The Inclination Correction to The Turbulence Structure Function of Thin Molecular Clouds and Its Application

The energy cascade rate of turbulence can be measured with the structure function. In practice, the 3D velocity of the gas in molecular cloud is hard to measure, which makes the measurement of structure function difficult. In the case of thin molecular clouds perpendicular to the line of sight, the structure function $S^2_{ tt}$ can be measured with core velocity dispersion (CVD), ${\rm CVD}^2=\frac{1}{2}S^2_{ tt}$. This method was extended to the case when the thin molecular cloud is not perpendicular to the line of sight, with intersection angle $\theta$, ${\rm CVD}^2=\frac{1}{2}S^2_{ tt}\left(1-\frac{1}{8}\cos^2\theta\right)R^{2/3}$, where $R$ can be expressed with elliptic integrals of the second kind $E(k,\varphi)$ as $R=\frac{2}{\pi}E(\cos\theta,\frac{\pi}{2})$.