  This paper investigates spheroidal galaxies comprising a self-interacting
dark matter halo (SIDM) plus de Vaucouleurs stellar distribution. These are
coupled only via their shared gravitational field, which is computed
consistently from the density profiles. Assuming conservation of mass, momentum
and angular momentum, perturbation analyses reveal the galaxy's response to
radial disturbance. The modes depend on fundamental dark matter properties, the
stellar mass, and the halo's mass and radius. The coupling of stars and dark
matter stabilises some haloes that would be unstable as one-fluid models.
However the centrally densest haloes are unstable, causing radial flows of SIDM
and stars (sometimes in opposite directions). Depending on the dark
microphysics, some highly diffuse haloes are also unstable. Unstable galaxies
might shed their outskirts or collapse. Observed elliptical galaxies appear to
exist in the safe domain. Halo pulsations are possible. The innermost node of
SIDM waves may occur within ten half-light radii. Induced stellar ripples may
also occur at detectable radii if higher overtones are excited. If any SIDM
exists, observational skotoseismology of galaxies could probe DM physics,
measure the sizes of specific systems, and perhaps help explain peculiar
objects (e.g. some shell galaxies, and the growth of red nuggets).
