Planet Energy Budget:
Solar energy absorbed by a Hemisphere with radius "r" after reflection and dispersion:
Jabs = Φ*πr²S (1-a) (W)
Total energy emitted to space from a whole planet:
Jemit = A*σΤe⁴ /(β*N*cp)¹∕ ⁴ (W)
Α - is the total planet surface (m²)
(β*N*cp)¹∕ ⁴ - dimensionless, is a Rotating Planet Surface Solar Irradiation Warming Ability
A = 4πr² (m²), where r – is the planet's radius
Jemit = 4πr²σTe⁴ /(β*N*cp)¹∕ ⁴ (W)
global Jabs = global Jemit
Φ*πr²S (1-a) = 4πr²σTe⁴ /(β*N*cp)¹∕ ⁴
Or after eliminating πr²
Φ*S*(1-a) = 4σTe⁴ /(β*N*cp)¹∕ ⁴
The planet average Jabs = Jemit per m² planet surface:
Jabs = Jemit
Φ*S*(1-a) /4 = σTe⁴ /(β*N*cp)¹∕ ⁴ (W/m²)
Solving for Te we obtain the effective temperature:
Te = [ Φ (1-a) S (β*N*cp)¹∕ ⁴ /4σ ]¹∕ ⁴ (K)
β = 150 days*gr*oC/rotation*cal – is a Rotating Planet Surface Solar Irradiation Absorbing-Emitting Universal Law constant
N rotations/day, is planet’s sidereal rotation period
cp – is the planet surface specific heat
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.
Here (β*N*cp)¹∕ ⁴ - is a dimensionless Rotating Planet Surface Solar Irradiation Warming Ability
σ = 5,67*10⁻⁸ W/m²K⁴, the Stefan-Boltzmann constant
Rotating Planet Spherical Surface Solar Irradiation Absorbing-Emitting Universal Law:
Jemit = σΤe⁴/(β*N*cp)¹∕ ⁴ (W/m²)
The year-round averaged energy flux at the top of the Earth's atmosphere is Sο = 1.362 W/m².
With an albedo of a = 0,3 and a factor Φ = 0,47 we have Te = 288,36 K or 15°C.
This temperature is confirmed by the satellites measured Tmean.earth = 288 K.