4.6 Spin-orbit coupling and geodetic precession

A complete discussion of GR effects in pulsar observations must mention geodetic precession, though these results are somewhat qualitative and do not (yet) provide a model-free test of GR. In standard evolutionary scenarios for double-neutron-star binaries (see, e.g., [22108]), both stellar spins are expected to be aligned with the orbital angular momentum just before the second supernova explosion. After this event, however, the observed pulsar’s spin is likely to be misaligned with the orbital angular momentum, by an angle of the order of 20° [14Jump To The Next Citation Point]. A similar misalignment may be expected when the observed pulsar is the second-formed degenerate object, as in PSR J1141–6545. As a result, both vectors will precess about the total angular momentum of the system (in practice the total angular momentum is completely dominated by the orbital angular momentum). The evolution of the pulsar spin axis S1 can be written as [4215]
dS1 ---- = Ωsp1in× S1, (38 ) dt
where the vector spin Ω 1 is aligned with the orbital angular momentum. Its magnitude is given by
spin 1 ( Pb) −5∕3 m2(4m1 + 3m2 ) 2∕3 Ω 1 = -- --- ------2-----------4∕3-T⊙ , (39 ) 2 2π (1 − e )(m1 + m2 )
where T⊙ ≡ GM ⊙ ∕c3 = 4.925490947 μs, as in Section 4.1. This predicted rate of precession is small; the three systems with the highest Ωspin 1 values are:

The primary manifestation of this precession is expected to be a slow change in the shape of the pulse profile, as different regions of the pulse emission beam move into the observable region.

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Figure 12: Changes in the observed pulse profile of PSR B1913+16 throughout the 1980s, due to a changing line-of-sight cut through the emission region of the pulsar. (Taken from [132]; used by permission.)
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Figure 13: Top: change in peak separation of the relativistic double-neutron-star binary PSR B1913+16, as observed with the Arecibo (solid points, [141Jump To The Next Citation Point]) and Effelsberg (open circles, [81Jump To The Next Citation Point]) telescopes. Bottom: projected disappearance of PSR B1913+16 in approximately 2025. (Taken from [81Jump To The Next Citation Point]; used by permission.)
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Figure 14: Hourglass-shaped beam for PSR B1913+16 derived from the symmetric-component analysis of [143Jump To The Next Citation Point]. (Taken from [143Jump To The Next Citation Point]; used by permission.)

Evidence for long-term profile shape changes is in fact seen in PSRs B1913+16 and B1534+12. For PSR B1913+16, profile shape changes were first reported in the 1980s [141], with a clear change in the relative heights of the two profile peaks over several years (Figure 12View Image). No similar changes were found in the polarization of the pulsar [33]. Interestingly, although a simple picture of a cone-shaped beam might lead to an expectation of a change in the separation of the peaks with time, no evidence for this was seen until the late 1990s, at the Effelsberg 100-m telescope [81Jump To The Next Citation Point], by which point the two peaks had begun to move closer together at a rather fast rate. Kramer [81] used this changing peak separation, along with the predicted precession rate and a simple conal model of the pulse beam, to estimate a spin-orbit misalignment angle of about 22° and to predict that the pulsar will disappear from view in about 2025 (see Figure 13View Image), in good agreement with an earlier prediction by Istomin [66] made before the peak separation began to change. Recent results from Arecibo [143] confirm the gist of Kramer’s results, with a misalignment angle of about 21°. Both sets of authors find there are four degenerate solutions that can fit the profile separation data; two can be discarded as they predict an unreasonably large misalignment angle of ∼ 180° – 22° = 158° [14], and a third is eliminated because it predicts the wrong direction of the position angle swing under the Rotating Vector Model [109]. The main area of dispute is the actual shape of the emission region; while Weisberg and Taylor find an hourglass-shaped beam (see Figure 14View Image), Kramer maintains that a nearly circular cone plus an offset core is adequate (see Figure 15View Image). In any event, it is clear that the interpretation of the profile changes requires some kind of model of the beam shape. Kramer [82Jump To The Next Citation Point83] lets the rate of precession vary as another free parameter in the pulse-shape fit, and finds a value of 1.2° ± 0.2°. This is consistent with the GR prediction but still depends on the beam-shape model and is therefore not a true test of the precession rate.

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Figure 15: Alternate proposed beam shape for PSR B1913+16, consisting of a symmetric cone plus an offset core. The red lines indicate an example cut through the emission region, as well as the predicted pulse peak ratio and separation as functions of time. (After [82], courtesy Michael Kramer.)

PSR B1534+12, despite the disadvantages of a more recent discovery and a much longer precession period, also provides clear evidence of long-term profile shape changes. These were first noticed at 1400 MHz by Arzoumanian [710] and have become more obvious at this frequency and at 430 MHz in the post-upgrade period at Arecibo [124]. The principal effect is a change in the low-level emission near to the main pulse (Figure 16View Image), though related changes in polarization are now also seen. As this pulsar shows polarized emission through most of its pulse period, it should be possible to form a better picture of the overall geometry than for PSR B1913+16; this may make it easier to derive an accurate model of the pulse beam shape.

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Figure 16: Evolution of the low-level emission surrounding the main pulse of PSR B1534+12, over a period of nearly 10 years, as measured with the Arecibo telescope [6]. (Stairs et al., unpublished.)

As for other tests of GR, the pulsar–white-dwarf binary PSR J1141–6545 promises interesting results. As noted by the discoverers [73Jump To The Next Citation Point], the region of sky containing this pulsar had been observed at the same frequency in an earlier survey [71], but the pulsar was not seen, even though it is now very strong. It is possible that interference corrupted that original survey pointing, or that a software error prevented its detection, but it is also plausible that the observed pulsar beam is evolving so rapidly that the visible beam precessed into view during the 1990s. Clearly, careful monitoring of this pulsar’s profile is in order.

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