# Harvard Astronomy 201b

## ARTICLE: A Theory of the Interstellar Medium: Three Components Regulated by Supernova Explosions in an Inhomogeneous Substrate

In Journal Club 2013 on March 15, 2013 at 11:09 pm

## Abstract (the paper’s, not ours)

Supernova explosions in a cloudy interstellar medium produce a three-component medium in which a large fraction of the volume is filled with hot, tenuous gas.  In the disk of the galaxy the evolution of supernova remnants is altered by evaporation of cool clouds embedded in the hot medium.  Radiative losses are enhanced by the resulting increase in density and by radiation from the conductive interfaces between clouds and hot gas.  Mass balance (cloud evaporation rate = dense shell formation rate) and energy balance (supernova shock input = radiation loss) determine the density and temperature of the hot medium with (n,T) = ($10^{-2.5}$, $10^{5.7}$) being representative values.  Very small clouds will be rapidly evaporated or swept up.  The outer edges of “standard” clouds ionized by the diffuse UV and soft X-ray backgrounds provide the warm (~$10^{4}$ K) ionized and neutral components.  A self-consistent model of the interstellar medium developed herein accounts for the observed pressure of interstellar clouds, the galactic soft X-ray background, the O VI absorption line observations, the ionization and heating of much of the interstellar medium, and the motions of the clouds.  In the halo of the galaxy, where the clouds are relatively unimportant, we estimate (n,T) = ($10^{-3.3}$, $10^{6.0}$) below one pressure scale height.  Energy input from halo supernovae is probably adequate to drive a galactic wind.

## The gist

The paper’s (McKee and Ostriker 1977) main idea is that supernova remnants (SNRs) play an important role in the regulation of the ISM.  Specifically, they argue that these explosions add enough energy that another phase is warranted: a Hot Ionized Medium (HIM)

A basic supernova explosion consists of several phases.  Their characteristic energies are on the order of $10^{51}$ erg, and indeed this is a widely-used unit.  For a fairly well-characterized SNR, see Cas A which exploded in the late 1600s.

Nearby supernova remnant Cassiopeia A, in X-rays from NuSTAR.

1. Free expansion
A supernova explosion begins by ejecting mass with a range of velocities, the rms of which is highly supersonic.  This means that a shock wave propagates into the ISM at nearly constant velocity during the beginning.  Eventually the density decreases and the shocked pressure overpowers the thermal pressure in the ejected material, creating a reverse shock propagating inwards.  This phase lasts for something on the order of several hundred years.  Much of the Cas A ejecta is in the free expansion phase, and the reverse shock is currently located at 60% of the outer shock radius.
2. Sedov-Taylor phase
The reverse shock eventually reaches the SNR center, the pressure of which is now extremely high compared to its surroundings.  This is called the “blast wave” portion, in which the shock propagates outwards and sweeps up material into the ISM.  The remnant’s time evolution now follows the Sedov-Taylor solution, which finds $R_s \propto t^{2/5}$.  This phase ends when the radiative losses (from hot gas interior to the shock front) become important.  We expect this phase to last about $10^3$ years.
3. Snowplow phase
When the age of the SNR approaches the radiative cooling timescale, cooling causes thermal pressure behind the shock to drop, stalling it.  This phase features a shell of cool gas around a hot volume, the mass of which increases as it sweeps up the surrounding gas like a gasplow.  For typical SNRs, this phase ends at an age of about $10^6$ yr, leading into the next phase:
Eventually the shock speed approaches the sound speed in the gas, and turns into a sound wave.  The “fadeaway time” is on the order of $10^{6}$ years.

## So why are they important?

To constitute an integral part of a model of the ISM, SNRs must occur fairly often and overlap.  In the Milky Way, observations indicate a supernova every 40 years.  Given the size of the disk, this yields a supernova rate of $10^{-13} pc^{-3} yr^{-1}$.

Here we get some justification for an ISM that’s a bit more complicated than the then-standard two-phase model (proposed by Field, Goldsmith, and Habing (1969) consisting mostly of warm HI gas).  Taking into account the typical fadeaway time of a supernova, we can calculate that on average 1 other supernova will explode within a “fadeaway volume” within that original lifetime.  That volume is just the characteristic area swept out by the shock front as it approaches the sound speed in the last phase.  For a fadeaway time of $10^6$ yr and a typical sound speed of the ISM, this volume is about 100 pc.  Thus in just a few million years, this warm neutral medium will be completely overrun by supernova remnants!  The resulting medium would consist of low-density hot gas and dense shells of cold gas.  McKee and Ostriker saw a better way…

## The Three Phase Model

McKee and Ostriker present their model by following the evolution of a supernova remnant, eventually culminating in a consistent picture of the phases of the ISM. Their model consists of a hot ionized medium with cold dense clouds dispersed throughout. The cold dense clouds have surfaces that are heated by hot stars and supernova remnants, making up the warm ionized and neutral media, leaving the unheated interiors as the cold neutral medium. In this picture, supernova remnants are contained by the pressure of the hot ionized medium, and eventually merge with it. In the early phases of their expansion, supernova remnants evaporate the cold clouds, while in the late stages, the supernova remnant material cools by radiative losses and contributes to the mass of cold clouds.

