Observing a column-dependent zeta in dense interstellar sources: the
  case of the Horsehead Nebula
  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.
