### 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.**

.

### Mars and Moon satellite measured mean temperatures comparison: 210 K and 220 K

**Mars and Moon satellite measured mean temperatures comparison:**

**210 K and 220 K**

**Let's see what we have here:**

**Planet Tsat.mean**

** measured**

**Mercury 340 K**

**Earth 288 K**

** Moon 220 Κ **

**Mars 210 K**

**Let’s compare then: Moon:**

**Tsat.moon = 220K**

**Moon’s albedo is amoon = 0,11**

**What is left to absorb is (1 – amoon) = (1 - 0,11) = 0,89**

**Mars:**

**Tsat.mars = 210 K**

**Mars’ albedo is amars = 0,25**

**What is left to absorb is (1 – amars) = (1 – 0,25) = 0,75**

**Mars /Moon satellite measured temperatures comparison:**

** Tsat.mars /Tsat.moon = 210 K /220 K = 0,9545**

**Mars /Moon what is left to absorb (which relates in ¼ powers) comparison,**

**or in other words the Mars /Moon albedo determined solar irradiation absorption ability:**

** ( 0,75 /0,89 )¹∕ ⁴ = ( 0,8427 )¹∕ ⁴ = 0,9581**

** Conclusions:**

**1. Mars /Moon satellite measured temperatures comparison**

** ( 0,9545 )**

**is almost identical with the Mars /Moon albedo determined solar irradiation absorption ability**

** ( 0,9581 )**

**2. If Mars and Moon had the same exactly albedo, their satellites measured temperatures would have been exactly the same.**

**3. Mars and Moon have two major differencies which equate each other.**

**The first major difference is the distance from the sun both Mars and Moon have.**

**Moon is at R = 1 AU distance from the sun and the solar flux on the top is So = 1.361 W/m² ( it is called the Solar constant).**

**Mars is at 1,524 AU distance from the sun and the solar flux on the top is S = So*(1/R²) = So*(1/1,524²) = So*1/2,32 .**

**(1/R²) = (1/1,524²) = 1/2,32**

**Mars has 2,32 times less solar irradiation intensity than Earth and Moon have.**

**Consequently the solar flux on the Mar’s top is 2,32 times weaker than that on the Moon.**

**The second major difference is the sidereal rotation period both Mars and Moon have.**

**Moon performs 1 rotation every 29,531 earth days.**

**Mars performs 1 rotation every ( 24,622hours / 24hours/day ) = 1,026 day.**

**Consequently Mars rotates 29,531 /1,026 = 28,783 times faster than Moon does.**

**So Mars is irradiated 2,32 times weaker, but Mars rotates 28,783 times faster.**

**And… for the same albedo, Mars and Moon have the same satellite measured mean temperatures.**

**Let’s take out the calculator now and make simple calculations:**

**The rotation difference fourth root**

** (28,783)¹∕ ⁴ = 2,316**

**The irradiation /rotation comparison**

** 2,32 /(28,783)¹∕ ⁴ = 2,32 /2,316 = 1,001625**

**It is only 0,1625 % difference**

** When rounded the difference is 0,16%**

** It is obvious now, the Mars’ 28,783 times faster rotation equates the Moon's 2,32 times stronger solar irradiaton.**

**That is why the 28,783 times faster rotating Mars has almost the same average satellites measured temperature as the 2,32 times stronger solar irradiated Moon.**

**Thus we are coming here again to the same conclusion:**

** The Faster a Planet Rotates, the Higher is the Planet's Average Temperature.**

.

### Earth, Moon and Mars - two very important observations - conclusions

**We are ready now to make two very important observations.**

** 1. Moon and Mars Moon's satellite measured Tsat.mean.moon = 220 K Mars' satellite measured Tsat.mean.mars = 210 K**

** These two observed temperatures on the different planets (Mars and Moon) are very close.**

** The solar flux on Moon is So = 1.361 W/m². The solar flux on Mars is S.mars = 586,4 W/m². **

**Thus we observe here that there can be planets with different solar irradiation fluxes, and yet the planets may have (for equal albedo) the same mean surface temperatures. **

**So we may have:**

** Many planets with different solar irradiation fluxes, and yet the planets may have (for equal albedo) the same mean surface temperatures. **

**Conclusion:**

** Many different solar fluxes (for equal albedo) can create the same mean surface temperatures.**

** 2. Moon and Earth Moon's satellite measured Tsat.mean.moon = 220 K Earth's satellite measured Tsat.mean.earth = 288 K **

**These two observed temperatures on the different planets (Moon and Earth) are very different. **

**The solar flux on Moon is So = 1.361 W/m². The solar flux on Earth is So = 1.361 W/m².**

** Thus we observe here that there can be planets with the same solar irradiation fluxes, and yet the planets may have (for equal albedo) very different mean surface temperatures.**

** So we may have: **

**Many planets with the same solar irradiation fluxes, and yet the planets may have (for equal albedo) different mean surface temperatures. **

**Conclusion:**

**Many different global temperature distributions (for equal albedo) may balance the same solar flux.**

**These two very important observations - conclusions lead us to the formulation of the Mean Surface Temperature Equation.**

.

