The LVG case

The Large Velocity Gradient (LVG) case (see section 3.2) shows that for a spherical infalling cloud the observed antenna temperature depends only on $T_K$, $n_{H_2}$ and $\frac{X}{\frac{dv}{dd}}$ (ie. the kinetic temperature, the hydrogen abundance and the relative abundance divided by the rate of change of velocity along the line of sight, respectively).

Theoretical output for a LVG model   [l] If these parameters are all chosen to be constant throughout the cloud then the antenna temperature will be constant over its entire velocity range and will produce a line shape as shown in figure 5.10. The line will be flat toppped and will extend from $-\frac{V}{d}$ to $\frac{V}{d}$ where $V$ is the infall velocity of the outer edge of the cloud and $d$ is dependent on the distance from the cloud centre that the line of sight takes as shown in figure 5.11.

Line of sight direction   [r] Setting $V=kr$ where $V$ is the infall velocity, $r$ is the radius and $k$ is a constant it is then clear from figure 5.11 that $v$, the velocity component along any given line of sight is given by

$$~ v=V \cos \theta$$ (5.5)

where $\theta$ is the angle between the line of sight and the line drawn from the centre of the cloud to the point where the line of sight exits the cloud as shown in figure 5.11. Also simple geometry gives

$$~ d=r \cos \theta$$ (5.6)

Dividing equation 5.5 by equation 5.6 then yields $\frac{v}{d}=\frac{V}{r}=k$ and thus

$$\frac{dv}{dd}=k$$

which, as required, is constant. In other words, with these conditions, the emission along the line of sight is evenly spread over the velocity range present along the line of sight. This velocity range has its extremes at the entry and exit points of the line of sight from the cloud which are given by $\pm V_e \cos \theta_{\rm min}=\pm \frac{k}{d}$ where $V_e$ is the velocity at the exit point which is $k$ if the radius is normalised and $\theta_{\rm min}$ is the value of $\theta$ at the exit point which gives $\cos \theta_{\rm min}\frac{r}{d}=\frac{1}{d}$ for a normalised $r$.

  LVG example output  

Figure 5.12 shows the output for a model with $V=2$ kms$^{-1}$. Output is shown at four positions which correspond to the centre of the cloud, half way to the edge of the cloud, $\frac{9}{10}$ of the way to the edge of the cloud and $\frac{99}{100}$ of the way to the edge of the cloud respectively (from left to right). It can be seen the the output is as predicted in figure 5.10 with the line always reaching the same maximum temperature but with a decreasing line width as the telescope looks closer to the edge of the cloud.