The Pressure Structure of Molecular Clouds


Abstract. Broadly, we seek to understand the role of pressure in star forming molecular clouds. We examine molecular line data of the Perseus region from the COMPLETE survey alongside radiative transfer-processed ‘observations’ of the turbulent simulations of S. Offner to try to (1) understand to what extent we can actually measure pressure through observations, and (2) study how pressure changes within a cloud’s substructure.


  1. Internal pressure in GMCs acts to resist collapse locally and is an important part of the force balance equation controlling the efficiency of star formation

  2. Pressure of the ambient ISM “external” to clouds may act to help confine them against dissipation on short timescales – i.e. clouds may be in pressurized virial equilibrium even if unbound according to simple virial analysis (Field, Blackman & Keto 2011).

  3. Clouds are complex, turbulent entities that contain hierarchical structure and both large- and small-scale features (in density, velocity, possibly magnetic fields).

  4. Molecular line observations of clouds reveal some of this structure but are limited to a small dynamic range in density (for a single tracer) and one dimension of the velocity field. Observations also represent only a snapshot in what is in reality a dynamic system. Simulations have the advantage of allowing the time evolution within clouds to be followed, and including the full (“true”) six-dimensional phase space information, but are heavily reliant on the (necessarily limited) input physics.

  5. Dendrograms represent a powerful tool for probing hierarchical structure and can be applied to real and simulated data to make progress on understanding the pressure structure within and external to clouds.

  6. We have analyzed data from COMPLETE and synthetic cubes from radiative transfer-processed turbulent simulations to attempt to clarify the role of pressure within clouds and in the star formation process.

\(^{13}\)CO integrated intensity map of IC348 in Perseus.

Expectations from Simple Scaling Laws


Pressure scaling: \[P \propto \rho \sigma_v^2\]

Density and velocity scale as power laws: \[\begin{aligned} \rho &\propto& L^{-a} \\ \sigma_v &\propto& L^b\end{aligned}\] Here \(\sigma_v\) is the 3d velocity dispersion, related to the 1d dispersion for isotropic fields by \(\sigma_{v,{\rm 1d}} = \sqrt{3} \sigma_{v,{\rm 3d}}\).

What values do we expect for the exponents \(a\) and \(b\)? Table \ref{tab:scaling} gives some possible examples. Combining these equations, we see that pressure scales as \[P \propto L^c; c=2b-a\] where \(-2 \lapprox c \lapprox 1\), with the lowest value for the exponent for thermal gas in a SIS envelope, and the highest value for turbulent, constant density gas. (Is the latter realistic? Probably not, limiting exponent to \(c < 0\)). Note Larson’s relations (updated for modern observations, and focusing on scales larger than core scales) give \(b \sim 0.5\) (Larson 1981, Solomon 198?, Goodman 199?, others?).

Upshot: we expect pressure to decrease toward small scales in clouds. We will test this with our data and simulations.

to do: think about: wouldn’t scaling expectations change as a function of size sampled? a cloud as a whole does NOT have the same density structure as a core – swiss cheese vs. bbs


Scaling exponents
condition \(a\) condition \(b\)
Singular Isothermal Sphere envelope 2 turbulent (Burger’s) 0.5
SIS inner free-falling region 1.5 turbulent (Kolmogorov) 0.33
Constant density 0 thermal 0

Data and Simulations


Observational data

We have analyzed molecular line data from the COMPLETE survey of star-forming regions. This work focuses on \(^{13}\)CO \((J=1\rightarrow0)\) observations of Perseus. These were taken with the FCRAO telescope, providing spatial resolution of 46\(\arcsec\) (Nyquist-sampled to 23\(\arcsec\) pixels) and a final Hanning-smoothed (I assume?) spectral resolution of 0.066 km s\(^{-1}\). \(^{13}\)CO traces densities from a few hundred to a few thousand cm\(^{-3}\) (check these numbers, and provide a reference) and should be within the inertial range of the turbulent cascade, following scales from a few pc down to 0.1 pc or so.

