Critical Density

The critical density ρ_crit is the density at which the coherence function transitions from quantum (C → 0) to classical (C → 1). It is per-system, not a universal constant: V_flat is the observed flat rotation velocity of each specific system, so each galaxy (or other gravitationally bound object) has its own ρ_crit. A is the universal proportionality constant; V_flat enters as an input. The rotation-curve fit is not zero-parameter — it requires an observed V_flat per galaxy.

The A Parameter

A is derived from the Jeans criterion (Session 53): α = λ_Jeans / R_halfis the dimensionless Jeans-length-to-galaxy-size ratio. Empirically, α ≈ 1.1 ± 0.2 across SPARC galaxies — the Jeans length approximately equals the galaxy half-light radius at the coherence boundary. With α = 1.0 (fiducial), G in galactic units, and R₀ = 8 kpc, the formula yields A ≈ 0.029, vs empirical 0.028 (5% agreement).

Note on the α symbol: α here is not the electromagnetic fine-structure constant (α_em ≈ 1/137). Earlier versions of this page made that error. The formula closes numerically only with α = O(1); α_em² ≈ 5×10⁻⁵ would make A ≈ 550 (km/s)⁻² — about 20,000× too large. See parameter derivations for the full Jeans-criterion derivation chain.

Physical Meaning

Every gravitationally bound system has a characteristic rotation velocity V_flat — the velocity at which its rotation curve flattens. This velocity encodes the total mass and size of the system. The critical density is where internal gravitational binding energy equals the coherence threshold.

Below ρ_crit: the system is under-dense, loosely bound, quantum effects persist. Above ρ_crit: the system is gravitationally coherent, classical behavior dominates.

Connection to MOND

From ρ_crit = A × V_flat² and the coherence function, two cosmological results emerge: