  It is well established that the central deficit of dark matter (DM) observed
in many dwarf galaxies disagrees with the cuspy DM haloes predicted in the
collision-less and cold DM (CDM) model. Plausible solutions to this problem are
based on an effective energy deposition into the central halo with an origin
that is either based on baryonic physics (e.g. supernova-driven gas blowouts;
SNF) or on new DM physics (e.g. self-interacting DM, SIDM). We argue that the
fundamental difference between the two is whether the process is impulsive or
adiabatic, and explore novel ways to distinguishing them by looking at the
response of stellar orbits. We perform idealised simulations of tracers
embedded in a $1.48\times 10^{10}{\rm M}_{\odot}$ spherical halo, and model the
creation of a $\sim1$kpc DM matter core in SIDM with $\sigma_T/m_{\chi} =
2\,{\rm cm}^2{\rm g}^{-1}$ and through SNF by impulsively removing a mass
equivalent to $\mathcal{O}(10\%)$ of the halo's potential energy. Choosing
idealised initial orbital configurations for the tracers, we find that radial
actions are conserved (changed) in the SIDM (impulsive) case. The adiabaticity
of the SIDM case prevents tracers from changing their orbital family during
core formation whereas SNF separates tracers of initially the same family to a
variety of orbits. We show that these key features remain in a cosmological
halo, albeit for a few dynamical timescales. The number density and velocity
dispersion profile of a Plummer sphere with $r_{1/2} = 500$ pc change only
marginally under adiabatic core formation, whereas SNF causes a substantial
expansion of the sphere driving it out of Jeans equilibrium. Our results point
towards promising ways of differentiating adiabatic from impulsive core
formation.
