The Central Concentration of Trojan Dust: Action-Space Transport, Poynting-Robertson Drag, and the Emergence of Density Structure

Theorem 6 in Doucette (2025) establishes a striking result for resonant dust trapped near Trojan libration points: weak Poynting-Robertson drag does not merely dissipate the cloud, but reorganizes it. Under assumptions of integrability, valid action-angle coordinates, negative secular drift in the action variable, no-flux boundary conditions, and monotone radial occupancy, the angle-averaged dust distribution obeys a one-dimensional transport equation in action space whose characteristics drift toward smaller actions. Because smaller actions correspond to more tightly bound libration orbits that spend more time near the libration center, the cumulative mass inside sufficiently small radii increases monotonically on secular timescales, while total mass is conserved. The result is a rigorous mechanism for central concentration and inward-steepening density gradients. This theorem is important mathematically because it shows how adiabatic invariants remain meaningful under weak non-Hamiltonian perturbations and how the full continuity equation can be reduced to a scalar transport law in action space. It is important physically because it predicts that Trojan dust clouds and related resonant dust populations should evolve toward centrally enhanced cores rather than remain diffuse. This article explains the statement and proof logic of Theorem 6, develops its mathematical and astrophysical significance, identifies empirical signatures that would support it, explores its uses in reduced-order modeling and observational interpretation, and outlines a wide agenda for future research extending from collisions and three-dimensional structure to inverse problems, optimal transport, and resonance migration. The theorem may be read as a general principle of dissipative organization in near-integrable celestial-mechanical systems.