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.

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 Tsat.mean measured by satellites:

Planet    Te.incompl  Te.compl  Tsat.mean

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          209,91 K    213,42 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…

Conclusions:

The complete formula produces remarkable results.

The calculated planets’ temperatures are almost identical with the measured by satellites.

The 288 K – 255 K = 33 oC

difference does not exist in the real world.

There are only traces of greenhouse gasses.

The Earth’s atmosphere is very thin.

There is not any measurable Greenhouse Gasses Warming effect on the Earth’s surface.

Moon rotates around its axis at a slow rate of 29,5 days.

The day on the Moon is 14,25 earth days long, and the night on the Moon is also 14,25 Earth days long.

Moon is in our immediate neighborhood. So Moon is at the same distance from the sun, as Earth, R=1 AU (astronomical unit).

The year average solar irradiation intensity on the top of atmosphere for Moon and Earth is the same So = 1362 W/m2.

They say on the top of the atmosphere, it means the solar intensity which reaches a celestial body and falls on it.

It is all right then, that during these 14,25 earth days long lunar day the Moon's surface gets warmed at much higher temperatures than the Earth.

It is well known that when a planet rotates faster its daytime maximum temperature lessens and the night time minimum temperature rises.

But there is something else very interesting happens.

When a planet rotates faster it is a warmer planet.

The understanding of this phenomenon comes from a deeper knowledge of the Stefan-Boltzmann Law.

It happens so because when rotating faster a planet's surface has a new radiative equilibrium temperatures to achieve.

So that is what happens:

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

Tmin↑→ T↑mean ← T↓max

when n2>n1 (it happens because Tmin↑ grows higher than T↓max goes down)

It happens in accordance to the Stefan-Boltzmann Law.

Let's explain:

Assuming a planet rotates faster and

Tmax1 -Tmax2 = 1°C.

Then, according to the Stefan-Boltzmann Law:

Tmin2 -Tmin1 > 1°C

Consequently Tmean2 > Tmean1.

Assuming a planet rotates faster (n2>n1).

If on the solar irradiated hemisphere we observe the difference in average temperature

Tsolar1 -Tsolar2 =1°C

Then the dark hemisphere average temperature

Tdark2 -Tdark1 >1°C

Consequently the total average

Tmean2 > Tmean1

So we shall have:

Tmin↑→ T↑mean ← T↓max

The faster a planet rotates (n2>n1) the higher is the planet’s average (mean) temperature T↑mean.

A numerical example:

Assuming a planet with

T1max = 200 K, and T1min = 100 K

Assuming this planet rotates faster,

so T2max = 199 K.

When rotating faster what is the planet's T2min?

J1emit.max ~ (T1max)⁴ ,

(200 K)⁴ = 1.600.000.000

J2emit.max ~ (T2max)⁴ ,

(199 K)⁴ = 1.568.000.000

J1emit.max - J2emit.max =

= 1.600.000.000- 1.568.000.000 = = 31.700.000

On the other hand on the dark side we should have a greater warming than a one degree

( 200 K - 199 K =1 oC ) cooling we had on the solar irradiated side.

J1emit.min ~ (T1min)⁴ ,

(100 K)⁴ = 100.000.000

J2emit.min ~ (T2min)⁴ ,

(107,126 K)⁴ = 131.079.601

J2emit.min - J1emit.min =

= 131.698.114 -100.000.000 =

= 31.698.114

The result is

T2min = 107,126 K

As we see in this numerical example, when rotating faster maximum temperature on the solar irradiated side subsides from

200 K to 199 K.

On the other hand the minimum temperature on the dark side rises from

100 K to 107,126 K.

So when the solar irradiated side gets on average cooler by 1 degree oC, the dark side gets on average warmer by 7,126 degrees oC.

And as a result the planet total average temperature gets higher.

That is how when a planet rotating faster the radiative equilibrium temperatures are accomplished.

It happens so because when rotating faster a planet's surface has a new radiative equilibrium temperatures to achieve.

Consequently, when rotating faster, the planet's mean temperature rises.

Thus when a planet rotates faster its mean temperature is higher.

Conclusion: Earth's faster rotation rate, 1 rotation per day, makes Earth a warmer planet than Moon.

Moon rotates around its axis at a slow rate of 1 rotation in 29,5 days.

Moon gets baked hard during its 14,25 earth days long lunar day.

