Constraining the Dust Opacity Law in Three Small and Isolated Molecular Clouds

FIR continuum observations were obtained from the Herschel Space Observatory (Pilbratt et al. 2010) which covers the wavelength range 100–500 μm using its two bolometers: the Photodetector Array Camera and Spectrograph (PACS; Poglitsch et al. 2010), observing at 100 and 160 μm (HPBW = 71 and 112, respectively); and the Spectral and Photometric Imaging Receiver (SPIRE; Griffin et al. 2010), observing at 250, 350, and 500 μm (HPBW = 182, 249, and 363, respectively).

The Herschel observations were taken from the Herschel Science Archive (HSA). We used the Level 2.5 data products for PACS and Level 2 products for SPIRE. All data were reduced with HIPE using version 14 of the the HSA pipeline. The PACS data were imaged using the Scanamorphos routine (Roussel 2013), whereas the SPIRE data were imaged using the Naive Mapper in HIPE with the optional correction for extended emission. More details about the HSA data reduction can be found in the Data Reduction Guides for both instruments.^{8 ,9}

Absolute sky fluxes were not measured by Herschel, and as a result, Planck and IRAS data have been often used to determine zero-point flux offsets for the Herschel bands (e.g., Lombardi et al. 2014). The Level 2 SPIRE maps from the Herschel archive apply a similar correction using Planck data at 857 and 545 GHz (see the SPIRE Handbook for more details). PACS data from the archive, however, have no such correction. Therefore, we estimate the zero-point flux offset in the PACS data using the all-sky Planck SED models in a manner similar to Lombardi et al. (2014). The Planck SED models provide maps of dust temperature and optical depth (at 353 GHz) at 5' resolution, and dust emissivity, β, at 30' resolution. With these SED parameters, we constructed modified blackbody functions for each Planck pixel and then determined the corresponding flux that would have been measured by the Herschel bands. Unlike Lombardi et al. (2014), we cannot simply convolve the Herschel data to a resolution similar to that of the Planck models (e.g., 5') and compare fluxes pixel by pixel, because the Herschel maps of the globules are relatively small. For example, the PACS maps are generally less than 10' in size. Instead, we compared azimuthal averages of the Planck-determined emission maps at each of the 160–500 μm Herschel bands to azimuthal averages of the observed Herschel data, where it is expected that the two radial profiles are equal at large radii.

The SPIRE maps were indeed in good agreement with those expected from Planck, illustrating the success of their zero-point corrections. The radial emission profiles at 250, 350, and 500 μm show good agreement with expectations from Planck at large radii (e.g., >300''). The PACS 160 μm profiles, however, had substantial deficits relative to the expected 160 μm profiles at the same large radii. Because we expect PACS data to also match Planck data at the same radii as SPIRE data, we apply a correction on the 160 μm data to bring the two profiles into agreement. The PACS 160 μm offsets were ∼45 MJy sr⁻¹ for CB 68, ∼160 MJy sr⁻¹ for L 429, and ∼73 MJy sr⁻¹ for L 1552. Thus, our radial profile correction technique provides a good, first-order estimate of the zero-point offsets at 160 μm. A full study of this method will be given in S. Sadavoy et al. (2017, in preparation).

The SPIRE data covered a larger area than those from PACS for all three cores. As a result, each core has a central region where all Herschel wavelengths were observed, and an outer region where only SPIRE wavelengths were observed. The optical depths in the SPIRE-only regions have much larger uncertainties due to the missing constraints on the fitted SED on the Wien side of the spectrum. For this reason, only the regions with both SPIRE and PACS coverage were considered in the following comparison.

Column densities from optically thin thermal dust emission will trace all the emission along the line of sight. In contrast, column densities from dust extinction will primarily trace the foreground component of dust associated with the nearby globules themselves. Thus, it is necessary to subtract a "background" column density level for the thermal dust emission so that these data better match the dust extinction observations. To be conservative, we use the median background emission associated with radii >300'' from the central core. This background measurement is likely overestimated, as it will include some material from the core as well. Extensive modeling to determine how to best correct the thermal dust emission, however, is beyond the scope of this paper. A discussion of the results of the analysis without applying this correction is provided in Appendix A.

