# Harvard Astronomy 201b

## CHAPTER: Thermodynamic Equilibrium

In Book Chapter on February 28, 2013 at 3:13 am

(updated for 2013)

Collisions and radiation generally compete to establish the relative populations of different energy states. Randomized collisional processes push the distribution of energy states to the Boltzmann distribution, $n_j \propto e^{-E_j / kT}$. When collisions dominate over competing processes and establish the Boltzmann distribution, we say the ISM is in Thermodynamic Equilibrium.

Often this only holds locally, hence the term Local Thermodynamic Equilibrium or LTE. For example, the fact that we can observe stars implies that energy (via photons) is escaping the system. While this cannot be considered a state of global thermodynamic equilibrium, localized regions in stellar interiors are in near-equilibrium with their surroundings.

But the ISM is not like stars. In stars, most emission, absorption, scattering, and collision processes occur on timescales very short compared with dynamical or evolutionary timescales. Due to the low density of the ISM, interactions are much more rare. This makes it difficult to establish equilibrium. Furthermore, many additional processes disrupt equilibrium (such as energy input from hot stars, cosmic rays, X-ray background, shocks).

As a consequence, in the ISM the level populations in atoms and molecules are not always in their equilibrium distribution. Because of the low density, most photons are created from (rare) collisional processes (except in locations like HII regions where ionization and recombination become dominant).

## 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: Relevant Velocities in the ISM

In Book Chapter on February 28, 2013 at 3:06 am

(updated for 2013)

Note: it’s handy to remember that 1 km/s ~ 1 pc / Myr.

• Galactic rotation: 18 km/s/kpc (e.g. 180 km/s at 10 kpc)
• Isothermal sound speed: $c_s =\sqrt{\frac{kT}{\mu}}$
• For H, this speed is 0.3, 1, and 3 km/s at 10 K, 100 K, and 1000 K, respectively.
• Alfvén speed: The speed at which magnetic fluctuations propagate. $v_A = B / \sqrt{4 \pi \rho}$ Alfvén waves are transverse waves along the direction of the magnetic field.
• Note that $v_A = {\rm const}$ if $B \propto \rho^{1/2}$, which is observed to be true over a large portion of the ISM.
• Interstellar B-fields can be measured using the Zeeman effect. Observed values range from $5~\mu {\rm G}$ in the diffuse ISM to $1 mG$ in dense clouds. For specific conditions:
• $B = 1~\mu{\rm G}, n = 1 ~{\rm cm}^{-3} \Rightarrow v_A = 2~{\rm km~s}^{-1}$
• $B = 30~\mu {\rm G}, n = 10^4~{\rm cm}^{-3} \Rightarrow v_A = 0.4~{\rm km~s}^{-1}$
• $B = 1~{\rm mG}, n = 10^7 {\rm cm}^{-3} \Rightarrow v_A = 0.5~{\rm km~s}^{-1}$
• Compare to the isothermal sound speed, which is 0.3 km/s in dense gas at 20 K.
• $c_s \approx v_A$ in dense gas
• $c_s < v_A$ in diffuse gas
• Observed velocity dispersion in molecular gas is typically about 1 km/s, and is thus supersonic. This is a signature of the presence of turbulence. (see the summary of Larson’s seminal 1981 paper)

## CHAPTER: Energy Density Comparison

In Book Chapter on February 26, 2013 at 3:04 am

(updated for 2013)

See Draine table 1.5. The primary sources of energy present in the ISM are:

1. The CMB ($T_{\rm CMB}=2.725~{\rm K}$
2. Thermal IR from dust
3. Starlight ($h\nu < 13.6 {\rm eV}$
4. Thermal kinetic energy (3/2 nkT)
5. Turbulent kinetic energy ($1/2 \rho \sigma_v^2$)
6. Magnetic fields ($B^2 / 8 \pi$)
7. Cosmic rays

All of these terms have energy densities within an order of magnitude of $1 ~{\rm eV ~ cm}^{-3}$. With the exception of the CMB, this is not a coincidence: because of the dynamic nature of the ISM, these processes are coupled together and thus exchange energy with one another.

