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

What I have discovered is the ROTATING PLANET SPHERICAL SURFACE SOLAR IRRADIATION ABSORBING-EMITTING UNIVERSAL LAW

 

Jemit = σΤmean⁴/(β*N*cp)¹∕ ⁴ (W/m²)

Planet Energy Budget:

Jabs = Jemit

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

Solving for Tmean we obtain the PLANET MEAN SURFACE TEMPERATURE EQUATION:

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

Rotating Planet Spherical Surface Solar Irradiation Absorbing-Emitting Universal Law

Planet Energy Budget:

Solar energy absorbed by a Hemisphere with radius "r" after reflection and dispersion:

Jabs = Φ*πr²S (1-a)      (W) 

What we have now is the following:

Jsw.incoming - Jsw.reflected = Jsw.absorbed

Φ = (1 - 0,53) = 0,47

Φ = 0,47

Φ is the planet's spherical surface solar irradiation accepting factor.

Jsw.reflected = (0,53 + Φ*a) * Jsw.incoming

And

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

Where

(0,53 + Φ*a) + Φ* (1-a) = 0,53 + Φ*a + Φ - Φ*a =

= 0,53 + Φ = 0,53 + 0,47 = 1

The solar irradiation reflection, when integrated over a planet sunlit hemisphere is:

Jsw.reflected = (0,53 + Φ*a) * Jsw.incoming

Jsw.reflected = (0,53 + Φ*a) *S *π r²

For a planet with albedo a = 0

we shall have

Jsw.reflected = (0,53 + Φ*0) *S *π r² =

= Jsw.reflected = 0,53 *S *π r²

The fraction left for hemisphere to absorb is:

Φ = 1 - 0,53 = 0,47

and

Jabs = Φ (1 - a ) S π r²

The factor Φ = 0,47 "translates" the absorption of a disk into the absorption of a 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 disk:

Jdirect = π r² S

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 part 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. The heavy cratered planets have only the albedo "a".

Another thing that I should explain is that planet's albedo actually doesn't represent a primer reflection. It is a kind of a secondary reflection ( a homogenous dispersion of light also out into space ).

That light is visible and measurable and is called albedo.

The primer reflection from a spherical hemisphere cannot be seen from some distance from the planet. It can only be seen by an observer being on the planet's surface.

It is the blinding surface reflection right in the observer's eye.

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 Planet Rotating Surface Solar Irradiation Absorbing-Emitting Universal Law: 

Jsw.incoming - Jsw.reflected = Jsw.absorbed

 

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

Total energy emitted to space from entire planet:

Jemit = A*σΤmean⁴ /(β*N*cp)¹∕ ⁴        (W)

Α - is the planet's surface (m²)

(β*N*cp)¹∕ ⁴ - dimensionless, is a Rotating Planet Surface Solar Irradiation Warming Ability

A = 4πr² (m²), where r – is the planet's radius

Jemit = 4πr²σTmean⁴ /(β*N*cp)¹∕ ⁴   (W)

global Jabs = global Jemit

Φ*πr²S (1-a) = 4πr²σTmean⁴ /(β*N*cp)¹∕ ⁴

Or after eliminating πr²

Φ*S*(1-a) = 4σTmean⁴ /(β*N*cp)¹∕ ⁴

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 Planet Mean Surface Temperature Equation:

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

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

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

cp – is the planet surface specific heat

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.

Here (β*N*cp)¹∕ ⁴ - is a dimensionless Rotating Planet Surface Solar Irradiation Warming Ability

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

Rotating Planet Spherical Surface Solar Irradiation Absorbing-Emitting Universal Law:

  Jemit = σΤmean⁴/(β*N*cp)¹∕ ⁴  (W/m²)

The year-round averaged energy flux at the top of the Earth's atmosphere is Sο = 1.361 W/m².

With an albedo of a = 0,306 and a factor Φ = 0,47 we have Tmean.earth = 287,74 K or 15°C.

This temperature is confirmed by the satellites measured Tmean.earth = 288 K.

 Jemit=σΤmean⁴/(β*N*cp)¹∕ ⁴ (W/m²)

http://www.cristos-vournas.com

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

Tmin→ T↑mean ← Tmax

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