Observations of dust extinction were obtained from the Canada–France–Hawaii Telescope (CFHT) Wide-field Infrared Camera (WIRCam; Puget et al. 2004)¹⁰ in the J, H, and K (1.25 μm, 1.63 μm, and 2.15 μm, respectively) NIR wavebands. CFHT WIRCam has a 20' × 20' field of view (FOV) well-suited to observe each of the cores with a single pointing, while allowing for simultaneous observations of surrounding sky fields with minimal extinction. The instrument is mounted on the 3.6 m CFHT, and has a focal plane consisting of four 2048 × 2048 pixel detectors, with a sampling of 03 pix⁻¹. The cores were imaged between 2014 September–October under program 14BC37, and 2015 May under program 15AC18 (P.I.: J. Di Francesco). A summary of the integration times, sizes, and depths of the final observations is listed in Table 2. The exposures were dithered to correct for the CCD chip gaps and bad pixels, and combined with the CFHT preprocessing pipeline IDL Interpreter of WIRCam Images ('I'iwi). A background image was produced from the stacked exposures, then subtracted from each exposure. Post-processing was performed at the Terapix¹¹ astronomical data reduction center at the Institut d'Astrophysique in Paris, where the images were astrometrically aligned and coadded. Scattered light from the dust in the densest regions, called "coreshine" (Steinacker et al. 2010), was observed in the cores of CB 68 and L 1552 (Stutz et al. 2009b; Lefèvre et al. 2014) and care was taken to not subtract it from the images. Using the coreshine observations to constrain the dust models is beyond the scope of this paper, however. Figure 1 shows RGB-color images (for J, H, and Ks) of the final core maps where the effect of extinction is seen in the reddening toward the center of each core.

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Aperture photometry was performed on the final images with the program Source Extractor (SExtractor, v2.19.5; Bertin & Arnouts 1996). Detections at a threshold of 7σ (minimum area threshold of 5 pix, and Gaussian filter) were matched to objects in the Two-Micron All-Sky Survey¹² (2MASS; Skrutskie et al. 2006) catalog to determine the zero-point-magnitude. The detection threshold was then lowered to 3.5σ for the same detection parameters, where anomalous detections were removed in the object cross-match in the catalogs of the three filters. Stellar objects were selected from the linear distribution of half-light radii as a function of magnitude, and the colors were calculated for objects in common between the three NIR bands.

The depth of the survey was assessed through the insertion of artificial stars, where the limiting magnitude was taken as the magnitude at which 50% of a sample of artificial stars was recovered by the detection pipeline (see Table 2).

The depth of the observations suffered from a known issue with the Direct Imaging Exposure Time Calculation (DIET)¹³ and were less deep than predicted for the observing conditions and exposure times requested in the CFHT proposal. Sufficient numbers of stars were detected, however, to achieve a resolution in the NIR extinction maps equal to the Herschel optical depth maps. The significance of not having the intended depth is discussed further in Section 4.5.

Some extended emission was observed in the SPIRE maps in the selected "unreddened" regions of L 429 and L 1552 (see Section 3.2). Therefore, to remove any bias introduced by an unclean background, we applied the same background subtraction procedure as used for the emission maps (discussed in Section 2.1). We note that the background extinction subtraction was insignificant in comparison with the background emission subtraction.

The emission maps were convolved to the beam of the SPIRE 500 μm map with the program MIRIAD,¹⁴ then regridded to 14'' pix⁻¹ using Nyquist sampling. The average 1σ uncertainties in the zero-point-corrected maps amounted to: 19% for CB 68, 14% in L 429, and 16% in L 1552.

Thus, S_ν, the flux density as a function of frequency, ν, in the data cubes, can be described as:

$$\begin{eqnarray}&&{S}_{\nu }={\rm{\Omega }}(1-{e}^{-{\tau }_{\nu }}){B}_{\nu }({T}_{d}),\end{eqnarray} \tag{ 1 }$$

where ${B}_{\nu }({T}_{d})$ is the Planck function, which is dependent on the dust temperature, T_d, and frequency, ν; Ω is the beam solid-angle; and ${\tau }_{\nu }$ is the optical depth through the cloud where

$$\begin{eqnarray}&&{\tau }_{\nu }={N}_{{\rm{H}}}\,{m}_{{\rm{H}}}\displaystyle \frac{{M}_{d}}{{M}_{{\rm{H}}}}{\kappa }_{{\rm{d}},\nu }.\end{eqnarray} \tag{ 2 }$$

