# Measuring the Lense-Thirring Orbital Precession and the Neutron Star Moment of Inertia with Pulsars
## 2 The Lense-Thirring Precession
### Spin-Orbit Coupling in Periastron Advance  
The spin-orbit coupling yields a contribution to the advance of periastron, \(\dot{\omega}\). According to [21], in GR, the total observed rate of periastron advance \(\dot{\omega}^{\rm obs}\), up to the second post-Newtonian (PN) order, can be written as  
\[\dot{\omega}^{\rm obs}=\frac{2\pi}{P_{\rm b}}\,k\,, \tag{2}\]  
with the dimensionless periastron advance parameter \(k\) written in the form [21; 58]:  
\[k=\frac{3\,\beta_{\rm O}^{2}}{1-e^{2}}\Big{(}1+f_{\rm O}\,\beta_{\rm O}^{2}-g_ {\rm S_{A}}\,\beta_{\rm O}\,\beta_{\rm S_{A}}-g_{\rm S_{B}}\,\beta_{\rm O}\, \beta_{\rm S_{B}}\Big{)}\,, \tag{3}\]  
where  
\[\beta_{\rm O} =\left(\frac{2\pi}{P_{\rm b}}T_{\odot}M\right)^{1/3}, \tag{4}\] \[\beta_{\rm S_{j}} =\frac{G\,S_{j}}{c^{5}\,T_{\odot}^{2}\,m_{j}^{2}}=\frac{G\,I_{j} \,2\pi v_{j}}{c^{5}\,T_{\odot}^{2}\,m_{j}^{2}}\,,\] (5) \[f_{\rm O} =\frac{1}{1-e^{2}}\bigg{(}\frac{39}{4}X_{\rm A}^{2}+\frac{27}{4}X _{\rm B}^{2}+15X_{\rm A}X_{\rm B}\bigg{)}-\bigg{(}\frac{13}{4}X_{\rm A}^{2}+ \frac{1}{4}X_{\rm B}^{2}+\frac{13}{3}X_{\rm A}X_{\rm B}\bigg{)},\] (6) \[g_{\rm S_{j}} =\frac{X_{j}\,(4X_{j}+3X_{j})}{6\,(1-e^{2})^{1/2}\sin^{2}i}\Big{[} (3\sin^{2}i-1)\,{\bf k}\cdot{\bf s}_{j}+\cos(i)\,{\bf K}\cdot{\bf s}_{j}\Big{]}\,. \tag{7}\]  
The subscripts A and B stand for the two bodies in the binary (in the case of the Double Pulsar, i.e., pulsar A and pulsar B). The terms with subscript O are associated with orbital contributions and with S are associated with spin contributions. The subscript \(j\) denotes one body (\(j=\) A, B), and \(j^{\prime}\) corresponds to the other body of the system. \(S_{j}\) is the magnitude of the spin of body \(j\), which equals the product of the Mol \(I_{j}\) with \(2\pi v_{j}\), where \(v_{j}\) is the spin frequency. The mass ratio of one mass component, \(m_{j}\), and the total mass of the system, \(M=m_{\rm A}+m_{\rm B}\), is indicated by \(X_{j}=m_{j}/M\). \For nearly edge-on systems (\(i\approx 90^{\circ}\)), this contribution is small as \(\cot i\ll 1\) and most likely cannot be detected, independently of the cause of \(\mathrm{d}i/\mathrm{d}t\).  
The \((\mathrm{d}i/\mathrm{d}t)^{\mathrm{LT}}\) term is given, in GR, by:  
\[\left(\frac{\mathrm{d}i}{\mathrm{d}t}\right)^{\mathrm{LT}}=\frac{3}{2}\left( \frac{P_{\mathrm{b}}}{2\pi}\right)^{-2}\frac{G}{c^{5}T_{\odot}M(1-e^{2})^{3/2} }\sum_{j}\left(\frac{1}{X_{j}}+\frac{1}{3}\right)\left(\mathbf{S}_{j}\cdot \mathbf{I}\right), \tag{9}\]  
where \(\mathbf{S}_{j}\) is the spin angular momentum (\"spin\") of body \(j\) and \(\mathbf{I}\) is a unit vector pointing from the centre of mass of the system to the ascending node.  
In binary pulsar systems, the total observed change in the projected semi-major axis of the pulsar orbit can be written as [59]:  
\[\dot{x}^{\mathrm{obs}}=\dot{x}^{\mathrm{PM}}+\dot{x}^{\mathrm{D}}+\dot{x}^{ \mathrm{GW}}+\dot{x}^{\mathrm{m}}+\dot{x}^{3\mathrm{rd}}+\dot{x}^{\mathrm{ \dot{e}}_{\mathrm{A}}}+\dot{x}^{\mathrm{LT}}\,, \tag{10}\]  
where the other contributions come from the proper motion of the system (\(\dot{x}^{\mathrm{PM}}\)), the changing radial Doppler shift (\(\dot{x}^{\mathrm{D}}\)), GW emission (\(\dot{x}^{\mathrm{GW}}\)), mass loss in the system (\(\dot{x}^{\mathrm{m}}\)), the presence of a hypothetical third body in the system \(\dot{x}^{3\mathrm{rd}}\), and a secular change in the aberration of the pulsar beam due to geodetic precession \(\dot{x}^{\mathrm{\dot{e}}_{\mathrm{A}}}\), respectively.