moon is a rocky planet, Moon’s surface solar irradiation accepting factor

Φmoon = 0,47

(Accepted by a Smooth Hemisphere with radius r sunlight is S* Φ*π*r²*(1-a), where Φ = 0,47)

cp.moon = 0,19cal/gr oC, moon’s surface is considered as a dry soil

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

N = 1/29,5 rotations per/ day

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

Moon’s Effective Temperature Complete Formula Te.moon: 

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

Te.moon = { 0,47 (1-0,136) 1.362 W/m² [150* (1/29,5)*0,19]¹∕ ⁴ /4*5,67*10⁻⁸ W/m²K⁴ }¹∕ ⁴ =

Te.moon = 221,74 Κ

The newly calculated Moon’s Effective Temperature differs only by 0,8% from that measured by satellites!

Tsat.mean.moon = 220 K, measured by satellites.

(β*N*cp)¹∕ ⁴ is a dimensionless Rotating Planet Surface Solar Irradiation Warming Ability

Mars' and Mercury's Effective Temperature Calculation

3. Mars’ Effective Temperature Calculation:

Te.mars

(1/R²) = (1/1,5²) = 1/2,25 Mars has 2,25 times less solar irradiation intensity than Earth has

Mars’ albedo: amars = 0,25

N = 1 rotations/per day, Planet Mars completes one rotation around its axis in about 24 hours

Mars is a rocky planet, Mars’ surface solar irradiation accepting factor: Φmars = 0,47

cp.mars = 0,18 cal/gr oC, on Mars’ surface is prevalent the iron oxide

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

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

So = 1.362 W/m² the Solar constant

Mar’s Effective Temperature Complete Formula is:

Te.mars = [ Φ (1-a) So (1/R²) (β*N*cp)¹∕ ⁴ /4σ ]¹∕ ⁴

Planet Mars’ Effective Temperature

Te.mars is:

Te.mars = [ 0,47 (1-0,25) 1.362 W/m²*(1/2,25)*(150*1*0,18)¹∕ ⁴ /4*5,67*10⁻⁸ W/m²K⁴ ]¹∕ ⁴ =

Te.mars = 215,23 K

The calculated Mars’ effective temperature Te.mars = 215,23 K is only by 2,4% higher than that measured by satellites

Tsat.mean.mars = 210 K !

4. Mercury’s Effective Temperature Calculation:

Te.mercury

N = 1/58,646 rotations/per day, Planet Mercury completes one rotation around its axis in 58,646 days.

Mercury average distance from the sun is R=0,387AU. The solar irradiation on Mercury is (1/R²) = (1AU/0,387AU)²= 2,584²= 6,6769 times stronger than that on Earth.

Mercury’s albedo is: amercury = 0,088

Mercury is a rocky planet, Mercury’s surface solar irradiation accepting factor: Φmercury = 0,47

Cp.mercury = 0,19 cal/gr oC, Mercury’s surface is considered as a dry soil

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

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

So = 1.362 W/m² the Solar constant

Mercury’s Effective Temperature Complete Formula is:

Te.mercury = [ Φ (1-a) So (1/R²) (β*N*cp)¹∕ ⁴ /4σ ]¹∕ ⁴

Planet Mercury’s effective temperature

Te.mercury is:

Te.mercury = { 0,47(1-0,088) 1.362 W/m²*6,6769*[150* (1/58,646)*0,19]¹∕ ⁴ /4*5,67*10⁻⁸ W/m²K⁴ }¹∕ ⁴ =

Te.mercury = 346,11 K

The calculated Mercury’s Effective Temperature Te.mercury = 346,11 K is only 1,80% higher than the measured by satellites

Tsat.mean.mercury = 340 K !

