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TPOINT a telescope pointing analysis system
Version 18.20, for Unix 10th December 2011

Published by:

Tpoint





Software

www.tpsoft.demon.co.uk

TPOINT is a trade mark.


Information in this document is sub ject to change without notice. c Copyright 2011 P. T. Wallace. All Rights Reserved. Reproduction, adaptation, or translation without prior written permission is prohibited, except as allowed under the copyright laws.


CONTENTS

iii

Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 INTRODUCTION 1.1 Installing TPOINT on your Unix 1.1.1 Standard distribution . . 1.1.2 Full distribution . . . . . 1.2 System overview . . . . . . . . 1.3 How TPOINT is used . . . . . . 1.4 The TPOINT philosophy . . . . 1.5 Running TPOINT . . . . . . . 1.6 An example TPOINT session . 1.7 Syntax . . . . . . . . . . . . . . 2 MODELING 2.1 The pointing terms . . . 2.2 Sizes of coefficients . . . 2.3 Saving models . . . . . . 2.4 RMS and PSD estimates v v 1 1 1 2 3 3 4 5 7 11 12 12 17 17 20

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3 GRAPHICS 21 3.1 Using the G. . . commands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 3.2 The E9 and A9 plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 4 POINTING DATA 4.1 Sky coverage and mapping order . . . 4.1.1 German equatorials . . . . . . 4.2 Making the observations . . . . . . . 4.2.1 Altazimuths and rotators . . . 4.3 Reading and editing the observations 4.4 Different sorts of coordinates . . . . . 4.5 INDAT data formats . . . . . . . . . 4.5.1 Caption record . . . . . . . . 4.5.2 Option records . . . . . . . . 4.5.3 Run parameters record . . . . 4.5.4 Observation records . . . . . . 4.5.5 Subset records . . . . . . . . . 4.6 Non-standard data formats . . . . . . 4.7 Star catalogs . . . . . . . . . . . . . 4.8 Coordinate ranges, and "beyond the p 4.9 Sample data . . . . . . . . . . . . . . 4.10 Simulated pointing tests . . . . . . . 30 30 30 33 33 35 36 37 38 38 38 39 41 42 42 43 44 46

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iv 5 ADVANCED TECHNIQUES 5.1 Building models . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Exposing unmodeled effects . . . . . . . . . . . . . . . . . . . . 5.3 Physical versus empirical . . . . . . . . . . . . . . . . . . . . . . 5.4 Command scripts . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 The UNFIT command . . . . . . . . . . . . . . . . . . . . . . . 5.6 Reducing multiple data sets . . . . . . . . . . . . . . . . . . . . 5.7 Combining data with UNFIT and OUTDAT . . . . . . . . . . . 5.8 Using data subsets . . . . . . . . . . . . . . . . . . . . . . . . . 5.8.1 Controlling subset membership ­ INDAT and SUBSET . 5.8.2 Model terms for data subsets ­ USE, LOSE, FIX, CHAIN 5.9 Order of terms . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.10 Rigorous fitting . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.11 The generalized gimbal mount . . . . . . . . . . . . . . . . . . . 5.12 Off-axis pointing origin . . . . . . . . . . . . . . . . . . . . . . . 5.13 Batch mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 HOW TO ALIGN YOUR POLAR AXIS

CONTENTS 48 48 49 50 50 51 52 52 53 54 54 55 56 57 57 59 60

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7 REFERENCE SECTION 63 7.1 Commands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 7.2 Pointing terms: names and sign conventions . . . . . . . . . . . . . . . . . . . 102 8 THE INITIALIZATION FILES 138 8.1 The TPOINT initialization file . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 8.2 The TPG (TPOINT graphics) initialization file . . . . . . . . . . . . . . . . . 140 9 COMMAND SUMMARY 143


v

Preface
TPOINT is a software system for analysing telescope pointing. It is a descendant of systems that have been in use at ma jor observatories for many years, including optical, IR, mm and radio telescopes. Its origins go back to the author's work on the Anglo-Australian 3.9 m telescope in the 1970s. The version to be described in this manual runs on Unix platforms; other versions exist which run on Microsoft Windows PCs.

Acknowledgements
The author is pleased to acknowledge all those who have given advice and encouragement during TPOINT's long history, especially Peter Gillingham, Joe Wampler, Don Morton, Steve Lee, Robert Laing, Ken Elliott, Hilton Lewis, Bob Kibrick, Fang Yanling, Russ Owen, Steve, Dan, Matt and Tom Bisque, Krister Wirenstrand, Wayne Rosing, Jeff Mangum and David Terrett.


vi


1

1

INTRODUCTION

TPOINT is a software system for analysing telescope pointing. It is used by observatories around the world: AAT, UKST, UKIRT, LPO, WIYN, ARC, Keck, JCMT, WHT, INT, NTT, VLT, Gemini, SOAR, GBT, LBT and many other ma jor telescopes use TPOINT routinely to maintain their pointing models and to monitor performance. Using TPOINT you can assess the potential for accurate pointing in your telescope and measure misalignments and flexures. The information TPOINT generates can be used in telescope control systems (TCSs) to improve setting accuracy and to enable good coordinates to be logged at any time. In addition, misalignments can be addressed mechanically, for example through adjustments to the polar axis. Using the sophisticated pointing models that can be developed with TPOINT, the best large telescopes deliver pointing accuracy below the 2 arcsecond level and some approach 1 arcsecond, roughly the diameter of the larger satellites of Jupiter. The best amateur telescopes can easily beat 1 arcminute, placing targets such as faint galaxies reliably in the center of even a high-power eyepiece.

1.1
1.1.1

Installing TPOINT on your Unix system
Standard distribution

The standard TPOINT package includes a makefile and all the files necessary to build an executable system, for a specific Unix. The only pre-requisites are the Unix operating system itself and the Tcl/Tk package, which is used for graphics. The package is in the form of separate files, either on a single ISO 9660 format CD-ROM or downloaded as a tar file. Installation is straightforward, and the makefile can easily be adapted should the default installation procedure not be ideal. This is how to install TPOINT: 1. If CD-ROM, mount it and either cd to it or copy the contents to a directory on the hard disk and cd there. If a downloaded tar file, extract its contents into a directory on the hard disk and cd there. 2. The documentation is supplied on a .pdf file. Copy it to an appropriate place on your system. 3. Review the makefile for compatibility with local conventions. By default, files will be placed in the directories $HOME/bin and $HOME/etc/tpoint, creating these directories if they do not already exist. If these destinations are not suitable, either edit the makefile, or alternatively when executing the makefile use make INSTALL_DIR= to override the $HOME default. 4. Invoke make.


2

1 INTRODUCTION 5. Make sure the script tpoint and the executable tpt are in your path. (The makefile puts them in $HOME/bin by default.) 6. Should de-installation become necessary, type make deinstall.

To run TPOINT, cd to the directory you wish to work from and type tpoint. If the program fails to start, or if error messages appear during the start-up phase, or if the graphics commands fail to work, the things to check first are (a) paths, (b) the location and contents of the tpoint.ini file, (c) the location and contents of the tpg.ini file and (d) the existence, location and version of the Tcl/Tk software. Details of how to use special versions of tpoint.ini and/or tpg.ini are given in Sections 8 and 8, near the end.

1.1.2

Full distribution

The full TPOINT distribution includes source code, not only for TPOINT itself but also for the SLALIB library that it uses. Each of these is supplied in source form as one or more tar files, on any appropriate media, for example CD-ROM. As for the executable system, the installation procedure is straightforward and not excessively automated: 1. Create separate directories for the various components: tpoint slalib TPOINT (library and freestanding application) Positional-astronomy library

2. As appropriate, and using conventional techniques, decompress and unpack the contents of the tar file(s) to the directories just created. 3. It is usually necessary to edit the makefiles to suit local directory structures and compiler options. Each makefile contains instructions on what to do. In most cases the changes are all near the beginning of the file. Note that make's environment variables (for example INSTALL_DIR and the compiler options) can be specified on the command line, removing the need to modify the makefiles. 4. For SLALIB, cd to the appropriate directory and invoke make. If you are not using the gcc compiler, refer to the comments near the start of each makefile to see how to invoke make to use your choice of compiler. Then execute a rehash. 5. Go to the directory containing the TPOINT package and invoke make to create both the library and the executable system. (The same advice on running make as in the previous step apply here.)


1.2 System overview

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6. Make sure the script tpoint and the executable tpt are in your path. (The makefile puts them in $HOME/bin by default.) 7. Should de-installation become necessary, type make deinstall. To run TPOINT, cd to the directory you wish to work from and type tpoint. If the program fails to start, or if error messages appear during the start-up phase, or if the graphics commands fail to work, the things to check first are (a) paths, (b) the location and contents of the tpoint.ini file, (c) the location and contents of the tpg.ini file and (d) the existence, location and version of the Tcl/Tk software. Details of how to use special versions of tpoint.ini and/or tpg.ini are given in Sections 8 and 8, near the end.

1.2

System overview

TPOINT is a straightforward, monolithic program which accepts commands from the keyboard. There is a simple but effective "command script" capability, and a log file can be produced, recording what operations were carried out and with what result. Pointing observations are input from ordinary ASCII files (as produced by a text editor) in flexible formats. Graphics output goes to a separate X window; TPOINT can be run on a non-X system but the graphics commands will not work. Some small star catalogs are provided, and some sample data. There is a straightforward text-based hierarchical help system.

1.3

How TPOINT is used

This is how to use TPOINT to investigate the pointing performance of your telescope: 1. Decide which place in the focal plane you want the pointing to refer to. Depending on the size of the telescope and the arrangements for viewing the field, this may be crosswires in an eyepiece, a particular pixel of a CCD, or an autoguider probe. 2. Point your telescope at a sequence of stars of known coordinates (so that the star image is on the crosswires etc.), and record the telescope readouts in a file. 3. Run TPOINT and invoke the INDAT command in order to read in the file of observations. 4. Invoke TPOINT's USE command to specify a likely pointing model for the telescope concerned. The extensive repertoire of TPOINT terms includes zero points, misalignments, flexures and so on as well as general-purpose harmonics and polynomials. There are standard models for both equatorial and altazimuth mounts which make it very easy to get started.


4

1 INTRODUCTION 5. TPOINT's FIT command automatically calculates the optimum size for each term in the model, such that the overall model is the best possible fit to the observations. 6. Using TPOINT's graphics commands, the remaining pointing errors can be inspected. 7. If obvious non-random errors remain, additional terms can be added to the pointing model ­ and removed as necessary ­ and the fit repeated.

The final result estimates how well the telescope is capable of pointing, and what corrections need to be made to achieve this result. The coefficients which TPOINT produces (i.e. the numbers specifying the size of each pointing term) can be fed into the telescope's computer control system to improve the pointing. On small telescopes where there is no computer, they can be used to prepare tables of corrections for various parts of the sky. Another use of TPOINT is to identify opportunities for mechanical adjustment, for example enabling the polar axis of an equatorial mount to be set up accurately even on telescopes which cannot for some reason see the pole.

1.4

The TPOINT philosophy

Many observatories and telescope users have their own ways of fitting models to pointing observations, and in a numerical sense TPOINT may not produce results that are different or better. However, TPOINT is unusual (a) in that the telescope model can be changed during the interactive session and (b) in the variety of its graphical displays. These two features allow rapid exploration of the pointing possibilities of any given telescope and time after time leads to a better model than the one formerly used and to new insights into the telescope's behavior. TPOINT's ability to analyse data from multiple observing sessions, even where zero points etc. have changed, is another unique advantage. The general approach taken by TPOINT is that as far as possible the telescope model should describe real effects (geometrical misalignments, well understood flexures, etc.), and empirical functions should be used only to mop up any remaining systematic errors. There is a school of thought which advocates using empirical functions (for example polynomials or spherical harmonics) for the whole job. However, the TPOINT approach has some advantages: · Simple geometrical misalignments ­ for example a miscentered instrument on a mount which is located at a Nasmyth focus but is not coincident with the elevation axis ­ might require very complicated empirical functions but are simple to deal with analytically. · Direct manipulation of certain geometrical terms while the telescope is in operation can be very useful. An important application of this technique is where a star is switched from one instrument aperture to another simply by changing the collimation parameters. Another example is where the polar axis of a wide field telescope is routinely raised and


1.5 Running TPOINT

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lowered as a function of declination, to minimize the field rotation effects of differential refraction; the polar axis elevation parameter in the pointing model can simply be changed by the same amount and accurate pointing is maintained. · A realistic model of a telescope frequently exposes mechanical deficiencies which can then be diagnosed and cured, or at least understood. Time- and temperature-dependent effects can be pinpointed and remedies sought. · A realistic model is likely to require fewer terms, and the number of stars observed in a pointing test can be correspondingly smaller. · Physically-based models are less likely to misbehave when extrapolating outside the area covered by the available test data. In any case, the models available with TPOINT include an extensive range of empirical terms, giving the TPOINT user the best of both worlds. A good example of this is the AAT model, the terms of which are built into TPOINT as an example of what may be needed to tame a large equatorial with stringent pointing requirements. Though firmly based on well-understood geometrical misalignments and plausible tube flexure corrections, the AAT model also contains complicated harmonics describing what are believed to be flexures in the horseshoe and center section. These flexure models are encapsulated in two terms called HFX and HFD, which are always left at their nominal sizes and are not fitted when individual pointing tests are reduced. HFX and HFD were determined by empirical fits to a very large sample (well over 1000 stars) formed by superimposing the residuals from tests carried out over several years.

1.5

Running TPOINT

To run the TPOINT system from the Unix shell (typically running in an xterm window): tpoint logfile initfile commands messages The four parameters ­ the log file, the initialization file, the command input device and the message output device ­ are all optional and are normally allowed to default, respectively, to the following files: logfile initfile commands messages /dev/null tpoint.ini stdin stdout


6

1 INTRODUCTION

On termination, the log file, if one has been specified, contains a full record of the session, and includes extra information (e.g. correlations between pointing terms) too voluminous to be displayed on the screen during the run. Each TPOINT command line consists of one or more fields separated by spaces. The first field, of up to 6 characters, is the command name, and specifies the action to be performed. The subsequent fields, if any, are the arguments. Where the arguments are numeric, freeformat decoding is performed, and numbers may be entered in a wide variety of formats. The precise way the arguments are interpreted depends on the command in question. TPOINT deals with all forms of input ­ both commands and the various sorts of data file ­ in a standardized way which allows for comments and blank lines to be used and provides consistent handling of lowercase information. Details of these conventions are given in Section 1.7. To abort a command, type Ctrl+C. The command will terminate cleanly as soon as it can, and control will be returned to the operator ready for the next command. To exit normally from TPOINT, use the END command. A full list of all TPOINT commands, in alphabetical order, is given in Section 7.1. A quick reference list can be found at the end. Sequences of TPOINT commands can be executed from a text file called a procedure library. Standard TPOINT syntax applies to such libraries, which may thus contain blank lines and comments (beginning with ! ) to enhance readability. In using procedures from a library, the following commands are involved: CALL x .x PROC x RETURN calls procedure x from the library same: shorthand for CALL x specifies the start of procedure x returns from and ends a procedure

Of these commands, CALL alone is permissible from the keyboard; PROC and RETURN are used only in library procedures. A command CALL x (or .x ) can optionally be followed by arguments, separated by spaces. The argument strings are then substituted into any macros $1, $2,. . . $n that are encountered during execution of the procedure, where $n is the n th string after the procedure name. The standard procedure library file is automatically loaded when TPOINT is started unless a different one is specified on the command line as described above. After loading the library, TPOINT automatically executes a `CALL INIT'; the INIT procedure supplied in the standard library does nothing. Any procedure library loaded as part of start-up must contain an INIT procedure.


