# The Planet Surface Rotational Warming Phenomenon

## The Planet Mean Surface Temperature Equation Tmean = [ Φ (1-a) S (β*N*cp)¹∕ ⁴ /4σ ]¹∕ ⁴

### Earth - Moon temperatures comparison - there is something else very interesting happens

The faster Earth's (compared to Moon) rotation smooths the average heat. The higher planet’s surface specific heat (oceanic waters vs dry regolith), also smooths the average heat. Consequently Earth's daytime surface temperature (compared to Moon) lessens, and Earth's nighttime  surface temperature (compared to Moon) rises.

Earth receives the same amount of solar heat (per unit area) from sun as Moon – for the same albedo. And Earth emits the same amount of solar heat, as the Moon does.

But there is something else very interesting happens.

Now there is a difference between Earth’s and Moon’s emitting temperatures.

At the daytime Earth’s surface is warmed at a much lower temperatures and therefore at the daytime Earth’s surface emits IR radiation at a much lower temperatures. So the intensity of Earth’s daytime IR radiation is much lower.

So there is a great amount of energy – compared to Moon – “saved” on Earth during the daytime emission...

This “saved” energy should be emitted during the nighttime then.

At the nighttime Earth’s surface is warmer (than Moon's) and therefore Earth’s surface emits at the nighttime IR radiation at a higher temperatures (than Moon's). So the intensity of Earth’s nighttime IR radiation is higher.

There is always a balance

The energy in = the energy out

But there is something else very interesting happens.

In order to achieve that balance Earth’s nighttime IR emitting intensity should be much higher than the nighttime IR emitting intensity of the Moon.

Now we should take notice of the nonlinearity of the Stefan-Boltzmann emission law. Consequently the nighttime temperatures on Earth rise higher (compared to Moon) than the daytime temperatures on Earth lessens.

So the average Earth’s surface temperature is warmer (compared to the Moon).

Thus Earth’s Tmean.earth = 288 K and Moon’s Tmean.moon = 220 K

The faster rotation and the higher specific heat does not make sun to put more energy in the Earth’s surface.

What the faster rotation and the higher specific heat do is to modify the way Earth’s surface emits, the same amount as Moon, of energy (per unit area).

Earth emits IR radiation at lower temperatures during the daytime and at higher temperatures at nighttime. Because of the nonlinearity of this process according to the Stefan-Boltzmann emission law Earth ends up to have on average warmer surface than Moon.

The nighttime temperatures on Earth rise higher (compared to Moon) than the daytime temperatures on Earth lessens.

Earth receives (for the same albedo and per unit area) the same as Moon amount of solar energy. This energy is “welcomed” on each planet and processed in a unique for each planet way.

To illustrate the above conclusions I’ll try to demonstrate on the Earth-Moon temperatures comparison rough example:

Surface temperatures

………….min……mean……max

..........Tmin↑↑→T↑mean ←T↓max

Moon…100 K...220 K …390 K

Δ………..+84 K +68 K….- 60 Κ

Earth…184K↑↑.288 K↑.330 K↓

So we shall have for Earth, compared to the Moon:

Tmin↑↑→ T↑mean ← T↓max

+84↑↑→ +68↑mean ← -60↓

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

To emphasize we should mention that Moon’s max and min temperatures are measured on Moon’s equator, and Earth’s max and min temperatures are not. Earth’s max and min temperatures are measured on continents, and not on oceanic waters.

Otherwise the Δmin would have been even bigger and the Δmax would have been much smaller.

This rough example nevertheless illustrates that for the faster rotating and covered with water (higher cp) Earth compared with Moon the average temperature should be higher. The planet’s faster rotation and the planet’s higher specific heat “cp” not only smooths, but also processes ( Δmin > Δmax ), the same incoming solar heat, but in a different emission pattern.