# How to Calculate Planetary Temperatures

In the second graph we have the Ratio of Planet Measured Temperature to the Corrected Blackbody Temperature (Tsat /Te.correct). Link [30] In this graph we use in (Tsat /Te.correct) the planet corrected blackbody temperatures – which are the planet effective temperatures Te.correct corrected by the use of the Φ -factor. The Φ = 0,47 for smooth surface planets and moons, and the Φ = 1 for the rough surface planets and moons. As we can see, in the second graph, the red dot planets and the green dot planets have stretched in a linear functional relation according to their Warming Factor = (β*N*cp)^1/16 values. The bigger is the planet’s or moon’s Warming Factor, the higher is the (Tsat /Te.correct) ratio. It is obviously a linearly related function.

On a recent comment thread at Climate Etc. Christos Vournas provided a link to his blog. After spending time reading his articles I made this post to introduce aspects of his studies and thinking that I find persuasive. His home page sets the theme The Planet Surface Rotational Warming Phenomenon. Below are just a few excerpts from Vournas’ blog in italics with my bolds.

[Note:  I have added two additional posts on Vournas findings Earthshine and Moonshine: Big Difference  and Beware Energy Balance Cartoons]

Introduction

My name is Christos J. Vournas, M.Sc. mechanical engineer, living in Athens Greece. 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.

Φ – the next discovery – 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 and the Planet’s Without-Atmosphere Mean Surface Temperature Equation.

##### The Planet Surface Rotational Warming Phenomenon

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

Tmin↑↑→ T↑mean ← T↓max

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.

##### A Planet Without-Atmosphere Mean Surface Temperature Equation

A Planet Without-Atmosphere Mean Surface Temperature Equation derives from the incomplete Te equation which is based on the radiative equilibrium and on the Stefan-Boltzmann Law.

Using the new equation, the new estimate Tmean closely matches the estimate surface temperatures from satellite observations:

 Planet Te.incomp Tmean Tsat.mean Mercury 437,30K 323,11K 340K Earth 255K 287,74K 288K Moon 271K 221,74K 220K Mars 209,91K 213,59K 210K

We have moved further from the incomplete effective temperature equation

Te = [ (1-a) S / 4 σ ]¹∕ ⁴

(which is in common use right now, but actually it is an incomplete planet Te equation and that is why it gives us very confusing results)

a – is the planet’s surface average albedo

S – is the solar flux, W/m²

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

We have discovered the Planet Without-Atmosphere Mean Surface Temperature Equation

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

The Planet Without-Atmosphere Mean Surface Temperature Equation is also based on the radiative equilibrium and on the Stefan-Boltzmann Law.

The Equation is being completed by adding to the incomplete Te equation the new parameters Φ, N, cp and the constant β.

Φ – is the dimensionless Solar Irradiation accepting factor

Φ – is the dimensionless Solar Irradiation accepting factor.  It is a realizing that a sphere’s surface absorbs the incident solar irradiation not as a disk of the same diameter, but accordingly to its spherical shape.  For a smooth spherical surface Φ = 0,47

N – rotations /day, is the planet’s axial spin

cp – cal /gr*oC, is the planet’s surface specific heat capacity

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

The Planet Without-Atmosphere Mean Surface Temperature Equation is also based on the radiative equilibrium and on the Stefan-Boltzmann Law.

But the New Equation doesn’t consider planet behaving as a blackbody, and the New Equation doesn’t state planet having a uniform surface temperature.

Interesting, very interesting what we see here:

 Planet Tsat mean Rotations Tmin Tmax Mercury 340 K 1/176 100K 700K Earth 288 K 1 Moon 220 Κ 1/29,5 100K 390K Mars 210 K 0,9747 130K 308K

Earth and Moon are at the same distance from the Sun R = 1 AU.

Earth and Mars have almost the same axial spin N = 1rotation /day.

Moon and Mars have almost the same satellite measured average temperatures 220 K and 210 K.

Mercury and Moon have the same minimum temperature 100 K.

Mars’ minimum temperature is 130 K, which is much higher than for the closer to the Sun Mercury’s and Moon’s minimum temperature 100 K.

The planet’s effective temperature old Te = [ (1-a) S /4σ ]¹∕ ⁴ incomplete equation gives very confusing results.

And the faster rotating Earth and Mars appear to be relatively warmer planets.

We ended up to the following remarkable results

To be honest with you, at the beginning, I was surprised myself with these results.

You see, I was searching for a mathematical approach…

##### We use more major parameters for the planet’s surface temperature equation.

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 planet’s axial spin N rotations/day.

4. The thermal property of the surface (the specific heat capacity cp).

5. The planet’s surface solar irradiation accepting factor Φ ( the spherical surface’s primer solar irradiation absorbing property ).