### A Planet Without-Atmosphere Mean Surface Temperature Equation

**A Planet Without-Atmosphere Mean Surface Temperature Equation derives from the incomplete Te equation which is based on the radiative equilibrium and on the Stefan-Boltzmann Law.**

** from the incomplete**

** Te = [ (1-a) S / 4 σ ]¹∕ ⁴**

** which is in common use right now, but actually it is an incomplete Te equation and that is why it gives us very confusing results.**

** A Planet Without-Atmosphere Surface Mean Temperature Equation is also based on the radiative equilibrium and on the Stefan-Boltzmann Law.**

** The Equation is being formulated by adding to the incomplete Te Equation the new parameters Φ, N, cp and the constant β.**

** to the Planet Without-Atmosphere Surface Mean Temperature Equation**

** Tmean.planet = [ Φ (1-a) S (β*N*cp)¹∕ ⁴ /4σ ]¹∕ ⁴ (1)**

** (.......)¹∕ ⁴ is the fourth root**

** S = So(1/R²), where R is the average distance from the sun in AU (astronomical units)**

** S - is the solar flux W/m²**

** So = 1.361 W/m² (So is the Solar constant)**

** Planet’s albedo: a**

** Φ - is the dimensionless solar irradiation spherical surface accepting factor**

** Accepted by a Hemisphere with radius r sunlight is S*Φ*π*r²(1-a), where Φ = 0,47 for smooth surface planets, like Earth, Moon, Mercury and Mars…**

** (β*N*cp)¹∕ ⁴ is a dimensionless Rotating Planet Surface Solar Irradiation Warming Ability**

** 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 spin**

** 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.**

** cp = 0,19 cal/gr*oC, for dry soil rocky planets, like Moon and Mercury.**

** Mars has an iron oxide F2O3 surface,**

** cp.mars = 0,18 cal/gr*oC**

** σ = 5,67*10⁻⁸ W/m²K⁴, the Stefan-Boltzmann constant**

** This Universal Formula (1) is the instrument for calculating a Planet-Without-Atmosphere Surface Mean Temperature. **

**The results we get from these calculations are almost identical with those measured by satellites.**

* Planet Te.incompl Tmean Tsat.mean*

* Mercury 439,6 K 325,83 K 340 Κ*

* Earth 255 K 287,74 K 288 K*

* Moon 270,4 Κ 223,35 Κ 220 Κ*

* Mars 209,91 K 213,21K **210 K*

.

### A Rotating Planet 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 entire planet:**

** Jemit = A*σΤmean⁴ /(β*N*cp)¹∕ ⁴ (W) **

**Φ - is a dimensionless Solar Irradiation accepting factor**

** (1-Φ + Φ*a) - is the reflected fraction of the incident on the planet solar flux **

**S - is a Solar Flux at the top of atmosphere (W/m²)**

** Α - is the total planet surface (m²)**

** Tmean - is a Planet's Surface Mean Temperature (K)**

** (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 = 4πr²σTmean⁴ /(β*N*cp)¹∕ ⁴ (W) **

**global Jabs = global Jemit**

** Φ*πr²S (1-a) = 4πr²σTmean⁴ /(β*N*cp)¹∕ ⁴**

** Or after eliminating πr² **

**Φ*S*(1-a) = 4σTmean⁴ /(β*N*cp)¹∕ ⁴**

** The planet average**

** Jabs = Jemit per m² planet surface:**

** Jabs = Jemit**

** Φ*S*(1-a) /4 = σTmean⁴ /(β*N*cp)¹∕ ⁴ (W/m²) **

**Solving for Tmean we obtain the planet's surface Mean temperature:**

** Tmean = [ 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 spin**

** 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**

** The year-round averaged energy flux at the top of the Earth's atmosphere is Sο = 1.361 W/m².**

** With an albedo a = 0,306 and a factor Φ = 0,47 we have:**

** Tmean.earth = 287,74 K or 15°C.**

** This temperature is confirmed by the satellite measured Tsat.mean.earth = 288 K.**

.