1.6 An example TPOINT session

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A private library can be loaded at any time during the TPOINT session by using the INPRO command. This supplements whatever has previously been loaded, so there is no need for private libraries to contain copies of the standard procedures ­ though this will do no harm. Indeed, a private library may contain only one procedure if desired, and each time it is reloaded that procedure will replace its previous version. The INIT procedure is not called when INPRO is used from the command line. An INPRO command without arguments restores the original library, without any subsequent additions. By default, procedure commands are not echoed on the screen; they can be made to appear by means of the ECHO ON command. The volume of messages output during the running of complicated procedures can be controlled by means of the MESLEV command. (This command can also be used from the keyboard, though suppressing messages during normal interactive use of TPOINT is not recommended.)

1.6

An example TPOINT session

A painless way to learn how TPOINT works is to reduce some real data, using one of the sample files. Proceed as follows. First start the system: tpoint There will be various announcements, followed by a * prompt. Read in the ukst.dat sample data file, which is from the 1.2 meter UK Schmidt Telescope at Siding Spring Observatory in Australia: INDAT ukst (You may need to specify a full pathname for the ukst.dat file.) The observations will be listed on the screen. Specify the standard geometrical model for an equatorial mount, and then fit the model to the data: USE IH ID NP CH ME MA FIT The values of the pointing coefficients are reported, as well as the root-mean-square (RMS) error on the sky and the population standard deviation (PSD ­ an estimate of what the RMS would be if the number of stars was much larger). There is also information about which observation happens to have the most influence on the model, and whether it looks abnormal.) Plot the residuals (the pointing errors which remain after the TPOINT model has been applied):


8

1 INTRODUCTION

Figure 1: UK Schmidt Telescope, after fitting the standard 6-term model for an equatorial. The central plot is residuals against h, and is characteristic of fork flexure.

CALL E9 (or its shorthand form .E9) The resulting set of graphs is shown in Fig. 1, where the same residuals are displayed in a variety of ways (see Section 3.2). In this particular case, conspicuous systematic effects are present in the central plot, which is dD (declination residual) versus H (hour angle). The term FO (fork flexure, = FO cos h) is a standard one for fork mounted telescopes and produces this characteristic pattern of residuals. Include the term in the model, fit again and plot the residuals: USE FO FIT .E9 See Fig. 2: that the RMS and PSD are much improved, and there are no obvious remaining systematic effects. This simple 7-term model is, in fact, the one used operationally.


1.6 An example TPOINT session

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Figure 2: UK Schmidt Telescope, after adding the fork flexure term FO to the model.

This illustrates the general strategy for modeling a telescope from scratch. Begin by suitably preparing the TPOINT system and inputting the data. Then create a preliminary model consisting of the basic set of geometrical terms, and perform a fit:

INDAT file USE IH ID NP CH MA ME or USE IA IE NPAE CA AW AN FIT

(for an equatorial) (for an altazimuth)

(Procedures for setting up the basic equatorial and altaz models are also provided in the standard library: use CALL EQUAT or CALL ALTAZ.) The fit can be repeated until the solution settles down. Make a note of the population standard deviation, which is an indication of the general quality of the telescope before any modeling of flexure etc. The next step is to display the residuals graphically. A useful way to start is procedure E9 or A9 in the standard library:


10 CALL E9 or CALL A9 (or use the shorthand form .E9 etc.).

1 INTRODUCTION 9 favorite plots for an equatorial 9 favorite plots for an altazimuth

By means of the USE and FIT commands, try out extra terms to reduce any systematic errors. The simple Hooke's-law tube flexure term TF is frequently required; for fork and yoke mounted equatorials try the fork flexure term FO, and for mounts with a declination axis supported at one end only (the English cross-axis and the German mountings are examples) try the declination axis flexure term DAF. When adding new terms, pay attention to the standard deviation of the new coefficient, the effect on other terms, and whether the population standard deviation has been reduced. Remove new terms if they are reported as being indistinguishable from existing ones, or if they lead to the appearance of messages warning that the fit is ill-conditioned. With the exception of the six standard geometrical terms, it is wise to reject any terms which are not at least 2 sigma in size. For each FIT, a table of the correlations between every pair of terms is output to the log file, for inspection after the TPOINT session; values close to unity show terms which are not easily distinguished given the current data. Obvious "outliers" ­ observations much too far out to believe ­ can be temporarily excluded from the fit by using the command MASK n, where n is the observation number. The FIT command itself identifies the most likely candidate, while the OUTL command allows multiple outliers to be identified and MASKed. More detailed advice is given in Section 2. But be sure to try TPOINT's auto-modeling facility: FAUTO In many cases, this gives better results than a purely manual approach, especially as finetuning by hand can always be used as a second step. The current settings of various internal parameters can be displayed as follows: SHOW The list produced by SHOW includes the command names required to change the parameters, and is a useful guide to some of the facilities available within TPOINT. Here is an example:
ADJUST: the model adjusts the telescope to fit the stars. APPEND: the next input file will overwrite the current data list. CAPT: caption plotting is enabled. CLRNG: the plotting zone is cleared before each plot.


1.7 Syntax
ECHO: library procedures execute silently. FITTOL: the ill-conditioning criterion is 0.001. FRAME: a frame will be plotted. MARKH: the marker height is 0.2. MESLEV: messages of all levels are displayed. messages of all levels are logged. PENS: current pens are 18, 5, 17, 12, 13, 2. PLTZ: the plotting zone is from (0.000, 0.000) to (1.000, 1.000). REPLEN: long reports are given in full. TXP: text precision is 2 (stroke). VT: the screen is assumed to be non-vt100. There are 42 observations in the data list, all active; equatorial mount has been specified.

11

1.7

Syntax

All TPOINT commands, procedure library records, star catalog records, model data records, and pointing data records, are sub jected on input to a preliminary vetting and conditioning, as follows: · Comments ­ records which are blank, or which have ! as the first non-space character ­ are logged and displayed if appropriate but are otherwise ignored. · Non-printing characters (TABs for instance) are replaced by single spaces. · Leading blanks are eliminated as appropriate. · Except within string arguments, lowercase characters are converted to uppercase. · Multiple input lines can be concatenated using the backslash continuation character: if an input line ends with a backslash, that backslash and the subsequent newline are ignored. String arguments are groups of characters delimited by pairs of either " or characters to show that they must stay lowercase. When a procedure library file is being input, string argument delimiters are left intact (to allow library commands to have lowercase arguments). For TPOINT commands, string argument delimiters are interpreted by the command routine itself. When star catalog files, pointing data files, and model data files are input, string argument delimiters are removed. File names containing spaces can be used if they are surrounded by quote characters. TPOINT does not support procedure arguments that contain spaces etc.


12

2 MODELING

2

MODELING

The pointing model is a sequence of terms selected from an internal repertoire. As well as explicitly formulated terms (geometrical effects for example) there is a generic type covering a wide range of polynomials and harmonics. Terms can be added to the model by means of the USE command. For example, to add the terms for polar axis misalignment: USE ME MA The USE command is also used to re-enable fitting after a FIX command. Terms can be removed from the model by means of the LOSE command. For example, to drop the term PDH2: LOSE PDH2 The value of the coefficient of a single term can be sp ecified by using the command: coeff val where coeff is the name of the term and val the value in arcseconds. Thus to set the HA index error to +20 . 7: IH 20.7 You would normally only do this for "FIXed" coefficients ­ ones excluded from fitting. The current value of the coefficient of a single term can be inquired by entering a command consisting simply of the name of the term. For example, to display the current value of the declination index error, simply use the command: ID The current model is fitted to the observations by means of the FIT command. Individual terms can be frozen at a particular value by means of the FIX command, or reinstated in the fitting by means of the USE command.

2.1

The p ointing terms

The pointing terms fall into five categories ­ equatorial, altazimuth, general, special and generic. The model for any particular telescope may be a mixture of several of these types. Some terms or efficiency harmonics), commands. are functionally identical but have different names for historical, convenience reasons. Most of them can be expressed using generic terms (polynomials and but at the expense of clarity and slightly slower execution of the FIT and UNFIT To find out what a given term means in a geometrical sense, refer to Section 7.2.


2.1 The pointing terms The following terms are most useful when modeling an equatorial telescope: IH ID NP CH ME MA FO DAF DAB HCES HCEC DCES DCEC DNP X2HC A1H A1X A1D A2H A2X A2D index error in HA index error in Dec nonperpendicularity of HA and Dec axes nonperpendicularity of Dec and pointing axes polar axis elevation error polar axis error east-west fork flexure flop of cantilevered Dec axis bending of cantilevered Dec axis HA centering error (sin component) HA centering error (cos component) Dec centering error (sin component) Dec centering error (cos component) dynamic nonperpendicularity cos(2h) term EW HA change supplied through auxiliary reading EW change supplied through auxiliary reading Dec change supplied through auxiliary reading HA change supplied through auxiliary reading EW change supplied through auxiliary reading Dec change supplied through auxiliary reading

13

1 1 1 2 2 2

The following terms apply to altazimuth telescopes: IE IA CA AN AW NPAE NRX NRY CRX CRY ACES ACEC ECES ECEC A1A A1S A1E index error in elevation index error in azimuth nonperpendicularity of elevation and pointing axes NS misalignment of azimuth axis EW misalignment of azimuth axis nonperpendicularity of azimuth and elevation axes horizontal displacement of Nasmyth rotator vertical displacement of Nasmyth rotator altazimuth coud´ NS displacement e altazimuth coud´ EW displacement e Az centering error (sin component) Az centering error (cos component) El centering error (sin component) El centering error (cos component) Az change supplied through auxiliary reading 1 LR change supplied through auxiliary reading 1 El change supplied through auxiliary reading 1


14 A2A A2S A2E Az change supplied through auxiliary reading 2 LR change supplied through auxiliary reading 2 El change supplied through auxiliary reading 2

2 MODELING

The following general terms may apply to any telescope: TF TX FLOP tube flexure ­ sin law tube flexure ­ tan law constant vertical displacement

There are several types of sp ecial term. The first type of special term is those belonging to a particular telescop e: the AAT in most cases, for historical reasons. These terms are included in TPOINT mainly as examples of what is likely to be involved in improving the pointing model for any large telescope as experience builds up: ZH ZE HF HGES HGEC DGES DGEC TFP HFX HFD CD4A CD4B CD5A CD5B AWGS AAT HA Z-gear effect in HA AAT HA Z-gear effect in polar axis elevation AAT main east-west horseshoe flexure 36m gear error in HA ­ sin 36m gear error in HA ­ cos 9 gear error in Dec ­ sin 9 gear error in Dec ­ cos AAT tube flexure ­ non-Hooke's-law term AAT residual horseshoe east-west AAT residual horseshoe north-south AAT coud´ 4 collimation error A component e AAT coud´ 4 collimation error B component e AAT coud´ 5 collimation error A component e AAT coud´ 5 collimation error B component e Gemini South azimuth axis tilt EW

The following special terms implement simplified approximate formulas and are included to match use of these formulas in control system software. An example is CA/CAL: the former is the nonperpendicularity of the telescope or antenna boresight to the elevation axis while the latter is just A = -CA sec E . ANL AWL CAL CHL MAL MEL NPL NPAEL TXL like like like like like like like like like AN but using simplified A, E formulas AW but using simplified A, E formulas CA but using a simplified A formula CH but using a simplified h formula MA but using simplified h, formulas ME but using simplified h, formulas NP but using a simplified h formula NPAE but using a simplified A formula TX but working in elevation rather than zenith distance


2.1 The pointing terms

15

Two special terms provide for the case where there is an instrument rotator and the pointing origin (i.e. the image location) is not at the rotator axis: POX POY the x-coordinate of a pointing origin on an instrument rotator the y -coordinate of a pointing origin on an instrument rotator

To be able to solve for the POX and POY coefficients (see Section 5.12) requires observations at a variety of rotator position angles. This happens naturally the case of an altazimuth telescope if the rotator is being driven to maintain a given field orientation. TPOINT supports rotators at difference foci: mounted directly on the telescope, such as at Cassegrain or prime docus, or at a Nasmyth or coud´ focus. The rotator angle is supplied in degrees as auxiliary e reading #1. The rotator location, and the telescope type, are specified by "option" records in the data file that is read in by the INDAT command. There are three types of generic (i.e. not individually named) term, where the name of the coefficient spells out the required formula. The most important generic terms are polynomials and harmonics. Polynomial terms have names of the form: Prc [ i [ c [ i ] ] ] For example, a term PXH2D1 would model an effect which produced an east-west shift (X) proportional to h2 (H2D1). The initial P identifies this generic term as a polynomial. The r field describes the result, and is one of the following: H X D U L P A S Z E N W V result result result result result result result result result result result result result is in hour angle is east-west is in declination tilts polar axis up/down tilts polar axis left/right changes HA/Dec nonperpendicularity is in azimuth is horizontal is in zenith distance (sky) is in elevation (mount) tilts azimuth axis north/south tilts azimuth axis east/west changes Az/El nonperpendicularity

(The distinction between mount and sky is important only for the "generalized gimbal" case. This will be dealt with in Section 5. The two ci fields indicate the independent variables and their powers. Each i is in the range 0­9; each c can be any of:


16

2 MODELING

H hour angle D declination A azimuth Z zenith distance (sky) E elevation (mount) Q parallactic angle Trailing ci or i fields, if omitted, default to unity. For example, a change in HA proportional to HA (i.e. a scale change, such as might be necessary with a roller-driven encoder) can be produced by using the term PHH. Harmonic terms have names Hrfc[i][fc[i or Hrfc[i[i[i]] For example, a term HXSH2 proportional to sine (S) of 2h of the forms: ]] ] would model an effect which produced an east-west shift (X) (H2).

The initial H identifies this generic term as a harmonic. The r and c fields describe the result and an independent variable respectively, and are as for polynomials, above. Each fci, fcii and fciii field indicates a function of an integer multiple of an independent variable c. The f field is either S for sine or C for cosine. Each i is normally in the range 0­9, making available frequencies from zero to 999 cycles per revolution in the case of simple one-coordinate harmonics and, in the case of compound two-coordinate harmonics, 0­99 for the first angle and 0­9 for the second. An omitted i defaults to unity, as does an omitted trailing fci. As a special case, the integer can appear as a single % character, denoting onehalf. This allows one-cycle-per-two-revolutions effects to be modeled, such as might occur in ball or roller bearings. Auxiliary terms, less commonly used than the other two types of generic term, simply apply a correction proportional to one of the auxiliary readings. They have names of the form: Anr For example, a term A2E would model an effect which produced a correction to mount elevation proportional to auxiliary reading #2. The initial A identifies this generic term as an auxiliary. The n and r fields identify which of the three auxiliary readings is to be used and what is to be adjusted. The decimal number (n ) is 1-99, and the same result codes (r ) are used as for polynomials and harmonics. A variant, called "vector auxiliaries", allows a single coefficient to be fitted using two auxiliary readings and delivery a vector result of some kind. For example, if basis function values for corrections in azimuth and elevation are supplied as auxiliary readings #3 and #4, the single term A3A4E can be used.


2.2 Sizes of coefficients

17

2.2

Sizes of co efficients

The mathematical expressions used by TPOINT are rather simple and designed to give results of adequate accuracy when not too close to the pole of the mounting and when the pointing errors of the telescope are smal l. Flexures and misalignments of up to a few arcminutes are unlikely to give problems; they will not interact significantly with one another, and will be calculated by TPOINT to sufficient accuracy. However, terms with larger coefficients may not be so well-behaved, especially near a pole. A particular source of problems in practice is collimation error, the non-perpendicularity of the chosen pointing axis with respect to the declination or elevation axis of the mounting. If the measuring device ­ eyepiece, autoguider probe, CCD, TV camera etc. ­ is offset too far, there may be unmodeled pointing errors due to such effects as optical distortion. The best plan is to ensure that the measuring device is offset no more than a few minutes of arc. If this is impossible, and alternative is to pre-process pointing observations to remove the bulk of the offset before the TPOINT run. This might involve rigorous calculation of the tangent-plane pro jection and optical distortions. These difficulties all go away if the "rigorous fitting" option is used. This will be covered later, in Section 5.10.

