The very POWERFUL the Solar Irradiated planet surface Rotational Warming Phenomenon: ( N*cp )^1/16

"Specular reflection from a body of water is calculated by the Fresnel equations.[6] Fresnel reflection is directional and therefore does not contribute significantly to albedo which primarily diffuses reflection."

https://en.wikipedia.org/wiki/Reflectance#Hemispherical_reflectance

Water-earth total specular reflection based on the Fresnel reflection

Specular reflection from a body of water is calculated by the Fresnel equations.[6] Fresnel reflection is directional and therefore does not contribute significantly to albedo which primarily diffuses reflection.

The radius of a planet is r

The height " h.i " of a spherical zone " i " at the point of solar flux's angular incidence θ°i is

In the above scheme we explain the spherical zone's height calculation method:

h.i = [ r*cos θ°i - r*cos θ°(i+1) ] =

= r [ cos θ°i - cos θ°(i+1) ]

Analysis of terms.

S - the incident on the planet solar flux (W/m²), perpendicular to the planet's cross-section

r - planet's radius (m)

πr² - planet's cross-section area perpendicular to the solar flux's beams (m²)

N - the normal to the surface

θ° - angle of solar flux's incidence

θ°i - angle of solar flux's incidence at i point

h.i = r [ cos θ°i - cos θ°(i+1) ] - the i spherical zone area height

A.i - spherical zone area m² at point i

A.i = 2πr * ( r*cos θ°i - r*cos θ°i+1 ) - spherical zone area at i point (m²)

A.i = 2πr² * ( cos θ°i - cos θ°i+1 ) (m²)

...........

Spec.i - ( specular reflectivity at point i ) ( specular reflectivity at point i ) taken from graph for the ( θ°i ) angle of incidence

A.i * Spec.i - the total incident on zone Ai area solar irradiation reflected portion m² *W/m² = W

Σ ( Α.ι* Spec.i ) - the Sum total incident on the entire hemisphere's surface solar irradiation reflected portion (W)

Σ ( Α.ι* Spec.i ) /πr² - the total specular reflection portion of the incident solar flux, averaged on the planet's cross-section disk (W/m²)

When substituting terms in the above sentence we would have:

Σ [ 2πr² * ( cos θ°i - cos θ°i+1 ) (m²) * Spec.i W/m² ] /πr² (m²)

When simplifying by eliminating the πr² term

Σ 2 *( cos θ°i - cos θ°i+1 ) * Spec.i (W/m²) - the sphere's total specular reflection portion of the incident solar flux, averaged on the planet's cross-section disk perpendicular to the incoming solar flux

or

2 * Σ [ ( cos θ°i - cos θ°i+1 ) ] * Spec.i (W/m²) (1)

let's symbolize the ( cos θ°i - cos θ°i+1 ) expression with Δcosθ°i term

So we shall write:

2 * Σ Δcosθ°i * Spec.i (W/m²)

Table of data (by 2,5° steps ) and the product ( Δcosθ°i * Spec.i ) results

Angle of

incidence..................cosθ°i - cosθ°i+1 ..graph data.......product

θ°ι...............cos θ°ι.......... Δcosθ°i........Spec.i....... Δcosθ°i * Spec.i

0°......................1.............0,00095..............0....................0

2,5°................0,99905.........0,00285........0,02...............0,00019 5°............0,99619.........0,004750............0,02...............0,000056

7,5°................0,99144.........0,006637......0,02............... 0,000091

10°..........0,98481..........0,0085117............0,02............ 0,000135

12,5°..............0,97630..........0,0104........0,02................ 0,000165 15°..........0,96593............0,01221...........0,02...............0,00020

17,5°..............0,95372...........0,0140...........0,02............. 0,00024

20°..........0,93969............0,0158...........0,02................0,00028

22,5°..............0,92388..........0,0176...........0,02.............. 0,00031 25°..........0,90631............0,0193...............0,02.............0,00035

