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Reconstructing the density and temperature structure of prestellar cores from $Herschel$ data: A case study for B68 and L1689B

Utilizing multi-wavelength dust emission maps acquired with $Herschel$, we reconstruct local volume density and dust temperature profiles for the prestellar cores B68 and L1689B using inverse-Abel transform based technique. We present intrinsic radial dust temperature profiles of starless cores directly from dust continuum emission maps disentangling the effect of temperature variations along the line of sight which was previously limited to the radiative transfer calculations. The reconstructed dust temperature profiles show a significant drop in core center, a flat inner part, and a rising outward trend until the background cloud temperature is reached. The central beam-averaged dust temperatures obtained for B68 and L1689B are 9.3 $\pm$ 0.5 K and 9.8 $\pm$0.5 K, respectively, which are lower than the temperatures of 11.3 K and 11.6 K obtained from direct SED fitting. The best mass estimates derived by integrating the volume density profiles of B68 and L1689B are 1.6 M_sol and 11 M_sol, respectively. Comparing our results for B68 with the near-infrared extinction studies, we find that the dust opacity law adopted by the HGBS project, $κ_λ =0.1(λ/300 μm)^{-2}$, agrees to within 50% with the dust extinction constraints