2.3

Saving mo dels

Commands are provided for writing the current model to a file, and for replacing the current model with one read from such a file: OUTMOD file [S] INMOD file write [session] model to a file read model from a file

where file is the name of the file containing the model. As well as providing long-term storage for models, the file may also be read by programs external to the TPOINT system. (For the case where the external program is a telescope control system, OUTMOD's S option omits terms not normally needed for that application, namely POX, POY and any data subset terms ­ see Sections 5.12 and 5.8.) In addition to the model information required by INMOD, the file contains the current caption and the basic statistics which relate the model to the current data list. The INMOD command ignores these latter items. The file consists of variable length formatted character records. The order of records within the file is:


18 caption record method, statistics and refraction record term records (one per term in the model) end record Here is an example file: AATf/151979/06/11 T491.120652.015-0.06242.0116 IH+174.75431.18854 =ZH+3.5100 ID+23.04640.35693 &=HFX+1.0000 &=HFD+1.0000 &HF-18.72830.48982 &X2HC-3.11220.26477 &NP+2.96360.79658 &CH-18.68891.20785 =ZE+0.7000 &ME+58.25040.43977 &MA+2.95180.24207 TF+8.96550.54629 &TFP+1.16860.47733 END

2 MODELING

The caption record can be up to 80 characters long. The metho d, statistics and refraction record can be read with a C scanf format string of "%c %5d %9lf %9lf %9lf %9lf" or a FORTRAN FORMAT specification of (A1, I5, F9.4, F9.3, 2F9.4. The fields are: method: T or S number of active observations sky RMS (arcseconds) refraction constant A (arcseconds) refraction constant B (arcseconds) sampled PSD (arcseconds, may be absent) A method of T means that the model corrects the telescope readings, whereas S means that the model is applied to star positions to predict the required telescope setting. Each term record can be read with a C scanf format string of "%c %c %8s %10lf %12lf" or a FORTRAN FORMAT specification of (2A1, A8, F10.4, F12.5). The fields are:


2.3 Saving models chained/parallel flag fixed/floating flag term name coefficient value coefficient sigma

19

The chained/parallel flag is either a space (for "chained" terms, ones which are computed from the position as affected by all previous terms; see Section 5) or an ampersand (for "parallel" terms: members of a group all computed from the same starting position). The fixed/floating flag is either a space (for coefficients that are to be fitted) or an equals sign (for coefficients of fixed value). The term name and the coefficient value and sigma have their normal meanings. The end record consists of the string END . Reading programs should also interpret end-of-file as equivalent to an end record. A further command: OUTMEX file writes the same information (plus some additions) but in the form of a single line of fields separated by | (vertical bar) characters. This can be read directly by spreadsheet applications such as Microsoft Excel. The technique is particularly suited to studying how the pointing performance (in particular the values of particular model coefficients) is changing over time. The order of fields in the OUTMEX output record is the same additional fields ouput by OUTMEX are the components of the (east-west, north-south, left-right and up-down) and a list of The four RMS components immediately precede the sky RMS record: method: T or S number of active observations EW RMS (arcseconds) NS RMS (arcseconds) LR RMS (arcseconds) UD RMS (arcseconds) sky RMS (arcseconds) refraction constant A (arcseconds) refraction constant B (arcseconds) sampled PSD (arcseconds, may be blank) At the end of the OUTMEX record is a list of MASKed stars, in place of the END record output by OUTMOD. (MASKed stars are ones you have elected to exclude from the fit.) as in the OUTMOD file. The sky RMS in the four directions the stars that were excluded. in the statistics portion of the


20

2 MODELING

2.4

RMS and PSD estimates

It is important to understand that the sky RMS figures which TPOINT produces using the FIT command (also OUTMOD and OUTMEX) are a posteriori and may not be realized in practice. They do not take into account long term stability, the effectiveness of any preobserving calibration procedures, the correctness of the implementation of the model within the telescope or antenna control software, or the ability of the user interface to give the astronomer access to the available performance. Moreover, the reported sky RMS values are optimistic in cases where a large model is fitted to a small number of observations. To take this into account, the FIT command also reports the more honest statistic "population standard deviation", obtained as follows. Writing n for the number of floating coefficients (ones not marked = ) in the pointing model, for the sky RMS and o for the number of active observations, the PSD is estimated as = 2 o/(o - n). The "sampled PSD" produced by the SAMSIG command gives an even more honest result, by using the residuals obtained from models where the observation concerned has been excluded from the fit (a procedure known as "jackknifing"). If SAMSIG has been run, this sampled PSD will be present in the OUTMOD and OUTMEX output.


21

3

GRAPHICS

TPOINT has the following repertoire of graphics commands: CAPT FRAME G GAM GC GDIST GHYST GMAP GSCAT GSMAP MARKH PENS PLTOF PLTON PLTZ enable/disable plotting of caption enable/disable plotting of graph frame plot residuals in one coordinate against another look for changing polar/azimuth axis misalignment special functions: clear screen, write hardcopy file plot error distributions look for hysteresis plot error vectors on cylindrical pro jection plot residuals as a scatter diagram plot error vectors on azimuthal pro jection specify marker size specify pens for lines and text close plotting window open plotting window specify plotting zone

The PLTON command opens a plotting window which can optionally be named. The various graph plotting commands are then available: G, GAM, GDIST, GHYST, GMAP, GSCAT and GSMAP. Sample graphs from each of these commands can be found in Figs. 3 and 4. PLTZ is useful for displaying several small graphs at once instead of one full-size graph; GC can be used to clear the whole screen beforehand. The PLTOFF command closes the current plotting window and reverts to the earlier one, if any.1 The remaining commands ­ CAPT, FRAME, MARKH, PENS ­ are used either for reducing plotting time, or for achieving special effects for demonstration and publication purposes. They can be ignored by novice users. Multiple plotting zones are supported, allowing several plots to be displayed at once. The current plotting zone is specified by the PLTZ command: PLTZ or PLTZ name or PLTZ x1 x2 y1 y2 use full display surface use named region use numerically specified region

The predefined zone names specify regular subdivisions of the display surface:

In most TPOINT sessions there is no need to use PLTON and PLTOFF; a graphics window is, if necessary, opened automatically if there is any plotting to be done, and closed again when TPOINT is terminated.

1


22

3 GRAPHICS

T L B R

TLQ

TRQ

TL CL

TC CC BC

TR CR BR

BLQ

BRQ

BL

If numeric parameters are supplied, they describe directly the region of the display surface which is to contain the plotting zone. The numbers are the two X extremes and the two Y extremes respectively, in units where the corresponding dimension of the display surface is unity. All four numbers are required. Irrespective of the shape of the zone and how it was specified, plotting will actually occur in a zone of the standard aspect ratio centered within this area.

3.1

Using the G. . . commands

The commands in TPOINT which plot graphs all have names beginning with G. The most important one for deciding what terms to add to the model is called simply G, and plots one component of the residuals against one coordinate. It has the following syntax: G ydata xdata [scale] The argument ydata specifies which component of the residuals is to be plotted: H X D P A S in hour angle east-west in declination corresponding to HA/Dec nonperpendicularity in azimuth horizontally


3.1 Using the G. . . commands

23

Figure 3: Example plots using the commands G D H (top left), GDIST (top right), GAM E (center left), GAM A (center right), GHYST E (bottom left) and GHYST A (bottom right).


24

3 GRAPHICS

Figure 4: Example plots using the commands GMAP E (top left), GMAP A (top right), GSCAT E (center left), GSCAT A (center right), GSMAP (bottom left) and GSMAP = (bottom right).


3.1 Using the G. . . commands E Z V R in elevation (mount) in zenith distance (sky) corresponding to Az/El nonperpendicularity total residual

25

The argument xdata specifies what coordinate to plot against:

H D A E Z Q N

hour angle declination azimuth elevation (mount) zenith distance (sky) parallactic angle observation order

The optional argument scale indicates the vertical plotting scale: it is the absolute value of the largest residual to be plotted. In default, a scale is chosen automatically to suit the data. The general strategy is to plot different components of the residuals against the various coordinates. Terms can then be added to the model (USE followed by FIT) to target any systematic effects that are seen. Some plots are more likely than others to have a mechanical interpretation, and attention should be confined to these, at least to start with. For example, an apparent relationship between polar axis elevation and azimuth, on the evidence of a G U A plot, should be treated with great skepticism (especially on an altazimuth mount), whereas systematic residuals on G N A and G W A plots (for an altazimuth) or on G U H and G L H plots (for an equatorial) are much more credible. Because files of pointing observations are normally written in time order, G ydata N plots are useful for exposing shifts or drifts that have happened during the pointing run. Other x-axis meanings can be contrived by appropriately sorting the observation file prior to input. The other plots have a variety of presentational and diagnostic roles. GSCAT plots residuals as a scatter diagram, like a shooting target; as well as being a good way of presenting the overall pointing performance, it exposes errant observations and abnormal distributions. GSMAP displays the residuals as error vectors on an azimuthal pro jection. There is a choice of the orthographic pro jection (a distant view of the celestial sphere looking down on the zenith from above) and an equal-area pro jection. The vectors (lines pro jecting from the square symbols which mark the stars) are the great-circle continuations of the pointing residuals. GMAP plots error vectors on a Cartesian Cylindrical pro jection, either HA/Dec or Az/El. GDIST plots histograms showing various sorts of error distribution, and is useful for testing whether the residuals are reasonably normal. GAM looks for changing polar/azimuth


26

3 GRAPHICS

axis misalignment. Each residual is interpreted as being due to a misalignment of the specified "roll" axis ­ either polar or azimuth as specified. A histogram is plotted to show if the residuals do, in fact, favor one particular direction. The mean direction is calculated and reported, and then the component of each residual in the direction of this misalignment is plotted. GHYST looks for hysteresis, by drawing the error vector at a position which indicates the direction from which the telescope has come, assuming that the original data file contains all the observations and in the correct order, and that any hysteresis is related to the direction of the large scale movements in cylindrical coordinates (either HA/Dec or Az/El). A numerical estimate of the hysteresis is made, by summing the residuals oriented relative to the telescope movements. Many of the plotting commands have arguments for specifying scales and selecting HA/Dec and Az/El options etc. Full details are given in Section 7.1. Hardcopy, in the form of a color PostScript file, can be generated by entering the command:

GC

P

file

The filename defaults to tpoint.ps in the standard TPOINT distribution. Some experimentation with pen and background colors (see Section 8, near the end) may be necessary to get the best results, especially on black-and-white printers. Alternatively, all the pen colors can be overridden to black by instead using the command: GC B file

3.2

The E9 and A9 plots

When developing a pointing model it is necessary to examine the residuals in several different ways, and it is usually most convenient to use ready-made scripts in the standard procedure library to accomplish this, in particular: .E9 or .A9 9 favorite plots for an equatorial 9 favorite plots for an altazimuth

In the equatorial case, the nine E9 plots are as follows; Fig. 5 is an example: TL TC The east-west (i.e. h cos ) residuals plotted against hour angle. The declination residuals plotted against declination.


3.2 The E9 and A9 plots

27

Figure 5: The nine plots produced by the command .E9

TR CL CC CR BL BC BR

Zenith-distance errors against zenith distance. The east-west residuals plotted against zenith distance. The declination residuals plotted against hour angle. The residuals are interpreted as changes in the h/ non-perpendicularity and plotted against hour angle. The "scatter" diagram, like an archery target; the inner circle shows the standard deviation of the radial error. Various histograms showing the error distributions in different coordinates. A cylindrical pro jection of the whole sky showing the star positions (blobs) and pointing residuals (lines).

In the altazimuth and generalized gimbal cases, the nine A9 plots are as follows; Fig. 6 is an example:


28

3 GRAPHICS

Figure 6: The nine plots produced by the command .A9

TL TC TR CL CC CR

The "left-right" (or "sideways" or "cross-elevation", i.e. A cos E ) residuals plotted against azimuth. The elevation residuals plotted against azimuth. The residuals are interpreted as changes in the az/el non-perpendicularity and plotted against azimuth. The left-right residuals plotted against zenith-distance (90 - E ). Zenith-distance errors against zenith distance. The residuals are interpreted as azimuth axis tilt, and plotted against azimuth; the superimposed histogram shows if the residuals are biased towards a particular tilt direction. The "scatter" diagram, like an archery target; the inner circle shows the standard deviation of the radial error. Various histograms showing the error distributions in different coordinates.

BL BC


3.2 The E9 and A9 plots BR

29

An orthographic pro jection of the whole sky showing the star positions (blobs) and pointing residuals (lines).


30

4 POINTING DATA

4
4.1

POINTING DATA
Sky coverage and mapping order

It is important to sample the whole sky and for the observations to evenly distributed. The order in which the observations are carried out can also be significant. Partial sky coverage means that the model must extrapolate into the unmapped areas. This is always dangerous, especially if any polynomial terms are used, because the unconstrained "end" of the region allows the correction function to diverge. Fitting across a gap, i.e. interpolating, is always safer. Sometimes limited sky coverage is unavoidable, either because of obstructions or interruption by bad weather, and in these cases the best policy is to restrict the model to terms of a harmonic character and of low spatial frequency. In fact a basic six-term geometric model plus a couple of flexure terms is relatively immune from the effects of limited sky coverage, to the extent that for an altazimuth telescope a respectable model can be obtained if only a single ob ject is observed, sampled periodically as it tracks from rise to set. Reasonably even distribution is important so that the model is not unduly weighted in a particular area of sky, and so that the residual plots are well-populated and unbunched. The small star catalogues that come with TPOINT meet this need. A common mistake is to observe stars on an [ h, ] or [ Az , E l ] grid. This not only biases the model to the polar or zenith region, but the residual plots become ugly and are difficult to interpret. Order matters if the mapping run is refining the operational pointing model in real time. A degree of randomness is also useful, to expose any hysteresis and to distinguish between spatial and temporal effects.

4.1.1

German equatorials

The German equatorial mount (GEM) is a design that is often used for small telescopes, where flexibility of telescope attachment is paramount and where the need for counterweights is acceptable. Its biggest disadvantage is that observations across the meridian must be interrupted and the telescope reversed in its position relative to the tripod or pier. Because this reversal has the effect of extending the declination range into the "beyond the pole" region (see Section 4.8), there are important implications for pointing calibration that must be borne in mind when deciding the mapping strategy. This is particularly true in cases where the mount is portable, and the mapping run is refining the pointing model in real time. A typical GEM scenario is where the detector has a limited field of view ­ a camera with a small chip say ­ and the mount's geometrical misalignments are relatively large. The


4.1 Sky coverage and mapping order

31

first star is found, one way or another, and a "sync" is performed to reset the encoder zero points. The problem then arises of finding a second star. Moving too far from the first star often means missing the new one altogether, so it is natural to be cautious and move only short distances at first. Gradual enlargement of the sampled area will eventually build up a sufficient data set to begin to estimate the polar axis misalignment (the ME and MA terms) and the nonperpendicularity between the telescope tube and the declination axis (the CH term). However, it is often the case that the first star following the first meridian flip will not be seen, because the sky orientation of the not-yet-accurately-known CH correction has reversed. A better mapping strategy is needed if good all-sky pointing is to be achieved early in the test, and if reliable estimates of polar axis misalignment are to be obtained speedily. The recommended procedure is as follows:

1. Find the first star, sync on it and make this the first star in the data set. Remove the pointing model or reset it to zeroes. 2. Find a second star, ideally but not necessarily some distance from the first. Execute the following commands: USE IH ID FIT This has the effect of establishing an "average sync" for the first two stars. 3. Find a third star, ideally but not necessarily some distance from the first two. Execute the following commands: USE IH ID CH MVET 99 USE ID FIT After the USE command, the model comprises IH ID CH, and if the three stars are sufficiently spread in declination, this may allow a preliminary estimate of CH. However, the MVET command may have other ideas: the resulting model will contain the ID term at least, but IH and/or CH may be absent if MVET has decided they have not been reliably detected. 4. Find a fourth star, this time on the opposite side of the meridian. Be prepared to use the finder or to scan; however, this should be the last difficult star to find. Execute the following commands:


32 USE IH ID CH ME MA MVET 99 USE ID FIT

4 POINTING DATA

Depending on MVET's assessment of statistical significance, the resulting model may consist of up to five terms. 5. Find a fifth star. There is no need for an additional meridian flip, and this time it should be possible to choose a star distant from the previous one. Execute the following commands (the same as last time): USE IH ID CH ME MA MVET 99 USE ID FIT Again, depending on MVET's assessment of statistical significance, the resulting model may consist of up to five terms. 6. Find a sixth star and execute the following commands: USE IH ID NP CH ME MA MVET 99 USE ID FIT By now, some indication of polar axis misalignment should be emerging. 7. As subsequent stars are acquired, it becomes feasible to look for other terms, such as TF and DAF. The MVET command will show when this stage has been reached, by removing currently ineffectual terms.