A schematic of the three phase model, showing how supernovae drive the evolution of the interstellar medium.

In the early phases of the supernova remnant, McKee and Ostriker focus on the effects of electron-electron thermal conduction. First, they cite arguments by Chevalier (1975) and Solinger, Rappaport, and Buff (1975) that conduction is efficient enough to make the supernova remnant’s interior almost isothermal. Second, they consider conduction between the supernova remnant and cold clouds that it engulfs. Radiative losses from the supernova remnant are negligible in this stage, so the clouds are evaporated and incorporated into the remnant. Considering this cloud evaporation, McKee and Ostriker modify the Sedov-Taylor solution for this stage of expansion, yielding two substages. In the first substage, the remnant has not swept up much mass from the hot ionized medium, so mass gain from evaporated clouds dominates. They show this mechanism actually modifies the Sedov-Taylor solution to a $t^{3/5}$ dependance. In the second substage, the remnant has cooled somewhat, decreasing the cloud evaporation, making mass sweep-up the dominant effect. The classic $t^{2/5}$ Sedov-Taylor solution is recovered.

The transition to the late stages occurs when the remnant has expanded and cooled enough that radiative cooling becomes important. Here, McKee and Ostriker pause to consider the properties of the remnant at this point (using numbers they calculate in later sections): the remnant has an age of 800 kyr, radius of 180 pc, density of $5 \times 10^{-3} cm^{-3}$, and temperature of 400 000 K. Then, they consider effects that affect the remnant’s evolution at this stage:

• When radiative cooling sets in, a cold, dense shell is formed by runaway cooling: in this regime, radiative losses increase as temperature decreases. This effect is important at a cooling radius where the cooling time equals the age of the remnant.
• When the remnant’s radius is larger than the scale height of the galaxy, it could contribute matter and energy to the halo.
• When the remnant’s pressure is comparable to the pressure of the hot ionized medium, the remnant has merged with the ISM.
• If supernovae happen often enough, two supernova remnants could overlap.
• After the cold shell has developed, when the remnant collides with a cold cloud, it will lose shell material to the cloud.

Frustratingly, they find that these effects become important at about the same remnant radius. However, they find that radiative cooling sets in slightly before the other effects, and continue to follow the remnant’s evolution.

The mean free path of the remnant’s cold shell against cold clouds is very short, making the last effect important once radiative cooling has set in. The shell condenses mass onto the cloud since the cloud is more dense, creating a hole in the shell. The density left behind in the remnant is insufficient to reform the shell around this hole. The radius at which supernova remnants are expected to overlap is about the same as the radius where the remnant is expected to collide with its first cloud after having formed a shell. Then, McKee and Ostriker state that little energy loss occurs when remnants overlap, and so the remnant must merge with the ISM here.

At this point, McKee and Ostriker consider equilibrium in the ISM as a whole to estimate the properties of the hot ionized medium in their model. First, they state that when remnants overlap, they must also be in pressure equilibrium with the hot ionized medium. Second, the remnants have added mass to the hot ionized medium by evaporating clouds and removed mass from the hot ionized medium by forming shells – but there must be a mass balance. This condition implies that the density of the hot ionized medium must be the same as the density of the interior of the remnants on overlap. Third, they state that the supernova injected energy that must be dissipated in order for equilibrium to hold. This energy is lost by radiative cooling, which is possible as long as cooling occurs before remnant overlap. Using supernovae energy and occurrence rate as well as cold cloud size, filling factor, and evaporation rate, they calculate the equilibrium properties of the hot ionized medium. They then continue to calculate “typical” (median) and “average” (mean) properties, using the argument that the hot ionized medium has some volume in equilibrium, and some volume in expanding remnants. They obtain a typical density of $3.5 \times 10^{-3} cm^{-3}$, pressure of $5.0 \times 10^{-13} cm^{-2} dyn$, and temperature of 460 000 K.

McKee and Ostriker also use their model to predict different properties in the galactic halo. There are fewer clouds, so a remnant off the plane would not gain as much mass from evaporating clouds. Since the remnant is not as dense, radiative cooling sets in later – and in fact, the remnant comes into pressure equilibrium in the halo before cooling sets in. Supernova thus heat the halo, which they predict would dissipate this energy by radiative cooling and a galactic wind.

