The Planet's Effective Temperature Complete Formula Te = [ Φ (1-a) S (β*N*cp)¹∕ ⁴ /4σ ]¹∕ ⁴

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Mars, Phobos and Deimos Effective Temperatures comparison

Enhanced-color image of Phobos from the Mars Reconnaissance Orbiter with Stickney crater on the right

An enhanced-color image of Deimos (MRO, 21 February 2009). Image: NASA/JPL-Caltech/University of Arizona

Mars, Phobos and Deimos Effective Temperatures comparison

Mars, Phobos and Deimos Effective Temperatures comparison

 

3. Mars’ Effective Temperature Calculation:

Te.mars

(1/R²) = (1/1,524²) = 1/2,32 Mars has 2,32 times less solar irradiation intensity than Earth has

Mars’ albedo: amars = 0,25

N = 0,9747 rotations/per day, Planet Mars completes one rotation around its axis in 24 hours 37 min 22 s.

Mars is a rocky planet, Mars’ surface irradiation accepting factor: Φmars = 0,47

cp.mars = 0,18cal/gr oC, on Mars’ surface is prevalent the iron oxide

β = 150 days*gr*oC/rotation*cal – it is a Rotating Planet Surface Solar Irradiation Absorbing-Emitting Universal Law constant

σ = 5,67*10⁻⁸ W/m²K⁴, the Stefan-Boltzmann constant

Mars' Effective Temperature Complete Formula is:

Te.mars = [ Φ (1-a) So (1/R²) (β*N*cp)¹∕ ⁴ /4σ ]¹∕ ⁴

Planet Mars’ Effective Temperature Te.mars is:

Te.mars = [ 0,47 (1-0,25) 1.362 W/m²*(1/2,32)*(150*0,9747*0,18)¹∕ ⁴ /4*5,67*10⁻⁸ W/m²K⁴ ]¹∕ ⁴ =

= ( 2.066.635.547,46 )¹∕ ⁴ = 213,21 K

Te.mars = 213,21 K

The calculated Mars’ effective temperature Te.mars = 213,21 K is only by 1,53% higher than that measured by satellites

Tsat.mean.mars = 210 K !

 

Phobos’ ( Mars' moon ) Effective Temperature Calculation:

Te.phobos

(1/R²) = (1/1,524²) = 1/2,32

Phobos has 2,32 times less solar irradiation intensity than Earth has

Phobos’ albedo: aphobos = 0,071

N = 24/7,7 rotations/per day, Phobos completes one rotation around its axis in about 7,7 hours

Phobos is a rocky planet, Phobos’ surface irradiation accepting factor: Φphobos = 0,47

cp.phobos = 0,19cal/gr oC, on Phobos’ surface is regolith

β = 150 days*gr*oC/rotation*cal – it is a Rotating Planet Surface Solar Irradiation Absorbing-Emitting Universal Law constant

σ = 5,67*10⁻⁸ W/m²K⁴, the Stefan-Boltzmann constant

Phobos' Effective Temperature Complete Formula is:

Te.phobos = [ Φ (1-a) So (1/R²) (β*N*cp)¹∕ ⁴ /4σ ]¹∕ ⁴

Phobos’ Effective Temperature Te.phobos is:

Te.phobos = { 0,47 (1-0,071) 1.362 W/m²*(1/2,32)*[(150*(24/7,7)*0,19]¹∕ ⁴ /4*5,67*10⁻⁸ W/m²K⁴ }¹∕ ⁴ =

Te.phobos = 242,14 K

Tsat.mean.phobos ≈ 233 K !

 

Deimos’ ( Mars' moon ) Effective Temperature Calculation:

Te.deimos

(1/R²) = (1/1,524²) = 1/2,32

Deimos has 2,32 times less solar irradiation intensity than Earth has

Deimos’ albedo: adeimos = 0,068

N = 24/30,3 rotations/per day, Deimos completes one rotation around its axis in about 30,3 hours

Deimos is a rocky moon, Deimos’ surface irradiation accepting factor: Φdeimos = 0,47

cp.deimos = 0,19cal/gr oC, on Deimos’ surface is regolith

β = 150 days*gr*oC/rotation*cal – it is a Rotating Planet Surface Solar Irradiation Absorbing-Emitting Universal Law constant

σ = 5,67*10⁻⁸ W/m²K⁴, the Stefan-Boltzmann constant

Deimos' Effective Temperature Complete Formula is:

Te.deimos = [ Φ (1-a) So (1/R²) (β*N*cp)¹∕ ⁴ /4σ ]¹∕ ⁴

Deimos’ Effective Temperature Te.deimos is:

Te.deimos = { 0,47 (1-0,068) 1.362 W/m²*(1/2,32)*[150*(24/30,3)*0,19]¹∕ ⁴ /4*5,67*10⁻⁸ W/m²K⁴ }¹∕ ⁴ =

Te.deimos = 222,98 K

Tsat.mean.deimos ≈ 233 K !