The optical depth is a function of the total hydrogen column density, ${N}_{{\rm{H}}}=2N({{\rm{H}}}_{2})+N({\rm{H}})$, the proton mass, m_H, the dust-to-hydrogen mass ratio, M_d/M_H (assumed to be 1/110 following for example, Sodroski et al. 1997), and the dust mass absorption coefficient (here called the opacity), ${\kappa }_{d,\nu }$. Given that the Herschel wavelength coverage is limited such that both the dust emissivity and temperature cannot simultaneously be robustly measured, we assumed a fixed dust grain model with a prescribed ${\kappa }_{d,\nu }$. Section 3.1.1 describes the four dust models tested in this paper.

A ${\chi }^{2}$ minimization technique was then used to fit an SED (describing a single temperature modified blackbody) to each pixel in the calibrated dust emission maps, where T_d and ${\tau }_{\nu }$ were free parameters. We remind the reader that four different dust models were used, and we note that the slope of ${\tau }_{\nu }$ varies with the assumed dust model. The individual fluxes in the SED were weighted by ${\sigma }^{-2}$. Here, the σ is calculated as the quadrature sum of (i) the uncertainty in the Planck-derived offset and (ii) the mean rms noise. Calibration errors and color correction errors were also accounted for using a Monte Carlo analysis.

In using 250 μm as the fiducial wavelength for our comparison of the optical depths derived from the FIR, the column density can be determined via

$$\begin{eqnarray}&&{N}_{{\rm{H}}}=(6.58\times {10}^{25}\,{\mathrm{cm}}^{-2}){\tau }_{250}/({\kappa }_{250}/1\,{\mathrm{cm}}^{2}\,{{\rm{g}}}^{-1}).\end{eqnarray} \tag{ 3 }$$

3.1.1. Dust Opacity Models

Dust models determine opacities for particular environments. It is therefore important to select an appropriate dust model for the conditions of an MC core. In fact, studies of coreshine (e.g., Lefèvre et al. 2014, 2016; Steinacker et al. 2014) suggest that a single dust opacity model is not appropriate. Rather, there is differential coagulation and ice mantle growth between the cold dense core interiors and the thin warm outer envelopes. Therefore, there can be large systematic uncertainties in the fits of the SEDs related to the dust opacity and its spectral dependence. This problem highlights the motivation of this paper: to test the accuracy of selected dust models used to measure column densities via comparison with supplementary NIR observations.

We tested four commonly used dust opacity models that cover a wide range of dust properties. We refer to these models as OH1a and OH5a from Ossenkopf & Henning (1994; OH)¹⁵ and Orm1 and Orm4 from Ormel et al. (2011, Orm).¹⁶

The first model, OH1a, describes interstellar medium (ISM)-like dust. The grains are of silicate-based carbonaceous composition with no ice mantles and no coagulation. Extinction optical depths were determined from the albedos, a, provided by Weingartner & Draine (2001) for R_V = 3.1,

$$\begin{eqnarray}&&{\kappa }_{\mathrm{ext}}={\kappa }_{\mathrm{abs}}/(1-a).\end{eqnarray} \tag{ 4 }$$

In contrast, OH5a grains were coagulated for ${10}^{5}$ years with gas densities of ${10}^{5}$ cm⁻³. This model was used in Launhardt et al. (2013) for CB 68. Again, the extinction opacities were used.

The Orm1 and Orm4 models describe mixed silicate and graphite grains of fraction 2:1 and ice mantle thickness coagulated for ${10}^{5}$ years. Orm1 has bare silicate and graphite grains, Orm4 has ice-silicate and graphite grains mixed at the level of individual aggregates. Both of these models describe the extinction optical depths, and are therefore consistent with the altered OH models.

Figure 2 shows the predicted dust opacities for each of these models from NIR to submm wavelengths, and Table 3 lists the values of the dust opacities at the NIR 1.25 μm (J) wavelength and the FIR 250 μm wavelength. Although the models have similar slopes in the 160–500 μm range, there are differences in the ratios of the NIR and FIR opacities. The ratio ${\kappa }_{J}/{\kappa }_{250}$ is $2.54\times {10}^{3}$ for OH1a, $1.48\times {10}^{3}$ for OH5a, $8.44\times {10}^{2}$ for Orm1, and $1.09\times {10}^{3}$ for Orm4. Four additional models, two each from OH and Orm, are discussed in Appendix B.

