The geometry2 subroutine

The geometry2 subroutine is called with 5 parameters: these are $thet$, $\phi$ and the 3 co-ordinates for the offset position for which the line of sight is to be calculated (x, y & z co-ordinates). It then proceeds to locate the ring in the cloud in which this offset position is located. Then, using the same method as in the geometry routine the line of sight from that position out of the cloud towards the observer is divided into segments, one for each ring, for which the length and relative velocity are returned. Unfortunately the vector describing the start point for this line of sight is not identical to the one used previously (equation 4.8) as the lines of sight were previously defined as starting on (what we are here calling) the y-axis, therefore $\theta=0$ in equation 4.8. Therefore the new equation for ${\bf X_l}$ is

$${\bf X_l}=(x_s,y_s,z_s)+b(\sin \theta_{los}, \cos \theta_{los}, z_{los})$$ (4.49)

where $x_s,y_s,z_s$ are the starting co-ordinates (in cartesians) of the line of sight. Note that as the x and y axes are now swapped round^4.19 compared to the previous use the $\cos \theta_{los}$ and $\sin \theta_{los}$components are also reversed. Thus equation 4.10 describing the two possible intersections with a particular cylinder is given by

$$b_c=-(x_s \sin \theta_l + y_s \cos \theta_l) \pm \sqrt{(x_s \sin \theta_l + y_s \cos \theta_l)^2-(x_s^2+y_s^2-r_c^2)}$$ (4.50)

The equation for finding the intersection with the disks is unchanged (equation 4.9).

For the same reasons the equations for finding the angles between the lines of sight and the velocity components are also changed. The vectors for the velocity components (as in equation 4.14) are

 

$${\bf V_r(x,y,z)} = (x_s+\frac{b}{\sqrt{1+z_{los}^2}}\sin \theta_l, y_s+\frac{b}{\sqrt{1+z_{los}^2}} \cos \theta_l,0)$$

$${\bf V_\theta(x,y,z)} = (-y_s-\frac{b}{\sqrt{1+z_{los}^2}} \cos \theta_l,x_s+\frac{b}{\sqrt{1+z_{los}^2}}\sin \theta_l,0)$$ (4.51)

$${\bf V_z(x,y,z)} = (0,0,1)$$

So now taking the dot product of each of these in turn with equation 4.49 yields the required three angles

$$\cos \theta_{r_2} = \frac{x_s \sin \theta_l + y_s \cos \theta_l+b} {\sqrt{x_s^2+y_s^2+2b(x_s \sin \theta_l + y_s \cos \theta_l) +b^2} \hspace*{2mm} \sqrt{1+z_l^2}}$$

$$\cos \theta_{\theta_2} = \frac{x_s \cos \theta_l-y_s \sin \theta_l} {\sqrt{x_s^2+y_s^2+2b(x_s \sin \theta_l + y_s \cos \theta_l) +b^2}\hspace*{2mm} \sqrt{1+z_l^2}}$$ (4.52)

$$\cos \theta_{z_2} = \frac{z_l}{\sqrt{1+z_l^2}}$$

There is also a midfind2 subroutine which performs the same functions as the midfind subroutine described in section 4.5.7. The equations are slightly modified, mainly just involving the swapping round of the $\cos \theta$ and $\sin \theta$ terms and to take into account the non zero value of the y-co-ordinate for the starting position of the line of sight. For example when calculating the point of closest approach to the $r=0$ line, equation 4.18 becomes $r=x_s^2+y_s^2+2b(x_s \sin \theta_l + y_s \cos \theta_l)+b^2$ and thus differentiating this to find the value of $b$ at the closest approach yields the equivalent of equation 4.19 to be

$$b_{close}=-(x_s \sin \theta_l - y_s \cos \theta_l)\times \sqrt{1+z_{los}^2}$$