The H I Column Density Distribution Function at z=0: the Connection to Damped Ly$\alpha $ Statistics

Martin A. Zwaan , Marc A. W. Verheijen , Frank H. Briggs, PASA, 16 (1), in press.
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How to determine $f(N_{\rm HI})$at z=0?

A simple but illustrative and instructive method is to take the analytical approach. This is illustrated in Figure 1. Here we represent the radial distribution of the neutral hydrogen gas in galaxies by both an exponential and a Gaussian model. The differential cross sectional area of an inclined ring with a column density in the range N to N + dN is given by

$d \Sigma (N,i) = 2 \pi r(N) dr \cos i$, where r(N) is the radius at which a column density N is seen, and i is the inclination of the ring. We assume that the luminosity function $\phi(M)$ of the local galaxy population can be described by a Schechter function as indicated in the upper right panel of figure 1. The local f(N) can be derived from $\phi(M)$ and the area function $d\Sigma(N)$ by taking the integral

\begin{displaymath} f(N) = \frac{c}{H_0} \frac{\int_{M_{\rm min}}^{M_{\rm max}} \phi(M) \langle d\Sigma (N) \rangle_i \, dM}{dN}, \end{displaymath} (1)

where the subscript i indicates an average over all inclinations. To evaluate this integral, the area function, or more generally the radial H I distribution, needs to be related to M. Here we adopt the relation

$\log \mbox{$M_{\rm HI}$}=A+B M_B$ (following Rao & Briggs 1993) and assume that the central gas surface density in disks is not dependent on morphological type or luminosity. The resulting f(N) for both models is shown in the lower left panel. The integral H I gas density in

$h_{100}\, \rm g \,cm^{-3}$ as a function of column density is shown in the lower right panel. This function can be calculated with

$\mbox{$\rho_{\rm HI}$}(N) = m_{\rm H} N \frac{H_0}{c}f(N) dN$, where $m_{\rm H}$ is the mass of the hydrogen atom.

The Gaussian models yield a CDDF of the form

$\mbox{$f(N)$}\propto N^{\alpha}$, where $\alpha=-1$ for N smaller than the maximum column density seen in a face-on disk ($N_{\rm max}$) and $\alpha=-3$ for the higher values of N. The exponential model gives a smoother function. The logarithmic slope is approximately -1.2 around

$N=10^{20}~\mbox{$\rm cm^{-2}$}$, slowly changing to -3 at higher column densities. In fact, it was shown already by Milgrom (1988) that

$\mbox{$f(N)$}\propto N^{-3}$ for $N>N_{\rm max}$ for any radial surface density distribution. The lower right panel clearly illustrates that an overwhelming part of the total H I mass in the local Universe is associated with column densities close to $N_{\rm max}$.

In addition to these simple models we also show the effect of disk truncation on the CDDF. The thin dashed line illustrates a Gaussian disk truncated at

$\mbox{$N_{\rm HI}$}=10^{19.5}~\mbox{$\rm cm^{-2}$}$, the level below which photo-ionization by the extragalactic UV-background is normally assumed to be important (e.g. Corbelli & Salpeter 1993, Maloney 1993). It appears that this truncation only seriously affects the CDDF below

$\mbox{$N_{\rm HI}$}=10^{19.5}~\mbox{$\rm cm^{-2}$}$. No significant changes occur at higher column densities.

Figure 1: Illustration of simple models for the CDDF. Upper left: Gaussian and exponential models for the radial distribution of neutral gas density in galactic disks. Upper right: Schechter function describing the local luminosity function. Lower left: Resulting CDDFs for the Gaussian and exponential radial profiles. Lower right: Integral neutral gas density in the local Universe as a function of column density for both models. The effect of disk truncation is indicated by the dashed line.
\begin{figure} \begin{center} \centerline{\psfig{file=modelfn.ps,height=11cm}} \end{center} \end{figure}

A more reliable method than this analytical approach is to determine $f(N_{\rm HI})$ by using observed H I distributions. 21cm maps of nearby galaxies routinely reach sensitivity limits comparable to column densities that typify DL$\alpha $ absorbers. It therefore seems natural to calculate

$\mbox{$f(N_{\rm HI})$}(z=0)$, simply by adding cross sectional areas as a function of $N_{\rm HI}$ for a large sample of galaxies for which 21cm synthesis observations are available. However, this approach is complicated by the fact that there is an enormous variation in sensitivity and angular resolution of the 21cm maps, and the problem of choosing a fair and complete sample of galaxies. Most galaxies that have been studied extensively in the 21cm line were selected on having either a large H I diameter, so that the rotation curve can be sampled out to large galactocentric radii, or on having peculiarities such as polar rings or warps. Thus, most samples for which 21cm synthesis data exist are not representative of the galaxy population of the local Universe and would likely be biased against dwarf and low surface brightness galaxies.

Next Section: The Ursa Major Cluster
Title/Abstract Page: The H I Column Density
Previous Section: Introduction
Contents Page: Volume 16, Number 1

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