# Harvard Astronomy 201b

## CHAPTER: Excitation Processes: Collisions

In Book Chapter on March 7, 2013 at 3:18 pm

(updated for 2013)

Collisional coupling means that the gas can be treated in the fluid approximation, i.e. we can treat the system on a macrophysical level.

Collisions are of key importance in the ISM:

• cause most of the excitation
• can cause recombinations (electron + ion)

Three types of collisions

1. Coulomb force-dominated ($r^{-1}$ potential): electron-ion, electron-electron, ion-ion
2. Ion-neutral: induced dipole in neutral atom leads to $r^{-4}$ potential; e.g. electron-neutral scattering
3. neutral-neutral: van der Waals forces -> $r^{-6}$ potential; very low cross-section

We will discuss (3) and (2) below; for ion-electron and ion-ion collisions, see Draine Ch. 2.

In general, we will parametrize the interaction rate between two bodies A and B as follows:

${\frac{\rm{reaction~rate}}{\rm{volume}}} = <\sigma v>_{AB} n_a n_B$

In this equation, $<\sigma v>_{AB}$ is the collision rate coefficient in $\rm{cm}^3 \rm{s}^{-1}. <\sigma v>_{AB}= \int_0^\infty \sigma_{AB}(v) f_v~dv$, where $\sigma_{AB} (v)$ is the velocity-dependent cross section and $f_v~dv$ is the particle velocity distribution, i.e. the probability that the relative speed between A and B is v. For the Maxwellian velocity distribution,

$f_v~dv = 4 \pi \left(\frac{\mu'}{2\pi k T}\right)^{3/2} e^{-\mu' v^2/2kT} v^2~dv$,

where $\mu'=m_A m_B/(m_A+m_B)$ is the reduced mass. The center of mass energy is $E=1/2 \mu' v^2$, and the distribution can just as well be written in terms of the energy distribution of particles, $f_E dE$. Since $f_E dE = f_v dv$, we can rewrite the collision rate coefficient in terms of energy as

$\sigma_{AB}=\left(\frac{8kT}{\pi\mu'}\right)^{1/2} \int_0^\infty \sigma_{AB}(E) \left(\frac{E}{kT}\right) e^{-E/kT} \frac{dE}{kT}$.

These collision coefficients can occasionally be calculated analytically (via classical or quantum mechanics), and can in other situations be measured in the lab. The collision coefficients often depend on temperature. For practical purposes, many databases tabulate collision rates for different molecules and temperatures (e.g., the LAMBDA databsase).

For more details, see Draine, Chapter 2. In particular, he discusses 3-body collisions relevant at high densities.

## CHAPTER: Introductory Remarks on Radiative Processes

In Book Chapter on February 28, 2013 at 3:10 am

(updated for 2013)

The goal of the next several sections is to build an understanding of how photons are produced by, are absorbed by, and interact with the ISM. We consider a system in which one or more constituents are excited under certain physical conditions to produce photons, then the photons pass through other constituents under other conditions, before finally being observed (and thus affected by the limitations and biases of the observational conditions and instruments) on Earth. Local thermodynamic equilibrium is often used to describe the conditions, but this does not always hold. Remember that our overall goal is to turn observations of the ISM into physics, and vice-versa.

The following contribute to an observed Spectral Energy Distribution:

• gas: spontaneous emission, stimulated emission (e.g. masers), absorption, scattering processes involving photons + electrons or bound atoms/molecules
• dust: absorption; scattering (the sum of these two -> extinction); emission (blackbody modified by wavelength-dependent emissivity)
• other: synchrotron, brehmsstrahlung, etc.

The processes taking place in our “system” depend sensitively on the specific conditions of the ISM in question, but the following “rules of thumb” are worth remembering:

1. Very rarely is a system actually in a true equilibrium state.
2. Except in HII regions, transitions in the ISM are usually not electronic.
3. The terms Upper Level and Lower Level refer to any two quantum mechanical states of an atom or molecule where $E_{\rm upper}>E_{\rm lower}$. We will use k to index the upper state, and j for the lower state.
4. Transitions can be induced by photons, cosmic rays, collisions with atoms and molecules, and interactions with free electrons.
5. Levels can refer to electronic, rotational, vibrational, spin, and magnetic states.
6. To understand radiative processes in the ISM, we will generally need to know the chemical composition, ambient radiation field, and velocity distribution of each ISM component. We will almost always have to make simplifying assumptions about these conditions.

