Projection Is All You Need: Interpreting Polarization Measurements in the Orion Clouds with Sub-Alfv\'enic MHD Simulations
Dust polarization observations are widely used to diagnose the relative importance of magnetic fields and turbulence in star forming molecular clouds, often through summary statistics such as the mean polarization direction $\mu$ and dispersion $\sigma$. Recent multi-scale polarization observations of the Orion Integral-Shaped Filament (ISF) reveal substantial diversity in polarization morphology among its dense cores, raising questions about the underlying Alfv\'enic nature of the cloud. In this work, we develop a statistical framework to compare polarization-based summary statistics from observations with those derived from projected three dimensional MHD simulations, explicitly accounting for projection effects. Using globally sub-Alfv\'enic simulations that naturally produce slightly super-Alfv\'enic dense cores, we show that modest deviations of core-scale magnetic fields from the parent cloud field, when combined with projection, can generate a wide range of plane-of-sky polarization dispersions. Applying hypothesis testing, we find that the observed $(\mu, \sigma)$ values in the Orion ISF are statistically consistent with sub-Alfv\'enic cloud models over a broad range of viewing angles. This broad degeneracy implies that $\mu$ and $\sigma$ alone cannot provide precise information about the three-dimensional magnetic-field distribution, and hence the Alfv\'en Mach number, of an individual cloud. While the observations can provide evidence against certain projection geometries, we demonstrate that polarization statistics based solely on $(\mu, \sigma)$ are insufficient to provide evidence against sub-Alfv\'enic cloud models. Our results highlight the necessity of explicitly incorporating projection effects when interpreting polarization observations of molecular clouds.