Saturn’s Mean Temperature Equation at 1 bar level Tmean.saturn.1bar is:

Tmean.saturn.1bar = [Φ (1-a) So (1/R²) (B*N)¹∕ ⁴ /4σ]¹∕ ⁴

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

N = 24h/10,56h rotations/per day

R = 9,5826 AU, 1/R² = 1/9,5826² = 0,01089 times lesser is the solar irradiation on Saturn than that on Earth.

So = 1.361 W/m² is Solar constant

Saturn’s albedo, asaturn = 0,342 

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

(Saturn has not 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.

B = 850 days/rotation – it is the Rotating Gaseous Planet at 1 bar level Solar Irradiation Absorbing-Emitting Universal constant

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

So we have: Saturn’s mean temperature at 1 bar level Tmean.saturn.1bar is:

Tmean.saturn.1bar = {1*(1-0,342)1.361*0,01089(W/m²) [850*(24h/10,56h)]¹∕ ⁴ /4*5,67*10⁻⁸(W/m²K⁴) }¹∕ ⁴ =

Tmean.saturn.1bar = [0,658*14,84(W/m²) (850*2,273)¹∕ ⁴ /4*5,67*10⁻⁸(W/m²K⁴) ]¹∕ ⁴ =

Tmean.saturn.1bar = [0,658*14,84(W/m²) 6,63 /4*5,67*10⁻⁸(W/m²K⁴) ]¹∕ ⁴ =

Tmean.saturn.1bar = (285.444.273,47)¹∕ ⁴ = 129,98 K = 130 K

Tmean.saturn.1bar = 130 K is the calculated.

And below is the measured by satellites

Tsat.mean.saturn = 134 K (at 1bar level)