  This paper describes an improved parameterization for proton-beryllium
production of secondary $K^{+}$ mesons for experiments with primary proton
beams from 8.89 to 24 GeV. The parameterization is based on Feynman scaling in
which the invariant cross section is described as a function of $x_{F}$ and
$p_{T}$. This method is theoretically motivated and provides a better
description of the energy dependence of kaon production at low beam energies
than other parameterizations such as the commonly used "Modified Sanford-Wang"
model. This Feynman scaling parameterization has been used for the simulation
of the neutrino flux from the Booster Neutrino Beam (BNB) at Fermilab and has
been shown to agree with the neutrino interaction data from the SciBooNE
experiment. This parameterization will also be useful for future neutrino
experiments with low primary beam energies, such as those planned for the
Project X accelerator.
