Harvard Astronomy 201b

ARTICLE: The “true” column density distribution in star-forming molecular clouds

In Uncategorized on March 24, 2011 at 3:03 am

Read the paper by A.A. Goodman, J.E. Pineda, and S.L. Schnee (2008)

Summary by Bekki Dawson

Abstract

We use the COMPLETE Survey’s observations of the Perseus star-forming region to assess and intercompare the three methods used for measuring column density in molecular clouds: near-infrared (NIR) extinction mapping; thermal emission mapping in the far-IR; and mapping the intensity of CO isotopologues. Overall, the structures shown by all three tracers are morphologically similar, but important differences exist among the tracers. We find that the dust-based measures (NIR extinction and thermal emission) give similar, log-normal, distributions for the full (~20 pc scale) Perseus region, once careful calibration corrections are made. We also compare dust- and gas-based column density distributions for physically meaningful subregions of Perseus, and we find significant variations in the distributions for those (smaller, ~few pc scale) regions. Even though we have used 12CO data to estimate excitation temperatures, and we have corrected for opacity, the 13CO maps seem unable to give column distributions that consistently resemble those from dust measures. We have edited out the effects of the shell around the B-star HD 278942 from the column density distribution comparisons. In that shell’s interior and in the parts where it overlaps the molecular cloud, there appears to be a dearth of 13CO, which is likely due either to 13CO not yet having had time to form in this young structure and/or destruction of 13CO in the molecular cloud by the HD 278942’s wind and/or radiation. We conclude that the use of either dust or gas measures of column density without extreme attention to calibration (e.g., of thermal emission zero-levels) and artifacts (e.g., the shell) is more perilous than even experts might normally admit. And, the use of 13CO data to trace total column density in detail, even after proper calibration, is unavoidably limited in utility due to threshold, depletion, and opacity effects. If one’s main aim is to map column density (rather than temperature or kinematics), then dust extinction seems the best probe, up to a limiting extinction caused by a dearth of sufficient background sources. Linear fits among all three tracers’ estimates of column density are given, allowing us to quantify the inherent uncertainties in using one tracer, in comparison with the others.

Goals

  • Assess three common tracers for measuring column density in molecular clouds: dust extinction, dust emission, and gas (13Co) emission.
  • Estimate true density distribution and compare to simulations

 

We use the COMPLETE Survey’s observations of the Perseus star-forming region to assess and intercompare the three methods used for measuring column density in molecular clouds: near-infrared (NIR) extinction mapping; thermal emission mapping in the far-IR; and mapping the intensity of CO isotopologues. Overall, the structures shown by all three tracers are morphologically similar, but important differences exist among the tracers. We find that the dust-based measures (NIR extinction and thermal emission) give similar, log-normal, distributions for the full (~20 pc scale) Perseus region, once careful calibration corrections are made. We also compare dust- and gas-based column density distributions for physically meaningful subregions of Perseus, and we find significant variations in the distributions for those (smaller, ~few pc scale) regions. Even though we have used 12CO data to estimate excitation temperatures, and we have corrected for opacity, the 13CO maps seem unable to give column distributions that consistently resemble those from dust measures. We have edited out the effects of the shell around the B-star HD 278942 from the column density distribution comparisons. In that shell’s interior and in the parts where it overlaps the molecular cloud, there appears to be a dearth of 13CO, which is likely due either to 13CO not yet having had time to form in this young structure and/or destruction of 13CO in the molecular cloud by the HD 278942’s wind and/or radiation. We conclude that the use of either dust or gas measures of column density without extreme attention to calibration (e.g., of thermal emission zero-levels) and artifacts (e.g., the shell) is more perilous than even experts might normally admit. And, the use of 13CO data to trace total column density in detail, even after proper calibration, is unavoidably limited in utility due to threshold, depletion, and opacity effects. If one’s main aim is to map column density (rather than temperature or kinematics), then dust extinction seems the best probe, up to a limiting extinction caused by a dearth of sufficient background sources. Linear fits among all three tracers’ estimates of column density are given, allowing us to quantify the inherent uncertainties in using one tracer, in comparison with the others.

