Random Velocity

Figure E.1: HWHM (half width of half maximum: the line width measured from the the center of the emission line to the point of the half intensity) and FWHM (full width of half maximum: the line width measured between the points of the half intensity).

Considering gas in Maxwellian velocity distribution, the distribution function for the velocity is as follows:

$$f(v_x)=A \exp\left(-\frac{v_x^2}{\sigma}\right).$$ (E.1)

This gives the one-dimensional random velocity as

$$<v_x^2> = \frac{ \int_0^\infty v_x^2\exp\left(-\frac{v_x^2}{\sigma^2}\right) dv_x }{ \int_0^\infty \exp\left(-\frac{v_x^2}{\sigma^2}\right) dv_x},$$

$$= \frac{\sigma^2}{2}.$$ (E.2)

If we observe emissions from such a gas, the emission line is broaden due to the Doppler shift. Using equation (E.2), the HWHM (half width of half maximum: the line width measured from the the center of the emission line to the point of the half intensity; see Fig.E.1) of the emission line is

$$\exp\left[-\left(\frac{v_{x,\rm HWHM}}{2<v_x^2>}\right)^2 \right]=\frac{1}{2},$$ (E.3)

which leads to

$$v_{x,\rm HWHM}=(2 \ln 2 <v_x^2>)^{1/2},$$ (E.4)

and

$$v_{x,\rm FWHM}=(2^3 \ln 2 <v_x^2>)^{1/2}.$$ (E.5)

Thus, if we assume isotropic distribution, three-dimensional random velocity of gas

$$<v_{\rm 3D}^2>=<v_x^2>+<v_y^2>+<v_z^2>=3<v_x^2>$$ (E.6)

is obtained with the line width as

$$<v_{\rm 3D}^2>^{1/2} = \left(\frac{3}{2^3\ln 2}\right)^{1/2}v_{x,\rm FWHM},$$

$$= \left(\frac{3}{2\ln 2}\right)^{1/2}v_{x,\rm FWHM}.$$ (E.7)