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 above on the box for more

The faster a planet rotates (n2>n1) the higher is the planet’s average (mean) temperature T↑mean

It is well known that when a planet rotates faster its daytime maximum temperature lessens and the night time minimum temperature rises.

But there is something else very interesting happens.

When a planet rotates faster it is a warmer planet.

(it happens because Tmin↑↑ grows higher than T↓max goes down)

The understanding of this phenomenon comes from a deeper knowledge of the Stefan-Boltzmann Law.

It happens so because when rotating faster a planet's surface has a new radiative equilibrium temperatures to achieve.

So that is what happens:

The faster a planet rotates (n2>n1) the higher is the planet’s average (mean) temperature T↑mean.

It happens in accordance to the Stefan-Boltzmann Law.

 

Let's explain:

Assuming a planet rotates faster and

Tmax2 -Tmax1 = -1°C.

Then, according to the Stefan-Boltzmann Law:

Tmin2 -Tmin1 > 1°C

Consequently Tmean2 > Tmean1.

 

Assuming a planet rotates faster (n2>n1).

If on the solar irradiated hemisphere we observe the difference in average temperature

Tsolar2-Tsolar1 = -1°C

Then the dark hemisphere average temperature

Tdark2 -Tdark1 >1°C

Consequently the total average

Tmean2 > Tmean1

So we shall have:

Tdark↑↑→ T↑mean ← T↓solar

The faster a planet rotates (n2>n1) the higher is the planet’s average (mean) temperature T↑mean.

A numerical example:

Assuming a planet with

Tsolar1 = 200 K, and Tdark1 = 100 K

Assuming this planet rotates faster, so

Tsolar2 = 199 K. What is the planet's Tdark2  then?

 

J1emit.solar ~ (T1solar)⁴ ,

(200 K)⁴ = 1.600.000.000

J2emit.solar ~ (T2solar)⁴ ,

(199 K)⁴ = 1.568.000.000

J2emit.solar - J1emit.solar =

= 1.568.000.000- 1.600.000.000 =

= - 31.700.000

So we have ( - 31.700.000 ) less emitting on the solar side. It should be compensated by the increased emission on the dark side ( + 31.700.000 ).

On the other hand on the dark side we should have a greater warming than a one degree

( 199 K - 200 K = -1 oC ) cooling we had on the solar irradiated side.

 

J1emit.dark ~ (T1dark)⁴ ,

(100 K)⁴ = 100.000.000

J2emit.dark ~ (T2dark)⁴ ,

(107,126 K)⁴ = 131.698.114

J2emit.dark - J1emit.dark =

= 131.698.114 -100.000.000 =

= 31.698.114

The dark side higher temperature to compensate the solar side cooler emission by ( - 31.700.000 ) would be

T2dark = 107,126 K

As we see in this numerical example, when rotating faster maximum temperature on the solar irradiated side subsides from

200 K to 199 K.

On the other hand the minimum temperature on the dark side rises from

100 K to 107,126 K.

So when the solar irradiated side gets on average cooler by 1 degree oC, the dark side gets on average warmer by 7,126 degrees oC.

And as a result the planet total average temperature gets higher.

That is how when a planet rotating faster the radiative equilibrium temperatures are accomplished. It happens so because when rotating faster a planet's surface has a new radiative equilibrium temperatures to achieve.

Consequently, when rotating faster, the planet's mean temperature rises.

Thus when a planet rotates faster its mean temperature is higher.

Conclusion:

Earth's faster rotation rate, 1 rotation per day, makes Earth a warmer planet than Moon.

Moon rotates around its axis at a slow rate of 1 rotation in 29,5 days.

.

http://www.cristos-vournas.com

  The faster a planet rotates (n2>n1) the higher is the planet’s average (mean) temperature T↑mean:

Tmin→ T↑mean ← Tmax

.