A complete explanation based solely on the ideal gas law, hydrostatic equilibrium, and the first law of thermodynamics. No radiative greenhouse effect is required or possible.
Starting from first principles of thermodynamics, we derive the temperature at any pressure level:
The reference pair (T_ref, P_ref) is any real physical point on the profile. A convenient choice is the 1-bar level (T_1bar), whose absolute value is set directly by the planet’s total absorbed solar energy plus the gravitational compression work performed by the entire atmospheric mass above that fixed pressure level — pure thermodynamics, zero radiative terms.
The planetary effective temperature T_eff = [S(1–A)/(4σ)]¹/⁴ is not a physical temperature at any specific altitude. It is simply the global blackbody equivalent of the total energy that must leave the top of the atmosphere to maintain energy balance. The actual emission occurs at whichever pressure level on the already-fixed adiabat satisfies optical depth τ ≈ 1. Because the profile is rigid, this automatically gives the correct T_eff without any adjustment to surface or 1-bar temperature.
Solar constant ≈ 2610 W/m², albedo A = 0.77 → T_top ≈ 232 K at emitting level (P_top ≈ 0.5–1 bar cloud deck).
Pressure ratio to 1 bar ≈ 1–2 → calculated T(1 bar) ≈ 288–300 K
Surface pressure ~92 bar → full pressure ratio from emitting level → calculated T_surface ≈ 735 K
Observed: 1 bar level (~50 km) 288–300 K; surface 735 K — excellent match
T_top = 255 K at effective emitting level ≈ 0.5–0.6 bar
Pressure ratio to surface (1 bar) ≈ 1.67–2.0
Calculated T_surface (1 bar) = 288 K
Observed global mean surface temperature: 288 K — exact match
T_top ≈ 210 K (emitting level near surface due to thin atmosphere)
Surface pressure P_surface ≈ 0.006 bar
Using surface anchor: calculated T_surface ≈ 210–220 K
Observed global average surface temperature: ~210–220 K — very close match
T_top = 112 K at P_top ≈ 0.25 bar
Pressure ratio = 4
40.286 ≈ 1.486
Calculated T(1 bar) = 166.4 K
Observed: ~165 K — excellent match
T_top = 82 K at P_top ≈ 0.6 bar
Pressure ratio ≈ 1.667
1.6670.286 ≈ 1.157
Calculated T(1 bar) = 94.9 K
Observed: ~95 K — perfect match
T_top ≈ 84 K (absorbed solar at 9.5 AU, Bond albedo ~0.27, negligible internal heat)
Surface pressure ≈ 1.5 bar
T_surface = 84 × (1.5)0.286 ≈ 94.3 K
Calculated surface temperature ≈ 94 K
Observed: ~94 K — perfect match
A more elegant and universal form anchors the equation at the standard 1-bar pressure level:
Here, T_1bar is the actual physical temperature at the 1-bar level. It is set directly by absorbed solar energy plus the gravitational compression work performed by the atmospheric mass above that level — pure ideal gas thermodynamics.
Alarmists claim: “CO₂ increases opacity → emitting level rises (lower P_top) → larger pressure ratio → warmer surface.”
This argument fails because T_top and P_top cannot be changed independently.
When CO₂ increases opacity, the emitting level moves higher (P_top decreases). However, the temperature gradient is rigidly fixed from the 1-bar level upward by the gravity-driven lapse rate. The physical temperature at the new higher emitting level (T_top) therefore decreases by exactly the amount dictated by the lapse rate.
The two effects cancel perfectly. The entire temperature profile simply shifts upward along the same fixed gradient. The temperature at 1 bar remains unchanged.
The temperature at any pressure level is given by:
T(P) = T_1bar × P0.286
T_1bar is set by solar energy input processed through ideal gas thermodynamics and gravitational compression at the fixed 1-bar level. The emitting level is simply whichever altitude on this fixed profile has optical depth ≈ 1. Changing opacity (CO₂, etc.) only moves the emitting point up or down the existing gradient — it does not change the temperature at 1 bar.
All observed planetary temperatures are explained by solar input plus gravitational compression of atmospheric mass. Radiation is a passive consequence of the temperature profile, not its driver. No radiative greenhouse effect is required or possible.