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We show that magnetized, radiation dominated atmospheres can support steadystate patterns of density inhomogeneity that enable them to radiate at farabove the Eddington limit, without suffering mass loss. The inhomogeneitiesconsist of periodic shock fronts bounding narrow, high-density regions,interspersed with much broader regions of low density. The radiation fluxavoids the regions of high density, which are therefore weighed down bygravity, while gas in the low-density regions is slammed upward into the shockfronts by radiation force. As the wave pattern moves through the atmosphere,each parcel of matter alternately experiences upward and downward forces, whichbalance on average. Magnetic tension shares the competing forces betweenregions of different densities, preventing the atmosphere from blowing apart.We calculate the density structure and phase speed of the wave pattern, andrelate these to the wavelength, the density contrast, and the factor by whichthe net radiation flux exceeds the Eddington limit. In principle, this factorcan be as large as the ratio of magnetic pressure to mean gas pressure, or theratio of radiation pressure to gas pressure, whichever is smaller. Although themagnetic pressure must be large compared to the mean gas pressure in order tosupport a large density contrast, it need not be large compared to theradiation pressure. These highly inhomogeneous flows could represent thenonlinear development of the "photon bubble" instability discovered by Gammie.We briefly discuss the applicability of these solutions to astrophysicalsystems.Comment: 11 pages, 1 figure, accepted for publication in The Astrophysical  Journa

Super‐Eddington Atmospheres That Do Not Blow Away