Legacy of the paper

Here are some statements attributed to this paper by the papers that cite it. While observers found the comparison of the three tracers useful (and unsettling), the theorists were cheered by the column density distribution’s consistency with log normal, as expected from turbulent models. At the end of this summary, we assess whether these citing statements are actually justified by the results.

Theoretical papers:

“The column-density distributions of both the cloud as a whole and that of smaller, few parsec ‘subregions’ are lognormal.” Tasis et al. 2010, Do lognormal column-density distributions in molecular clouds imply supersonic turbulence?, MNRAS

“Due to the filamentary, fractal structure of the interstellar medium, and due to the large density contrasts, star formation in turbulent molecular clouds proceeds in multiple regions at the same time in parallel, reflecting the hierarchical and fractal nature of the gas density probability distribution.” Federrath et al. 2010,
Modeling Collapse and Accretion in Turbulent Gas Clouds: Implementation and Comparison of Sink Particles in AMR and SPH, ApJ.

Observational papers:

“Near-infrared extinction mapping is the best tracer of the ‘real’ column density of a cloud, based only on the assumption of a constant gas to dust ratio.” Froebrich and Rowles, 2010, The structure of molecular clouds – II. Column density and mass distributions, MNRAS.

“A larger dynamical range can be achieved by thermal dust emission observations, which are used to trace intermediate to high column densities, and dust extinction mapping, which is more sensitive at low to intermediate column density range.” Schmalzl et al. 2010, Star Formation in the Taurus Filament L 1495: From Dense Cores to Stars, ApJ

“The commonly used methods for deriving column densities, i.e. measurements of CO line emission, thermal dust emission, and dust extinction, suffer from various model-dependent effects, and often probe only narrow ranges of physical conditions.” Kainulainen et al. 2009, Probing the evolution of molecular cloud structure. From quiescence to birth, A&A.

Comparing the three tracers

Dust Extinction

Dust extinction is measured by the reddening of stars:

intrinsic star color + dust = observed star color

Thus from the stellar reddening, we can infer the amount of dust present. Assuming a gas-to-dust ratio, we can convert this measurement to a column density. The NICER algorithm is an approach to measure the stellar reddening in units of visual extinction magnitudes Av (see the Interstellar Reddening section of Properties of Stars for a tutorial on visual extinction magnitudes). The greater the extinction magnitude, the greater the column density.

NICER = Near-Infrared Color Excess Revisited) (Lombardi and Alves 2001)

where N is the number of angularly close stars, (H-K)iobs is the color of the ith star, and <(H-K)tr> is the average color of the stars in the control field.

Here is a schematic I made of how the NICER algorithm works:

The box outlined in blue is the control field, selected because the stars in that field are blue enough that we think they are not significantly reddened by dust along the line of sight. The region of interest is divided into a grid, and the average color of stars in each grid is computed. Because the reddening by dust is typically much larger than any intrinsic variability in the color of stars, the difference of the average color of the stars in each cell from the control field yields the extinction magnitude. Abnormally blue stars (bottom left) are assumed to be foreground stars and are removed from the calculation. See Lombardi and Alves 2001 for more details on the NICER algorithm.

Pros and cons of using dust extinction as a tracer:

Pros:

  • Colors of stars are similar enough in the IR that the difference can, to a good approximation, be attributed to dust, not the range of star temperatures and IR has suitable dynamical range for star-forming regions (which would appear almost completely opaque in the optical)
  • Only assumption is gas-to-dust ratio

Cons:

  • Trade-off between uncertainty and resolution
  • Resolution is 10 times poorer than other maps and thus harder to compare cloud and core (Schnee et al. 2009).
  • Need to assume gas-to-dust ratio (and it varies along line of sight)

Dust Emission

The second tracer considered was emission by dust. The column density is derived by creating a color/temperature map from flux maps at two wavelengths, calibrating with the extinction to map to determine the density of dust, deriving the total density by assuming a gas-to-dust ratio, and then determining the column density.

The data for the dust emission map were taken from the Improved Reprocessing of the IRAS Survey (IRIS).