## CHAPTER: Measuring States in the ISM

In Book Chapter on February 26, 2013 at 3:00 am

(updated for 2013)

There are two primary observational diagnostics of the thermal, chemical, and ionization states in the ISM:

1. Spectral Energy Distribution (SED; broadband low-resolution)
2. Spectrum (narrowband, high-resolution)

#### SEDs

Very generally, if a source’s SED is blackbody-like, one can fit a Planck function to the SED and derive the temperature and column density (if one can assume LTE). If an SED is not blackbody-like, the emission is the sum of various processes, including:

• thermal emission (e.g. dust, CMB)
• synchrotron emission (power law spectrum)
• free-free emission (thermal for a thermal electron distribution)

#### Spectra

Quantum mechanics combined with chemistry can predict line strengths. Ratios of lines can be used to model “excitation”, i.e. what physical conditions (density, temperature, radiation field, ionization fraction, etc.) lead to the observed distribution of line strengths. Excitation is controlled by

• collisions between particles (LTE often assumed, but not always true)
• photons from the interstellar radiation field, nearby stars, shocks, CMB, chemistry, cosmic rays
• recombination/ionization/dissociation

Which of these processes matter where? In class (2011), we drew the following schematic.

A schematic of several structures in the ISM

Key

A: Dense molecular cloud with stars forming within

• $T=10-50~{\rm K};~n>10^3~{\rm cm}^{-3}$ (measured, e.g., from line ratios)
• gas is mostly molecular (low T, high n, self-shielding from UV photons, few shocks)
• not much photoionization due to high extinction (but could be complicated ionization structure due to patchy extinction)
• cosmic rays can penetrate, leading to fractional ionization: $X_I=n_i/(n_H+n_i) \approx n_i/n_H \propto n_H^{-1/2}$, where $n_i$ is the ion density (see Draine 16.5 for details). Measured values for $X_e$ (the electron-to-neutral ratio, which is presumed equal to the ionization fraction) are about $X_e \sim 10^{-6}~{\rm to}~10^{-7}$.
• possible shocks due to impinging HII region – could raise T, n, ionization, and change chemistry globally
• shocks due to embedded young stars w/ outflows and winds -> local changes in Tn, ionization, chemistry
• time evolution? feedback from stars formed within?

B: Cluster of OB stars (an HII region ionized by their integrated radiation)

• 7000 < T < 10,000 K (from line ratios)
• gas primarily ionized due to photons beyond Lyman limit (E > 13.6 eV) produced by O stars
• elements other than H have different ionization energy, so will ionize more or less easily
• HII regions are often clumpy; this is observed as a deficit in the average value of $n_e$ from continuum radiation over the entire region as compared to the value of ne derived from line ratios. In other words, certain regions are denser (in ionized gas) than others.
• The above introduces the idea of a filling factor, defined as the ratio of filled volume to total volume (in this case the filled volume is that of ionized gas)
• dust is present in HII regions (as evidenced by observations of scattered light), though the smaller grains may be destroyed
• significant radio emission: free-free (bremsstrahlung), synchrotron, and recombination line (e.g. H76a)
• chemistry is highly dependent on nT, flux, and time

C: Supernova remnant

• gas can be ionized in shocks by collisions (high velocities required to produce high energy collisions, high T)
• e.g. if v > 1000 km/s, T > 106 K
• atom-electron collisions will ionize H, He; produce x-rays; produce highly ionized heavy elements
• gas can also be excited (e.g. vibrational H2 emission) and dissociated by shocks

D: General diffuse ISM

• ne best measured from pulsar dispersion measure (DM), an observable. ${\rm DM} \propto \int n_e dl$
• role of magnetic fields depends critically on XI(B-fields do not directly affect neutrals, though their effects can be felt through ion-neutral collisions)

## ARTICLE: The Physical State of Interstellar Hydrogen

In Journal Club 2013 on February 12, 2013 at 9:57 pm

The Physical State of Interstellar Hydrogen by Bengt Strömgren (1939)

Summary by Anjali Tripathi

Abstract

In 1939, Bengt Strömgren published an analytic formulation for the spatial extent of ionization around early type stars.  Motivated by new H-alpha observations of sharply bound “diffuse nebulosities,” Strömgren was able to characterize these ionized regions and their thin boundaries in terms of the ionizing star’s properties and abundances of interstellar gas.  Strömgren’s work on these regions, which have come to be eponymously known as Strömgren spheres, has found longstanding use in the study of HII regions, as it provides a simple analytic approach to recover the idealized properties of such systems.

Background: Atomic Physics in Astronomy & New Observations

Danish astronomer Bengt Strömgren (1908-87) was born into a family of astronomers and educated during a period of rapid development in our understanding of the atom and modern physics.  These developments were felt strongly in Copenhagen where Strömgren studied and worked for much of his life.  At the invitation of Otto Struve, then director of Yerkes Observatory, Strömgren visited the University of Chicago from 1936 to 1938, where he encountered luminaries from across astrophysics, including Chandrasekhar and Kuiper.  With Struve and Kuiper, Strömgren worked to understand how photoionization could explain observations of a shell of gas around an F star, part of the eclipsing binary $\epsilon$ Aurigae (Kuiper, Struve and Strömgren, 1937).  This work laid out the analytic framework for a bounded region of ionized gas around a star, which provided the theoretical foundation for Strömgren’s later work on HII regions.

