What is the significance of Hydrogen column density? I have always read in the textbooks about star formation and ${\rm H}_2$ column densities. What is the relationship between the two? How is star formation affected by ${\rm H}_2$ column densities?
 A: Star formation requires gas to collapse gravitationally, and gas can only collapse if it is cold enough, or if it cools as it collapses. Cold, dense hydrogen will form hydrogen molecules, ${\rm H}_2$. The presence of ${\rm H}_2$ therefore correlates with favourable star formation conditions and can act as a tracer of the star formation rate. Since ${\rm H}_2$ has no strong emission lines (cold gas in general doesn't, really), it must be measured via the absorption spectrum of a background source. Even then, ${\rm H}_2$ can be tricky to detect, and often ${\rm CO}$ is used as a tracer of ${\rm H}_2$ via some assumption for the ${\rm CO}$ to ${\rm H}_2$ ratio. Since absorption can occur anywhere along the line of sight, a column density rather than a (local) volume density is measured.
A: Hydrogen molecules are the most common molecules in the interstellar medium (ISM). H$_2$ regions in interstellar space are quite cold, because hydrogen molecules dissociate at $T>100$ K. The cold ISM is roughly in pressure equilibrium with other (warmer) phases of the ISM. Thus molecular hydrogen also traces the densest part of the ISM.
These facts are salient to star formation because star formation can only occur in gas clouds that are dense and cold. The reason for this can be seen when considering the Jeans instability. A cloud may collapse if its mass exceeds the Jeans mass, which is $M_J \sim 3\times 10^{4}(T^3/n)^{1/2} M_{\odot}$, where $n$ is the number of particles per cubic centimetre.
Thus if $T$ is low, then for a fixed pressure (and hence $nT$) the Jeans mass is lower and a cloud can collapse and fragment. In a giant molecular cloud, typically $n > 10^{5}$ cm$^{-3}$, temperatures can be as low as 10 K and thus the cloud can collapse and fragment into cluster-sized objects, which, if they can cool as they collapse, will then fragment into stellar-sized objects.
Molecular H$_2$ plays two roles here. First it is a measure of how dense the gas is. Unfortunately, density cannot be measured directly. In general we get tracers of gas density measured along a line of sight. i.e.
$$ {\rm H}_2\ {\rm column\ density} = \int n(H_2)\ dl$$
In the case of molecular hydrogen what is usually measured is the column density of CO molecules along the line of sight (the H$_2$ molecule itself has almost no observable signature since it has no electric dipole moment), which is then converted into a corresponding H$_2$ column density.
Second, although the H$_2$ molecules cannot efficiently radiate away the thermal energy of a collapsing cloud, they do accelerate the collapse of a stellar core by dissociating as the cloud heats up and removing a large fraction of the pressure support.
Because of these connections it has been hypothesised, and to some extent empirically shown, that the column density of molecular hydrogen is correlated with the rate of star formation per unit area (i.e. the mass of gas that is being turned into stars per unit time per square parsec). This is usually applied to external galaxies or parts of galaxies and is known as the Kennicutt-Schmidt law, though gas column densities would normally be traced with CO or similar because, as I mentioned above, the H$_2$ gas, although a very major mass component, is essentially invisible.
