Context: Observations of small carbon-bearing molecules such as CCH, C4H, c-C3H2, and HCO in the Horsehead Nebula have shown these species to have higher abundances towards the edge of the source than towards the center. Aims: Given the determination of a wide range of values for zeta (s-1), the total ionization rate of hydrogen atoms, and the proposal of a column-dependent zeta(N_H), where N_H is the total column of hydrogen nuclei, we desire to determine if the effects of zeta(N_H) in a single object with spatial variation can be observable. We chose the Horsehead Nebula because of its geometry and high density. Method: We model the Horsehead Nebula as a near edge-on photon dominated region (PDR), using several choices for zeta, both constant and as a function of column. The column-dependent zeta functions are determined by a Monte Carlo model of cosmic ray penetration, using a steep power-law spectrum and accounting for ionization and magnetic field effects. We consider a case with low-metal elemental abundances as well as a sulfur-rich case. Results: We show that use of a column-dependent zeta(N_H) of 5(-15) s-1 at the surface and 7.5(-16) s-1 at Av = 10 on balance improves agreement between measured and theoretical molecular abundances, compared with constant values of zeta.
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