The observational basis for Strömgren’s 1939 paper came from new spectroscopic measurements taken by Otto Struve.  Using the new 150-Foot Nebular Spectrograph (Struve et al, 1938) perched on a slope at McDonald Observatory, pictured below, Struve and collaborators were able to resolve sharply bound extended regions “enveloped in diffuse nebulosities” in the Balmer H-alpha emission line (Struve and Elvey, 1938).  This emission line results from recombination when electrons transition from the n = 3 to n = 2 energy level of hydrogen, after the gas was initially ionized by UV radiation from O and B stars.  Comparing these observations to those of the central parts of the Orion Nebula led the authors to estimate that the number density of hydrogen with electrons in the n=3 state is $N_3 = 3 \times 10^{-21} cm^{-3}$, assuming a uniform concentration of stars and neglecting self-absorption (Struve and Elvey, 1938).  From his earlier work on $\epsilon$ Aurigae, Strömgren had an analytic framework with which to understand these observations.

Instrument used to resolve HII Regions in H-alpha (Struve et al, 1938)

Putting it together – Strömgren’s analysis

To understand the new observations quantitatively, Strömgren worked out the size of these emission nebulae by finding the extent of the ionized gas around the central star.  As in his paper with Kuiper and Struve, Strömgren considered only neutral and ionized hydrogen, assumed charge neutrality, and used the Saha equation with additional terms:

${N'' N_e \over N'} = \underbrace{{(2 \pi m_e)^{3/2} \over h^3} {2q'' \over q'} (kT)^{3/2}e^{-I/kT}}_\text{Saha} \cdot \underbrace{\sqrt{T_{el} \over T}}_\text{Temperature correction} \cdot \underbrace{R^2 \over 4 s^2}_\text{Geometrical Dilution}\cdot \underbrace{e^{-\tau_u}}_\text{Absorption}\\ N': \text{Neutral hydrogen (HI) number density}\\ N'':\text{Ionized hydrogen (HII) number density}\\ N_e:\text{Electron number density, }N_e = N''\text{ by charge neutrality}\\ x: \text{Ionization fraction}, x = N''/(N'+N'')$

Here, the multiplicative factor of $\sqrt{T_{el} \over T}$ corrects for the difference between the stellar temperature($T$) and the electron temperature($T_{el}$) at a distance $s$ away from the star.  The dilution factor ${R^2 \over 4 s^2}$, where $R$ is the stellar radius and $s$ is the distance from the star, accounts for the decrease in stellar flux with increasing distance.  The factor of $e^{-\tau_u}$, where $\tau_u$ is the optical depth, accounts for the reduction in the ionizing radiation due to absorption.  Taken together, this equation encapsulates the physics of a star whose photons ionize surrounding gas.  This ionization rate is balanced by the rate of recombination of ions and electrons to reform neutral hydrogen.  As a result, close to the star where there is abundant energetic flux, the gas is fully ionized, but further from the star, the gas is primarily neutral.  Strömgren’s formulation allowed him to calculate the location of the transition from ionized to neutral gas and to find the striking result that the transition region between the two is incredibly sharp, as plotted below.

Plot of ionization fraction vs. distance for an HII Region (Values from Table 2 of Strömgren, 1939)

Strömgren found that the gas remains almost completely ionized until a critical distance $s_0$, where the ionization fraction sharply drops and the gas becomes neutral due to absorption.  This critical distance has become known as the Strömgren radius, considered to be the radius of an idealized, spherical HII region.  The distance over which the ionization fraction drops from 1 to 0 is small (~0.01 pc), corresponding to one mean free path of an ionizing photon, compared to the Strömgren radius(~100pc).  Thus Strömgren’s analytic work provided an explanation for sharply bound ionized regions with thin transition zones separating the ionized gas from the exterior neutral gas.

Strömgren also demonstrated how the critical distance depends on the total number density $N$, the stellar effective temperature $T$, and the stellar radius $R$:

$\log{s_0} = -6.17 + {1 \over 3} \log{ \left( {2q'' \over q'} \sqrt{T_{el} \over T} \right)} - {1 \over 3} \log{a_u} - {1 \over 3} \frac{5040K}{T} I + {1 \over 2} \log{T} + {2 \over 3} \log{R} - {2 \over 3} \log{N},$

where $a_u$ is the absorption coefficient for the ionizing radiation per hydrogen atom (here assumed to be frequency independent) and $s_0$ is given in parsecs.  From this relation, we can see that for a given stellar radius and a fixed number density, $s_0 \propto T^{1/2}$, so that hotter, earlier type stars have larger ionized regions.  Plugging in numbers, Strömgren found that for a total number density of $3~cm^{-3}$, a cluster of 10 O7 stars would have a critical radius of 100-150 parsecs, in agreement with estimates made by the Struve and Elvey observations.

