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Appendix A: Surface brightness profiles and the outer radius of B68

The top panel of Fig. A.1 shows the circularly averaged intensity profiles of the B68 core at SPIRE and PACS wavelengths, which were used to construct the column density and temperature profiles shown in Fig. 2. The bottom panel of Fig. A.1 shows the logarithmic slope of the column density profile (black line), which is defined as s ≡ d   ln   N_H2/d   ln   r (dimensionless), as a function of radius. The measured logarithmic slope is s = 0 near the core center due to flat inner density profile and the finite resolution of the observations. The logarithmic slope profile reaches a minimum value s_min ~ −1.5 at r ~ 10⁴ AU and goes back to s ~ 0 at the outer boundary, where the core merges with a slowly-varying background. For comparison, a spherical core with outer density profile ρ ∝ r^-2 would have s = −1 at large radii. In the same plot, we show the logarithmic slope of the 500 μm surface brightness profile (blue line), which has a shallower slope due to the additional effect of the positive outward temperature gradient. Inspection of the intensity and slope profiles shown in Fig. A.1 allows us to select an appropriate upper integration radius in the right-hand side of Eq. (2) when the density and temperature structure of B68 (see Sect. 3) are reconstructed. Since beyond a radius ~25 000 AU, larger radii do not contribute to the integral of Eq. (2). In practice, we adopt an upper integration radius R_up = 37   500 AU for B68, as shown by the dashed vertical line in Fig. A.1. The reconstruction results, however, are insensitive to the precise choice of R_up as long as R_up ≳ 25   000 AU. The adjacent dot-dashed vertical line shows the effective radius of ~27 000 AU (or ~200″), as derived by the getsources source-finding algorithm for the “footprint” of B68. The getsources algorithm (Men’shchikov et al. 2012) is the source extraction method used by the HGBS consortium to produce the first-generation catalogs of dense cores found by Herschel in the regions covered by the HGBS survey. The footprint of a core corresponds to the area just outside, of which getsources estimates the local background emission and over which it integrates the background-subtracted emission to derive the total flux densities of the core. In the case of B68, the results automatically derived by getsources are in excellent agreement with those obtained through a detailed radial profile analysis (cf. Fig. A.1).

Fig. A.1 a) Circularly averaged radial surface brightness profiles of B68 at 70, 160, 250, 350, 500 μm derived from Herschel/SPIRE and PACS data (after adding Planck offsets). b) Logarithmic slopes of the circularly averaged column density profile (solid black curve) and 500 μm intensity profile (solid blue curve) of B68 as a function of radius. The horizontal dotted line is the logarithmic slope s = −1 expected for the column density profile of a core with a ρ ∝ r-2 density profile. The vertical dashed line marks the integration upper bound adopted when integrating the right-hand side of Eq. (2) to perform the Abel reconstruction of the density and temperature profiles (see Sect. 3). The vertical dot-dashed line shows the radius of the footprint automatically derived for B68 by the getsources source-finding algorithm.

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Appendix B: Tests of the Abel inversion method using simple models

Appendix B.1: Spherically symmetric core model

To test the performance level of our Abel inversion scheme and quantify the robustness of the reconstruction, we applied the algorithm as described in Sect. 3 to synthetic images corresponding to model starless cores of known density and temperature distributions. First, we considered a spherically-symmetric core model with a Plummer-type density distribution for r ≤ R_out, (B.1)which is parameterized by physical parameters that are approximately similar to the derived properties of B68 (see Sect. 4.1): central H₂ number density n_c ≡ ρ_c/μ_H2m_H = 8 × 10⁴ cm^-3, flat inner radius R_flat= 5 × 10³ AU, and outer radius R_out = 1.5 × 10⁴ AU. The surface density profile of such a model core has an analytical form of (B.2)where p represents the impact parameter from core center in the plane of the sky and N_H2(p) = Σ(p)/μ_H2m_H is the H₂ column density profile. The intrinsic density profile of the model is shown as a black solid curve in Fig. B.1a and the corresponding column density profile as a black solid curve in Fig. B.1b. The synthetic dust temperature profile is shown as a black solid curve in Fig. B.1c and was obtained for a solar-neighborhood ISRF (G₀ = 1) using an analytic approximation formula that reproduces a grid of spherically symmetric models performed with the dust radiative transfer code MODUST (Bouwman et al., in prep.; see Bouwman et al. 2001; and André et al. 2003).

