### 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].
The proximity of this system means that the orbital motion of the Earth changes the apparent
inclination angle 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 of the pulsar’s orbit, written as

where is the time-dependent vector from the centre of the Earth to the SSB, and is a vector on
the plane of the sky perpendicular to the line of nodes. A second contribution to the observed
and hence comes from the pulsar system’s transverse motion in the plane of the sky [78]:
where is the proper motion vector. By including both these effects in the model of the pulse arrival
times, both the inclination angle and the longitude of the ascending node can be determined [139].
As is equivalent to the shape of the Shapiro delay in GR (PK parameter ), 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 in GR). The variations in the timing residuals are well explained by a
companion mass of (Figure 11). The measured value of , together with , also
provide an estimate of the companion mass, , 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.