The Timing Noise of PSR 0823+26, PSR 1706-16, PSR 1749-28, PSR 2021+51 and The Anomalous Braking Indices

We have investigated the stability of the pulse frequency second derivatives ($\ddot \nu $) of PSR 0823+26, PSR 1706-16, PSR 1749-28, PSR 2021+51 which show significant quadratic trends in their pulse frequency histories in order to determine whether the observed second derivatives are secular or they arise as part of noise processes. We have used TOA data extending to more than three decades which are the longest time spans ever taken into account in pulse timing analyses. We investigated the stability of pulse frequency second derivative in the framework of low resolution noise power spectra (Deeter 1984) estimated from the residuals of pulse frequency and TOA data. We have found that the $\ddot \nu $ terms of these sources arise from the red torque noise in the fluctuations of pulse frequency derivatives which may originate from the external torques from the magnetosphere of pulsar.