To estimate the hydrogen number density from the H-alpha observations, Strömgren also considered the excitation of the n=3 energy levels of hydrogen.  Weighing the relative importance of various mechanisms for excitation – free electron capture, Lyman-line absorption, Balmer-line absorption, and collisions – Strömgren found that their effects on the number densities of the excited states and electron number densities were comparable.  As a result, he estimated from Struve’s and Elvey’s $N_3$ that the number density of hydrogen is 2-3 $cm^{-3}$.

Strömgren’s analysis of ionized regions around stars and neutral hydrogen in “normal regions” matched earlier theoretical work by Eddington into the nature of the ISM (Eddington, 1937).  “With great diffidence, having not yet rid myself of the tradition that ‘atoms are physics, but molecules are chemistry’,” Eddington wrote that “presumably a considerable part” of the ISM is molecular.  As a result, Strömgren outlined how his analysis for ionization regions could be modified to consider regions of molecular hydrogen dissociating, presciently leaving room for the later discovery of an abundance of molecular hydrogen.  Instead of the ionization of atomic hydrogen, Strömgren worked with the dissociation of molecular hydrogen in this analysis.   Given that the energy required to dissociate the bond of molecular hydrogen is less than that required to ionize atomic hydrogen, Stromgren’s analysis gives a model of a star surrounded by ionized atoms, which is surrounded by a sharp, thin transition region of atomic hydrogen, around which molecular hydrogen remains.

In addition to HI and HII, Strömgren also considered the ionization of other atoms and transitions.  For example, Strömgren noted that if the helium abundance was smaller than that of hydrogen, then most all of the helium will be ionized out to the boundary of the hydrogen ionization region.  From similar calculations and considering the observations of Struve and Elvey, Strömgren was able to provide an estimate of the abundance of OII, a ratio of $10^{-2}-10^{-3}$ oxygen atoms to each hydrogen atom.

Strömgren Spheres Today

Strömgren’s idealized formulation for ionized regions around early type stars was well received initially and  has continued to influence thinking about HII regions in the decades since.  The simplicity of Strömgren’s model and its assumptions, however, have been recognized and addressed over time.  Amongst these are concerns about the assumption of a uniformly dense medium around the star.  Optical and radio observations, however, have revealed that the surrounding nebula can have clumps and voids – far from being uniformly dense (Osterbrock and Flather, 1959).  To address this, calculations of the nebula’s density can include a ‘filling factor’ term.  Studies of the Orion Nebula (M42), pictured below, have provided examples of just such clumpiness.  M42 has also been used to study another related limitation of Strömgren’s model – the assumption of a central star surrounded by spherical symmetry.

Orion Nebula, infrared image from WISE. Credit: NASA/JPL/Caltech

Consideration of the geometry of Strömgren spheres has been augmented by blister models of the 1970s whereby a star ionizes surrounding gas but the star is at the surface or edge of a giant molecular cloud (GMC), rather than at the center of it.  As a result, ionized gas breaks out of the GMC, like a popping blister, which in turn can prompt “champagne flows” of ionized gas leaching into the surrounding medium.  In a review article of Strömgren’s work, Odell (1999) states that due to observational selection effects, many HII regions observed in the optical actually are more akin to blister regions, rather than Strömgren spheres, since Strömgren spheres formed at the center or heart of a GMC may be obscured so much that they are observable only at radio wavelengths.

In spite of its simplifying assumptions, Strömgren’s work remains relevant today.   Given its abundance, hydrogen dominates the physical processes of emission nebulae and, thus, Strömgren’s idealized model provides a good first approximation for the ionization structure, even though more species are involved than just atomic hydrogen.  Today we can enhance our understanding of these HII regions using computer codes, such as CLOUDY, to calculate the ionization states of various atoms and molecules.  We can also computationally model the  hydrodynamics  of shocks radiating outwards from the star and use spectral synthesis codes to produce mock spectra.  From these models and the accumulated wealth of observations over time, we have come to accept that dense clouds of molecular gas, dominated with molecular hydrogen, are the sites of star formation.  Young O and B-type stars form out of clumps in these clouds and their ionizing radiation will develop into an emission nebula with ionized atomic hydrogen, sharply bound from the surrounding neutral cloud.  As the stars age and the shocks race onwards, the HII regions will evolve.  What remains, however, is Strömgren’s work which provides a simple analytic basis for understanding the complex physics of HII regions.

