The vortex shadow term

RSI adds a second term to the equation for orbital speed. It is drawn from the kinetic part of the RSI Lagrangian, the master formula that describes the field's energy.

Contributions to orbital speed add as squares. The square of a star's speed, v(r)², is the sum of two pieces. The first is the familiar Newtonian term, GM / r, set by the shadow of the visible mass M inside the orbit. The second, v_Φ², comes from the extra depth of the galaxy's Vortex Shadow, the low pressure at the heart of its swirl, with v_Φ = β(∇ × Φ). The curl, ∇ × Φ, measures how strongly the field swirls.

Near the center, the Newtonian term dominates, and the gravity of the central bulge sets the motion. Far out, the Newtonian term fades toward zero, but the vortex term holds steady. RSI proposes that the vortex's pressure deficit keeps v_Φ steady. This creates a constant velocity floor, which matches the flat rotation curves astronomers observe. One piece is still missing. A swirl that simply conserves its angular momentum slows as 1/r, which would not give a flat curve. A flat curve needs the medium itself to turn at nearly constant speed, which in a superfluid means the quantized vortex lines thin out as 1/r with distance.

RSI proposes how this could happen. In an ordinary frictionless fluid, a moving body disturbs the medium only while it passes; the disturbance travels with the body and leaves no lasting spin behind. A superfluid has one exception, seen in the laboratory: matter moving faster than a critical speed creates quantized vortex lines, and those lines can pin to matter, as they do on solid surfaces in laboratory helium and, by the leading account of pulsar glitches, in the crust of a neutron star. RSI proposes that as a galaxy formed, its rotating matter seeded and pinned vortex lines until the medium turned along with it. Feynman showed that in a superfluid the density of vortex lines tracks the local spin of the flow. For a medium turning at constant speed, the spin falls as 1/r, so the line density falls as 1/r too, which is exactly the shape a flat curve needs. RSI must still show that the stars and the medium settle into this state together, and that today's planets and stars move too slowly relative to the surrounding medium to keep making vortex lines, since that would show up as a drag their orbits do not have.

The coefficient β still has to be derived from the medium's tension and density, so that rotation curves can be predicted for real galaxies without adjusting anything galaxy by galaxy.