SUBMILLIMETER POLARIZATION OF GALACTIC CLOUDS: A COMPARISON OF 350 μm AND 850 μm DATA

The Hertz instrument, its observing strategy, and data analysis are described in detail elsewhere (Kirby et al. 2005; Dowell et al. 1998; Schleuning et al. 1997; Platt et al. 1991). Here we briefly review the aspects relevant to the present work. Hertz incorporates two separate detector arrays which simultaneously measure the two orthogonal modes of linear polarization, modulated by a cryogenic half-wave plate (HWP). The 6 × 6 pixel² arrays have pixel center-to-center spacings of 178 and a beam size of approximately 20'' FWHM. The observing strategy involves rotating the instrument to follow the sky rotation throughout the night, as well as steps of order the array size to map areas larger than the 2' × 2' field of view. The rotation allowed a single pixel to continue observing the same patch of sky throughout the night. Additionally, step sizes were typically chosen to be an integer number of pixels; only rarely were maps made with samples spaced more closely than the 178 pixel pitch. As a result, these data do not meet the Nyquist criterion for a fully sampled map and we have, therefore, made no attempt to generate polarization maps at increased spatial sampling. All the polarization data reported by Dotson et al. (2010) and used here maintain a spatial resolution of 20''. (Measured beam profiles are given here in Figure 2 and in Figure 3 of Dowell et al. 1998.)

The SCUBA camera and polarimeter is described in detail elsewhere (Greaves et al. 2003; Jenness et al. 2000; Holland et al. 1999). Briefly, the 850 μm SCUBA-pol instrument consists of a single detector array with 37 pixels arranged on a hexagonal grid with a 23 field of view; the polarization is measured by inserting a warm wire grid, modulated by a stepped HWP. Fully sampled polarimetric and photometric maps are generated by "jiggle mapping" which moves the array by sub-pixel steps. The data analysis involves combining the individual jiggle maps and re-sampling them onto an output grid with a 6'' pitch. Generation of fully sampled maps in this manner also alleviates the need to follow sky rotation throughout the night. While the intrinsic SCUBA beam size at 850 μm is close to the 14'' diffraction limit of the JCMT, this map-making process results in an effective beam size closer to 20'' (Figure 2).

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As noted above, the resultant spatial resolution for both Hertz at 350 μm and SCUBA-pol at 850 μm is ≈20''. We refer to this resolution as the effective beamwidth, which should not be confused with the diffraction limited resolution of the respective telescopes. Direct comparisons of the two beams measured on Uranus are shown in Figure 2. (The size of Uranus at the time of each measurement was 37.) Gaussian fits to these profiles yield 195 ± 01 for SCUBA-pol and 19'' ± 2'' for Hertz; the reported statistical uncertainties follow from a formal nonlinear fit to the Gaussian profile. These beam sizes are consistent with the measurements reported by Dowell et al. (1998) for Hertz (20'' ± 2'') and Di Francesco et al. (2008) for SCUBA (a "primary" beam of 195). Given this beam similarity we have made no correction for different spatial resolutions when comparing these data sets.

The Hertz data are undersampled. However, since the SCUBA data are fully sampled, there is sufficient information to estimate the SCUBA intensity and polarization at the same sky locations of the Hertz data. This is accomplished by reducing the SCUBA-pol data in the same manner as presented in Matthews et al. (2009) but choosing to output the data to grids and map-center locations which match the Hertz data set. Table 1 lists the objects observed by both Hertz and SCUBA-pol at 350 and 850 μm; Table 2 (in the electronic version only) gives a more complete list of the locations within each of the clouds. Table 2 also includes data for all points at both wavelengths for the polarization magnitudes and their ratio P(850)/P(350), position angles and their difference ϕ(850)–ϕ(350), intensity values and their ratio F(850)/F(350), and uncertainties on all values. All polarization magnitudes in Table 2 have been corrected for positive bias (Section 2.2). The best estimates of those values are sometimes zero; as a result the ratio P(850)/P(350) is reported as Nan for cases in which P(850) = P(350) = 0, Inf for cases in which only P(350) = 0, and equal to zero when only P(850) = 0.

