## 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.*

.

*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**

**The Φ - solar irradiation accepting factor - how it "works". It is not a planet specular reflection coefficient itself.**

**There is a 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.**

** Albedo is defined as the ratio of the scattered SW to the incident SW radiation, and it is very much precisely measured (the planet Bond Albedo).**

** 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.**

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

**For a planet with albedo a = 0 (completely black surface planet) we would have**

** Jsw.reflected = [1 - Φ*(1-a)]*S *π r² =**

** Jsw.reflected = (1 - Φ) *S *π r² **

** For a planet which captures the entire incident solar flux (a planet without any outgoing specular reflection) we would have Φ = 1**

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

** Jsw.reflected = a *Jsw.incoming**

** And For a planet with Albedo a = 1 , a perfectly reflecting planet**

** Jsw.absorbed = 0 (no matter what is the value of Φ)**

** In general: **

**The fraction left for hemisphere to absorb is **

**Jabs = Φ (1 - a ) S π r²**

**We have Φ for different planets' surfaces varying**

** 0,47 ≤ Φ ≤ 1**

** And we have surface average Albedo "a" for different planets' varying**

** 0 ≤ a ≤ 1 **

**Notice:**

** Φ is never less than 0,47 for planets (spherical shape). Also, the coefficient Φ is "bounded" in a product with (1 - a) term, forming the Φ(1 - a) product cooperating term. **

**So Φ and Albedo are always bounded together. The Φ(1 - a) term is a coupled physical term.**

**The Φ(1 - a) term "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 = Φ*( 1 - a) π r² S **

**It happens because a smooth hemisphere of the same radius "r" absorbs only the Φ*(1 - a)S portion 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 Φ*(1 - a)S portion 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". **

**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**

** Energy in = Φ(1 - a) **

**left side of the Planet Radiative Energy Budget.**

** Conclusively, the Φ -Factor is not the planet specular reflection portion itself.**

** The Φ -Factor is the Solar Irradiation Accepting Factor (in other words, Φ is the planet surface shape and roughness coefficient).**

.

### Instead of Te, the Planet Corrected Effective Temperature Te.correct

**At a first approach, when without the Rotational Warming phenomenon implementation, I use instead of Te, the Planet Corrected Effective Temperature Te.correct.**

** The formula is: **

** [ Φ(1-a) /4σ ]¹∕ ⁴**

** Φ = 0,47**

** (the 0,47 is for smooth surface planets without atmosphere, the factor Φ accounts for the smooth planet surface specular reflection) **

**Table of results for Te and Te.corrected**

** Planet........ Te..........Te.correct**

** Mercury.....440 K.......364 K**

** Moon.........270 K......224 K**

** Earth.........255 K.......210 K**

** Mars,,,,,,,,,,210 K......174 K **

.

### 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. **

**Φ is the planet solar irradiation accepting factor (the spherical shape and roughness coefficient). **

**Φ = 0,47 for a smooth surface sphere**

** What we had till now: **

**Jsw.incoming - Jsw.reflected = Jsw.absorbed **

**and**

** 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**

** And **

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

**For Planet Earth (smooth surface planet Φ = 0,47)**

** 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**

.

**https://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 ← T↓max**