# 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

### Calisto is the warmest Jupiter's satellite. Calisto /Io satellite Tmean temperatures comparison 134 K and 110 K

Tsat.mean.io = 110 K

Let's calculate Io's effective temperature old blackbody equation:

Te.io = [ (1-a) So (1/R²) /4σ ]¹∕ ⁴

Τe.io = [ (1-0,63)1.362 W/m² *0.0369 /4*5,67*10⁻⁸ W/m²K⁴ ]¹∕ ⁴ =

= (81.990.238,1)¹∕ ⁴ = 95,16 K

Tsat.mean.europa = 102 K

Let's calculate Europa's effective temperature old blackbody equation:

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

Τe.europa = [ (1-0,63)1.362 W/m² *0.0369 /4*5,67*10⁻⁸ W/m²K⁴]¹∕ ⁴ =

= (81.990.238,1)¹∕ ⁴ = 95,16 K

So here it is what happens:

Io and Europa have the same albedo a = 0,63

They both are at the same distance from the sun.

So Te.io = Te.europa = 95,16 K …. logical is not it?

Tsat.mean.io = 110 K

Tsat.mean.europa = 102 K

Why Io is warmer than Europa then?

Let's look closer, we would understand even more, look how close the Te = 95,16 K is to the Europa's Tsat.mean.europa =102 K

And compare it with the Tsat.mean.io = 110 K ...

Ganymede and Calisto don't fit in the play, because they have different albedo a.ganymede = 0,43 and a.calisto = 0,22

Also Ganymede and Calisto are at larger distance from Jupiter.

But still Tsat.mean.ganymede = 110 K

Tsat.mean.calisto = 134 K ± 11

Let's check it too, let's calculate Ganymede's effective temperature old blackbody equation:

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

Τe.ganymede = [ (1-0,43)1.362 W/m² *0.0369 /4*5,67*10⁻⁸ W/m²K⁴ ]¹∕ ⁴ =

= (126.309.285,7)¹∕ ⁴ = 106,01 K

Te.ganymede = 106,01 K

Tsat.mean.ganymede = 110 K

Again very close fit Te.ganymede = 106 K

and Tsat.mean.ganymede = 110 K

Let's calculate Calisto's effective temperature old blackbody equation:

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

Τe.calisto = [ (1-0,22)1.362W/m² *0.0369 /4*5,67*10⁻⁸ W/m²K⁴ ]¹∕ ⁴ =

= (172.844.285,7)¹∕ ⁴ = 114,66 K

Tsat.mean.calisto = 134 K ± 11

Calisto does not fit, that is why the 134 K ± 11

There is a big difference of

134 K - 114,66 K = 19,34

So Calisto is warmer, no matter what.

Calisto is at the outmost distance from the Jupiter, so it cannot be warmed from the planet's IR, also because of the distance it has the lowest tidal effect.

Calisto rotates 10 times less than Io, but Calisto has cp =1, compared to Io having cp = 0,145 also Calisto has (1 - 0,22) = 0,78 and Io has (1 - 0,63) = 0,37

That means Calisto "absorbs" twice as much solar energy and the (β*N*cp)¹∕ ⁴ for Calisto is (150*0,0599 *1)¹∕ ⁴ = 1,7313

the (β*N*cp)¹∕ ⁴ for Io is (150*0,5559 *0,145)¹∕ ⁴ = 1,8647

Io coefficient is ( 0,37 * 1,8647 )¹∕ ⁴ = 0,6899¹∕ ⁴ = 0,91137

Calisto coefficient is ( 0,78 * 1,7313 )¹∕ ⁴ = 1,3504¹∕ ⁴ =  1,07799

Calisto coeff /Io coeff = 1,07799 /0,91137 = 1,1828

Tsat.mean.calisto /Tsat.mean.io = 134 K /110 K = 1,2181

1,2181 /1,1828 = 1,029 or only 2,9 % difference !

Io and Calisto have Φ = 1

And that explains the reasons Calisto is the warmest Jupiter's satellite.

Canymede and Europa have Φ = 0,47 .

A smooth planet has Φ = 0,47 so the old blackbody equation Te temperature (calculated without inserting the actual Φ = 0,47) is closer to the satellite measured Tmean.

That is how the tragic coincidences happen, like - the Mars' Te.old tragic coincidence, when Tsat.mean.mars = 210 K and Te.old.mars = 209,8 K appeared almost equal, which led to the wrong conclusion that the planets' Tmean = Te.

Actually the Tsat.mean.mars = 210 K should be compared with the Te.correct.mars = 174 K.