### We can confirm now with great confidence

**So, we can confirm now with great confidence, that a Planet or Moon Without-Atmosphere Surface Mean Temperature Equation, according to the Stefan-Boltzmann Law, is:**

** Tmean.planet = [ Φ (1-a) So (1/R²) (β*N*cp)¹∕ ⁴ /4σ ]¹∕ ⁴ (1) **

**We have collected the results now:**

** Comparison of results**

** the planet's Te calculated by the Incomplete Equation,**

** the planet's surface Tmean calculated by the Equation,**

** and the planet Tsat.mean measured by satellites:**

** Te. incompl Tmean Tsat.mean**

* Mercury 439,6 K 325,83 K 340 Κ*

* Earth 255 K 287,74 K 288 K*

* Moon 270,4 Κ 223,35Κ 220 Κ*

* Mars 209,91 K 213,21 K 210 K*

** These data, the calculated by a Planet Without-Atmosphere Surface Mean Temperature Equation and the measured by satellites are almost the same, very much alike.**

** They are almost identical, within limits, which makes us conclude that the Planet's Without-Atmosphere Surface Mean Temperature Equation**

** Tmean.planet = [ Φ (1-a) So (1/R²) (β*N*cp)¹∕ ⁴ /4σ ]¹∕ ⁴ (1)**

** can calculate the planets' mean temperatures.**

** It is a situation that happens once in a lifetime in science. Although the evidences existed, were measured and remained isolated information so far.**

** It was not obvious one could combine the evidences in order to calculate the planet’s surface mean temperature.**

** A planet without-atmosphere effective temperature equation**

** Te = [ (1-a) S / 4 σ ]¹∕ ⁴**

** is incomplete because it is based only on two parameters:**

** 1. On the average solar flux S W/m² on the top of a planet’s atmosphere and**

** 2. The planet’s average albedo a.**

** We use more major parameters for the planet's surface temperature equation.**

** Planet is a celestial body with more major features when calculating planet effective temperature to consider. The planet without-atmosphere effective temperature calculating formula has to include all the planet’s basic properties and all the characteristic parameters.**

** 3. The planet's axial spin N rotations/day.**

** 4. The thermal property of the surface (the specific heat cp).**

** 5. The planet's surface solar irradiation accepting factor Φ ( the spherical surface’s primer solar irradiation absorbing property ).**

** Altogether these parameters are combined in the Planet's Without-Atmosphere Surface Mean Temperature Equation:**

** Tmean.planet = [ Φ (1-a) So (1/R²) (β*N*cp)¹∕ ⁴ /4σ ]¹∕ ⁴ (1)**

** A Planet Without-Atmosphere Surface Mean Temperature Equation produces very reasonable results:**

** Tmean.earth = 287,74 K, **

**calculated with the Equation, which is identical with the**

** Tsat.mean.earth = 288 K, **

**measured by satellites.**

** Tmean.moon = 223,35 K,**

** calculated with the Equation, which is almost the same with the**

** Tsat.mean.moon = 220 K,**

** measured by satellites.**

** A Planet Without-Atmosphere Surface Mean Temperature Equation gives us a planet surface mean temperature values very close to the satellite measured planet mean temperatures.**

** It is a Stefan-Boltzmann Law Triumph! And it is a Milankovitch Cycle coming back! And as for NASA, all these new discoveries were possible only due to NASA satellites planet temperatures precise measurements!**

.

### Observations resulting in Equation- the perfect fitting !

**Observations –**

**the planets’ measured temperatures,**

**the planets’ surface specific heat cp,**

**the planets’ sidereal rotation period,**

**the distance from the sun,**

**the measured by space crafts planets’ albedo,**

**the planets’ smooth or heavy cratered surface.**

**The discovery of the “The faster a planet rotates (n2>n1) the higher is the planet’s average temperature:**

**Tmin ↑↑→T↑mean←T↓max,**

**because Tmin grows faster”.**

**The understanding that a planet’s surface does not behave as a blackbody surface and it does not emit as a blackbody.**

**The understanding that:**

**(1 - Φ + Φ*a)S - is the reflected fraction of the incident on the planet solar flux**

**And**

**Φ(1 - a)S - is the absorbed fraction of the incident on the planet solar flux **

**All these observations together led to the discovery of the Rotating Planet Spherical Surface Solar Irradiation Absorbing-Emitting Universal Law:**