It is not necessary to stick precisely to this scheme, as long as there is a meridian flip early on, preferably before the fourth star. There will probably be two stars that are difficult to acquire, namely the first star of all and the first star after the meridian flip. These are both dealt with early on, and the improved pointing after the first few stars will speed the remainder of the test. At the expense of a few extra minutes, it is helpful to jump around the sky during the mapping, including multiple meridian flips. This will exercise the machinery more, and help expose any hysteresis and drifts.


4.2 Making the observations

33

4.2

Making the observations

The preferred method for making the observations is to move the telescope so that the image of the specified star (or other calibration target) falls at the reference point (the nominated place in the focal plane ­ the center of the CCD perhaps). This is completely unambiguous and avoids various difficulties of interpretation, such as knowing which way up is north. However, many users prefer instead ­ and on some telescopes there may be no choice ­ to take an exposure at what is hoped to be the position of the star, to measure where the image actually appeared, and from the offset to deduce the (, ) of the reference point. Working in this way is appealing, because it eliminates a time-consuming and delicate adjustment of the telescope, and the image analysis phase can be put off until later. However, for the method to work, it is essential to have a complete understanding of astrometric details, in particular the image scale and the orientation of the picture. In practice this is not always the case. For example, the CCD may not be aligned accurately with declination, so that an assumed simple mapping between pixel (x, y ) coordinates and sky (, ) is in fact not achieved. Another common blunder is to assume that an x-offset of a particular size always means the same change in the right ascension coordinate , when in fact there is a cos factor to take into account: near the pole a given x-offset will mean a bigger change than at low declinations. And on the most precise telescopes, subtle errors may be introduced by small position-angle errors caused by the imperfect pointing model that was in service when the test was carried out. One variant on this technique, i.e. later, is however completely safe. to enable a complete astrometric predict the (, ) of the reference at exactly the desired point. making an exposure and recovering the pointing information It requires the exposures to contain enough reference stars fit of the recorded field. The solution can then be used to point, thereby creating what is in effect an invisible target

Analogous remarks may be made about radio telescopes. To keep the TPOINT picture simple it is always best to "peak up" on the calibration source, using standard techniques such as fitting Gaussians to cross-scans or using the "5-point" method. However, for some radio telescopes (and some targets, such as the Sun) it may be necessary instead to make raster scans for later analysis. Doing so requires great care to be taken with sign conventions, scales and cos E factors.

4.2.1

Altazimuths and rotators

On altazimuth-mounted optical/IR telescopes there is almost always a rotatable instrument mount (or equivalently an image rotator). This introduces complications to do with the interaction between the TPOINT model and where in the rotating focal plane the target star is to be imaged, and these affect the strategy for obtaining the pointing observations. Several choices are available.


34

4 POINTING DATA

One approach is simply to fix the rotator and pretend it isn't there, so that the pointing model refers to where in the non-rotating focal plane the camera or guider etc. happens to be. But it is best to define the reference point as the rotator axis, the only place that is stationary in both the rotating and non-rotating focal planes. If, during the pointing test, the telescope is adjusted to place the stars on the rotator axis, the fitted IE and CA coefficients will be those that correspond to that spot, and this will be true whether the rotator is fixed or is allowed to rotate to maintain a specified field orientation. But if the reference point is somewhere else, say the middle of a CCD that itself is not exactly on-axis, or worse still a guide probe at the edge of the field, difficulties arise. If the rotator remains fixed during the pointing test, the resulting IE and CA coefficients will be for that spot in the non-rotating focal plane, and as soon as the rotator angle is changed the pointing will be incorrect in the rotating focal plane. On the other hand, if the rotator is active during the test, to stabilize the field, pointing terms will be introduced that depend on the rotator angle. The usual manifestation of this is that on the GSCAT A scatter plot the points lie on a circle. The classic solution is to center the detector on the rotator axis, which is found by taking exposures with the rotator angle moved 180 . It then should not matter whether the rotator is active or not, though it is probably best to keep the rotator stationary so that any error in the axis determination disturbs only the IE and CA coefficients and not the rest of the model. A crude but effective alternative is to carry out the pointing test with the rotator active and to include in the pointing model the equatorial collimation terms ID and CH. The IE and CA coefficients in such a model automatically refer to the the rotator axis, while the ID and CH terms provide information about the displacement between the "pointing origin" (the CCD center, say) and the rotator axis. The method depends on the same field orientation being maintained throughout, and becomes inaccurate if the detector is a long way off axis ­ more than a few arcminutes. The ID and CH terms would normally be discarded, though if they are used in the operational pointing model they will to first order reproduce the conditions of the pointing test and make the star images fall on where the detector was rather than the rotator axis, as long as the same field orientation is used. The most general solution is to allow the instrument rotator to be active, to record its orientation as the first auxiliary reading, and to include in the pointing model the terms POX and POY, the position of the pointing origin in the rotating focal plane. This is discussed in Section 5.12. These options are summarized in the following table:


4.3 Reading and editing the observations detector rotator anywhere stopped terms IE CA remarks The IE and CA terms refer to wherever in the non-rotating focal plane the detector was placed ­ ideally the rotator axis. This method relies on the rotator axis having been accurately calibrated. The same same field orientation must be maintained throughout. The rotator angle (in degrees) must be supplied through Aux #1. IE and CA refer to the rotator axis; POX and POY may be used to calibrate the focal plane or simply discarded.

35

on-axis slightly off-axis anywhere

active active active

IE CA IE CA ID CH IE CA POX POY

4.3

Reading and editing the observations

The TPOINT command for reading a set of pointing observations is called INDAT. Several different formats are accepted; all are converted into a standard internal form, called the data list. The command is: INDAT file [subset] [subset]. . . where file is the name of the file of pointing observations and subset (if present) associates the observations with the specified "data subset" (see Section 5.8). The new observations may either replace or be appended to any controlled by a flag armed by the APPEND command. The flag is have been appended. In the append case the caption is reset and data file is ignored. The new latitude replaces the previous one; it unusual to combine the data from several telescopes. previous ones. This is reset once the new data the one supplied in the would, however, be very

The MASK and UNMASK commands allow specified subsets of the observations to be suppressed temporarily. They are useful for verifying the independence of the solution in these different subsets. An important application is where observations from only a specified region of the sky are included. MASK and UNMASK can also remove and reinstate individual observations or sequences of observations. One common use of MASK is to suppress individual outliers, observations where the total residual is so large that something must be seriously amiss ­ telescope malfunction or star misidentification for example. However, it is unwise to remove points that are just a few sigma off without some a priori justification. Possible outliers can be identified by using the SLIST command with the optional argument, the threshold radial error in arcseconds. Another clue to whether an observation is suspect


36

4 POINTING DATA

comes at the end of the FIT report where the "least typical" observation, i.e. the one that has the largest influence on the fitted model, is identified. A more rigorous test, but one that consumes more processing time, is performed by the OUTL command, which not only identifies the best candidate but also reports its "badness" quantity. For normal points, the badness should be less than unity; points whose badness exceeds a specified value can be MASKed automatically by typing OUTL v n, where v is the desired cut-off value and n is the maximum number of terms to MASK, defaulting to one. Observations flagged as abnormal by either FIT or OUTL are usually plain outliers, and immediately visible as such on a GSCAT plot, but sometimes they are wrong in a more subtle way related to position in the sky. A quick way to check is to type MASK n, where n is the observation number, then MARK 2, then .A9 or .E9 as appropriate to plot the residuals: the suspect star will show up in red. (The MARKed stars can be reset to the MASKed state by typing UNMARK and are then excluded from the next FIT.) The PURGE command removes completely the MASKed observations, so that they can be restored only by rereading the data file. Genuinely erroneous observations (for example misidentified stars) should be removed outside TPOINT simply by using a text editor.

4.4

Different sorts of co ordinates

In the internal form produced by the INDAT command, each observation consists of (i) the true direction of the star, which is known as the "observed place",2 and (ii) the raw mechanical HA/Dec of the telescope. The raw telescope HA/Dec normally means either the mechanical HA/Dec directly read from dials or encoders,3 or, in the case of non-equatorial mounts, the HA/Dec obtained by applying the appropriate standard transformation to the actual readings. Az/El data, for example, are transformed by the standard text-book Az/El to HA/Dec rotation, using the longitude and geodetic latitude of the observatory.4 In order to avoid having to log anything which changes rapidly, the equatorial coordinates accepted by INDAT are in the form of RA/Dec rather than HA/Dec. The telescope HA is
Observed is a technical term, meaning the direction of the incoming light ray, i.e. where a perfect theodolite would observe the star to be. It is a common mistake to assume that "observed" means the logged encoder readings. 3 For a servo-controlled mount where the encoders are used to close the position loops, the correct data to log are the demands to the servos, not the encoder readings themselves. This means that any offsets or lags in the servos will automatically be taken care of in the pointing model. It also means that noise in the encoder, for example due to irregular resolution or even occasional bad readings, does not pollute the data used for the pointing analysis. Moreover, the servo demands are often much easier to obtain than the encoder readings, especially if the position servos are implemented in local processors or specialized controllers. 4 and, for the utmost accuracy, with polar motion taken into account.
2


4.5 INDAT data formats

37

deduced simply by subtracting the given RA from the given sidereal time. This applies to all but one of the data formats, where star and telescope azimuths and elevations are supplied directly and must all correspond to one instant of time. Starting from the star positions supplied in the INDAT file, the "observed" HA/Dec positions used by TPOINT internally are obtained by allowing as necessary for proper motion, precession, nutation, annual aberration, and light deflection, to obtain geocentric apparent place, followed by corrections for diurnal aberration and atmospheric refraction. It is also possible to supply the star data as "true" RA/Dec, by omitting the temperature, pressure, etc. in order to disable the refraction corrections. This would be done in cases where the TCS which generated the pointing data has already applied these corrections (either correctly, or at least in a way which will be consistent with any subsequent implementation of the pointing model obtained from TPOINT). In cases where the TCS neglects the small corrections for diurnal aberration, it is possible to include an optional record which disables the diurnal aberration corrections in TPOINT to match. The modeling procedures available elsewhere in this package have the job of expressing the relationship between these two sets of coordinates. It is thus the responsibility of the TCS itself to perform accurate mean to apparent place transformations, and to allow for Earth polar motion, diurnal aberration, refraction, etc., before applying the pointing model as determined by using this package. Further information on the mean-to-observed place transformation can be found in Section 7.2.

4.5

INDAT data formats

The INDAT input file consists of free-format records up to 500 characters in length, as follows:

CAPTION record OPTION records (optional) RUN PARAMETERS record OBSERVATION records (plus any subset or rotator records) END or end-of-file

Blank lines and comments (which begin with a ! character) can be included anywhere. Long lines can be supplied piecemeal by ending partial lines with a backslash character. The individual fields in the record descriptions below are separated by spaces. (In some cases other forms of whitespace will work, also commas.) All numeric fields are free format.


38 4.5.1 Caption record

4 POINTING DATA

The CAPTION record is simply an alphanumeric string. The first 80 characters are used; trailing blanks are eliminated.

4.5.2

Option records

There are several different OPTION records, all optional:



: : : : : : : : : : :

NODA ALLSKY equinox EQUAT ALTAZ GIMBAL z y x ROTTEL ROTNL ROTNR ROTCL ROTCR



disables correction for diurnal aberration disables horizon/zenith/pole checks required if telescope positions are mean places mount is an equatorial mount is an altazimuth mount is a generalized gimbal locates rotator on OTA (Cass, PF etc.) locates rotator at Nasmyth left locates rotator at Nasmyth right locates rotator at coud´ left e locates rotator at coud´ right e

The EQUAT, ALTAZ and GIMBAL options are covered in Section 5. The correction for diurnal aberration (maximum 0 . 3) control system itself omits it. The ALLSKY option is or another, the observations exceed the normal zenith positions are mean places, only one equinox (probably and applies to the whole file. should be disabled if the telescope useful in cases where, for one reason distance range. Where the telescope B1950 or J2000 ) may be specified,

The rotator locator options, ROT. . . are used when the pointing origin (e.g. a guide probe) is on a rotating instrument mount and is not on-axis. They can also appear later in the file, allowing such observations made at different foci to be mixed. See Section 5.12.

4.5.3

Run parameters record

The RUN PARAMETERS record is a single line of text containing the following:


4.5 INDAT data formats latitude ( , , ) UTC date (y,m,d) temperature ( C) pressure (mB = hPa) height above sea level (meter, default determined from pressure) relative humidity (range 0­1, default 0.5) observing wavelength (micrometer, default 0.55) tropospheric lapse rate ( K/meter, default 0.0065)

39

Which of these parameters are needed depends on the chosen observation record format (see below). The latitude is mandatory, and the only one used in the case of Format 4, where the observation consists of simply the star and telescope Az/Els. If the UTC date is omitted, the star and telescope data must not be mean places. If the refraction data (temperature and so on) are omitted, no corrections for refraction are made. If either the pressure or the height is omitted, it is estimated from the other. Optical refraction is computed for wavelengths below 100 micrometers, and radio refraction for wavelengths longer than this figure. In the radio case, it is important to specify the humidity accurately. Note that the pressure is actual pressure at the site (QFE), not the equivalent sea-level pressure (QNH). For a site at sea level it will be around 1000 hPa (millibars); for a high altitude site it will be a much lower figure.

