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Observations and theory suggest that line driven winds from hot stars andluminous accretion disks adopt a unique, critical solution which corresponds tomaximum mass loss rate. We analyze the numerical stability of the infinitefamily of shallow wind solutions, which resemble solar wind breezes, and theirtransition to the critical wind. Shallow solutions are sub-critical withrespect to radiative (or Abbott) waves. These waves can propagate upstreamthrough shallow winds at high speeds. If the waves are not accounted for in theCourant time step, numerical runaway results. The outer boundary condition isequally important for wind stability. Assuming pure outflow conditions, as isdone in the literature, triggers runaway of shallow winds to the criticalsolution or to accretion flow.Comment: 10 pages, 1 figure. Accepted for publication in Astrophysical Journa

Runaway Acceleration of Line‐driven Winds: The Role of the Outer Boundary