# The Planet Surface Rotational Warming Phenomenon

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

### Why Calisto (Jupiter's satellite) and Saturn have the same Tsat = 134 K?

Since Calisto is Jupiter's satellite, on Calisto the Solar flux is S = 50,37 W/m², the same Solar flux as on Jupiter.

Jupiter and... Calisto are at R = 5,20 AU (astronomical units) distance from the Sun.

Saturn is at the distance of R = 9,58 AU from the Sun, and the solar flux on Saturn is S = 14,84 W/m².

Let's compare the solar fluxes:

S.calisto = 50,37 W/m²

S.saturn = 14,84 W/m²

It is a very significant difference; the solar flux on Saturn is more than three times lesser.

Why Saturn having the same satellite measured Tmean temperature, as Calisto?

Why Tsat.mean.calisto = Tsat.mean.saturn = 134 K?

It was a scientific mystery. It was a mystery, but it is not a mystery any more...

Let's assemble all the available data for both Calisto and Saturn. And we have much more data to compare now.

Calisto’s albedo: acalisto = 0,22

Saturn’s albedo, asaturn = 0,342

Calisto’s surface consists of water ice crust (by estimation), so Cp.calisto = 1cal/gr*oC

Calisto is an ice-crust planet (rocky) without atmosphere, but Calisto has a heavy cratered surface (golf ball surface) so Calisto’s surface irradiation accepting factor Φcalisto = 1

Saturn is a gaseous planet, Saturn’s surface irradiation accepting factor Φsaturn = 1

(Saturn has not a surface to reflect the incident sunlight. Accepted by a Gaseous Hemisphere with radius r sunlight is S*Φ*π*r²(1-a), where Φ = 1)

Atmosphere composition: 96,3% ± 2,4% H₂, 3,25% ± 2,4% He, 0,45% ± 0,2% CH₄.

Cp.H₂ = 3,1388 cal/gr. oC , H₂ specific heat at 175 K,

Cp.He = 1,243 cal/gr. oC , He specific heat,

Cp.CH₄ = 0,531 cal/gr. oC , CH₄ specific heat.

Cp.saturn = 96,5%*Cp.H₂ + 3,5%*Cp.He = 0,965*3,14 + 0,035*1,243 = 3,030 + 0,0435 = 3,074 cal/gr.oC

Calisto’s orbital period is 16,689.0184 days

Calisto’s sidereal rotation period is synchronous

N.calisto = 0,05992 rotations /day

Saturn’s sidereal rotation period is10 h 33 min 38 sec, or 10,56 h

N.saturn = 2,27273 rotations /day

Ok, we see something now,

N.saturn /N.calisto = 2,27273 /0,05992 = 37,93 times faster Saturn rotates than Calisto

Cp.saturn /Cp.calisto = 3,074 /1 = 3,074 times Saturn's specific heat capacity is higher than Calisto's

Let's correspond now to the Planet's Surface Tmean Temperature New Equation:

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

Φ = 1 for both planets

Tmean.calisto = [ (1 - 0,22) 50,37 W/m² (β*0,05992*1)¹∕ ⁴ /4σ]¹∕ ⁴

Tmean.saturn = [ (1 - 0,342) 14,84 W/m² (β*2,27273*3,074)¹∕ ⁴ /4σ]¹∕ ⁴

Tmean.calisto = [ (0,78) 50,37 W/m² (β*0,05992*1)¹∕ ⁴ /4σ]¹∕ ⁴

Tmean.saturn = [ (0,658) 14,84 W/m² (β*2,27273*3,074)¹∕ ⁴ /4σ]¹∕ ⁴

Tmean.calisto = [ 39,287 W/m² (β*0,05992)¹∕ ⁴ /4σ]¹∕ ⁴

Tmean.saturn = [ 9,765 W/m² (β*6,9864)¹∕ ⁴ /4σ]¹∕ ⁴

Tmean.calisto = [ 39,287 W/m² 0,4948*(β*)¹∕ ⁴ /4σ]¹∕ ⁴

Tmean.saturn = [ 9,765 W/m² 1,6258*(β)¹∕ ⁴ /4σ]¹∕ ⁴

Tmean.calisto = [ 19,4376 W/m² (β*)¹∕ ⁴ /4σ]¹∕ ⁴                        Tmean.saturn = [ 15,8759 W/m² (β)¹∕ ⁴ /4σ]¹∕ ⁴

Tmean.calisto = 2,0997*[ W/m² (β*)¹∕ ⁴ /4σ]¹∕ ⁴                         Tmean.saturn = 1,9961*[ W/m² (β)¹∕ ⁴ /4σ]¹∕ ⁴

Tmean.calisto /Tmean.saturn =

= 2,0997 /1,9961 = 1,052

or 5,2 % difference only !

Now we know why the satellite measured surface mean temperatures Tsat.mean.calisto = Tsat mean.saturn = 134 K

Conclusion:

There is not any inner Saturnine warming phenomenon causing Saturn's atmosphere at 1 bar level being Tmean.1bar = 134 K.

The three times lesser solar flux on Saturn, compared to Calisto, is compensated by the 37,93 times faster Saturn's spin and the 3,074 times higher Saturn's specific heat capacity.