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

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Suggestion:

Have you "tuned" any parameters in derivation of these closely agreeing temperatures with the satellite measured ones?

And could the effect of an atmosphere be "hiding" in some of these parameters?

Answer:

These data, the calculated with a Planet Without-Atmosphere Surface MeanTemperature Equation and the measured by satellites are almost the same, very much alike.

They are almost identical, within limits, which makes us conclude that the Planet Without-Atmosphere Surface Mean Temperature Equation

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

can calculate a planet mean temperatures.

It is a situation that happens once in a lifetime in science. Although the evidences existed, were measured and remained isolated information so far.

It was not obvious one could combine the evidences in order to calculate the planet’s temperature.

A planet-without-atmosphere effective temperature equation

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".

Those two parameters are not enough 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 period 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 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 Surface Mean Temperature Equation:

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

The Planet Without-Atmosphere Surface Mean Temperature Equation produces very reasonable results:

Tmean.earth = 288,36 K, calculated with the Complete Formula, which is identical with the

Tsat.mean.earth = 288 K, measured by satellites.

Tmean.moon = 221,74 K, calculated with the Complete Formula, which is almost the same with the

Tsat.mean.moon = 220 K, measured by satellites.

A Planet Without-Atmosphere Surface Mean Temperature Equation gives us a planet mean temperature values very close to the satellite measured planet mean temperatures. 

 

Suggestion:

Is there a difference for Earth having an ocean either than just beeing a dry rocky planet?

Answer:

Yes there is a big difference. Earth’s Surface Mean Temperature Equation Tmean.earth:

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

Τmean.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 =

Τmean.earth = [ 0,47(1-0,30)1.362 W/m²(150*1*1)¹∕ ⁴ /4*5,67*10⁻⁸ W/m²K⁴ ]¹∕ ⁴ =

Tmean.earth = 288,36 Κ

Moon’s Surface Mean Temperature Equation Tmean.moon:

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

Tmean.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 =

Tmean.moon = 221,74 Κ

cp.earth = 1 cal/gr*oC

cp.moon = 0,19 cal/gr*oC

The cp.earth is 5,263 times higher.

If Earth was not a Planet ocean, but a pure rocky planet, then:

Tmean.rocky.earth = 288,36 Κ * [(0,19)¹∕ ⁴ ]¹∕ ⁴ =

Tmean.rocky.earth = 288,36 Κ * 0,9014 = 259,93 K

If the Earth was a rocky planet the Tmean.earth would be

Te.rocky.earth = 259,93 = 260 K

 

Suggestion:

Isn't the Equation an adjustment on the already satellites measured planet mean temperatures?

Answer:

A Planet Without-Atmosphere Effective Temperature Calculating Equation, the Te equation, 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 equation and that is why it gives us very confusing results.

Comparison of results the planet Te calculated by the Incomplete Equation, the planet Te calculated by the Surface Mean Temperature Equation, and the planet Tsat.mean measured by satellites:

Planet or   Te.incomplete  Tmean Tsat.mean

moon          equation      equation   measured

Mercury        437,30 K      323,11 K      340 K

Earth             255 K           288,36 K      288 K

Moon            271 K            221,74 K      220 K

Mars             209,91 K       213,59 K      210 K 

 

The Planet Without-Atmosphere Surface Mean Temperature Equation:

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

is not a product of adjustments.

A Planet Without-Atmosphere Surface Mean Temperature Equation:

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

is based on a newly discovered Rotating Planet Surface Solar Irradiation Absorbing-Emitting Universal Law.

The planet average Jabs = Jemit, per m² planet surface:

Jabs = Jemit 

Φ*S*(1-a) /4 = σTmean⁴ /(β*N*cp)¹∕ ⁴ (W/m²)

Solving for Tmean we obtain the surface mean temperature:

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

 

Jemit = σΤmean⁴/(β*N*cp)¹∕ ⁴ (W/m²)

 

It is obvious now that the planet without-atmosphere effective temperature incomplete formula:

Te = [ (1-a) S / 4 σ ]¹∕ ⁴

should not be in use anymore.

 

The satellites measured Planet Mean Temperatures  we should relay on.

 

Suggestion:

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

For a very high N, will you get a very hot planet?"

Answer:

For the final Tmean result N (rotations/day) value is operated twice in forth root .

Example: Let's say N = 100.000.000

[ ( 100.000.000 )¹∕ ⁴ ]¹∕ ⁴ = ( 100 )¹∕ ⁴ = 3,1623

And for N = 1000.000.000 it is 3,6525

 

But for N = 10 it is 1,1548

If Earth were rotating 10 times as fast, Earth's effective temperature would be:

288 K * 1,1548 = 332,58 K 

.

http://www.cristos-vournas.com

The faster a planet rotates (n2>n1) the higher its mean temperature T↑mean.

Tmin↑→ Tmean Tmax

.