  Longitudinal oscillations in prominences are common phenomena on the Sun.
These oscillations can be used to infer the geometry and intensity of the
filament magnetic field. Previous theoretical studies of longitudinal
oscillations made two simplifying assumptions: uniform gravity and
semi-circular dips on the supporting flux tubes. However, the gravity is not
uniform and realistic dips are not semi-circular. To understand the effects of
including the nonuniform solar gravity on longitudinal oscillations, and
explore the validity of the pendulum model with different flux-tube geometries.
We first derive the equation describing the motion of the plasma along the flux
tube including the effects of nonuniform gravity, yielding corrections to the
original pendulum model. We also compute the full numerical solutions for the
normal modes, and compare them with the new pendulum approximation. We have
found that the nonuniform gravity introduces a significant modification in the
pendulum model. We have also found a cut-off period, i.e. the longitudinal
oscillations cannot have a period longer than 167 minutes. In addition,
considering different tube geometries, the period depends almost exclusively on
the radius of curvature at the bottom of the dip. We conclude that nonuniform
gravity significantly modifies the pendulum model. These corrections are
important for prominence seismology, because the inferred values of the radius
of curvature and minimum magnetic-field strength differ substantially from
those of the old model. However, we find that the corrected pendulum model is
quite robust and is still valid for non-circular dips.
