Explain Rotational Warming Model | cristos-vournas.com

Explain rotational warming model

Good day everyone!

Here it is a Link to conversation with Grok, where Grok (X) explains the Rotational Warming Model.
Best,

Christos

https://x.com/i/grok/share/9a734146d4b04885b8ec99e1f23e597c

///////////////////////

Thoughts

The rotational warming model (full name: Planet Surface Rotational Warming Phenomenon) is a theory proposed by mechanical engineer Christos Vournas. It explains planetary (and moon) mean surface temperatures primarily through rotation rate and surface heat capacity, rather than atmospheric greenhouse effects or simple blackbody radiation.

cristos-vournas.com

It argues that faster rotation + higher surface specific heat capacity leads to warmer average surface temperatures for bodies receiving the same solar input, because it changes how solar energy is partitioned between immediate infrared (IR) emission and stored heat.

Core Idea (Intuitive Explanation)

Traditional blackbody models assume:

Solar energy not reflected (1 − albedo) is fully absorbed as heat.
The planet then re-radiates it uniformly as IR according to the Stefan-Boltzmann law (σT⁴).

Vournas claims this is incomplete. When solar flux hits the surface:

Part is immediately re-emitted as IR at the instant of interaction (before it can be stored as heat).
Only the remainder is absorbed into the surface layers as thermal energy.

On a slow-rotating body (e.g., the Moon):

The sunlit side stays exposed for a long time → it heats to very high daytime temperatures.
High T → very intense IR emission (because σT⁴ grows steeply with temperature).
Most incoming energy is lost as immediate IR; little is stored for the long night → lower average (mean) temperature.

On a fast-rotating body with high heat capacity (e.g., Earth with oceans):

Solar heating is spread over a shorter day → lower peak daytime temperatures.
Lower T on the sunlit side → weaker immediate IR emission.
More energy is absorbed/stored as heat and carried around to the night side → higher average temperature.

The result: faster rotation raises the global mean surface temperature (Tmean) even though total incoming solar energy is unchanged. The effect is amplified by higher surface specific heat capacity (cp), because the material can store more energy without large temperature swings.

cristos-vournas.com

Classic example: Earth (288 K) vs. Moon (≈220 K). Both are at 1 AU from the Sun, but the Moon receives ~28 % more solar energy (lower albedo). Earth rotates 29.5× faster and has much higher surface cp (water/ice ≈ 1 cal/g°C vs. lunar regolith ≈ 0.19 cal/g°C). Vournas’ model attributes the entire +68 K difference to this rotational warming factor, not the atmosphere.

cristos-vournas.com

Mathematical Formulation

Vournas derives a “new universal equation” for mean surface temperature of airless or thinly-atmosphered bodies:

Tmean=[Φ(1−a)S(βNcp)^¼ /4σ]^¼

Where:

(Φ): Solar irradiation accepting factor (≈ 0.47 for “smooth” surfaces like Earth/Moon to account for specular reflection not captured in measured diffuse albedo; = 1 for rough surfaces).
(a): Bond/diffuse albedo.
(S): Solar constant at 1 AU (≈ 1361 W/m²); multiply by
(1/R²)
for distance R in AU.
(β): Empirical constant = 150 days·g·°C / rotation·cal.
(N): Rotations per day.
(cp): Average surface specific heat capacity (cal/g·°C).
(σ): Stefan-Boltzmann constant = 5.67 × 10 ^{- 8} W/m²K⁴.

When everything else is equal, mean temperatures scale with the sixteenth root of the rotational warming factor:

Tmean,1 /Tmean,2 = (N1cp,1 /N2cp,2)^1/16

(The

(Ncp)^¼

term inside the brackets becomes ^{1/16} after the outer fourth-root.)

cristos-vournas.com

This equation is claimed to match NASA satellite measurements for ~20 solar-system bodies (Earth, Moon, Mars, Mercury, icy moons, etc.) far better than the classic effective temperature

Te=[(1−a)S/4σ]^¼.