(IRAS – Infrared Astronomical Satellite)

Pros:

  • Resolution limited only by detector size
  • Ability to derive temperature

Cons:

  • Need to assume gas-to-dust ratio
  • Need flux maps at two wavelengths to derive a dust temperature
  • Need extinction maps to convert from optical depth to column density
  • Not sensitive to diffuse extended structures
  • L.o.s. variations in temperature (but can be corrected for)

Gas Emission

The H2 gas which dominates the gas density is too dense to trace directly. Therefore 13CO is used a tracer for H2 gas, which yields the column density.

Problem: The amount of emission depends on collision rate (which depends on density), temperature (Trot → kinetic energy from rotational transition), and the optical depth.

The “special trick”:
Use 12CO (optically thin) to get the temperature
Use 13CO (optically thick) to get the optical depth
→ Combine with line profile to get density

Pros:

  • Ability to kinematically separate regions along l.o.s.
  • May be better on small scales (<0.1 pc) where dust traces gas less well

Cons:

  • Excitation threshold not always met
  • Chemically depleted at high densities
  • Fails to work at high optical depths, even when corrections for opacity are made
  • Thus a lack of dynamical range: Underestimates at low column density (high temperature) and at high column density (high optical depth)
  • Variations in amount of with respect to hydrogen gas throughout cloud: doesn’t trace hydrogen perfectly

Equation 1 for converting from gas emission to equivalent extinction magnitudes:

Comparison of the three tracers

Figure 1 of the paper compares the maps from the three tracers:

Figure 1 (caption taken from paper): Maps in the left column show column density as traced by: extinction (a); Dust Emission (c); and Gas Emission (e). Figure 1b shows the small uncertainties associated with the NICER extinction map in Figure1a. Dust color temperatures based on 60 and 100 μm data are shown in Figure 1d. Note the warm dust associated with the shell around HD278942. The opacity of 13CO, which is correlated with column density, is shown in Figure 1f. In all panels, only data points with detections of 13CO with S/N ≥5 are shown, and the resolution is 5′. The IC348 and NGC1333 regions have been excised from the datasets, because all three techniques are biased in cluster regions. The single pink contour surrounds the apparently heated material around HD278942, and the single light blue contour outlined the “overdense” area presumably created by HD278942’s shell, showing the dearth of 13CO indicated most clearly in Figure 3.

Figure 3 caption (from paper): Intercomparison of measured column density distributions, all shown in units of AV (see the text). The horizontal dotted lines in the top and bottom panels show the minimum column density measurable with 13CO corresponding to the additive constant in Equation 1. Note that the gray scale color of the data points shows the “third” measure of column density not plotted on the x or y axis for each plot, so that any points that do not look to be part of a smooth gray gradient are “outliers” in that third measure. Points colored blue and pink correspond to the blue-outlined shell exterior and the pink-outlined shell interior in Figure 1. The pink and blue point are shown here for illustrative value, but they are not included in the histograms in Figures 1 and 5, or in any fits to column density distributions, as explained in the text. The 45 degree lines are not fits: they simply show a 1:1 relationship that might reasonably be expected given that the IRAS-based and 13CO-based measures have been calibrated to best match the 2MASS-based AV distribution overall (see the text).

Sass the gas

I have annotated a panel of Fig. 3 to highlight the weaknesses of the gas tracer method:

Schematic of the origin of the discrepancies

Figure 4 caption (from paper): Schematic diagram showing various lines of sight through various conditions. A, B, and C, are views as seen from our vantage point on Earth. In A, we have the favorable situation where 13CO is in near-LTE (shaded purple), and all of the low-density material around it, some of which does not emit in 13CO at all (shaded gray), but still shows up in dust measures, is associated with the cloud of interest. In B, the situation is as in A, but an additional region which emits in dust but not in 13CO (either due to low density or due to a dearth of 13CO) is included. In C, the densest gas the line of sight passes through (shown as marbled gray) is dense enough to excite some 13CO, but not at a level truly indicative of the full column density present, because the collisions at this “subcritical” density are too infrequent. In D, we see an “alien’s view” of the same cloud, which passes through the “dust-only” zones but no 13CO emitting regions, even though it crosses A, B, and C.

