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

About author and about the discovery


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

I live in Athens Greece.

My e-mail address is:

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;

The method I use is the ”Planet Surface Temperatures Comparison Method”

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.

Φ - the next discovery - it is the dimensionless Solar Irradiation accepting factor - very important.


The further studies led me to discover the

Rotating Planet Spherical Surface Solar Irradiation Absorbing-Emitting Universal Law


the Planet Without-Atmosphere Mean Surface Temperature Equation.

A Planet Universal Law Equation


As you know, to maintain a Planet Universal Law Equation 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 Planet's Mean Surface Temperature Equation has the wonderful ability the calculated results closely matching to the measured by satellites planets' mean temperatures. This New Universal Equation can be applied to all the without atmosphere planets and moons in a solar system.

The more we compare the planets' surface temperatures, the more we understand the planets' surface warming phenomenon.

Flux is not heat... The incident solar flux cannot be averaged over the planet surface

The “Effective Temperature” removes any effect of spin. It is the temperature a body would reach if input energy were uniformly spread over the entire surface. It negates the effect of spin. Slow spinning can result in surface temperatures below the effective temperature but not above."

Effective temperature is a mathematical abstraction. Every planet has a calculated effective temperature no matter how fast planet rotates.

Why should we consider the planet effective temperature to be the highest mean surface temperature a real planet mean surface temperature can theoretically achieve?

The Stefan-Boltzmann emission law states:

Jemit = σ*T⁴ W/m² where T is a uniform surface emission temperature.

The amount of energy emitted from a surface with A m² area is:

energy out = A (m²) * σ*T⁴ W/m² = A * σ*T⁴ W

or energy out = A * Jemit

The Stefan-Boltzmann blackbody surface already has the uniform temperature T, because it is uniformly warmed to that T temperature.

Planet surface cannot be considered as a blackbody uniformly warmed surface. The incident solar flux cannot be averaged over the planet surface.

Flux is not heat... The fact that planet receives a solar flux, does not mean its energy first warms planet surface and only then the warmed surface emits the same incident amount of solar energy as IR EM emission.

EM radiation is not a heat transfer process, like the heat conduction is...

EM radiation is a flux-surface-matter interaction process.


Heat is the transfer of molecular kinetic energy.

The darkside temperature of Moon is determined by thermal conductance through the regolith, and is mostly a constant temperature.

That is why the night-time temperature in the Graph appears as linear. Because the night-time IR emission intensity is ruled by conductance through the regolith, and conductance is a linear function…


1. Planet cannot reach uniform surface temperature because it is solar irradiated from one side only.

2. The planet blackbody Te equation Te = [ (1-a) S /4σ ]¹∕ ⁴ is mistaken because it is based on concept that the not reflected portion of incident solar flux's SW EM energy warms planetary surface (gets absorbed in form of heat) and then the heat is distributed evenly over the entire planetary surface.

In fact only a small fraction is transformed to heat. The Te equation is a mathematical abstraction.


Φ – is the Planet Surface Solar Irradiation Accepting Factor.

It is a New and very important concept for the correct estimation of planetary “energy in” (not reflected) SW EM energy estimation.

I have proposed to the scientific community… because Φ is a key parameter in the planetary Radiative Energy Budget.


Let’s continue with this mathematical abstraction: The planet blackbody Te equation

Te = [ (1-a) S /4σ ]¹∕ ⁴

the Φ -factor for planet Earth is Φ=0,47

The Planet Corrected effective temperature (which still remains a mathematical abstraction) Te.correct for Earth is:

Te = [ Φ(1-a) Sο /4σ ]¹∕ ⁴

When substituting values The corrected mathematical abstraction Te for planet Earth is Te = 210 K.

The measured Albedo for planet smooth spherical surface does not entirely cover the smooth spherical surface specular reflection issue.


When I got myself "involved", I started from the very first term - it was the planetary average surface Albedo.

In my understanding of the reflection, the measured Albedo for planet smooth spherical surface does not entirely cover the smooth spherical surface specular reflection issue...

I searched in the Internet for publications describing the parallel incident radiation on the smooth spherical surface reflection - there was none.

Then it was when I looked for the analog the Drag Coefficient for the smooth sphere in the parallel fluid flow.

The Drag Coefficient for sphere is Cd = 0,47

Drag Coefficient is a measured value. It is the portion of the parallel fluid flow's incident on the sphere energy, the sphere's resistance.

When a sphere is hit by 1 Newton it has to resist with 0,47 Newton for the sphere to stay there... So the 0,47 is the portion of energy left for the sphere to deal with (it is the "absorbed" or, in other words, the accepted portion of the incident parallel flow's Total interacting with sphere's surface energy).

Back to planet surface not-reflected portion of the incident solar flux.

For a smooth surface planet with Albedo a=0 the not-reflected portion of the incident solar flux S would be as an analog to the drag coefficient Cd=0,47 as:

Φ*S where Φ = 0,47

and in the general case:

Φ(1- a)S is the not-reflected portion of the incident solar flux S.


Thus the theoretical (the mathematical abstraction) planet blackbody effective temperature formula:

Te = [ (1-a) S /4σ ]¹∕ ⁴

is corrected as:

Te.corrected = [ Φ (1-a) S /4σ ]¹∕ ⁴


We have, theoretically, the Φ for different surface roughness ratio varying

0,47 ≤ Φ ≤ 1

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

0 ≤ a ≤ 1


Φ is never less than 0,47 for planets (it is because of the 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.

