Limits on the amplitude of r-mode oscillations of neutron stars through X-ray observations

R-modes are the oscillations observed especially in stars rotating at high speeds, where the dominant force is the Coriolis force. Since the dominant force in these modes is the Coriolis force, the frequency of the mode is independent of the internal structure of the star but is a function of its angular velocity. Gravitational radiation drives these modes towards the zone of instability in all rotating stars, while the viscous forces inside the star play a destructive role in this oscillation. If the star is spinning slowly, the time scale for gravitational radiation is always much larger and the star is stable. In fast rotating stars, the instability of the star against these oscillations is defined as a function of temperature, since the viscous forces depend on the temperature. The X-ray spectroscopy is ideal for limiting the amplitude of these oscillations because the damping of these oscillations causes an extra temperature increase in the star. Then the r-mode instability stops and the star slowly starts to cool again. This temperature rise in the star can be examined by X-ray spectroscopy. This instability is expected to occur in fast-spinning systems, for example, newborn neutron stars, or members of low-mass X-ray binaries (LMXBs), or millisecond pulsars. r-mode instability can affect any system. According to the observations of the LMXBs, many sources are located in the theoretically expected r-mode instability window. Since the heat transport and cooling characteristics in neutron stars or LMXBs are dependent on their internal structure, they provide us with important clues about the internal structure of stars. For instance; rotation frequency - the boundary of the r-mode zone of instability in the temperature plane, and especially the minimum that determines the final rotation frequency of the star, is very different for stars with different internal compositions. In this context, potential X-ray millisecond pulsars or LMXBs with high rotation frequency from the literature will be selected and studied within the scope of the thesis. By analyzing the X-ray data taken from the archive, it was attempted to set boundaries on the surface temperatures of these sources. As a result of the analyzes to be made, it is aimed to examine the r-mode oscillations and internal structure of the studied sources.