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

What is the 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.

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.

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

.

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

"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 absorption.

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:

Φ*S*(1-a) = 4σTmean⁴ /(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 dispersion:

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 a dimensionless Solar Irradiation accepting factor

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

S - is a 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

Jabs = Φ (1- a ) S W/m² sunlit hemisphere

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 )

What is going on here is that instead of

Jabs.earth = 0,694* 1.361 π r² ( W )

we should consider

Jabs.earth = 0,326* 1.361 π r² ( W ).

 

 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*1.361 W/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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