Fig. B.1 Comparison between intrinsic (black curves) and reconstructed (red and blue curves) volume density a), column density b), and dust temperature c) profiles for a spherically symmetric core model with a Plummer-like density distribution (see Eq. (B.1) and text for model parameters). The cross symbols show the results obtained by applying the Abel reconstruction scheme on the synthetic 160–500 μm intensity profiles with “infinite” resolution. The red and blue curves show the reconstruction results obtained from synthetic emission maps smoothed to HPBW resolutions of 363 and 249, respectively, corresponding to the resolutions of Herschel 500 μm and 350 μm observations. Note the good agreement between the reconstructed profiles and the intrinsic profiles beyond the the half power beam radius of 363/2 (500 μm resolution) or 249/2 (350 μm resolution), as marked by the red and blue vertical dotted lines, respectively. The error bars in each panel show the 1σ uncertainties due to random noise and calibration errors.

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A set of synthetic emission maps was created by line-of-sight integration of this model core at all Herschel wavelengths, assuming optically thin dust emission (see Eq. (1)) and the same dust opacity law as given in Sect. 3.1. The density and temperature profiles of the model core were then reconstructed as described in Sect. 3 from the circularly averaged radial intensity profiles of the model emission. The cross symbols overlaid on the model density, column density, and temperature profiles in Fig. B.1a–c show the Abel-reconstructed profiles that would be obtained with “infinite” angular resolution (and in the absence of noise). It can be seen in Fig. B.1 that, the reconstruction is perfect in this case, demonstrating the validity of our Abel-inversion code. The overplotted red curves in Fig. B.1a–c show the reconstructed volume density, column density, and temperature profiles that result from the Abel-inversion method after convolution of the model images to a common resolution of 363. This corresponds to the Herschel resolution at 500 μm. Likewise, the overplotted blue curves in Fig. B.1a–c show the results obtained at a resolution of 249 using the synthetic data convolved to the Herschel resolution at 350 μm and ignoring the 500 μm data. It can be seen that the profiles reconstructed at the Herschel resolution remain in excellent (1%) agreement with the intrinsic profiles in the outer part of the core. Although the reconstruction becomes somewhat inaccurate below the Herschel resolution limit (marked by vertical dotted lines in Fig. B.1), the reconstructed column density and temperature profiles still agree with the corresponding intrinsic profiles to within 20% and 9%, respectively, at 500 μm resolution. The accuracy of the results at small radii improves to 11% and 5% when the reconstruction is performed at 350 μm resolution. (Although in the presence of noise with real data, the statistical measurement uncertainties are somewhat larger at 350 μm resolution.) At both resolutions, the Abel-reconstructed temperature and column density profiles coincide within 1% with the corresponding intrinsic profiles that are convolved with the effective beam resolution. The reconstructed central temperature and column density thus provide excellent estimates of the beam-averaged central temperature and column density in the model. The total mass that is estimated by integrating the reconstructed column density profile agrees with the model mass more accurate than 0.1% even at 500 μm resolution.

Fig. B.2 a) Synthetic column density image of a prolate ellipsoidal core model with an aspect ratio of 1.3 and a Plummer-like density distribution (see Eq. (B.4) and the text for model parameters). The horizontal dotted line shows the axis of symmetry in the plane of sky, and the contour levels are the same as in Fig. 3. b) Comparison between the intrinsic (solid curves) and the reconstructed (crosses) column density profiles of the model. c) Comparison between the intrinsic (solid curves) and the reconstructed (crosses) dust temperature profiles of the model. The blue and red curves represent the intrinsic radial profiles along and perpendicular to the axis of symmetry, respectively. The green curves represent the intrinsic circularly averaged radial profiles. The blue and red crosses display the results of the Abel-inversion method applied to the synthetic 160–500 μm intensity profiles of the model, which are convolved to 363 resolution and taken along the major and minor axes, respectively. The green crosses show the reconstruction results by using the circularly averaged intensity profiles of the model as inputs. Note the good agreement between the reconstructed profiles and the intrinsic profiles beyond the half power beam radius of 363/2, as marked by the vertical dotted line.

