# Measuring the Lense-Thirring Orbital Precession and the Neutron Star Moment of Inertia with Pulsars
## 3 Evolution of Binary Pulsars
### Types of Binary Pulsars  
In Figure 5, we show in a very schematic way the two main formation channels for binary pulsars (for a detailed review, see [61]). For simplicity, we start with a binary system consisting of two main-sequence stars. This is a common occurrence, especially for massive stars, the majority of which form in binary and multiple systems [62]. As the more massive star (the primary) evolves, it eventually explodes as an SN, forming a \"normal\" pulsar. The system will then very likely disrupt owing to the kick and mass loss associated with this SN; we know this not only from the aforementioned fact that most \"normal\" pulsars are \"single\", but also from the observation that surviving systems (consisting of a pulsar and a massive main-sequence star) have very high orbital eccentricities (generally of order 0.9, e.g., PSR B1259\(-\)63 with \(e=0.87\)[64], PSR J0045\(-\)7319 with \(e=0.808\)[65], PSR J1638\(-\)4724 with \(e=0.955\)[66], and PSR J2032\(+\)4127 with \(e=0.96\)[67]), indicating a near disruption. Eventually, in some of these systems, the secondary will also evolve and become a giant star. Its large size will cause several effects, the first being tidal circularisation of the orbit. When the secondary fills its Roche lobe, the transfer of matter to the NS starts, and the system becomes an X-ray binary. This mass transfer can become unstable, leading to a common envelope phase. At this stage, the NS is slowly spun up, and its spin will become aligned with the orbital angular momentum. This process is thought to ablate the magnetic field of the NS. What happens next depends on the mass of the secondary:  
* If it is massive enough, it will also go through an SN explosion; the system then either disrupts (a likely occurrence) or forms an eccentric DNS system, which will consist  
Figure 5: Schematic diagram showing the evolution scenario of various pulsar systems, based on the concepts in [63]. of a recycled pulsar and a short-lived \"normal\" pulsar. The high risk of disruption makes these systems relatively rare (\(<\)1%). These systems are necessarily eccentric because of the mass loss and the kick associated with the SN (the average kick for young isolated radio pulsars is 400-500 km s\({}^{-1}\)[68; 69]; this is likely to be slightly less in binary systems). Because of this kick, the orbital plane after the SN might become very different from what it was before the explosion, causing a misalignment between the spin of the recycled pulsar and the orbital angular momentum. The most compact of these systems will coalesce in a Hubble time, forming a light black hole (BH) (for an extensive discussion on these systems, see [70]). * If the secondary is light, then it will slowly evolve into a WD star. In this case, the system will retain a low orbital eccentricity and the alignment between the spin and orbital angular momentum. As we can see in Figure 4, the recycled pulsars with low-mass WD companions (commonly known as \"millisecond pulsars\" (MSPs)) have spin frequencies one order of magnitude larger than the recycled pulsars with NS companions. The reason is that, for NSs with low-mass companions, they spend much longer (up to \(\sim\)10\({}^{8}\) yr) as X-ray binaries because their evolution is much slower. The resulting high spin frequencies give MSPs high timing precision. The DNS systems have a known range of eccentricities from 0.06 to 0.83. These eccentricities are determined by a range of factors, like the masses of the components of the system before the second SN, the orbital period of that binary, and the mass loss and the magnitude and orientation of the kick that happen during the second SN. The observations indicate that the dominant factor appears to be the mass of the second-formed NS: if it is smaller than 1.35 M\({}_{\odot}\), then \(e<0.3\); if it is larger, then \(e>0.55\)[70]. This suggests that SN kicks are linked to the mass of the newly formed NS. Among the DNSs with lower mass companions, these seems to be a hard floor for the eccentricities.