Professor Goodman noted that this may be the first picture of an alien to appear in ApJ.

The column density distribution

Entire region

Figure 2 caption (from paper): Pixel-by-pixel column density histograms for the full Perseus COMPLETE data set shown in Figure 1: Note that all tracers are smoothed to a common 5′ resolution, and only points where 13CO is reliably detected are included (see the text). Each distribution shows the column density as labeled on the plot, converted to units of AV, as explained in the text. In every panel, the solid smooth blue curve shows the Gaussian fit to the 2MASS/NICER-based distribution, for reference. The red smooth curve in the middle panel, and the green smooth curve in the lower panel, show Gaussian fits to the IRAS- and 13CO-implied distributions, respectively. The gray shaded histogram in the bottom panel shows W(13CO) converted to units of AV, but the fit shown is only for the curve just labeled “13CO,” which gives total column density calculated using Equation 1. Fit parameters for all four column density distributions shown in this Figure are given in Table 1. The dashed vertical line extending through all three panels shows the cutoff (lowest) value of column density measurable with 13CO, according to Equation 1. The short horizontal bar centered on the dashed line in the top pane shows the 1σ spread in distribution of |AV(2MASS) – AV(13CO)|/AV(13CO) and for the middle pane the same dispersion but for |AV(IRAS) – AV(13CO)]/AV(13CO)|. Figure 5 shows the regional breakdown of these same histograms. Figure 3 shows a point-to-point comparison of the different tracers.

Subregions

Note that some tracers (and some subregions) appear more consistent with lognormal than others. For example, in the shell region (middle left panel), the extinction tracer appears to follow a lognormal distribution but the emission distribution is skewed/asymmetric and the gas tracer distribution appears more like a power law.

Figure 5 caption (from paper): Regional variations in column density distributions. Each panel shows the same assortment of measured distributions as in Figure 2 only without W(13CO). As explained in more detail in Figure 2’s caption and in the text, the vertical dotted line shows the minimum AV measurable with 13CO and the short horizontal bars show indicate the characteristic scatter about the relationship between each dust tracer and 13CO. Parameters for Gaussian fits to normalized versions of these distributions are given in Table 2. The short vertical lines hanging from the top axis indicate the mean value of extinction for each distribution (by color), and the means of the normalized extinction are given in Table 2. Note that we use “frequency” on the y-axis in these plots, rather than “number” as in Figure 2, to facilitate inter-comparison of the various regions, not all of which include the same total number of positions (see Table 2).

Consistency with turbulence model

Figure 6 caption (from paper): Comparison of the fit parameters listed in Table 2 with each other, and with the 1:1 line predicted, for a log-normal, by Equation 5.

Note: x is the normalized column density

By the properties of logarithms, a log normal distribution should obey the following relation:

Thus the subregions generally appear consistent with log-normal, indicating a self-similarity on a range of scales, as we would expect from turbulence.

List of conclusions from paper

(Note: these are copied directly from the paper, which included a helpful summary of its conclusions.)

  1. The column density distribution of material in the full Perseus star-forming region, with 1 < AV < 12 mag, is roughly log-normal, when it is not directly affected by embedded clusters or young stellar outflows (bipolar or spherical).
  2. Dust is superior to molecular lines for tracing out the “full” mass distribution over the range of extinction studied, because it does not require a threshold density to “excite” and it does not die out at high densities due to high opacity or chemical depletion, the way 13CO does.
  3. The dearth of molecular gas (13CO) in the region corresponding the shell created by the B-star HD 278942 suggests that either CO has not yet formed in this young structure, and/or that the existing molecular gas has been dissociated by Shell’s interaction with the cloud.
  4. When Perseus is dissected into smaller “subregions,” the column density distributions become more ragged, as is predicted by simulations for samples that are statistically small. However, we find that the subregions distributions are still log-normal-like, in that the relationship between their normalized means and their dispersions follows a trend consistent with log-normal distribution.
  5. In comparing observations of column density (or mass) distributions with each other, and/or with simulations, it is perhaps more important than has been previously appreciated to account for the effects of biases due to dust temperature variations, abundance variations, opacity effects, and observing strategies.

