  The self-force acting on a (scalar or electric) charge held in place outside
a massive body contains information about the body's composition, and can
therefore be used as a probe of internal structure. We explore this theme by
computing the (scalar or electromagnetic) self-force when the body is a
spherical ball of perfect fluid in hydrostatic equilibrium, under the
assumption that its rest-mass density and pressure are related by a polytropic
equation of state. The body is strongly self-gravitating, and all computations
are performed in exact general relativity. The dependence on internal structure
is best revealed by expanding the self-force in powers of 1/r, with r denoting
the radial position of the charge outside the body. To the leading order, the
self-force scales as 1/r^3 and depends only on the square of the charge and the
body's mass; the leading self-force is universal. The dependence on internal
structure is seen at the next order, 1/r^5, through a structure factor that
depends on the equation of state. We compute this structure factor for
relativistic polytropes, and show that for a fixed mass, it increases linearly
with the body's radius in the case of the scalar self-force, and quadratically
with the body's radius in the case of the electromagnetic self-force. In both
cases we find that for a fixed mass and radius, the self-force is smaller if
the body is more centrally dense, and larger if the mass density is more
uniformly distributed.
