Estimating the birth period of pulsars through GLAST/LAT observations of their wind nebulae

In this paper we show that the high energy $gamma$-ray flux in the GeV domain from mature pulsar wind nebulae (PWN) scales as the change in rotational kinetic energy $I(Omega_0^2-Omega^2)/2$ since birth, rather than the present day spindown power $IOmegadot{Omega}$. This finding holds as long as the lifetime of inverse Compton emitting electrons exceeds the age of the system. For a typical $gamma^{-2}$ electron spectrum, the predicted flux depends mostly on the pulsar birth period, conversion efficiency of spindown power to relativistic electrons and distance to the PWN, so that first order estimates of the birth period can be assessed from {it GLAST/LAT} observations of PWN. For this purpose we derive an analytical expression. The associated (``uncooled) photon spectral index in the GeV domain is expected to cluster around $sim 1.5$, which is bounded at low energies by an intrinsic spectral break, and at higher energies by a second spectral break where the photon index steepens to $sim 2$ due to radiation losses. Mature PWN are expected to have expanded to sizes larger than currently known PWN, resulting in relatively low magnetic energy densities and hence survival of GeV inverse Compton emitting electrons. Whereas such a PWN may be radio and X-ray quiet in synchrotron radiation, it may still be detectable as a {it GLAST/LAT} source as a result of the relic electrons in the PWN.