What is Airmass, atmospheric extinction coefficient?

The airmass $X$ of an observation is the optical depth of the line of sight divided by the optical depth at the zenith: $X = \tau(z)/\tau_0$. It equals 1 at the zenith and, for a plane-parallel atmosphere, $X = \sec z$. The atmospheric extinction coefficient $k$ in a given band is the dimming, in magnitudes, produced by one airmass, so that a star of magnitude $m_0$ above the atmosphere is observed at