Pulsar current revisited

The pulsar current, in the $P$--${\dot P}$ plane where $P$ is the pulsar period and ${\dot P}$ is the period derivative, is used as a supposedly ``model free'' way to estimate the pulsar birthrate from statistical data on pulsars. We reconsider the derivation of the kinetic equation on which this is based, and argue that the interpretation of the pulsar current is strongly model dependent, being sensitive to the form of the assumed evolution law for pulsars. We discuss the case where the trajectory of a pulsar is assumed to be of the form ${\dot P}=KP^{2-n}$ with $K$ and $n$ constant, and show that (except for $n=2$) one needs to introduce a pseudo source term in order to infer the birthrate from the pulsar current. We illustrate the effect of this pseudo source term using pulsar data to estimate the birthrate for different choices of $n$. We define and discuss an alternative ``potential'' class of evolution laws for which this complication is avoided due to the pseudo source term being identically zero.