  Gravitational waves (GWs) have enabled direct detections of compact binary
coalescences (CBCs). However, their poor sky localisation and the typical lack
of observable electromagnetic (EM) counterparts make it difficult to
confidently identify their hosts, and study the environments that nurture their
evolution. In this work, we show that $\textit{detailed}$ information of the
host environment (e.g. the mass and steepness of the host potential) can be
directly inferred by measuring the kinematic parameters (acceleration and its
time-derivatives) of the binary's center of mass using GWs alone, without
requiring an EM counterpart. We consider CBCs in various realistic environments
such as globular clusters, nuclear star clusters, and active galactic nuclei
disks to demonstrate how orbit and environment parameters can be extracted for
CBCs detectable by ground- and space-based observatories, including the LIGO
detector at A+ sensitivity, Einstein Telescope of the XG network, LISA, and
DECIGO, $\textit{on a single-event basis}$. These constraints on host stellar
environments promise to shed light on our understanding of how CBCs form,
evolve, and merge.
