  We propose an improved analytical model for the horizon-absorbed
gravitational-wave energy flux of a small body in circular orbit in the
equatorial plane of a Kerr black hole. Post-Newtonian (PN) theory provides an
analytical description of the multipolar components of the absorption flux
through Taylor expansions in the orbital frequency. Building on previous work,
we construct a mode-by-mode factorization of the absorbed flux whose Taylor
expansion agrees with current PN results. This factorized form significantly
improves the agreement with numerical results obtained with a frequency-domain
Teukolsky code, which evolves through a sequence of circular orbits up to the
photon orbit. We perform the comparison between model and numerical data for
dimensionless Kerr spins $-0.99 \leq q \leq 0.99$ and for frequencies up to the
light ring of the Kerr black hole. Our proposed model enforces the presence of
a zero in the flux at an orbital frequency equal to the frequency of the
horizon, as predicted by perturbation theory. It also reproduces the expected
divergence of the flux close to the light ring. Neither of these features are
captured by the Taylor-expanded PN flux. Our proposed absorption flux can also
help improve models for the inspiral, merger, ringdown of small mass-ratio
binary systems.
