Volume 584, December 2015
|Number of page(s)||134|
|Section||Interstellar and circumstellar matter|
|Published online||26 November 2015|
Modelling was used to assess the possible impact that radiative transfer effects have on the observed spectral index values. The model consists of a cylindrical filament with a Plummer-like radial density profile (A.1)where r is the distance from the symmetry axis, n0 is the central density, and RC determines the extent of the inner flatter part of the density profile. The filament is illuminated externally by interstellar radiation field (Mathis et al. 1983). We use the Ossenkopf & Henning (1994) dust model for coagulated grains with thin ice mantles accreted over 105 yr in a density of n = 106 cm-3. To remove any ambiguity from the comparison of the observed and the intrinsic β values, the model is modified so that β is equal to 1.9 at all wavelengths λ> 100 μm. Note that this is the intrinsic value of the dust opacity spectral index. Because of line-of-sight temperature mixing, the apparent value estimated from surface brightness data can be expected to be lower (Shetty et al. 2009b; Juvela & Ysard 2012b).
The filament is observed in a direction perpendicular to its symmetry axis. The radiative transfer calculations are used to solve the three-dimensional temperature structure of the filament and to calculate surface brightness maps at 160, 250, 350, and 500 μm. The surface brightness data are used to derive maps of observed colour temperature and spectral index. In the calculation we use 10% and 2% relative uncertainties for the PACS channel and the SPIRE channels, the same as in the case of actual observations. The maps consist of 256 × 256 pixels corresponding to the 2563 cell discretisation of the 3D model cloud. The models are characterised by their total optical depth in V-band that, measured through the filament in the perpendicular direction, is either 10.0 or 50.0. To make the model more concrete, we assume for the models a linear size of 10 pc and a distance of 400 pc. With this scaling, the parameter RC is equal to 0.1 pc and n0 is determined by the selected optical depths. Each pixel corresponds to ~20″ and the final maps are convolved to a resolution of 40.0″ before analysis. Figure A.1 shows cross sections of the modelled filament in column density (V-band optical depth), dust temperature, and the colour temperature and the spectral index estimated from the surface brightness data.
For the τ(V) = 10 model, the apparent spectral index decreases to 1.75, 0.15 units below the intrinsic β value of 1.9. For the τ(V) = 50 model, the minimum observed value is only 1.10. This demonstrates the difficulty of making a direct connection between the apparent spectral indices and the intrinsic dust properties in the case of optically thick clouds. However, because the line-of-sight temperature variations always imply a decrease in the apparent spectral index, this increases the significance of the observed correlation between column density and spectral index. At least in some regions along the line-of-sight, the actual dust opacity spectral index should be much higher than the observed values.
Cross sections of the cylindrical radiative transfer models. The peak optical depths are τV = 10.0 in the left and τV = 50.0 in the right panel. Each frame shows the τV profile, the actual dust temperature in the model midplane (T3D), and the colour temperature TC and spectral index β derived from the simulated surface brightness maps.
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We calculated two additional cases where the dust properties were modified in the central regions with density exceeding 25% of its maximum value. The modification is applied only to wavelengths λ> 40 μm and it is determined by two criteria. First, the dust opacity is set to twice its original value at 250 μm and, secondly, the long wavelength spectral index β′ is set to a value of either 1.7 or 2.2. These are ad hoc models where the dust opacity has discontinuity at 40 μm. This has little practical importance because grains absorb energy at wavelengths shorter than this limit and correspondingly emit the energy at much longer wavelengths.
The first result is that the observed (i.e., values derived from surface brightness data) β values towards the filament centre are lower in both cases of modified dust. We interpret this as an indication that the increased line-of-sight temperature variations associated with the enhanced submillimetre opacity outweigh the effect of Δβ ~ 0.2 changes of the intrinsic spectral index. For the τV = 10 model, the minimum β values were 1.41 and 1.67 for the β′ = 1.7 and β′ = 2.2 cases, respectively. Both are lower than the previous apparent value of 1.75. For the τV = 50 model, the corresponding minimum values are 0.57 and 0.67, much lower than the 1.10 obtained before modifications to the dust model.