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

We can confirm now with great confidence

So, we can confirm now with great confidence, that a Planet or Moon Without-Atmosphere Effective Temperature Complete Formula, according to the Stefan-Boltzmann Law, is:

Te.planet = [ Φ (1-a) So (1/R²) (β*N*cp)¹∕ ⁴ /4σ ]¹∕ ⁴        (1)

We have collected the results now:

Comparison of results the planet Te calculated by the Incomplete Formula, the planet Te calculated by the Complete Formula, and the planet Te (Tsat.mean) measured by satellites:

Planet or     Te. incomplete            Te.complete               Te (sat.mean)

Moon           calculated by              calculated by               Measured by

                 Incomplete formula      Complete formula          satellites 

Mercury        437 K            346,11 K             340 K

Earth          255 K            288,36 K             288 K

Moon          271 Κ            221,74 Κ             220 Κ

Mars           211,52 K      215,23 K             210 K

 

These data, the calculated with a Planet Without-Atmosphere Effective Temperature Complete Formula and the measured by satellites are almost the same, very much alike.

They are almost identical, within limits, which makes us conclude that the Planet-Without-Atmosphere Effective Temperature Complete Formula

Te.planet = [ Φ (1-a) So (1/R²) (β*N*cp)¹∕ ⁴ /4σ ]¹∕ ⁴        (1)

can calculate a planet mean temperatures.

It is a situation that happens once in a lifetime in science. Although the evidences existed, were measured and remained isolated information so far. 

It was not obvious one could combine the evidences in order to calculate the planet’s temperature.

A planet-without-atmosphere effective temperature calculating formula

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

is incomplete because it is based only on two parameters:

1. On the average solar flux S W/m² on the top of a planet’s atmosphere and

2. The planet’s average albedo a.

Those two parameters are not enough to calculate a planet effective temperature. Planet is a celestial body with more major features when calculating planet effective temperature to consider.

The planet-without-atmosphere effective temperature calculating formula has to include all the planet’s major properties and all the characteristic parameters.

3. The sidereal rotation period N rotations/day

4. The thermal property of the surface (the specific heat cp)

5. The planet surface solar irradiation accepting factor Φ (the spherical surface’s primer geometrical quality).

Altogether these parameters are combined in a Planet-Without-Atmosphere Effective Temperature Complete Formula:

Te.planet = [ Φ (1-a) So (1/R²) (β*N*cp)¹∕ ⁴ /4σ ]¹∕ ⁴          (1)

A Planet-Without-Atmosphere Effective Temperature Complete Formula produces very reasonable results:

Te.earth = 288,36 K, calculated with the Complete Formula, which is identical with the Tsat.mean.earth = 288 K, measured by satellites. 

Te.moon = 221,74 K, calculated with the Complete Formula, which is almost the same with the Tsat.mean.moon = 220 K, measured by satellites.

A Planet-Without-Atmosphere Effective Temperature Complete Formula gives us a planet effective temperature values very close to the satellite measured planet mean temperatures (the satellite measured planet effective temperatures).

Thus we have to conclude here that the satellites measured planet mean temperatures should be considered as the satellite measured Planet Effective Temperatures.

It is a Stefan-Boltzmann Law Triumph! And it is a Milankovitch Cycle coming back! And as for NASA, all these new discoveries were possible only due to NASA satellites planet temperatures precise measurements!

The Fast Rotating Planet Earth

So far we came to the end of this presentation. Its topic was to present the Planet-Without-Atmosphere Effective Temperature Complete Formula:

             Te = [ Φ (1-a) So (1/R²) (β*N*cp)¹∕ ⁴ /4σ ]¹∕ ⁴  (K)

This Formula is based on the incomplete effective temperature formula:                                    

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

And also it is based on the discovered of the Rotating Planet Spherical Surface Solar Irradiation Absorbing-Emitting Universal Law:

                  Jemit = σΤe/(β*N*cp)¹∕ ⁴    (W/m²)

 

Here the (β*N*cp)¹∕ ⁴ is a dimensionless Rotating Planet Surface Solar Irradiation Warming Ability.

Φ - is the dimensionless solar irradiation spherical surface accepting factor.

