The Planet Surface Rotational Warming Phenomenon

The Planet Mean Surface Temperature Equation Tmean = [ Φ (1-a) S (β*N*cp)¹∕ ⁴ /4σ ]¹∕ ⁴

Φ -Factor is an analogue of the well known Drag Coefficient Cd=0,47

Φ -Factor is an analogue of the well known Drag Coefficient Cd=0,47 for smooth sphere in the parallel fluid flow.

And it is about the by sphere’s surface the portion of incident energy acceptance!

From Wikipedia, the free encyclopedia

“In fluid dynamics, the drag coefficient (commonly denoted as: Cd, Cx or Cw) is a dimensionless quantity that is used to quantify the drag or resistance of an object in a fluid environment, such as air or water. It is used in the drag equation in which a lower drag coefficient indicates the object will have less aerodynamic or hydrodynamic drag. The drag coefficient is always associated with a particular surface area.[3]

The drag coefficient of any object comprises the effects of the two basic contributors to fluid dynamic drag: skin friction and form drag.”

Planet is a sphere and it reflects and absorbs like a sphere

Planet does not reflect and absorb as a disk. Planet reflects and absorbs as a sphere.


The by a smooth spherical body solar irradiation absorption

Jabs = Φ (1 - a ) S π r²

Φ = 0,47

Φ factor explanation

Φ factor explanation

The Φ - solar irradiation accepting factor - how it "works". It is not a planet specular reflection coefficient itself.

There is a need to focus on the Φ factor explanation.

Φ factor emerges from the realization that a sphere reflects differently than a flat surface perpendicular to the Solar rays.

Φ – is the dimensionless Solar Irradiation accepting factor

"Φ" is an important factor in the Planet's Surface Mean Temperature Equation:

Tmean.planet = [ 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 reflection.

There is the specular reflection and there is the diffuse reflection.

The planet's surface Albedo "a" accounts for the planet's surface diffuse reflection.

Albedo is defined as the ratio of the scattered SW to the incident SW radiation, and it is very much precisely measured (the planet Bond Albedo).

So till now we didn't take in account the planet's surface specular reflection.

A smooth sphere, as some planets are, have the invisible from the space and so far not detected and not measured the specular reflection.

The sphere's specular reflection cannot be seen from the distance, but it can be seen by an observer situated on the sphere's surface. Thus, when we admire the late afternoon sunsets on the sea we are blinded from the brightness of the sea surface glare. It is the surface specular reflection what we see then.

Jsw.not-reflected = Φ*(1-a) *Jsw.incoming

For a planet with albedo a = 0 (completely black surface planet) we would have

Jsw.reflected = [1 - Φ*(1-a)]*S *π r² =

Jsw.reflected = (1 - Φ) *S *π r²

For a planet which captures the entire incident solar flux (a planet without any outgoing specular reflection) we would have Φ = 1

Jsw.not-reflected = Φ*(1-a) *Jsw.incoming

Jsw.reflected = a *Jsw.incoming

And For a planet with Albedo a = 1 , a perfectly reflecting planet

Jsw.not-reflected = 0 (no matter what is the value of Φ)

In general:

The fraction left for hemisphere not-reflected is

Jnot-reflected = Φ (1 - a ) S π r²

We have Φ for different planets' surfaces varying

0,47 ≤ Φ ≤ 1

And we have surface average Albedo "a" for different planets' varying

0 ≤ a ≤ 1


Φ is never less than 0,47 for planets (spherical shape). Also, the coefficient Φ is "bounded" in a product with (1 - a) term, forming the Φ(1 - a) product cooperating term.

So Φ and Albedo are always bounded together. The Φ(1 - a) term is a coupled physical term.

The Φ(1 - a) term "translates" the not-reflected of a disk into the not-reflected of a smooth 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 the disk:

Jdirect = π r² S

But the not-reflected Solar energy by the hemisphere's area of 2π r² is:

Jnot-reflected = Φ*( 1 - a) π r² S

It happens because a smooth hemisphere of the same radius "r" not-reflects the Φ*(1 - a)S portion 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 not-reflects only the Φ*(1 - a)S portion of the directly incident on the disk Solar irradiation.

Jnot-reflected = 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".

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

Energy in = Φ(1 - a)

left side of the Planet Radiative Energy Budget.

Conclusively, the Φ -Factor is not the planet specular reflection portion itself.

The Φ -Factor is the Solar Irradiation Accepting Factor (in other words, Φ is the planet surface shape and roughness coefficient).


