# 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 \(\dot{\omega}\)  
4.2.1. PSR J0737\(-\)3039A/B, the Unique Double-Pulsar System  
The Double-Pulsar system PSR J0737\(-\)3039A/B was discovered in 2003 [76], with a pulsar in a compact orbit with an NS, which was also confirmed to be a pulsar shortly afterwards [77]. The system is composed of a 23 ms recycled pulsar \"A\" and a 2.8 s young pulsar \"B\" in a highly relativistic 2.45 h orbit. This remains the only known system comprising two pulsars, although pulsar B disappeared from our view since March 2008 due to geodetic precession [87]. The low orbital eccentricity (\(e=0.088\)), small transverse velocity of the system (\(\sim\)10 km s\({}^{-1}\)) [88], and small misalignment angle between A's spin vector \(\mathbf{s}_{\rm A}\) and the orbital angular momentum vector \(\mathbf{k}\) (\(\delta^{\rm SO}_{\rm A}<3.2^{\circ}\)) [89; 90; 91] all support the opinion that the second-born pulsar \"B\" was formed in a low-kick SN event [92; 70; 93]. Moreover, the rotation of A has been confirmed to be prograde using the modulation of B's emission by the interaction with A's pulsar wind [94]. The unique high inclination angle between the orbital angular momentum and the line of sight towards the pulsar system (\(i=89.36^{\circ}\pm 0.03^{\circ}\) or \(180^{\circ}-i\) from timing analysis) enabled the studies of an aberrational light bending effect [58; 79], thus independently confirming the prograde rotation of A and providing additional evidence for a low-kick SN. 2.2 Measure the Neutron Star Moment of Inertia with \(\dot{\omega},s\), and \(\dot{P}_{\rm b}\)  
Since PSR J0737\(-\)3039A/B is almost edge-on to our line of sight, the LT contribution in \(\dot{x}\) is insignificant. Therefore, the only practicable way to access the Mol of pulsar A, \(I_{\rm A}\), is through the LT contribution in \(\dot{\omega}\). Given that the spin of A is practically aligned with the orbital angular momentum, assuming \(\mathbf{s}_{\rm A}=\mathbf{k}\), for pulsar A, the expression of the spin contribution in Equation (7) can be simplified to [21]  
\[\dot{g}^{\parallel}_{\rm S_{A}}=\frac{1}{(1-e^{2})^{1/2}}\left(\frac{1}{3}X_ {\rm A}^{2}+X_{\rm A}\right). \tag{11}\]  
As pulsar B spins about 122-times slower than pulsar A, the LT contribution of pulsar B is negligible [85]. Therefore, the total observed rate of periastron advance in the Double-Pulsar system can be written as  
\[\dot{\omega}^{\rm obs}=\dot{\omega}^{\rm 1 PN}+\dot{\omega}^{\rm 2 PN}+\dot{ \omega}^{\rm LT}_{\rm A}, \tag{12}\] \begin{table}
\begin{tabular}{l c c} \hline \hline
**Pulsar Name** & \(\dot{\omega}^{\rm LT}\) **(deg yr\({}^{-1}\))** & \(\dot{\boldsymbol{x}}^{\rm LT}\) **(lt-s s\({}^{-1}\))** \\ \hline PSR J0737\(-\)3039A & \(-\)0.000377 \(\times\)\(I_{\rm A}^{(45)}\)\({}^{*}\)[85] & - \\ PSR J1946+2052 ** & \(-\)0.000677 \(\times\)\(I_{\rm A}^{(45)}\) & - \\ PSR J1757\(-\)1854 & - & \(|\dot{x}^{\rm LT}|\leq 2.01\times 10^{-15}\times I_{\rm A}^{(45)}\)[34] \\ PSR J0514\(-\)4002E & - & \(\dot{x}^{\rm LT}\lesssim 1.7\times 10^{-13}\)[86] \\ \hline \hline \end{tabular} \({}^{*}I_{\rm A}^{(45)}=I_{\rm A}/(10^{45}\rm gcm^{2})\). ** Mass is assumed to be 1.25 M\({}_{\odot}\) for both NSs.  
\end{table} 
Table 1: The LT contribution to the rate of periastron advance in four aforementioned promising pulsar systems. where the 1 PN and higher-order corrections due to 2 PN effects and the LT contribution of pulsar A are  
\[\dot{\omega}^{\rm 1 PN} =3\left(\frac{P_{\rm b}}{2\pi}\right)^{-5/3}\frac{T_{\odot}^{2/3}M^{ 2/3}}{1-e^{2}}\,, \tag{13}\] \[\dot{\omega}^{\rm 2 PN} =3\left(\frac{P_{\rm b}}{2\pi}\right)^{-7/3}f_{\rm O}\,\frac{T_{ \odot}^{4/3}M^{4/3}}{1-e^{2}}\,,\] (14) \[\dot{\omega}_{\rm A}^{\rm LT} =-3\left(\frac{P_{\rm b}}{2\pi}\right)^{-2}\frac{G}{c^{5}T_{\odot }M(1-e^{2})^{3/2}}\bigg{(}\frac{1}{X_{\rm A}}+\frac{1}{3}\bigg{)}\,I_{\rm A} \,2\pi\
u_{\rm A}\,, \tag{15}\]  
where \(\
u_{\rm A}\) is the spin frequency of pulsar A.