Accepted by a Hemisphere with radius r sunlight is S*Φ*π*r²(1-a), where Φ = 0,47 for smooth surface planets, like Earth, Moon, Mercury and Mars…

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

N rotations/day, is planet’s sidereal rotation period

cp cal/gr oC – is the planet’s surface specific heat

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

The Rotating Planet Surface Solar Irradiation Absorbing-Emitting Universal Law is based on a simple thought. It is based on the thought, that physical phenomenon which distracts the black body surfaces from the instant emitting the absorbed solar radiative energy back to space, warms the black body surface up.

In our case those distracting physical phenomena are the planet’s sidereal rotation, N rotations/day, and the planet’s surface specific heat, cp cal/gr oC.

Thus we have the measured by satellites Earth’s Tmean.earth = 288 K to be the same as the calculated by the effective temperature complete formula Te.earth = 288,36 K.

Those physical phenomena distracting Earth from the instant emitting back to space are the Earth’s rotation around its axis and the Earth’s surface specific heat.

Also we should mention here, that a smooth surface spherical body, as the planet Earth is, doesn’t accept and absorb all the solar radiation falling on the hemisphere. Only the 0,47*So of the solar energy’s amount is accepted by the hemisphere. The rest 0,53*So is reflected back to space. That is why Φ= 0,47 what is left for surface to absorb.

Now we have to say about the planet’s albedo (a). The planet’s albedo describes the dispersed on the surface secondary reflection to space fraction of the falling on the hemisphere solar light.

Thus a planet’s surface absorbs only the Φ*(1– a) fraction of the incident on the hemisphere solar energy. That is why we have the Φ (1-a) So (1/R²) expression in the complete effective temperature formula.

Conclusions:

We had to answer those two questions:

1. Why Earth’s atmosphere doesn’t affect the Global Warming?

It is proven now by the Planet Effective Temperature Complete Formula calculations. There aren’t any atmospheric factors in the Complete Formula. Nevertheless the Planet-Without-Atmosphere Effective Temperature Complete Formula produces very reasonable results:

Te.earth = 288,36 K, calculated by the Complete Formula, which is the same as the Tsat.mean.earth = 288 K, measured by satellites.

Te.moon = 221,74 K, calculated by the Complete Formula, which is almost identical with the Tsat.mean.moon = 220 K, measured by satellites.

Earth has a very thin atmosphere; Earth has a very small greenhouse phenomenon in its atmosphere and it doesn’t warm the planet.

2. What causes the Global Warming then?

The Global Warming is happening due to the orbital forcing. It is not happening because of the atmosphere. We have the prove - a newly discovered for the Rotating Planet Surface Solar Irradiation Absorbing-Emitting Universal Law:

                  Jemit = σΤe/(β*N*cp)¹∕ ⁴  (W/m²)

And knowing that      Jemit = Jabs     

And          Jabs = [ Φ (1-a) So (1/R²) /4 ]   (W/m²)

Solving for Te we obtain the Planet without Atmosphere Effective Temperature Complete Formula:

    Te.earth = [ Φ (1-a) So (1/R²) (β*N*cp)¹∕ ⁴ /4σ ]¹∕ ⁴ = 288,36 K

The calculations made by the Planet without Atmosphere Effective Temperature Complete Formula also correspond to the next conclusion:

The measured by satellites Earth’s mean temperature T = 288 K is the Earth’s surface radiative equilibrium temperature.

And… what keeps the Earth warm at Te.earth = 288 K, when the Moon is at Te.moon = 220 K? Why Moon is on average 68 oC colder? It is very cold at night there and it is very hot during the day…

Earth is warmer because Earth rotates faster and because Earth’s surface is covered with water.

Does the Earth’s atmosphere act as a blanket that warms Earth’s surface?

No, it does not.

