For those few binaries that remain bound, and in which the companion is sufficiently massive to evolve into a giant and overflow its Roche lobe, the old spun-down neutron star can gain a new lease of life as a pulsar by accreting matter and angular momentum at the expense of the orbital angular momentum of the binary system . The term “recycled pulsar” is often used to describe such objects. During this accretion phase, the X-rays produced by the frictional heating of the infalling matter onto the neutron star mean that such a system is expected to be visible as an X-ray binary. Two classes of X-ray binaries relevant to binary and millisecond pulsars exist: neutron stars with high-mass or low-mass companions. Detailed reviews of the X-ray binary population, including systems likely to contain black holes, can be found elsewhere [38, 312].
In a high-mass X-ray binary, the companion is massive enough that it also explodes as a supernova, producing a second neutron star. If the binary system is lucky enough to survive the explosion, the result is a double neutron star binary. Nine such systems are now known, the original example being PSR B1913+16  – a 59-ms radio pulsar which orbits its companion every 7.75 hr [359, 360]. In this formation scenario, PSR B1913+16 is an example of the older, first-born, neutron star that has subsequently accreted matter from its companion.
For many years, no clear example was known where the second-born neutron star was observed as a pulsar. The discovery of the double pulsar J0737–3039 [50, 242], where a 22.7-ms recycled pulsar “A” orbits a 2.77-s normal pulsar “B” every 2.4 hr, has now provided a dramatic confirmation of this evolutionary model in which we identify A and B as the first and second-born neutron stars respectively. Just how many more observable double pulsar systems exist in our Galaxy is not clear. While the population of double neutron star systems in general is reasonably well understood (see Section 3.4.1), a number of effects conspire to reduce the detectability of double pulsar systems. First, the lifetime of the second born pulsar cannot be prolonged by accretion and spin-up and hence is likely to be less than one tenth that of the recycled pulsar. Second, the radio beam of the longer period second-born pulsar is likely to be much smaller than its spun-up partner making it harder to detect (see Section 3.2.3). Finally, as discussed in Section 4.4, the PSR J0737–3039 system is viewed almost perfectly edge-on to the line of sight and there is strong evidence  that the wind of the more rapidly spinning A pulsar is impinging on B’s magnetosphere which affects its radio emission. In summary, the prospects of ever finding more than a few 0737-like systems appears to be rather low.
A notable recent addition to the sample of double neutron star binaries is PSR J1906+0746, a 144-ms pulsar in a highly relativistic 4-hr orbit with an eccentricity of 0.085 . Timing observations show the pulsar to be young with a characteristic age of only 105 yr. Measurements of the relativistic periastron advance and gravitational redshift (see Section 4.4) constrain the masses of the pulsar and its companion to be and respectively . We note, however, that the pulsar exhibits significant amounts of timing noise (see Section 4.3), making the analysis of the system far from trivial. Taken at face value, these parameters suggest that the companion is also a neutron star, and the system appears to be a younger version of the double pulsar. Despite intensive searches, no pulsations from the companion star (expected to be a recycled radio pulsar) have so far been observed . This could be due to unfavourable beaming and/or intrinsic radio faintness. Continued searches for radio pulsations from these companions are strongly encouraged, however, as geodetic precession may make their radio beams visible in future years.
The companion in a low-mass X-ray binary evolves and transfers matter onto the neutron star on a much longer timescale, spinning it up to periods as short as a few ms . Tidal forces during the accretion process serve to circularize the orbit. At the end of the spin-up phase, the secondary sheds its outer layers to become a white dwarf in orbit around a rapidly spinning millisecond pulsar. This model has gained strong support in recent years from the discoveries of quasi-periodic kHz oscillations in a number of low-mass X-ray binaries , as well as Doppler-shifted 2.49-ms X-ray pulsations from the transient X-ray burster SAX J1808.4–3658 [397, 67]. Seven other “X-ray millisecond pulsars” are now known with spin rates and orbital periods ranging between 185 – 600 Hz and 40 min – 4.3 hr respectively [396, 264, 176, 206].
Numerous examples of these systems in their post X-ray phase are now seen as the millisecond pulsar–white dwarf binary systems. Presently, 20 of these systems have compelling optical identifications of the white dwarf companion, and upper limits or tentative detections have been found in about 30 others . Comparisons between the cooling ages of the white dwarfs and the millisecond pulsars confirm the age of these systems and suggest that the accretion rate during the spin-up phase was well below the Eddington limit .
Further support for the above evolutionary scenarios comes from two correlations in the observed sample of low-mass binary pulsars. Firstly, as seen in Figure 8, there is a strong correlation between orbital period and eccentricity. The data are in very good agreement with a theoretical relationship which predicts a relic orbital eccentricity due to convective eddy currents in the accretion process . Secondly, as shown in Panel b of Figure 9, where companion masses have been measured accurately, through radio timing (see Section 4.4) and/or through optical observations , they are in good agreement with a relation between companion mass and orbital period predicted by binary evolution theory . A word of caution is required in using these models to make predictions, however. When confronted with a larger ensemble of binary pulsars using statistical arguments to constrain the companion masses (see Panel a of Figure 9), current models have problems in explaining the full range of orbital periods on this diagram .
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