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

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.