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One often hears that $\text{CO}$ is a good tracer for $\text{H}_2 .$ How are they correlated? How can you deduce from the (measurable) $\text{CO}$ the amount of the (unmeasurable) $\text{H}_2$ in the interstellar medium?

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    $\begingroup$ "One always reads and hears that CO is a good tracer for H2." - I have never read or heard that in my life, and I don't even know what it means. Also, what does "ISM" mean? $\endgroup$ – ACuriousMind Dec 15 '15 at 13:55
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    $\begingroup$ @ACuriousMind Standard astro parlance for the interstellar medium. The ratio is denoted $X_\mathrm{CO}$ and is notorious for being both unjustified and extensively used. See e.g. arxiv.org/abs/1301.3498 $\endgroup$ – user10851 Dec 15 '15 at 16:21
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Answering this question inevitably leads to stumble upon one of the most controversial topics in astronomy. First let’s give some context here by reminding why tracers of $\mathrm{H_2}$ are required in the first place:

Molecular hydrogen – or $\mathrm{H_2}$ – is the most abundant molecule in the universe, hence it embodies what is called the dense molecular phase of the interstellar medium (ISM), in which matter aggregates into massive, cold dark clouds. Thus, the distribution of $\mathrm{H_2}$ is a key observable when astronomers want to characterize these clouds. 

So what’s the issue ?

Unfortunately, the most profuse molecule of the universe is almost invisible. Indeed, $\mathrm{H_2}$ is a simple, homonuclear molecule with no permanent dipole moment, thus only quadrupole rotational transitions can occur. These transitions of relatively high energy $h \nu$ allow only to probe warm regions (Considering that $h \nu/k = 514~\mathrm{K}$ for the first pure rotational transition of $\mathrm{H_2}$, and $h \nu/k = 1200~\mathrm{K}$ for the second one) which account only for a tiny proportion of the total mass present in the interstellar medium.

This means that virtually no photons are emitted by molecular hydrogen lying in cold dark clouds, because there are no transitions available at their typical temperature of $10$ to $30~\mathrm{K}$.

Note: alternatively, $\mathrm{H_2}$ can be directly observed by absorption at Far-UV wavelengths, but this is only possible in the diffuse interstellar medium, along sight lines toward nearby stars.

This is the very reason why tracers are required to determine column densities of $\mathrm{H_2}$ in the interstellar medium.

Then, why use $\mathrm{CO}$ ?

Carbon monoxide happens to be the second most abundant molecule in the universe, right behind $\mathrm{H_2}$. Unlike molecular hydrogen, it does have a dipolar moment, hence it is easily observed thanks to its numerous pure rotational transitions, in particular thanks to the $J=1-0$ transition which is associated with a temperature $h \nu/k=5,53~\mathrm{K}$, allowing to trace the cold phase of the gas. Its transitions, lying in millimeter wavelengths, can be observed and mapped with good precision with a large number of ground based radio-telescopes. Therefore, $\mathrm{CO}$ can be considered a very astronomer-friendly molecule when it comes to probe the most condensed parts of the interstellar medium, and it is indeed extensively used.

But how does this relate to $\mathrm{H_2}$ ?

This is where questionable assumptions start to show up. Since the formation of molecular species like $\mathrm{CO}$ occurs under conditions favorable for $\mathrm{H_2}$ formation, the idea is to guess that a measure of $\mathrm{CO}$ column density can be directly converted into a $\mathrm{H_2}$ column density for a given region, using some empirically determined $\mathrm{CO} / \mathrm{H_2}$ ratio.

Over the years, many reviews and papers concerning the relative abundances of $\mathrm{H_2}$ and $\mathrm{CO}$ in various regions permitted to propose a measure of this ratio, believed to be around the so-called canonical value $10^{-4}$. This remarkable result allows to directly relate $N_{\mathrm{H_2}}$ to $N_{\mathrm{CO}}$, thanks to the following equation:

$$N_{\mathrm{H_2}} = {10}^{4} N_{\mathrm{CO}}$$

However, the reliability of this quantitative correlation is highly debated, and remains an active research topic both in astrochemistry (eg. this paper) and observational astrophysics (e.g. this paper, or this paper). In particular, there are concerns about the possibility that there might be a strong discrepancy in environments different from the Milky Way, where the result has been established.

In fact, there are many observational evidence suggesting that the value adopted for the $\mathrm{CO} / \mathrm{H_2}$ ratio is not valid (at all) everywhere. (e.g. this paper) This makes sense as there are no reasons to believe that the ratio would remain constant under different environmental conditions in which a molecular cloud is placed (local density, metallicity, cosmic ray ionisation rate, interstellar radiation field leading to photodissociation, magnetized turbulence, etc.)

Anyway, astronomers use extensively this ratio to infer the local density of the molecular gas from observations of $\mathrm{CO}$ transitions, because it is so convenient when a quick estimate is required.

In fact, this is a widely spread issue in astronomy, as most of what we know about interstellar molecules comes from observations of so-called « tracer » species, in which a lot of trust is put.

Addendum : Could we predict this ratio, using methods other than empirical determination by direct observations ?

To some extent, the $\mathrm{CO}/\mathrm{H_2}$ ratio could be expected to be found in the scope of $10^{-4}$ using what is known about abundances of the chemical elements in the interstellar medium.

Indeed, the abundances of $\mathrm{C}$ and $\mathrm{O}$ relatively to $\mathrm{H}$ are well known by astronomers:

1) On one hand, the abundance of hydrogen can be determined considering the production of nuclei during primordial nucleosynthesis.

2) On the other hand, abundances of oxygen and carbon can be measured from spectroscopic observations toward statistical samples of stars, where stellar nucleosynthesis constrains the cosmic abundances of heavier elements.

In the literature typical abundances of the chemical elements are the following for carbon and oxygen:

$$n_{\mathrm{C}}/n_{\mathrm{H}} \approx 3 \cdot 10^{-4}$$

$$n_{\mathrm{O}}/n_{\mathrm{H}} \approx 5 \cdot 10^{-4}$$

Hence, assuming that for each couple of hydrogen atoms forming $\mathrm{H_2}$ an atom of carbon and an atom of oxygen form $\mathrm{CO}$, you would find a $\mathrm{CO}/\mathrm{H_2}$ ratio that is in agreement with what is found empircally.

Warning : Is that really so simple ? No ! This is a very simplistic view, and by no mean a legitimate justification or proof that the $\mathrm{CO}/\mathrm{H_2}$ ratio should always be around $10^{-4}$. But this gives an idea of what can be done to infer the ratio from abundances of chemical elements.

A proper analysis would rely on taking into account the different rates of formation of $\mathrm{H_2}$ and $\mathrm{CO}$ both in the gas phase and on dust grain surfaces, and also the destruction rates of these molecules by photodissociation, chemical reactions and depletion of the gas into the solid phase of dust grains. This implies a full physical description of the different processes and use of a complete chemical network. Obviously, these methods are hard to establish because the results are very sensitive to the accuracy and completeness with which physical mechanisms are modelled. Nonetheless, what is found with these models is that the $\mathrm{CO}/\mathrm{H_2}$ varies largely with local conditions such as metallicity and ionization rates (see this paper for a clear example of that kind of astrochemical study.)

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    $\begingroup$ Interesting. But there should be other molecules (CHx, OHx) with which one could check if the estimate is more or less reliable? $\endgroup$ – Pieter Nov 25 '18 at 22:15

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