The dependence of polarization fraction p on total intensity I in polarized submillimeter emission measurements is typically parameterized as p \propto I ^ { - \alpha } ( \alpha \leq 1 ) , and used to infer dust grain alignment efficiency in star-forming regions , with an index \alpha = 1 indicating near-total lack of alignment of grains with the magnetic field .
In this work we demonstrate that the non-Gaussian noise characteristics of polarization fraction may produce apparent measurements of \alpha \sim 1 even in data with significant signal-to-noise in Stokes Q , U and I emission , and so with robust measurements of polarization angle .
We present a simple model demonstrating this behavior , and propose a criterion by which well-characterized measurements of polarization fraction may be identified .
We demonstrate that where our model is applicable , \alpha can be recovered by fitting the p - I relationship with the mean of the Rice distribution , without statistical debiasing of polarization fraction .
We apply our model to JCMT BISTRO Survey POL-2 850 \mu m observations of three clumps in the Ophiuchus Molecular Cloud , finding that in the externally-illuminated Oph A region , \alpha \approx 0.34 , while in the more isolated Oph B and C , despite their differing star formation histories , \alpha \sim 0.6 - 0.7 .
Our results thus suggest that dust grain alignment in dense gas is more strongly influenced by incident interstellar radiation field than by star formation history .
We further find that grains may remain aligned with the magnetic field at significantly higher gas densities than has previously been believed , thus allowing investigation of magnetic field properties within star-forming clumps and cores .