The equilibrium of over-pressurised polytropes

We investigate the impact of an external pressure on the structure of self-gravitating polytropes for axially symmetric ellipsoids and rings. The confinement of the fluid by photons is accounted for through a boundary condition on the enthalpy $H$. Equilibrium configurations are determined numerically from a generalised "Self-Consistent-Field"-method. The new algorithm incorporates an intra-loop re-scaling operator ${\cal R}(H)$, which is essential for both convergence and getting self-normalised solutions. The main control parameter is the external-to-core enthalpy ratio. In the case of uniform rotation rate and uniform surrounding pressure, we compute the mass, the volume, the rotation rate and the maximum enthalpy. This is repeated for a few polytropic indices $n$. For a given axis ratio, over-pressurization globally increases all output quantities, and this is more pronounced for large $n$. Density profiles are flatter than in the absence of an external pressure. When the control parameter asymptotically tends to unity, the fluid converges toward the incompressible solution, whatever the index, but becomes geometrically singular. Equilibrium sequences, obtained by varying the axis ratio, are built. States of critical rotation are greatly exceeded or even disappear. The same trends are observed with differential rotation. Finally, the typical response to a photon point source is presented. Strong irradiation favours sharp edges. Applications concern star forming regions and matter orbiting young stars and black holes.