4.5.4

Observation records

Each OBSERVATION record is a single line of text corresponding to one star observed; there is a choice of four formats: Observation record Format 1:

star geocentric apparent RA/Dec (h,m,s, , , ) telescope raw RA/Dec (h,m,s, , , ) local apparent sidereal time (h,m) [up to 2 auxiliary readings]

Observation record Format 2:


40 star mean RA/Dec (h,m,s, , , , µ ,µ ,equinox) telescope raw RA/Dec (h,m,s, , , ) local apparent sidereal time (h,m) [up to 2 auxiliary readings] Observation record Format 3: * identifier from star catalog telescope raw RA/Dec (h,m,s, , , ) local apparent sidereal time (h,m) [up to 2 auxiliary readings] Observation record Format 4:


4 POINTING DATA

star "observed" Az/El ( , ) ­ the true direction telescope raw Az/El ( , ) ­ where the mount was commanded to point [up to 2 auxiliary readings] Notes: · Although star positions in Format 1 are usually apparent right ascension and declination, together with local apparent sidereal time (LAST), it is equally valid to provide instead celestial intermediate reference system (CIRS) right ascension and declination, together with Earth rotation angle (ERA). The LAST or ERA is used only to transform the supplied right ascensions to hour angles; indeed, it may be more convenient to supply hour angles directly and to set LAST/ERA to zero. · Using Format 4 requires precise times to be extracted from the telescope control system in order to predict the exact Az/El of the star, whereas Formats 1-3 use instead the telescope RA/Dec, which is unlikely to vary rapidly during tracking of a star and is therefore easier to get right. Moreover, precisely consistent Az/El and time information may simply be unavailable if special provision has not been made in the control software.5 However, in the case of Formats 1-3, if sidereal time of high precision is available,
A word of advice: do not take the line of least resistance and supply star positions that contain the corrections applied by the current operational model. It is very tempting to do this, because all that needs to be logged is where the telescope was told to point and the small adjustments that proved necessary to achieve final alignment. The disadvantage of this approach is that the resulting TPOINT model is in addition to rather than instead of the current operational model. If, at a later date, it is necessary to go back and re-analyze old data, only if meticulous records have been kept of what model was in use for each test will re-analysis be possible. In practice this doesn't happen. There is really only one right way to do it: the telescope control software must be able to log the true star direction and the raw telescope demands while at the same time implementing an operational pointing model (without which acquisition of the calibration stars would be at best slow, at worst impossible).
5


4.5 INDAT data formats

41

this information can, and should, be supplied, by appending decimals to the sidereal time minutes. · In Format 2, the units for µ and µ are, respectively, seconds and arcseconds per Julian year. (Parallax is neglected.) Note that in Format 2, although the star positions are mean place the telescope positions are still "raw", not a mean-place representation. · In Format 2, the equinox is likely to be J2000 and the star data ICRS rather than being referred to a dynamical system as in the past. The astrometric algorithms used by TPOINT in fact always apply the small ( 23 mas) offset between ICRS and mean J2000, and so by specifying equinox J2000 the star coordinates are interpreted as ICRS positions. · Following the normal convention, the azimuths in Format 4 are reckoned from north through east (though internally within the TPOINT software they go from south through east). · In the "generalized gimbal" case, both the star and the telescope [ Az , E l ] are indeed [ Az , E l ], not raw mount coordinates. In other words, the telescope measurements must be converted from their raw form into their [ Az , E l ] equivalents. In "beyond the pole" cases, appropriate changes must be made to ensure that the recorded [ Az , E l ] reflects the condition. · Each format permits given record format. ampersand character maximum number of up to two auxiliary readings to follow the mandatory fields for the If more than two are required, they can be included by placing an & between the mandatory fields and the auxiliary readings. The auxiliary readings is 99.

· If auxiliary readings are not supplied, a sequence number and its square are inserted into the first two. The sequence number is the number of the observation record divided by 100. The feature is useful for fitting terms that are time dependent, as for evenly-spaced observations the stored number is a surrogate for time. · If a mount type has not already been specified, through an appropriate option record, the first "beyond the pole" observation in Formats 1, 2 or 3, or "beyond the zenith" observation in Format 4 will summarily impose a selection of EQUAT or ALTAZ.

4.5.5

Subset records

Individual observations may be flagged as members of "data subsets", with access to their own pointing terms. Full details are given in Section 5.8. Subset membership is controlled through "D-records", interspersed with the observation records. The following options are available:


42 D/subset D subset D+subset D-subset DD D/ D+ subsequent subsequent subsequent subsequent subsequent subsequent subsequent subsequent observations observations observations observations observations observations observations observations

4 POINTING DATA in specified subset only in specified subset only in specified subset also not in specified subset not in last-mentioned subset not subsetted not subsetted in last-mentioned subset also

D-records override any subset name(s) specified on the INDAT command.

4.6

Non-standard data formats

Two approaches are available for reading "foreign" data formats. The first is to implement a local TPOINT command which accepts the foreign format and converts it directly into the internal form. The second approach, which does not involve modifications to TPOINT, is to write a freestanding program which translates the foreign format into one of the INDAT formats. Such translation programs can be tricky to write, and frequently involve applying positional astronomy adjustments that will all be undone again when INDAT reads the file. Nevertheless, this approach is probably the best in most cases.

4.7

Star catalogs

Several small star catalogs are supplied with the TPOINT system, as follows: stars20b.dat stars20f.dat stars15.dat stars10.dat starssch.dat stars.dat 102 FK5 stars, 5m and brighter, 20 spacing. 102 Tycho-2 stars, 7.0m -7.5m ,20 spacing. 182 Tycho-2 stars, 7.0m -7.5m , 15 spacing. 411 Tycho-2 stars, 7.0m -7.5m , 10 spacing. UK Schmidt calibrators default

stars.dat is loaded automatically when the system is initialized. TPOINT is delivered with stars.dat containing a concatenation of stars20b.dat and starssch.dat. The catalogs are needed for cases where, in a file of pointing observations, the stars' catalog numbers are given rather than their full [ , ], proper motions etc. However, they are also valuable for actually carrying out pointing tests; using the chosen catalog and observing every star visible on the night concerned, a set of observations will be obtained where the samples


4.8 Coordinate ranges, and "beyond the pole"

43

are evenly spread over the sky, and where the plots of the results display minimal "bunching." Regarding the latter effect, it is a common mistake to select stars on a straightforward [ , ] grid; because the stars all lie on meridians or small circles, they are bunched together on the equatorial plots, making it harder to see trends. A private star catalog can be substituted, either at startup time via the command line or during a TPOINT session via the INST command. The original catalog can be restored at any time by means of an INST command with no arguments. A star catalog is a sequential file of up to 180-character records. Here is an example record: 072852 10 02 54.793 -04 43 52.01 +0.0042 -0.103 2000.0

The first field of each record is an alphanumeric star identifier. The remainder of the record is the mean RA/Dec (hours, minutes, seconds, degrees, arcminutes, arcseconds), RA/Dec proper motions (seconds and arcseconds per year) and equinox (optionally preceded by B or J to distinguish between the FK4 and FK5 systems6 ). Two optional fields may be appended, respectively the parallax in arcseconds and the radial velocity in km/s (+ = receding), though only in extremely rare cases will this affect the pointing analysis perceptibly. The equinox field may be omitted and defaults to J2000/ICRS. Free-format input decoding is used. The file is terminated by END or end-of-file. The standard TPOINT syntax rules apply to such catalog files, which may thus contain blank lines and comments (beginning with ! ) to enhance readability. n.b. The star identifier is not case-dependent unless enclosed by quotes (double or single). However, spaces are not allowed even if quotes are used.

4.8

Co ordinate ranges, and "b eyond the p ole"

Most equatorial mountings can be configured so that the mechanical declination range extends "beyond the pole". The simple and obvious case is a horseshoe or fork mounting that allows access to regions of sky underneath the pole, where circumpolar sources pass through lower culmination. Less obvious, but a fruitful source of confusion, is what happens with asymmetric mounts: cross-axis or, in particular German equatorials. In this case, in order to cross the meridian the telescope has to pass through the pole to the declination range beyond, while HA is offset by 12h . TPOINT accepts data containing a mixture of observations, and fits models where the various pointing terms change appropriately as the telescope switches into the other configuration. To
6

TPOINT neglects the tiny distinctions between J2000/FK5/ICRF.


44

4 POINTING DATA

obtain this behavior, telescope [ , ] coordinates in the INDAT data file must be "mechanical" rather than "celestial". This means that in one of the two configurations, declinations beyond the pole should appear (for example +123 at a northern-hemisphere site, -99 at a southern-hemisphere site) paired with hour angles that are 12h from the celestial value. Such observations are flagged b on the SLIST and FLIST output; the [ h, ] coordinates reported appear in the conventional ranges. It is usual for mount controllers on amateur telescopes to work entirely in celestial coordinates, and not uncommon for the vital "beyond the pole" (also known as "side of pier") state to be invisible to TPOINT. The symptoms of this problem are (i) all logged telescope positions have in the normal range (i.e. none where | | > 90 ), (ii) TPOINT reports a very poor RMS result, (iii) the dX (= h cos ) versus h residuals have a discontinuity on the meridian, and (iv) the operational pointing is poor, and may differ markedly on the two sides of the meridian. When this happens, the workaround is to fit separate models for the two sides of the meridian and have the TCS switch between them when the meridian flip maneuver is carried out. Note that the "rigorous fitting" method (Section 5.10), where the mount type is explicitly specified, is mandatory for coordinates outside the standard range. This can be enabled either by an appropriate choice of option record in the INDAT file or will be automatically invoked when the first beyond the pole/zenith observation is encountered. Implementation of a pointing model should include provision for detecting the "beyond the pole" case; in general, the mechanical coordinates should be used in the formulas, and the corrections applied to the mechanical coordinates before the latter are transformed back into celestial ones. Analogous remarks apply to altazimuth mounts and generalized gimbal mounts. Note however that it is a very bad idea to take an altazimuth through the zenith, as any slightly loose mechanical and optical components will slump into a different state and cause pointing errors.

4.9

Sample data

Four examples of authentic pointing data are provided: from the UK Schmidt Telescope (1.2 meter aperture, fork equatorial); from the Anglo-Australian Telescope (3.9 meter, horseshoe equatorial) in the f/15 Cassegrain configuration; from the Palomar 5 meter telescope (horseshoe equatorial) in the f/17 IR configuration; and from the central reference telescope of the pre-upgrade Multiple Mirror Telescope (4.5 meter equivalent aperture, altazimuth). The UKST file refers to the star catalog starssch.dat (the same stars are included in the default stars.dat file; the other three files use explicit star positions and are therefore selfcontained). The filenames are as follows:


4.9 Sample data UK Schmidt AAT Palomar MMT ukst.dat aat15.dat hale.dat mmt.dat

45

(When using the INDAT command it is not necessary to specify the .dat filename extension. If the filename doesn't contain a period you can append a forward slash / to stop the .dat being added.) The recommended analysis procedure, which was described in Section 1.6, earlier, begins as follows: INDAT file CALL EQUAT or CALL ALTAZ FIT CALL E9 or CALL A9

(equatorial) (altazimuth) (equatorial) (altazimuth)

It is then usual to start trying obvious extra terms via the USE command ­ tube flexure TF, fork flexure FO, etc., as appropriate. The credibility of the added terms can be judged partly by the significance of the fitted coefficient (the ratio of the coefficient value to its estimate) and partly by its influence on the population standard deviation for the fit as a whole. New terms should have at least a 2 significance, and should reduce the PSD. Another guide is the MVET command, which ranks the model terms in decreasing order of credibility and suggests which terms may be doing more harm than good. Example models (which in the case of the UKST and AAT are the ones actually in service) are supplied as procedures, yielding the results shown in the following table: datafile UKST AAT15 HALE MMT procedure UKST AAT HALE MMT / terms 6.17 7 1.26 10 fitted + 4 fixed 2.23 11 0.55 13

To reproduce, for example, the AAT result, you would simply type: INDAT aat15 CALL AAT To display this AAT result as a scatter diagram, type:


46

4 POINTING DATA

Figure 7: Anglo-Australian Telescope, f/15 focus, after fitting the TPOINT model.

GSCAT

The resulting graphic is illustrated in Fig. 7. The circle in the middle of the diagram is roughly the size of a dime seen from a distance of one mile (or a Euro at 2 km).

4.10

Simulated p ointing tests

When developing the pointing-model part of a telescope control system, refer to The Pointing Terms in Detail, later, for information on the mathematical form of TPOINT's various pointing terms. The operational implementation of the terms may well involve techniques which differ in some way from TPOINT's ­ perhaps by being more elaborate and general. Any substantial inconsistencies between TPOINT and the telescope software will manifest themselves as pointing errors, and telescope users will not experience the levels of performance which the TPOINT results appear to offer.


4.10 Simulated pointing tests

47

In order to check that the model as implemented in the telescope system is consistent with TPOINT, a "dummy" pointing test is recommended. This is like a real pointing test except that the telescope is left where it is after each blind setting rather than being guided onto the star. When the dummy pointing test is reduced with TPOINT, a perfect fit should be reported, and the coefficients should exactly match the values used by the control software. Without a test of this sort, errors of scaling and (especially) sign may go unnoticed. Note that the dummy pointing test just described is a check on the software alone, with the telescope itself playing no part. Consequently, it is usually best to arrange that the control software can operate in a "simulation" mode, capable of generating a dummy pointing test offline. It is strongly recommended that dummy pointing tests be carried out from time to time, to verify that any revisions of the telescope software and TPOINT have not led to incompatibilities. When TPOINT is used to process data from dummy pointing tests, it should be borne in mind that only if the model is implemented in exactly the same way is it is in TPOINT will the residuals be identically zero. For example, if spherical trigonometry formulas are used to apply corrections for which TPOINT uses a vector formulation, this may have detectable effects, especially if the affected terms have large coefficients and particularly for stars near the pole and/or zenith. Another possible cause for slightly imperfect fits is where all the terms in the model are "parallel" rather than "chained"; although TPOINT computes all the corrections from the same starting-point, it accumulates them sequentially, and the telescopecontrol code should follow suit. Grouping the terms sensibly and then chaining the groups will, in general, give better results, especially if any of the coefficients (for example encoder zero-points) are large. Even better results are achievable by using the "rigorous fitting" option (Section 5.10), with matching telescope control software. This is described later.


48

5 ADVANCED TECHNIQUES

5

ADVANCED TECHNIQUES

TPOINT allows a straightforward model for a reasonably well-behaved equatorial or altazimuth telescope to be generated easily and without great cunning or insight. It also provides the tools more experienced and determined users will need when conducting especially searching analyses and tackling difficult cases.

5.1

Building mo dels

As already described, the general strategy for modeling a telescope from scratch is to input a data file with INDAT, to set up a basic geometrical model using CALL EQUAT or CALL ALTAZ, and to fit using the FIT command, repeating as necessary until the coefficients settle down. With the aid of the G... commands, plots of residuals can then be made and inspected for signs of systematic error. Plausible flexure terms can be tried (USE TF etc.), and any others that the particular telescope design suggests. TPOINT provides expert help with modeling, in the form of the FAUTO command and the .EFIT and .AFIT procedures. In many cases, after the observations have been input using INDAT, all that is needed is the command: FAUTO Quite good results are usually obtained, and the resulting model can form the basis for further fine-tuning by hand. A similar strategy to that used by FAUTO is employed by the library procedures .EFIT (for an equatorial) or .AFIT (for an altazimuth). Unlike FAUTO, which automatically reverts to simpler models when the number of observations is limited, these procedures will always attempt an exhaustive analysis, which can be informative. Moreover, the supplied scripts form a good template for further experimentation, perhaps tailored to a particular telescope or antenna. The scripts search for statistically meaningful terms up to 8th harmonic; the FAUTO command accepts an optional argument that allows the highest harmonic to be specified. When adding new terms, pay attention to the standard deviation of the new coefficient, the effect on other terms, and whether the population standard deviation has gone down; a new term will almost always reduce the RMS but the PSD is a better indication of whether the improvement is real.7 An impersonal assessment of the credibility of the individual terms is made by the command MVET. This performs a series of fits with the term temporarily omitted, to see if there is significant degradation. Ineffectual terms are flagged; if the command
You can reassure yourself that PSD is the more honest figure by using the command SAMSIG. This makes an independent estimate of the PSD by carrying out a separate fit for every observation, with that observation excluded ("jackknifing"). In most cases the agreement is extremely close, within a couple of percent.
7


5.2 Exposing unmodeled effects

49

is followed by the optional argument n the worst n of these (if any) are automatically removed. If you try to use two terms which are more or less indistinguishable (for example CH and PXH0, which are identical), the condition will be reported; if this happens, get rid of one of them via the LOSE command. In such ill-conditioned cases, the singular value decomposition (SVD) methods used by FIT will normally return a workable model, with the "true" value of a duplicated term being partitioned between the two (or more) coefficients involved. Two common cases are (i) if two inseparable terms are present from the start, they each receive an equal share of the value, and (ii) where a new term is added to an already fitted model, and that term is inadequately distinguished from something already present, then the new term will be left close to zero. This behavior can be controlled by means of the FITTOL command, which allows the SVD ill-conditioned criterion to be set. A command FITTOL 0 will switch off the detection of ill-conditioning, whereupon FIT will return a conventional least squares fit, and any highly correlated terms will head off towards cancelling infinities. A FITTOL argument of between 10-3 and 10-2 will give a useful degree of control over ill-conditioning; the value is set to 10-3 initially. With FITTOL arguments much bigger than 10-2 , there is increasing danger that respectable terms will be excluded from full fitting. However, whenever FIT decides to take action over ill-conditioning, it reports that this has occurred. Do not try to extract more information from a given sample of data than the size of the sample justifies. A preliminary assessment of a telescope's capabilities can be made using perhaps 2030 stars. Routine fitting of a typical operational telescope model, consisting of six geometrical terms, two or three flexures, and perhaps a few scale adjustments, centering corrections, etc., calls for 50-100 stars, well-distributed over the whole observable sky (see Section 4.1). A campaign to expose high-frequency flexures and bumps should not be embarked upon without a sample of 500-1500 stars, obtained by combining multiple tests by means of the APPEND command and the "data subsets" feature (see Section 5.8).

