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Close-orbit low-mass X-ray binaries (LMXBs), radio binary millisecond pulsars(BMSPs) with extremely low-mass helium WDs (ELM He~WDs) and ultra-compact X-raybinaries (UCXBs) are all part of the same evolutionary sequence. It istherefore of uttermost importance to understand how these populations evolvefrom one specie to another. Moreover, UCXBs are important gravitational wave(GW) sources and can be detected by future space-borne GW observatories.However, the formation and evolutionary link between these three differentpopulations of neutron star (NS) binaries are not fully understood. Inparticular, a peculiar fine-tuning problem has previously been demonstrated forthe formation of these systems. In this investigation, we test a newlysuggested magnetic braking prescription and model the formation and evolutionof LMXBs. We compute a grid of binary evolution models and present the initialparameter space of the progenitor binaries which successfully evolve all theway to produce UCXBs. We find that the initial orbital period range of LMXBs,which evolve into detached NS+ELM~He~WD binaries and later UCXBs, becomessignificantly wider compared to evolution with a standard magnetic brakingprescription, and thus helps to relieve the fine-tuning problem. However, wealso find that formation of wide-orbit BMSPs is prohibited for strong versionsof this new magnetic braking prescription, which therefore calls for a revisionof the prescription. Finally, we present examples of the properties of UCXBs asGalactic GW sources and discuss their detection by the LISA, TianQin and Taijiobservatories.

Formation of millisecond pulsars with helium white dwarfs, ultra-compact X-ray binaries and gravitational wave sources