Strömgren, “The Physical State of Interstellar Hydrogen”, ApJ (1939)

Kuiper et al, “The Interpretation of $\epsilon$  Aurigae”, ApJ (1937)

Struve et al, “The 150-Foot Nebular Spectrograph of the McDonald Observatory”, ApJ (1938)

Eddington, “Interstellar Matter”, The Observatory (1937)

Osterbrock and Flather, “Electron Densities in the Orion Nebula. II”, ApJ (1959)

O’Dell, “Strömgren Spheres”, ApJ (1999)

## CHAPTER: Topology of the ISM

In Book Chapter on February 6, 2013 at 9:57 pm

(updated for 2013)

A grab-bag of properties of the Milky Way

• HII scale height: 1 kpc
• CO scale height: 50-75 pc
• HI scale height: 130-400 pc
• Stellar scale height: 100 pc in spiral arm, 500 pc in disk
• Stellar mass: $5 \times 10^{10} M_\odot$
• Dark matter mass: $5 \times 10^{10} M_\odot$
• HI mass: $2.9 \times 10^9 M_\odot$
• H2 mass (inferred from CO): $0.84 \times 10^9 M_\odot$
• HII mass: $1.12 \times 10^9~M_\odot$
• -> total gas mass $= 6.7 \times 10^9~M_\odot$ (including He).
• Total MW mass within 15 kpc: $10^{11} M_\odot$ (using the Galaxy’s rotation curve). About 50% dark matter.

So the ISM is a relatively small constituent of the Galaxy (by mass).

The Milky Way is a very thin disk (think a CD with a ping-pong ball in the middle) In class (2011), we drew the following schematic of the “topology” of phases in the ISM.

A schematic of several structures in the ISM

## CHAPTER: Bruce Draine’s List of Constituents of the ISM

In Book Chapter on February 5, 2013 at 9:09 pm

(updated for 2013)

1. Gas
2. Dust
3. Cosmic Rays*
4. Photons**
5. B-Field
6. Gravitational Field
7. Dark Matter

*cosmic rays are highly relativistic, super-energetic ions and electrons

**photons include:

• The Cosmic Microwave Background (2.7 K)
• starlight from stellar photospheres (UV, optical, NIR,…)
• $h\nu$ from transitions in atoms, ions, and molecules
• “thermal emission” from dust (heated by starlight, AGN)
• free-free emission (bremsstrahlung) in plasma
• synchrotron radiation from relativistic electrons
• $\gamma$-rays from nuclear transitions

His list of “phases” from Table 1.3:

1. Coronal gas (Hot Ionized Medium, or “HIM”): $T> 10^{5.5}~{\rm K}$. Shock-heated from supernovae. Fills half the volume of the galaxy, and cools in about 1 Myr.
2. HII gas: Ionized mostly by O and early B stars. Called an “HII region” when confined by a molecular cloud, otherwise called “diffuse HII”.
3. Warm HI (Warm Neutral Medium, or “WNM”): atomic, $T \sim 10^{3.7}~{\rm K}$. $n\sim 0.6 ~{\rm cm}^{-3}$. Heated by starlight, photoelectric effect, and cosmic rays. Fills ~40% of the volume.
4. Cool HI (Cold Neutral Medium, or “CNM”). $T \sim 100~{\rm K}, n \sim 30 ~{\rm cm}^{-3}$. Fills ~1% of the volume.
5. Diffuse molecular gas. Where HI self-shields from UV radiation to allow $H_2$ formation on the surfaces of dust grains in cloud interiors. This occurs at $10~{\rm to}~50~{\rm cm}^{-3}$.
6. Dense Molecular gas. “Bound” according to Draine (though maybe not). $n >\sim 10^3 ~{\rm cm}^{-3}$. Sites of star formation.  See also Bok Globules (JC 2013).
7. Stellar Outflows. $T=50-1000 {\rm K}, n \sim 1-10^6 ~{\rm cm}^{-3}$. Winds from cool stars.

These phases are fluid and dynamic, and change on a variety of time and spatial scales. Examples include growth of an HII region, evaporation of molecular clouds, the interface between the ISM and IGM, cooling of supernova remnants, mixing, recombination, etc.

## CHAPTER: Composition of the ISM

In Book Chapter on February 5, 2013 at 9:03 pm

(updated for 2013)

• Gas: by mass, gas is 60% Hydrogen, 30% Helium. By number, gas is 88% H, 10% He, and 2% heavier elements
• Dust: The term “dust” applies roughly to any molecule too big to name. The size distribution is biased towards small (0.2 $\mu$m) particles, with an approximate distribution $N(a) \propto a^{-3.5}$. The density of dust in the galaxy is $\rho_{\rm dust} \sim .002 M_\odot ~{\rm pc}^{-3} \sim 0.1 \rho_{\rm gas}$
• Cosmic Rays: Charged, high-energy (anti)protons, nuclei, electrons, and positrons. Cosmic rays have an energy density of $0.5 ~{\rm eV ~ cm}^{-3}$. The equivalent mass density (using E = mc^2) is $9 \times 10^{-34}~{\rm g cm}^{-3}$
• Magnetic Fields: Typical field strengths in the MW are $1 \mu G \sim 0.2 ~{eV ~cm}^{-3}$. This is strong enough to confine cosmic rays.

