The Distribution of Column Densities and b Values in the Lyman-Alpha Forest

We describe the properties of the Ly$\alpha$ forest in the column density range $\nhi \geq 2\times10^{12}$ cm$^{-2}$ based on 1056 lines in the wavelength range 4300--5100\AA measured in extremely high S/N, $R=36,000$ spectra of four quasars. The column density distribution is well described by a -1.5 power law to $2\times10^{12}$ cm$^{-2}$, below which limit confusion becomes too severe to measure a spectrum of individual clouds. The distribution of $b$ values shows a well-defined lower envelope with a cutoff at $b=20\kms$ corresponding to a cloud temperature of 24,000 K. There is only a very small fraction (less than 1%) of narrow line clouds which cannot be identified with metal-lines. From modeling the Ly$\alpha$ absorption lines as complexes of clouds each with thermal broadening corresponding to $b_c$ we find the $b$ distribution can be understood if there is a mean of 3.25 clouds per absorption line with a spread in velocity centroids characterized by a dispersion of $10.75\kms$.