Newtonian and 1PN Orbital Dynamics from a Superfluid Defect Toy Model

We construct and analyze a superfluid defect toy model that reproduces both Newtonian (0PN) gravity and the leading post-Newtonian (1PN) perihelion precession of a test body in a central field. The model consists of a homogeneous superfluid background and sink-like defects whose effective gravitational potential splits into two scalar pieces: an instantaneous Poisson sector ΦP and a finite-speed “lag” sector ΦL governed by a wave equation with propagation speed cs. For a static central defect in the test-mass limit we show that the retarded scalar solution collapses exactly to the Poisson solution, so that ΦL vanishes and the near-zone potential is strictly Newtonian, Φtot(r) = −μ/r with no 1/cs2 corrections. As a result the scalar sector generates no 1PN perihelion precession: the entire 1PN correction is encoded kinematically in a position-dependent effective mass meff(r) = m1 + σ(r) with σ(r) = β μ/(cs2 r). Matching the Schwarzschild 1PN precession for cs = c and μ = GM requires β = 3. We interpret this coefficient as a sum of three hydrodynamic contributions, β = κρ + κadd + κPV, coming respectively from density depletion in the cavitation region, classical added mass of entrained fluid, and internal pressure–volume inertia of a compressible throat. The first two pieces are derived quantitatively from the toy model and the associated Mathematica calculations, yielding κρ = 1 and κadd = ½. The remaining piece is fixed by the 1PN matching condition to be κPV = 3/2, which we view as an effective field theory constraint on the bulk equation of state and throat compressibility. This provides a concrete hydrodynamic target for future microphysical derivations.