**Jabs=Φ*S*(1-a)/4=**

**=Jemit=σΤmean⁴/(β*N*cp)¹∕ ⁴ (W/m²)**

**And only then, solving for Tmean we obtain the Planet's Surface Mean Temperature Equation:**

**Tmean.planet = [ Φ (1-a) So (1/R²) (β*N*cp)¹∕ ⁴ /4σ ]¹∕ ⁴ (1)**

**and then we calculate**

**Tmean.earth = 287,74 K,**

**calculated with the Equation, which is identical with the**

**Tsat.mean.earth = 288 K,**

**measured by satellites.**

**Tmean.moon = 223,35 K,**

**calculated with the Equation, which is almost the same with the**

**Tsat.mean.moon = 220 K,**

**measured by satellites.**

**The calculated planet’s temperatures were confirmed by space crafts’ measurements.**

**It is the Rotating Planet Spherical Surface Solar Irradiation Absorbing-Emitting Universal Law:**

**Jabs=Φ*S*(1-a)/4=**

**=Jemit=σΤmean⁴/(β*N*cp)¹∕ ⁴ (W/m²)**

**confirmation.**

**There is a lot of physics here to consider.**

**It is a new Universal Law. That is why it fits in observations.**

**A Universal Law has to fit in observations.**

**A Universal Law has the ability to describe the observations.**

**And has the ability to explain the observations.**

**A new Universal Law becomes then a powerful instrument for the further scientific research.**

.

### The surface responses so fast, we can skip the absorption phase and go straight to the emission

**It is common to believe that the irradiated surface first absorbs and warms and only then emits according to the Stefan-Boltzmann Law**

* No, it is not like this. It does not happen like this.*

** Conduction and convection are very slow energy transfer processes.**

** The radiation is very fast, it “works” at the very instant.**

** When irradiated the surface responses at the very instant.**

** It does not absorb first, rise the temperature and then emit.**

* The surface responses so fast, we can skip the absorption phase and go straight to the emission.*

** Surface emission’s intensity determines surface temperature rise.**

** The temperature rise creates a higher temperature gradient with the soil or the water.**

** It is then when the absorption starts.**

**There is a very small amount energy absorbed and it depends on the surface properties.**

.

### Earth's Without-Atmosphere Mean Surface Temperature Equation

**Tmean.earth**

**So = 1.361 W/m² (So is the Solar constant)**

**Earth’s albedo: aearth = 0,306**

**Earth is a rocky planet, Earth’s surface solar irradiation accepting factor Φearth = 0,47 (Accepted by a Smooth Hemisphere with radius r sunlight is S*Φ*π*r²(1-a), where Φ = 0,47)**

**β = 150 days*gr*oC/rotation*cal – is a Rotating Planet Surface Solar Irradiation Absorbing-Emitting Universal Law constant**

**N = 1 rotation /per day, is Earth’s sidereal rotation spin**

**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.**

**σ = 5,67*10⁻⁸ W/m²K⁴, the Stefan-Boltzmann constant**

**Earth’s Without-Atmosphere Mean Surface Temperature Equation Tmean.earth is:**

* Tmean.earth = [ Φ (1-a) So (β*N*cp)¹∕ ⁴ /4σ ]¹∕ ⁴*

**Τmean.earth = [ 0,47(1-0,306)1.361 W/m²(150 days*gr*oC/rotation*cal *1rotations/day*1 cal/gr*oC)¹∕ ⁴ /4*5,67*10⁻⁸ W/m²K⁴ ]¹∕ ⁴ =**

**Τmean.earth = [ 0,47(1-0,306)1.361 W/m²(150*1*1)¹∕ ⁴ /4*5,67*10⁻⁸ W/m²K⁴ ]¹∕ ⁴ =**

**Τmean.earth = ( 6.854.897.370,96 )¹∕ ⁴ = 287,74 K**

** Tmean.earth = 287,74 Κ**

**And we compare it with the**

** Tsat.mean.earth = 288 K, measured by satellites.**

**These two temperatures, the calculated one, and the measured by satellites are almost identical.**

**Conclusions:**

**The equation produces remarkable results.**

**The calculated planets’ temperatures are almost identical with the measured by satellites.**

*Planet...Te. incompl....Tmean..Tsat.mean*

*Mercury…440 K…….325,83 K……..340 K*

*Earth…….255 K……..287,74 K……..288 K*

*Moon……270.4 Κ…...223,35 Κ……..220 Κ*

*Mars…..209,91 K……..213,21 K…….210 K*

** The 288 K – 255 K = 33 oC difference does not exist in the real world.**

**There are only traces of greenhouse gasses.**

**The Earth’s atmosphere is very thin. There is not any measurable Greenhouse Gasses Warming effect on the Earth’s surface.**

.