Molecular Lines

What are the molecules are where are they

Molecular clouds are interstellar gas structures that are sufficiently dense to shield their interior against interstellar UV radiation. Inside molecular clouds, molecular hydrogen (H₂) can stably exist rather than being photo-dissociated or photo-ionized (e.g., to atomic hydrogen or H⁺). The total mass of a molecular cloud is dominated by molecular hydrogen. Typically, the molecular gas volume number-density n_H2, i.e., the number of H₂ molecules in a cubic centimeter, is higher than 10³ cm^-3. The gas temperature is usually around 20 K (e.g., Lu et al. 2014; Lin et al. 2017). In some dense starless cores, the gas temperature can be as low as ~10 K (e.g., Crapsi et al. 2007; Lin et al. 2022b). Close to some forming young stars the gas temperature can become higher than >100 K due to radiative heating (e.g., Lin et al. 2022a).

In addition to H₂, various other molecules may form, such as CO, OH, H₂O, NH₃, SO, CH₄, N₂H⁺, HCO⁺, HCN, CS, their isotopologues (e.g., ¹³CO, C¹⁸O, C¹⁷O), and so on. In certain environments, there can also be carbon chain molecules (e.g., HC₃H, HC₇H) and/or organic molecules (e.g., CH₃OH, CH₃OCH₃, etc).

You might have notice that many of them are combinations of the H, and C, N, O atoms. H is the most abundant element since the early universe when all the other elements are rare. C, N, and O can be produced by the nuclear reactions in stars; they can be released to the interstellar medium during the expanding phase of the evolved stars. C, N, and O are involved in relative important reactions and are significantly enriched. Heavier elements can be produced by the nuclear reactions in high-mass stars (i.e., those with stellar masses larger than ~8 solar mass) and then be released to the interstellar medium through supernova explosions.

How to form these molecules is an interesting astrochemical question by itself. Environments with different temperatures, gas density, radiation field, cosmic ray field, and elemental/molecular abundances are prone to form/destruct different molecules. As a result, high angular resolution observations sometimes can resolve a chemical segregation (e.g., Ungerechts et al. 1997; Feng et al. 2015; Feng et al. 2020). For example, in <20 K regions, heavy molecules such as CO may be preferentially absorbed onto the surfaces of interstellar dust. We call that CO is depleted in this case. When CO is depleted in a low-temperature molecular cloud, the volatile molecule N₂H⁺ can be relatively abundant in the gas phase. In a lukewarm region where CO is sublimated and released back to the gas phase, they can interact with N₂H⁺ to form HCO⁺. This chemical reaction reduces the abundance of N₂H⁺ and enhances the abundance of HCO⁺ in the gas phase. With the typical physical condition in a molecular cloud, this chemical reaction has a short enough characteristic timescale to be effective and has significant enough effects to be detected in astronomical observations of molecular lines.

The formation and destruction of the complex organic molecules are tightly related to the origin of life. For a review of this topic, you may check Herbst and van Dishoeck (2009). If you are a physicist who hates chemistry, I found this book Yamamoto (2018) to be very readable (I completed it and it was a very good reading experience for me). To see the connect to the origin of life, you may check Pearce et al. (2017) and references therein.

Besides, if we believe that the Big Bang Nuclear Synthesis (BBNS) theory can derive the abundance of all atomic species precisely, then we have to accept that we have not found all of the carriers of sulfur and lithium. This is a long-standing puzzle. It is relatively difficult to observe lithium from ground-based observatories. It has been popular to observe the sulfur-bearing molecules using the ground-based millimeter observatories and discuss their chemical networks. There used to be debates like whether or not OCS forms on the surfaces of the dust grains is only released to the gas phase when the dust grains are heated or are sputtered in shocks, e.g., Charnley (1997). One of the earliest (to my knowledge) clean cases that spatially resolved enriched OCS in shocks is the Submillimeter array (SMA) observations reported by Liu et al. (2012). The follow-up observations using the Atacama Large Millimeter/submillimeter Array (ALMA) have considerably improve the quality of this image (Minh et al. 2016).