5.2

Exp osing unmo deled effects

When first analysing a new telescope, it is not uncommon to see obvious systematic (i.e. visibly non-random) residuals on the simple graphs of error in one coordinate plotted against that or another one ­ plotting error in zenith distance against zenith distance (G Z Z) for example. However, once these straightforward effects have been treated, systematic errors can still hide in what looks like noise. Some of these can be exposed by using more specialized plots ­ zenith distance errors against parallactic angle (G Z Q), errors in HA/Dec nonperpendicularity plotted against HA (G P H), and so on. However, certain effects which depend simultaneously on two coordinates may prove more stubborn; given a data set of adequate size, the solution is to use the MASK and UNMASK commands to select bands in one coordinate, allowing the residuals for that one band to be plotted.


50

5 ADVANCED TECHNIQUES

At various stages, the GHYST command should be tried, to assess whether the current pointing accuracy seems to be limited by hysteresis. Do not overlook the option of plotting residuals against observation number (G x N). As long as the original data file contained the observations in the order in which they were made, such a plot will expose effects which are varying with time. For example if an encoder has suddenly jumped by some amount, a not uncommon fault due either to electronic malfunction or operator error, this will be very apparent from the appropriate G x N plot. Drifts due to temperature changes can also be revealed using such plots. All the other graphs will have shuffled the data and any glitch or drift will not, as a rule, be visible.

5.3

Physical versus empirical

When pursuing low-level flexures and irregularities, difficult questions arise. To what extent is the model physically meaningful? Has adding empirical terms affected the supposedly well-understood underlying model? Is it worth trying to keep a grip on physical reality, or wouldn't it be just as good to fit a massive set of polynomials or harmonics? An approach that seems to preserve the considerable advantages of having a model firmly based on mechanical reality, yet which allows all the power of empirical modeling to be unleashed, is to construct a model which includes both physical and empirical terms, but at no stage to leave both sets free to float at once during fitting.

5.4

Command scripts

Advanced modeling will require repeated application of complex and subtle actions. This will take time and be prone to mistakes. A good way to proceed is to pre-program the key steps using a private procedure library. Such a library can be edited during the TPOINT run in another window (or by means of the SPAWN command) and then re-input with the INPRO command. A private library is also an option when running in batch mode; instead of specifying TPOINT commands in the batch control file, the latter can simply start TPOINT, read the procedure library with INPRO, and execute the required library routine by means of an appropriate CALL. When constructing a private library, it is a good plan to start with the standard library procs.dat as the procedures it contains may be useful as templates. However, it is possible (and commonplace) for a private library to contain a mere handful of procedures, possibly just one.


5.5 The UNFIT command

51

5.5

The UNFIT command

It is sometimes necessary to express the pointing residuals in the context of a model that differs from that in effect when the data were obtained. This can be accomplished using the UNFIT command. UNFIT applies a pointing model in reverse, taking the current adjusted positions and predicting the observations that produce those adjusted positions when the model is applied. Thus an adopted standard model can be applied to a set of pointing residuals to yield new "observations" which can be combined for further analysis. Two modes of operation are provided: to apply the current model to apply a null model (coeffs all zero) to reset the residuals to zero

or or

UNFIT UNFIT N UNFIT Z

If the star-to-telescope modeling option has been chosen (via the ADJUST command), the star positions are set and the telescope positions left alone. If the telescope-to-star option has been chosen, the reverse is true. To understand UNFIT, think first of what the FIT command does. Assume that the telescopeto-star modeling option has been chosen (the default). Under these circumstances, the FIT command takes the raw telescope positions and computes what coefficient values in the current model will give the smallest residuals, then applies the model to the "raw telescope" positions to give the "adjusted telescope" positions. The differences between these "adjusted telescope" positions and the "star observed" positions are the pointing residuals. UNFIT leaves these pointing residuals at their current values, but applies the pointing model in reverse to generate fictitious "observations" (in this case raw telescope positions) which are consistent with both the current model and the current residuals. If the current residuals and coefficient values are both simply the result of fitting, UNFIT will not have any significant effect. However, if one or more of the coefficients is set to a different value, UNFIT will produce a changed set of observations which, if fitted with the same model, will reproduce the changed coefficient values. This is how data from different runs can be combined ­ as will be seen in Section 5.7. Very often, a null model (all coefficients zero) is required, and here the UNFIT N command is useful. UNFIT N has essentially the same effect as setting all the coefficients to zero by hand and then executing an UNFIT command, except that it leaves the original coefficient values intact. UNFIT N is useful for recording pure residuals and eliminating the need for fitting with the original model when new terms are being investigated. The remaining mode, UNFIT Z, produces a result that is consistent with the current coefficient values and zero residuals. One use is investigating any consequences to model-fitting of the particular sky distribution of the current set of observations.


52

5 ADVANCED TECHNIQUES

5.6

Reducing multiple data sets

Observations made at different times cannot necessarily be reduced using a single pointing model. The optical configurations may be different, and there can be small drifts and changes in zero points caused by temperature variations or mechanical variations. (By the same token, it is likely that some sort of ad hoc change to the model will be required in the operational case in order to find the first star.) When the modeling possibilities of any one set of observations have been exhausted it will be necessary to combine the results of several runs if further systematic effects are to emerge from the noise. There are two ways of accomplishing this: by using UNFIT (Section 5.7) and by means of data subsets (Section 5.8). The two methods will give the same end results; the former avoids simultaneous fitting of very large data sets and models, while the latter, which is preferable, is more direct and does not require iteration.

5.7

Combining data with UNFIT and OUTDAT

One way to combine data acquired on different occasions is to normalize each run to a standard state before combination and fitting. The procedure is as follows:

1. Fit all the data to a model that comprises a fixed "core" and a variable "specific to run" portion. 2. In each case, set the fitted model terms to standard values (typically zeroes) and perform an UNFIT followed by OUTDAT to a series of scratch files. 3. Concatenate the resulting data sets using INDAT and APPEND. 4. Fit the entire model, including the "core" terms. Those terms that were allowed to vary in the previous fits should stick at close to their standard values (e.g. zeroes). 5. Fix the core terms and repeat from Step 1. 6. Continue until no further improvement occurs.

Within a few iterations, the core terms will have stabilized and the analysis is complete. The operational model will consist of these core terms plus other terms that can be recalibrated on a regular basis, using a relatively small number of observations.


5.8 Using data subsets

53

5.8

Using data subsets

A more direct method of dealing with heterogeneous data, that leaves the original observations intact throughout the modeling process, is to use data subsets. This is one of TPOINT's most powerful capabilities. Its principles are as follows: · A data subset is identified by a user-specified string. · Any observation is a member of zero or more data subsets. · A pointing correction term can be linked to a specified subset when the USE command adds it to the model; there can be multiple instances of the same correction term, each applying to a specified subset, and the model can contain a mixture of normal (i.e. "global" terms) and subsetted terms. · The FIT command accepts simultaneously fits all the terms in the model, matching subsetted terms and observations appropriately.

A typical application of the technique would be when combining several runs that share a common flexure model but have different encoder zero points or misalignment terms, or where something has changed abruptly during the run. A particularly common case is an equatorially mounted telescope without absolute encoders, where in preparing for an observing session it is usual to "sync" on a bright star. This has the effect of introducing index corrections IH and ID that are different for each session. The fitting procedure would be:

1. Use INDAT and APPEND to produce a concatenated data file, placing each set of records in subsets A, B, C etc. 2. With the USE command, create this model: IH/A ID/A IH/B ID/B IH/C ID/C ... TF NP CH ME MA 3. Execute a FIT command.

The fit will produce a single set of core coefficients (TF NP CH ME MA) but multiple independent index errors (IH/A ID/A IH/B ID/B IH/C ID/C. . . ). The latter can be discarded, leaving the fixed "core" model in service, to be supplemented by the start-of-session sync. There are certain restrictions on subset names. They must not contain spaces or other blanks, nor ! , nor start with * or = . They are case-independent, and appear on reports in uppercase.


54 5.8.1

5 ADVANCED TECHNIQUES Controlling subset memb ership ­ INDAT and SUBSET

The INDAT command provides two methods for controlling subset membership of the observations in the file it reads. The first method is to append one or more optional arguments to the INDAT command: INDAT file subset subset. . . where file is the name of the file of pointing observations and subset the name of a data subset to which the observations will belong. The alternative is to embed within the data file "D-records" ­ see Section 4.5.5. The subset membership of individual observations can be controlled during data analysis by means of the SUBSET command. This works in conjunction with the MASK and UNMASK commands, that can be used to select specific records or ranges of records. Once the required observations have been so identified, the command: SUBSET / subset will cause each observation that is currently in the MASKed state to become members of the specified data subset. If the argument is omitted, the observation(s) lose all subset memberships. Once the SUBSET command has been applied, the selected observations can be brought into use with the UNMASK commend.

5.8.2

Mo del terms for data subsets ­ USE, LOSE, FIX, CHAIN and PARAL

Selected pointing terms can, as appropriate, be brought to bear on specified data subsets by appending a forward slash and the subset name to the term name. The resulting ensemble, for example CA/060401 , becomes in all respects just like a modeling term of that name, in particular when using the USE, LOSE etc. commands. For example, typing NPAE CA/A CA/B AN AW would identify five terms: a CA term applying to observation subset A, another CA term but instead applying to observation subset B, and NPAE, AN and AW terms applying globally. To reduce the amount of typing, limited defaulting and wildcarding is supported:


5.9 Order of terms term term/= term/subset term/* * */= */subset /subset term1 term2. . . term that applies to all observations term that applies to the latest subset term that applies to specified subset all current terms of specified name all current terms of whatever subset all current terms of latest subset all current terms of specified subset multiple terms that apply to the specified subset

55

The "latest subset" is that of the previous term in the current command line. It is initially the empty string and becomes empty again if a global term is specified; otherwise it is the last subset to be specified. For example, typing USE IH/RUN3 ID/= CH/= would be the same as typing USE IH/RUN3 ID/RUN3 CH/RUN3 .

5.9

Order of terms

To match different styles of telescope control system, the model can either start with raw telescope positions and predict the corresponding observed8 place, or alternatively can start with observed place and predict the required raw telescope position. The selection is made with the ADJUST command. The sign convention for the coefficients is the same for both options, so that the two options should deliver coefficients that are substantially the same. When the ADJUST command is used to change the option, the order of the terms in the model is reversed so that more or less the same coefficient values will be produced if a FIT is carried out (following a RESET to cancel the previous set of corrections). Normally, each term in the model is chained to the previous one, the input position for each term being the output position from the previous term. Sometimes this may not be appropriate, and a set of terms can instead share one input position and their adjustments be added in as a group; in this case the terms are said to be paral lel. The commands CHAIN and PARAL select which option applies to a given term, the first in a parallel group being marked "chained" and the rest marked "parallel" to indicate they share the former's starting point. In typical cases the coefficients will not be much altered by changing the order of terms, or switching terms between chained and parallel. Significant changes are likely only if some coefficients are large, or if high order polynomials or harmonics are employed. It is nevertheless reassuring if the model being fitted matches the way the pointing corrections are implemented
As already pointed out in Section 4.4, "observed" has the specific technical meaning of the true lineof-sight direction, and hence is the predicted position of the star taking into account everything up to and including diurnal aberration and refraction.
8


56

5 ADVANCED TECHNIQUES

in the telescope control system itself, in direction, order, formulation, and chaining. The purest match is achieved with the "rigorous fitting" option, which is about to be described, and compatible control software. A plausible ordering and choice of chained/parallel can be applied to the current model by using the command MODOR.

5.10

Rigorous fitting

TPOINT models are most simply viewed as a series of more or less independent terms. During fitting, each term is treated separately, apart from the input to a given term being slightly affected, in the "chained" case, by its predecessor. Similarly, most telescope control systems that implement TPOINT terms do so by taking each one and applying it in sequence. For small corrections this works well, and in practice it takes large collimation or polar axis errors to cause any noticeable mismatch between the two models (i.e. as fitted and as applied). However, TPOINT provides an alternative fitting strategy that removes the distinction between the fitted and applied models. This is called the "rigorous fitting" method, and it is brought into play by including in the INDAT file one of the option records : EQUAT , : ALTAZ or : GIMBAL z y x. The rigorous fitting method works as follows. All TPOINT terms are implemented internally as adjustments to a basic 7-term model. This model, which applies to both equatorial and altazimuth mounts (and the "generalized gimbal" mount, further consideration of which will be postponed until the next section), consists of the six basic geometrical terms plus vertical deflection. The seven terms are, in telescope-to-star order, the two index errors, the vertical error, the two non-perpendicularities, and the two mount misalignment angles. In the equatorial case, the terms are -IH, ID, FLOP, -CH, -NP, -MA, ME. For an altazimuth, the terms are IA, IE, FLOP, CA, NPAE, AW, AN. TPOINT includes routines which apply the 7-term model in exactly the same way that a telescope control system would;9 if the mount type has been specified, through the appropriate INDAT option record, the basic 7-term model is used during the fitting process in a close emulation of operational conditions. The POX and POY pointing terms (Section 5.12), which allow for an off-axis pointing reference such as a guiding chip or autoguider probe, require rigorous fitting and hence the mount type to be declared in the INDAT file. n.b. In special circumstances it is necessary for TPOINT to impose selection of the rigorous fitting method. If no mount type has been explicitly specified, and if observations beyond the
In fact the TPOINT routines are specifically designed for use in a telescope control system. The entire suite of modeling routines can be used in the "pointing kernel" portion of such a TCS, establishing an endto-end tautology between the model that is fitted and the model that is used operationally. The scheme is implemented in the Tpoint Software product TCSpk.
9


5.11 The generalized gimbal mount

57

pole/zenith are included, either EQUAT (INDAT format 1-3) or ALTAZ is assumed. Only the first such record will caused the override, and then only if no explicit option record was included in the INDAT file.

