ON THE VERIFICATION OF THE PLANETARY SYSTEM AROUND PSR 1257+12

It has been pointed out that if the quasiperiodic variations in the times of pulse arrival of the millisecond pulsar PSR 1257 + 12 are indeed due to the reaction of the pulsar to the orbital motion of two planets, the mutual perturbations of the planets will alter the variations in a characteristic, predictable way. Unambiguous detection of the consequences of these mutual perturbations on the pulsar motion would be unassailable proof of the planets' existence. The planets are inferred from quasiperiodic variations of the residuals in the times of pulse arrival (TOA) where the residuals are obtained by subtracting the TOA predicted by the best-fit constant period model from the observed TOA. Here we define the problem of detecting the perturbations by determining the nature and magnitude of the additional residuals in the TOA. The additional residuals are represented by the TOA residuals calculated numerically for the orbital motion including the perturbations less the TOA residuals calculated from analytic representations of the orbital motion with orbital parameters fixed at averaged values. The latter averaged values represent the least-squares best-fit orbital parameters. The TOA residual differences so obtained oscillate with periods comparable to the orbital periods with the oscillations varying in amplitude as a function of epoch within any given observational period. For the minimum possible values of the two planetary masses, the TOA residual differences reach maximum amplitudes of about 10 mus for observational intervals exceeding 1000 days, and increase approximately as 1/sin i for 1/sin i less than or similar to 5. Here i is the inclination of the orbital plane to that of the sky. The TOA residual differences due to the perturbations are in practice superposed on those due to systematic and random noise, so the observable signature of the perturbations is a quasiperiodic modulation of TOA residual differences obtained from the data by removing the effects of the best-fit orbits. The modulation changes phase and, to a lesser extent, amplitude as a function of the observational interval. Another signature of the perturbations is the differences in the times of particular zero crossings of the TOA residuals for perturbed and unperturbed orbits. The apparent advantage of the greater accumulation of phase differences between the two cases for these zero crossings is offset by a relatively large uncertainty in determining the actual times of the zero crossings. The current observationally determined TOA residual differences of about 10 mus peak amplitude appear to already rule out values of 1/sin i greater than or similar to 4, and the signature of the perturbations at this noise level will probably not be detectable for likely values of 1/sin i until the observational interval exceeds 1000 days.