Figure A.2 shows selected SEDs from the τV = 50 models described above. The uppermost three SEDs are observed towards the filament centre for the three different dust models. The two lower SEDs are obtained at different radial offsets, from a region that is unaffected by the β′ changes that were applied to the inner part of the filament. We use arrows to indicate the difference between the fitted modified blackbody curves and the actual data points. Towards the optically thick centre, the data do not follow a single modified blackbody spectrum. The errors are largest at 160 μm, because of its smaller weight in the fit (assuming 10% uncertainty instead of the 2% at the longer wavelengths). However, the fits also systematically overestimate the 250 μm intensity and underestimate the 350 μm intensity, this being related to the low values of the fitted spectral index. The SEDs for both modified dust cases are clearly colder than the original SED because of the higher opacity at wavelengths where the dust grains emit most of their energy.
The spectral index was modified at λ> 40 μm. In the 100–500 μm range, the β difference corresponds to a factor of two difference of relative opacity. Nevertheless, the observed spectra for the β′ = 1.7 and β′ = 2.2 cases are practically on top of each other over the whole wavelength range probed by Herschel observations. After the opacity increase, the central part of the filament is very cold with dust temperatures close to 5 K. As a result, even at 500 μm the emission per unit dust mass is more than a factor of ten lower than for the ~17 K dust on the cloud surface. This means that the β changes at the cloud centre are masked by the warmer, more strongly emitting envelope. In the model the difference of intrinsic β values in the filament centre becomes evident only at mm wavelengths. In the Herschel range, the spectral index determination is clearly very sensitive to temperature variations. If the 160 μm data point were omitted from the fits, the spectral index values of 1.16, 0.52, and 0.62 would increase to 1.37, 0.75, and 0.93 respectively, the three values corresponding to the three models: one with original dust, one with β′ = 1.7 dust in the centre, and one with β′ = 2.2 dust in the centre. The difference between the observed and intrinsic β (for given column density) would also decrease if the column density peak were less steep or if the observations were averaged over a larger beam. At lower column densities, at larger offsets from the symmetry axis of the cylindrical filament, the dust SED is well-described by a single modified blackbody curve, with the observed β approaching the intrinsic β value.
Selected SEDs from τV = 50.0 models. The uppermost three spectra are towards the centre of the filament, using the original dust model (black line) or using β = 1.7 dust (blue line) or β = 2.2 dust (red line) in the central part of the filament. The lower spectra (magenta and green) correspond to larger offsets (4.4 and 8.7 arcmin) from the symmetry axis that are not directly affected by the dust modifications. The differences between the measured points and the modified blackbody curves are indicated with vertical arrows. The length of the arrow is ten times the difference between the data point and the fitted modified blackbody curve.
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In spite of such a strong effect in the above model, the observations rarely show any decrease of β towards the dense regions. A weak anti-correlation between β and column density was seen only in a few fields. These include G82.65-2.00 where, based on the observed dust emission, the column density exceeds N(H2) = 2 × 1022 cm-2 in many places (Juvela et al. 2015). WISE 12 μm data show clear absorption along the full filament, also suggesting that the visual extinction is at least AV ~ 20 mag. The average properties of the filament would therefore be between the two models above but on average closer to the τV = 10.0 case. The large scale decrease of β towards the filament is at the level of Δβ = 0.1 and could easily be explained by temperature variations within the beam. However, even in G82.65-2.00 the
T−β relation shows an anti-correlation at the smallest scales towards the densest clumps, contrary to the behaviour in the models.
The temperature mixing in regions like G82.65-2.00 is probably very complex compared to the simple models above and the net effect on the observed spectral index can be qualitatively different. The models above showed a maximal effect because we compared emission between the cold centre and the fully illuminated cloud surface. In real clouds the dense regions are shielded by diffuse envelopes whose extinction decreases temperature contrasts at small scales. The drop in the observed β would be smaller if the column density profile were less steep or if the data were smoothed by a larger beam. If the emission region is inhomogeneous, elongated, or consists of completely separate clumps along the line-of-sight, the radiative transfer effects would be reduced. Internal heating of some clumps is not able to explain the general anti-correlation between T and β (Malinen et al. 2011; Juvela & Ysard 2012b). Therefore, it seems that we are observing real changes in the intrinsic dust properties that, remarkably, are strong enough to be seen above the opposite effects caused by temperature variations.
