We calculate the evolution of close binary systems (CBSs) formed by a neutron star (behaving as a radio pulsar) and a normal donor star, evolving either to helium white dwarf (HeWD) or ultra short orbital period systems. We consider X-ray irradiation
feedback and evaporation due to radio pulsar irradiation. We show that irradiation feedback leads to cyclic mass transfer episodes, allowing CBSs to be observed in-between as binary radio pulsars under conditions in which standard, non-irradiated models predict the occurrence of a low mass X-ray binary. This behavior accounts for the existence of a family of eclipsing binary systems known as redbacks. We predict that redback companions should almost fill their Roche lobe, as observed in PSR J1723-2837. This state is also possible for systems evolving with larger orbital periods. Therefore, binary radio pulsars with companion star masses usually interpreted as larger than expected to produce HeWDs may also result in such {it quasi - Roche Lobe Overflow} states, rather than hosting a carbon-oxygen WD. We found that CBSs with initial orbital periods $mathrm{P_{i}<1}$ day evolve into redbacks. Some of them produce low mass HeWDs, and a subgroup with shorter $mathrm{P_{i}}$ become black widows (BWs). Thus, BWs descent from redbacks, although not all redbacks evolve into BWs. There is mounting observational evidence favoring that BW pulsars are very massive ($mathrm{gtrsim 2; M_{odot}}$). As they should be redback descendants, redback pulsars should also be very massive, since most of the mass is transferred before this stage.
It has been recently proposed that helium white dwarfs may provide promising conditions for the occurrence of the Bose-Einstein condensation. The argument supporting this expectation is that in some conditions attained in the core of these objects, t
he typical De Broglie wavelength associated with helium nuclei is of the order of the mean distance between neighboring nuclei. In these conditions the system should depart from classical behavior showing quantum effects. As helium nuclei are bosons, they are expected to condense. In order to explore the possibility of detecting the Bose-Einstein condensation in the evolution of helium white dwarfs we have computed a set of models for a variety of stellar masses and values of the condensation temperature. We do not perform a detailed treatment of the condensation process but mimic it by suppressing the nuclei contribution to the equation of state by applying an adequate function. As the cooling of white dwarfs depends on average properties of the whole stellar interior, this procedure should be suitable for exploring the departure of the cooling process from that predicted by the standard treatment. We find that the Bose-Einstein condensation has noticeable, but not dramatic effects on the cooling process only for the most massive white dwarfs compatible with a helium dominated interior (approx 0.50 M_odot) and very low luminosities (say, Log(L/L_odot) < -4.0). These facts lead us to conclude that it seems extremely difficult to find observable signals of the Bose-Einstein condensation. Recently, it has been suggested that the population of helium white dwarfs detected in the globular cluster NGC 6397 is a good candidate for detecting signals of the Bose-Einstein condensation. We find that these stars have masses too low and are too bright to have an already condensed interior.
