High precision pulsar timing and spin frequency second derivatives

We investigate the impact of intrinsic, kinematic and gravitational effects on high precision pulsar timing. We present an analytical derivation and a numerical computation of the impact of these effects on the first and second derivative of the pulsar spin frequency. In addition, in the presence of white noise, we derive an expression to determine the expected measurement uncertainty of a second derivative of the spin frequency for a given timing precision, observing cadence and timing baseline and find that it strongly depends on the latter ($\propto t^{-7/2}$). We show that for pulsars with significant proper motion, the spin frequency second derivative is dominated by a term dependent on the radial velocity of the pulsar. Considering the data sets from three Pulsar Timing Arrays, we find that for PSR J0437$-$4715 a detectable spin frequency second derivative will be present if the absolute value of the radial velocity exceeds 33 km s$^{-1}$. Similarly, at the current timing precision and cadence, continued timing observations of PSR J1909$-$3744 for about another eleven years, will allow the measurement of its frequency second derivative and determine the radial velocity with an accuracy better than 14 km s$^{-1}$. With the ever increasing timing precision and observing baselines, the impact of the, largely unknown, radial velocities of pulsars on high precision pulsar timing can not be neglected.