## 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: Dark Nebulae, Globules, and Protostars

In Journal Club, Journal Club 2013 on February 19, 2013 at 10:45 pm

### Dark Nebulae, Globules, and Protostars by Bart Bok (1977)

Summary by George Miller

### Introduction

In Bart Bok’s 1977 paper Dark nebulae, globules, and protostars (Bok 1977), largely based on a lecture given upon acceptance of the Astroomical Society of the Pacific’s Bruce Medal, he presents two fundamentally different pictures of star formation. The first, constituting the majority of the paper’s discussion, occurs in large Bok globules which are compact, rounded and remarkably well-defined regions of high-extinction ranging from 3′ to 20′.  The globules show a strong molecular hydrogen and dust component and relatively little signs of higher neutral HI concentrations than its surroundings. In contrast, Bok briefly examines star formation in the Magellanic Clouds which show a vast amount of neutral atomic hydrogen and a comparatively small amount of cosmic dust. In this review, I will summarize a number of key points made by Bok, as well as provide additional information and modern developments since the paper’s original publishing.

### Large Bok Globules

#### A history of observations

In 1908, Barnard drew attention to “a number of very black, small, sharply defined spots or holes” in observations of the emission nebula Messier 8 (Barnard 1908).  39 years later Bok published extensive observations of 16 “globules” present in M8 as well others in $\eta$ Carinae, Sagittarius, Ophiuchus and elsewhere, making initial estimates of their distance, diameter and extinction (Bok & Reilly 1947). He further claimed that these newly coined “globules” were gravitationally contracting clouds present just prior to star formation, comparing them to an “insect’s cocoon” (Bok 1948). As we will see, this bold prediction was confirmed over 40 years later to be correct. Today there over 250 globules known within roughly 500 pc of our sun and, as Bok claims in his 1977 paper, identifying more distant sources is difficult due to their small angular diameter and large number of foreground stars.  There are currently four chief methods of measuring the column density within Bok Globules: extinction mappings of background stars, mm/sub-mm dust continuum emission, absorption measurements of galactic mid-IR background emission, and mapping molecular tracers.  See Figure 1 for a depiction of the first three of these methods.  At the time Bok published his paper in 1977, only extinction mapping and molecular tracer methods were readily available, thus I will primarily discuss these two.  For a more in depth discussion, see Goodman et. al. 2009 and the subsequent AST201b Journal Club review.

Figure 1.  Three methods of determining column density of starless molecular cores or Bok globules. (a) K-band image of Barnard 68 and plot of the $A_K$ as a function of radius from the core.  This method measures the H–K excess, uses the extinction law to convert into $A_V$, and then correlated to the $H_2$ column density from UV line measurements, parameterized by f. (b) 1.2-mm dust continuum emission map and ﬂux versus radius for L1544.  $\kappa_{\nu}$ is the dust opacity per unit gas mass, ρ is the dust density, and m the hydrogen mass (corrected for He). (c) 7-μm ISOCAM image and opacity versus radius for ρ Oph D.  In this method the absorbing opacity is related to the hydrogen column via the dust absorption cross section, $\sigma_{\lambda}$.  Figure taken from Bergin & Tafalla 2007.

#### Measuring photometric extinction

Measuring the photometric absorption, and thus yielding a minimum dust mass, for these globules is itself an arduous process. For globules with $A_v<10$   mag, optical observations with large telescopes can be used to penetrate through the globules and observe the background stars.  Here $A_{\lambda} \equiv m_{\lambda}-m_{\lambda, 0} = 2.5 \, log(\frac{F_{\lambda,0}}{F_{\lambda}})$.  Thus an extinction value of $A_v=10$ mag means the flux is decreased by a factor of $10^4$.  By using proper statistics of the typical star counts and magnitudes seen within a nearby unobstructed field of view, extinction measurements can be made for various regions.  It is important to note that the smaller an area one tries to measure an extinction of, the greater the statistical error (due to a smaller number of background stars).  This is one of the key limitations of extinction mappings.  For the denser cores or more opaque globules with $10 < A_V < 20$ mag, observations in the near infrared are needed (which is relatively simple by today’s standards but not so during Bok’s time). This is further complicated due to imprecisely defined BVRI photometric standard sequences for fainter stars, a problem still present today with various highly-sensitive space telescopes such as the HST. Bok mentions two methods. In the past a Racine (or Pickering) prism was used to produce fainter companion images of known standards, yet as discussed by Christian & Racine 1983 this method can produce important systematic errors. The second, and more widely used, method is to pick an easily accessible progression of faint stars and calibrate all subsequent photographic plates (or ccd images) from this. See Saha et. al. 2005 for a discussion of this problem in regards to the Hubble Space Telescope.