Finally, McKee and Ostriker find the properties of the cold clouds in their model, starting from assuming a spectrum of cloud sizes. They use Hobbs’s (1974) observations that the number of clouds with certain column density falls with the column density squared, adding an upper mass limit from when the cloud exceeds the Jeans mass and gravitationally collapses. A lower mass limit is added from considering when a cloud would be optically thin to ionizing radiation. Then, they argue that the majority of the ISM’s mass lies in the cold clouds. Then using the mean density of the ISM and the production rate of ionizing radiation, they can find the number density of clouds and how ionized they are.

## Parker Instability

The three-phase model gives little prominence to magnetic fields and giant molecular clouds. As a tangent from McKee and Ostriker’s model, the Parker model (Parker 1966) will be presented briefly to showcase the variety of considerations that can go into modelling the ISM.

The primary motivation for Parker’s model are observations (from Faraday rotation) that the magnetic field of the Galaxy is parallel to the Galactic plane. He also assumes that the intergalactic magnetic field is weak compared to the galactic magnetic field: that is, the galactic magnetic field is confined to the galaxy. Then, Parker suggests what is now known as the Parker instability: that instabilities in the magnetic field cause molecular cloud formation.

Parker’s argument relies on the virial theorem: in particular, that thermal pressure and magnetic pressure must be balanced by gravitational attraction. Put another way, field lines must be “weighed down” by the weight of gas they penetrate: if gravity is too weak, the magnetic fields will expand the gas it penetrates. Then, he rules out topologies where all field lines pass through the center of the galaxy and are weighed down only there: the magnetic field would rise rapidly towards the center, disagreeing with many observations. Thus, if the magnetic field is confined to the galaxy, it must be weighed down by gas throughout the disk.

He then considers a part of the disk, and assumes a uniform magnetic field, and shows that it is unstable to transverse waves in the magnetic field. If the magnetic field is perturbed to rise above the galactic plane, the gas it penetrates will slide down the field line towards the disk because of gravity. Then, the field line has less weight at the location of the perturbation, allowing magnetic pressure to grow the perturbation. Using examples of other magnetic field topologies, he argues that this instability is general as long as gravity is the force balancing magnetic pressure. By this instability, he finds that the end state of the gas is in pockets spaced on the order of the galaxy’s scale height. He suggests that this instability explains giant molecular cloud formation. The spacing between giant molecular clouds is of the right order of magnitude. Also, giant molecular clouds are too diffuse to have formed by gravitational collapse, whereas the Parker instability provides a plausible mode of formation.

In today’s perspective, it is thought that the Parker instability is indeed part of giant molecular cloud formation, but it is unclear how important it is. Kim, Ryu, Hong, Lee, and Franco (2004) collected three arguments against Parker instability being the sole cause:

• The formation time predicted by the Parker instability is ~10 Myr. However, looking at giant molecular clouds as the product of turbulent flows gives very short lifetimes (Ballesteros-Paredes et al. 1999). Also, ~10 Myr post T Tauri stars are not found in giant molecular clouds, suggesting that they are young (Elmegreen 2000).
• Adding a random component to the galactic magnetic field can stabilize the Parker instability (Parker & Jokipii 2000, Kim & Ryu 2001).
• Simulations suggest that the density enhancement from Parker instability is too small to explain GMC formation (Kim et al. 1998, 2001).

## Does it hold up to observations?

The paper offers several key observations justifying the model.  First, of course, is the observed supernova rate which argues that a warm intercloud medium would self-destruct in a few Myr.  Other model inputs include the energy per supernova, the mean density of the ISM, and the mean production rate of UV photons.

They also cite O VI absorption lines and soft X-ray emission as evidence of the three-phase model.  The observed oxygen line widths are a factor of 4 smaller than what would be expected if they originated in shocks or the Hot Ionized Medium, and they attribute this to the idea that the lines are generated in the conductive surfaces of clouds — a key finding of their model above.  If one observes soft X-ray emission across the sky, a hot component of T ~ $10^{6.3}$ K can be seen in data at 0.4-0.85 keV, which cannot be well explained just with SNRs of this temperature (due to their small filling factor).  This is interpreted as evidence for large-scale hot gas.

## So can it actually predict anything?

Sure!  Most importantly, with just the above inputs — the supernova rate, the energy per supernova, and the cooling function — they are able to derive the mean pressure of the ISM (which they predict to be $3700 K cm^-3$, very close to the observed thermal pressures).

## Are there any weaknesses?

The most glaring omission of the three-phase model is that the existence of large amounts of warm HI gas, seen through 21cm emission, is not well explained; they underpredict the fraction of hydrogen in this phase by a factor of 15!  In addition, observed cold clouds are not well accounted for; they should disperse very quickly even at temperatures far below that of the ISM that they predict.

## 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