Gravity vs Pressure: The g = GM/R² Law and Why a Pressure-Proportional Weight Model Fails Everywhere Off Earth

Scope

This document establishes Newtonian surface gravity, g = GM/R², as the governing law of weight on a planetary body, and decomposes the competing hypothesis that surface gravity is set by atmospheric pressure (g = k·P). It tests the pressure model by calibrating it on Earth and predicting g on six other Solar-System bodies. It also records the "D-value" (Dobson Unit) only as a geometric column index, with no role in setting gravity. This is a cross-domain falsification guard, carried so that no "atmosphere = life-or-physics substrate" narrative can smuggle in a pressure-replaces-gravity claim. It is not ozone chemistry.

Equations (preserved exactly)

$$g = \frac{GM}{R^2}, \qquad v_{esc} = \sqrt{\frac{2GM}{R}}$$

$$g = \frac{GM}{R^2} \quad\text{(holds, <=0.36%)} \qquad vs \qquad g \stackrel{?}{=} k,P,; k=\frac{g_\oplus}{P_\oplus} \quad\text{(FAILS)}$$

Variables and units

Dimensional check

GM/R² = [m³ kg⁻¹ s⁻²][kg]/[m²] = [N kg⁻¹] = m s⁻². ✓ v_esc = √([m³ kg⁻¹ s⁻²][kg]/[m]) = √[m² s⁻²] = m s⁻¹. ✓ k·P = [m s⁻² Pa⁻¹][Pa] = m s⁻² — dimensionally valid, but empirically false off-Earth (the failure is in the ratios, not the units). [Class A]

Worked example (with numbers)

Titan, by g = GM/R²: μ = GM = 8978.13710 km³ s⁻², R = 2574.76 km. g = 8978.13710 / 2574.76² = 8978.137 / 6.6294×10⁶ = 1.3543×10⁻³ km s⁻² = 1.354 m s⁻². v_esc = √(2·8978.137/2574.76) = √6.9737 = 2.6408 km s⁻¹. Both match the reference. [Class A/B]

The pressure model g = k·P, calibrated on Earth (k = 9.82 / 1.014×10⁵):

Near-vacuum bodies (Moon, Mercury) keep full gravity; Venus has 92× Earth's pressure yet lower g than Earth; Titan has a thicker-than-Earth atmosphere yet the lowest g of the set. Gravity is set by mass and radius, not by surface pressure. [Class A — reproduced this session]

Evidence class per claim

• g = GM/R² reproduces all seven verified bodies (Earth, Moon, Mars, Venus, Mercury, Jupiter, Titan) to ≤0.36% (binding residual = Moon, 1.6258 vs 1.62 reference): Class A, derivable + reproduced. (Note: the tighter ≤0.26% sometimes quoted holds only at full-precision NSSDCA masses; from rounded factsheet values the bound is ≤0.36%.)
• g = k·P calibrated on Earth fails by up to ~14 orders of magnitude (Moon) and overshoots Venus ~100×: contradicted-by-test (Class X).
• The D-value (Dobson Unit) is a column number-density index (2.69×10¹⁶ molecules cm⁻² per DU), with zero role in setting gravity: Class B (geometric definition only).

Falsification conditions

The Newtonian law would be contradicted-by-test if any body's measured g departed from GM/R² beyond rotational/oblateness corrections (sub-percent to few-percent). The pressure model g = k·P is already falsified: it requires g→0 as P→0, but the Moon and Mercury hold g = 1.62 and 3.70 m s⁻² in near-vacuum; and it requires g to rise monotonically with P, contradicted by Venus (high P, modest g) and Titan (thick atmosphere, lowest g). One such body suffices; four independently confirm.

Outside this model

This document concerns Newtonian surface gravity only. It says nothing about ozone, life, biosignatures, or water synthesis — those are decomposed in their own domains. It does not assert "atmospheric pressure replaces gravity" (that is Class X, contradicted-by-test), nor that a thick atmosphere strengthens gravity to retain water. g = GM/R² is itself a weak-field Newtonian approximation, exact enough across these bodies; it carries no claim about general relativity, planetary interiors, or the origin of mass. No result here proves any broad framework, and "a simulation proves reality" remains outside-model-scope.