Chandrasekhar-Fermi Method

Chandrasekhar & Fermi (1953) obtained the strength of interstellar magnetic field. Applying the same method to the molecular cloud, Crutcher et al. (2004) obtained the magnetic field strength in the plane of the sky in three prestellar clouds as $\simeq 80\mu {\rm G}$ (L183), $\simeq 140\mu {\rm G}$ (L1544), and $\simeq 160\mu {\rm G}$ (L43). The method is as follows: Using the phase velocity of transverse Alfvén wave, $V_{\rm A}=B/(4\pi \rho)^{1/2}$, the wave is described as

$$y=A\cos\left[k(x-V_{\rm A}t)\right],$$ (4.38)

where $y$ is a displacement in the plane of the sky and perpendicular to the mean magnetic field. This gives

$$V_{\rm A}^2\overline{\left(\frac{\partial y}{\partial x}\right)^2}=\overline{\left(\frac{\partial y}{\partial t}\right)^2},$$ (4.39)

where $\overline{X}$ represents the time average. The rhs of the equation, which is the one-dimensional velocity dispersion in $y$ direction, seems to be equal to the velocity dispersion in the line-of-sight $\Delta V$ as

$$\overline{\left(\frac{\partial y}{\partial t}\right)^2}=\Delta V^2.$$ (4.40)

The lhs of the equation is given with a dispersion of polarization direction $\Delta \phi$ (rad)

$$\overline{\left(\frac{\partial y}{\partial x}\right)^2}=\Delta \phi^2,$$ (4.41)

in which the lhs is measured by the wave-pattern of the Alfvén wave. Therefore, the magnetic field strength is given as

$$B_\perp=(4\pi\rho)^{1/2}\frac{\Delta V}{\Delta \phi}$$ (4.42)