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Table 3.  Dust Optical Depth Values for the Models Considered. Values Are Interpolated

Model	${\kappa }_{J}$ $(\times {10}^{4}$ cm2 g−1)	${\kappa }_{250}$ (cm2 g−1)
OH1a	1.722	6.77
OH5a	2.098	14.16
Orm1	1.545	18.29
Orm4	1.728	15.84

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One caveat to using the above models is that the densities describe the central cores of the sources, whereas our analysis is limited to the envelope where densities are ${10}^{3}$–${10}^{4}$ cm⁻³. At lower densities, dust will take longer to coagulate, such that the dust sizes may be overestimated, which affects the determined optical properties. None of the models described in Ossenkopf & Henning (1994) or Ormel et al. (2011), however, include models at these lower densities.

The extinction of background stars was used to independently derive the optical depth from NIR observations. The NIR color excess technique from Lada et al. (1994) was used to combine the color information for the NIR bands with a standard reddening law to obtain a robust measure of the extinction in the cores. The extinction is proportional to the color excess that describes the observed flux ${S}_{\lambda }$ in one band λ and a second ${\lambda }^{{\prime} }$, for a reddened (observed) and unreddened (intrinsic) field,

$$\begin{eqnarray}&&{A}_{\lambda }=E({S}_{\lambda }-{S}_{{\lambda }^{{\prime} }})/{R}_{\lambda }={({S}_{\lambda }-{S}_{{\lambda }^{{\prime} }})}^{\mathrm{obs}}-{({S}_{\lambda }-{S}_{{\lambda }^{{\prime} }})}^{\mathrm{intr}}.\end{eqnarray} \tag{ 5 }$$

The color variations in a population of reddened and un-reddened stars in each core map are shown in Figure 3. The effect of extinction is to shift the color–color distribution along the reddening vector, indicated by the arrow.

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The proportionality constant, ${{\rm{R}}}_{\lambda }$, is dependent on the environment along the LoS. Empirical studies (see the review by Mathis 1990) have shown that, at visual wavelengths, R_V = 3.1 for the ISM and can be as high as 5.5 in dense regions, such as MCs. Recent studies, such as Schlafly et al. (2016), however, suggest not all MCs have high R_V values. A higher R_V is generally associated with dust grain growth via accretion and coagulation (e.g., Weingartner & Draine 2001; Flaherty et al. 2007). In the NIR, however, reddening laws (such as Rieke & Lebofsky 1985; Cardelli et al. 1989; Mathis 1990) are independent of the environment such that A_J ∼ constant.

Applying the reddening law of Cardelli et al. (1989), for R_V = 3.1,

$$\begin{eqnarray*}&&{\tau }_{K}=0.600\,{\tau }_{H}=0.405\,{\tau }_{J},\end{eqnarray*}$$

where ${\tau }_{K}=0.114\,{\tau }_{V}$. There is only a 15.6% difference in ${\tau }_{K}/{\tau }_{V}$ if R_V = 5.5 is used instead.

The NIR color excess method of (Lombardi 2009, NICEST) was used to map the extinctions, where the technique has been tailored from those of Lada et al. (1994) and Lombardi & Alves (2001) to correct for the inhomogeneous distribution of background stars and its effect on smoothing small-scale structure in the optical depths. The application of smoothing parameters related to the distribution of stars corrects for the bias toward regions of low extinction by a factor of the sample size.

Table 4 lists the color variations for stars in these control regions. Intrinsic color variations were determined from regions with minimal reddening. Although extended emission was observed in these control regions in the SPIRE maps of L 429 and L 1552, the effect of the bias is completely dominated by the approximation of the background subtracted from both the emission and extinction maps. Refer to Appendix A for a discussion of the significance of this bias when no background subtraction is performed.

Table 4.  Parameters Used for the Calculation of the Extinction with NICEST

Source	${(J-H)}^{\mathrm{intr}}$	${(H-K)}^{\mathrm{intr}}$	max AJ	med σAJ
			(mag)	(mag)
CB 68	0.58 ± 0.19	0.11 ± 0.18	4.44	0.093
L 429	0.91 ± 0.14	0.25 ± 0.09	16.68	0.034
L 1552	0.77 ± 0.17	0.22 ± 0.17	3.95	0.180

Note. Intrinsic J–H, and H–K scatter of the control (minimal extinction) region stars, maximum J-band extinction, and median of the error in the J-band extinction.

