# Harvard Astronomy 201b

## Why is CO an important coolant in the (very) cold ISM?

In Uncategorized on March 2, 2011 at 9:56 pm

Cooling mechanisms are very important for facilitating the collapse of molecular clouds, the formation of stars, and radiative equilibrium in the ISM. As discussed in class, molecular hydrogen is a poor radiator as a homonuclear molecule with no dipole moment. This means that H2 can only radiate through forbidden transitions so the rates are too low. Compared to atomic hydrogen, because molecular hydrogen doesn’t even have a 21cm analogue, cooling rates rates for H2 are lower than that for atomic Hydrogen. However as discussed in class, at high temperatures (i.e. shocks) its high abundance makes H2 the dominant coolant. Consequently, heavy molecules play in important role transforming thermal energy to radiation that can escape the region through collisions (mostly with hydrogen) and various emission mechanisms. For the low temperatures of the cold ISM, the potential energy imparted to a CO molecule from an inelastic collision is only enough to excite a rotational transition.

Well, it is both abundant and able to radiate from inelastic collisions at low temperatures and densities.

In general, the cooling rate of a specific is determined by the amount of material, the level populations, and the physics of the transitions:

$\Lambda_{CO} = X_{CO} (2J+1) \frac{e^{-E_J/(k_B T)}}{Z} A_{J,J-1} \Delta E_{J,J-1}$

At the most general level we might try to understand the equation by area of study. Abundances, $X_{CO}$ are determined by the chemistry of the environment as various processes conspire with the temperature, densities, etc. to form molecules. The level populations, $(2J+1) \frac{e^{-E_J/(k_B T)}}{Z}$,are covered by statistical mechanics. To calculate the transition energy, $\Delta E_{J,J-1} = E_J - E_{J-1}$, and Einstein A coefficient, $A_{J,J-1}$, we need to use quantum mechanics.

To answer the question at hand let’s focus on the chemistry and quantum mechanics by first looking at the transition energy. Treating the diatomic molecule CO as a rigid rotator the energy of a particular level, $J$ is set by the rotation constant $B$:

$\Delta E_{J,J-1} = hBJ(J+1) - hBJ(J-1) \propto B$

where $B = \hbar^2/2m_r r_0^2$ where $m_r$ is the reduced mass of the molecule and $r_0$ is the bond length. We’re working in the very cold ISM so we’ll assume that the CO stays in the least energetic vibrational state.

Furthermore, the Carbon and Oxygen molecules in CO are connected by a triple bond with a dissociate energy of  11.2eV (1120 $\AA$.) The strength of this bond means that most starlight won’t break up a CO molecule and so often CO will be the most abundant heavy molecule (Solmon & Klemperer, 1972). Furthermore, CO’s reduced mass is large compared to that of, say, molecular hydrogen. This means that transition energies for these rotational levels are small and so they can be excited at low temperatures (think tens of Kelvin). See sections 5.1.4 and 5.1.7 of Draine’s book for a nice discussion.

Even with the small dipole moment of CO (0.112 Debye), optical depth effects are important. When a line is optically thick the photons just bounce from molecule to molecule and there is no net cooling effect. However, turbulence in the medium and velocity gradients will Doppler shift the photons so they may escape through the line wings. In addition, the geometry of the local medium now matters in the optically thick regime since photons emitted near to the cloud’s ‘boundary’ will be more likely to escape.

In brief, the reasons that CO is such a dominant coolant in the cold ISM is because it is abundant and also because it can radiate at low densities and temperatures.

But what happens when you heat the ISM just a bit?

Molecules besides CO play an important role in line cooling. For instance, water has a much larger dipole moment (1.85 D) than CO (0.112 D) and consequently at higher temperatures water is a dominant cooling mechanism. This can be seen from dipole dependence, $\mu^2$ in the the equation from the Einstein A coefficient:

$A_{J+1,J} = \frac{512 \pi^4 B^3 \mu^2}{3 h c^3} \frac{(J+1)^4}{2J+1}$

Complicating this analysis is the chemistry since the abundance of a species will depend strongly on the local conditions. Of course this complication can be useful when coupling observations and models of the chemistry (i.e. Jiminez-Serra, I et al. 2009).

Abundances of molecules with Oxygen as a function of magnitude for a model of a photon dominated region that includes surface grain chemistry. Figure from Kaufman 2009.

In brief, ion-neutral reactions dominate at the lowest temperatures so the most abundant Oxygen bearing molecules (besides CO) are O, OH, O2, and H2O. At a transition temperature of ~400 K, neutral-neutral reactions produce lots of water:

$O + H_2 \longrightarrow OH + H$
$OH + H_2 \longrightarrow H_2O + H$

By 500 K most of gas-phase oxygen is either in H2O or CO. So in a slightly warmer environment the cooling will be dominated by water.

Cooling rates per molecular Hydrogen molecule for gas temperatures of 40 and 100 K. (Figure 2 from Neufeld, Lepp, and Melnick 1995)

Calculations show that cooling from CO dominates in the low temperature, low density regime of molecular clouds.

Another way of plotting the fractional contribution of various coolants are through contour plots over temperature and density. Effectively this calculated takes the previous figure and expands it to two dimensions. Looking at the figure below the dominance of CO in the low temperature and density regime as well as the transition to water cooling becomes apparent.

Fractions of the total cooling rate from important coolants. Image from the SWASS science page, adapted from Neufeld, Lepp, and Melnick 1995.

So what might this mean to me?

The moral of the story is that if you are doing a calculation involving cooling or heating processes of gas, you’ll want to make sure to treat include coolants such as CO and treat the chemistry and radiative transfer correctly. That is you want the abundances and cooling rate to be right or your answers might be wrong! Some current topics include the formation of stars in low-metallicity regions (i.e. Jappsen et al. 2009) and modeling CO chemistry in Giant Molecular Clouds (i.e. Glover & Clark 2011).

Hopefully this discussion help to answer the question although you’ll have to find the answers to any it brought up your own. Some places to start searching are listed below
References