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

## Plus the introduction to the Reversed Milankovitch Cycle. Click above on the box for more

### sciencereview18 | April 18, 2020 at 5:28 pm | Brief overview at Climate Etc. Dr. Judith Curry's blog

Christos Vournas | April 16, 2020 at 7:31 pm | Reply

Everything started with error.

And here is why.

The planet’s old incomplete effective temperature formula:

Te = [ (1-a) S / 4 σ ]¹∕ ⁴ is defined as a planet’s equilibrium temperature in the absence of atmosphere.

When calculated, the Earth’s Te = 255 K, instead of the satellites measured actual Te = 288 K the cause was obvious.

Earth’s surface was considered warmer by +Δ33oC because of the planet Earth’s atmosphere.

It was error.

http://www.cristos-vournas.com

sciencereview18 | April 18, 2020 at 5:28 pm | Reply

https://www.cristos-vournas.com/

Here is a brief overview:

Christos Vournas proposes an improved formula for estimating the planet’s effective temperature, without atmosphere. Using the new formula, the new estimate of the Earth’s effective temperature 288K closely matches the estimate from satellite observations, so that there is then no missing 33C to attribute to some factor such as the alleged greenhouse gas effect.

The improved formula accounts for the fact that “when a planet rotates faster the nighttime temperature rises higher than the daytime temperature lessens” so that “a faster rotating planet is a warmer planet”, and the “understanding of this phenomenon comes from a deeper knowledge of the Stefan-Boltzmann Law.”

So, what was the error in the old formula used by everyone until now?

Christos Vournas said, “The old formula considers planet absorbing solar energy as a disk and not as a sphere. “

### A short summary

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.

Interesting, very interesting what we see here:

Planet..Tsat.mean..Rotations..Tmin..Tmax

............measured.....per day...................

Mercury..340 K....1/175,938..100 K...700 K

Earth.......288 K..........1..........................

Moon......220 Κ.......1/29,5...100 K...390 K

Mars......210 K.......0,9747...130 K...308 K

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 satellites 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) So (1/R²) /4σ ]¹∕ ⁴ incomplete formula gives very confusing results.

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

Let’s begin:

1. 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 temperature lessens and the night time temperature rises.

But there is something else very interesting happens.

When a planet rotates faster it is a warmer planet. (It happens because Tdark.side↑↑ grows higher than T↓solar.side goes down).

It happens because, when a planet rotates faster, the night time temperature rises higher, than the daytime temperature lessens.

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 new radiative equilibrium temperatures to achieve.

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.

2. The planets’ satellite measured mean temperatures relate according to comparison coefficient:

[ (1-a) (1/R²) (N)¹∕ ⁴ ]¹∕ ⁴

Where

a – the planet’s surface average albedo

R – the semi-major axis in AU (Astronomical Units)

N – the planet’s rotational spin in (rotations /day)

The planets were being separated in groups according to their similar surface specific heat cp (Jupiter, Saturn and Neptune – H2), (Mercury, Moon and Mars - regolith), and (Earth with Europa – H2O).

In every separate group the satellite measured planets’ mean temperatures relate according to the comparison coefficient:

[ (1-a) (1/R²) (N)¹∕ ⁴ ]¹∕ ⁴

That is why the planet's axial spin (rotations per day) "N" should be considered in the Tmean.planet Equation in the fourth root:

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

So = 1.362 W/m² - is the solar constant

Φ - is the dimensionless Solar Irradiation accepting factor

cp – is the planet's surface specific heat

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

3. Planet reflects solar irradiation like a sphere and not like a disk.

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 fraction 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. And the heavy cratered planets have only the albedo "a".

Φ - is the dimensionless Solar Irradiation accepting factor

4. We ended up to the following remarkable results:

Comparison of results the planets' Te calculated by the Incomplete Equation:

Te = [ (1-a) So (1/R²) / 4 σ ]¹∕ ⁴

the planets' Tmean calculated by the Tmean Equation:

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

and the planets' Tsat.mean measured by satellites:

Planet…Te. incomp…...Tmean..…Tsat.mean

…………..…equation..…equation…..measured

Mercury…….439,6 K……325,83 K……..340 K

Earth………..255 K…….…287,74 K……..288 K

Moon………..271 Κ……….221,74 Κ……..220 Κ

Mars…….…209,91 K……..213,42 K…...210 K

The planet's surface Mean Temperature Equation has the wonderful ability to calculate the Planets’ Surface Mean Temperatures getting almost the same results as the measured by satellites planets’ mean temperatures.

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

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