We model the polarized thermal dust emission from protostellar cores that are assembled by super–sonic turbulent flows in molecular clouds .

Self–gravitating cores are selected from a three dimensional simulation of super–sonic and super–Alfvénic magneto–hydrodynamic ( MHD ) turbulence .
The polarization is computed in two ways .
In model A it is assumed that dust properties and grain alignment efficiency are uniform ; in model B it is assumed that grains are not aligned at visual extinction larger than A _ { V, 0 } = 3 mag , consistent with theoretical expectations for grain alignment mechanisms .
Instead of using a specific set of grain properties , we adopt a maximum degree of polarization P _ { max } = 15 % .
Results are therefore sensitive mainly to the topology of the magnetic field ( model A ) and to the gas distribution that determines the distribution of A _ { V } ( model B ) .
Furthermore , the radiative transfer in the MHD model is solved with a non–LTE Monte Carlo method , to compute spectral maps of the J=1–0 transition of CS .
The CS spectral maps are used to estimate the turbulent velocity , as in the observations .

The main results of this work are : i ) Values of P between 1 and 10 % ( up to almost P _ { max } ) are typical , despite the super–Alfvénic nature of the turbulence ; ii ) A steep decrease of P with increasing values of the sub-mm dust continuum intensity I is always found in self–gravitating cores selected from the MHD simulations , if grains are not aligned above a certain value of visual extinction A _ { V, 0 } ( model B ) ; iii ) The same behavior is hard to reproduce if grains are aligned independently of A _ { V } ( model A ) ; iv ) The Chandrasekhar–Fermi formula , corrected by a factor f \approx 0.4 , provides an approximate estimate of the average magnetic field strength in the cores .

Sub–mm dust continuum polarization maps of quiescent protostellar cores and Bok globules have recently been obtained .
They always show a decrease in P with increasing value of I consistent with the predictions of our model B .
We therefore conclude that sub–mm polarization maps of quiescent cores do not map the magnetic field inside the cores at visual extinction larger than A _ { V, 0 } \approx 3 mag .
The use of such maps to constrain models of protostellar core formation and evolution is questionable .
This conclusion suggests that there is no inconsistency between the results from optical and near–IR polarized absorption of background stars , and the observed polarization of sub-mm dust continuum from quiescent cores .
In both cases , grains at large visual extinction appear to be virtually unaligned .