The Planet's Effective Temperature Complete Formula

Te.complete = [ Φ (1-a) S (β*N*cp)¹∕ ⁴ /4σ ]¹∕ ⁴ gives a wonderful result even for the closest to the sun planet Mercury.

Te.complete.mercury = 346,11 K

This result is almost identical to the measured by satellites

Tsat.mean mercury = 340 K

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

Mercury………….437 K……….346,11 K……..340 K

It is time to abandon the old

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

.

### 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 live in Athens Greece.**

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

** The date is October 11, 2019**

**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;**

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*.

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.*

.

### 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 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…

## .

### Moon is in our immediate neighborhood

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.

.

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

**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.**

(it happens because **Tmin↑↑** grows higher than **T↓max** goes down)

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:**

It happens in accordance to the Stefan-Boltzmann Law.

Let's explain:

Assuming a planet rotates faster and

**Tmax2 -Tmax1 = -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

**Tsolar2-Tsolar1 = -1°C**

Then the dark hemisphere average temperature

**Tdark2 -Tdark1 >1°C**

Consequently the total average

**Tmean2 > Tmean1**

So we shall have:

**Tdark↑↑→ T↑mean ← T↓solar**

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

**Tsolar1 = 200 K,** and **Tdark1 = 100 K**

Assuming this planet rotates faster,

so **Tsolar2 = 199 K.**

When rotating faster what is the planet's **Tdark2 ?**

**J1emit.solar ~ (T1solar)⁴ ,**

**(200 K)⁴ = 1.600.000.000**

**J2emit.solar ~ (T2solar)⁴ ,**

**(199 K)⁴ = 1.568.000.000**

**J2emit.solar - J1emit.solar =**

**= 1.568.000.000- 1.600.000.000 =**

** = - 31.700.000**

So we have (** - 31.700.000** ) less emitting on the solar side. It should be compensated by the increased emission on the dark side (** + 31.700.000** ).

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

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

**J1emit.dark ~ (T1dark)⁴ ,**

**(100 K)⁴ = 100.000.000**

**J2emit.dark ~ (T2dark)⁴ ,**

*(107,126 K)⁴* = 131.698.114

**J2emit.dark - J1emit.dark =**

**= 131.698.114 -100.000.000 =**

** = 31.698.114**

The dark side higher temperature to compensate the solar side cooler emission by

( **- 31.700.000** ) would be

**T2dark = 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.

.

### And it becomes very cold on the Moon at night

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

**Mars and Moon **

** 210 K and 220 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.**

### 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.

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 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 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.

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

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.**

.

### A Planet Without-Atmosphere Effective Temperature Complete Formula

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. Earth's Without-Atmosphere Effective Temperature Calculation

**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.

.

### 2. Moon’s Effective Temperature Calculation

**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.

.

### 3. Mars’ Effective Temperature Calculation

**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 !**

.

### 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's 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*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

Planet 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 !**

.

### We can confirm now with great confidence

**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!

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

## .

### 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 = 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.

## .

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

Conclusions:

We had to answer these 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 = σΤe⁴/(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 =**

**Te.earth = 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.

.

The by a smooth spherical body solar irradiation absorption

### A Blackbody Planet Surface Equilibrium Temperature (A Blackbody Effective Temperature)

**A Blackbody Planet Surface Equilibrium Temperature (A Blackbody Effective Temperature)**

A blackbody planet surface is meant as a classical blackbody surface approaching.

Here are the blackbody's properties:

1. Blackbody absorbs the entire incident on the blackbody's surface radiation.

2. Blackbody is considered only as a blackbody's surface physical properties.

3. Blackbody does not consist from any kind of a matter. Blackbody has not mass. Thus blackbody has not a specific heat. Blackbody's cp = 0.

4. Blackbody has a surface dimensions. So blackbody has the radiated area and blackbody has the emitting area. The whole blackbody's surface area is the blackbody's emitting area.

5. The blackbody's surface has an infinitive conductivity. All the incident on the blackbody's surface radiative energy is instantly and evenly distributed upon the whole blackbody's surface.

6. The radiative energy incident on the blackbody's surface the same very instant the blackbody's surface emitts this energy away.

7. The emittance temperature the blackbody's surface has according to the Stefan-Boltzmann Law is:

**Te = (Total incident W /Total area m² *σ)¹∕ ⁴ K**

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

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### A Real Planet's case

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

**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.**

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

### 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.

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.

### 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.