  In this paper we investigate the formation of narrow planetary rings such as
those found around Uranus and Saturn through the tidal disruption of a weak,
gravitationally bound satellite that migrates within its Roche limit. Using
$N$-body simulations, we study the behaviour of rubble piles placed on circular
orbits at different distances from a central planet. We consider both
homogeneous satellites and differentiated bodies containing a denser core. We
show that the Roche limit for a rubble pile is closer to the planet than for a
fluid body of the same mean density. The Roche limit for a differentiated body
is also closer to the planet than for a homogeneous satellite of the same mean
density. Within its Roche limit, a homogeneous satellite totally disrupts and
forms a narrow ring. The initial stages of the disruption are similar to the
evolution of a viscous fluid ellipsoid, which can be computed
semi-analytically. On the other hand, when a differentiated satellite is just
within the Roche limit only the mantle is disrupted. This process is similar to
Roche-lobe overflow in interacting binary stars and produces two narrow rings
on either side of a remnant satellite. We argue that the Uranian rings, and
possibly their shepherd satellites, could have been formed through the tidal
disruption of a number of protomoons that were formed inside the corotation
radius of Uranus and migrated slowly inwards as a result of tidal interaction
with the planet.