By definition, the polarization amplitude is a positive-definite quantity. As a result, a noisy measurement of a truly zero-polarization source will result in a mean positive polarization. While there is no exact analytical method to correct for this bias many authors use the formula: P_c ≈ (P²_m − σ²_p)^1/2, where P_c is the bias-corrected polarization, P_m is the measured polarization, and σ_p is the measured polarization uncertainty. Vaillancourt (2006; also Simmons & Stewart 1985; Quinn 2012) showed this was a good estimator when P_m ≳ 3σ_p but that P_c = 0 was a better estimator if $P_\mathrm{m}/\sigma _p \le \sqrt{2}$. Matthews et al. (2009) applied the formula for high signal to noise to all their 850 μm data while Dotson et al. (2010) applied no corrections to their 350 μm data. For the data presented in this work we set P_c = 0 in cases where $P_\mathrm{m}/\sigma _p \le \sqrt{2}$ and apply the above formula for data with $P_\mathrm{m}/\sigma _p > \sqrt{2}$. When computing signal-to-noise cuts on P/σ_p in Table 1 and the following sections we use the corrected values P_c as described above. The conclusions drawn in Sections 3 and 4 use only the P_c ≳ 3σ_p sample and are therefore not affected by the fact that the correction does not provide the best estimate within the regime $\sqrt{2} \le P/\sigma _p < 3$.

The corrections on P have no effect on estimates of the polarization position angle; that is in the sense that there is no bias in the angle estimate as there is for the polarization amplitude. However, there is necessarily an effect on the angle uncertainty (e.g., Naghizadeh-Khouei & Clarke 1993). For example, in cases where the $P_\mathrm{m}/\sigma _p \le \sqrt{2}$ the best estimate is P_c = 0 and thus any angle measurement is meaningless. We have made no attempt to estimate "corrected" angle uncertainties in Table 2. Such corrections are small for high signal-to-noise data (Naghizadeh-Khouei & Clarke 1993) and therefore they will have little effect on our analysis and discussions below which use only data with P_c ≳ 3σ_p.

One of the most obvious aspects of the polarizations presented in the maps is the agreement, or disagreement, of the position angles between the two wavelengths. Figure 3 compares these angles across the entire data set and within three clouds with the largest number of data points. All the distributions peak near angle differences of 0°, that is Δϕ ≡ ϕ(850)–ϕ(350) = 0°. For all nonzero-polarization data (i.e., data where both P(350) > 0 and P(850) > 0) the median angle difference is 1° with a standard deviation of 39°. This agreement is stronger (i.e., has a smaller deviation) if we limit ourselves to only points with P ⩾ 3σ_p (398 points in the entire data set). In this case the median angle difference is 4° with a standard deviation of 28°.

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To rule out the possibility that the width of the angle distribution is strongly dependent on the measurement uncertainties we calculate the χ²_r value of the angle differences following Equation (1). If most data points were consistent with the median angle difference within their uncertainties then we would expect χ²_r ∼ 1 (especially for N − 1 = 397 degrees of freedom for the entire 3σ data set). However, for the complete data set with P ⩾ 3σ_p we find χ²_r = 14. From this we conclude the distribution's width is intrinsic and not a result solely of data uncertainties.

Most of the clouds in our sample have insufficient data to perform this analysis separately on individual clouds. Exceptions to this point are OMC-1, OMC-3, and DR 21, whose angle distributions are also shown in Figure 3. For data satisfying the P ⩾ 3σ_p criterion in those three clouds the median angle differences and standard deviations are 3° ± 25°, 4° ± 19°, and 10° ± 22°, with χ²_r values of 18, 5, and 7, respectively. Therefore, as was observed for the entire data set above, the width of the angle-difference distribution (Δϕ) in these individual clouds is real, in the sense that they are not a result solely of the measurement uncertainties.

Lastly, we should note that the position angle rotations observed in these clouds are unlikely to be the result of Faraday rotation. In typical interstellar cloud conditions, at these wavelengths, Faraday rotation is generally much smaller than that observed here (e.g., Schleuning et al. 2000; Matthews et al. 2001).

Table 1 lists the total number of locations where measurements were made at both wavelengths. For the best comparisons we typically choose to include only data satisfying the signal-to-noise criterion P ⩾ 3σ_p; this criterion is applied after the de-biasing correction discussed in Section 2.2. Figure 4(a) shows the distribution of these points. This figure shows all data satisfying the 3σ_p criterion, including outliers as high as P(850)/P(350) = 31; the inset concentrates on the main peak.