## ARTICLE: Turbulence and star formation in molecular clouds

In Journal Club, Journal Club 2013, Uncategorized on February 5, 2013 at 4:43 pm

### Read the Paper by R.B. Larson (1981)

Summary by Philip Mocz

### Abstract

Data for many molecular clouds and condensations show that the internal velocity dispersion of each region is well correlated with its size and mass, and these correlations are approximately of power-law form. The dependence of velocity dispersion on region size is similar to the Kolmogorov law for subsonic turbulence, suggesting that the observed motions are all part of a common hierarchy of interstellar turbulent motions. The regions studied are mostly gravitationally bound and in approximate virial equilibrium. However, they cannot have formed by simple gravitational collapse, and it appears likely that molecular clouds and their substructures have been created at least partly by processes of supersonic hydrodynamics. The hierarchy of subcondensations may terminate with objects so small that their internal motions are no longer supersonic; this predicts a minimum protostellar mass of the order of a few tenths of a solar mass. Massive ‘protostellar’ clumps always have supersonic internal motions and will therefore develop complex internal structures, probably leading to the formation of many pre-stellar condensation nuclei that grow by accretion to produce the final stellar mass spectrum. Molecular clouds must be transient structures, and are probably dispersed after not much more than $10^7$ yr.

How do stars form in the ISM? The simple theoretical picture of Jeans collapse — that a large diffuse uniform cloud starts to collapse and fragments into a hierarchy of successively smaller condensations as the density rises and the Jeans mass decreases — is NOT consistent with observations. Firstly, astronomers see complex structure in molecular clouds:  filaments, cores, condensations, and structures suggestive of hydrodynamical processes and turbulent flows. In addition, the data presented in this paper shows that the observed linewidths of regions in molecular clouds are far from thermal. The observed line widths suggest that ISM is supersonically turbulent on all but the smallest scales. The ISM stages an interplay between self-gravity, turbulent (non-thermal) pressure, and feedback from stars (with the fourth component, thermal pressure, not being dominant on most scales). Larson proposes that protostellar cores are created by supersonic turbulent compression, which causes density fluctuations, and gravity becomes dominant  in only the densest (and typically subsonic) parts, making them unstable to collapse. Larson uses internal velocity dispersion measurements of regions in molecular clouds from the literature to support his claim.

### Key Observational Findings:

Data for regions in molecular clouds with scales $0.1 pc follow:

(1) A power-law relation between velocity dispersion σ  and the size of the emitting region, $L$:

$\sigma \propto L^{0.38}$

Such power-law forms are typical of turbulent velocity distributions. More detailed studies today find $\sigma\propto L^{0.5}$, suggesting compressible, supersonic Burger’s turbulence.

(2) Approximate virial equilibrium:

$2GM/(\sigma^2L)\sim 1$

meaning the regions are roughly in self-gravitational equilibrium.

(3) An inverse relation between average molecular hydrogen $H_2$ density, $n$, and length scale $L$:

$L \propto n^{-1.1}$

which means that the column density $nL$ is nearly independent of size, indicative of 1D shock-compression processes which preserve column densities.

Note These three laws are not independent. They are algebraically linked: that is, any one law can be derived from the other two. The three laws are consistent.

### The Data

Larson compiles data on individual molecular clouds, clumps, and density enhancements of larger clouds from previous radio observations in the literature. The important parameters are:

• $L$, the maximum linear extent of the region
• variation of the radial velocity $V$ across the region
• variation of linewidth $\Delta V$ across the region
• mass M obtained without the virial theorem assumption
• column density of hydrogen $n$

Larson digs through papers that investigate optically thin line emissions such as $^{13}$CO to determine the variations in $V$ and $\Delta V$, and consequently calculate the three-dimensional velocity dispersion σ  due to all motions present (as indicated by the dispersions $\sigma(V)$ and $\sigma(\Delta V)$) in the cloud region (assuming isotropic velocity distributions). Both $\sigma(V)$ and $\sigma(\Delta V)$ are needed to obtain the three-dimensional velocity dispersion for a length-scale because the region has both local velocity dispersion and variation in bulk velocity across the region. The two dispersions add in quadrature: $\sigma = \sqrt{\sigma(\Delta V)^2 + \sigma(V)^2}$.

To estimate the mass, Larson assumes that the ratio of the column density of $^{13}$CO to the column density of $H_2$ is $2\cdot 10^{-6}$ and that $H_2$ comprises 70% of the total mass.

Note that for a molecular cloud with temperature 10 K the thermal velocity dispersion is only 0.32 km/s, while the observed velocity dispersions $\sigma$, are much larger, typically 0.5-5 km/s.

### (1) Turbulence in Molecular clouds

A log-log plot of velocity dispersion $\sigma$ versus region length $L$ is shown in Figure 1. below:

Figure 1. 3D internal velocity dispersion versus the size of a region follows a power-law, expected for turbulent flows. The $\sigma_s$ arrow shows the expected dispersion due to thermal pressure only. The letters in the plots represent different clouds (e.g. O=Orion)

The relation is fit with the line

$\sigma({\rm km~s}^{-1}) = 1.10 L({\rm pc})^{0.38}$

which is similar to the $\sigma \propto L^{1/3}$ power-law for subsonic Kolmogorov turbulence. Note, however, that the motion in the molecular clouds are mostly supersonic. A characteristic trait of turbulence is that there is no preferred length scale, hence the power-law.

