Temporal Properties of Accreting Millisecond X-ray Pulsars via Maximum Likelihood Technique

Low Mass X-ray Binaries (LMXBs) are the binary systems consist of a compact object -a black hole or a neutron star (NS) and a low mass main sequence companion star. Pulsations, expected to be produced from the pole of the NS are not detected in the most of LMXB systems with NS components. Accreting Millisecond X-ray Pulsars (AMXPs) differ from LMXB population by showing pulsations in their X-ray light curve. This subset of LMXBs is very convenient lab to study the pulse properties, pulse profile evolution and the orbital parameters of LMXBs. Maximum likelihood (ML) technique is a timing method to obtain the most probably phase offset for a set of data without to get the pulse itself, by using a template pulse profile and the best pulse parameters; the pulse period or the frequency and the epoch. The time evolution of the most probably phase offset provides the information about orbital parameters and the correction on the input parameters and the first derivative (and maybe second) of pulse frequency. ML is more sensitive than the classical Fourier Techniques but the handicap of the method is that one must have the pulse profile for using as template. The template profile is converted to the probability density function to be used in ML methods. In the case of LMXBs, the pulse profile cannot be obtained because of the absence of pulsations though ML method cannot be performed. On the other hand, AMXPs allow us to use ML to study the temporal properties of this class. Differently than the other AMXPs, intermittent pulsations, switching from pulse-on to pulse-off stages or vice versa, are detected from three sources; SAX J1748.9-2021, HeTe J1900.1-2455 and Aql X-1. Pulse characteristic of these sources can be in three different scenarios; (i) The pulsations might be intermittent and the transition between pulse-on -- pulse-off stages happens instantly. (ii) The pulsations might be intermittent but the amplitude of the pulsation changes smoothly between detectable and undetectable levels. (iii) The pulsation might be continuous with changing amplitude. These scenarios need to be checked via more sensitive detectors or methods. In this research, we show the details of ML method to be used as timing tool on AMXPs, including intermittent ones. We obtained the pulse profile from a chosen sample set of AMXP population (SAX J1808.4-3658, IGR J1749.8-2921, XTE J1751 305 etc.) by using classical Fourier method. We get the pulse frequency evolution and the orbital parameters of the systems by using the obtained profiles as probability density function. We also followed the similar procedure for Intermittent AMXPs and searched for possible pulsations around the pulse-on stages published in the literature. Throughout our result, we discuss the scenarios given above.