The turbulence spectrum of molecular clouds in the galactic ring survey: A density-dependent principal component analysis calibration

Turbulence plays a major role in the formation and evolution of molecular clouds. Observationally, turbulent velocities are convolved with the density of an observed region. To correct for this convolution, we investigate the relation between the turbulence spectrum of model clouds, and the statistics of their synthetic observations obtained from principal component analysis (PCA). We apply PCA to spectral maps generated from simulated density and velocity fields, obtained from hydrodynamic simulations of supersonic turbulence, and from fractional Brownian motion (fBm) fields with varying velocity, density spectra, and density dispersion. We examine the dependence of the slope of the PCA pseudo-structure function, α_PCA, on intermittency, on the turbulence velocity (β_v) and density (β_n) spectral indexes, and on density dispersion. We find that PCA is insensitive to β_n and to the log-density dispersion σ_s, provided σ_s ≤ 2. For σ_s > 2, α_PCA increases with σ_s due to the intermittent sampling of the velocity field by the density field. The PCA calibration also depends on intermittency. We derive a PCA calibration based on fBm structures with σ_s ≤ 2 and apply it to 367 ¹³CO spectral maps of molecular clouds in the Galactic Ring Survey. The average slope of the PCA structure function, 〈α_PCA〉 = 0.62 0.2, is consistent with the hydrodynamic simulations and leads to a turbulence velocity exponent of 〈β_v〉 = 2.06 0.6 for a non-intermittent, low density dispersion flow. Accounting for intermittency and density dispersion, the coincidence between the PCA slope of the GRS clouds and the hydrodynamic simulations suggests β_v ≃ 1.9, consistent with both Burgers and compressible intermittent turbulence.