Possible subdominant effects modifying the velocity dispersion include stellar winds, supernova explosions, and expansion of HII regions, which may explain some of the scatter in Figure 1.

### (2) Gravity in Molecular Clouds

A plot of the ratio $2GM/(\sigma^2 L)$ for each cloud region, which is expected to be ~1 if the cloud is in virial equilibrium, is shown in Figure 2. below:

Figure 2. Most regions are near virial equilibrium ($2GM/(\sigma^2L)\sim 1$). The large scatter is mostly due to uncertainties in the estimates of physical parameters.

Most regions are near virial equilibrium. The scatter in the figure may be large, but expected due to the simplifying assumptions about geometric factors in the virial equilibrium equation.

If the turbulent motions dissipate in a region, causing it to contract, and the region is still in approximate virial equilibrium, then $L$ should decrease and $\sigma$ should increase, which should cause some of the intrinsic scatter in Figure 1 (the $L$$\sigma$ relation). A few observed regions do have unusually high velocity dispersions in Figure 1, indicating significant amount of gravitational contraction.

### (3) Density Structure in Molecular Clouds

The $\sigma \propto L^{0.38}$ relation implies smaller regions need higher densities to be gravitationally bound (if one also assumes $\rho \sim M /L^3$ and virial equilibrium $2GM/(\sigma^2L)\sim 1$ then these imply $\rho \propto L^{-1.24}$). This is observed. The correlation between the density of $H_2$ in a region and the size of the region is shown in Figure 3 below:

Figure 3. An inverse relation is found between region density and size

The relationship found is:

$n({\rm cm}^{-3}) = 3400 L(pc)^{-1.1}$

This means that the column density $nL$ is proportional to $L^{-0.1}$, which is nearly scale invariant. Such a distribution could result from shock-compression processes which preserve the column density of the regions compressed. Larson also suggested that observational selection effects may have limited the range of column densities explored (CO density needs to be above a critical threshold to be excited for example). Modern observations, such as those by Lombardi, Alves, & Lada (2010), show that that while column density across a sample of regions and clouds appears to be constant, a constant column density does not described well individual clouds (the probability distribution function for column densities follows a log-normal distribution, which are also predicted by detailed theoretical studies of turbulence).

### Possible Implications for Star Formation and Molecular Clouds

• Larson essentially uses relations (1) and (2) to derive a minimum mass and size for molecular clouds by setting the velocity dispersion $\sigma$ to be subsonic. Smallest observed clouds have $M\sim M_{\rm \odot}$ and $L\sim 0.1$ pc and subsonic internal velocities. These clouds may be protostars. The transition from supersonic to subsonic defines a possible minimum clump mass and size: $M\sim 0.25M_{\rm \odot}$ and $L\sim 0.04$ pc, which may collapse with high efficiency without fragmentation to produce low mass stars of comparable mass to the initial cloud. Hence the IMF should have a downward turn for masses less than this minimum mass clump. Such a downturn is observed. Simple Jeans collapse fragmentation theory does not identify turnover at this mass scale.
• Regions that would form massive stars have a hard time collapsing due to the supersonic turbulent velocities. Hence their formation mechanism likely involves accretion/coalescence of clumps. Thus massive stars are predicted to form as members of clusters/associations, as is usually observed.
• The above two arguments imply that the low-mass slope of the IMF should be deduced from cloud properties, such as temperature and magnitude of turbulent velocities. The high-mass end is more complicated.
• The associated timescale for the molecular clouds is $\tau=L/\sigma$. It is found to be $\tau({\rm yr}) \sim 10^6L({\rm pc})^{0.62}$. Hence the timescales are less 10 Myr for most clouds, meaning that molecular clouds have short lifetimes and are transient.

Larson brings to attention the importance of turbulence for understanding the ISM in this seminal paper. His arguments are simple and are supported by data which are clearly incongruous with the previous simplified picture of Jeans collapse in a uniform, thermally-dominated medium. It is amusing and inspiring that Larson could dig through the literature to find all the data that he needed. He was able to synthesize the vast data in a way the observers missed to build a new coherent picture. As most good papers, Larson’s work fundamentally changes our understanding about an important topic but also provokes new questions for future study. What is the exact nature of the turbulence? What drives and/or sustains it? In what regimes does turbulence no longer become important? ISM physics is still an area of active research.

Many ISM research to this day has roots that draw back to Larson’s paper. One of the few important things Larson did not explain in this paper is the role of magnetic fields in the ISM, which we know today contributes a significant amount to the ISM’s energy budget and can be a source of additional instabilities, turbulence, and wave speeds. Also, there was not much data available at the time on the dense, subsonic molecular cloud cores in which thermal velocity dispersion dominates and the important physical processes are different, and so Larson only theorizes loosely about their role in star formation.

