A Planet Effective Temperature Complete Formula Te = [ Φ (1-a) S (β*N*cp)¹∕ ⁴ /4σ ]¹∕ ⁴

Plus the introduction to the Reversed Milankovitch Cycle. Click on the box for more

Rotating Planet Spherical Surface Solar Irradiation Absorbing-Emitting Universal Law

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 = 4πr²σTe /(β*N*cp)¹∕ ⁴   (W)

global Jabs = global Jemit

Φ*πr²S (1-a) = 4πr²σTe /(β*N*cp)¹∕ ⁴

Or after eliminating πr²

Φ*S*(1-a) = 4σTe /(β*N*cp)¹∕ ⁴

The planet average Jabs = Jemit per m² planet surface:

Jabs = Jemit

Φ*S*(1-a) /4 = σTe /(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 = σΤe/(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.