Independent geometrical information: PSR J0437

4.5 Independent geometrical information: PSR J0437–4715

A different and complementary test of GR has recently been permitted by the millisecond pulsar PSR J0437–4715 [139

]. At a distance of only 140 pc, it is the closest millisecond pulsar to the Earth [70 ], and is also extremely bright, allowing root-mean-square timing residuals of 35 ns with the 64-m Parkes telescope [107

], comparable to or better than the best millisecond pulsars observed with current instruments at the 300-m Arecibo telescope [6

Figure 11: Solid line: predicted value of the Shapiro delay in PSR J0437–4715 as a function of orbital phase, based on the observed inclination angle of 42° ± 9°. For such low-eccentricity binaries, much of the Shapiro delay can be absorbed into the orbital Roemer delay; what remains is the $∼ Pb∕3$ periodicity shown. The points represent the timing residuals for the pulsar, binned in orbital phase, and in clear agreement with the shape predicted from the inclination angle. (Reprinted by permission from Nature [139

The proximity of this system means that the orbital motion of the Earth changes the apparent inclination angle $i$ of the pulsar orbit on the sky, an effect known as the annual-orbital parallax [77 ]. This results in a periodic change of the projected semi-major axis $x$ of the pulsar’s orbit, written as

where $r⊕(t)$ is the time-dependent vector from the centre of the Earth to the SSB, and $Ω ′$ is a vector on the plane of the sky perpendicular to the line of nodes. A second contribution to the observed $i$ and hence $x$ comes from the pulsar system’s transverse motion in the plane of the sky [78 ]:

where $μ$ is the proper motion vector. By including both these effects in the model of the pulse arrival times, both the inclination angle $i$ and the longitude of the ascending node $Ω$ can be determined [139

]. As $sin i$ is equivalent to the shape of the Shapiro delay in GR (PK parameter $s$ ), the effect of the Shapiro delay on the timing residuals can then easily be computed for a range of possible companion masses (equivalent to the PK parameter $r$ in GR). The variations in the timing residuals are well explained by a companion mass of $0.236 ± 0.017 M ⊙$ (Figure 11

). The measured value of $˙ω$ , together with $i$ , also provide an estimate of the companion mass, $0.23 ± 0.14 M ⊙$ , which is consistent with the Shapiro-delay value.

While this result does not include a true self-consistency check in the manner of the double-neutron-star tests, it is nevertheless important, as it represents the only case in which an independent, purely geometric determination of the inclination angle of a binary orbit predicts the shape of the Shapiro delay. It can thus be considered to provide an independent test of the predictions of GR.