# Planet's Without-Atmosphere Mean Surface Temperature New Equation: Tmean = [ Φ (1-a) S (β*N*cp)¹∕ ⁴ /4σ ]¹∕ ⁴ (K)

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

Jemit = 4πr²σΤmean⁴ /(β*N*cp)¹∕ ⁴ (W)

Planet Energy Budget:

Jnot.reflected = Jemit

πr²Φ*S*(1-a) = 4πr²σTmean⁴ /(β*N*cp)¹∕ ⁴ (W)

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)

When interacting with planet surface the solar SW radiative energy on the same very instant does the following: 1). Gets partly SW reflected (diffusely and specularly). 2). Gets partly transformed into the LW radiative emission - the IR emission energy. 3). Gets partly transformed into HEAT, which is accumulated in the inner layers.

### In short...

The Rotating Planet Spherical Surface Solar Irradiation Interacting-Emitting Universal Law

Here it is the ENTIRE planet surface IR emittance Universal Law

Jemit = 4πr²σΤmean⁴ /(β*N*cp)¹∕ ⁴ (W)

The solar irradiated rotating sphere (planet) does not emit as a uniform temperature sphere in accordance to the classical Stefan-Boltzmann emission law.

4πr²σΤmean⁴ (W)

No, the solar irradiated rotating sphere (planet) emits as a rotating planet in accordance with both, the classical Stefan-Boltzmann emission law and the Newly discovered Planet Surface Rotational Warming Phenomenon.

4πr²σΤmean⁴ /(β*N*cp)¹∕ ⁴ (W)

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Let's continue...

Planet Energy Budget:

Jnot.reflected = Jemit

πr²Φ*S*(1-a) = 4πr²σTmean⁴ /(β*N*cp)¹∕ ⁴ (W)

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 Interacting-Emitting Universal Law

Planet Energy Budget:

The Amount of Solar energy for further INTERACTION on a Hemisphere with radius "r", after some of the incident energy  instantly reflected is:

Jnot.reflected = Φ*πr²S (1-a)      (W)

What we have now is the following:

Jsw.incoming - Jsw.reflected = Jsw.not.reflected

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.not.reflected = 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 INTERACT WITH is:

Φ = 1 - 0,53 = 0,47

and

Jnot.reflected = Φ (1 - a ) S π r²

The factor Φ = 0,47 "translates" the "not reflected" of a disk into the "not reflected" 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 "not reflected" Solar energy by the hemisphere's area of 2π r² is:

Jnot.reflected = 0,47*( 1 - a) π r² S

It happens because a hemisphere of the same radius "r" "not reflects" 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 "not reflects" only the 0,47 part of the directly incident on the disk Solar irradiation.

Jnot.reflected = 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.not.reflected

Jsw.not.reflected = 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:

Jnot.reflected = 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 INTERACTING-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 Interacting-Emitting Universal Law:

Jemit = 4πr²σΤmean⁴/(β*N*cp)¹∕ ⁴  (W)

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.

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When interacting with planet surface the solar SW radiative energy on the same very instant does the following:

1). Gets partly SW reflected (diffusely and specularly).

2). Gets partly transformed into the LW  radiative emission - the IR emission energy.

3).  Gets partly transformed into HEAT, which is accumulated in the inner layers.