The modulation of an extra-solar signal can, if working in terms of signal phase, be expressed as a time modulation, e.g. for a phase evolution given by
(1)
\begin{eqnarray}
\phi (t) = \phi _0 + 2\pi f_0\left( t - t_0 + \Delta \tau (t) \right),
\end{eqnarray}
where t is the time of arrival of the signal at the observer, and ϕ0 and f0 are an initial phase and frequency at the epoch t0 in a reference frame at rest with respect to the source, the time modulation term is Δτ(t). Assuming, for now, that the source is at rest with respect to the SSB, the time modulation can be expressed as a combination of terms
(2)
\begin{eqnarray}
\Delta \tau = \Delta _{\rm R} + \Delta _{\rm E} - \Delta _{\rm S},
\end{eqnarray}
where ΔR (the Roemer delay) is a geometric retardation term, ΔE (the Einstein delay) is a relativistic frame transformation term taking into account relativistic time dilation, and ΔS (the Shapiro delay) is the delay due to passing through curved space–time. These terms are discussed in, for example, chapter 5 of Lyne & Graham-Smith (1998), while Edwards, Hobbs & Manchester (2006) provide more detailed discussion of time delays accounting for more effects with particular relevance to pulsar observations. Here, for each of the terms we use the sign conventions given in the source code for the pulsar timing software tempo21 (Hobbs, Edwards & Manchester 2006) and in the LALSuite gravitational wave software library functions (LIGO Scientific Collaboration 2017), rather than those used in the equation of Edwards et al. (2006).2 The Roemer delay is given by
(3)
\begin{eqnarray}
\Delta _{\rm R} = \frac{\boldsymbol {r}\cdot \hat{\boldsymbol {s}}}{c},
\end{eqnarray}
where $\boldsymbol {r}$ is a vector giving the position of the observer with respect to the SSB, and $\hat{\boldsymbol {s}}$ is a unit vector pointing from the observer to the source. The Einstein delay (see e.g. equations 9 and 10 of Edwards et al. 2006) converts to a new time coordinate frame, and depends on the choice of frame you want, i.e. Barycentric Coordinate Time (TCB), in which the effect of the presence of the Sun's gravitational potential is removed. The Shapiro delay (for which we will only consider the contribution from the Sun) is to first order given by
(4)
\begin{eqnarray}
\Delta _{\rm S} \equiv \Delta _{\rm S_{\odot }} = -\frac{2G {\rm M}_{\odot }}{c^3}\ln {\left({\boldsymbol {r}}_{\rm se}\cdot \hat{\boldsymbol {s}} + |\boldsymbol {r}_{\rm se}| \right)}
\end{eqnarray}
for waves passing around the Sun, where $\boldsymbol {r}_{\rm se} = \boldsymbol {r}_{{\oplus}} - \boldsymbol {r}_{\odot }$ is the vector from the centre of the Sun to the geocentre.3 Unlike electromagnetic waves, gravitational waves will pass through matter, and therefore a different term is required for a wave passing through the Sun, i.e. when $\left|\boldsymbol {r}_{\rm se}\right|^2 - \left(\boldsymbol {r}_{\rm se}\cdot \hat{\boldsymbol {s}}\right)^2 < R_{\odot }^2$ and $\boldsymbol {r}_{\rm se}\cdot \hat{\boldsymbol {s}} < 0$, giving
(5)
\begin{eqnarray}
\Delta _{\rm S_{\odot }} &amp;=&amp; -\frac{2G {\rm M}_{\odot }}{c^3}\Bigg [\ln {\left(\boldsymbol {r}_{\rm se}\cdot \hat{\boldsymbol {s}} + \sqrt{R_{\odot }^2 + \left(\boldsymbol {r}_{\rm se}\cdot \hat{\boldsymbol {s}}\right)^2} \right)} \nonumber \\
&amp;&amp;- \,2\left(1 - \frac{\sqrt{\left|\boldsymbol {r}_{\rm se}\right|^2 - \left(\boldsymbol {r}_{\rm se}\cdot \hat{\boldsymbol {s}}\right)^2}}{R_{\odot }}\right)\Bigg]\!,
\end{eqnarray}
where R⊙ is the radius of the Sun.
Citation: Hobbs, Edwards & Manchester 2006
Citation function: Uses