  We present a theory for interpreting the sodium lines detected in
transmission spectra of exoplanetary atmospheres. Previous analyses employed
the isothermal approximation and dealt only with the transit radius. By
recognising the absorption depth and the transit radius as being independent
observables, we develop a theory for jointly interpreting both quantities,
which allows us to infer the temperatures and number densities associated with
the sodium lines. We are able to treat a non-isothermal situation with a
constant temperature gradient. Our novel diagnostics take the form of
simple-to-use algebraic formulae and require measurements of the transit radii
(and their corresponding absorption depths) at line center and in the line wing
for both sodium lines. We apply our diagnostics to the HARPS data of HD
189733b, confirm the upper atmospheric heating reported by Huitson et al.
(2012), derive a temperature gradient of $0.4376 \pm 0.0154$ K km$^{-1}$ and
find densities $\sim 1$ to $10^4$ cm$^{-3}$.
