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

## 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 Mean Surface Temperatures comparison

3. Mars’ Mean Surface 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' Mean Surface Temperature Equation is:

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

Planet Mars’ Mean Surface Temperature Tmean.mars is:

Tmean.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

Tmean.mars = 213,21 K

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

Tsat.mean.mars = 210 K !

Phobos’ ( Mars' moon ) Surface Mean Temperature Calculation:

Tmean.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' Mean Surface Temperature Equation is:

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

Phobos’ Surface Mean Temperature Tmean.phobos is:

Tmean.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 =

Tmean.phobos = 242,14 K

Tsat.mean.phobos ≈ 233 K !

Deimos’ ( Mars' moon ) Surface Mean Temperature Calculation:

Tmean.deimos

(1/R²) = (1/1,524²) = 1/2,32

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

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' Surface Mean Temperature Equation is:

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

Deimos’ Surface Mean Temperature Tmean.deimos is:

Tmean.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 =

Tmean.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:

Tmean.phobos = 242,14 K

Tmean.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

Tm.mars : Tm.deimos : Tm.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 Surface Mean  Temperatures were calculated with the Planet Surface Mean Temperature Equation:

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

Tmean.phobos = 242,14 K

Tmean.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 Equation:

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

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

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

Tmean.deimos = 222,98 K.