I dedicate this work to the great mathematician and astronomer of the 20th century Milutin Milankovitch

The Original Milankovitch Cycle

According to Milankovitch Ice Ages are generally triggered by minima in high-latitude Northern Hemisphere summer insolation, enabling winter snowfall to persist through the year and therefore accumulate to build Northern Hemisphere glacial ice sheets. Similarly, times with especially intense high-latitude Northern Hemisphere summer insolation, determined by orbital changes, are thought to trigger rapid deglaciations, associated climate change and sea level rise. But, at second thought, I concluded that Earth cannot accumulate heat on the continents’ land masses. Earth instead accumulates heat in the oceanic waters.

The Reversed Milankovitch Cycle

Milankovitch’s main idea was that the glacial periods are ruled by planet’s movements forcing. At the right we have the Reversed Milankovitch cycle. The minimums in the reversed Milankovitch cycle are the maximums in the original. These two cycles, the original Milankovitch cycle and the reversed differ in time only by a half of a year. According to the reversed Milankovitch cycle there are long and very deep glacial periods and small and very short interglacial. The reversed cycle complies with the paleo geological findings. As we can see in the reversed Milankovitch cycle, we are getting now to the end of a long and a slow warming period. What we are witnessing as a Global Climate Change are the culmination moments at the end of that warming period.

The NASA planets surface temperatures measurements are all we have to work with.

And we believe in NASA measurements, because they are very precise and very professionally performed.

And I underline here again, we have to rely only on the NASA measurements. The NASA planets surface temperatures measurements are all we have to work with.

Resume

A Planet-Without-Atmosphere Effective Temperature Calculating Formula, the Te formula which is based on the radiative equilibrium and on the Stefan-Boltzmann Law, and which is in common use right now:          

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

is actually an incomplete Te formula and that is why it gives us very confusing results.

A planet-without-atmosphere effective temperature calculating formula  

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

is incomplete because it is based only on two parameters:

1. On the average solar flux S W/m² on the top of a planet’s atmosphere and

2. The planet’s average albedo a.

Those two parameters are not enough to calculate a planet effective temperature. Planet is a celestial body with more major features when calculating planet effective temperature to consider.

The planet-without-atmosphere effective temperature calculating formula has to include all the planet’s major properties and all the characteristic parameters.

3. The sidereal rotation period N rotations/day

4. The thermal property of the surface (the specific heat cp)

5. The planet surface solar irradiation accepting factor Φ (the spherical surface’s primer geometrical quality). For Mercury, Moon, Earth and Mars without atmosphere Φ = 0,47.

Earth is considered without atmosphere because Earth’s atmosphere is very thin and it does not affect Earth’s Effective Temperature.

Altogether these parameters are combined in a Planet-Without-Atmosphere Effective Temperature Complete Formula:

    Te.planet = [ Φ (1-a) So (1/R²) (β*N*cp)¹∕ ⁴ /4σ ]¹∕ ⁴ (1)

A Planet-Without-Atmosphere Effective Temperature Complete Formula produces very reasonable results:

Te.earth = 288,36 K, calculated with the Complete Formula, which is identical with the Tsat.mean.earth = 288 K, measured by satellites.

Te.moon = 221,74 K, calculated with the Complete Formula, which is almost the same with the Tsat.mean.moon = 220 K, measured by satellites.

A Planet-Without-Atmosphere Effective Temperature Complete Formula gives us a planet effective temperature values very close to the satellite measured planet mean temperatures (the satellite measured planet effective temperatures).

Thus we have to conclude here that the satellites measured planet mean temperatures should be considered as the satellite measured Planet Effective Temperatures.

We have collected the results now:

Comparison of results the planet Te calculated by the Incomplete Formula, the planet Te calculated by the Complete Formula, and the planet Te (Tsat.mean) measured by satellites:

Planet or  Te.incomplete   Te.complete    Te.satellites

moon             formula             formula         measured

Mercury        437,30 K           346,11 K            340 K

Earth             255 K                288,36 K            288 K

Moon            271 Κ                221,74 Κ            220 Κ

Mars             211,52 K           215,23 K            210 K

          

As you can see Te.complete.earth = 288,36 K.

That is why I say in the real world the Δ 33 oC difference does not exist.