See also subsequent analyses.












 (17) 



(18)  



(19) 
subroutine Model(Az,Alt,dAz,dAlt,p) c RT32 model for azimuth (Az) and altitude (Alt) pointing c offsets, dAz, dAlt. Parameters p are in degrees, so are the c offsets while Az and Alt must be supplied in radians. c It is more accurate version of Model3A with exact analytical c expressions for effects of telescope tilt, elevation axis c skew and main lobe offset. implicit real*8 (ah,oz) real*4 Az,dAz,Alt,dAlt,p(12),Hjump data pi/3.141592653589793d0/ arcsin(arg)=dasin(dmin1(1.d0,dmax1(1.d0,arg))) xi=p(5)*pi/180d0 ! tilt out (toward Az=0 deg) zeta=p(6)*pi/180d0 ! tilt over (toward Az=90 deg) sigma=p(3)*pi/180d0 ! skew beta=p(4)*pi/180d0 ! beam (box) offset sh=dsin(Alt) ch=dcos(Alt) c Tilt se=dsign(dsqrt(dsin(xi)**2+dsin(zeta)**2),ch) ce=dsqrt(1se*se) alfa=datan2(dsin(zeta),dsin(xi)) AT=datan2(ch*dsin(alfaAz),sh*sech*ce*dcos(alfaAz)) *datan2(dsin(alfa),ce*dcos(alfa)) hT=dasin(ce*sh+se*ch*dcos(alfaAz)) AT_Az=AT  Az if(ch.lt.0d0) then AT_Az=piAT_Az hT=pihT endif c Skew & box offset together (good also for hT > pi/2) dAz=arcsin( (dsin(sigma)*dsin(hT)+ *dsin(beta))/(dcos(hT)*dcos(sigma)) ) hb=datan2(dsin(hT)*dcos(sigma) *+cos(hT)*sin(sigma)*dsin(dAz),dcos(hT)*dcos(dAz)) dAz=dmod((dAz+AT_Az)*180/pi,360d0) *+p(1) ! encoder offset *+p(9)*sin(2*Az) ! ad hoc *+p(10)*cos(2*Az) ! ad hoc if(dabs(dAz).gt.180d0) dAz=dAzdsign(360d0,dAz) dAlt= (hbAlt)*180/pi *+p(2) ! encoder offset *+p(7)*ch ! sag *+p(8)*sh ! ad hoc *+p(11)*sin(2*Az) ! ad hoc if(Hjump(Az,Alt).gt.0.) dAlt=dAlt+p(12) end 
subroutine ModelA(Az,Alt,dAz,dAlt,p) c RT32 model for azimuth and altitude offsets; parameters p are c in degrees. Should be good for Alt > pi/2 since then p5 and c p6 both get opposite signs relative to pi  Alt direction and c assuming p9 and p10 are connected with wheels position on c the azimuth track. real*4 p(12) data pi/3.141593/ c ********* Model for azimuth offset ********** t=tan(Alt) dAz=p(1) ! encoder offset *+p(3)*t ! axis skew *+p(4)/cos(Alt) ! box offset *+p(5)*sin(Az)*t ! tilt out (toward Az=0 deg) *p(6)*cos(Az)*t ! tilt over (toward Az=90 deg) *+p(9)*sin(2*Az) ! ad hoc *+p(10)*cos(2*Az) dAz=amod(dAz,360.) if(abs(dAz).gt.180.) dAz=dAzsign(360.,dAz) c ********** Model for altitude offset ********** A=Az if(Alt.gt.pi/2) A=pi+A dAlt= p(2) ! encoder offset *+p(5)*cos(A) ! tilt out (toward Az=0 deg) *+p(6)*sin(A) ! tilt over (toward Az=90 deg) *+p(7)*cos(Alt) ! sag *+p(8)*sin(Alt) ! ad hoc *+p(11)*sin(2*A) ! ad hoc if(Hjump(Az,Alt).gt.0.) dAlt=dAlt+p(12) end 
RT32 Model3 parameters (without refraction) All ~5600 data collected in 2003 and 2004 were analysed and 3684 points fitted with RMS of 12.8" = 0.003550 deg p Parameter (deriv.) Value 
Although the above table may indicate quite nice fit of the model and data, we would like to point out that individual parameters are not to be relied upon too much and the errors given are overoptimistic. This conlusion is based on comparison with earlier fits to parts of data included in this most complete collection. We believe this is due to high correlation between some of the 12 parameters. In particular, the correlation of the parameter No 1 in the table (azimuth encoder offset) with the 3^{rd} and 4^{th} is as high as about 0.95 and still higher correlation exists between the parameters 2, 7 and 8. For our purposes, however, the noted property is not of real moment, since we are interested primarily in stability and goodness of the overall fit.
As seen in Fig. 3, the residua from this model, which was fitted to about 66 % of the data, are generally small on the whole AltAz plane. Among 5615 (2758 altitude offsets and 2857 azimuth offsets) plotted points we have found as many as 4987 (2376 plus 2611), i.e. about 89 %, with residua smaller than 0.01 deg. There are, however, whole sequences of measurement points with evidently systematically high departures. These that lie in close proximity of apparently good measurements may be assumed to be spurious and resulting e.g. from some error connected with particular observing session. More fundamental seems the abrupt fall of the altitude offsets when moving the telescope to higher altitudes and modeled as the 'jump'. The red line in the plot approximates the location of this jump.Fig. 3: Residuals from RT32 pointing model fitting. Majority of them lie well below 0.01 degree in magnitude (a scale bar of +0.02 degree is shown near to the upper right corner), however there are areas of larger and systematic mismatching, most notably in the altitude case (lower figure) around the red curve, which marks a discontinuity (the jump in the model) of 0.02 degree. These need further investigation (a working hypothesis is that they might be due to a mechanical problem related to the subreflector) 
Fig. 4: Smoothed residuals from Model3. The vertical bars correspond to the altitude offsets and the horizontal ones to the azimuth offsets. Each smoothed value is an average of residuals within 5° off the point, weighted with cosine function of the angular distance scaled to π/2 at the edge of this circular area. Averaged residuals were the same as in Fig. 3 except that those that exceeded 0.01° in absolute value have been suppressed. 
Fig. 5: Distribution of residuals from Model3 along the azimuth (left panels) and along altitude coordinate (right panels). The upper panels show the residuals of azimuth offsets, the lower panels show the residuals of altitute offsets. 
270 5 0.0257069 0.0722200 270 4 0.0189086 0.0713727 ............................... 270 89 0.0668883 0.3294362 269 5 0.0252313 0.0723455 . . . 270 88 0.0668030 0.2543178 270 89 0.0669250 0.3300445 
Acknowledgements:
This document is based on years of efforts
of the TRAO staff. I would like to especially thank my colleagues,
Andrzej Kus, Marcin Gawroński and Radosław Zajączkowski, whose help and
contribution were essential to finalizing the presented work.
Translated from T_{E}X by T_{T}H, v. 3.59 on 16 May 2004, 19:55  Last modified: 2004.06.24 