  Hartle's model describes the equilibrium configuration of a rotating isolated
compact body in perturbation theory up to second order in General Relativity.
The interior of the body is a perfect fluid with a barotropic equation of
state, no convective motions and rigid rotation. That interior is matched
across its surface to an asymptotically flat vacuum exterior. Perturbations are
taken to second order around a static and spherically symmetric background
configuration. Apart from the explicit assumptions, the perturbed configuration
is constructed upon some implicit premises, in particular the continuity of the
functions describing the perturbation in terms of some background radial
coordinate. In this work we revisit the model within a modern general and
consistent theory of perturbative matchings to second order, which is
independent of the coordinates and gauges used to describe the two regions to
be joined. We explore the matching conditions up to second order in full. The
main particular result we present is that the radial function $m_0$ (in the
setting of the original work) of the second order perturbation tensor, contrary
to the original assumption, presents a jump at the surface of the star, which
is proportional to the value of the energy density of the background
configuration there. As a consequence, the change in mass needed by the
perturbed configuration to keep the value of the central energy density
unchanged must be amended. We also discuss some subtleties that arise when
studying the deformation of the star.