27,5°..............0,88701...........0,0210........0,02............... 0,00036

30°..........0,86603...........0,0226.............0,02...............0,00042 32,5°..............0,84339...........0,0242..........0,02...............0,000452 35°..........0,81915...........0,0258..............0,02.................0,000484 37,5°.............0,79335...........0,0273.........0,02...............0,000516 40°..........0,76604...........0,0288.............0,02...............0,000546 42,5°.............0,73728............0,0302..........0,023.............0,000662 45°..........0,70711...........0,0315...............0,025...........0,000755 47,5°.............0,67559............0,0329...........0,031............0,00098 50°..........0,64279...........0,0340.................0,035.............0.00115 52,5°..............0,60876...........0,035.............0,037............0,00126 55°..........0,57358..........0,0361.....................0,040.........0,00141 57,5°..............0,53730...........0,0373............0,055............0,00200 . 60°..........0,5....................0,0383...............0,065...........0,00243 62,5°..............0,46175...........0,0391............0,085............0,00325 65°..........0,42262............0,0399.................0,1..............0,003913 67,5°..............0,38268...........0,0407...............0,17...........0,00679 70°..........0,34202.............0,0413................0,22..............0,00895

72,5°..............0,30071..........0,0419...............0,27...........0,01115

75°..........0,25882.......... 0.0424 ...............0,30.............0,01257 77,5°..............0,21644...........0,0428............0,39............0,01653 80°..........0,17365...........0,0431................0,45................0.01926 82,5°..............0,13053...........0,0434.............0,60..............0,02587 85°..........0,08716...........0,0435...............0,70................0,03036 87,5°..............0,04362..........0,0436..............0,82............0,03570 90°...................0...........................................1.............0,04362

Σ....................................................................................0,217

When summarizing from the Table the Δcosθ°i * Spec.i the product results we shall have

Σ Δcosθ°i * Spec.i = 0,217

and multiplying times 2 according to the equation (1) above

2 * Σ Δcosθ°i * Spec.i = 0,217 * 2 = 0,434

- it is the specularly reflected portion of the incident solar flux It is the sphere's total specular reflection portion of the incident solar flux, which is averaged on the planet's cross-section disk.

When considering an oceanic-like planet Earth total reflected energy the diffuse a*So + specular 0,434 *So =

= 0,3 * 1.362W/m² + 0,434 * 1.362W/m² =

= 0,734 * 1.362 W/m² = 999,71 W/m² REFLECTED

and only 1.362 - 999,71 = 362 W/m² "ABSORBED"

This result (362 W/m² "ABSORBED") is in a satisfactory magnitude accordance with the smooth planet surface the solar incident flux's "absorption"  ( 444 W/m² not-reflected).

Φ(1 - a)So = 0,47(1 - 0,306)1362 W/m² =  444 W/m² not-reflected

when compared with the blackbody theory the 1.362 * (1 - 0,306) =

= 945 W/m² "ABSORBED"

The difference is more than twice as much!

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0,434*S specularly reflected from the sea waters...

Now, let's see:

Φ(1- a) *S is the not reflected portion of the incident solar flux.

The sea water Albedo a=0,08

Φ=0,47 for the smooth surface spheres (planets without or with a thin atmosphere)

Φ(1-a)*S = 0,47(1 - 0,08)*S = 0,47*0,92*S = 0,4324*S

it is the not reflected portion of the incident solar flux.

"Because light specularly reflected from water does not usually reach the viewer, water is usually considered to have a very low albedo <b>in spite of its high reflectivity at high angles of incident light</b>.”
(emphasis added)

Link:
https://en.wikipedia.org/wiki/Albedo

Here it is the key point:

"... in spite of its high reflectivity <b>at high angles of incident light</b>."

(early morning, late afternoon, and near the poles)

Sun shines on the Globe all the time. At every given moment there is only one point on the Globe where the angle of incidence is zero.

And, at every given moment there are always the high angles of incident light on the Globe.

So, every given moment the most of the Globe's surface area is at high angles of incident light.
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The Distinguishly Directional Reflective Constituent

We have multiplied the Φ with the (1 -a)*S ,

so it is Φ*(1- a)*S W/m² 

Where:

Φ - is the solar irradiation accepting factor (dimensionless)

a - is the satellite measured Albedo (dimensionless)

S - is the solar flux at the planet's or moon's distance from the sun (W/m²)

The  Φ*(1- a)*S   (W/m²)  is the not reflected portion of the incident on planet or moon surface solar SW EM energy.