Instead of Te, the Planet Corrected Effective Temperature Te.correct

At a first approach, when without the Rotational Warming phenomenon implementation, I use instead of Te, the Planet Corrected Effective Temperature Te.correct.

The formula is:

[ Φ(1-a) /4σ ]¹∕ ⁴

Φ = 0,47

(the 0,47 is for smooth surface planets without atmosphere, the factor Φ accounts for the smooth planet surface specular reflection)

Table of results for Te and Te.corrected

Planet........ Te..........Te.correct

Mercury.....440 K.......364 K

Moon.........270 K......224 K

Earth.........255 K.......210 K

Mars,,,,,,,,,,210 K......174 K



The Mercury’s Φ = 0,47 Paradigm Confirmation

We now have chosen Mercury for its very low albedo a=0,088 and for its very slow rotational spin N=1/175,938 rotations/day.

Mercury is most suitable for the incomplete effective temperature formula definition - a not rotating planet, or very slow rotating. Also it is a planet where albedo plays little role in energy budget.

These (Tmean, R, N, and albedo) parameters of the planets are all satellite measured. These parameters of the planets are all observations.


Tsat.mean.340 K….220 K…210 K

R…......0,387 AU..1 AU..1,525 AU


N…1 /175,938..1 /29,531..0,9747



Let’s calculate, for comparison reason, the Mercury’s effective temperature with the old incomplete equation:

Te.incomplete.mercury = [ (1-a) So (1/R²) /4σ ]¹∕ ⁴

We have

(1-a) = 0,932

1/R² = 6,6769

So = 1.362 W/m² - it is the Solar constant ( the solar flux on the top of Earth’s atmosphere )

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

Te.incomplete.mercury = [ 0,932* 1.362 W/m² * 6,6769 /4*5,67*10⁻⁸ W/m²K⁴ ]¹∕ ⁴ =

Te.incomplete.mercury = ( 37.369.999.608,40 )¹∕ ⁴ = 439,67 K

Te.incomplete.mercury = 439,67 K

And we compare it with the

Tsat.mean.mercury = 340 K - the satellite measured Mercury’s mean surface temperature

Amazing, isn’t it? Why there is such a big difference between the measured Mercury’s mean surface temperature, Tmean = 340 K, which is the correct, ( I have not any doubt about the preciseness of satellite planets' temperatures measurements ) and the Mercury's Te by the effective temperature incomplete formula calculation Te = 439,67 K?

Let’s put these two temperatures together:

Te.incomplete.mercury = 439,67 K = 440 K

Tsat.mean.mercury = 340 K

Very big difference, a 100°C higher!

But why the incomplete effective temperature equation gives such a wrongly higher result?

The answer is simple – it happens because the incomplete equation assumes planet absorbing solar energy as a disk and not as a sphere.

We know now that even a planet with a zero albedo reflects the [1 - Φ(1-a)]S portion of the incident solar irradiation.

Imagine a completely black planet; imagine a completely invisible planet, a planet with a zero albedo. This planet still reflects the [1 - Φ(1-a)]S portion of the incident on its surface solar irradiation.

The satellite measurements have confirmed it. The Mercury’s Φ = 0,47 Paradigm has confirmed it:

Φ - the dimensionless planet surface solar irradiation accepting factor Planet reflects the (1-Φ + Φ*a) portion of the incident on the planet's surface solar irradiation.

Here "a" is the planet's average albedo. So we always have:

Jreflected = (1-Φ + Φ*a)S Jabsorbed = Φ(1-a)S


What is Earth's averaged on the entire surface absorbed solar SW radiation?

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's dependent value.

2. The planet's spherical shape.

Φ is the planet solar irradiation accepting factor (the spherical shape and roughness coefficient).

Φ = 0,47 for a smooth surface sphere

What we had till now:

Jsw.incoming - Jsw.reflected = Jsw.absorbed


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


Jsw.reflected = a* Jsw.incoming


What we have now is the following:

Jsw.incoming - Jsw.reflected = Jsw.not-reflected


Jsw.not-reflected = Φ* (1-a) * Jsw.incoming


For Planet Earth (smooth surface planet Φ = 0,47)

Jsw.not-reflected = 0,47*(1-a)*1.361 W/m² =

= 0,47*0,694*1.362W/m² = 444,26 W/m²

Averaged on the entire Earth's surface we obtain:

Jsw.not-reflected.average = [ 0,47*(1-a)*1.361 W/m² ] /4 =

[ 0,47*0,694*1.361W/m² ] /4 = 444,26 W/m² /4 =

= 111,07 W/m²

Jsw.not-reflected.average = 111 W/m²


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

Tmin→ T↑mean ← Tmax