Fortunately, for our research, the planets' multibillion years the surface History has shaped planet surface either for smooth Φ=0,47 version, or for the rough version Φ=1, and there is only the Triton (Neptune's satellite) which has the value of Φ somewhere in between 0,47 ≤ Φ ≤ 1.

Table of results for Te and Te.corrected compared to Tsat and to Rotations/day for smooth surface planets and moons with Φ=0,47

Planet....... Te.....Te.corrected.....Tsat...Rot/day

Mercury.....440 K.......364 K.....340 K...0,00568

Moon.........270 K......224 K......220 K...0,0339

Earth.........255 K.......210 K.....288 K....1

Mars..........210 K......174 K.....210 K...0,9747

Europa.......95,2 K.....78,8 K...102 K...0,2816

Ganymede..107,1 K..88,6 K...110 K....0,1398


As we can see, for the very slow rotating Mercury (0,00568 Rot./day) and for the slow rotating Moon (0,0339 Rot./day) the Corrected theoretical blackbody temperature is

Te.correct > Tsat

Mercury 364 K > 340 K

Moon .... 224 K > 220 K

But for the faster rotating

Earth .... 210 K < 288="">

Mars .....174 K < 210="">

Europa .. 78,8 K < 102="">

Ganymede..88,6 K < 110="">


Te.correct <>

- - - - - - -

Here are the rest of the planets and moons in our solar system. It is those which have rough surface, and, therefore, the Φ=1. The blackbody temperature for these planets and moons is not corrected, because Φ=1.

Planet........ Te...........Tsat......Rot/day

Ιο.............95,16 K.....110 K....0,5559

Calisto....114,66 K...134±11 K...0,0599

Enceladus...55,97 K...75 K......0,7299

Tethys.........66,55 K.....86 ± 1 K....0,52971

Titan...........84,52 K.....93,7 K......0,06289

1).Triton..(Φ=1)..35,4 K...38 K.....0,17021

2).Triton..(Φ=0,47) .29,29 K .38 K..0,17021

Pluto...........37 K............44 K.....0,1565

Charon...... 41,90 K.......53 K......0,1565


We see that for the rest of the planets and moons the theoretical blackbody temperature Te is also smaller than the satellite measured mean surface temperature  Tsat.

Thus for them it is also:

Te <>

for planets it is usually either Φ= 0,47 or Φ= 1


Thus the theoretical (the mathematical abstraction) planet blackbody effective temperature formula:

Te = [ (1-a) S /4σ ]¹∕ ⁴

is corrected as:

Te.corrected = [ Φ (1-a) S /4σ ]¹∕ ⁴



0,47 ≤ Φ ≤ 1 but for planets it is usually 

either Φ= 0,47

or   Φ= 1

That is why the planet blackbody temperature Te is a mathematical abstraction!


“The blackbody temperature — is the same no matter how fast you spin the planet. It doesn’t change with spin.”

That is why the planet blackbody temperature Te is a mathematical abstraction!

We can compare those planets with different rotational spin !!!



 > "Let’s say you are right and the average planetary temperature increases with rotation. That means that when I compare a non-rotating planet to a rotating planet, the rotating planet has a higher temperature. That means the rotating planet is radiating more energy than a non-rotating planet. Where does the additional energy come from? It’s not coming from the rotation the planet."

The answer:

We do not have tidally locked to sun planets or moons - every planet and moon in solar system is a rotating celestial body. So we do not have measured data for non-rotating planets to compare with rotating.

But we have measured data for existing rotating planets and moons, some rotating faster and some rotating slower.

So we can compare those planets with different rotational spin the measured surface temperatures - and we have observed, and we have discovered the Planet Surface Rotational Warming Phenomenon


Let's compare a faster rotating Earth (N = 1 rotation/day) with the slower rotating Moon (N = 1 /29,53 rotation/day = 0,03386 rotation/day).

Earth has higher than Moon average surface Albedo (a =0,306 vs a = 0,11). As a result Moon has to "absorb" 28 % larger amount of the solar SW incident energy than Earth.

It means, on the average surface area, Moon's surface emits 28 % more IR EM outgoing emission energy than Earth's surface.

Nevertheless, Earth's measured average surface temperature is 288 K. Moon's average surface temperature is 220 K.

The very big 288 K - 220 K= 68C difference is explained by the Earth's higher rotational spin (29,53 times higher) plus by the Earth's surface higher average specific heat (5 times higher).

Therefore, the Earth's higher than Moon's (N*cp) product is what makes Earth a much warmer planet than Moon. Earth (N*cp) /Moon (N*cp) ratio is 29,53*1/0,19 = 155,42 times higher!

What more illustrative example !!!

Moon IR radiates 28 % more IR outgoing EM energy than Earth, but, nevertheless, Moon's measured average surface temperature is 68C lower than that of Earth.

To conclude with:

Earth has 155,42 times higher (N*cp) product, and Earth has 68C higher average surface temperature.

And, not to forget, Earth "absorbs" 28 % less incident SW EM solar energy.

It is obvious, that we are ABSOLUTELY right, and the average planetary temperature increases with rotation