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As can be seen in Figure 4 the distributions are non-normal in nature and often contain outliers away from the main peaks. Therefore, unlike the polarization angle distributions in Section 3.1, none of these distributions are well characterized by a simple sample mean or a sample standard deviation. As robust descriptions of the distributions' central tendencies and width we use the samples' medians and median absolute deviations (MADs). The MAD of a set of measurements x is defined as the median value of the residuals, where the residuals are also calculated with respect to the sample median; that is

$$\begin{equation} \mathrm{MAD}\equiv \mathrm{median}\left(\vert x - x_\mathrm{m} \vert \right), \end{equation} \tag{ 2 }$$

where x_m is the median value of the measurements x. For a normal distribution the MAD is significantly smaller than its standard deviation (σ) with an expectation value of σ/1.48. However, given the small number of comparison points in many of the clouds in Table 1 and that the P(850)/P(350) distribution is not expected to be symmetric we report only the MADs there.

The complete 3σ data set contains 398 points with a median P(850)/P(350) value of 1.9, MAD = 0.7, and χ²_r = 9. The χ²_r value implies that the distribution's width is intrinsic and not a result solely of data uncertainties. We reach the same conclusion examining the distributions for OMC-1, OMC-3, and DR21. (Medians are shown in Table 1, χ²_r = 12, 5, and 3.)

We note that many of the peaks in the distributions of Figure 4 are clearly different from the medians listed in Table 1, this is mostly driven by some large outliers in the distribution. An alternate estimate of this peak is the value which minimizes the MAD. For the entire 3σ data set in Figure 4(a) this alternative peak estimate is 1.5 with MAD = 0.6. If the median in Equation (1) is replaced with this peak then χ²_r = 6. These peaks, new MADs, and χ²_r of OMC-1, OMC-3, and DR21 are given in Table 3. Using these peaks yields χ²_r > 1 for all three clouds and, therefore, does not change the conclusion that the distributions' widths are intrinsic and not a result solely of data uncertainties. Also, the difference between the median and the peak is less than the MADs in all cases (i.e., the whole data set and the three specific clouds); none of the discussion in Section 4 relies strongly on these precise values.

In using the polarization ratio to study grain alignment (e.g., Section 4.2) we want to ensure that data at both wavelengths are sampling the same regions of the cloud, both along the LOS and across the POS. The latter criterion is simply met due to the fortuitous ability to beam match the 350 and 850 μm data. If the emission sources are the same for radiation at both 350 and 850 μm then the former criterion would also be met. Meeting this criterion is more difficult, but we try to limit its effect by choosing data with little to no position angle rotation between the two wavelengths. For this reason our analysis is often limited to data points where |Δϕ| ⩽ 10°. Note that σ_ϕ ⩽ 10° corresponds to data points with P ≳ 3σ_p.

To understand this particular data cut, consider the case where the magnetic field changes its orientation along the LOS (and within the cloud depths sampled by at least one wavelength). This may result in a wavelength-dependent change in both the polarization position angle (which follows the change of the field's projected orientation) and the polarization level (due to the field's changing inclination angle). In interpreting the polarization spectrum in terms of grain alignment physics, we wish to eliminate the changing inclination angle as the cause of any change in the polarization level (which can also result from other grain/cloud properties; see Section 4.2). Since a changing field orientation must occur for any data with a wavelength-dependent angle, we can eliminate regions where this occurs by removing such data. While this does not ensure that points without wavelength-dependent angles arise from a single source it is unlikely, in the statistical sense, that the LOS field angles can change for many points in our large sample without an accompanying POS rotation.