Altogether these parameters are combined in the Planet’s Without-Atmosphere Surface Mean Temperature Equation:

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

Earth’s Without-Atmosphere Mean Surface Temperature Equation
Tmean.earth

So = 1.361 W/m² (So is the Solar constant)

Earth’s albedo: aearth = 0,306

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 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

Earth’s Without-Atmosphere Mean Surface Temperature Equation Tmean.earth is:

Tmean.earth = [ Φ (1-a) So (β*N*cp)¹∕ ⁴ /4σ ]¹∕ ⁴

Τmean.earth = [ 0,47(1-0,306)1.361 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 =

Τmean.earth = [ 0,47(1-0,306)1.361 W/m²(150*1*1)¹∕ ⁴ /4*5,67*10⁻⁸ W/m²K⁴ ]¹∕ ⁴ =

Τmean.earth = ( 6.854.897.370,96 )¹∕ ⁴ = 287,74 K

Tmean.earth = 287,74 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 equation produces remarkable results.

A Planet Without-Atmosphere Surface Mean Temperature Equation gives us a planet surface mean 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!

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.

The air density is some 1,23 kg/m³, and it is a very thin atmosphere of 1 bar at sea level.… In Earth’s very thin atmosphere  there are on average 1% H₂O and 0,04% CO₂.  Those two are trace gases in Earth’s very thin atmosphere. H₂O and CO₂ very tiny contents in earth’s atmosphere are not capable to absorb the alleged huge “absorbed by atmosphere 70%-85% outgoing IR radiation” portion.

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

##### Postscript:  Reversed Milankovitch Cycle

Of course climate changes.  And of course the planet’s rotational spin is almost constant.  Also Earth has a very thin atmosphere; Earth has a very small greenhouse phenomenon in its atmosphere and it doesn’t warm the planet.

The cause of climate change is not the Earth’s atmosphere. The cause of climate change is orbital.  Milutin Milankovitch has explained everything 100 years ago.

The ( Ṃ ↓ ) represents the Original Milankovitch Cycle grapheme.  And the ( Ẇ ↑ ) represents the Reversed Milankovitch Cycle grapheme.

( Ṃ ↓ ) – supposedly this is the Original Milankovitch Cycle. Please take notice of the dot under ( Ṃ ↓ ).  The dot’s position represents the present time, when Planet Earth is in Original Milankovitch Cycle Minima:  The Original Milankovitch Cycle shows a cooling trend.

( Ẇ ↑ ) The Reversed Milankovitch Cycle shows a warming trend.

Milankovitch had to reverse his cycle to match the instrumental data. But he didn’t have time.  It was a critical mistake in Milankovitch’s assumptions.  Now it is time for us to make the necessary correction. 100 years have passed, Milankovitch agrees, if it is necessary, for us to make a correction.

When comparing with the Perihelion point, which is at January 2, the solar irradiance Earth receives now is 7% less. As a result we have at the North Hemisphere much cooler summers and much warmer winters.  In 10.000 (ten thousand) years from now, Earth’s axis will be pointing at star Vega, instead of Polaris at which it points now. So in 10.000 years the Winter Solstice will occur when Earth is in Aphelion (it happens now with Earth in Perihelion).

As a result in 10.000 years we would have at the North Hemisphere much warmer summers and much cooler winters. A shift of 7% in the Hemispheres’ insolation intensity will happen.  Instead of the Southern Hemisphere (as it happens now) with its vast oceans accumulative capacity… there would be a +7% stronger insolation on the North Hemisphere’s plethora of continental areas.

We know continents do not accumulate heat so much effectively as oceans do, thus Earth will gradually cool down, until a New Ice Age commences!

As for the current warming phase – we still receive the +7% solar energy onto Southern Hemisphere’s oceans… and oceans willingly accumulate the excess solar energy…It happens so during the current Winter Solstices, when Earth is still tilted towards sun with its Southern Hemisphere’s vast oceanic waters.

The warming trend we observe now started some 6.500 years ago. It is a very slow process. The MWP ( the Medieval Warm Period ) is a confirmation of the existence of a long warming trend.  The LIA ( the Little Ice Age ) was observed as a colder atmosphere and more snowy winters. Also the glaciers were increasing.

On the other hand oceans continued accumulating heat.  It is a very long cycle. We are observing the Reversed Milankovitch Cycle culmination period. It will last about a millennia and a half and then there will be a cooling trend.

Right now Planet Earth is in an orbital forced warming trend. And these are culmination times.  The very slow warming trend will continue for about a 1,5 millennia on. Then slowly and gradually the Global Temperatures will become cooler.

1. · July 22

Ron, thank you for doing a synopsis of my findings at your blog.
We are capable now to theoretically calculate planet mean surface temperatures.

Ron, you have created a very informative blog. I will regularly visit and read your blog on the daily basis!
Thank you again,
Christos

https://www.cristos-vournas.com

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• · July 22

Thank you Christos for your rigorous math approach to this issue. I appreciated browsing your blog and wanted to summarize what I understood. Discovering a fresh viewpoint makes my day.

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2. · July 23

Reblogged this on Climate Collections.

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7. · August 3

Great article. Congratulations!

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• · August 3

Thanks ren. Christos did the work, and I learned a lot.

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