Context : Observations of small carbon-bearing molecules such as CCH , { C _ { 4 } H } , c– { C _ { 3 } H _ { 2 } } , 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 .

Methods : 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 \times 10 ^ { -15 } s ^ { -1 } at the surface and 7.5 \times 10 ^ { -16 } s ^ { -1 } at A _ { V } = 10 on balance improves agreement between measured and theoretical molecular abundances , compared with constant values of \zeta .

Conclusions :