  Compact stars satisfy certain no-hair relations through which their multipole
moments are given by their mass, spin and quadrupole moment. These relations
are approximately independent of their equation of state, relating pressure to
density. Such relations are similar to the black hole no-hair theorems, but
these possess event horizons inside which information that led to their
formation can hide. Compact stars do not possess horizons, so whether their
no-hair relations are related to the black hole ones is unclear. We investigate
how the two relations are related by studying relations among multipole moments
for compact stars with anisotropic pressure as a toy model, which allows such
stars to be more compact than those with isotropic pressure. We here show
numerically that the compact star no-hair relations approach the black hole
ones as the compactness approaches that of a black hole. We also prove
analytically that the current dipole moment exactly reaches the black hole
limit quadratically in compactness as strongly-anisotropic stars approach the
black hole limit. We moreover show that compact stars become progressively
oblate in this limit, even if prolate at low compactness due to strong
anisotropies.