As you can see none of my calculations are the same as the satellites measured. Both Phobos and Deimos have the same approximate Tsat.mean ≈ 233 K

Also I realize this:

Te.phobos = 242,14 K 

Te.deimos = 222,98 K

Tsat.mean.phobos ≈ 233 K

Tsat.mean.deimos ≈ 233 K !

Also it is obvious that two different celestial bodies on the same R = 1,5 AU distance from the sun and with Phobos rotating 4 times faster than Deimos there should be a substantial difference in temperatures.

 

Let’s compare the calculations

Te.mars : Te deimos : Te phobos

213,21 K : 222,98 K : 242,14 K

0,9747 rotation/day : 24/30.3 rotation/day : 24/7,7 rotation/day

amars = 0,25 : adeimos = 0,068 : aphobos = 0,071

Mars and Deimos have close rotational spin but Mars has higher albedo

Phobos and Deimos have close albedo but Phobos has higher rotational spin.

 

The Phobos and Deimos Effective Temperatures were calculated with the Planet Effective Temperature Complete Formula:

Te.compl = [ Φ (1-a) So (1/R²) (β*N*cp)¹∕ ⁴ /4σ ]¹∕ ⁴

Te.phobos = 242,14 K

Te.deimos = 222,98 K

On the other hand the satellite measured

Tsat.mean.phobos ≈ 233 K

Tsat.mean.deimos ≈ 233 K

were actually calculated with the  Effective Temperature Incomplete Formula:

Te.incompl = [ (1-a) So (1/R²) /4σ ]¹∕ ⁴

 

Lets do the calculation for Phobos

Te.phobos.incompl = [ (1-0,071) 1.362 W/m²*(1/2,32) /4*5,67*10⁻⁸ W/m²K⁴ }¹∕ ⁴ =

Te.phobos.incompl ≈ 233 K

 

Lets do the calculation for Deimos

Te.deimos.incompl = [ (1-0,068) 1.362 W/m²*(1/2,32) /4*5,67*10⁻⁸ W/m²K⁴ }¹∕ ⁴ =

Te.deimos.incompl ≈ 233 K

 

Thus we conclude now that the Phobos and Deimos Effective Temperatures  calculated with the Planet Effective Temperature Complete Formula are the correct Phobos' and Deimos' effective temperatures 

Te.compl = [ Φ (1-a) So (1/R²) (β*N*cp)¹∕ ⁴ /4σ ]¹∕ ⁴

Te.phobos = 242,14 K  ( which is the faster rotating ) and

Te.deimos = 222,98 K.

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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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1. Earth's Without-Atmosphere effective temperature ( equilibrium temperature ) calculation formula

1. Earth’s Without-Atmosphere Effective Temperature Calculation:

So = 1.362 W/m² (So is the Solar constant)

Earth’s albedo: aearth = 0,30

Earth is a rocky planet, Earth’s surface solar irradiation accepting factor Φearth = 0,47 (Accepted by a Smooth Hemisphere with radius r sunlight is S*Φ*π*r²(1-a), where Φ = 0,47)

β = 150 days*gr*oC/rotation*cal – is a Rotating Planet Surface Solar Irradiation Absorbing-Emitting Universal Law constant

N = 1 rotation /per day, is Earth’s sidereal rotation period

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.

σ = 5,67*10⁻⁸ W/m²K⁴, the Stefan-Boltzmann constant

Earth’s Without-Atmosphere Effective Temperature Complete Formula Te.earth is:

 

Te.earth = [ Φ (1-a) So (β*N*cp)¹∕ ⁴ /4σ ]¹∕ ⁴

 

Τe.earth = [ 0,47(1-0,30)1.362 W/m²(150 days*gr*oC/rotation*cal *1rotations/day*1 cal/gr*oC)¹∕ ⁴ /4*5,67*10⁻⁸ W/m²K⁴ ]¹∕ ⁴ =

Τe.earth = [ 0,47(1-0,30)1.362 W/m²(150*1*1)¹∕ ⁴ /4*5,67*10⁻⁸ W/m²K⁴ ]¹∕ ⁴ =

Τe.earth = ( 6.914.170.222,70 )¹∕ ⁴ =

Te.earth = 288,36 Κ

And we compare it with the

Tsat.mean.earth = 288 K, measured by satellites.

These two temperatures, the calculated one, and the measured by satellites are almost identical.

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