# Measuring the Lense-Thirring Orbital Precession and the Neutron Star Moment of Inertia with Pulsars
## 7 Summary and Prospects  
The extraordinary compactness of NSs makes them fascinating objects to probe the behaviour of matter under extreme conditions that cannot be replicated on Earth. This can be achieved by measuring the macroscopic quantities of NSs, namely the mass, radius, tidal deformability, Mol, and quadrupolar moment. Since NS masses can be precisely measured via radio pulsar timing, some of the most massive ones have provided crucial constraints on the EOS.However, measuring the NS radius is incredibly challenging. Many of the current methods, such as X-ray observations, could suffer from strong model dependencies. In Section 2, we reviewed the method to measure the LT precession, which gives access to the Mol of NSs, and in Section 3, we discussed which types of binary pulsars are suitable for measuring this effect. Section 4 discussed five binary pulsars that have already showed evidence of LT precession (such as PSR J1141\(-\)6545) or are promising for the measurement of this effect, either as a precession of periastron \(\dot{\omega}\) or a precession of the projected semi-major axis \(\dot{x}\). Table 1 summarises the estimated values for the LT contributions in four of these systems. A detailed description of measuring the Mol using \(\dot{\omega}\) for DNS systems was given in Section 4.2.2, and the first measurements are expected to come soon from the Double Pulsar, adding complementary constraints to other methods [85]. In Section 5, we discussed how, depending on the nature of its companion, timing measurements of LT precession in the PSR J0514\(-\)4002E system could lead to the measurement of the spin of a BH formed in a merger system. Such measurements have the potential to shed light on the origins of the BHs in these systems and test the cosmic censorship hypothesis. As demonstrated in Section 6, DNS systems with orbital periods shorter than those currently known can yield drastically higher precision Mol measurements, which would greatly enhance our understanding of the behaviour of dense matter. Such precise measurements of the Mol would represent an exciting development for many reasons besides studies of dense matter. Two are especially relevant here. First, if they are measured for NSs near 1.4 M\({}_{\odot}\) (such as PSR J0737\(-\)3039A and the NSs seen in GW mergers like GW170817), they allow direct comparisons with future improvements of the NS tidal deformability to emerge from ground-based measurements of GW mergers via the I-Love-Q relation. This would represent a fundamental test of our understanding of the structure of compact objects. Secondly, the Mol of NSs has a very direct application in tests of GR. As shown in the study with 16 yr Double-Pulsar data, the test of the radiative properties in leading order and next-to-leading order of GR also requires the knowledge of the MoI of the pulsar [58; 85], as the mass loss contribution depends on the MoI and the masses measured from \(\dot{\omega}\) and \(s\) have to account for LT precession. This calculation currently utilises the multimessenger constraint on the radius of an NS as outlined in [6] (pink bar in Figure 1), which are then converted into the Mol of pulsar A using the radius-Mol relationship given in [4]. Without further improvements in measuring the MoI, the capability to enhance the tests of orbital decay due to GW damping will remain limited. Furthermore, the resulting knowledge of the NS structure and the EoS is fundamental for a precise interpretation of tests of alternative theories of gravity performed with pulsar timing. In such theories, the universality of free fall (UFF) does not apply to massive gravitating objects, because their gravitational properties depend on their internal structure (e.g., [108; 115]). This is especially true for very compact objects like NSs; for them, the detailed internal structure can only be calculated with the EoS of dense nuclear matter.