other parameters that result in present day distributions that are broadly consistent with observations. We note that in the CZ01 model the spatial distribution of the injected $e^+e^-$ will depend somewhat on the energy range of interest if pulsars older than about million years contribute significantly to that energy range. To test this, we repeat the modeling of CZ01, including a description of motion of pulsars in the galactic potential, and compute the final spatial and energy distribution of positrons. Our result for the energy distribution of the positrons agrees with CZ01. In addition, we find that the mean age of the pulsars, weighted by the positron ejection rate, is of order $10^5$ years for the energy range of interest. Given the birth velocities, these ages imply that typical pulsars (contributing significantly to the positron flux) have only traveled $\sim 100$ parsecs from their birth place. We thus use the simple approximation that the spatial distribution of these pulsars is the same as the initial pulsar distribution, which in turn tracks that of the young stars in the stellar disk. Specifically, we adopt [@Pulsardist; @sturner96] $$\begin{aligned}
\label{eq:rho}
\rho(\vec{x}) & = & N^{-1} e^{-r/r_0-|z|/z_0},\quad {\rm where}\\
N & = & 4 