Perseus is a “low-mass” star-forming region but contains a broad range of physical environments, from recently formed young clusters (IC 348, NGC 1333), to quiescent but perhaps imminently star-forming gas (B5, L1451). We focus on sub-regions within Perseus in order to investigate whether environment plays a role in the pressure structure and external pressure.

IC 348 parameters derived by Pineda et al. (2008) (all for ^13CO):
\(T_{B,\rm max}=0.93\) K
\(v_{\rm LSR}=8.99\) km s\(^{-1}\)
\(\sigma_v=0.95\) km s\(^{-1}\)
\(T_{\rm ex}=12\) K

to do: add data for additional regions


In order to assess the accuracy to which we can recover physical parameters from the observations, we perform a hydrodynamic simulation based on the inferred average properties of IC348. Factors of two differences between the simulated and true densities and temperatures can lead to significant discrepancies between the simulated and observed molecular emission. Consequently, we converge on the final simulation parameters by iterating within the limits of the observed uncertainties.

To construct the model clouds we follow a procedure similar to that of \citet{Offner13}. The simulations assume a periodic box of side length 4 pc. Thus, the domain represents a piece of a molecular cloud, a reasonable approximation given that the target region, IC 348, lies within the larger Perseus cloud complex. The gas is assumed to be a constant temperature of 15 K, which is comparable to the slightly warmer inferred temperatures for the region \citep{citation}. The gas density is assumed to be \(n_{{\rm H}_2} = 850\) cm\(^{-3}\) so that the total simulated gas mass is 3780 \(\msun\). The observed CO emission and dust extinction provide mass estimates that differ by a factor of 2 \citep{citation}. We adopt the higher value for the simulation mass, and hence mean density, because it yields a better match to the measured CO excitation temperatures. The 3D Mach number is 10. This gives a velocity dispersion, \(\sigma=1.4\) kms\(^{-1}\), which is slightly higher than the typical linewidth-size relation at 4pc (e.g. \citealt{MandO07}) but matches the IC348 linewidths well .

To obtain initial conditions, we begin with a uniform density and apply velocity perturbations for three cloud crossing times. The perturbations have a flat power spectrum between \(k=1-2\) and represent large-scale turbulent energy injection. Following the driving phase, we turn on self-gravity and allow the gas to collapse and form stars. The stars are represented by sink particles according to the method of \citet{krumholz04}. The calculation has a 256\(^3\) basegrid and 4 levels of refinment, which corresponds to a maximum cell resolution of 200 AU. Turbulent driving continues throughout the calculation to ensure the velocity dispersion remains constant. We perform the comparison with observations when the calculation has reached half a freefall time, at which point the star formation efficiency is 3.2%. Simulations that do not include feedback, such as this one, tend to overestimate the stellar mass by a factor of three or more at one freefall time (e.g., \citealt{hansen12,krumholz13}).

The simulation does not include magnetic fields, which may influence the dynamics of the region. However, observations suggest that magnetic fields do not dominate on core to cloud scales \citet{crutcher12}. We have no direct information about the mean field strength or distribution in IC348 specifically.

