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, the day-time planet surface emission temperatures are lower and the HEAT ACCUMULATION is higher.
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
Here it is the improved numerical example which proves, the Tmean > Te when the planet rotates fast enough:
As you will see in the numerical example, which I have shown below, when planet rotates faster, on the planet's solar irradiated side the Te.solar temperature subsides from 200 K to 199 K.
On the other hand on the planet's dark side, when planet rotates faster, the Te.dark temperature rises from 100 K to 107,126 K.
So when the solar irradiated side gets on Te.solar cooler by -1 degree °C, the dark side gets on Te.dark warmer by +7,126 degrees °C.
And as a result the planet's total Te temperature gets higher. It happens so because when rotating faster (n2>n1) the planet's surface has emission temperatures Te the new distribution to achieve.
It happens so, because we have assumed planet emitting as a blackbody with two separate hemispheres.
The solar hemisphere emitting some of the absorbed incident solar flux's energy, and the dark hemisphere emitting the rest of the absorbed solar flux's energy.
Also, when the two hemispheres blackbody planet rotating faster
the energy in = energy out
balance should be met.
The faster rotation does not change the real planet's energy balance.
Also, the real planet never achieves uniform temperature on both sides, because it receives the solar flux only on the sunlit side.
Because we consider the faster rotating real planet at the same distance from the sun, with the same albedo and Φ factor -
energy in = energy out
balance should be met.
The numerical example:
Assuming a planet with two hemispheres' Te temperatures
Te.solar1 = 200 K, and Te.dark1 = 100 K
Assuming this planet rotates somehow faster (n2 > n1), so assuming the new Te.solar2 average temperature resulting
Te.solar2 = 199 K.
What would be the planet's Te.dark2 then?
Jemit.solar1 = σ*(Te.solar1)⁴ ,
(200 K)⁴ = 1.600.000.000*σ for (n1) rot/day
Jemit.solar2 = σ*(Te.solar2)⁴ ,
(199 K)⁴ = 1.568.000.000*σ for (n2) rot/day
Jemit.solar2 - Jemit.solar1 =
= 1.568.000.000*σ - 1.600.000.000*σ =
= - 31.700.000*σ is the difference in the Te solar side emitting intensity when (n2>n1) and 199 K - 200 K = - 1°C
So we have ( - 31.700.000*σ ) less emitting intensity on the solar side (2) when n2>n1.
It should be compensated by the increased emission on the dark side ( + 31.700.000*σ ) for the energy balance equation to get met:
Jemit.dark1 = σ*(Te.dark1)⁴ ,
(100 K)⁴ = 100.000.000
Jemit.dark2 = σ*(Te.dark2)⁴ ,
(Te.dark2)⁴ = (100.000.000 + 31.700.000) = 131.700.000
The dark side higher temperature (2) to compensate the solar side cooler emission (2) by ( - 31.700.000 ) would be
Te.dark2 = (131.700.000)¹∕ ⁴ = 107,126 K
As we see in this numerical example, when the planet rotating faster (n2>n1) the Te temperature on the solar irradiated side subsides from 200 K to 199 K.
On the other hand the Te temperature, when planet rotating faster (n2>n1) on the dark side rises from 100 K to 107,126 K.
So when rotating faster (n2>n1) the solar irradiated planet's side gets on Te cooler by -1 degree °C, the planet's dark side gets on Te warmer by +7,126 degrees °C.
And as a result the planet's total Te temperature gets higher.
It happens so because when rotating faster (n2>n1) the planet's surface has emission temperatures Te the new distribution to achieve.
Consequently, when rotating faster, the planet's mean temperature rises.
Because Te.solar2-Te.solar1 = -1°C Then the dark hemisphere's Te temperature Te.dark2 -Te.dark1 = +7,126°C
And Tmean = [ Φ (1 - a) S (β*N*cp)¹∕ ⁴ /4σ ]¹∕ ⁴
Tmean = [ Te⁴ (β*N*cp)¹∕ ⁴ ]¹∕ ⁴
Tmean.solar2 = [ (Te.solar1 -1°C)⁴ (β*N*cp)¹∕ ⁴ ]¹∕ ⁴
Tmean.dark2 = [ (Te.dark1 +7,126°C)⁴ (β*N*cp)¹∕ ⁴ ]¹∕ ⁴
(Tmean.solar1 + Tmean.dark1) / 2 < (tmean.solar2="" +="" tmean.dark2)="">
Consequently the total average
Tplanet.mean2 > Tplanet.mean1
So we shall have: when n2>n1
Tmean.dark↑↑→ T↑mean ← T↓mean.solar
The faster a planet rotates (n2>n1) the higher is the planet’s average (mean) temperature T↑mean.
Because Te.dark2↑↑ grows higher (+7,126°C) than the T↓e.solar2 lessens (-1°C).
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.
Notice:
In the above numerical example we assumed a rotating planet blackbody with two hemispheres' Te.solar and Te.dark.
It was an assumption, because the blackbody by definition has a uniform temperature on its entire surface.
Also we assumed, that the blackbody somehow had accumulated some of the daytime solar energy.
In this numerical example we have a combination of the blackbody and the real planet emitting behavior. And it is also an assumption.
Real planet does not emit according to the exact Stefan-Boltzmann emission law. Real planet emits exactly according to the New Universal law:
Jemit.planet = 4σ Tmean⁴ /(β*N*cp)¹∕ ⁴ .
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The faster a planet rotates (n2>n1) the higher is the planet’s average (mean) temperature T↑mean:
Tmin↑→ T↑mean ← T↓max
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