The equation is true for all planets and moons ib solar system

Φ = 0,47  is for the smooth surface planets and moons without-atmosphere or with a thin atmosphere, Earth included:

( Mercury, Moon, Earth, Mars, Europa, Ganymede ) 

those planets and moons have a distinguishly directional reflective constituent, which cannot be "seen" and measured by satellites' sensors.

Φ = 1 is for the rest planets and moons - the rough surface planets and moons, which do not have a distinguishly directional reflective constituent.

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Opponent:

"Fresnel reflection will not be specular unless it is taking place on a specular surface.”

Answer:

A specular surface… Do you consider a surface specular only when it is polished to a level someone sees his image in the surface, the way we are used to see ourselves in mirrors? The mirror-polished surface?

Well, when we look in the mirror, we do not see any specular reflection of direct light.
When we look in the mirror, we see what light is being diffuselly reflected from our face towards the mirror, and then specularly reflected into our eyes.

On the other, the specular reflection of direct solar light is blinding!
The specular reflection of direct solar light from any surface is blinding.

When an illuminated surface exhibits diffuse reflection, we cannot see our face in that surface, because our face also exhibits a diffuse light.

The not polished surface (a matte surface, which, when illuminated, exhibits some level of diffuse reflection), that surface also reflects specularly – that surface has a specular constituent, which is directionally oriented to the opposite from the source of light, and at angles of the law of specular reflection.

And when the source of light is at higher angles of incidence, the specular reflection is always strong and blinding.

So, it is the angles of incidence that make a surface strongly reflecting.

Planets and moons are spheres illuminated from sun 24/7. Planets and moons always have vast areas on globe with high angles of incidence. In addition, because of planets and moons sphericity, the areas of the highest angles of solar rays incidence are the largests.

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Mars’ diffuse reflection (a=0,250)

Therefore:
Φ(1-0,250) = 0,47*0,75 = 0,3525 it is the not reflected portion.

1-0,3525 = 0,6475 it is the Mars’ Bond Albedo.

the 1-Φ = 1 -0,47 =0,53 is the Bond Albedo for a smooth planet without diffuse reflection

Earth's average surface Albedo

(a = 0,306) is a satellite measured diffuse reflection.

Earth's Bond Albedo (= 0,6738). 

Bond Albedo is defined as =

= (entire reflected)/(total incident) 

Therefore, Earth's Bond Albedo is more than twice as much as Earth's diffuse Albedo.

For smooth surface planets and moons (Mercury, Moon, Earth, Mars, Europa and Ganymede} surface strong specular reflection constituents were ignored.

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To become more specific:

Rough surfaces also have specular reflection directional constituents.

Not only the polished, not only the mirror-like surfaces have specular reflection, but the rough surfaces have too.

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At the microscopical (infinitesimal) level, there is only specular reflection. The surface diffuse reflection consists from multiple infinitesimal specular reflections.

The surface diffuse reflection is a macroscopical observation!

The reflection's nature is specular!

When you look on fresh asphalt with the sun in front of you, the observer's eyes are spontaneously narrowing.

Eyes spontaneous narrowing phenomenon happens because the from asphalt the strong specular reflection constituent is blinding. So eyes, to protect themselves, automatically respond by narrowing.

Solar light is directional, because it is coming from a large distance. When reflected on a micro level it is diffuse reflection, but because of solar light's directional nature, the rough surface's diffuse reflection is not an isotropical reflection.

The solar light reflected from planets with smooth surfaces has strong directional constituents, which are directed to the opposite from sun direction.

Of course there are planets with a dipper multiple cracks on surface, where the directional nature of the specular constituent is captured and faded out. For those planets and moons, the diffuse reflection is more isotropical, so there is much less, there is almost none specular reflection constituents.

The planets and moons with strong specular reflection constituents in Solar System are six (6),namely:

Mercury, Earth, Moon, Mars, Europa (of Jupiter) and Ganynede (of Jupiter).

For the rest planets and moons in Solar System the surface specular reflection constituent is very much small, so it is insignificant. For the rest planets and moons the planetary surface reflection is almost isotropically-like diffuse reflection.

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Solar energy, the portion which is not reflected, what it does is to interact with matter. And, what is not reflected, is not entirelly absorbed as heat. Only a small portion, of the not reflected solar energy, only a small portion is absorbed as Heat

The radiative energy balance is not what it is thought to be.

When solar irradiated, surface exhibits a strong Immediate IR Emission, which is  ignored.