Larson’s estimates for molecular lifetimes (10 Myr) is relatively short compared to galaxy lifetimes and much shorter than what most models at the time estimated. This provoked a lot of debate in the field. Old theories which claim molecular clouds are built up by random collisions and coalescence of smaller clouds predict that Great Molecular Clouds take over 100 Myr to form. Turbulence speeds up this timescale, Larson argues, since turbulent motion is not random but systematic on larger scales.

##### The Plot Larson Did Not Make

Larson assumed spherical geometry of the molecular cloud regions in this paper to keep things simple. Yet he briefly mentions a way to estimate region geometry. He did not apply this correction to the data and unfortunately does not list the relevant observational parameters ($\sigma (V)$, $\sigma (\Delta V)$) for the reader to make the calculation. But such a correction would likely have reduced the scatter of the region size $L$ and have steepened the $\sigma$ vs $L$ relation, closer to what we observe today.  His argument for geometrical correction, fleshed out in more detail here, goes like this.

Let’s say the region’s shape is some 3D manifold, M. First, lets suppose M is a sphere of radius 1. Then, the average distance between points along a line of sight through the center is:

$\langle \ell\rangle_{\rm los} = \frac{\int |x_1-x_2|\,dx_1\,dx_2}{\int 1 \,dx_1\,dx_2}= 2/3$

where the integrals are over $x_1=-1,x_2=-1$ to $x_1=1,x_2=1$.

The average distance between points inside the whole volume is:

$\langle \ell\rangle_{\rm vol} =\frac{\int \sqrt{(x_1-x_2)^2+(y_1-y_2)^2+(z_1-z_2)^2}r_1^2 r_2^2 \sin\theta_1\sin\theta_2 dr_1 dr_2 d\theta_1 d\theta_2 d\phi_1 d\phi_2}{\int r_1^2 r_2^2 \sin\theta_1\sin\theta_2 dr_1 dr_2 d\theta_1 d\theta_2 d\phi_1 d\phi_2}= 36/35$

where the integrals are over $r_1=0,r_2=0,\theta_1=0,\theta_2=0,\phi_1=0,\phi_2=0$ to $r_1=1,r_2=1,\theta_1=\pi,\theta_2=\pi,\phi_1=2\pi,\phi_2=2\pi$.

Thus the ratio between these two characteristic lengths is:

$\langle \ell\rangle_{\rm vol}/\langle \ell\rangle_{\rm los}=54/35$

which is a number Larson quotes.

Now, if the velocity scales as a power-law $\sigma \propto L^{0.38}$, then one would expect:

$\sigma = (\langle \ell\rangle_{\rm vol}/\langle \ell\rangle_{\rm los})^{0.38} \sigma(\Delta V)$

We also have the relation

$\sigma = \sqrt{\sigma(\Delta V)^2 + \sigma(V)^2}$

These two relations above allow you to solve for the ratio

$\sigma(V)/\sigma(\Delta V) = 0.62$

Larson observes this ratio for regions of size less than 10 pc, meaning that the assumption that the are nearly spherical is good. But for larger regions, Larson sees much larger ratios ~1.7. This ratio can be expected for more sheetlike geometries. For example, if the geometry is 10 by 10 wide and 2 deep, one can calculate that the expected ratio is $\sigma (V)/\sigma (\Delta V)= 2.67$.

It would have been interesting to see a plot of $\sigma (V)/\sigma (\Delta V)$ as a function of $L$, which Larson does not include, to learn about how geometry changes with length scale. The largest regions have most deviation from spherical geometries, which is perhaps why Larson did not include large ~1000pc regions in his study.

~~~

The North America Nebula Larson mentions in his introduction. Complex hydrodynamic processes and turbulent flows are at play, able to create structures with sizes less than the Jeans length. (Credit and Copyright: Jason Ware)

Reference:

Larson (1981) – Turbulence and star formation in molecular clouds

Lombari, Alves, & Lada (2010) – 2MASS wide field extinction maps. III. The Taurus, Perseus, and California cloud complexes

## CHAPTER: Density of the Milky Way’s ISM

In Book Chapter on January 14, 2013 at 8:46 pm

(updated for 2013)

How do we know that $n \sim 1 ~{\rm cm}^{-3}$ in the ISM? From the rotation curve of the Milky Way (and some assumptions about the mass ratio of gas to gas+stars+dark matter), we can infer

$M_{\rm gas} = 6.7 \times 10^{9} M_\odot$

Maps of HI and CO reveal the extent of our galaxy to be

$D = 40$ kpc

$h = 140$ pc (scale height of HI)

This applies an approximate volume of

$V = \pi D^2 h / 4 = 5 \times 10^{66} ~{\rm cm}^{3}$

Which, yields a density of

$\rho = 2.5 \times 10^{-24} ~{\rm g cm}^{-3}$