Theoretical model of HD 163296 presently forming in-situ planets and comparison with the models of AS 209, HL Tau, and TW Hya

We fit an isothermal oscillatory density model to the disk of HD 163296 in which planets have presumably already formed and they are orbiting at least within the four observed dark gaps. This 156 AU large axisymmetric disk shows various physical properties comparable to those of AS 209, HL Tau, and TW Hya that we have modeled previously; but it compares best to AS 209. The disks of HD 163296 and AS 209 are comparable in size and they share similar values of the power-law index $k\approx 0$ (a radial density profile of the form $\rho(R)\propto R^{-1}$), the rotational parameter $\beta_0$ (to within a factor of 3); a relatively small inner core radius (although this parameter for HD 163296 is exceptionally small, $R_1\simeq 0.15$ AU, presumably due to unresolved planets in the inner 50 AU); the scale length $R_0$ and the Jeans gravitational frequency $\Omega_J$ (to within factors of 1.4); the equation of state ($c_0^2/\rho_0$) and the central density $\rho_0$ (to within factors of 2); and the core angular velocity $\Omega_0$ (to within a factor of 4.5). In the end, we compare all six nebular disks that we have modeled so far.