provided us with the basic features of the positron injection spectrum. We find that the spatial distribution should track that of the young stars in the disk, with an energy spectrum that is less steep than $E^{-2}$ – specifically $E^{-1.6}$ for the model of CZ01 – and a total normalization that requires about $10\%$ of the spin-down power of ${\cal O}(10^5{\rm
  yr})$ pulsars to be injected into positrons.

Pulsars as a Source for the Haze \[sec:pulsarmodel\]
====================================================

The positrons pumped into the ISM will lose energy and diffuse outwards, and as they do so, they will produce the synchrotron background. To model this we use the standard diffusion equation that describes the propagation and energy loss for a charged particle: $$\frac{\partial}{\partial t} {\frac{dn}{dE_e}}(\vec{x},t,E)= K(E)\nabla^2{\frac{dn}{dE_e}}+ \frac{\partial}{\partial E} B(\vec{x},E){\frac{dn}{dE_e}}+ Q(\vec{x},t,E)\,,$$ where $Q(\vec{x},t,E)$ is the source function, i.e., rate of production of positrons per unit energy with energy $E$ at time $t$ and location $\vec{x}$. It is a sum over all the pulsars in the galaxy. We have assumed that the diffusion coefficient $K(E)$ is spatially constant, as there is no evidence in the cosmic ray or diffuse galactic gamma-ray data to the contrary. In addition, we note that very little is known 