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The final extinction map (A_J) was determined from an unbiased and minimum variance maximum-likelihood combination of the individual extinctions, binned to match the resolution of the FIR emission-derived optical depth maps. The sensitivities of the extinction maps are directly related to the number of background stars in each pixel and the uncertainty of the measured magnitude. Measurements in locations with few background stars, due to high extinctions or an inhomogeneous background stellar distribution, are reported with higher relative error. A minimum of 10 stars per pixel was applied to ensure robust measurements. The smoothing parameter used to produce the extinction maps was selected to match the sampling (14'' pix⁻¹)¹⁷ of the binned Herschel optical depth maps, with the consequence of having several locations with an insufficient number of background stars to calculate the extinction with confidence. Hence, there are pixels in the final maps without a measure of the extinction.

The optical depth maps were then determined from the simple relation ${\tau }_{\lambda }={A}_{\lambda }/1.086.$

SED fits to thermal dust emission are sensitive to the estimates of dust temperature, as discussed in Section 3.1. In Equation (1), the modified blackbody function is fit with a single temperature, which is representative of the average temperature along the LoS. These fits do not take into account any temperature gradients. For example, the cores will be heated by the interstellar radiation field. CB 68, in particular, is protostellar, so it has a central heating source. Radiative transfer models of filaments and spherical clouds with central sources by Ysard et al. (2012) show that temperatures were overestimated by single-temperature SED fits, compared to the true dust temperatures. Several other studies (e.g., Pagani et al. 2015; Lippok et al. 2016; Steinacker et al. 2016) also point out issues in modeling dense cores with single-temperature blackbody functions. For diffuse media, the column density can be underestimated by up to a factor of 1.7 in Herschel bands and 2.2 in Planck bands. Such a bias would be most significant where the gradient is strongest and least significant where the temperatures are more uniform. Moreover, it will vary from source to source (e.g., the factor will vary with the magnitude of the gradient and the distribution of material).

In the context of this paper, an overestimation of dust temperature up to a factor of ∼1.7 diminishes the measured ${\tau }_{250}$, based on analysis by Ysard et al. (2012). The optical depth will primarily be underestimated toward the center of the core. This material, however, is not traced in the extinction maps and was excluded from our analysis. The following comparisons of optical depths are therefore reasonable, despite the simplifying assumption of a constant LoS temperature.

Figure 4 shows the dust temperature maps derived from the SED fitting procedure for the three cores. L 429 has the steepest temperature gradient from its outermost edges to its center. By comparison, CB 68 has a relatively small gradient, even though it is the only protostellar core in our sample.

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Figure 5 shows the distributions of extinction and associated uncertainties in each core. Table 4 lists the maximum A_J and median rms sensitivities of the cores. The map rms sensitivities range from σA_J ∼ 0.064–1.030 mag in CB 68, 0.027–0.592 mag in L 429, and 0.111–0.784 mag in L 1552. The median relative uncertainties of the resulting optical depth maps (${\tau }_{J}$) are 33.1%, 12.1%, and 28.4%, respectively.

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In combination with the emission-derived column density map uncertainties (which were 19% for CB 68, 14% in L 429, and 16% in L 1552; see Section 3.1), the median uncertainties in the comparison analysis were therefore 25.2%, 16.2%, and 23.6%, respectively.

Figure 6 shows the extinction-derived optical depth maps (${\tau }_{J}$, left) and emission-derived optical depth maps at the fiducial wavelength 250 μm (${\tau }_{250}$, center-left) for the three cores. The center-right column shows the ratio of the two optical depth maps scaled by the theoretical opacity ratio for OH5a, $({\tau }_{J}/{\tau }_{250,\mathrm{OH}5{\rm{a}}}){({\kappa }_{J}/{\kappa }_{250})}_{\mathrm{OH}5{\rm{a}}}^{-1}$, and the right column describes the consistency of the scaled comparison as discussed in Section 4.4. Filamentary extensions from the central regions are apparent in all three cores. As discussed in Launhardt et al. (2010), there is a weak comet-like tail around CB 68, apparent also in the RGB color image in Figure 1. Similar to CB 68, the cores L 429 and L 1552 have structures which extend away from their central cores, although they are more diffuse and broad. L 429's tail primarily extends to the south and northwest of the core, whereas L 1552's tail extends to the north.

