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<h1>Astrometry</h1>
<p class="abstract">
An astrometry calibration of FITS frames.
</p>
<h2>Synopsis</h2>
<code>
munipack astrometry [.. parameters ..] file(s)[,result(s)]
</code>
<h2>Description</h2>
<p>
Astrometry action derives astrometry calibration of FITS frames
(<a href="astoverview.html">Overview</a>).
</p>
<p>
Astrometry is naturally separated on two parts:
</p>
<ul>
<li>Matching where the correspondence between stars on frame
and in catalogue is established. The matching can be done
without knowledge of any transformation, when very general
presumptions are supposed (the coordinates are Euclidean).
</li>
<li>
Astrometry transformation is computed by using of robust algorithms.
</li>
</ul>
<h3>Reccomendations</h3>
<p>The matching process is searching for a mutual correspondece
in two list of coordinates. The mathcing is generaly slow and
complex process. To increase of its reliability and speed,
is is important to compare (and prepare) the list by such way
so that the coordinates as well as magnitues overlaps.
The overlap in coodinates means to use of catalogues selection
with very similar centre and radius as the frames. Also
magnitude overlap must corresponds with non-saturated (and non-faint)
stars on images. (If both the conditions will not meet, the
matching will probbaly also sucessufll.)
</p>
<p>
There two ways how to realise. The coordinate correspondece
can be make with known object position and field of view (FOV)
of telescope (it means angular diameter of the cone on sky)
in degress. If the telescope has FOV 0.5 deg and the observed
objects is Crab nebulae than the optimal selection from
UCAC4 cataloguje is:
</p>
<pre>
munipack cone -r 0.2 -- 83.63 22.01
</pre>
<p>
The magnitude selection is simliar. It depends on all telescope,
exposure time and filter. The mathcing (as well as photometry)
will sucesfull only with non-saturated bright stars. If we have
an experinece that the stars are saturated at 12, than
the catalogue selection can be improved (syntax is
column_name=[some constrain]):
</p>
<pre>
munipack cone -r 0.2 --par 'f.mag=>12' -- 83.63 22.01
</pre>
<p>
Alternatively, the <a href="https://heasarc.gsfc.nasa.gov/docs/software/fitsio/c/c_user/node97.html">filtration capabailities</a> of FITSIO can be used
</p>
<pre>
munipack astrometry -c 'cone.fits[1][$f.mag$>12]' crab.fits
</pre>
<p>
We are selecting only non-saturated stars fainter than magnitude 12
in column f.mag. The usual linear range of 16-bit CCD cameras is
about 3-5 magnitudes.
</p>
<p>
Parameter --maxmatch slices the catalogue on parts with similar
magnitudes (magnitudes in certain range). This is due to sorting
of both catalpogue and frame by magnitudes.
</p>
<p>
Parameter --minmatch sets minimal length of match sequence
which is supposet to by complete (sucesfull match). For sparse
fileds, the defaults are usually satisfactyory, but very dense
fields like SMC, LMC needs increase this parameters. Munipack
sets the default of 5. If the mean surface density exceeds
the limit (?), the parameter is increased. The small values
on crowded fiedls can produce false match (because probability
of miss-match is getting appreciable).
</p>
<table>
<caption>Suitable parameters for astrometric calibration</caption>
<tr><th>Case</th><th>Solution</th></tr>
<tr><td>sparse field, < 1 m telescope</td><td>defaults</td></tr>
<tr><td>crowded field, < 1 m telescope</td><td>--minmatch=7, --maxmatch=44</td></tr>
<tr><td>sparse field, > 1 m telescope</td><td>-c 'cone.fits[1][$f.mag$>14]'</td></tr>
<tr><td>crowded field, > 1 m telescope</td><td><samp>--minmatch=7, --maxmatch=44, -c 'cone.fits[1][$f.mag$>14 .and. $f.mag$<20]'</samp></td></tr>
</table>
<h2>Algorithm</h2>
<p>
Lets we denote the reference star coordinates as the catalogue and
coordinates of detected stars as the frame.
</p>
<p>
The following algorithm is used for matching:
</p>
<ul>
<li>Catalogue coordinates are projected to rectangular. The projection
can be set with <samp>-p,--projection</samp> switch. As the centre
of projection all stars computed as arithmetical mean is selected.
</li>
<li>
Frame coordinates are pre-scaled to lie in interval 0 .. 1 and to have origin
in centre of frame (the projected and measured coordinates has similar
scale and origin after this transformation to improve numerical precision).
</li>
<li>
Twines of stars in catalogue and in frame are get as the starting step
of matching algorithm.
</li>
<li>
To get a next star in sequence, the following
function (in a meta-language) illustrates the principle:
<pre>
Function Sequence for next star
For all unused stars in catalogue:
Compute u1,v1
For all unused stars on image:
Compute u2,v1
Is Acceptable and distance({u1,u2} and {v1,v2}) < limit?
Has the sequence required length?
Got Solution!
Call Sequence for next+1 star
Did fail the Sequence?
