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We fit an isothermal oscillatory density model to two ALMA/DSHARP-observeddisks, AS 209 and HL Tau, in which planets have presumably already formed andthey are orbiting within the observed seven dark gaps in each system. Theselarge disks are roughly similar to our solar nebula, albeit they exhibit milderradial density profiles and they enjoy lower centrifugal support. We findpower-law density profiles with index $k=0.0$ (radial densities $\rho(R)\propto R^{-1}$) and centrifugal support against self-gravity so small that itguarantees dynamical stability for millions of years of evolution. The scalelengths of the models differ only by a factor of 1.9, but the inner cores ofthe disks are very different: HL Tau's core is 8.0 times larger and 3.6 timesdenser than the core of AS 209. This results in four dark gaps having formedwithin the core of HL Tau, whereas no dark gap is found in the core of AS 209.On the other hand, the Jeans frequencies and the angular velocities of thecores are comparable to within factors of 1.9 and 1.6, respectively.

Theoretical models of the protostellar disks of AS 209 and HL Tau presently forming in-situ planets