Obtaining an accurate photometric extinction for various regions within the globule is valuable as it leads an estimate of the dust density. Bok reports here from his previous Nature paper (Bok et. al. 1977) that the extinction $A_v$ within the Coalsack Globule 2 varies inversely as the square of distance, thus also implying the dust density varies inversely as the cube of distance from the core.  Modern extinction mappings, as seen in Figure 1(a) of Barnard 68,  show that at as one approaches the central core the extinction vs. distance relation actually flattens out nearly to $r^{-1}$.  This result was a key discovery, for the Bonnor-Ebert (BE) isothermal sphere model predicts a softer power law at small radii.  In his paper, Bok remarks “The smooth density gradient seems to show that Globule 2 is […] an object that reminds one of the polytropic models of stars studied at the turn of the century by Lane (1870) and Emden (1907)”.  It is truly incredible how accurate this assessment was.  The Bonnor-Ebert sphere is a model derived from the Lane-Emden equation for an isothermal, self-gravitating sphere which remains in hydrostatic equilibrium.  Figure 2 displays a modern extinction mapping of Barnard 68 along with the corresponding BE sphere model, showing that the two agree remarkably well.  There are, however, a number of detractors from the BE model applied to Bok globules.  The most obvious is that globules are rarely spherical, implying that some other non-symmetric pressure must be present.  Furthermore, the density gradient between a globule’s core and outer regions often exceeds 14 ($\xi_{max} > 6.5$) as required for a stable BE sphere (Alves, Lada & Lada 2001).

Figure 2.  Dust extinction mapping for Barnard 68 plotted against an isothermal Bonnor-Ebert sphere model.  Figure taken from Alves, Lada & Lada 2001.

#### Using CO as a tracer

Important tracer molecules, such as CO, are used to study the abundance of $H_2$, temperatures and kinematics of these globules. Because the more common $^{12}CO$ isotope tends to become optically thick and saturate in regions of higher column density such as globules, the strength of $^{13}CO$ emission is usually used to indicate the density of H2.  The conversion factor of $N_{H_2} = 5.0 \pm 2.5 \times 10^5 \times N_{13},$ from Dickman 1978 has changed little in over three decades. The column density of $H_2$, combined with its known mass and radius of the globule, can then be used to estimate the globule’s total mass. Furthermore, the correlation of $^{13}CO$ density with photometric extinction, $A_v = 3.7 \times 10^{-16} \times N_{13},$ is another strong indication that $^{13}CO$ emission is an accurate tracer for H$_2$ and dust. Further studies using $C^{17}O$ and $C^{18}O$ have also been used to trace even higher densities when even $^{13}CO$ can become optically thick(Frerking et. al. 1982).  As an example, Figure 3 shows molecular lines from the central region of the high-mass star forming region G24.78+0.08.  In the upper panel we can see the difference between the optically thick $^{12}CO$ and thin $C^{18}O$.  The $^{12}CO$ line shows obvious self-absorption peaks associated with an optically thick regime, and one clearly can not make a Gaussian fit to determine the line intensity.   $^{12}CO$, due to the small dipole moment of its $J=1 \rightarrow 0$ transition and thus ability to thermalize at relatively low densities, is also used to measure the gas temperature within globules. These temperatures usually range from 7K to 15K. Finally, the width of CO lines are used to measure the velocity dispersion within the globule. As Bok states, most velocities range from 0.8 to 1.2 km/s. This motion is often complex and measured excess line-widths beyond their thermal values are usually attributed to turbulence (Bergin & Tafalla 2007). Importantly, the line-width vs. size relationship within molecular clouds first discovered by Barnard 1981 does not extend to their denser cores (which have similar velocity motions as Bok globules).  Instead, a “coherence” radius is seen where the non-thermal component of a linewidth is approximately constant (Goodman et. al. 1998).  In the end, as Bok surmises, the subsonic nature of this turbulence implies it plays a small role compared to thermal motions.

