# Measuring the Lense-Thirring Orbital Precession and the Neutron Star Moment of Inertia with Pulsars
## 4 Measuring the LT Precession via Pulsar Timing
### Lense-Thirring Precession in \(\dot{\omega}\)
For the Double Pulsar, the magnitude of the LT contribution is comparable to the 2 PN order contribution, but of opposite sign [84; 95]. Figure 8 shows the influence of the LT effect on \(\dot{\omega}\). If the contribution of \(\dot{\omega}_{\rm A}^{\rm LT}\) can be isolated from the total periastron advance, then \(I_{\rm A}\) can be measured, as suggested by [PERSON]. [77; 88]. Even though the uncertainty of the observed advance of periastron is much below the contribution of \(\dot{\omega}_{\rm A}^{\rm LT}\)[58], one has to determine the 1 PN and 2 PN contributions, which depend on the masses of the two pulsars \(m_{\rm A}\) and \(m_{\rm B}\). These masses can be measured very precisely and independently of \(\dot{\omega}\) using two other post-Keplerian (PK) parameters: the Shapiro delay parameter \(s\equiv\sin i\) and the orbital period decay parameter \(\dot{P}_{\rm b}\)[58; 85]. Figure 8: Mass–mass diagram of the Double Pulsar based on the assumption of GR, plotted for six PK parameters and the mass ratio \(R\) with pairs of curves showing \(\pm 1\sigma\). The insert on the upper right corner shows an enlarged view of the region of interest, where the influence of the LT effect is shown. The solid black \(\dot{\omega}\) line includes the LT contribution, whereas the dashed \(\dot{\omega}_{\rm 0}\) line (i.e., \(\dot{\omega}^{\rm 1 PN}+\dot{\omega}^{\rm 2 PN}\)) ignores it. Reproduced from Figure 13 in [58] under a Creative Commons Attribution 4.0 International license. Earlier studies on \(I_{\rm A}\) of PSR J0737\(-\)3039A concluded that a \(\sim\)10% or higher accuracy MoI measurement is possible by 2030 with the [PERSON] telescope and the Square Kilometre Array (SKA) [84; 96], without accounting for the contribution of the spin-down mass-loss of pulsar A to the orbital period derivative. This contribution, however, is proven to be crucial, as it also depends on the MoI [85]:  
\[\left(\frac{\dot{P_{\rm b}}}{P_{\rm b}}\right)^{\dot{m}_{\rm A}}=\frac{8\pi^{2} G\,I_{\rm A}\dot{P_{\rm A}}}{c^{5}T_{\odot}MP_{\rm A}{}^{3}} \tag{16}\]  
The quantities \(P_{\rm A}\) and \(\dot{P}_{\rm A}\) are the spin period and spin period derivative of pulsar A. In addition, the measurement of \(\dot{P_{\rm b}}\) is contaminated by two kinematic effects that have to be corrected for: the difference in Galactic acceleration of the pulsar system and the Solar System [97] and the proper motion of the pulsar, known as the Shklovskii effect [98]. Both are proportional to the orbital period \(P_{\rm b}\). A good knowledge of the distance and proper motion of the pulsar, as well as the Galactic parameters is required (for the details, see, e.g., [85]). Thus, the contribution that needs to be taken into account in the observed value of \(\dot{P_{\rm b}}\) is  
\[\dot{P_{\rm b}}^{\rm obs}=\dot{P_{\rm b}}^{\rm GW}+\dot{P_{\rm b}}^{\dot{m}_{ \rm A}}+\dot{P_{\rm b}}^{\rm Gal}+\dot{P_{\rm b}}^{\rm Shk}\,. \tag{17}\]  
The leading-order contribution from GW emission enters the equations of motion at 2.5 PN order (i.e., \(v^{5}/c^{5}\)) [29; 99; 100]:  
\[\dot{P_{\rm b}}^{\rm 2.5 PN} =-\frac{192\pi}{5}\left(\frac{P_{\rm b}}{2\pi}\right)^{-5/3}\frac{ T_{\odot}^{5/3}}{(1-e^{2})^{7/2}}\,f_{\rm e}\,\frac{m_{\rm A}m_{\rm B}}{M^{1/3}}\,, \tag{18}\] \[f_{\rm e} =1+\frac{73}{24}e^{2}+\frac{37}{96}e^{4}\,. \tag{19}\]  
Therefore, a self-consistent approach to determining the MoI is to account for the extrinsic contributions and spin-down mass-loss in \(\dot{P_{\rm b}}\) and solve for the three unknowns \((m_{\rm A},m_{\rm B},\,I_{\rm A})\) jointly using \(\dot{P_{\rm b}}(m_{\rm A},m_{\rm B},I_{\rm A})\), \(\dot{\omega}(m_{\rm A},m_{\rm B},I_{\rm A})\), and \(s\,(m_{\rm A},m_{\rm B})\). Based on the timing precision of data from early MeerKAT observations, Ref. [85] simulated timing observations with MeerKAT and the SKA and found that a \(\sim\)10% accuracy is likely to be achieved by 2030, given improvements on the Galactic parameters in the coming years.