1. Telescope pointing models Gregor project meeting AIP, 14 th /15 th Oct, 2010 T. Granzer

2. What is it for?  Describes miss-alignments in the two principal telescope axis, alt-az or ra-de etc..  Used to improve initial pointing ( RMS point  1")  Improves open-loop tracking ( RMS track  1"/h)

3. Classic pointing model 7-parameter model, follows from pure spherical trigonometry of an imperfectly aligned telescope: A N,A E … tilt of az -axis against N,E N PAE … non-perpendicularity of alt to az axis B NP … non-perpendicularity of opt. axis to alt axis T F …tube flexure (alt/az mount)

4. Tracking relevance Differentiate with respect to time: (five parameter classic, only two parallactic: ) Use open-loop tracking errors instead of pointing offset, will of course result in different values for the parameters

5. Quality limits Basis functions in classic model not orthogonal, i.e. Will lead to high correlation between N PAE and B NP, in particular

6. Quality limits (cont’d) Reach orthogonality with constant terms by parameter normalisation: Replace a i : e.g. Allows for quality assessment of the PM solution by calculating the correlation coefficients and the condition number.

7. Or: pure harmonic model Use complete set of orthogonal polynomials on the sphere (better hemisphere): Y lm …spherical harmonics Guarantees orthonormality (+quality assessment!): … but may require lots of parameters

8. Often used: mixed models Classic plus low-order terms of harmonic model e.g.

9. Quality requisitions  singular point at E=90° for Alt/az mounts,  =90° for parallactic mounts  Tube flexure: h->0  'Wrap around' Az>360° ?  Hysteresis? Distribution of stars on hemisphere important

10. Deriving Tile the sky in equal-sized tiles, acquire (bright) stars and measure offset, least square fits. From 18 th Feb to 17 th March 2007, we obtained 28 pointing models, each with N  500

11. Classic model An f(3az) correlation visible

12. Harmonic model (h>20°)  l=2 (9 constants) bad in az, in alt as good as classic  Problem: rapid increase in parameters for high l  l=6(az), l=5(alt), 22 constants RMS<2”

13. Mixed model Include f(3az), Y 11, Y 1-1  In az, an f(2az) might still be present  No systematic in alt clearly detectable.

14. Results on all sets High correlation on B NP and N PAE A N,E (az)  A N,E (alt) A N,E (az)  A N,E (alt)

15. RMS (classic model)  Altitude axis less than azimuth  …but no influence of N

16. Covariance Decreasing covariance in N PAE and B NP measures quality of fit

17. RMS comparison  Classic model sufficient, if RMS  10”  Harmonic model fit well, but requires many measures  Mixed good compromise

18. Bootstrap From model,  on parameters, RMS Classic bootstrap: Duplication of measures, fit several times, variance of solution

19. Bootstrapping analysis …gives a good hint on quality

20. Bootstrapping vs. N bootstrapping  's better correlated to N then RMS

21. Temperature Dependency Temperature span limited (10°) Temperature drift in A N,E (tilt of telescope az)

22. Consequences  A stable mount is required for good pointing.  Temperature drifts in some parameters possible, but may be a simple time drift (sagging of telescope).  Pointing model can be checked during normal observations.  …but use of science observations may introduce bias.