Conservation of radial actions in time-dependent spherical potentials
  In slowly evolving spherical potentials, $\Phi(r,t)$, radial actions are
typically assumed to remain constant. Here, we construct dynamical invariants
that allow us to derive the evolution of radial actions in spherical central
potentials with an arbitrary time dependence. We show that to linear order,
radial actions oscillate around a constant value with an amplitude $\Delta J_r
\propto \dot{\Phi}/\Phi\,P(E,L)$. Using this result, we develop a diffusion
theory that describes the evolution of the radial action distribution of
ensembles of tracer particles orbiting in generic time-dependent spherical
potentials. Tests against restricted $N$-body simulations in a varying Kepler
potential indicate that our linear theory is accurate in regions of phase-space
in which the diffusion coefficient $\tilde{D}(J_r) < 0.01\,J_r^2$. For
illustration, we apply our theory to two astrophysical processes. We show that
the median mass accretion rate of a Milky Way (MW) dark matter (DM) halo leads
to slow global time-variation of the gravitational potential, in which the
evolution of radial actions is linear (i.e. either adiabatic or diffusive) for
$\sim 84$ per cent of the DM halo at redshift $z=0$. This fraction grows
considerably with lookback time, suggesting that diffusion may be relevant to
the modelling of several Gyr-old tidal streams in action-angle space. As a
second application, we show that dynamical tracers in a self-interacting DM
(SIDM) dwarf halo (with $\sigma/m_\chi = 1\,{\rm cm^2g^{-1}}$) have invariant
radial actions during the formation of a cored density profile.