Table 1 shows the total number of data points satisfying both the P/σ_p ⩾ 3 and |Δϕ| ⩽ 10° data cuts in each cloud, along with their median polarization ratios and MADs. After making this second data cut the number of surviving data points drops to 141, only 35% of the 3σ data set. The resulting median ratio is P(850)/P(350) = 1.7 with MAD = 0.6 and χ²_r = 11 (see Figure 4(a)). The χ²_r values for OMC-1, OMC-3, and DR21 are also large using this data cut (χ²_r = 19, 5, and 3, respectively), again implying that the distributions' widths are not a result solely of data uncertainties. There are not as many outliers in the distributions after the |Δϕ| cut as their were in the 3σ-only cut. These do have some effect on the measured MADs (as discussed earlier in this section); Table 3 reports the distribution peaks, MADs, and revised values of χ²_r for the |Δϕ| cut in the case where we have minimized the MADs. These small changes in the distribution widths still result in values of χ²_r ≳ 1 meaning that our conclusion, that the widths are intrinsic and not a result solely of data uncertainties, still holds.

Figure 5 shows an updated version of the polarization spectrum from Vaillancourt (2002) and Vaillancourt et al. (2008). All the data in this figure satisfy the criteria P ⩾ 3σ_p and |Δϕ| ⩽ 10°. Here we plot the median value of DR21 (Main), rather than DR21, as the data at other wavelengths (1300 μm) only cover that region of the cloud. While all the data in Figure 5 show P(850) > P(350) we would draw the reader's attention to the range of ratios plotted in the distributions of Figure 4. For example, the median ratio for all clouds in this work is plotted at P(850)/P(350) = 1.7 but has a relatively large MAD (0.6).

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The previous work comparing Hertz and SCUBA-pol performed by Vaillancourt (2002) and Vaillancourt et al. (2008) was based on slightly different data sets than we use here. First, the earlier results were obtained using the data prior to the systematic re-analyses performed by Dotson et al. (2010; see also Kirby et al. 2005) and Matthews et al. (2009). Second, those results attempted to match the Hertz and SCUBA beams by smoothing the SCUBA data to match Hertz's presumably larger beam size and re-sampled at a rate of 5 arcsec pixel⁻¹. Given the pointing accuracies of Hertz (4''–6''; Dowell et al. 1998) and SCUBA (2'')³ the 5'' re-sampling was not unreasonable. However, as shown in Figure 2, the smoothing step was likely unnecessary. Despite these analysis differences the median results are in good agreement. For data satisfying P ⩾ 3σ_p and |Δϕ| ⩽ 10° the previous work found P(850)/P(350) medians and standard deviations of 1.4 ± 0.6 and 1.7 ± 2.7 for OMC-3 and DR 21 (Main), respectively (Vaillancourt 2002). Here we find medians and MADs of 1.6 ± 0.6 and 1.6 ± 0.4 for those two clouds. The large difference in the DR21 (Main) standard deviations (=1.5 for the current work) likely lies in the fact that the earlier 850/350 data comparison used a different set of DR21 data. The analysis by Matthews et al. (2009) used additional observations not available at the time of Vaillancourt (2002) and the analysis resulted in better rejection of noisy data.

For W51, Vaillancourt et al. (2008) found 1.8 ± 2.4 (median and standard deviation). This is consistent with the W51 results shown in Table 1, 6.0 ± 4.2 (median and MAD), but only because of the large deviations. These data are not plotted in Figure 5 as only two data points survive the data cuts.

The P(850)/P(350) ratio for OMC-1 is calculated in this work for the first time. Figure 5 also includes P(450)/P(350) (solid triangle; Vaillancourt et al. 2008) and P(100)/P(350) (open triangle; Vaillancourt 2002) data points. The OMC-1 cloud is thus the only cloud in our data set which includes data both above and below the 350 μm minimum in the spectrum.

The original work by Hildebrand et al. (1999) made it clear that the simplest model, isothermal dust populations all with the same polarization and/or alignment properties, cannot reproduce a spectrum like that in Figure 5. In fact such a model yields a polarization spectrum independent of wavelength beyond 50 μm. In order to explain such a spectrum we consider that a number of physical mechanisms are responsible for the absolute polarization level observed in dust emission. Foremost among these are the efficiency with which dust grains become aligned with magnetic fields and variations in the inclination of that field to the LOS. Ideally, our |Δϕ| ⩽ 10° data cut (Section 3.2) has eliminated the field inclination as a variable in the observed spectrum, leaving alignment efficiency as the key variable.