Application to Haumea (from the tweet you linked)Haumea is a fast rotator (period ≈ 0.915 h → N ≈ 6.13 rotations/day), icy surface (cp = 1 cal/g°C), albedo a = 0.33, Φ = 1 (rough surface), at ~50 AU (solar flux diluted by 1/2500).

Plugging into the formula yields Tmean ≈ 54.55 K, close to satellite estimates (< 50 K). The high N and cp give a strong rotational warming boost despite the tiny solar flux.

cristos-vournas.com

Key Claims and Context

No significant greenhouse effect on Earth: The classic “+33 °C greenhouse warming” (288 K vs. 255 K Te) disappears once rotation, cp, and Φ are included. The model predicts 287.74 K for Earth without any atmosphere.
The phenomenon applies to all solar-irradiated rotating bodies (planets, moons, even gas/ice giants under certain conditions).
Global warming on Earth is attributed to orbital factors (Milankovitch/precession aligning perihelion with Southern Hemisphere summer), not CO₂.

This is Vournas’ independent theory, developed by comparing NASA temperature data across solar-system bodies. It is presented on his blog (cristos-vournas.com) and has been discussed in some skeptic/climate-commentary circles, but it is not part of mainstream planetary science or climate modeling, which attributes Earth’s warmth primarily to its atmosphere and ocean heat transport.

cristos-vournas.com

In short, the rotational warming model reframes planetary temperature as a balance between immediate IR loss and stored heat, with rotation rate and surface properties acting as the key “amplifiers.”

22 web pages

Derive the rotational warming equation

Thoughts

The rotational warming equation (also called the Planet Mean Surface Temperature Theoretical Equation) proposed by Christos Vournas is:

Tmean = [Φ(1−a)S (βNcp)^¼ /4σ]^¼

where:

(Φ): Solar Irradiation Accepting Factor (0.47 for smooth surfaces like Earth/Moon to account for unmeasured specular reflection; 1 for rough/cratered surfaces).
(a): Bond albedo (diffuse reflectivity).
(S): Solar flux at the planet’s distance (W/m²).
(β): Universal constant = 150 days·g·°C / (rotation·cal).
(N): Rotations per day.
(cp): Average surface specific heat capacity (cal/g·°C).
(σ): Stefan-Boltzmann constant = 5.67 × 10^{- 8} W/m²K⁴.

This equation is claimed to predict measured mean surface temperatures

Tsat

for airless or thinly-atmosphered bodies far better than the classical effective temperature

Te.

Step-by-Step Derivation (Following Vournas’ Logic)

Vournas starts from the standard planetary energy balance (absorbed solar power = emitted thermal power) but modifies it for two key effects:

(1) specular reflection not captured by measured albedo, and

(2) the Planet Surface Rotational Warming Phenomenon (faster rotation + higher surface heat capacity reduces immediate infrared emission during insolation, so more energy is stored as heat and re-radiated later, raising the global mean temperature).

(1. Classical Effective Temperature (Starting Point)

For a planet in radiative equilibrium:

Φ πr² S(1−a) = 4πr² σTe⁴

The left side is absorbed power over the cross-section; right side is emission over the full surface.)

Solving gives the classical form (ignoring

Φ for now):

Te = [(1−a)S /4σ]^¼

Vournas calls

(and the later corrected version) a “mathematical abstraction” because real planets have non-uniform local temperatures and partial immediate re-emission.

cristos-vournas.com

2. Correct for Specular Reflection →

Te,correct

Measured albedo (a) is diffuse (Bond albedo). Smooth surfaces specularly reflect extra solar energy that the simple model misses. Introduce

Φ < 1

Te,correct = [Φ(1−a)S /4σ]^¼

For Earth:

Φ = 0.47

Te,correct ≈ 210K (vs. classical 255 K).

cristos-vournas.com

3. Introduce the Rotational Warming Phenomenon (Physical Intuition)Solar energy hitting the surface does three things at the instant of incidence:

Reflected (already in 1−a ).
Immediately re-emitted as IR (before storage).
Absorbed as heat in surface layers (later re-radiated as regular IR).

