What mass distribution M(r) is mathematically required for the orbital speed v to remain constant (v \approx constant)?
M(r) must continue to increase linearly with the radius r (M(r) \propto r).
The relationship between orbital velocity and enclosed mass is fundamentally linked. In a system governed by simple gravity, if the velocity $v$ is constant, the gravitational force must remain constant at that radius. This constant force requires that the total mass enclosed within the orbit, $M(r)$, must increase in direct proportion to the radius, $r$. If $M(r)$ were not increasing linearly as the radius expands, the velocity would necessarily begin to drop off according to Keplerian dynamics. Therefore, the observed flatness of the rotation curve implies that mass continues to accumulate outward in a linear fashion, which is the physical justification for inferring the presence of unseen mass components.
