Gravitational instability of solar prominence threads: curved magnetic fields with dips

Recent studies have examined the gravitational instability of prominence threads---dense plasma embedded along coronal magnetic field lines---in curved magnetic topologies without magnetic dips. Because most prominence models, guided by observations, assume that dense plasma is supported in magnetic dips, we extend those results by including a dip in the equilibrium magnetic configuration. We derive analytical expressions for the admissible equilibrium (fixed) points and the corresponding oscillation frequencies, and we survey a wide parameter space that includes the flux-tube and dip lengths, the thread density and length, and the thread initial position. The system can admit up to five fixed points, which may be stable or unstable. For dips longer than the thread (i.e. longer than the dense segment), a stable equilibrium appears near the dip centre and two additional stable/unstable pairs arise through bifurcations. The resulting bifurcation diagrams depend qualitatively on the initial thread position, which acts as an imperfection parameter. When the thread starts at the dip centre, the frequency of the central equilibrium separates naturally into a pressure-driven slow-mode contribution plus the pendulum-model frequency. For dips shorter than the thread, the central equilibrium can become ''fitfully'' stable or unstable, switching at a sequence of critical points. These fitful points characterise the transition between threads in curved fields without dips and dipped configurations. Together with the no-dip case, our bifurcation-theory framework provides new criteria for when prominence plasma can remain in equilibrium and when it is expected to drain or drive flows.