An application of an optimal statistic for characterizing relative orientations

Abstract

We present the projected Rayleigh statistic (PRS), a modification of the classic Rayleigh statistic, as a test for non-uniform relative orientation between two pseudovector fields. In the application here this gives an effective way of investigating whether polarization pseudo-vectors (spin-2 quantities) are preferentially parallel or perpendicular to filaments in the interstellar medium. For example, there are other potential applications in astrophysics, e.g., when comparing small-scale orientations with largerscale shear patterns. We compare the efficiency of the PRS against histogram binning methods that have previously been used for characterising the relative orientations of gas column density structures with the magnetic field projected on the plane of the sky. We examine data for the Vela C molecular cloud, where the column density is inferred from Herschel submillimetre observations, and the magnetic field from observations by the Balloon-borne Large-Aperture Submillimetre Telescope in the 250-, 350-, and 500-µm wavelength bands. We find that the PRS has greater statistical power than approaches that bin the relative orientation angles, as it makes more efficient use of the information contained in the data. In particular, the use of the PRS to test for preferential alignment results in a higher statistical significance, in each of the four Vela C regions, with the greatest increase being by a factor 1.3 in the South-Nest region in the 250-µm band.

Using Herschel and Planck observations to delineate the role of magnetic fields in molecular cloud structure

“…We evaluated the uncertainty that a single V measurement reflects the dispersion of the relative angles following the description presented in Jow et al (2018). Assuming that the angles φ i are independently and uniformly distributed, it follows that every cos 2φ k in Eq.…”

Relative alignment between dense molecular cores and ambient magnetic field: the synergy of numerical models and observations

“…Ophiuchus also shows a very different magnetic field morphology, where the high column density structure has very little preference for aligning either parallel or perpendicular to the Planck-scale magnetic field (see, e.g., Planck Collaboration Int. XXXV 2016; Jow et al 2018). This could indicate the existence of super-Alfvénic turbulence.…”

“…Detailed studies of Planck maps of nearby star-forming MCs have previously shown that high column density structures often tend to have a preference for alignment perpendicular to the magnetic field, while low column density structures preferentially align parallel to the magnetic field (Planck Collaboration Int. XXXV 2016;Jow et al 2018;Soler et al 2019). This change in relative orientation within the cloud has been interpreted as the signature of dynamically important magnetic fields, and an indication that gas accumulation onto dense filaments and cores is preferentially occurring in the direction parallel to the magnetic field.…”

Analysis of Galactic molecular cloud polarization maps: a review of the methods

“…[23], [24]), ζ = A 0 −A 90 A 0 +A 90 , where A 0 is a measure of the total number of points in the quartile 0 • < φ < 22 • .5, and A 90 a measure of the same quantity in the quartile 67 • .5 < φ < 90 • . Smoother and more accurate definitions of ζ have been investigated by [28]. For this Rayleigh statistics Z is used to test whether a given set θ i of n independent angles are uniformly distributed within the range[0, 2π].…”

“…For this Rayleigh statistics Z is used to test whether a given set θ i of n independent angles are uniformly distributed within the range[0, 2π]. Using the relation θ = 2φ, where φ is the relative orientation angle discussed above, [28] showed that the Projected Rayleigh Statistic (PRS) of Z: Z x = Σ n ind i cosθ i / √ n ind /2, where n ind is the number of independent data samples in the map, can be used to test for a preference for perpendicular or parallel alignment between the magnetic field orientations and the iso-contours. The variables ζ or Z x are often calculated on samples of data lying in different ranges of intensity of the material tracer map, and used to explore variations of the relative orientations between the magnetic field structures and the ISM morphology structures as a function of the column density or number density parameters.…”