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

Φ factor explanation

Planet is a sphere and it reflects and absorbs like a sphere

Planet does not reflect and absorb as a disk. Planet reflects and absorbs as a sphere.

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The by a smooth spherical body solar irradiation absorption

Jabs = Φ (1 - a ) S π r²

Φ = 0,47

Φ factor explanation

Φ factor explanation

There is need to focus on the Φ factor explanation.

Φ factor emerges from the realization that a sphere reflects differently than a flat surface perpendicular to the Solar rays.

Φ – is the dimensionless Solar Irradiation accepting factor

"Φ" is an important factor in the Planet's Surface Mean Temperature Equation:

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

It is very important the understanding what is really going on with by planets the solar irradiation reflection.

There is the specular reflection and there is the diffuse reflection.

The planet's surface albedo "a" accounts for the planet's surface diffuse reflection.

So till now we didn't take in account the planet's surface specular reflection.

A smooth sphere, as some planets are, have the invisible from the space and so far not detected and not measured the specular reflection.

The sphere's specular reflection cannot be seen from the distance, but it can be seen by an observer situated on the sphere's surface.

Thus, when we admire the late afternoon sunsets on the sea we are blinded from the brightness of the sea surface glare. It is the surface specular reflection what we see then.

When we integrate the specular reflection from the parallel solar rays hitting the disk of radius "r" and the cross-section "π r²" over the sunlit hemisphere of radius "r", the result is 0,53π r² S.

Thus the 0,53π r²*S is the specular reflected fraction of the incident on the smooth planet's spherical surface solar flux.

Φ = 1 - 0,53 = 0,47

Φ = 0,47

Thus the 0,47π r²S - is The What Is Left Fraction of the incident solar flux for the planet's smooth spherical surface to absorb because of the spherical surface's specular reflection.

What we have now is the following:

Jsw.incoming - Jsw.reflected = Jsw.absorbed

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 (the specular plus the diffuse) over the planet's 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 (completely black surface planet) we would have

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

= Jsw.reflected = 0,53 *S *π r²

For a planet without any outgoing specular reflection we would have Φ =1

And

Jsw.reflected = a *S *π r² 

In general:

The fraction left for hemisphere to absorb is

Jabs = Φ (1 - a ) S π r²

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

Jdirect = π r² S

But 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 smooth 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:

Φ*(1-a)*Sπ r² = 4π r²*σTmean⁴ /(β*N*cp)¹∕ ⁴

And they are also cooperate in the Planet's Surface Mean Temperature Equation:

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

 

Planet Energy Budget:

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

Jabs = Φ*πr²S (1-a) ( W )

Total energy emitted to space from a whole planet:

Jemit = A*σΤmean⁴ /(β*N*cp)¹∕ ⁴ ( W )

Φ - is the dimensionless Solar Irradiation accepting factor

(1 - Φ) - is the reflected fraction of the incident on the planet solar flux

S - is the Solar Flux at the top of atmosphere ( W/m² )

Α - is the total planet surface area ( m² )

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

Tmean - is the Planet's Surface Mean Temperature ( K )

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

Thus

energy in = energy out

Φ*(1-a)*Sπ r² = 4π r²*σTmean⁴ /(β*N*cp)¹∕ ⁴

 

For Earth's surface we would have:

Jabs.earth = 0,47 ( 1 - 0,306 ) So π r² =

= 0,47*0,694 * 1.361* π r² ( W ) = Jabs.earth = 0,326 So π r² =

= 0,326* 1.361 π r² =

= 444,26 π r² ( W )

Jabs.earth = 444,26 π r² ( W )

I am not averaging Jabs.earth = 444,26 π r² ( W ) over the entire Earth's surface, because it is a wrong approach.

It is a misleading mistake to average the "absorbed" incident solar flux's fraction over the entire earth's surface.

Also I have put the word "absorbed" in brackets, because it is not exactly absorbed, but instantly emitted back to space as an IR radiation.

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What is Earth's averaged on the entire surface absorbed solar SW radiation?

A planet reflects incoming short wave solar radiation.

A planet's surface has reflecting properties.

1. The planet's albedo "a". It is a surface quality's dependent value.

2. The planet's spherical shape. For a smooth planet the solar irradiation reflection is (0,53 + Φ*a)*Jincoming.

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

Φ = 0,47 for a smooth sphere

Thus 1 - Φ = 0,53

0,53 is for smooth sphere's specular reflection and "a" albedo is for diffuse reflection.

 

What we had till now:

Jsw.incoming - Jsw.reflected = Jsw.absorbed

Here

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

And

Jsw.reflected = a* Jsw.incoming

 

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.

(1 - Φ + Φ*a)S - is the reflected fraction of the incident on the planet solar flux

Φ(1 - a)S - is the absorbed fraction of the incident on the planet solar flux

or

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

Conclusion:

A planet's absorbed fraction of the SW incoming radiation in total is:

Jsw.absorbed = 0,47*(1-a)*Jsw.incoming

For Planet Earth

Jsw.absorbed = 0,47*(1-a)*1.361 W/m² =

= 0,47*0,694*1.362W/m² = 444,26 W/m²

Averaged on the entire Earth's surface we obtain:

Jsw.absorbed.average = [ 0,47*(1-a)*1.361 W/m² ] /4 =

= [ 0,47*0,694*1.361W/m² ] /4 = 444,26 W/m² /4 =

= 111,07 W/m²

Jsw.absorbed.average = 111 W/m²

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  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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