  We present a simple model for the response of a dissipationless spherical
system to an instantaneous mass change at its center, describing the formation
of flat cores in dark matter haloes and ultra-diffuse galaxies (UDGs) from
feedback-driven outflow episodes in a specific mass range. This model
generalizes an earlier simplified analysis of an isolated shell into a system
with continuous density, velocity and potential profiles. The response is
divided into an instantaneous change of potential at constant velocities due to
a given mass loss or gain, followed by energy-conserving relaxation to a new
Jeans equilibrium. The halo profile is modeled by a two-parameter function with
a variable inner slope and an analytic potential profile (Dekel et al. 2017),
which enables determining the associated kinetic energy at equilibrium. The
model is tested against NIHAO cosmological zoom-in simulations, where it
successfully predicts the evolution of the inner dark-matter profile between
successive snapshots in about 75% of the cases, failing mainly in merger
situations. This model provides a simple understanding of the formation of
dark-matter halo cores and UDGs by supernova-driven outflows, and a useful
analytic tool for studying such processes.
