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Measuring the spin of Accreting Neutron Stars is important because it can provide constraints on the Equation of State of ultra-dense matter. Particularly crucial to our physical understanding is the discovery of sub-millisecond pulsars, because this will immediately rule out many proposed models for the ground state of dense matter. So far, it has been impossible to accomplish this because, for still unknown reasons, only a small amount of Accreting Neutron Stars exhibit coherent pulsations. An intriguing explanation for the lack of pulsations is that they form only on neutron stars accreting with a very low average mass accretion rate. I have searched pulsations in the faintest persistent X-ray source known to date and I found no evidence for pulsations. The implications for accretion theory are very stringent, clearly showing that our understanding of the pulse formation process is not complete. I discuss which sources are optimal to continue the search of sub-ms pulsars and which are the new constraints that theoretical models need to explain to provide a complete description of these systems
Accreting Millisecond X-Ray Pulsars (AMXPs) are astrophysical laboratories without parallel in the study of extreme physics. In this chapter we review the past fifteen years of discoveries in the field. We summarize the observations of the fifteen kn
The maximum mass of a neutron star has important implications across multiple research fields, including astrophysics, nuclear physics and gravitational wave astronomy. Compact binary millisecond pulsars (with orbital periods shorter than about a day
An understanding of spin frequency ($ u$) evolution of neutron stars in the low-mass X-ray binary (LMXB) phase is essential to explain the observed $ u$-distribution of millisecond pulsars (MSPs), and to probe the stellar and binary physics, includin
In millisecond pulsars the existence of the Coriolis force allows the development of the so-called Rossby oscillations (r-modes) which are know to be unstable to emission of gravitational waves. These instabilities are mainly damped by the viscosity
Accreting millisecond X-ray pulsars are known to provide a wealth of physical information during their successive states of outburst and quiescence. Based on the observed spin-up and spin-down rates of these objects it is possible, among other things