Figure 3.  Spectra taken from the core of the high-mass star forming region G24.78+0.08.  The solid line corresponds to $^{12}CO (1\rightarrow 0)$, $^{12}CO (2\rightarrow 1)$, and $C^{32}S (3\rightarrow 2)$, the dashed line to $^{13}CO (1\rightarrow 0)$$^{13}CO (2\rightarrow 1)$, and $C^{34}S (3\rightarrow 2)$ and the dotted line to $C^{18}O (1\rightarrow 0)$.  From the top panel, one can clearly see the difference between the optically thick, saturated $^{12}CO (1\rightarrow 0)$ line and the optically thin $C^{18}O (1\rightarrow 0)$ transition.  Figure taken from Cesaroni et. al. 2003.

#### The current status of Bok globules

Today, the majority of stars are thought to originate within giant molecular clouds or larger dark cloud complexes, with only a few percent coming from Bok globules. However, the relative simplicity of these globules still make them important objects for studying star formation. While an intense debate rages today regarding the influence of turbulence, magnetic fields, and other factors on star formation in GMCs, these factors are far less important than simple gravitational contraction within Bok globules. The first list of candidate protostars within Bok globules, obtained by co-adding IRAS images, was published in 1990 with the apropos title “Star formation in small globules – Bart Bok was correct” (Yun & Clemens 1990).  To conduct the search, Yun & Clemens first fit a single-temperature modified blackbody model the the IRAS 60 and 100 μm images (after filtering out uncorrelated background emission) to obtain dust temperature and optical depth values.  This result was then used as a map to search for spatially correlated 12 and 25 μm point sources (see Figure 4.).  More evidence of protostar outflows (Yun & Clemens 1992), Herbig-Haro objects due to young-star jets (Reipurth et al. 1992) and the initial stages of protostar collapse (Zhou et. al. 1993) have also been detected within Bok Globules. Over 60 years after Bok’s pronouncement that these globules were “insect cocoons” encompassing the final stages of protostar formation, his hypothesis remains remarkably accurate and validated. It is truly “pleasant indeed that globules are there for all to observe!”

Figure 4.    (a) Contour map of the dust temperature $T_{60/100}$ of the Bok Globule CB60 derived from 60 and 100 μm IRAS images.  (b) 12 μm IRAS image of CB60 after subtracting background emission using median-filtering.  This source is thought to be a young stellar object or protostar located within the globule.  The other 12 μm field sources seen in (b) are thought not to be associated with the globule. Figure taken from Yun & Clemens 1990.

### Magellanic Cloud Star Formation

At the end of his paper, Bok makes a 180 degree turn and discusses the presence of young stars and blue globulars within the Magellanic Clouds. These star formation regions stand in stark contrast to the previously discussed Bok globules; they contain a rich amount of HI and comparatively small traces of dust, they are far larger and more massive, and they form large clusters of stars as opposed to more isolated systems. Much more is known of the star-formation history in the MCs since Bok published this 1977 paper. The youngest star populations in the MCs are found in giant and supergiant shell structures which form filamentary structures throughout the cloud. These shells are thought to form from supernova, ionizing radiation and stellar wind from massive stars which is then swept into the cool, ambient molecular clouds. Further gravitational, thermal and fluid instabilities fragment and coalesce these shells into denser star-forming regions and lead to shell-shell interactions (Dawson et. al. 2013). The initial onset of this new ($\sim$ 125 Myr) star formation is thought to be due to close encounters between the MCs, and is confirmed by large-scale kinematic models (Glatt et al. 2010).