to do: put this in table form


To obtain the CO abundance distribution and gas temperature variation, we post-process the simulation with 3d-pdr (Bisbas et al. 2012). Given some input illumiating UV field 3d-pdr uses a HEAL_pix ray-tracing method to solve for the UV field at each point in the cloud \citep{gorski05}. It subdivides the cloud into ionized (\(n_H<50\)cm\(^{-3}\)), molecular (\(n_H>10^4\)cm\(^{-3}\)), and PDR (\(50\leq n_H \leq 10^4\) cm\(^{-3}\)) regions. It then iterates the chemical networks to convergence in the PDR region, assuming that the gas achieves chemical equillibirum at a specified time. Towards high densities the PDR region abundances limit to the those in molecular region, which are assumed to be constant. We adopt 12 HEAL-pix rays and a chemical time of 10 Myr. The chemical reaction rates are supplied by the UMIST2012 database \citep{mcelroy13}, and the chemical network contains 215 species and more than 3000 reactions Need to check with Thomas about ray number, and reaction network. Several bright stars are radiating IC348 from above and potentially enhance the CO excitation. Consequently, we adopt an isotropic 1 Draine field at the clound boundaries plus an additional plane-parallel 10 Draine UV field along the \(x\) direction, where “1 Draine" is the standard stellar radiation field \citep{draine78}. We find that a cloud irradited in this manner is slightly warmer and has slightly higher brightness temperatures (see below) than an identical cloud irradiated with a completely uniform 1 Draine field.

Emission Line Modelling

We use the non-LTE radiative transfer code radmc-3d1 to produce a \(^{13}\)CO(1-0) emission cube. radmc-3d requires the three-dimensional gas density, velocity, and temperature fields as inputs. Isotopologues are not included in the 3d-pdr network, so we compute the \(^{13}\)CO abundance from the \(^{12}\)CO abundance by assuming an constant isotopic ratio of [12CO]/[13CO]=60 \citep{langer93}. radmc-3d solves the equation of radiative statistical equillbrium to obtain the level populations and then computes the resulting line emission using the large velocity gradient (LVG) approximation of the radiative transfer equation \citep{shetty11}. We compute the excitation assuming that the only collisional partners of CO are H and \(H_2\), where these number densities are provided by the 3d-pdr chemical network.

In the final step of the process, we account for observational resolution. First, we assume the simulated cloud has a distance of 320 pc, the distance of IC 348, which sets the physical resolution for the synthetic observations. Next, we convolve the spectral cube with a 46(FWHM) beam, rebin to the observational pixel resolution of \(\Delta x= 23\), and add Gaussian noise with an rms dispersion of \(\sigma_{\rm rms}=0.15\) K (this is the sigma-clipped standard deviation of the off-channels in the observational cube). Finally, we rebin the velocity channels to match the COMPLETE spectral resolution of 0.066 km s\(^{-1}\).

Observing Pressure

Pressure \(P=\rho \sigma_v^2\) is computed at each level of each leaf of the dendrogram. \(\rho\) is not a direct observable, and is calculated as follows. Consider a set of pixels belonging to a given level in a given leaf, which we will call a contour. First, the CO intensity is summed over the contour and multiplied by the cube pixel size (in pc\(^2\) km s\(^{-1}\)) to obtain the ‘3-d integrated’ CO intensity \(I_{\rm 3d}\) (note that this is effectively a discrete integration). This is then converted to a total molecular gas mass \(M_{\rm mol}\) via

\[M_{\rm mol} = 6.7\, \mu \, m_{\rm H} \, I_{\rm 3d} \, X_{\rm CO},\]

where \(\mu\) is the mean molecular weight (taken to be 2.715 to account for H\(_2\) and He) and m\(_{\rm H}\) is the hydrogen atomic mass. The factor of 6.7 corrects for the relative underabundance of \(^{13}\)CO with respect to \(^{12}\)CO. We use the ‘standard’ X-factor of \(2.0~\times~10^{20}\) cm\(^{-2}\) (K km s\(^{-1}\))\(^{-1}\).

\(M_{\rm mol}\) is the total mass within the contour. To convert to a density, we estimate the spatial extent of the contour using the second moments of the set’s pixel distribution and then assume that the (unseen) third dimension is identical for simplicity. In particular, we define the ‘size’ as \(R=1.91 \, \sqrt{\sigma_x \, \sigma_y}\), where \(\sigma_i\) is the intensity-weighted standard deviation of the pixel distribution in direction \(i\), converted to physical units. \(R\) is essentially a characteristic spread in the \(xy\) pixel distribution for the set of pixels in a given contour. \(\rho\) is then simply \(M_{\rm mol} / V\), where the volume \(V = \pi^{3/2} \, R^3\), and we have assumed quasi-spherical geometry.

