  Prominence threads are dense and cold structures lying on curved magnetic
fields that can be suspended in the solar atmosphere against gravity. The
gravitational stability of threads, in the absence of non-ideal effects, is
comprehensively investigated in the present work by means of an elementary but
effective model. Based on purely hydrodynamic equations in one spatial
dimension and applying line-tying conditions at the footpoints of the magnetic
field lines, we derive analytical expressions for the different feasible
equilibria and the corresponding frequencies of oscillation. We find that the
system allows for stable and unstable equilibrium solutions subject to the
initial position of the thread, its density contrast and length, and the total
length of the magnetic field lines. The transition between the two types of
solutions is produced at specific bifurcation points that have been determined
analytically in some particular cases. When the thread is initially at the top
of the concave magnetic field, that is at the apex, we find a supercritical
pitchfork bifurcation, while for a shifted initial thread position with respect
to this point the symmetry is broken and the system is characterised by an
S-shaped bifurcation. The plain results presented in this paper shed new light
on the behaviour of threads in curved magnetic fields under the presence of
gravity and help to interpret more complex numerical magnetohydrodynamics (MHD)
simulations about similar structures.