### References

Alves, J. F., Lada, C. J., & Lada, E. A. 2001, Nature, 409, 159
Barnard, E. E. 1908, Astronomische Nachrichten, 177, 231
Bergin, E. A., & Tafalla, M. 2007, ARA&A, 45, 339
Bok, B. J. 1948, Harvard Observatory Monographs, 7, 53
—. 1977, PASP, 89, 597
Bok, B. J., & Reilly, E. F. 1947, ApJ, 105, 255
Bok, B. J., Sim, M. E., & Hawarden, T. G. 1977, Nature, 266, 145
Cesaroni, R., Codella, C., Furuya, R. S., & Testi, L. 2003, A&A, 401, 227
Christian, C. A., & Racine, R. 1983, PASP, 95, 457
Dawson, J. R., McClure-Griffiths, N. M., Wong, T., et al. 2013, ApJ, 763, 56
Dickman, R. L. 1978, ApJS, 37, 407
Frerking, M. A., Langer, W. D., & Wilson, R. W. 1982, ApJ, 262, 590
Glatt, K., Grebel, E. K., & Koch, A. 2010, A&A, 517, A50
Goodman, A. A., Barranco, J. A., Wilner, D. J., & Heyer, M. H. 1998, ApJ, 504, 223
Goodman, A. A., Pineda, J. E., & Schnee, S. L. 2009, ApJ, 692, 91
Reipurth, B., Heathcote, S., & Vrba, F. 1992, A&A, 256, 225
Saha, A., Dolphin, A. E., Thim, F., & Whitmore, B. 2005, PASP, 117, 37
Yun, J. L., & Clemens, D. P. 1990, ApJ, 365, L73
—. 1992, ApJ, 385, L21
Zhou, S., Evans, II, N. J., Koempe, C., & Walmsley, C. M. 1993, ApJ, 404, 232

## CHAPTER: Chemistry

In Book Chapter on February 13, 2013 at 10:04 pm

See Draine Table 1.4 for elemental abundances for the proto-solar environment. H:He:C = $1:0.1:3 \times 10^{-4}$ by number. $1:0.4:3.5 \times 10^{-3}$ by mass However, these ratios vary by position in the galaxy, especially for heavier elements (which depend on stellar processing). For example, the abundance of heavy elements (Z > Carbon) is twice as low at the sun’s position than in the Galactic center. Even though metals account for 1% of the mass, they dominate most of the important chemistry, ionization, and heating/cooling processes. They are essential for star formation, as they allow molecular clouds to cool and collapse. Generally, it is easier (i.e. requires less energy) to dissociate a molecule than to ionize something. The lower the electronic state you are trying to ionize, the more energy is needed. The Lyman Limit is the minimum photon energy needed to ionize Hydrogen from the ground state (13.6 eV, 912 Angstrom).

## CHAPTER: Hydrogen Slang

In Book Chapter on February 12, 2013 at 10:02 pm

Lyman limit: the minimum energy needed to remove an electron from a Hydrogen atom. A “Lyman limit photon” is a photon with at least this energy.

$E = 13.6 {\rm eV} = 1~ {\rm Rydberg} = hcR_{\rm H}$,

where $R_{\rm H}=1.097 \times 10^{7} {\rm m}^{-1}$ is the Rydberg constant, which has units of $1/\lambda$. This energy corresponds to the Lyman limit wavelength as follows:

$E = h\nu = hc/\lambda \Rightarrow \lambda=912 \AA$.

Lyman series: transitions to and from the n=1 energy level of the Bohr atom. The first line in this series was discovered in 1906 using UV studies of electrically excited hydrogen gas.

Balmer series: transitions to and from the n=2 energy level. Discovered in 1885; since these are optical transitions, they were more easily observed than the UV Lyman series transitions.

There are also other named series corresponding to higher n. Examples include Paschen (n=3), Brackett (n=4), and Pfund (n=5). The wavelength of a given transition can be computed via the Rydberg equation

$\frac{1}{\lambda}=R_{\rm H} \big(\frac{1}{n_f^2}-\frac{1}{n_i^2}\big)$.

