# Measuring the Lense-Thirring Orbital Precession and the Neutron Star Moment of Inertia with Pulsars
## 4 Measuring the LT Precession via Pulsar Timing
### Lense-Thirring Precession in \(\dot{x}\)  
#### 4.1.1 PSR J1141\(-\)6545, a Young Pulsar with a Massive White Dwarf Companion  
PSR J1141\(-\)6545 is an atypical system among pulsar-WD systems, in the sense that the pulsar is not recycled: its spin period of 396 ms and large spin period derivative place it firmly among the \"normal\" pulsars (see Figure 4). Furthermore, unlike most other pulsar-WD systems, its orbit has a significant eccentricity (\(e=0.17\)), and its spin axis is misaligned with the orbital angular momentum, as deduced from the variation of its pulse profile [39]. That the companion is a massive WD was established optically [81]. Thus, the system displays all the characteristics typical of DNS systems, in particular the characteristics of the pulsar are similar to those of PSR J0737\(-\)3039B, the second-formed pulsar in the \"Double-Pulsar\" system (see Section 4.2.1 for details). Why is the primary, then, a WD? The explanation was published in 2000 by [PERSON] and [PERSON] [82]. At the start of the evolution of the system, the primary lost so much mass to the secondary during its evolution that it went under the mass limit for the formation of an NS, forming instead a massive WD. The secondary, after accreting all the mass from the primary, became massive enough to undergo an SN, forming then an NS, now visible as a normal pulsar. A consequence of this evolution is that, before the SN explosion, the progenitor to the pulsar filled its Roche lobe and started transferring matter to the WD, thus spinning it up and aligning its angular momentum with the orbital angular momentum. The mass and kick associated with the subsequent SN made the system eccentric and changed the orbital plane, as in a DNS system. Since then, the direction of the spin of the WD has been, like that of the pulsar, precessing around the total angular momentum via geodetic precession. The orbital precession is, in this case, overwhelmingly due to the spin of the WD, not the NS, which has an Mol \(\sim\)10\({}^{5}\)-times smaller than the WD, but only spins about \(\sim\)10\({}^{2}\)-times faster. The large Mol of the fast-spinning WD explains why it was, in this pulsar, which has a relatively low timing precision, that the spin-orbit coupling was seen [75]. The effect was detected as a variation of the projected semi-major axis (\(\dot{x}\)) caused by the variation of the inclination angle of about 14 arcseconds per year. The effect could only be detected by assuming that GR is the correct gravity theory, so it cannot be interpreted as a test of the theory. Unlike in the case of NSs, where the \(\dot{x}\) caused by spin-orbit coupling (\(\dot{x}^{\rm SO}\)) comes entirely from the LT precession (\(\dot{x}^{\rm LT}\), see Equation (8)), and unlike in main-sequence stars, where this effect would arise from classical spin-orbit coupling due to the quadrupolar moment of the star (\(\dot{x}^{\rm QPM}\)), in massive, fast-spinning WDs, both contributions to \(\dot{x}^{\rm SO}\) are of a similar order of magnitude. What percentage of the \(\dot{x}^{\rm SO}\) arises from \(\dot{x}^{\rm LT}\) and \(\dot{x}^{\rm QPM}\) depends on the unknown spin period of the WD (\(P_{\rm WD}\)), the unknown misalignment angle between the spin of the WD and the orbital plane (\(\dot{\delta}^{\rm SO}_{\rm C}\)), and a precession phase \(\Phi_{\rm C}\); see Figure 7. Again, these contributions were calculated strictly within the framework of GR, which was used to detect \(\dot{x}^{\rm SO}\) in the first place. Figure 7 highlights some of the limitations of the measurements of the LT effect based on \(\dot{x}\), which is compounded in this case by the fact that the spin period of the companion WD is unknown. In a DNS system, we have a few advantages: (a) for NSs, \(\dot{x}^{\rm LT}\) is several orders of magnitude larger than \(\dot{x}^{\rm QPM}\), which means that the latter term can be ignored; (b) we generally know the spin period of the recycled pulsar.