Neutron stars are the most compact stars that can be directly observed, which makes them ideal laboratories to study physics at extreme densities. Neutron stars in low-mass X-ray binaries accrete hydrogen and helium from a lower-mass companion star through Roche lobe overflow. This matter undergoes thermonuclear burning in the neutron star envelope, creating carbon and heavier elements. The fusion process may proceed in an unstable manner, resulting in a thermonuclear runaway. Within one second the entire surface is burned, which is observable as a sharp rise in the emitted X-ray flux: a type I X-ray burst. Afterwards the neutron star surface cools down on a timescale of ten to one hundred seconds. During these bursts the surface of an accreting neutron star can be observed directly, which makes them instrumental for studying this type of stars. We have studied rare kinds of X-ray bursts. One such rare burst is the superburst, which lasts a thousand times longer than an ordinary burst. Superbursts are thought to result from the explosive burning of a thick carbon layer, which lies deeper inside the neutron star, close to a layer known as the crust. A prerequisite for the occurrence of a superburst is a high enough temperature, which is set by the temperature of the crust and the heat conductivity of the envelope. The latter is lowered by the presence of heavy elements that are produced during normal X-ray bursts. Using a large set of observations from the Wide Field Camera's onboard the BeppoSAX satellite, we find that, at high accretion rate, sources which do not exhibit normal bursts likely have a longer superburst recurrence time, than the observed superburst recurrence time of one burster. We analyze in detail the first superburst from a transient source, which went into outburst only 55 days before the superburst. Recent models of the neutron star crust predict that this is too small a time to heat the crust sufficiently for superburst ignition, indicating that the models need to be extended with a new heat source. Another rare phenomenon is the occurrence of bursts with recurrence times of less than 30 minutes. In a long set of observations of the source EXO 0748-676 we find for the first time triple bursts, where three bursts occur within 30 minutes. This time is too short to accrete new fuel for the next burst, which suggests that not all hydrogen and helium is burned during the first burst. Finally, using a hydrodynamic stellar evolution code we create a multi-zone numerical model of the neutron star envelope. For the first time we include mixing due to rotation and a rotationally induced magnetic field. We find that thermonuclear burning proceeds in a stable manner at a lower heat flux of the crust for models including mixing. This may explain the observed transition of stable to unstable burning at a lower mass accretion rate than models previously predicted
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