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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}](img1570.png) |
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}](img1571.png) |
(E.1) |
This gives the one-dimensional random velocity as
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}](img1575.png) |
(E.3) |
which leads to
![\begin{displaymath}
v_{x,\rm HWHM}=(2 \ln 2 <v_x^2>)^{1/2},
\end{displaymath}](img1576.png) |
(E.4) |
and
![\begin{displaymath}
v_{x,\rm FWHM}=(2^3 \ln 2 <v_x^2>)^{1/2}.
\end{displaymath}](img1577.png) |
(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}](img1578.png) |
(E.6) |
is obtained with the line width as
Next: About this document ...
Up: Star Formation1
Previous: Radiative Hydrodynamics
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Kohji Tomisaka
2007-07-08