A study of dynamical effects in the observed second time-derivative of the spin or orbital frequencies of pulsars

Abstract The observed values of the time-derivatives of the spin or orbital frequency of pulsars are affected by their dynamical properties. We derive thorough analytical expressions for such dynamical contributions in terms of the Galactic coordinates, the proper motion, the pulsar distance, and the radial velocity. We find that the effects of the dynamical terms in the second-derivative of frequencies or parameters based on such second derivatives, e.g., braking index, are usually negligible. However, unique pulsars for which the effects of the dynamical terms are significant can exist. In particular, dynamical effects can make the magnitude of the observed value of the braking index to be in the order of thousand while the true value of it is close to the theoretically expected value three, especially if the pulsars lie close to the Galactic centre. Dynamics can also affect the value of the second derivative of the orbital frequency of a binary pulsar at the first decimal place. We also emphasize the fact that our expressions provide more accurate results than pre-existing approximate ones that exclude some of the terms. Comparison with a set of pulsars showed that the median value of the difference between the results obtained by our method and a pre-existing method is about 50 percent.