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

               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         209,91 K    213,21 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 rotates around its axis at a slow rate of 29,5 days.

The day on the Moon is 14,75 earth days long, and the night on the Moon is also 14,75 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.

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

It is all right then, that during these 14,75 earth days long lunar day the Moon's surface gets warmed at much higher temperatures than the Earth.

Moon gets baked hard during its 14,75 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

A warm object in space loses heat via emission. The hotter is the object, the faster it loses heat. 

So there is not much energy left to emit during the 14,75 earth days long lunar night.

And it becomes very cold on the Moon at night.

It is in our Earth's immediate neighborhood happens.

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 planets mean temperatures.

This Complete Formula can be applied to all the without atmosphere planets and moons in a solar system.

R = 1 AU, is the Earth's distance from the sun in astronomical units

Earth’s albedo: aearth = 0,30

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

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

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

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

So = 1.362 W/m² (So is the Solar 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.

The satellites measured:

Surface temp..Tmin..Tmean..Tmax

Kelvin.............100.K...220.K..390.K

Moon’s albedo: amoon = 0,136

R = 1 AU, is the Moon's distance from the sun in astronomical units

Moon’s sidereal rotation period is 27,32 days. But Moon is Earth’s satellite, so the lunar day is 29,5 days

Moon is a smooth rocky planet, Moon’s surface solar irradiation accepting factor Φmoon = 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 the 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

So = 1.362 W/m² (So is the Solar 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.

The satellites measured:

Surface temp..Tmin..Tmean..Tmax

Kelvin.............130.K...210.K..308.K

R = 1,524 AU, is the Mars' distance from the sun in astronomical units

(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 ( 24,622hours / 24hours/day ) = 1,026 day

N = 1 /1,026 = 0,9747 rotations /day

Mars is a smooth 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 the 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

Mars' Effective Temperature Complete Formula is:

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

Te.mars = [ 0,47 (1-0,25) 1.362W/m²*(1/2,32)*(150*0,9747*0,18)¹∕ ⁴ /4*5,67*10⁻⁸ W/m²K⁴ ]¹∕ ⁴ =

= ( 2.066.635.457,46 )¹∕ ⁴ = 213,21 K

Te.mars = 213,21 K

The calculated Mars’ effective temperature

Te.mars = 213,21 K is only by 1,53% higher than that measured by satellites

Tsat.mean.mars = 210 K !

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 the entire planet:

Jemit = A*σΤe⁴ /(β*N*cp)¹∕ ⁴  (W)

Φ - is the dimensionless Solar Irradiation accepting factor

Φ = 0,47 for a smooth spherical surface

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

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

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

Te - is the Planet's Effective Temperature (K)

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

Φ*S*(1-a) = 4σTe⁴ /(β*N*cp)¹∕ ⁴

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 the Rotating Planet Surface Solar Irradiation Absorbing-Emitting Universal Law constant

N rotations/day, is the planet’s sidereal rotation spin 

cp – is the planet's surface specific heat

cp.earth = 1 cal/gr*oC, it is because Earth has a vast ocean. Generally speaking almost the entire 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

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.

So far we came to the end of this presentation. Its topic was to present the Planet's 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)

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.

A planet reflects incoming short wave solar radiation.

A planet's surface has reflecting properties.

1. The planet's albedo "a".

It is a surface quality dependent value.

2. The planet's spherical shape.

For a smooth planet the solar irradiation reflection is 0,53*Jincoming.

What we had till now:

Jsw.incoming - Jsw.reflected = Jsw.absorbed

Here

Jsw.absorbed = (1-a) * Jsw.incoming

And

Jsw.reflected = a * Jsw.incoming

What we have now is the following:

Jsw.incoming - Jsw.reflected = Jsw.absorbed

Here 

Jsw.reflected = 0,53 * a * Jsw.incoming

And

Jsw.absorbed = Φ* (1-a) * Jsw.incoming

Where

Φ = (1-0,53) = 0,47 Φ = 0,47

Φ is the planet's spherical surface solar irradiation accepting factor.

Conclusion: The planet's absorbed fraction of the SW incoming radiation in total is:

Jsw.absorbed = 0,47 * (1-a) * Jsw.incoming

Total energy emitted to space from the entire planet:

Jemit = A*σΤe⁴ /(β*N*cp)¹∕ ⁴ (W)

Α - is the planet's surface (m²)

For Planet Earth

Jsw.absorbed = 0,47 * (1-a) * 1.362 W/m² =

= 0,47*0,7*1.362W/m² = 448,098 W/m²

Averaged on the entire Earth's surface we obtain:

Jsw.absorbed.average = [ 0,47*(1-a)*1.362 W/m² ] /4 =

= [ 0,47*0,7*1.362W/m² ] /4 = 448,098 W/m² /4 =

= 112,029 W/m²

Jsw.absorbed.average = 112,029 W/m²

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 spin 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.

The planet Earth's and the planet Mars' faster rotation creates the necessary level of the "solar irradiation - planet surface" interaction phenomenon...

which results in the day-time much lower surface temperatures and, consequently, in much lower day-time surface infrared radiation emissions.

