# Measuring the Lense-Thirring Orbital Precession and the Neutron Star Moment of Inertia with Pulsars
## 2 The Lense-Thirring Precession  
In GR, the gravitational field of a body has contributions from the mass currents associated with the body's proper rotation. The rotation of a massive body, such as the Sun, causes a dragging on the spacetime and, hence, makes the orbits of nearby masses (i.e., planets) precess. According to a 2007 historical study by [PERSON] [44], such a dragging effect was first calculated by [PERSON] in 1913 in the so-called Einstein-Besso manuscript (see pp. 344-473 in [45]) for the tensorial Entwurt theory [46]. In 1918, [PERSON] and [PERSON], with substantial help from [PERSON], calculated the frame-dragging effect of rotating central bodies on the motion of planets and moons in GR [47; 48; 49], which is currently well known as the \"Lense-Thirring effect\" or \"Lense-Thirring precession\".  
Experimental tests of the LT precession were first proposed in the 1970s by launching two counter-orbiting drag-free satellites in polar orbit around the Earth [50; 51]. Later, after the LASer GEOdynamics Satellite of NASA (LAGEOS) was launched in 1976, [PERSON] proposed a new method of measuring the LT effect using a second LAGEOS satellite with an opposite orbital orientation [52]. LAGEOS 2 was then launched by the Italian Space Agency (ASI) and NASA in 1992, followed by another laser-ranged satellite of ASI launched in 2012, the LASer RElativity Satellite (LARES) [53]. Based on data collected with these three satellites, the LT precession test in Earth's gravity field has reached an accuracy of about 2% [54; 55].  
Another manifestation of the frame dragging, which should not be confused with the LT precession, is the \"Schiff precession\", where, on top of the effect of the geodetic precession on the spin of a test mass, there is a secondary effect caused by the spin of the massive body [56]. This resulted in the proposal for Gravity Probe B, which was eventually launched and measured both the geodetic and Schiff precessions; the latter matches the GR  
Figure 3: Orbital geometry of the system with all vectors shifted to the centre of mass of the system. \(\mathbf{S}\) is the spin angular momentum of the pulsar (or companion), and \(\mathbf{L}\) is the orbital angular momentum, which is perpendicular to the orbital plane and inclined at an angle \(i\) to the line of sight vector, \(\mathbf{K}\). The total angular momentum vector \(\mathbf{J}=\mathbf{L}+\mathbf{S}\), and \(\delta^{\rm SO}\) is the misalignment angle between \(\mathbf{L}\) and \(\mathbf{S}\). As a result of this misalignment, both the spin and orbit precess around \(\mathbf{J}\). The unit vector \(\mathbf{I}\) points from the centre of mass to the ascending node.  
prediction well within its 19% relative uncertainty [57]. This is the first measurement of \"spin-spin\" coupling.  
Similar to the weak-field scenario (i.e., the Solar System), the spins of compact objects (e.g., pulsars) in relativistic binaries are gravitationally coupled to the orbital motion of the systems [26]. This relativistic spin-orbit coupling leads to an additional contribution to the rate of change of periastron \(\dot{\omega}\) and a precession of the orbital plane about the direction of the system's total angular momentum (see Figure 3). In the latter case, it induces a change in the inclination angle \(i\). In the following sections, we recapitulate these two cases, highlighting the detailed predictions for the motions for binary pulsars, as these are the only ones where we might detect LT precession caused by the rotations of NSs.