Accurate measurement of the H i column density from H i 21 cm absorption-emission spectroscopy

We present a detailed study of an estimator of the H i column density, based on a combination of H i 21 cm absorption and H i 21cm emission spectroscopy. This 'isothermal' estimate is given by N-HI,N- ISO = 1.823 x 10(18) integral [tau(tot) x T-B / [ 1 - e(tot)(-tau)]dV, where tau(tot) is the total H i 21cm optical depth along the sightline and T-B is the measured brightness temperature. We have used a Monte Carlo simulation to quantify the accuracy of the isothermal estimate by comparing the derived N-HI,N- ISO with the true H i column density N-HI. The simulation was carried out for a wide range of sightlines, including gas in different temperature phases and random locations along the path. We find that the results are statistically insensitive to the assumed gas temperature distribution and the positions of different phases along the line of sight. The median value of the ratio of the true H i column density to the isothermal estimate, N-HI/N-HI,N- ISO, is within a factor of 2 of unity while the 68.2 per cent confidence intervals are within a factor of approximate to 3 of unity, out to high H i column densities, < 5 x 10(23) cm(-2) per 1 km s(-1) channel, and high total optical depths, < 1000. The isothermal estimator thus provides a significantly better measure of the H i column density than other methods, within a factor of a few of the true value even at the highest columns, and should allow us to directly probe the existence of high H i column density gas in the Milky Way.