How to observe the interstellar molecules

Here I just superficially mention a few frequently adopted techniques. I may make deeper discussion about some of them in independent chapters.

(*Under construction*)

The formation of molecular lines

Usually, I do not worry so much about chemistry. But even if you do not care about chemistry at all, it is still very important to learn about molecular line emission. The H₂ molecule has no dipole moment such that it cannot emit efficiently. In spite that H₂ dominates the molecular cloud mass, it is not possible for us to directly probe the physical properties of a molecular cloud by observing H₂. Instead, we observe various molecular lines and regard them as tracers of physical quantities in a molecular cloud.

How this works? For a beginner, a good (free) introduction may be Evans (1999)?

A specific molecule in the space (let’s call it X molecule for this moment, which can be CO, SO, CH₃OH, CH₃CN, etc) has a range of energy states (e.g., different in electron configuration or rotational/vibrational motions). It can be pumped from a lower energy state to a higher energy state upon (i) collisions with other molecules, or (ii) by absorbing photons at some specific wavelengths. When an excited X molecule is stimulated to or spontaneously cascade from a high energy state (E=E_up) to a low energy state (E=E_low), it emits a photon that has energy equal to E_up-E_low. These processes are often introduced in the undergraduate-level quantum mechanics classes with a simplified, two-level system and with the Einstein A/B coefficients (e.g., see the wiki page). The emitted photon is what we can observe.

Now, if you have an ensemble of X molecules (i.e., a very large number of them), and if you know what are the energy states of an X molecule, for example, by a quantum mechanical recipe that is similar to the way you calculated the energy states of a helium atom (not that simple of, course), then you can describe how the ensemble of X molecules are populating in the energy states, with an (energy) distribution function. In the language of statistical physics, what we need to know here is the partition function of the X molecule. Knowing the partition function, if we observe multiple line transitions of the X molecule, it may be possible to invert the observed line intensity ratios of these transitions to depict the (energy) distribution function (with some approximation or assumptions, usually).

Deciphering molecular line data

There is a chance that you can make a further step to infer the local physical condition (e.g., gas temperature, volume-density, molecular abundances) from the depicted distribution function (more below). A typical (and fair) assumption is that the dominant collision partners for the X molecule are the molecular and atomic hydrogen gas. Neglecting the collisions between the X molecules and the molecular species other than hydrogen, the energy distribution function of molecule X serves as a proxy to probe the macroscopic properties of the molecular and atomic hydrogen gas mixture (still, dominated by molecular hydrogen in a molecular cloud).

To illustrate how we may infer the local physical condition, we start with an unrealistically simple extreme condition: there is no local radiation field, and n_H2 is arbitrarily close to zero. In this case, all of the X molecules in your ensemble will be in the ground state. These ground state X molecules do not emit. They look dark in the observations. However, if there is strong background continuum emission, you can detect the absorption lines of the X molecule against the background continuum emission. To illustrate how absorption lines may look like, you may check Figure 8 of Liu et al. (2016) which shows the absorption of ground-state atomic carbon against the continuum emission of the Galactic supermassive black hole, Sgr A*. Well, this is not really a molecular line, but an atomic line instead. But the principle is the same. I found this case to be pedagogically elusive since these absorption lines show the same linear polarization as the background light source, the continuum emission of the Sgr A*.

