### 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 planet's surface ****has emission temperatures the new distribution 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**

**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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