Title: The equilibrium of over-pressurised polytropes,
Abstract: 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.