In order to generate wavelength-dependent polarization spectra the alignment efficiencies must be correlated (or anti-correlated) with changes in the grains' emission. In order to explain spectra like those in Figure 5, Hildebrand et al. (1999) considered simple emission laws of the form F(ν)∝ν^βB_ν(T), where ν is the observed frequency, β is the spectral index, and B_ν(T) is the Planck function at temperature T. For such models the required change in emission can take the form of differences in temperature, differences in the spectral index, or a combination of the two (see also Vaillancourt 2002, 2007). Below we discuss two physical models of the ISM and molecular clouds, both of which result in grain populations with the different alignment properties and temperatures/spectral indices which lead to wavelength-dependent polarization spectra.

Theoretical models of grain alignment have a long history (see reviews by Lazarian 2003, 2007; Hildebrand 1988) with detailed observational tests possible only very recently (e.g., Lazarian et al. 1997; Matsumura & Bastien 2009; Andersson & Potter 2010; Andersson et al. 2011; Matsumura et al. 2011). In one of the most recent models, that of "radiative alignment torques" (RAT; Cho & Lazarian 2005; Lazarian & Hoang 2007; Hoang & Lazarian 2008, 2009a, 2009b), stellar and interstellar photons provide the necessary torques to align the spin axes of dust grains parallel to the local magnetic field. Bethell et al. (2007) simulated a molecular cloud containing aspherical graphite and silicate grains with a typical interstellar grain-size distribution (i.e., Mathis et al. 1977) and radii of 0.005–0.5 μm. The RAT model results in alignment only of grain sizes larger than some cutoff, the exact value of which is dependent on properties like the gas density and radiation field and can vary throughout the simulated cloud (Cho & Lazarian 2005). As the larger grains are more efficient emitters they reach cooler temperatures than the smaller grains in equilibrium. This yields an anti-correlation between grain temperature and alignment; the small warm grains are unaligned while the large cool grains are aligned. The resulting polarization spectrum rises from 100 to 400 μm, with little variation at longer wavelengths. For the wavelengths of interest here they find P(850)/P(350) ∼ 1.0–1.1.

Draine & Fraisse (2009) also present models composed of aspherical silicate grains and spherical graphite grains (here we discuss only their Model numbers 1 and 3). The grain-size distribution and the relative silicate–graphite mix are constrained by the observed interstellar extinction. Additionally, rather than model any physical alignment mechanism (such as RAT), their grain alignment is empirically constrained by the typical interstellar polarization spectrum spanning near-optical wavelengths (i.e., the "Serkowski law"). The result is similar to the work of Bethell et al. (2007) in the sense that larger grains are both cooler and better aligned than the smaller grains and produces a steep polarization spectrum in the 40–400 μm range. However, the Draine & Fraisse (2009) spectrum continues to rise beyond 400 μm such that P(850)/P(350) ∼ 1.2–1.3. This long-wavelength behavior is a combination of (1) the cooler silicate grains being aligned, whereas the warmer spherical graphite grains are not and (2) the shallower spectral indices of silicates compared to graphites.

The median P(850)/P(350) values presented in this work (Figure 5 and Table 1) are clearly steeper than the model estimates just discussed (∼1.7 for the "all clouds, P > 3σ, |Δϕ| < 10°" sample). However, the data distributions are large and, therefore, cannot strongly rule out either model. Additionally, the models are calculated using physical conditions and constraints which likely do not prevail in the real clouds studied here. The Draine & Fraisse (2009) model is constrained by data from the very low density ISM (A_V ≲ a few) whereas our sample of clouds is flux limited to some very bright, dense Galactic regions (A_V > 20). While modeling a "molecular cloud," the Bethell et al. (2007) model bathes the cloud rather uniformly in a typical interstellar radiation field which may be quite different from real clouds containing embedded stars.

The RAT mechanism predicts that grains exposed to stronger radiation sources will be more efficiently aligned. From this we might expect to see systematic trends in the polarization with distance from stellar sources embedded in molecular clouds. Such a trend is hinted at in 60 μm polarization observations toward the W3A H ii region (Schleuning et al. 2000). The most prominent embedded sources in OMC-1 are a group of sources coincident with the BNKL intensity peak and the Trapezium stars in the visible Orion Nebula (Figure 6). Using the Midcourse Space Experiment (MSX) point-source catalog⁴ (Price et al. 2001; Egan et al. 2003) we have also identified a number of embedded sources in the DR21 cloud (Figure 7). However, the proximity of the sources to each other, coupled with the fairly low spatial resolution of the polarization maps, does not allow a careful quantitative study of the strength of the polarization (at either wavelength) or the polarization ratio as a function of distance from these sources.

