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).
\begin{figure}\centering\leavevmode
\epsfxsize =.45\columnwidth \epsfbox{eps/FWHM.ps}\end{figure}

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

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

This gives the one-dimensional random velocity as
$\displaystyle <v_x^2>$ $\textstyle =$ $\displaystyle \frac{
\int_0^\infty v_x^2\exp\left(-\frac{v_x^2}{\sigma^2}\right) dv_x
}{
\int_0^\infty \exp\left(-\frac{v_x^2}{\sigma^2}\right) dv_x},$  
  $\textstyle =$ $\displaystyle \frac{\sigma^2}{2}.$ (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
\begin{displaymath}
\exp\left[-\left(\frac{v_{x,\rm HWHM}}{2<v_x^2>}\right)^2 \right]=\frac{1}{2},
\end{displaymath} (E.3)

which leads to
\begin{displaymath}
v_{x,\rm HWHM}=(2 \ln 2 <v_x^2>)^{1/2},
\end{displaymath} (E.4)

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

Thus, if we assume isotropic distribution, three-dimensional random velocity of gas
\begin{displaymath}
<v_{\rm 3D}^2>=<v_x^2>+<v_y^2>+<v_z^2>=3<v_x^2>
\end{displaymath} (E.6)

is obtained with the line width as
$\displaystyle <v_{\rm 3D}^2>^{1/2}$ $\textstyle =$ $\displaystyle \left(\frac{3}{2^3\ln 2}\right)^{1/2}v_{x,\rm FWHM},$  
  $\textstyle =$ $\displaystyle \left(\frac{3}{2\ln 2}\right)^{1/2}v_{x,\rm HWHM}.$ (E.7)

Kohji Tomisaka 2012-10-03