The solar energy interacts with surface, the not reflected solar energy is impossible to average.

A smooth planet, like Earth, has a strong Specular Reflection Constituent, which is also ignored.

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Next very important issue:

They still use Stefan-Boltzmann Emission law as the radiative energy Absorption Law.
It is a non scientifical approach. It is not acceptable.

The Stefan-Boltzmann Emission law is not EM energy absorption law.

Thus the S-B  formula cannot be used backwards when theoretically estimating surface temperature.

the use of S-B formula backwards, in order to derive surface temperature from the known (measured) incident on surface EM energy flux J (W/m²) is impossible, because deriving surface temperature from:

" T = (J /σ)^1/4 K " is a great mistake.

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And also, another thing with S-B, it is very important to take into consideration, because it is very much obvious - the Stefan-Boltzmann emission law formula

J = σT⁴ (W/m²)

doesn't apply to terrestrial temperatures..

The 255K (-18°C) doesn't emit 240 W/m², because the (-18°C) is very cold.

Like-wise, the 288K (15°C) doesn't emit 390 W/m² - absolutely NOT.

Actually, the planetary surface Regular IR Emission intensity is very weak!

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Radiative Energy Balance.

We have a parallel solar flux of rays falling on a spherical surface. There is always only one point on the sphere where the sun's rays are perpendicular to the surface - the zenith point of globe.

At the zenith point, the specular reflection is minimal, so the maximum interaction of electromagnetic energy with surface's matter, and the maximum Immediate IR Emission.

As you get closer to the circle of the terminator, there is a gradual rise in specular reflection, there is less electromagnetic energy to interact with matter - and Immediate IR emission decreases.

The electromagnetic energy in heat conversion and heat absorption at the zenith point of globe have reached a maximum, and as you approach the circle of the terminator, heat absorption decreases.

Thus, the solar flux with an energy of So*pi*r2 Watts falls on the sphere at each moment of time.

On the sunny side, the heat absorption zone is mostly concentrated around the zenith point of globe.

In addition, around the zenith point of globe, it is the intense zone of immediate IR emission.

On the other hand, the closest to terminator's circle zone is the zone of twilight, it is the zone of the weakest insolation, because it is the zone of maximum Specular reflection.

In addition, the terminator's circle zone is the largest area on solar lit side of globe. Because terminator's circle is the largest solar lit circle - the diameter of Earth's.

Therefore, zones with sun lower at Horizont, they are zones covering the largest areas on globe.

So, generally speaking, there are much more areas on globe, where sun is lower, so those are the areas with less and less of the initial incident solar energy Watts per square meter (W/m²) - less incident solar energy per unit area - and, also, in addition, those are the areas where the specular reflection is the highest.

Thus, there are the larger areas,

having the higher Specular Reflection constituent!!!

Fresnel's specular reflection graph, on the graph the highest specular reflection values are at small area with the angles of incidence at highest. 

Fresnel's graph is the Specular Reflection geometrical degrees profile measurements.

It is a flat graph.

"Reflectivity vs Incident Angle from Normal, Deg"

Also view our calculations:

"The total amount of the specularly reflected portion of solar flux"

Link: Φ = 0,47 and FRESNEL

Because the areas on globe with the highest specular reflection are the largests!!!

The areas on globe with lower sun - the mornings, the evenings, the higher latitudes - they are always there, every moment of the time, and they are much larger (all those, the less solar irradiated- the areas with the highest specular reflection) than the intensively insolated, the around Zenith Point Zone much smaller areas.

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There are two important points to be necessarily consider. They formulate a completely different planetary radiative energy budget, from what it was used to be.

1). Planets and moons of smooth surfaces have a strong specular reflection constituent, which is ignored.

So, that strong specular reflection constituent - which abandons surface, and which is lost in space, shouldn't be considered in the radiative energy budget as absorbed heat.

2). The same with the Immediately emitted IR energy - which also abandons surface, and which also is lost in space, shouldn't be considered in the radiative energy balance as absorbed heat.

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The stored heat is the source of energy for the planet's regular IR emission.

Regular IR emission is distributed unevenly over the non-sunlit part of the sphere because there are differences in latitude temperatures, as well as differences that occur as the surface rotates from the zenith point to the circumference of the terminator and through the night.

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