The slightly more complicated case is that you assume the local radiation field is negligible. However, the mean-free-path of the X molecule is so small (i.e., the collisional cross-section is very large, and the hydrogen gas density is very high), such that your ensemble of X molecule reaches thermal equilibrium with the hydrogen gas rapidly. Here, rapid means that the characteristic timescale to reach thermal equilibrium is shorter than the characteristic timescale to cascade from high to low energy states. In this case, we expect the energy distribution function of X molecule to be described by a Maxwellian distribution, of which the characteristic temperature (namely the excitation temperature of the X molecule, T_ex) is identical to the kinetic temperature (T_k) of the hydrogen gas. In such cases, the intensity ratios of the line transitions of the X molecule will have a simple relation with T_ex=T_k. However, in this case, the intensity ratios cannot tell us anything about n_H2. Thermalization means some information has been washed out. We refer to such conditions as local thermodynamic equilibrium (LTE). This is often a good approximation when you are observing the very high-density regions of a molecular cloud.

For a Y molecule which the collisional cross-section (with hydrogen) is not so big, and when hydrogen gas density is not so high, the energy distribution function of the Y molecule may not saturate to a Maxwellian distribution. In this case, the system is not in local thermodynamic equilibrium. We say it is non-LTE. If the collisional cross-section is known, the energy distribution function of the Y molecule can be related to the temperature and density of the hydrogen gas by solving the differential equations that describe the collisional excitation/de-excitation and the radiative transitions (i.e., absorption and spontaneous/stimulated emission) of the Y molecule. Usually, there are a set of such differential equations. Out of these equations, what we are looking for is the equilibrium energy distribution function: for each energy state, the rate of removing Y molecules from that state (e.g., by collisional and radiative excitations to higher energy states, and by spontanous, stimulate, or collisional de-excitations to lower energy states) is equal to the rate of transiting to that state from other energy states.

Given a certain hydrogen gas temperature, it is less easy to pump a molecule from a high excitation state to a further higher excitation state through collisions. Therefore, the higher excitation (E_up) transitions of a molecule are usually less close to LTE, thus can provide more information about the hydrogen gas density. By simultaneously observing multiple transitions, it is sometimes possible to constrain hydrogen gas temperature and volume density.

Note that in the non-LTE systems, the molecular line emission emitted from a X molecule can pump an adjacent X molecule to a high excitation state if the relative velocity between these two X molecules is small. If their relative velocity is large, then the emitted photon from one X molecule will be too redshifted/blueshifted to be absorbed by the other. In this case, the emitted photon from the X molecule can escape to a larger distance. To determine the probability for a photon emitted by a X molecule to escape a local parcel of a molecular cloud, we need the information of velocity gradient or turbulent linewidth in that parcel. There often needs some assumptions to approximately evaluate the escape probability of photons at certain wavelengths, e.g., Sobolev (1957).

In principle, the non-LTE fittings of the observed line intensity ratios (of one or multiple molecules) can derive the hydrogen gas temperature, density, and linewidth. The derived linewidths not only have to reproduce the observed line intensity ratios, but also need to be consistent with the actually observed spectral line profiles, for example, see Figure 13 of Lin et al. (2022). In the applications to the observations towards high-mass star-forming regions, this is an ultra-complicated global radiative transfer problem which is illustrated by Figures 1 and 3 of the exquisite work of Lin et al. (2022).

This task is further complicated by that each of the molecular species is a biased tracer for certain chemical/physical condition. To reconstruct the 3D distribution of gas temperature/density in a star-forming region, we still need some knowledge about astrochemistry and perhaps some experiences. Figure 11 of Giannetti et al. (2017) may be a good illustration of what CH₃CN, CH₃OH, CO, and NH₃ are tracing around a high-mass star-forming core.

Finally, by far, we actually do not know the detailed energy states of all of the interstellar molecules. There need some quantum mechanical calculations and laboratory measurements. The energy states of NH₃ can be calculated exactly thus the observations of NH₃ have been widely applied since the late 1970s (for a review see Ho and Townes 1983). Linear molecules like CO, CH₃CN, and CH₃CCH are also relatively well understood. Examples of observing some of these molecules in high-mass star-forming regions are Liu et al. (2015), and Law et al. (2021). In contrast, it became possible to utilize CH₃OH as a tracer only very recently, e.g., . Leurini et al. (2004).