Assessment of citations

Are the statements attributed to this paper actually justified?

“The column-density distributions of both the cloud as a whole and that of smaller, few parsec ‘subregions’ are lognormal.” Tasis et al. 2010, Do lognormal column-density distributions in molecular clouds imply supersonic turbulence?, MNRAS Assessment: The distributions are consistent with being lognormal. A thorough assessment of which form best matches the distribution was beyond the scope of this paper.
“Due to the filamentary, fractal structure of the interstellar medium, and due to the large density contrasts, star formation in turbulent molecular clouds proceeds in multiple regions at the same time in parallel, reflecting the hierarchical and fractal nature of the gas density probability distribution.” Federrath et al. 2010,
Modeling Collapse and Accretion in Turbulent Gas Clouds: Implementation and Comparison of Sink Particles in AMR and SPH, ApJ.
Assessment: the distributions are indeed consistent with lognormal on the range of scales considered, but the evidence is weaker for the smaller regions.

Observational papers:

“Near-infrared extinction mapping is the best tracer of the ‘real’ column density of a cloud, based only on the assumption of a constant gas to dust ratio.” Froebrich and Rowles, 2010, The structure of molecular clouds – II. Column density and mass distributions, MNRAS. Assessment: Extinction mapping indeed appears to be the most robust tracer, but it also disadvantages, like lack of resolution.
“A larger dynamical range can be achieved by thermal dust emission observations, which are used to trace intermediate to high column densities, and dust extinction mapping, which is more sensitive at low to intermediate column density range.” Schmalzl et al. 2010, Star Formation in the Taurus Filament L 1495: From Dense Cores to Stars, ApJ Assessment: It may not be quite fair to relegate extinction mapping to low column densities; in fact, it has a large dynamical range and can be used at high densities as well. (Readers, do you agree?)
“The commonly used methods for deriving column densities, i.e. measurements of CO line emission, thermal dust emission, and dust extinction, suffer from various model-dependent effects, and often probe only narrow ranges of physical conditions.” Kainulainen et al. 2009, Probing the evolution of molecular cloud structure. From quiescence to birth, A&A. Assessment: This citation makes it seem like all of the tracers are useless. In fact, the three tracers show good agreement on several important aspects, including the consistency of the column density distribution with lognormal, and dust extinction is a particularly robust tracer.

Questions for Discussion

Unfortunately we didn’t have time for a detailed discussion of this paper. Here are some discussion questions. Feel free to chime in with a comment!

  • Why the authors decide not to “fix” gas by using the curve-of-growth and emission by accounting for l.o.s. temperature variations in this comparison? (as they did in their other paper, Pineda et al. 2008)
  • Can you think of other pros and cons to any of the tracers?
  • Are there other reasons the distribution might appear log-normal besides turbulence?
  • For the distribution of column densities, come up with your own shape and propose a physical explanation. High-density cores were found to follow a power-low distribution: could the gas be interpreted this way, or does the distribution derived from the gas tracer look that way due to the effects described in this paper?
  • What are the consequences of assuming a constant reddening-to-extinction ratio for the gas?
  • If need extinction map to convert from emission to column density, why not just use the extinction map itself? If you can just assume an emissivity, how does the quality of the conversion from emission to column density map change?
  • Does dust + gas win over just dust? What’s the best way to combine the three measurements?
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  1. […] However, recall that we should trust the dust and sass the gas. […]

  2. Re: Kainulainen’s citation – I don’t think this citation makes the tracers “seem useless”, but is instead a fair assessment of the paper’s results: all methods have limitations. Kainulainen uses extinction mapping in his own works, so this sentence is probably intended to demonstrate a familiarity with the limitations of the methods he employs.

    • I see your point that Kainulainen et al. may be intending to emphasize the limitations of these tracers, not dismiss them entirely. Thanks for weighing in!

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