The models can also be used to examine the dependence on the wavelengths used. Figure A.3 shows temperature and spectral index profiles of the τ(V) = 10 model that are estimated using five different wavelength combinations. As expected, β values are least biased when long wavelengths, 250–850 μm, are used. However, in the case of this model, even the combination of PACS and SPIRE bands 160–500 μm results in values that are lower by no more than Δβ = 0.07. This suggests that over most of the observed fields the bias resulting from temperature variations is not significantly different for the different combinations of frequency bands.
Temperature and spectral index profiles of the τ(V) = 10 model observed using different wavelength combinations. In order of decreasing colour temperature these are: 100, 350, 550, and 850 μm (black curves), 160, 250, 350, and 500 μm (green curves), 160, 350, 550, and 850 μm (blue curves), 250, 350, and 500 μm (cyan curves), 250, 350, 550, and 850 μm (red curves).
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Figure B.1 show results of modified blackbody fits of the 100 μm IRAS data and the Planck 857–353 GHz observations. The bottom frames show the 217 GHz residuals (observation minus the prediction of the fitted model) and the difference in spectral index when the lowest included frequency is 353 GHz or 217 GHz. For the field G300.86-9.00 the plot was already shown in Fig. 5.
Modified blackbody fits to Planck and IRIS data in fields G0.02+18.02 and G0.49+11.38. The frames a), c), and d) show 250 μm surface brightness, colour temperature, and spectral index fitted to 3000 GHz (IRAS 100 μm), 857 GHz, 545 GHz, and 353 GHz data, respectively. Frame b) shows the higher resolution Herschel τ(250 μm) map. Frame e) is the surface brightness residual between the 217 GHz observation and the above fit. The last frame shows the difference between spectral indices derived with fits to 3000–353 GHz and to 3000–217 GHz data. The fields in this and the following figures are in order of increasing galactic longitude.
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Figure C.1 show the colour temperature and spectral index maps calculated with Herschel data. The figures include results derived with both 160–500 μm and 250–500 μm data.
Temperature and spectral index fits of field G0.02+18.02 using Herschel data. Frames are: colour temperature and spectral index with SPIRE data and χ2 minimisation (frames a)–b)), 250μm optical depth and colour temperature for SPIRE data and β = 1.8 (frames c)–d)), colour temperature and spectral index with SPIRE data and MCMC calculations (frame e)–f)) and the corresponding error maps (frames g)–h)), MCMC results for 160–500 μm fits (frames i)–j)) with corresponding error estimates (frames k)–l)). The last frames show the effect of zero point uncertainty in the three-band fits (frames m)–n)) and four-band fits (frame o)–p)) (χ2 fits) based on Monte Carlo simulation using χ2 fits. Temperature maps have contours drawn at intervals of 2.0 K, starting at 10.0 K. Spectral index maps have contours at intervals of 0.2, starting at 1.0.
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The joint fit of Herschel (250–500 μm) and Planck (857–217 GHz) data was carried out only for the 53 fields with estimated distances d ≤ 400 pc (see Sect. 3.5). For comparison, the figures also include 353 GHz and 217 GHz residuals from fits where we assume twice the default CO correction (see Sect. 2.2). The larger correction could in some cases eliminate or even reverse the sign of the 217 GHz residual. However, for example in the case of field G6.03+36.73, ground-based observations confirm that the line ratios assumed in the default CO correction are of the correct magnitude (see Sect. 2.2).
Modified blackbody fits in field G0.02+18.02 using the combination of Herschel and Planck data. The uppermost frames show the fitted intensity at 250 μm, and the colour temperature and spectral index maps. The relative residuals (observation minus model, divided by model prediction) are shown in frames d) and f) for the Planck bands of 353 GHz and 217 GHz. Frame f) shows the median values of the residuals for the three Herschel bands (red symbols) and the four Planck bands (blue symbols). The error bars correspond to the median error estimate of the surface brightness over the map (mainly the assumed calibration errors and the uncertainties of the CO corrections). The lower data points at 850 μm and 1380 μm (Planck bands 353 GHz and 217 GHz; cyan symbols) correspond to twice the default CO correction.
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© ESO, 2015
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