We have presented the principles and framework for calculating the radio signal from a PSR in an EMRB. We restrict our study to the extreme mass ratio of EMRB systems and so do not consider PSRs in stellar-mass BH binaries with finite mass ratios (e.g. Blanchet 2014; Liu et al. 2014). We account for both relativistic and astrophysical effects and the convolution between the two. This includes gravitational and relativistic time dilation and energy shift, gravitational light bending, complex orbital dynamics induced by spin couplings, temporal variation and distortion of the pulse profile due to spin axis precession and relativistic aberration, second-order pulses due to gravitational bending, and dispersions (temporal and spatial) induced by the material along the line of sight. We have demonstrated that within our framework we are able to determine the time–frequency behaviour accounting for all these effects. The framework also applies for any orbital configuration, e.g. we are not restricted to orbital motion in the equatorial plane or beaming confined to the orbital plane. The methods used are entirely covariant and general relativistic, rather than working under any post-Newtonian approximation and so are inherently more accurate. Indeed, the post-Newtonian method is an explicitly weak-field method, and the validity of its application to strong-field dynamical regimes is unclear (Will 2011). Whilst working explicitly in the Kerr metric means that we are unable to independently probe either alternative gravitational theories or extensions to Kerr (e.g. Kerr space–time with an arbitrary mass quadrupole; see Bini et al. 2009), our framework provides the basis for a theoretical timing model that can then be compared with observations for tests of strong field GR. We approximate the PSR body as a perfect sphere. However, due to the spin of the PSR the true shape is more oblate. This will ultimately influence the pitch angle of the ray with the neutron star surface. This effect is considered to be minor, but the method could easily be extended to account for this oblateness (see Nättilä & Pihajoki 2018). We neglect the effects of hydrodynamic drag due to the plasma that surrounds that BH since at compact radii (≲ 104rg) the gravitational and relativistic effects dominate (Psaltis 2012). We also do not take account of any potential Newtonian perturbations on the motion of the pulsar (e.g. Merritt et al. 2011) due to the presence of other masses (e.g. stars, other compact objects etc.) since these factors are likely negligible for the orbital periods considered in this work (≲0.3 yr; Liu et al. 2012). Indeed, the potential for external perturbations to hamper tests of strong-field GR necessitates that an ideal PSR–EMRB systems should have orbital periods on the order of 0.1 yr (or better), or else observations should be taken close to periapsis (see discussion in Psaltis, Wex & Kramer 2016). These are precisely the regions where the space–time curvature and orbital acceleration is greatest, further stressing the importance of a strong-field timing model. We also neglect any influence of gravitational radiation on the orbit or the ray trajectory. The neglect of gravitational radiation is justified since in the extreme mass ratio limit, the time-scale for orbital decay due to gravitational wave emission is (Misner, Thorne & Wheeler 1973)
(45)
\begin{eqnarray*}
\tau _{\mathrm{ GW}} \sim \frac{5 r^4}{96mM(m+M)} f(e)^{-1} \ ,
\end{eqnarray*}
where M is the mass of the BH, m the pulsar mass, and r the orbital separation. The eccentricity function is
(46)
\begin{eqnarray*}
f(e) = (1-e^2)^{-7/2} \left(1 + \frac{73}{24} e^2 + \frac{37}{96} e^4\right) \ .
\end{eqnarray*}
If we take the PSR orbital period P to be Keplerian, then for a pulsar with mass $1.4 \, \mathrm{M}_{\odot }$ on an eccentric (e = 0.8), P = 0.1 yr orbit around a BH with mass $4.3 \times 10^6 \, \mathrm{M}_{\odot }$(47)
\begin{eqnarray*}
\frac{\tau _{\mathrm{ GW}}}{P} \sim 10^9 \gt \gt 1 \ ,
\end{eqnarray*}
and so the effects of gravitational radiation can be neglected. Even for smaller radii and more eccentric orbits the space–time is well approximated as stationary (e.g. τGW/P  ∼ 105 for e = 0.9, r = 100 M). Whilst the effects of gravitational radiation are then not important for a single orbit, for observations over longer periods of time the effect of gravitational emission on the orbit and hence the timing solution will need to be considered. The PSR may also emit a gravitational wave burst during passage through periastron (Berry & Gair 2013a,b). The influence of this gravitational radiation on both the PSR trajectory and the photon ToA is highly non-trivial and not considered here.
Citation: Nättilä & Pihajoki 2018
Citation function: Future Work