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To describe the consistency between ${\tau }_{250}$ from thermal dust emission and ${\tau }_{J}$ from dust extinction, we define the parameter ξ as:

$$\begin{eqnarray}&&\xi =\displaystyle \frac{1-\left(\tfrac{{\tau }_{J}}{{\tau }_{250}}/\tfrac{{\kappa }_{J}}{{\kappa }_{250}}\right)}{\sigma \left(\tfrac{{\tau }_{J}}{{\tau }_{250}}/\tfrac{{\kappa }_{J}}{{\kappa }_{250}}\right)},\end{eqnarray} \tag{ 6 }$$

where ${\kappa }_{J}/{\kappa }_{250}$ comes from the adopted dust models (see Section 3.1.1) and $\sigma \left(\tfrac{{\tau }_{J}}{{\tau }_{250}}/\tfrac{{\kappa }_{J}}{{\kappa }_{250}}\right)$ is the formal uncertainty on the scaled ratio (hereafter, σ). This parameter represents a weighted scaled ratio for the optical depth maps for a given dust model. A value of ξ = 0 means the extinction maps and thermal dust emission maps have perfect agreement for a particular dust model.

Figure 6 shows histograms of ξ for all three cores, assuming OH5a dust. Table 5 lists the median ξ value for all four dust models, as well as the percentage of ξ values within 1σ of ξ = 0 (hereafter, %$\xi \lt 1\sigma$). We use this percentage to assess the consistency of the individual dust models across all three sources.

Table 5.  Comparison of Measured Extinction- and Emission-derived Optical Depth Ratios, ${\tau }_{J}/{\tau }_{250}$, with Dust Opacity Ratios for Several Theoretical Models

Model	Slope ${\tau }_{J}$/$({\tau }_{250}({\kappa }_{J}/{\kappa }_{250}))$	Median ξ	% < 1σ
	(1)	(2)	(3)
CB 68
OH1a	0.252 ± 0.006	5.4 ± 9.3	16.8
OH5a	0.63 ± 0.02	1.3 ± 3.2	43.0
Orm1	0.75 ± 0.02	0.4 ± 2.5	49.3
Orm4	0.53 ± 0.01	1.4 ± 3.9	37.1
L 429
OH1a	0.273 ± 0.002	9.9 ± 8.0	4.4
OH5a	0.743 ± 0.006	0.6 ± 1.4	68.0
Orm1	0.822 ± 0.006	1.3 ± 2.3	32.2
Orm4	0.576 ± 0.004	3.0 ± 3.4	5.6
L 1552
OH1a	0.360 ± 0.005	3.8 ± 3.5	22.1
OH5a	1.07 ± 0.02	-0.2 ± 0.8	89.0
Orm1	1.08 ± 0.02	-0.2 ± 0.8	86.7
Orm4	0.74 ± 0.01	0.7 ± 1.3	55.7

Note. A visual representation of the data is shown in Figures 6 and 7. (1) Uncertainty-weighted slope of the ratio of the extinction- and emission-derived optical depth maps scaled by the expected slope. Errors correspond to the standard deviation of the fits determined from the covariance matrix. (2) Median of the comparison statistic (Equation (6)), where the error is the standard deviation. (3) Percent of the ξ distribution within 1σ-level.

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The top row of Figure 7 shows comparisons of the extinction-derived optical depth with the scaled emission-derived optical depth for each cloud and model on a pixel-by-pixel basis, where the data were binned with bin widths of 0.25 ${\tau }_{J}$. Only bins with more than five data points were considered, and the uncertainties on the data points correspond to one standard deviation in the values of each bin. A black dashed line of unity is shown for reference. A slope (zero intercept) was fit to ${\tau }_{J}/({\tau }_{250}({\kappa }_{J}/{\kappa }_{250}))$ for each model via orthogonal distance regression weighted by both uncertainty values, and is shown with a width corresponding to the error calculated from the covariance matrix of the fit. The fit was restricted to ${\tau }_{J}\gt 1$, to reduce the bias of the low optical depth data.