The 3-d velocity dispersion that enters into the pressure is calculated using the line-of-sight velocity and the assumption of an isotropic velocity field. For a given contour, \(\sigma_v\) is the intensity-weighted standard deviation of the (line-of-sight) velocity distribution of the pixels in that contour. are we missing a factor of sqrt(3)??.

Pressure is then put into the ‘standard’ units of \(P/k_{\rm B}\) in K cm\(^{-3}\).

\(^{13}\)CO integrated intensity of Perseus from COMPLETE, with the regions investigated in this work highlighted.

Pressure of a Singular Isothermal Sphere

Shu (1977) gives the analytic predictions for a a singular isothermal sphere (SIS). To see if we recover the expected dependences, we have conducted synthetic \(^{13}\)CO observations of a simulation of a SIS. The scale of the data cube was 0.42 pc x 0.42 pc x +-6 km/s. The peak density is 10\(^5\) cm\(^{-3}\). A small amount of noise was added to the data post-radiative transfer. (nb: 2 smaller SIS’s were also examined (0.052 and 0.22 pc\(^3\)), and due to the higher densities leading to partial or complete depletion of CO, the structures we recover are correspondingly skewed).

TO DO: Compare the density and pressure profiles of the SIS to individual cores in the simulation and observation.

Pressure of observed SIS (0.42 pc\(^3\) box). We expect \(\rho(r) \propto r^{-3/2}\) and \(v \propto r^{-1/2}\). But note that this \(v\) is NOT \(\sigma_v\), the velocity dispersion. \(v\) is purely infall (aside from possible noise? I assume no turbulence here?), so if a contour has a constant radius, \(\sigma_v\) should approach 0 as the width of the contour shrinks. There should be some thermal velocity, though, so at minimum, \(\sigma_v = \sigma_{\rm th} = \sqrt{k_B \, T/(\mu \, m_H)}\).

Note also that for isothermal gas with an ideal equation of state \(P \propto \rho\) and so \(P \propto r^{-3/2}\).




summary of dendrogram procedure and references to e.g. Rosolowsky et al. 2008)

“Observing” Pressure

description of pressure calculation, choice of dendrogram parameters, and dendrogram plot overlay

to do: fill in text for this section



to do: decide on final dendro parameters, match obs, sim ppv, sim ppp. Pick another region from Perseus. Make table of bulk properties of each region (size, mean density, X-factor, etc) from Pineda et al. 2008.

Show plots of density, velocity, pressure vs. size?

Comparison of observations and simulations

Comparison of raw simulation output (ppp) and synthetic data cubes (ppc)

to do: ppp levelprops?

Dendrograms of IC 348 with pressure overlaid in color scale (add a 4th region). The y-axis is main beam-corrected \(^{13}\)CO brightness temperature.

As above for B5
As above for L1448.

Dendrogram of the synthetic \(^{13}\)CO data cube (z-axis view) with pressure overlaid. Dendrogram parameters were the same as for the observed cubes. The y-axis is synthetic \(^{13}\) CO brightness temperature.

Dendrogram of 3d density field from simulation. The y-axis is gas density in g cm\(^{-3}\).



External pressure in molecular clouds

The pressure at the surface of the outermost structure of the regions we study here is \(P/k_B \approx 1-5\times 10^4\) K cm\(^{-3}\).

Pressure, velocity dispersion, and virial parameter vs. size scale in the raw simulation output (no PDR, no RT). Pressure appears to decrease with scale, though with significant scatter. (do not use)

For each leaf, we can look at the change in velocity dispersion \(\sigma_v\) as a function of size for various contours within the leaf. Both are computed as described above.

Size-linewidth relation for individual leaves. Did we compute the slope including all contours, including below where leaves merge?

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