telescope pointing models gregor project meeting aip, 14 th /15 th oct, 2010 t. granzer
DESCRIPTION
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)TRANSCRIPT
Telescope pointing models Gregor project meeting
AIP, 14th/15th Oct, 2010
T. Granzer
What is it for?
Describes miss-alignments in the two principal telescope axis, alt-az or ra-de etc..Used to improve initial pointing (RMSpoint 1")Improves open-loop tracking (RMStrack 1"/h)
Classic pointing model7-parameter model, follows from pure spherical trigonometry of an imperfectly aligned telescope:
ETAAAAEEEBENEAAEAAAA
cossincossectantancostansin
FENoff
NPPAEENoff
AN,AE… tilt of az-axis against N,ENPAE… non-perpendicularity of alt to az axisBNP…non-perpendicularity of opt. axis to alt axisTF… tube flexure
(alt/az mount)
Tracking relevanceDifferentiate with respect to time:
EEEA
AEE
EEAA
AAA
(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
cos~sin~tansin~tancos~
EN
EN
AA
AA
Quality limits
Basis functions in classic model not orthogonal, i.e.
EEEdEdA sectancos2/
0
2
0
Will lead to high correlation between NPAE and BNP
ijji dEAfEAf 2/
),(),( , in particular
Quality limits (cont’d)Reach orthogonality with constant terms by parameter normalisation:
Replace ai: 2/),(~2/
dEAfaa iii
e.g. 2/2/sec~NP
2/NPNP
BdEBB
Allows for quality assessment of the PM solution by calculating the correlation coefficients and the condition number.
Or: pure harmonic modelUse complete set of orthogonal polynomials on the sphere (better hemisphere):
)EA,(
)EA,(
,lm
mllm
l,mlmlm
YEE
YAA
Ylm… spherical harmonics
Guarantees orthonormality (+quality assessment!):
''*
'' ),(),( mmllmllm dEAYEAY
… but may require lots of parameters
Often used: mixed modelsClassic plus low-order terms of harmonic model
mllmlm
mllmlm
YEEE
YAAA
,class
,class
)EA,(
)EA,(
AAAAAA cossin 1010class
e.g.
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
Deriving
Tile the sky in equal-sized tiles, acquire (bright) stars and measure offset, least square fits.
From 18th Feb to 17th March 2007, we obtained 28 pointing models, each with N 500
Classic model
An f(3az) correlation visible
Harmonic model (h>20°)l=2 (9 constants) bad in az, in alt as good as classicProblem: rapid increase in parameters for high ll=6(az), l=5(alt), 22 constants RMS<2”
Mixed model Include f(3az), Y11, Y1-1
In az, an f(2az) might still be present
No systematic in alt clearly detectable.
Results on all setsHigh correlation on BNP and NPAE
High correlation on BNP and NPAE
AN,E (az) AN,E (alt)AN,E (az) AN,E (alt)
Covariance
Decreasing covariance in NPAE and BNP measures quality of fit
RMS comparisonClassic model sufficient, if RMS10”
Harmonic model fit well, but requires many measuresMixed good compromise
Bootstrap
From model, on parameters, RMS
Classic bootstrap: Duplication of measures, fit several times, variance of solution
Bootstrapping analysis
…gives a good hint on quality
Bootstrapping vs. N
bootstrapping 's better correlated to N then RMS
Temperature DependencyTemperature span limited (10°)
Temperature drift in AN,E (tilt of telescope az)
ConsequencesA 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.