The planet Earth’s and the planet Mars’ faster rotation creates the necessary level of the "solar irradiation - planet surface" interaction phenomenon...

which results in higher planet surface 24-hours average temperatures.

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's higher temperature to compensate the solar side's 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.

Φ factor explanation

"Φ" is an important factor in the Planet's Effective Temperature Complete Formula:

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

It is very important the understanding what is really going on with by planets the solar irradiation absorption.

The solar irradiation reflection, when integrated over a planet sunlit hemisphere is:

0,53 [ ( a ) S ] π r²

The fraction left for hemisphere to absorb is:

Φ = 1 - 0,53 = 0,47

or Jabs = Φ (1 - a ) S π r²

The factor Φ = 0,47 "translates" the absorption of a disk into the absorption of a hemisphere with the same radius.

When covering a disk with a hemisphere of the same radius the hemisphere's surface area is 2π r².

The incident Solar energy on the hemisphere's area is the same as on disk:

Jdirect = π r² S

The absorbed Solar energy by the hemisphere's area of 2π r² is:

Jabs = 0,47*( 1 - a) π r² S

It happens because a hemisphere of the same radius "r" absorbs only the 0,47 part of the directly incident on the disk of the same radius Solar irradiation.

In spite of hemisphere having twice the area of the disk, it absorbs only the 0,47 part of the directly incident on the disk Solar irradiation.

Jabs = Φ (1 - a ) S π r² , where Φ = 0,47 for smooth without atmosphere planets. and Φ = 1 for gaseous planets, as Jupiter, Saturn, Neptune, Uranus, Venus, Titan.

Gaseous planets do not have a surface to reflect radiation. The solar irradiation is captured in the thousands of kilometers gaseous abyss. The gaseous planets have only the albedo "a".

And Φ = 1 for heavy cratered planets, as Calisto and Rhea ( not smooth surface planets, without atmosphere ). The heavy cratered planets have the ability to capture the incoming light in their multiple craters and canyons. The heavy cratered planets have only the albedo "a".

Another thing that I should explain is that planet's albedo actually doesn't represent a primer reflection. It is a kind of a secondary reflection ( a homogenous dispersion of light also out into space ).

That light is visible and measurable and is called albedo. The primer reflection from a spherical hemisphere cannot be seen from some distance from the planet. It can only be seen by an observer being on the planet's surface.

It is the blinding surface reflection right in the observer's eye.

That is why the albedo "a" and the factor "Φ" we consider as different values.

Both of them, the albedo "a" and the factor "Φ" cooperate in the Planet Rotating Surface Solar Irradiation Absorbing-Emitting Universal Law:

Φ*S*(1-a) = 4σTe⁴ /(β*N*cp)¹∕ ⁴

And they are also cooperate in the Planet's Effective Temperature Complete Formula:

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

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's surface area ( m² )

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

Te - is a Planet Effective Temperature ( K )

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

Jabs = Φ (1- a ) S W/m² sunlit hemisphere

Jabs.earth = 0,47 ( 1 - 0,30 ) So π r² =

= 0,47*0,70 * 1.362* π r² ( W ) =

Jabs.earth = 0,329 So π r² =

= 0,329* 1.362 π r² =

= 448,10 π r² ( W )

What is going on here is that instead of

Jabs.earth = 0,7* 1.362 π r² ( W )

we should consider

Jabs.earth = 0,329* 1.362 π r² ( W ).

Averaged on the entire Earth's surface we obtain:

Jsw.absorbed.average = [ 0,47*(1-a)*1.362 W/m2 ] /4 =

= [ 0,47*0,7*1.362W/m2 ] /4 = 448,098 W/m2 /4 = 112,029 W/m2

Jsw.absorbed.average = 112,029 W/m2

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 equate 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 29,531 earth days.

Mars performs 1 rotation every ( 24,622hours / 24hours/day ) = 1,026 day.

Consequently Mars rotates 29,531 /1,026 = 28,783 times faster than Moon does.

So Mars is irradiated 2,32 times weaker, but Mars rotates 28,783 times faster.

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

We shall use now from the Planet's Effective Temperature Complete Formula Te = [ Φ (1-a) So (1/R²) (β*N*cp)¹∕ ⁴ /4σ ]¹∕ ⁴

In the Planet's Effective Temperature Complete Formula the N is in fourth root

the (β*N*cp)¹∕ ⁴

The rotation difference's fourth root is

(28,783)¹∕ ⁴ = 2,3162 

And the irradiating /rotating comparison 

2,32 /(28,783)¹∕ ⁴  = 2,32 /2,3162 = 1,001625

It is only 0,1625% difference

When rounded the difference is 0,16 %

It is obvious now, the Mars’ 28,783 times faster rotation equates the Moon’s 2,32 times higher solar irradiation.

That is why the 28,783 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.

And

The planet's rotations per day "N" should be considered in the Te.planet formula in the fourth root :

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

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.