  We consider the periapsis shifts in dark matter distribution with a dense
core. We model the dark matter distribution as an isotropic gas sphere, the
Emden polytropic sphere of index 5 in general relativity. This model has a
parameter range where all the energy conditions are satisfied in the entire
region. Within the parameter range, the asymptotic analysis for stellar motion
allows us to identify two competing effects on the bounded motion: the
general-relativistic effect and a local-density effect of matter. Furthermore,
using nearly circular bound orbits, we demonstrate that retrograde periapsis
shifts occur near the center, where the local-density effect dominates over the
general-relativistic effect, whereas prograde periapsis shifts occur in the far
region, where the general-relativistic effect dominates over the local-density
effect. This result means that a natural explanation for the retrograde
periapsis shifts is not the existence of exotic objects (e.g., naked
singularities or wormholes) but the local distribution of physically reasonable
matter on the stellar orbit. Furthermore, it also implies that the periapsis
shift plays a crucial role in distinguishing black hole alternatives, such as
dark matter cores, from a pure black hole.