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The bottom row of Figure 7 shows that the consistency of the dust models differs substantially between the three cores. On the bottom horizontal axis, the slope of the binned and scaled ratio is shown with error bars corresponding to the standard deviation of the data. Although none of the models reach %$\xi \lt 1\sigma$ values above 50% for CB 68 (Orm1 is only slightly below this cutoff), OH5a is above 50% for L 429, and OH5a, Orm1, and Orm4 are each above 50% for L 1552. Only Orm1 has a slope consistent with unity within one standard deviation for L 429, however, and only Orm1 and OH5a have consistent slopes for L 1552. Of particular note, OH1a is the least consistent of the models considered for each source.

Appendix B introduces four additional models, two from Ossenkopf & Henning (1994) and two from Ormel et al. (2011), that describe properties similar to the present four models. In considering the full set of models, the results suggest that the optical properties of dust in the envelopes of the cores are best described by either silicate and bare graphite grains (like Orm1), or carbonaceous grains with some coagulation and either thin or no ice mantles (like OH5a). It is not unexpected that models with minimal ice formation are found to be most consistent in the envelopes of these cores. As noted in Section 3.1.1, the models assume densities of ${10}^{5}$ cm⁻³, whereas the material traced is probably closer to ${10}^{3}$–${10}^{4}$ cm⁻³, where it will take longer for grains to coagulate.

Our results demonstrate the complex nature of determining the pre-stellar conditions of cores. Although we selected well-isolated and simply structured cores, models typically assumed to provide reasonable estimates of the dust properties of such objects do not consistently lead to correct total gas masses. Given that the models describe the center of the cores (whereas our comparison was restricted to the envelope), the consistency of some of the models was better than expected. However, the comparison highlights the limited capability of the SED fitting method (as discussed by others, e.g., Pagani et al. 2015). Alternatively, multiple temperature-modified blackbody functions may be required to model the dust emission accurately. For example, differential coagulation and ice mantle growth between the cold dense core interiors and the thin warm outer envelopes may mean that single functions are not applicable (as shown by studies of coreshine, e.g., Lefèvre et al. 2014, 2016; Steinacker et al. 2014).

The analysis in this study could be improved through the reduction of the formal uncertainties in the optical depth maps, which could be achieved with more sophisticated SED modeling or deeper NIR observations. The uncertainties in the extinction values are a function of the intrinsic color scatter of the stellar population, photometric error, and the number of stars within a weighted Gaussian (Lombardi & Alves 2001). Increasing the depth of the NIR observations would reduce the uncertainties twofold: it would reduce the photometric error and increase the number of stars observed. Moreover, observations of a dedicated "off" position or larger maps that measure extinction-free locations would also help improve the NIR optical depth measurements. With smaller errors, an improved distinction could be made between the OH and Ormel models.

The potential improvements given 2 mag deeper observations were determined from artificially adding stars into our images with the appropriate number density, photometric errors, and color offset. The artificial photometry files were then used to produce extinction maps with the NICEST algorithm, and the errors were compared to the original maps. The simulated results had a ∼40% improvement in the uncertainties.

There has been a significant effort recently to constrain dust properties in star-forming regions. Although at resolutions too low to observe individual cores, observations with the Planck telescope (such as Planck Collaboration et al. 2011; Juvela et al. 2015; Ysard et al. 2015) have yielded interesting results on the large-scale distribution of dust in many environments and their implications for dust properties. In addition, many individual MCs have been studied. Suutarinen et al. (2013) studied the Corona Australis cloud and found a median extinction ratio of ${A}_{250}/{A}_{J}=(1.4\pm 0.2)\times {10}^{-3}$, or equivalently ${\tau }_{J}/{\tau }_{250}=714\pm 102$. Of the four dust models considered, this result is consistent with Orm1 only. Similarly, a study by Juvela et al. (2015) of 20 Herschel fields found a median ${\tau }_{J}/{\tau }_{250}=625\pm 78$. With the Bohlin et al. (1978) column density relation and Cardelli et al. (1989) extinction curve, the Planck Collaboration et al. (2014) result ${\tau }_{250}/{N}_{{\rm{H}}}\sim 0.55\times {10}^{-25}$ cm² H⁻¹ corresponds to ${\tau }_{J}/{\tau }_{250}\,=2439$. Each of these studies, however, employs the relation $\kappa \propto {\nu }^{2}$ and thus they are not entirely equivalent to our results. Regardless, the variance in values indicates the complex nature of this problem due to changes in dust properties at different densities within MCs and cores.