  Proton interactions with $^{12}$C nuclei are a frequent nuclear interaction
leading to secondary radiation in tissues for space radiation and cancer
therapy with protons or $^{12}$C beams. The fragmentation of $^{12}$C by
protons produces a large number of heavy ion (A>4) target or projectile
fragments often with high ionization density. Here we develop an analytical
model of energy dependent proton-$^{12}$C cross sections for isotopic nuclei
production. Using experimental data and a 2nd order optical model an accurate
formula for the p-$^{12}$C absorption cross section from <10 MeV/n to >10 GeV/N
is obtained. The energy dependence of the elemental and isotopic cross sections
is modeled as multiplicities scaled to absorption cross section with average
isotopic fractions estimated from experimental data. We show that this approach
results in an accurate analytic formula model over the full energy range in
Hadron therapy and space radiation protection studies.