5.11

The generalized gimbal mount

TPOINT is applied as a rule to straightforward equatorial or altazimuth mounts. Where the mounting is unconventional, for example involving HET- or Arecibo-like arrangements or some form of coelostat, the harmonic terms may allow appropriate expressions to be formulated. An important intermediate case is that of the "generalized gimbal", where the mount is a conventional two-axis system but aligned neither to the pole nor the zenith. One example is the alt/alt mount, where the polar axis is horizontal; another is the type of tracking dish which can be tilted so that the target will bypass its blind spot. TPOINT provides support for the generalized gimbal case, essentially by treating it as a variety of altazimuth. The generalized gimbal is specified by including a : GIMBAL z y x option record in the INDAT file. The three (mandatory) arguments z y x are the Euler angles (in degrees) which specify the nominal mount orientation with respect to Az/El. The triad is right-handed, with its its z-axis pointing towards the positive pole and the x-axis pointing to zero longitude. The starting point is with the z-axis pointing upwards and the x-axis pointing south. The rotations, first about the z-axis, then y and finally x, are positive when anticlockwise as seen looking back towards the origin from the positive region of the axis. The result of specifying the mount orientation in this way is that all the pointing terms and graphs that refer to azimuth or elevation in fact refer instead to the corresponding mount angles, "roll" and "pitch". (The exception is zenith distance, which always means with respect to the sky.) This means that effects in collimation, azimuth axis tilt, nonperpendicularity and so on are modeled and plotted using the ordinary Az/El nomenclature, thus avoiding the need for extra terms and plots. (The one "missing" option sacrificed by this ruse is that any effects in true azimuth cannot be modeled or plotted; however, this will rarely matter.) Note that an equatorial mount could be analyzed by declaring it as a gimbal with angles of 0, 90 - , 0. The effect would be that all the altazimuth graphs would resemble the corresponding equatorial ones.

5.12

Off-axis p ointing origin

Almost all optical/IR altazimuth telescopes, and some equatorials, have instrument rotators. The most convenient choice of pointing model in such cases is one that refers to the rotator axis (see Section 4.2.1), so that for a given [ , ] the required [ Az , E l ] or [ h, ] demands are unaffected by rotator angle. TPOINT will deliver such a model if the pointing-test stars have


58

5 ADVANCED TECHNIQUES

been centered on the rotator axis. This normally requires a preliminary calibration, involving rotating the mount through 180 and adjusting the crosswires, camera or guide probe, or deducing which point in the field of the viewing device stays fixed on the sky. (n.b. Locating the rotator axis can be tricky, and always relies on accurate open-loop tracking.) The viewing device, even if nominally central, may be not adjustable. The field center may be obscured ­ a gap between CCD chips perhaps. The viewing device may lie permanently well off-axis. In all such cases it will be necessary to conduct the pointing test with the viewing device at some unknown location in the rotating focal place. This corresponds to the final option in the table at the end of Section 4.2.1, and introduces two special model terms called POX and POY, which are the [ x, y ] coordinates of the pointing origin in the rotating focal plane. In order for TPOINT to determine [ POX, POY ], the mount type must Section 5.10), and the observations must cover a range of rotator angles. two angles, ideally 180 apart, but for an altazimuth it would be usual for active in order to stabilize the field orientation, and as long as stars all over observed this will supply the necessary range of angles automatically. be specified (see The minimum is the rotator to be the sky have been

The rotator angle (in degrees) is included in the INDAT records as the first auxiliary reading, and must be present in all records where POX and POY apply. Incorrect and confusing results will be obtained if no angle is supplied. When the model is fitted ab initio, it will typically require an extra iteration or two as the interplay between [ POX, POY ] and the rest of the model converges and the model settles down. The resulting [ POX, POY ] coordinates are reported in units that correspond to arcseconds on the sky at the center of the assumed tangent-plane pro jection. As a rule, they will simply be discarded, having played their part in obtaining the rotator-axis-based model but having no intrinsic interest. However, if the viewing device ­ a guide probe, say ­ has [ x, y ] readouts or a known location, then comparison with the TPOINT-fitted [ POX, POY ] is potentially very useful. A pointing test involving multiple pointing origins (next paragraph) and/or multiple field orientations may provide enough data to establish the mapping between the TPOINT/TCSpk and nominal [ x, y ] systems. Using data subsets (Section 5.8), it is possible to include observations from multiple pointing origins. For example, observations from a pair of probes or guiding chips might be included in the INDAT file as subsets A and B, and their respective fitted coordinates would then be POX/A, POY/A, POX/B and POY/B. It is even possible to combine off-axis observations made at different foci (OTA, Nasmyth, coud´ requiring different ROT. . . option records e), to be included in the INDAT file at appropriate places. This would, of course, also require separate IE and CA (or ID and CH) terms for each of the rotators.


5.13 Batch mode

59

5.13

Batch mo de

It is possible to run TPOINT entirely without operator input, by including in the working directory a copy of the file procs.dat where the procedure INIT, which in the supplied procs.dat is empty, contains appropriate TPOINT commands to perform the required processing and ends with two QUIT commands. In a typical case this special INIT procedure would include one or more INDAT commands to read in observations and one or more OUTMOD commands to write fitted models to files. A special tpoint.ini file (Section 8), also in the working directory, is normally also needed, containing the commands short_report 2 to suppress all output and script_abort 0 to prevent aborts if errors occur. An example of such an INIT "batch" script would be one that fits an input file to a series of models of increasing elaboration, writing a model file for each. The same script will then work for input files containing any number of observations from 1 upwards. For a single observation, a model comprising only zero-points (IH and ID for an equatorial, IA and IE for an altazimuth) can be fitted, and the other models (that will have failed) ignored. For two observations the mount misalignment terms (ME, MA or AN, AW) can also be added, and so on. By means of this technique automatic self-alignment capabilities can be developed.


60

6 HOW TO ALIGN YOUR POLAR AXIS

6

HOW TO ALIGN YOUR POLAR AXIS

The polar axis of an equatorial mount is nominally parallel to the axis of rotation of the Earth, and therefore points towards the celestial pole. However, refraction causes the celestial pole to appear slightly higher in the sky than it would in the absence of an atmosphere; it turns out that it is best to align the polar axis to this "refracted pole", rather than the true pole. This particular choice of alignment minimizes field rotation effects in the polar region and restores the standard 15 per sidereal second tracking rate for the zenith region. A perfect equatorial mounting, with the polar axis pointing at the true, unrefracted, pole, would correspond to a TPOINT model where all the coefficients are zero. Consequently, if the polar axis is raised slightly to align it with the refracted pole, the TPOINT model will have a non-zero ME coefficient. The following table shows what value of ME to aim for; it lists how far, in arcseconds, the celestial pole is raised by refraction, for observing sites at different latitudes and heights above sea level.
latitude 85 80 75 70 65 60 55 50 45 40 35 30 25 20 15 10 degrees 0m 5 10 15 20 26 32 39 47 56 66 80 96 119 152 205 305 0ft 250m 5 10 15 20 25 31 38 46 54 65 78 94 116 148 200 298 820ft 500m 5 9 14 19 25 31 37 45 53 63 76 92 113 145 195 290 1640ft 750m 5 9 14 19 24 30 36 43 52 62 74 89 110 141 190 283 2460ft 1000m 4 9 14 18 24 29 35 42 50 60 72 87 108 138 185 276 3280ft 1250m 4 9 13 18 23 28 34 41 49 59 70 85 105 134 181 270 4100ft 1500m 4 8 13 17 22 28 34 40 48 57 68 83 102 131 176 263 4920ft 1750m 4 8 13 17 22 27 33 39 47 56 67 81 100 128 172 257 5740ft 2000m 4 8 12 17 21 26 32 38 46 54 65 79 98 125 168 250 6560ft 2500m 4 8 12 16 20 25 30 36 43 52 62 75 93 118 160 238 8200ft 3000m 4 7 11 15 19 24 29 35 41 49 59 71 88 113 152 227 9840ft 4000m 3 7 10 14 17 22 26 31 37 45 53 65 80 102 138 206 13120ft

In the northern hemisphere, the ideal ME value for a particular telescope is minus the figure in the above table. For example, an equatorial telescope mount at 40 latitude and 500 meters above sea level would ideally have ME= -63 . If TPOINT estimated that ME was about +150 , the correct action to take would be to raise the polar axis by 150 - (-63) = 213 arcseconds. In the southern hemisphere, the ideal ME value for a particular telescope is plus the figure in the above table. For example, an equatorial telescope mount at -35 latitude and 250 meters above sea level would ideally have ME= +78 . If TPOINT estimated that ME was about +181 , the correct action to take would be to lower the polar axis by 181 - 78 = 103 arcseconds. The other polar axis misalignment term, MA, is more straightforward: the ideal value is always zero. In either hemisphere, an MA value of (say) +93 could be eliminated by rotating the mount 93 / cos anticlockwise (as seen from above), where is the latitude of the site. Conversely, a negative MA value would require a clockwise adjustment.


61 The recommended procedure for adjusting the polar axis of a permanently-mounted equatorial telescope mount is therefore as follows:

1. Carry out a pointing test. Half a dozen stars, scattered around the sky, will be enough for a preliminary assessment, but a full test of 30-50 stars will give a more accurate final result. 2. Use TPOINT to analyze the test data. For a preliminary assessment the sequence of commands will be simply: INDAT file .EQUAT FIT Read the pointing data file. Standard model for an equatorial. Fit the standard 6-term model.

3. Look up the ideal ME value from the above table, paying attention to the sign (minus in the northern hemisphere). 4. Subtract the ideal ME value from the value which TPOINT reported, and raise (northern hemisphere) or lower (southern hemisphere) the polar axis by this amount. 5. Rotate the mount by MA/ cos arcseconds, anticlockwise if MA has a plus sign, otherwise clockwise. 6. Repeat Steps 1 and 2 to confirm that ME is now close to the desired value and MA is close to zero.

For telescopes that are not permanently mounted, the procedure can be adapted and simplified to allow rapid re-alignment of the polar axis prior to each observing session:

1. First characterize your telescope by doing a full TPOINT analysis, based on at least one comprehensive test of 30-50 stars. This will identify and calibrate and geometrical, cyclic and flexure terms. These terms would not be expected to change much with time, in contrast to the polar axis errors, ME and MA, the zero-point errors, IH and ID, and the collimation error, CH. 2. At the start of each observing session, make a preliminary alignment of the polar axis and then carry out a very short pointing test. In principle just three stars are enough. Use TPOINT to analyze the results, but with the model found in Step 1 as a fixed starting point. For example, if your pointing model included the terms HHSH, HHCH, TF and FO as well as the standard 6-term geometrical model (IH, ID, NP, CH, ME, MA), the sequence of commands will be:


62 INDAT file .EQUAT USE HHSH HHCH TF FO NP -103 TF 38 ... FIX USE IH ID CH ME MA FIT

6 HOW TO ALIGN YOUR POLAR AXIS Read the pointing data file. Standard model for an equatorial. The rest of the model. Set NP to the value found in Step 1. Set TF to the value found in Step 1. . . . . . similar commands for FO, HHSH etc. Prevent the terms being changed. . . . . . except for these five. Fit the terms that change from night to night.

(This can all be preprogrammed into your procs.dat procedure library to avoid unnecessary typing.) 3. Take the ME value which TPOINT reported (in arcseconds), and raise (northern hemisphere) or lower (southern hemisphere) the polar axis by this amount. 4. Rotate the mount by MA/ cos arcseconds, anticlockwise if MA has a plus sign, otherwise clockwise. 5. If you wish, repeat Step 2, to confirm that ME and MA are now close to zero. (The difference between the true pole and refracted pole has been neglected in this simplified procedure.)


63

7
7.1

REFERENCE SECTION
Commands

This section contains a description of every TPOINT command, in alphabetical order. A quick reference list is given at the end of the document.

coeff
FUNCTION :

set/inquire coefficient value

coeff

The value of a coefficient in the current model may be set or inquired by using its name as the command. COMMAND : coeff value where coeff is the name of the coefficient, and value is the value. NOTES : If no argument is supplied, the coefficient is unaltered ­ merely logged. In most cases value is in arcseconds; a few terms have dimensionless coefficients. The coefficient name is usually just the name of a pointing term but may optionally be followed by a forward slash and a data subset name (see Section 5.8). The resulting string behaves as a coefficient name in its own right, so that coefficients AW , AW/060331 and AW/060401 are separate modeling terms and may all be present in the model at the same time.

ADJUST
FUNCTION :

select model direction

ADJUST

ADJUST allows one of two methods of applying the pointing model to be selected: whether the telescope positions are to be adjusted to fit the star positions or vice versa. COMMAND : ADJUST T or ADJUST S to adjust the telescope positions to adjust the star positions


64 NOTES :

7 REFERENCE SECTION

If no argument is supplied, the method is unaltered ­ merely logged. If the ADJUST command results in the direction being changed, the order of the terms in the pointing model is reversed to match. A given set of data will not produce precisely the same coefficients if the model is reversed. The differences will be small unless one or more of the coefficients is very large ­ hundreds of arcseconds or more. Under these circumstances there is in any case a danger that TPOINT's formulation of the pointing terms does not match exactly the way the model is implemented in the control software of the telescope concerned. Large collimation corrections, in particular, may require more rigorous geometry than TPOINT normally offers (unless the "rigorous fitting" option is used), and detailed knowledge of field distortions. The results of any previous FIT are not changed when ADJUST is used and, if a fresh FIT is to be carried out, the RESET command should be used first in order to eliminate the previous corrections.

APPEND
FUNCTION :

prepare to append observations

APPEND

APPEND prepares to concatenate the next set of pointing observations with the ones already input. COMMAND : APPEND ON or APPEND OFF next set of observations will be appended cancel previous APPEND request

NOTES : The argument defaults to ON. The new data will be appended when the next INDAT command is executed. At the time an APPEND request is made, a set of pointing observations must already be present in the data list.


7.1 Commands

65

CALL
FUNCTION :

call a library procedure

CALL

CALL calls a TPOINT library procedure. COMMAND : CALL procedure where procedure is the name of the procedure to be called (1­10 characters). An alternative, shorthand, way of calling a procedure is to type: .procedure with or without spaces between the period and the procedure name.

CAPT
FUNCTION :

enable/disable captions

CAPT

CAPT enables or disables plotting of graph captions. COMMAND : CAPT ON or CAPT OFF enable plotting of captions disable plotting of captions

NOTES : The argument defaults to ON. The motivation for disabling caption plotting may be to save time, or to avoid having the same caption repeated on a multi-plot display.


66

7 REFERENCE SECTION

CHAIN
FUNCTION :

apply terms sequentially

CHAIN

CHAIN is used to indicate which terms are to be evaluated in sequence to their immediate predecessor in the model, each such term then being a function of the position as adjusted by the preceding term. COMMAND : CHAIN coeff1 coeff2 etc. where the arguments are the names of the coefficients to be CHAINed to their predecessors. NOTES : If no arguments are supplied, all the terms in the model are CHAINed. If any argument is unrecognized, the entire command is rejected. The converse command is PARAL, which allows more than one term to be calculated from a common starting position.

CLIST
FUNCTION :

list pointing coefficients

CLIST

CLIST lists the names of the current coefficients, their values, and whether they are fixed or floating. Here is an example: 1 2 3 4 5 6 7 8 9 10 11 12 13 IA IE IE IE HASA HACA HESE HECE HECA2 HESA3 NPAE CA CA -257.859 +383.760 +431.220 +429.013 -1.514 -4.918 -27.618 +31.412 -0.577 +0.928 -14.722 +20.339 +191.418 fixed 1.674 1.660 1.647 0.223 0.206 1.234 1.275 0.149 0.141 1.232 1.670 1.660

/A /B /C

/A /B


7.1 Commands 14 15 16 17 18 19 20 CA AN AN AN AW AW AW +190.927 -25.265 -32.479 -30.550 +14.751 +16.128 +10.674 1.651 0.275 0.195 0.204 0.303 0.204 0.186 /C /A /B /C /A /B /C

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The model includes one fixed term, IA, the remaining terms being available for fitting. Seven of the terms (HASA, HACA, HESE, HECE, HECA2, HESA3 and NPAE) form a "core" that is common to all the observations. Four additional terms (IE, CA, AN and AW) appear in separate instances for three data subsets A, B and C (see Section 5.8).