And Moon also  emits from its very hot surface hard.

What else the very hot surface does but to emit hard, according to the Stefan-Boltzmann emission Law.

The very hot surface emits in fourth power of its very high absolute temperature.

Jemit ~ T⁴

So there is not much energy left to emit during the 14,25 earth days long lunar night.

And it becomes very cold on the Moon at night.

It is in our Earth's immediate neighborhood happens.

Mars and Moon satellite measured mean temperatures comparison:

210 K and 220 K

Let's see what we have here:

Planet or     Tsat.mean

moon           measured

Mercury         340 K

Earth             288 K

Moon            220 Κ

Mars             210 K

Let’s compare then:

Moon: Tsat.moon = 220K

Moon’s albedo is amoon = 0,136

What is left to absorb is (1 – amoon) = (1- 0,136) = 0,864

Mars: Tsat.mars = 210 K

Mars’ albedo is amars = 0,25

What is left to absorb is (1 – amars) = (1 – 0,25) = 0,75

Mars /Moon satellite measured temperatures comparison:

Tsat.mars /Tsat.moon = 210 K /220 K = 0,9545

Mars /Moon what is left to absorb (which relates in ¼ powers) comparison, or in other words

the Mars /Moon albedo determined solar irradiation absorption ability:

( 0,75 /0,864 )¹∕ ⁴ = ( 0,8681 )¹∕ ⁴ = 0,9652

Conclusions:

1. Mars /Moon satellite measured temperatures comparison

( 0,9545 )

is almost identical with the Mars /Moon albedo determined solar irradiation absorption ability

( 0,9652 )

2. If Mars and Moon had the same exactly albedo, their satellites measured temperatures would have been exactly the same.

3. Mars and Moon have two major differences which eliminate each other:

The first major difference is the distance from the sun both Mars and Moon have.

Moon is at R = 1 AU distance from the sun and the solar flux on the top is So = 1.362 W/m²

( it is called the Solar constant ).

Mars is at 1,524 AU distance from the sun and the solar flux on the top is

S = So*(1/R²) = So*(1/1,524²) = So*1/2,32 .

(1/R²) = (1/1,524²) = 1/2,32

Mars has 2,32 times less solar irradiation intensity than Earth and Moon have.

Consequently the solar flux on the Mar’s top is 2,32 times weaker than that on the Moon.

The second major difference is the sidereal rotation period both Mars and Moon have.

Moon performs 1 rotation every 29,5 earth days.

Mars performs 1 rotation every ( 24,25hours / 24hours/day ) = 1,0104 day.

Consequently Mars rotates 29,5 /1,0104 = 29,1964 times faster than Moon does.

So Mars is irradiated 2,32 times weaker, but Mars rotates 29,1964 times faster.

And… for the same albedo, Mars and Moon have the same satellite measured mean temperatures.

Let’s take out the calculator now and make simple calculations:

The rotation difference's fourth root is

(29,1964)¹∕ ⁴ = 2,3245 

And the rotating /irradiating comparison 

(29,1964)¹∕ ⁴ /2,32 = 2,3245 /2,32 = 1,00195 

It is only 0,195 % difference

When rounded the difference is 0,20 %

It is obvious now, the Mars’ 29,1964 times faster rotation equals the Moon’s 2,32 times higher solar irradiation.

That is why the 29,1964 times faster rotating Mars has almost the same average satellites measured temperature as the 2,32 times stronger solar irradiated Moon.

Thus we are coming here again to the same conclusion:

The Faster a Planet Rotates, the Higher is the Planet's Average Temperature.

Now we are ready to continue for the 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 entire 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 a = 0,3 and a factor Φ = 0,47 we have:

Te.earth = 288,36 K or 15°C.

This temperature is confirmed by the satellites measured

Tsat.mean.earth = 288 K.

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.

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 incomplete

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.

Planet Te.incompl Te.compl Tsat.mean

Mercury   437 K    346,11 K     340 Κ

Earth        255 K    288,36 K     288 K

Moon       271 Κ    221,74 Κ     220 Κ

Mars      209,91 K  213,42 K    210 K

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.

Te.earth 

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 =

Τe.earth = ( 6.914.170.222,70 )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.

Conclusions:

The complete formula produces remarkable results.

The calculated planets’ temperatures are almost identical with the measured by satellites.