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We also assessed the contribution of background fluctuations and calibration errors to the uncertainties in the derived parameters (N_H2and T_d). To do so, we considered 500 realizations of synthetic skies, which included a random Gaussian noise component⁷, σ_ν, and a random multiplicative calibration factor, (1 + g): (B.3)where g is a Gaussian random number with a mean zero and standard deviation of 10% and 15% at SPIRE and PACS wavelengths, respectively. (We assumed 100% correlated calibration errors at SPIRE wavelengths and an independent calibration error in the PACS 160 μm band.) The net uncertainties in n_H2, N_H2, and T_d were estimated to be 15%, 12%, and 5%, respectively. The resulting 1σ errors in the derived parameters are displayed in Fig. B.1.

Appendix B.2: Ellipsoidal core model

As real cores, such as L1689B, are often elongated and thus not strictly spherically symmetric (see Sect. 4.2), we also tested the reliability of our Abel inversion scheme using a simple non-spherical model with an ellipsoidal Plummer-like density distribution for r ≤ R_out and z ≤ Z_out with cylindrical symmetry about the z axis (assumed to lie in the plane of the sky): (B.4)where R_flat and Z_flat are the radii of the flat inner core region perpendicular and parallel to the z axis of symmetry (see Fig. B.2a), respectively. We considered both the prolate (Z_flat > R_flat) and the oblate (Z_flat<R_flat) configuration but are primarily describing the prolate case here, as it is more likely for cores embedded within filaments such as L1689B (see Fig. 3). The synthetic temperature distribution was also assumed to be cylindrically symmetric about the z axis and was constructed using the same grid of MODUST radiative transfer models as in Appendix B.1. The synthetic temperature profiles along both the z axis and the radial (r) direction are shown in Fig. B.2. For direct comparison

with L1689B (see Figs. 3 and B.2a), we adopted physical parameters approximately consistent with the observed characteristics of the L1689B core (see Sect. 4.2): central H₂ number density n_c = 2 × 10⁵ cm^-3; aspect ratio Z_flat/R_flat = Z_out/R_out = 1.3; flat inner radius along the minor axis R_flat = 4000 AU; and outer radius along the minor axis R_out = 6.7 × R_flat = 26   800 AU.

Because of the lack of spherical symmetry, we applied our Abel reconstruction scheme to three sets of intensity profiles: 1) the profiles measured along the major axis of the model (intrinsic profiles shown as blue curves and results as blue crosses in Fig. B.2); 2) the profiles measured along the minor axis of the model (intrinsic profiles shown as red curves and results as red crosses in Fig. B.2); and 3) circularly averaged intensity profiles (intrinsic profiles shown as green curves and results as green crosses in Fig. B.2). Here, again, it can be seen that the reconstruction results are very satisfactory (2% agreement) beyond the beam radius (marked by the vertical dotted line in Fig. B.2). The reconstruction performed perpendicular to the axis of symmetry, i.e., along the minor axis for a prolate core, is more accurate (1%) than the reconstruction performed along the axis of symmetry (4%). In particular, the best estimate of the central dust temperature is obtained from the reconstruction performed along the minor axis. The reconstruction along the major axis nevertheless provides better estimates of the column density and temperature at large radii along the major axis. The central column density reconstructed at 500 μm resolution slightly underestimates but still agrees to within 20% with the true column density at core center. The best estimate of the total core mass, as obtained by using the results of the reconstruction performed on the circularly averaged intensity profiles, agrees more accurate than 4% with the model core mass. Even for a more elongated core model with an aspect ratio of 2 (instead of 1.3), the reconstructed core mass still agrees with the model mass to within 5%.

We also performed similar simulations for an oblate core model observed edge-on. The accuracy of the reconstruction results was found to be essentially the same as for the prolate case. Again, the reconstruction performed perpendicular to the axis of symmetry, i.e., along the major axis in this case, was found to be more accurate than the reconstruction performed along the axis of symmetry. The best estimate of the total core mass was again obtained from reconstructing the circularly averaged intensity profiles.

© ESO, 2014