CLOBBER
FUNCTION :

control file overwrite prompting

CLOBBER

CLOBBER controls whether attempts to overwrite an existing file require operator confirmation. COMMAND : CLOBBER ON or CLOBBER OFF NOTES : The argument defaults to ON. The state when TPOINT is started is CLOBBER OFF. The command is useful if a pre-programmed set of responses is being used, avoiding unforeseen requests for confirmation. overwrite silently don't overwrite unless operator confirms

ECHO
FUNCTION :

enable/disable command echoing

ECHO

ECHO enables or disables the echoing of commands to the screen during execution of library procedures. COMMAND : ECHO ON or ECHO OFF NOTES : The argument defaults to ON. However, echoing is disabled when TPOINT is started. Echoing is useful for debugging, but can clutter the screen at other times. enable command echoing disable command echoing


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END
FUNCTION :

terminate the TPOINT session

END

END terminates TPOINT. NOTE : Additionally, Q, QUIT and QU are interpreted as END commands.

FAUTO
FUNCTION :

automatic modeling

FAUTO

FAUTO carries out a search for terms which improve the pointing for the current data set, in the process detecting and MASKing any outliers among the observations. COMMAND : FAUTO [ n [ p ] ] where n is the highest harmonic to be considered (default 8) and p allows plotting to be enabled (p = Y, the default) or disabled (p = N). NOTES : For small numbers of observations, simple models are tried, using basic terms only. This enables sensible results to be obtained for a handful of stars, even just one or two. Above a few dozen stars, an exhaustive search, up to the specified harmonic, is carried out. Formal statistical tests are applied, to reject (i) terms of dubious benefit and (ii) observations that are inconsistent with the emerging model. The criteria are quite conservative, such that terms and observations to which a human modeler might give the benefit of the doubt are rejected. In all cases, any MASKed observations are reactivated before the modeling takes place; if this behaviour is not desired, a PURGE must be applied as a preliminary step. After the modeling has taken place, any observations considered to be outliers are left in a MASKed state. The full modeling procedures (i.e. when the number of points exceed a few dozen) accept any pre-existing model as a starting point. The original terms may or may not survive, depending on how they fare when tested against the range of terms tried out during the modelling process. This feature makes it worthwhile to repeat the FAUTO, in the hope that the presence of a preliminary model will make the evaluation of subsequently added terms more reliable. By the same token, if a pre-existing model is available, it may be helpful to load it before invoking FAUTO. However, these tactics don't always lead to an improvement, and quite often the first FAUTO, starting from scratch, is as good as it gets.


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69

FIT
FUNCTION :

fit model to observations

FIT

FIT computes the coefficient values for the current pointing model which fit the observations best, and updates either the adjusted telescope coordinates or the adjusted star coordinates depending on which of these two options is in force (see the ADJUST command). COMMAND : FIT or FIT N NOTES : If two terms in the model are too highly correlated to be reliably distinguished, this is reported by FIT so that one of them can be eliminated from the model. The FITTOL command provides control over the treatment of such ill-conditioned cases, allowing the acceptance criterion used by the "singular value decomposition" fitting method used in FIT to be specified. If this criterion is set to zero, by means of a FITTOL 0 command, the fit is simply a conventional least-squares approximation, and any highly correlated terms will be awarded meaningless, large and mutually cancelling values, accompanied as a rule by a poor overall RMS result. With a FITTOL parameter of (for example) 0.005, such correlated terms are kept under control and the model is sound. If all the coefficients have been FIXed, the result of the FIT command is simply to recompute the residuals, and in this case FIT and FIT N are equivalent. to perform a normal fit to apply the current model without fitting

FITTOL
FUNCTION :

set ill-conditioning tolerance

FITTOL

FITTOL specifies and reports the acceptance criterion for the singular value decomposition algorithm used in the FIT command. COMMAND : FITTOL v where v is the tolerance used by the FIT command to decide whether action is to be taken to keep a poorly determined solution under control.


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A tolerance value v between zero and 0.01 is recommended. The initial value is 0.001. A zero tolerance value means that FIT produces a conventional least-squares result, with no protection against ill-conditioning. However, it can be useful to try this, as it may help to expose highly correlated terms. If no argument is supplied, the current tolerance is reported but not changed.

FIX
FUNCTION :

fix the specified terms

FIX

FIX excludes one or more terms from the fit but not from the model. COMMAND : FIX coeff1 coeff2 etc. where the arguments are the names of the coefficients to be frozen at their current values. NOTES : If no arguments are supplied, the whole model is fixed. If any argument is unrecognized, the entire command is rejected.

FLIST
FUNCTION :

export results

FLIST

FLIST exports to a file a list of the pointing observations and the current residuals. This may useful for further graphics and analysis outside TPOINT. COMMAND : FLIST outputfile where the argument is the name of the file to be written. NOTES : The FLIST output is intended to be useful in cases where further analysis is to take place outside TPOINT. Each record (apart from the concluding END record) corresponds to one active observation. Its fields, with the corresponding C scanf conversion specification, are as follows:


7.1 Commands %4d %c %2d %3d %8.4f %c %2d %3d %7.3f %9.3f %8.3f %9.3f %9.3f %9.3f %9.3f %9.3f observation number below pole flag space star h hours star h minutes star h seconds space star sign star degrees star arcminutes star arcseconds star A (degrees, N thru E) star (degrees, observed) dX = east-west error dD = north-south error dS = left-right error dZ = up-down error dR = radial error

71

The listing produced is similar to that displayed by the SLIST command, except that the resolution is higher, there are no headers or footers, inactive observations are not listed and subset membership is not marked. The SLIST command options (error threshold, paged output) have no FLIST counterparts.

FRAME
FUNCTION :

enable/disable frames

FRAME

FRAME enables or disables the plotting of graph frames. A frame is a rectangular box drawn just inside the edge of the plotting zone. COMMAND : FRAME ON or FRAME OFF NOTE : If no argument is supplied, the default is frames enabled. enable plotting of frames disable plotting of frames


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G
FUNCTION :

plot residuals

G

G plots the pointing residuals as the component in one coordinate against that or another coordinate. COMMAND : G ydata xdata scale where ydata is one of the following: H X D P A S Z E V R pointing error pointing error pointing error pointing error pointing error pointing error pointing error pointing error pointing error total pointing in HA EW in Dec corresponding to HA/Dec nonperpendicularity in azimuth horizontally in zenith distance (sky) in elevation (mount) corresponding to Az/El nonperpendicularity error

and xdata is one of the following: H D A Z E Q N hour angle declination azimuth zenith distance (sky) elevation (mount) parallactic angle observation number

The optional argument scale indicates the vertical plotting scale: it is the absolute value of the largest residual to be plotted. In default, a scale is chosen automatically to suit the data. NOTES : The residuals are in the sense telescope minus "observed". For the generalized gimbal case, "azimuth" and "elevation" refer to the mount coordinates. However, "zenith distance" always means with respect to the vertical. Fig. 3 includes an example of a G plot.


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73

GAM
FUNCTION :

look for axis misalignment

GAM

GAM plots the pointing residuals interpreted as a misalignment of either the polar or the azimuth axis. Two superimposed graphs are plotted, showing respectively (i) the orientation and (ii) the amount of misalignment of the polar or azimuth axis that would produce each of the residuals. Graph (i) consists of the orientations calculated for each observation plotted in histogram form. Graph (ii) shows the component of pointing residual for each observation in the direction of the mean orientation. The x-axis of the two graphs is the hour angle or azimuth. COMMAND : GAM type scale The type is E or Q for equatorial and A or Z for Az/El. The equatorial plots have hour angle as their x-axis and look for misalignment of a polar axis. The Az/El plots have azimuth as their x-axis and look for misalignment of an azimuth axis. The Q and Z options suppress an informational message which is normally sent to the screen. This argument defaults to E. The scale indicates the plotting scale for the amount-of-misalignment plot, and is the absolute value of the largest residual to be plotted. In default, a scale is chosen automatically to suit the data. NOTES : When constructing the histogram and computing the mean orientation, each individual orientation is weighted by the size of the supposed axis misalignment. Fig. 3 contains examples of GAM plots, equatorial and altazimuth.

GC
FUNCTION :

graphics device control

GC

GC provides special control functions for the graphics device. Three functions are supported: GC [ C ] clears the display surface GC P [ file ] writes a PostScript file (colored pens) GC B [ file ] writes a PostScript file (all pens black)


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For the `P' and `B' options, the filename defaults to that specified in the initialization file tpoint.ini (Section 8). Using GC P or GC B when displaying multiple plots may produce unexpected black rectangles in the PostScript output. The cause is to do with the way individual regions of the plotting window are cleared. The cure is to precede the plotting with the command CLRNG OFF and then CLRNG ON afterwards.

GDIST
FUNCTION :

plot error distributions

GDIST

GDIST plots distributions of the pointing residuals. NOTES : The histograms labelled X, D, S and Z are the components east-west, north-south, left-right and up-down; the histogram labelled R is for the total pointing errors. Fig. 3 contains an example of a GDIST plot.

GHYST
FUNCTION :

hysteresis plot

GHYST

GHYST attempts to expose hysteresis by assuming that the direction (and perhaps distance) from the current observation to the previous observation determine the direction (and perhaps size) of the effect. The format is a polar plot, with north at the top if the equatorial option has been selected, or up at the top if the altazimuth option has been selected. Each observation (except the first) is plotted as a square marker, from the center of which comes a line indicating the pointing residual. The position of the marker shows in what direction the telescope had to move in order to travel from the previous star, and how far (on a logarithmic scale). COMMAND : GHYST type scale The type is E or Q for equatorial and A or Z for Az/El. The equatorial plot has north at the top, whereas the Az/El plot has up at the top. The Q and Z options suppress an informational message which is normally sent to the screen. This argument defaults to E. The scale is the residual in arcseconds to give a vector of nominal maximum length. This defaults to the largest actual residual providing it is reasonable.


7.1 Commands NOTES :

75

A numerical estimate of the hysteresis is made and reported. It is obtained by rotating each pointing residual by the orientation of the preceding telescope movement, accumulating vectorially, and finally dividing by the number of active observations. Note that various important but dubious assumptions are being made: · The order of observations in the data list is the same as in reality. · The telescope travelled directly from one star to the next, along a straight line in Cartesian cylindrical coordinates. · Any hysteresis comes from the large scale movements between observations and is unaffected by small scale adjustments during acquisition. Residuals are in the sense telescope minus "observed". For equatorial mounts, there may be hysteresis in both HA/Dec and Az/El, and it is worth trying both. Altazimuth mounts are not likely to display any HA/Dec effects, and the (default) E option is not appropriate for this case. Because most observing sequences tend to favor particular telescope movement directions the graph produced by GHYST is cluttered, with many points bunched together. The appearance may be improved by using the MARKH command to reduce the size of the markers. Fig. 3 contains examples of GHYST plots, equatorial and altazimuth.

GMAP
FUNCTION :

Cartesian cylindrical plot

GMAP

GMAP draws a map of the pointing residuals as error vectors on a Cartesian Cylindrical pro jection. COMMAND : GMAP type scale The type is E for equatorial and A for Az/El. This argument defaults to E. The scale is the residual in arcseconds to give a vector of nominal maximum length. This defaults to the largest actual residual providing it is reasonable. NOTE : Fig. 4 contains examples of GMAP plots, equatorial and altazimuth.


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GSCAT
FUNCTION :

scatter plot

GSCAT

GSCAT plots the pointing residuals as a scatter diagram. The plot resembles a view of the telescope field, with points showing where each star would have appeared had the telescope been set blind, using the current pointing model. The field can be one in which north-south is at a fixed orientation, or one in which up-down (or in the special case of a generalized gimbal the "pitch" coordinate) is fixed. COMMAND : GSCAT type radius The type is E for equatorial and A for Az/El. The equatorial plot is relative to east-west and north-south axes, whereas the Az/El plot is relative to left-right and up-down axes. This argument defaults to E. The radius is that of the scatter graph in arcseconds and must lie in the range 0.1­ 9999.9. It defaults to a suitable scale to display the data concerned. NOTE : Fig. 4 contains examples of GSCAT plots, equatorial and altazimuth.

GSMAP
FUNCTION :

azimuthal map

GSMAP

GSMAP draws a map of the pointing residuals as error vectors on an azimuthal pro jection, either orthographic or Lambert equal-area. COMMAND : GSMAP scale = where scale is the plotting scale, the residual in arcseconds to give a vector of nominal maximum length, and = is the character = , which selects the equal-area pro jection. NOTES : The scale defaults to the largest actual residual providing it is reasonable. If both arguments are used (selecting the equal-area pro jection as well as specifying a scale) it doesn't matter whether the = character comes first or second. Fig. 4 contains examples of GSMAP plots, orthographic and equal-area.


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77

H
FUNCTION :

search for harmonics

H

H (or H1) searches for harmonic terms by fitting to a specified component of the pointing residuals functions C0 + C1 sin nA + C2 cos nA, where n is an integer and A is a specified angle. COMMAND : H r a n1 n2

where r is one of the following components of pointing error: H X D P A S Z E V p p p p p p p p p ointing ointing ointing ointing ointing ointing ointing ointing ointing error error error error error error error error error in HA EW in Dec corresponding to HA/Dec nonperpendicularity in azimuth horizontally in zenith distance (sky) in elevation (mount) corresponding to Az/El nonperpendicularity

a is one of the following angles: H D A Z E Q hour angle declination azimuth zenith distance (sky) elevation (mount) parallactic angle

and the optional arguments n1 and n2 are integers in the range 1­999. NOTES : The search consists of trying out terms Hrfan, where the character r represents a component of pointing error, f is S or C , standing for "sine" or "cosine", the character a represents the argument of the harmonic term and n is the sequence of integers from n1 to n2 inclusive. Each trial fit includes a constant term (not reported) and the S and C terms for the r, a and n combination under test. The two numbers n1 and n2 can be in either order. The default for n1 is 1; the default for n2 is n1 + 8 (up to the maximum).


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7 REFERENCE SECTION The report includes the amplitudes of the sine and cosine terms for the frequency under test, and an estimate of the standard deviation. Terms of higher statistical significance are flagged as possible additions to the current model. The H command should be used with circumspection. It is normally wise to try out only one new harmonic frequency at a time, and even then it is common to find that a promising term loses much of its statistical significance once it takes its place in the model and competes with other terms. The command is most useful as an adjunct to the G command, to follow up suspicions of periodic effects gleaned from visual inspection of the various graphs.

H2
FUNCTION :

search for and list cross-harmonics

H2

H2 searches for harmonic terms by fitting to a specified component of the pointing residuals functions C0 + C1 sin na + C2 cos na + C3 sin N A + C4 cos N A, where n and N are integers and a and A are specified angles. (The H2A command ­ see below ­ does the same but, instead of listing the terms, adds the one with the biggest amplitude to the current model.) COMMAND : H2 r a A n1 n2 N1 N2

where r is one of the following components of pointing error:

H X D P A S Z E V

p p p p p p p p p

ointing ointing ointing ointing ointing ointing ointing ointing ointing

error error error error error error error error error

in HA EW in Dec corresponding to HA/Dec nonperpendicularity in azimuth horizontally in zenith distance (sky) in elevation (mount) corresponding to Az/El nonperpendicularity

a and A are, in each instance, one of the following angles:


7.1 Commands H D A Z E Q hour angle declination azimuth zenith distance (sky) elevation (mount) parallactic angle

79

and the optional arguments n1, n2 range of frequencies to be searched ranges are 1-99 for n1 and n2, and that if either of n1 and n2 is more NOTES :

and N1, N2 are pairs of integers that specify the for the angle argument in question. The acceptable 1­9 for N1 and N2, with the additional restriction than 9 then N1 and N2 must both be 1.

The search consists of trying out terms HrfanFAN, where the character r