Sternberg Summary
By Elijah Bernstein-Cooper, March 6, 2015, 0 comments.

This post is a continuation of this post, but don’t bother with the previous post, it’s a poor, short, summary.

## Summary of Sternberg et al. (2014)

The main predictor of $\Sigma_{HI}$ and $\Sigma_{\rm H_2}$ is the ratio of the free space FUV field intensity to the gas density, i.e. dissociation rate / $H_2$ formation rate. Next predictors are metallicity and dust-to-gas mass ratio.

### Breakdown:

The HI to $H_2$ transition profiles and column densities are controlled by the dimensionless parameter $\alpha G$. $\alpha G$ determines the Lyman-Werner optical depth due to dust associated with HI (described as HI-dust). They derive this optical depth as

$$\tau_{1,{\rm tot}} = {\rm log}[\frac{\alpha G}{2} + 1]$$

$\alpha$ is the ratio of free space dissociation rate to the $H_2$ formation rate. G is the cloud-averaged self-shielding factor. Together $\alpha G$ is a measure of the dust-absorption efficiency of the $H_2$-dissociating photons. G depends on the competition of $H_2$ line absorption and $H_2$ dust absorption.

### $H_2$ dissociation

$H_2$ absorbs LW photons, excites from ground electronic state to excited state. Rapid decays occur to either ro-vibrational or continuum state. Decays to continuum state result in dissociation. In the end, the mean dissociation probability is dependent on the total incident LW flux, the dissociation bandwidth, and the mean flux density weighted by dissociation transitions.

### Absorption

The local dissociation rate $D$ at any cloud depth is

where $f_{sheild}$ is the $H_2$ self_shielding function which quantifies the reduction of the total dissociation rate due to opacity in all of the absorption lines. $D_0$ is the free-space dissociation rate. $N_2$ is the column density of H. $\tau_g = \sigma_g N$ is the dust optical depth, where $\sigma_g$ is the dust-gran LW-photon absorption cross section per H nucleon.

Assuming that the DGR mass ratio is proportional to metallicity

where $\phi_g$ depends on the grain composition, $\sim 1$. $Z^\prime$ is the metallicity relative to solar.

### Balancing Absorption + Dissociation = HI + $H_2$ column densities

They assume a formation rate coefficient $R$ per volume for $H_2$ formation on grains, which depends on gas temperature and metallicity. They pair $R$ with the dissociation rate from Equation \ref{eq:diss_rate}. The volume density of $H$, $n$, can be written in terms of the $H_2$ volume density, $n_2$ and HI volume density, $n_1$ as $n = n_1 + 2 n_2$. Balancing absorption with dissociation we get

The relationship between the volume densities does not make sense to me. If we plug in $n$, we get this

so the dissociation rate is scaled by $n_1$? They integrate over the volume densities to get

This shows a key insight that the dust opacities associated with the atomic and molecular columns can be considered separately, despite the mixing of $HI$ and $H_2$.

They define the dimensionless parameter

where $F_\nu$ mean flux density weighted by dissociation transitions of $H_2$ and $\sigma_d^{\rm tot}$ is the total dust cross section. Then they define the dimensionless “G-integral”

where $W_g$ is the $H_2$-dust dissociating bandwidth. We can relate this to the formation rate of