Note that the Lyman (or Balmer, Paschen, etc.) limit can be computed by inserting $n_i=\infty$.

Lyman continuum corresponds to the region of the spectrum near the Lyman limit, where the spacing between energy levels becomes comparable to spectral line widths and so individual lines are no longer distinguishable.

## 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: Stromgren Sphere: An example “chalkboard derivation”

In Book Chapter on February 12, 2013 at 8:31 pm

(updated for 2013)

The Stromgren sphere is a simplified analysis of the size of HII regions. Massive O and B stars emit many high-energy photons, which will ionize their surroundings and create HII regions. We assume that such a star is embedded in a uniform medium of neutral hydrogen. A sphere of radius r around this star will become ionized; is called the “Stromgren radius”. The volume of the ionized region will be such that the rate at which ionized hydrogen recombines equals the rate at which the star emits ionizing photons (i.e. all of the ionizing photons are “used up” re-ionizing hydrogen as it recombines)

The recombination rate density is $\alpha n^2$, where $\alpha$ is the recombination coefficient (in $\mathrm{cm}^3~\mathrm{s}^{-1})$ and $n=n_e=n_\mathrm{H}$ is the number density (assuming fully ionized gas and only hydrogen, the electron and proton densities are equal). The total rate of ionizing photons (in photons per second) in the volume of the sphere is $N^*$. Setting the rates of ionization and recombination equal to one another, we get

$\frac43 \pi r^3 \alpha n^2 = N^*$, and solving for r,

$r = ( \frac {3N^*} {4\pi\alpha n^2})^{\frac13}$

Typical values for the above variables are $N^* \sim 10^{49}~\mathrm{photons~s}^{-1}$, $\alpha \sim 3\times 10^{-13}\; \mathrm{cm}^3 \; \mathrm s^{-1}$ and $n \sim 10\; \mathrm {cm}^{-3}$, implying Stromgren radii of 10 to 100 pc. See the journal club (2013) article for discussion of Stromgren’s seminal 1939 paper.

## ARTICLE: On the Dark Markings in the Sky

In Journal Club, Journal Club 2013 on February 8, 2013 at 2:46 pm

On the Dark Markings in the Sky by Edward E. Barnard (1919)

Summary by Hope Chen

#### Abstract

By examining photographic plates of various regions on the sky, Edward E. Barnard concluded in this paper that what he called “dark markings” were in fact due to the obscuration of nearby nebulae in most cases. This result had a significant impact on the debate regarding the size and the dimension of the Milky Way and also the research of the interstellar medium, particularly work by Vesto Slipher, Heber Curtis and Robert Trumpler. The publication of  Photographic Atlas of Selected Regions of the Milky Way after Barnard’s death, which included many of the regions mentioned in the paper, further provided a new method of doing astronomy research. In this paper and the Atlas, we are also able to see a paradigm very different from that of today.

It is now well-known that the interstellar medium causes extinction of light from background stars. However, think of a time when the infrared imaging was impossible, and the word “photon” meant nothing but a suspicious idea. Back in such a time in the second decade of the twentieth century, Edward Edison Barnard, by looking at hundreds of photographic plates, proposed an insightful idea that “starless” patches of the sky were dark because they are obscured by nearby nebulae. This idea not only built the foundation of the modern concept of the interstellar medium, but also helped astronomers figure out that the Universe extended so much farther beyond the Milky Way.

#### Young Astronomer and His Obsession of the Sky

In 1919, E. E. Barnard published this paper and raised the idea that what he called “dark markings” are mostly obscuration from nebulae close to us. The journey, however, started long before the publication of this paper. Born in Nashville, Tennessee in 1857, Barnard was not able to receive much formal education owing to poverty. His first interest, which became important for his later career, was in photography. He started working as a photographer’s assistant at the age of nine, and the work continued throughout most of his teenage years. He then developed an interest in astronomy, or rather, “star-gazing,” and would go watch the sky almost every night with his own telescope. He took courses in natural sciences at Vanderbilt University and started his professional career as an astronomer at the Lick Observatory in 1888. He helped build the Bruce Photographic Telescope at the Lick Observatory and there he started taking pictures of the sky on photographic plates. He then moved on to his career at the Yerkes Observatory at Chicago University and worked there until his death in 1922. (Introduction of the Atlas, Ref. 2)

