JCMT Pointing Model - Parameter 3 - notes

The diagram from the previous document pertains :

Date    : Tue, 9 Sep 1997 
From    : Iain Coulson
To      : Firmin Oliveira
Subject : parameter 3

1. Although the elevation axis, and hence the nominal zenith,
   are tilted away from their ideal position(s), the rotation
   of the antenna is still performed using the old (unchanged)
   azimuth axis of rotation.

2. Hence P' (or P with a suffix 1 in the diagram )and P are
   indeed on a plane perpendicular to the old azimuth axis OZ.

3. In the triangle APP' the sine law gives

   sin(PP')/sin(theta) = sin(e')/sin(90)  ie
  
   sin(PP') = sin(theta)*sin(e'), which for small theta gives
       PP'  =     theta*sin(e')

4. This is equivalent to an azimuth change of PP'/cos(e'), ie
                                             theta*tan(e'),
   but the distance on the sky is just
                  theta*sin(e').

5. Again in the triangle APP', the cosine rule gives

   cos(e') = cos(e)*cos(PP') + sin(e)*sin(PP')*cos(APP')

   the last term is cos(90) = 0 so

   cos(e') = cos(e)*cos(PP')

   All of this appears on the sheet.

6. This last expression can be rewritten using 3 as 

   cos(e') = cos(e)*cos(theta*sin(e'))
           = cos(e)*(1 - x**2)

   where x = theta*sin(e').

   For small theta, x = 1 - 0.5*theta**2*sin**2(e'), which is 1
                    to order theta**2.

7. So cos(e') = cos(e) to order theta**2 
   so     e'  =     e  to this order also.

8. Hence 4. becomes 
   PP' (the azimuth error on the sky in arcseconds) = theta*sin(e)




Note added later :

Also : if   e = 90 - theta  and  e'= 90 - y'       
       then 6. becomes
      sin(y') = sin(theta)*cos(theta*y')
          y'  =     theta*cos(theta)
              =     theta   to order theta**3  when theta << 1
   i.e.   e'  =  e          ........ theta**3 




imc @ jach.hawaii.edu


Updated: 23 July 1998