A minimal derivation of the Roche tidal disruption limit from a gravitational fixed-point condition

We present a three-step derivation of the Roche tidal disruption limit 
from a single fixed-point condition: a satellite is disrupted when the 
tidal acceleration gradient of the host planet equals the self-gravitational 
acceleration gradient at the satellite's surface. The derivation uses only 
Newton's law of gravitation and yields the result

    d_R = R_M × (2 ρ_M / ρ_m)^(1/3)

with no free parameters, where R_M is the planet radius, ρ_M the planet 
mean density, and ρ_m the satellite density. Numerical verification against 
Saturn's ring system (5.0% agreement) and the 1992 tidal disruption of 
Comet Shoemaker-Levy 9 at Jupiter confirms the formula. The fixed-point 
framing makes transparent why the threshold depends only on the density 
ratio and not on the absolute sizes or masses of the bodies.