  We develop an axial model for single steadily propagating positive streamers
in air. It uses observable parameters to estimate quantities that are difficult
to measure. More specifically, for given velocity, radius, length and applied
background field, our model approximates the ionization density, the maximal
electric field, the channel electric field, and the width of the charge layer.
These parameters determine the primary excitations of molecules and the
internal currents. Our approach is to first analytically approximate electron
dynamics and electric fields in different regions of a uniformly-translating
streamer head, then we match the solutions on the boundaries of the different
regions to model the streamer as a whole, and we use conservation laws to
determine unknown quantities. We find good agreement with numerical simulations
for a range of streamer lengths and background electric fields, even if they do
not propagate in a steady manner. Therefore quantities that are difficult to
access experimentally can be estimated from more easily measurable quantities
and our approximations. The theoretical approximations also form a stepping
stone towards efficient axial multi-streamer models.
