A0
azimuth encoder offset
P3 =
c2
collimation of mount
P4 =
 Za sin Aa
azimuth axis offset(EW)
P5 =
za cos Aa
azimuth axis offset (NS)
P6 =
 e0
elevation encoder offset
P7 =
 b
gravitational bending
P8 =
r
refraction
With these we can write the pointing corrections as cos e DA = P1 + P2 cos e + P3 sin e + P4 sin e cos A + P5 sin e sin A
D e = P6 + P7 cos e  P4 sin A + P5 cos A + P8 cot e. (5.47)
Special studies have been made of the atmospheric refraction and as a result this term is well known (see Ch. 6.2.5). Therefore it normally is kept outside the pointing model calculation and applied separately. Thus there remain 7 parameters to be determined when an antenna needs to update its pointing model. As said before this is accomplished by measuring the differences DA and De for a large number of sources, distributed over the entire sky and solving by least squares methods the set of equations (5.47) for the seven constants. These equations with their proper values for the constants are then incorporated in the servo control software to apply the necessary correction to the demanded celestial position to force the beam to be pointed to that position.
The approach to establishing the pointing model, as illustrated above, is now an inherent part of every antenna and telescope control system. Rigorously speaking it is based on approximations which are valid for small errors and positions not too close to the zenith. For the most accurate work with current radio and optical telescopes, a refinement of the analysis might be required.
These are provided by the software package TPOINT, written by Patrick Wallace (1997) and marketed by Tpoint Software. This package is in use at several observatories, among them ALMA and the VLT of ESO. TPOINT is compatible with the somewhat differently structured pointing algorithms described in Wallace (2002). These are implemented as the proprietary software TCSpk, also available from Tpoint Software, which can be used to build antenna control systems in which the pointing analysis and control are closely linked. The ESA 35m deep space antenna in Western Australia (Fig. 1.4) uses both TPOINT and TCSpk, as do a number of optical/IR telescopes.
The notations used by Wallace are different from those used above. For ease of comparison we summarise in Table 5.4 the two correction systems. Note that we have listed in the table the DA correction terms, while in Eq. (5.47) AAcos e is used. An example of the basic output of TPOINT is presented in Fig. 5.12. The distribution of the pointing sources over the sky is indicated in the lower right. The final scatter of the measurements around the nominal direction of the viewing angle is shown in the lower left. The upper two rows of scatter are the deviations of several parameters with respect to azimuth and elevation directions, while the distribution of deviations is shown in the middle lower panel.
While the 7parameter pointing model will account for the majority of the corrections, it is possible that the residuals after the leastsquares fitting procedure exhibit characteristics which point to further systematic errors in the telescope system. Examples of this are harmonic terms caused by encoder and bearing runout, azimuth bearing level variations due to azimuthal changes in the pedestal support stiffness, etc. These can be fitted to the appropriate functions and additional pointing model constants can be obtained and henceforth incorporated in the pointing system.
Table 5.4. Basic pointing model terms in the notation of Stumpff and Wallace  
St 
Wa 
Corr. formula 
Corr. formula 
Cause 

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