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A trend in polarization efficiency with distance from a radiation source also implies a correlation between the observed polarization and dust temperature. A careful measure of the dust temperature requires spectral energy distribution (SED) measurements over a wide range of wavelengths, a task which is beyond the scope of the present work (e.g., Vaillancourt 2002). To some extent, one can consider the intensity or flux density ratio, F(850)/F(350), as a proxy for the temperature. Figures 8(a) and (b) compare the intensity ratio in OMC-1 and DR21 to the polarization at both 350 and 850 μm. The polarization in both clouds generally drops with increasing intensity ratio. If we interpret these ratios as color temperatures then the polarization increases with increasing temperature, as would be expected for grains aligned via RAT. However, we caution against overinterpretation of this result as defining a color temperature is problematic in the dense clouds for at least two reasons. First, the range plotted in Figure 8 corresponds to unrealistically large temperatures; assuming β = 2 then T = 13 K for the largest ratio F(850)/F(350) = 0.1 is reasonable but T > 100 K for ratios F(850)/F(350) < 0.033. Second, the ratio may also be the result of changes in grain emissivity (i.e., spectral index) and column density as well as the temperature.

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Using the intensity ratio as a proxy for temperature also has the advantage that it is independent of distance, allowing us to combine the relatively sparse data in individual clouds into a larger data set. Combining the remaining data in clouds other than DR21 or OMC-1 results in Figure 8(c). The same trend of falling polarization is seen at both wavelengths. We emphasize that we are comparing the polarization with the intensity ratio and are not discussing the "polarization-hole" effect which is often observed when comparing the polarization to absolute intensity at any given wavelength (e.g., Schleuning 1998; Matthews et al. 2002).

One difficulty in using the absolute polarization values in Section 4.3 above is that the observed polarization magnitude is also a function of the parameters like the magnetic field's LOS inclination angle, grain cross-section, and turbulence, all of which may vary spatially across the cloud. The inclination angle effect is mitigated somewhat by our choice to limit the data set to those points with |Δϕ| ⩽ 10° (see Section 3.2). These effects can be further mitigated by using the ratio P(850)/P(350). If the same grains are responsible for the polarized emission at both wavelengths then those "polarization reduction" factors effectively cancel in the polarization ratio (Hildebrand et al. 1999).

Figures 6 and 7 show the spatial distributions of the polarization ratio in OMC-1 and DR21, respectively. Most of the mapped areas are characterized by polarizations which are larger at 850 μm than 350 μm. Notable exceptions are intensity peaks in both objects (BNKL at the origin of the OMC-1 map and DR21-OH(Main) at [+33, 0] in the DR21 map). The OMC-1 peak has also shown differences from the rest of the cloud in other polarization work (e.g., Rao et al. 1998; Vaillancourt 2002; Vaillancourt et al. 2008) so we will omit data within 20'' (one beam) of the peak in the analysis below.

The most direct tests of the grain alignment models using submillimeter data require comparisons between the measured polarization ratio and the dust temperatures, spectral indices, and/or radiation environment of the aligned grains. A careful measure of those parameters requires SEDs measured over a wide range of wavelengths, a task which is beyond the scope of the present work (e.g., Vaillancourt 2002). However, we can again use the intensity ratio, F(850)/F(350), as a proxy for the temperature or spectral index. Very different trends are observed when comparing this ratio to the polarization ratio in OMC-1 and DR21 (Figures 9(a) and (b)). The trend is an increase in P(850)/P(350) in OMC-1 but a decrease in DR21. As before the intensity ratio is independent of distance, allowing us to combine the remaining data in clouds other than DR21 or OMC-1 (Figure 9(c)). No strong trend between the intensity ratio and polarization ratio is observed.

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The large amount of scatter in these observations is not unexpected. The dense clouds studied here are certainly composed of multiple temperature components covering a wide range (e.g., ∼20–80 K). Therefore, the intensity ratio is a measure not only of the components' dust temperatures, but also their different spectral indices and relative column densities. Presumably, this effect is partly the cause of the large scatter observed in the data of Figure 9.