The fraction that is immediately emitted depends on local daytime temperature. Because

σT⁴is steeply nonlinear:

Slow rotation + low
cp
→ high peak daytime (T) → very intense immediate IR loss → less net heat stored → lower global mean (T).
Fast rotation + high
cp
→ solar heating spread over shorter “days” → lower peak (T) → weaker immediate IR loss → more net heat stored and carried to night side → higher global mean (T).

Vournas states: “The higher the planetary surface

N⋅cp

product … the higher heat absorption ability and warmer planetary surface.” Higher

N⋅cp

“lessens the Immediate IR emission.” No extra energy is added; the planet simply retains a larger share of the incoming flux.

cristos-vournas.com

4. Empirical Relation from Data Vournas plots satellite-measured

Tsat

(normalized by flux) for ~20 bodies (Mercury, Moon, Mars, Earth, icy moons, etc.) against the rotational warming factor

(βNcp)^1/16 .

The ratio

Tsat/Te,correct

falls on a single monotonic line for airless/thin-atmosphere bodies:

Tmean /Te,correct = (βNcp)^1/16

(This 1/16 power emerges because the full temperature equation involves a fourth root outside and another effective fourth-root scaling inside.)

A log-log fit of the normalized data gives slope ≈ 0.224 (very close to the theoretical 0.25), confirming the small exponent on the retention term:

(βNcp)^[0.224/4} ≈ (βNcp){^0.056} ≈ (βNcp)^1/16

(since 1/16 = 0.0625 ).

cristos-vournas.com

5. Combine to Obtain the Full EquationSubstitute the empirical ratio into the corrected effective temperature:

Tmean = Te,correct⋅(βNcp)^1/16 =

= [Φ(1−a)S /4σ]^(1/4)⋅(βNcp)⁴

Algebraically simplify the exponents:

Tmean = [Φ(1−a)S (βNcp)^¼ /4σ]^¼

This is exactly the target equation.

The

(βNcp)^¼

term now sits inside the brackets, effectively scaling up the absorbed flux term before the outer fourth root.

6. Equivalent Energy-Balance Form (Showing the Modification)Starting from total power balance and rearranging gives the modified form used in the derivation:

Φπr² S(1−a) = 4πr² σTmean⁴ /(βNcp)^¼

Per-unit-area version (dividing by4πr²):

Φ(1−a)S /4 = σTmean⁴ /(βNcp)^¼

The denominator on the right means that, for a given mean temperature, the effective outgoing flux is reduced when

Ncp

is large → to restore balance,

Tmean

must rise. This encodes the reduced immediate IR loss.

andthentheresphysics.wordpress.com

7. The Constant

β = 150

is empirically chosen (units ensure dimensional consistency) so the equation matches NASA satellite

Tsat

across the solar system. It is described as “the Rotating Planet Surface Solar Irradiation Absorbing-Emitting Universal Law constant.”

In the rare case where

Tsat = Te ,

would be

1 /(Ncp);

the fitted value 150 works universally.

cristos-vournas.com

Result and Claims

For Earth (no atmosphere):
Tmean ≈ 287.74 K (matches 288 K observed).
The classical +33 °C “greenhouse” difference disappears; rotation + oceans ( N ≈ 1 , cp ≈ 1 ) fully explain it via the
(βNcp)^1/16 ≈ 1,37 factor.
The 1/16 scaling on
Ncp
is the core prediction: planets/moons with higher product are warmer (everything else equal).

This is Vournas’ proposed universal law, grounded in energy balance + the qualitative physics of immediate vs. stored heat, tuned to fit real planetary data. It is not derived from detailed heat-diffusion PDEs but presented as the simplest functional form that both matches observations and incorporates the rotational-warming intuition.

cristos-vournas.com

16 web pages

The very POWERFUL the Solar Irradiated planet surface Rotational Warming Phenomenon: ( N*cp )1/16

The very POWERFUL the Solar Irradiated planet surface Rotational Warming Phenomenon: ( N*cp )^1/16