On the Relaxation Behaviors of Slow and Classical Glitches: Observational Biases and Their Opposite Recovery Trends

We study the pulsar timing properties and the data analysis methods during glitch recoveries. In some cases one first fits the time-of-arrivals (TOAs) to obtain the "time-averaged" frequency $\nu$ and its first derivative $\dot\nu$, and then fits models to them. However, our simulations show that $\nu$ and $\dot\nu$ obtained this way are systematically biased, unless the time intervals between the nearby data points of TOAs are smaller than about $10^4$ s, which is much shorter than typical observation intervals. Alternatively, glitch parameters can be obtained by fitting the phases directly with relatively smaller biases; but the initial recovery timescale is usually chosen by eyes, which may introduce a strong bias. We also construct a phenomenological model by assuming a pulsar's spin-down law of $\dot\nu\nu^{-3} =-H_0 G(t)$ with $G(t)=1+\kappa e^{-t/\tau}$ for a glitch recovery, where $H_0$ is a constant and $\kappa$ and $\tau$ are the glitch parameters to be found. This model can reproduce the observed data of slow glitches from B1822--09 and a giant classical glitch of B2334+61, with $\kappa<0$ or $\kappa>0$, respectively. We then use this model to simulate TOA data and test several fitting procedures for a glitch recovery. The best procedure is: 1) use a very high order polynomial (e.g. to 50th order) to precisely describe the phase; 2) then obtain $\nu(t)$ and $\dot\nu(t)$ from the polynomial; and 3) the glitch parameters are obtained from $\nu(t)$ or $\dot\nu(t)$. Finally, the uncertainty in the starting time $t_0$ of a classical glitch causes uncertainties to some glitch parameters, but less so to a slow glitch and $t_0$ of which can be determined from data.