Fig. 1 One of the many plates in the Atlas including the region around Rho Ophiuchii, which was constantly mentioned in many of Barnard’s works. (Ref. 2)

Fig. 1 is one of the many plates taken at the Yerkes Observatory. It shows the region near Rho Ophiuchii, which was a region constantly and repetitively visited by Barnard and his telescope. Barnard noted in his description of this plate, “the [luminous] nebula itself is a beautiful object. With its outlying connections and the dark spot in which it is placed and the vacant lanes running to the East from it, … it gives every evidence that it obscures the stars beyond it.” Numerous similar comments spread throughout his descriptions of various regions covered in A Photographic Atlas of Selected Regions of the Milky Way (hereafter, the Atlas). Then finally in his 1919 paper, he concluded, “To me these are all conclusive evidence that masses of obscuring matter exist in space and are readily shown on photographs with the ordinary portrait lenses,” although “what the nature of this matter may be is quite another thing.” The publication of these plates in the Atlas (unfortunately after his death, put together by Miss Mary R. Calvert, who was Barnard’s assistant at the Yerkes Observatory and helped publish many of Barnard’s works after his death) also provided a new way of conducting astronomical research just as the World Wide Telescope does today. The Atlas for the first time allowed researchers to examine the image and the astronomical coordinates along with lists of dominant objects at the same time.

Except quoting Vesto Slipher’s work on spectrometry measurements of these nebulae, most of the evidences in Barnard’s paper seemed rather qualitative than quantitative. So, as of today’s standard, was the “evidence” really conclusive? Again, the question cannot be answered without knowing the limits of astronomical research at the time. Besides an immature understanding of the underlying physics, astronomers in the beginning of the twentieth century were limited by the lack of tools on both the observation and analysis fronts. Photographic plates as those in the Atlas were pretty much the most advanced imaging technique at the time, on which even a quantitative description of “brightness” was not easy, not to mention an estimation of the extinction of these “dark markings.” However, this being said, a very meaningful and somewhat “quantitative” assumption was drawn in Barnard’s paper: the field stars were more or less uniformly distributed. Barnard came to this assumption by looking at many different places, both in the galactic plane and off the plane, and observing the densities of field stars in these regions. Although numbers were not given in the paper, this was inherently similar to a star count study. Eventually, this assumption lead to what Barnard thought as the conclusive evidence of these dark markings being obscuring nebulae instead of “vacancies.” Considering the many technical limits at the time, while the paper might not seem to be scientific in today’s standard, this paper did pose a “conclusion” which was strong enough to sustain many of the more quantitative following examinations.

#### The “Great Debate”

Almost at the same time, perviously mentioned Vesto Slipher (working at the Lowell Observatory) began taking spectroscopic measurements of various clouds and tried to understand the constituents of these clouds. Although limited by the wavelength range and the knowledge of different radiative processes (the molecular transition line emission used largely in the research of the interstellar medium today was not observed until half a century later in 1970, by Robert Wilson, who, on a side note, also discovered the Cosmic Microwave Background), Slipher was able to determine the velocities of clusters by measuring the Doppler shifts and concluded that many of these clusters move at a faster rate than the escape velocity of the Milky Way (Fig. 2). This result, coupled with Barnard’s view of intervening nebulae, revolutionized the notion of the Universe in the 1920s.

Fig. 2 The velocity measurements from spectroscopic observations done by Vesto Slipher. (Ref. 3)

On April 26, 1920 (and in much of the 1920s), the “Great Debate” took place between Harlow Shapley (the Director of Harvard College Observatory at the time) and Curtis Heber (the Lick Observatory, 1902 to 1920). The general debate concerned the dimension of the Universe and the Milky Way, but the basic issue was simply whether distant “spiral nebulae” were small and lay within the Milky Way or whether they were large and independent galaxies. Besides the distance and the velocity measurements, which suffered from large uncertainties due to the technique available at the time, Curtis Heber was able to “win” the debate by claiming that dark lanes in the “Great Andromeda Nebula” resemble local dark clouds as those observed by Barnard (Fig. 3, taken in 1899). The result of the debate then sparked a large amount of work on “extragalactic astronomy” in the next two decades and was treated as the beginning of this particular research field.

