Pulsars and neutron stars/Modelling timing residuals

Glitch events
In tempo/tempo2, glitch events are modelled as an offset in pulse phase, $$\Delta \phi$$ and the permanent increments in the pulse frequency $$\Delta \nu_p$$ and first frequency time derivative $$\Delta \dot{\nu}_p$$. In addition it is sometimes necessary to in a transient frequency increment $$\Delta \nu_d$$ which decays exponentially to zero with a timescale of $$\tau_d$$. The phase model (caused by the glitch) is therefore taken as:

$$\phi_g = \Delta \phi + \Delta \nu_p(t-t_g) + \frac{1}{2}\Delta\dot{\nu}_p(t-t_g)^2 + \left[ 1 - e^{-(t-t_g)/\tau_d}\right]\Delta\nu_d \tau_d$$

where $$t_g$$ is the glitch epoch and $$t$$ is the current time.