# Measuring the Lense-Thirring Orbital Precession and the Neutron Star Moment of Inertia with Pulsars
## 1 Introduction  
### The Equation of State of Dense Matter  
Since the time of the first detailed calculation of the structure of neutron stars (NSs), in 1939 [1], it has been recognised that the main uncertainty regarding their bulk properties (mass and maximum mass possible, radius, tidal deformability, moment of inertia (MoI), and quadrupolar moment) is a lack of knowledge of the microscopic properties of cold nuclear matter at the densities one finds in NS cores, which are well above that of the atomic nucleus. Especially important for the calculation of the NS structure is the relation between the pressure and density of this type of matter--its equation of state (EoS). The EoS had large uncertainties in 1939, and to some extent, these uncertainties persist today [2; 3; 4; 5; 6]. This results in a wide range of NS mass-radius relations (see Figure 1). This situation happens for several reasons, all related to the properties of the strong nuclear force: (a) Given the gluon self-interaction, perturbative methods do not allow precise theoretical calculations of the behaviour of the strong nuclear force, in particular the state and composition of matter at high densities; (b) laboratory experiments, in particular ion-ion collisions, can reach similar densities to those found in NSs, but for very short amounts of time and with very high kinetic energies per nucleon, a situation that is very different from what one finds in NS cores [2]. For this reason, the measurement (via astronomical observations) of bulk NS properties plays an important role in constraining the EoS and understanding the state of cold, dense neutron matter, which is a major research problem in fundamental physics and astrophysics [7]. The most straightforward way to constrain the dense matter EoS is via the measurement of NS masses. These masses can be measured through relativistic orbital effects in binary pulsars using pulsar timing; they can also be potentially combined with optical information from the companion in the case of a white dwarf (WD). In particular, the central matter must be incompressible enough (i.e., the EoS must be \"stiff\" enough) to support the maximum observed NS mass. The most massive pulsar, PSR J0740\(+\)6620 (\(m_{\rm p}=2.08\pm 0.07\) M\({}_{\odot}\)[8]; see Section 1.1 for the definition of the nominal solar mass M\({}_{\odot}\).) has excluded a number of soft EoSs that cannot support a 2-M\({}_{\odot}\) NS (see Figure 1). There are hints of more massive NSs from the measurements of stellar companions other than WDs (PSR J0952\(-\)0607 [10] and PSR J2215\(+\)5135 [11]), but these mass measurements depend, to a large extent, on the interpretation of stellar spectra and on detailed models of stellar surfaces and have given divergent results in the past (e.g., [12; 13]). Therefore, as we can see in Figure 1, quite a few EoSs survive. The reason is that this method only probes one dimension of the parameter space shown in Figure 1; as we see, the surviving EoSs have a wide range of radii. This means that any method where two bulk parameters can be probed simultaneously for the same NS, such as mass and radius or mass and MoI, would represent an ever-more powerful constraint. Measuring the extremely small radii of NSs (\(\sim\)10 km) is a challenging task. This has been attempted using a wide variety of methods with X-ray observations [7; 14], such as pulse profile modelling using the Neutron Star Interior Composition Explorer (NICER) [15; 16; 17; 18; 19]. These methods provide valuable radius measurements and are even more precise in cases where the mass is measured independently from radio timing measurements, e.g., PSR J0740\(+\)6620, a rare high-mass NS that will likely not be so often present in GW mergers. However, they depend on very detailed and difficult modelling of X-ray light curves. A more recent method uses LIGO's detailed gravitational wave (GW) measurements of the GW170817 NS-NS merger [20].