Fig. 3 The photographic plate of the “Great Andromeda Nebula” taken in 1988 by Isaac Roberts.

#### The Paper Finally Has a Plot

Then after the first three decades of the twentieth century, astronomers were finally equipped with a relatively more correct view of the Universe, the idea of photons and quantum theory. In 1930, Robert J. Trumpler (the namesake of the Trumpler Award) published his paper about reddening and reconfirmed the existence of local “dark nebulae.” Fig. 4 shows the famous plot in his paper which showed discrepancies between diameter distances and photometric distances of clusters. In the same paper, Trumpler also tried to categorize effects of the ISM on light from background stars, including what he called “selective absorption” or reddening as it is known today. This paper, together with many of Trumpler’s other papers, is one of the first systematic research results in understanding the properties of Barnard’s dark nebulae, which are now known under various names such as clouds, clumps, and filaments, in the interstellar medium.

Fig. 4 Trumpler’s measurements of diameter distances v. photometric distances for various clusters.

#### Moral of the Story

As Alyssa said in class, it is often more beneficial than we thought to understand what astronomers knew and didn’t know at different periods of time and how we came to know what we see as common sense today, not only in the historically interesting sense but also in the sense of better understanding of various ideas. In this paper, Barnard demonstrated a paradigm which we may call unscientific today but made a huge leap into what later became the modern research field of the interstellar medium.

## CHAPTER: The Sound Speed

In Book Chapter on February 7, 2013 at 10:00 pm

(updated for 2013)

The speed of sound is the speed at which pressure disturbances travel in a medium. It is defined as

$c_s \equiv \frac{\partial P}{\partial \rho}$,

where $P$ and $\rho$ are pressure and mass density, respectively. For a polytropic gas, i.e. one defined by the equation of state $P \propto \rho^\gamma$, this becomes $c_s=\sqrt{\gamma P/\rho}$. $\gamma$ is the adiabatic index (ratio of specific heats), and $\gamma=5/3$ describes a monatomic gas.

For an isothermal gas where the ideal gas equation of state $P=\rho k_B T / (\mu m_{\rm H})$ holds, $c_s=\sqrt{k_B T/\mu}$. Here, $\mu$ is the mean molecular weight (a factor that accounts for the chemical composition of the gas), and $m_{\rm H}$ is the hydrogen atomic mass. Note that for pure molecular hydrogen $\mu=2$. For molecular gas with ~10% He by mass and trace metals, $\mu \approx 2.7$ is often used.

A gas can be approximated to be isothermal if the sound wave period is much higher than the (radiative) cooling time of the gas, as any increase in temperature due to compression by the wave will be immediately followed by radiative cooling to the original equilibrium temperature well before the next compression occurs. Many astrophysical situations in the ISM are close to being isothermal, thus the isothermal sound speed is often used. For example, in conditions where temperature and density are independent such as H II regions (where the gas temperature is set by the ionizing star’s spectrum), the gas is very close to isothermal.

## CHAPTER: Energy Density Comparison

In Book Chapter on February 6, 2013 at 10:06 pm

See Draine table 1.5 The main kinds of energy present in the ISM are:

1. The CMB
2. Thermal IR from dust
3. Starlight
4. Thermal kinetic energy (3/2 nKT)
5. Turbulent kinetic energy ($1/2 \rho \sigma_v^2$)
6. B fields ($B^2 / 8 \pi$)
7. Cosmic rays

All of these terms have energy densities within an order of magnitude of $1 ~{\rm eV ~ cm}^{-3}$. With the exception of the CMB, this is not a coincidence. Because of the dynamic nature of the ISM, these processes are coupled together and thus exchange energy with one another.

## 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