# Measuring the Lense-Thirring Orbital Precession and the Neutron Star Moment of Inertia with Pulsars
## 1 Introduction
### The Equation of State of Dense Matter
The resulting estimates of tidal deformability have excluded the very stiffest EoSs considered by [3], including the MS1, MS1b, and H4 EoSs, as they would cause observable tidal effects in the GW waveform that are not consistent with the observation. This method has great potential for the future, with more  
Figure 1: The NS mass–radius relation for different EoSs listed in [2]. The horizontal lines in yellow represent the 1-\(\sigma\) mass ranges for the most massive NS known, PSR J0740\(+\)6620 [8]. The pink bar in the middle shows the multimessenger constraint on the radius of a 1.4-M\({}_{\odot}\) NS [6] at 90% confidence, and the cyan bar shows an updated radius constraint at 95% confidence [9]. The EoSs plotted in dashed lines are excluded by the lower limit of the maximum NS mass and the constraint of the radius range in [6]. Figure courtesy of [PERSON]. sensitive GW detectors and louder events, but for the moment, the constraints have rather large uncertainties. The method we will discuss here is to measure the MoI of binary pulsars via very small effects of relativistic spin-orbit coupling on their orbital motion, the _Lense-Thirring (LT) precession_, which was first suggested by [PERSON] and [PERSON] in 1988 [21]. Using radio pulsar timing (and our knowledge of the system geometry), we attempt to measure simultaneously the mass _and_ the MoI of an NS. This is, in principle, equivalent to simultaneously measuring the mass and radius, because, as we can see in Figure 2, each EoS gives a unique prediction for the relation between the mass and MoI. However, it might in effect be even more valuable, as it would provide direct constraints on the tidal deformability measured by ground-based GW detectors and the quadrupole moment of NSs via the universal (EoS-independent) I-Love-Q relation [22]. Measuring the MoI can be performed because of the high precision of pulsar timing. This method is intrinsically _accurate_, because the timing techniques are model-independent. In addition, as the MoI of an NS relates to its mass and radius as \(I_{\rm NS}\propto M_{\rm NS}R_{\rm NS}^{2}\), the range of the MoI for different EoS models is relatively larger than the range of the radius (see Figure 2) [23]. Also, the relative accuracy required for the MoI measurement is less than that of the radius as \(\Delta I_{\rm NS}/I_{\rm NS}=2\,\Delta R_{\rm NS}/R_{\rm NS}\). These make the measurement of the MoI easier than that of the radius. However, they are not yet _precise_, because of the small magnitude of the effects being measured. This article aims to provide a review of measuring the MoI of an NS using LT precession with the latest knowledge on promising pulsar systems. In Section 2, we provide a brief overview of the LT precession and how it can be tested using pulsar timing. In Section 3, we explain the evolution of different types of binary pulsar systems and the suitability of double-neutron star (DNS) systems for testing the LT precession. Several binary pulsars are  
Figure 2: The change of the MoI with mass for different EoSs, coloured as in Figure 1. The yellow band represents the mass range of PSR J0740\(+\)6620. The EoSs marked with labels survive the current lower limit of the maximum NS mass and multimessenger radius constraint. Figure courtesy of [PERSON]. discussed in detail in Section 4, including PSR J1141\(-\)6545, PSR J1757\(-\)1854, the Double Pulsar PSR J0737\(-\)3039A, and PSR J1946\(+\)2052. A discussion on the recent study of the massive binary pulsar J0514\(-\)4002E is given in Section 5. We discuss the future discoveries of ultra-compact DNS systems and the significance of LT precession measurements in relation to orbital period and eccentricity in Section 6. Finally, in Section 7, we provide a summary and outlook for the future. However, before we move on, we must say a few words about geodetic precession, which has been observed in binary pulsars and is intimately linked with LT precession.