# Measuring the Lense-Thirring Orbital Precession and the Neutron Star Moment of Inertia with Pulsars
## 1 Introduction
### Spin-Orbit Coupling and Geodetic Precession  
The coupling of the orbital and spin dynamics produces two observable effects. The change of the direction of the spin is known as _geodetic precession_. The corresponding change of direction of the orbital angular momentum, which is necessary to conserve the total angular momentum of the system, is the LT precession. The magnitudes of these vectors stay the same [24].  
Any vector undergoing parallel transport in a curved spacetime experiences a change in direction as seen from distant observers. If the spin axis of a pulsar is misaligned with the total angular momentum of the system \(\mathbf{J}\), it will slowly precess around \(\mathbf{J}\) at a rate of [25; 26]  
\[\Omega^{\rm geod}=\left(\frac{P_{\rm b}}{2\pi}\right)^{-5/3}\,\frac{T_{\odot}^ {2/3}}{1-e^{2}}\frac{m_{\rm c}(4m_{\rm p}+3m_{\rm c})}{2M^{4/3}} \tag{1}\]  
as predicted by General Relativity (GR). An illustration of the orbital geometry is shown in Figure 3. \(P_{\rm b}\) is the orbital period and the constant \(T_{\odot}\equiv(\mathcal{GM})_{\odot}^{\rm N}/c^{3}\), with \((\mathcal{GM})_{\odot}^{\rm N}\equiv 1.327124\,4\times 10^{26}\,{\rm cm}^{3}\, {\rm s}^{-2}\) being the nominal solar mass parameter defined by the IAU 2015 Resolution B3 [27] and \(c\) the speed of light. \(m_{\rm p}\) and \(m_{\rm c}\) are the masses of the pulsar and the companion, and \(M=m_{\rm p}+m_{\rm c}\) is the total mass of the system. In this and all the following equations with \(T_{\odot}\), the masses are dimensionless, being specified in nominal solar masses by taking the ratio \(GM_{\rm object}/(\mathcal{GM})_{\odot}^{\rm N}\), where \(G\) is the gravitational constant, and the nominal solar mass is defined as \(\mathrm{M}_{\odot}=(\mathcal{GM})_{\odot}^{\rm N}/G\). The time eccentricity \(e\) corresponds to the observed eccentricity included in the timing model [28].  
As the angle between the pulsar's spin and our line of sight varies with time, different parts of the emitted beam are observed and a temporal evolution of the pulse shape and polarisation is expected [24; 29]. The first evidence of this relativistic spin precession was provided by the narrowing of the pulse profile observed in the Hulse-Taylor pulsar PSR B1913\(+\)16 [30]. The full geometry of the system was determined by fitting a cone-like beam model, and the misalignment of the pulsar spin to the orbital momentum vector was measured. Later, [PERSON] and [PERSON] confirmed these results using new high-precision Arecibo data [31].  
Henceforth, the geodetic precession has been detected through profile variations in other recycled pulsars, such as PSR B1534\(+\)12 [32], PSR B2127\(+\)11C [33], and more recently, in PSR J1757\(-\)1854 [34] and PSR J1946\(+\)2052 [35]. As we will discuss in more detail below, in these systems, the orbital plane is no longer aligned with the spin axis of the recycled pulsar (thus creating the misalignment between the spin axis of the pulsar and \(\mathbf{J}\)), and the reason for this is the kick associated with the second supernova (SN) in the system.  
In fact, geodetic precession has been discovered in all binary systems where the precession rate is expected to be sufficiently high (see [36; 37] for a review). These also include pulsars that are the second-formed compact object in a binary system: PSR J0737\(-\)3039B [38], PSR J1141\(-\)6545 [39], and PSR J1906\(+\)0746 [40; 41]. In particular, the precession rate of PSR J0737\(-\)3039B has been measured from the observed changes of the eclipse light curve of PSR J0737\(-\)3039A when B passes in front of it [42; 43]. Indeed, for these pulsars, there is no special reason why their spin axes should be aligned with \(\mathbf{J}\), because they were never recycled by matter coming from the companion (which has angular momentum in the orbital plane). These younger pulsars provide the best measurements of \(\Omega^{\rm good}\)[41; 43] that match GR's prediction (Equation (1)) within the measurement uncertainties.