Planet or…..Te. incomplete….Te.complete…Tsat.mean

Moon

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…………….209,91 K……..213,42 K……..210 K

The 288 K – 255 K = 33 oC difference does not exist in the real world.

There are only traces of greenhouse gasses. The Earth’s atmosphere is very thin. There is not any measurable Greenhouse Gasses Warming effect on the Earth’s surface.

Te.moon

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 = 221,74 K

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.

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.

Te.mars

(1/R²) = (1/1,524²) = 1/2,32

Mars has 2,32 times less solar irradiation intensity than Earth has

Mars’ albedo: amars = 0,25

Mars performs 1 rotation every ( 24,25hours / 24hours/day ) = 1,0104 day

N = 1 /1,0104 = 0,98971 rotations /day

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) 1.362W/m²*(1/2,32)*(150*0,98971*0,18)¹∕ ⁴ /4*5,67*10⁻⁸ W/m²K⁴ ]¹∕ ⁴ =

=( 2.074.546.264,23 )¹∕ ⁴ = 213,42 K

Te.mars = 213,42 K

The calculated Mars’ effective temperature

Te.mars = 213,42 K is only by 1,63% higher than that measured by satellites

Tsat.mean.mars = 210 K !

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 Tsat.mean measured by satellites:

           Te. incompl  Te.compl  Tsat.mean

Mercury   437 K      346,11 K    340 Κ

Earth        255 K      288,36 K    288 K

Moon       271 Κ      221,74Κ     220 Κ

Mars       209,91 K  213,42 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.

We use more major parameters for the planet effective temperature calculating formula.

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 basic 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 solar irradiation absorbing property).

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.

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!

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 = σΤe⁴/(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.

These 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. 

A Real Planet's Case

When a planet is considered as a blackbody's properties celestial body we are having very confusing results.

A real planet, not a theoretical one, but a real planet "in flesh and blood" is very much different from a blackbody's properties celestial object.

The Real Planet's Surface Properties:

1. The planet's surface has not an infinitive conductivity. Right the opposite takes place. The planet's surface conductivity is very small, when compared with the solar irradiation intensity and the planet's surface infrared emissivity intensity.

2. The planet's surface has thermal behavior properties. The planet's surface has a specific heat, cp.

3. The incident on the planet solar irradiation does not being distributed instantly and evenly on the entire planet's surface area.

4. Planet does not accept the entire solar irradiation incident in planet's direction. Planet accepts only a small fraction of the incoming solar irradiation. This happens because of the planet's albedo, and because of the planet's smooth and spherical surface reflecting qualities, which we refer to as "the planet's solar irradiation accepting factor Φ".

5. Planet's surface has not a constant intensity solar irradiation effect. Planet's surface rotates under the solar flux. This phenomenon is dicisive for the planet's surface infrared emittance distribution.

The real planet's surface infrared radiation emittance distribution intensity is a planet's rotational speed dependent physical phenomenon.

Planet's Surface Radiative Equilibrium Temperature

What solar irradiated planet's surface does is to develop the incoming energy to achieve a radiative equilibrium.

It is an equation: energy in = energy out.

The mechanisms planet has to achieve the surface radiative equilibrium:

1. The negative feedbacks.

a). The rising precipitation, due to warming from the excess energy, rises the Earth's cloud cover. This in turn magnifies the planet's albedo. So some less energy reaches the surface.

b). The loss of Arctic oceanic ice cover, due to warming from the excess energy, opens the Arctic oceanic waters. This in turn magnifies the Arctic oceanic surface emissivity (water has higher emissivity compared to the ice, and water has a much higher emissivity compared to the snow covered Arctic oceanic ice fields). So some more energy is emitted to space from the Earth's surface to come closer to the radiative equilibrium (radiative balance).

2. The heat accumulation and the rise of the planet's average temperature.

Some of the solar energy that is not emitted out to space forms the accumulated heat, mostly in the oceanic waters and also in the land masses. The heat accumulation rises the planet's temperature. When the planet's temperature risen planet's infrared radiation energy emissions rise too.

Thus the Planet's Surface Radiative Equilibrium Temperature being formatted.

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.

We use much more parameters 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.

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 Tsat.mean measured by satellites:

Planet   Te.incompl. Te.compl.Tsat.mean

or 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           209,91 K  213,42 K   210 K

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

That is why I say in the real world the 288 K - 255 K = Δ 33 oC difference does not exist.

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.