Skip the star
End function Sequence
</pre>
Variables u1,v1 and u2,v2 are coordinates of in a triangle space
of a triplet of stars. The distance is distance in triangle space.
</li>
<li>
Implemented algorithm includes additional tests to be sequence acceptable:
<ul>
<li>the sequence length and limit for objects used to matching
can be changed by <samp>--minmatch, --maxmatch</samp> parameters</li>
<li>The limit is set via <samp>--sig</samp> parameter which sets <i>σ</i>
expected error in the coordinates of stars. The parameter is also
used for estimation of tolerances of angles.</li>
<li>Angles computed between all stars in a real space must be inside
tolerance.
</li>
<li>the sequence must have first and last stars connected (all implemented
tests must be passed).</li>
<li>To prevent use of degenerated triangles (for sides <i>a+b≈c</i>),
only triangles with the <i>v > 1 + σ - u</i> are used.
</ul>
</li>
<li>
The first sequence, which is successful found, stops matching. When
<samp>--full-match</samp> is used, it is scan throughout
all possible twines. In this case, the sequence with the best
parameters is used. The full scan can take a lot of time.
</li>
</ul>
<p>
An alternative sequence can be constructed from known transformation
and the astrometry is just more precise.
</p>
<p>
The known sequence of stars can be used to determine astrometry transformation:
</p>
<ul>
<li>The coordinates of catalogue stars are corrected for its proper motion
(only when <samp>--col-pm-ra, --col-pm-dec</samp>) is set</li>
<li>Initial estimation of parameters is computed by minimising of absolute
deviations.</li>
<li>The corresponding stars in catalogue and frame are found by searching
for stars in close neighbourhood (practically, the <samp>--match nearly</samp>
is performed.</li>
<li>The transformation is determined with all possibles stars by the robust
minimisation.</li>
</ul>
<p>
The matching needs at least of tree stars (to construct of a triangle. The
astrometry algorithm needs at least 5 stars (for 4 parameters) but
has lower-count alternative with minimum of two stars. Generally,
the astrometry has recommended minimum of 7 stars, but ideal is over 20.
Any modern (like Munipack default UCAC5) catalogue has more reference
objects on medium crowded fields.
</p>
<h2>Prerequisites</h2>
<p>
The calibration needs both detected stars and instrumental photometry.
</p>
<h2>Input And Output</h2>
<p>
On input, list of frames containing the table with already detected stars
and photometry is expected.
</p>
<p>
On output, the WCS calibration in the header of primary FITS is created
(or updated) for all input images.
</p>
<h2>Parameters</h2>
<h3>Modes of Calibration:</h3>
<dl>
<dt><samp>-m, --mode [=mode]</samp></dt><dd>Mode of calibration:
<ul>
<li><samp>match</samp> (default),</li>
<li><samp>sequence</samp></li>
<li><samp>manual</samp></li>
</ul>
</dd>
</dl>
<h3>Reference sources:</h3>
<dl>
<dt><samp>-c, --cat file.fits</samp></dt><dd>reference catalogue in FITS format,
if none of <samp>-c, -r, -R</samp> is presented, the default
<samp>cone.fits</samp> is used as
-c cone.fits</dd>
<dt><samp>-r, --ref file.fits</samp></dt><dd>reference frame (already calibrated frame)</dd>
<dt><samp>-R, --rel file.fits</samp></dt><dd>relative to the reference frame (no projection)</dd>
<dt><samp>--col-ra label</samp></dt><dd>Right Ascension column in catalogue,
default: <samp>RAJ2000</samp></dd>
<dt><samp>--col-dec label</samp></dt><dd>Declination column in catalogue,
default: <samp>DEJ2000</samp></dd>
<dt><samp>--col-pm-ra label</samp></dt><dd>Proper motion in Right Ascension
column in catalogue, default: <samp>pmRA</samp></dd>
<dt><samp>--col-pm-dec label</samp></dt><dd>Proper motion in Declination column
in catalogue, default: <samp>pmDE</samp></dd>
<dt><samp>--col-mag label</samp></dt><dd>Magnitude-like column in catalogue,
default: <samp>f.mag</samp></dd>
</dl>
<h3>Manual Calibration Parameters:</h3>
<dl>
<dt><samp>-p, --projection [=type]</samp></dt><dd>projection:
<ul>
<li>none,</li>
<li>gnomonic (default)</li>
</ul>
</dd>
<dt><samp>--xcen xxx.y</samp></dt><dd>centre of projection on chip [pix] (default: width/2)</dd>
<dt><samp>--ycen xxx.y</samp></dt><dd>centre of projection on chip [pix] (default: height/2)</dd>
<dt><samp>--rcen ddd.ddd</samp></dt><dd>centre of projection in Right Ascension [deg]</dd>
<dt><samp>--dcen ddd.ddd</samp></dt><dd>centre of projection in Declination [deg]</dd>
<dt><samp>--scale ssss.s</samp></dt><dd>scale [deg/pix]</dd>
<dt><samp>--angle aa.aaa</samp></dt><dd>angle of rotation [deg], clockwise positive</dd>
<dt><samp>--reflex</samp></dt><dd>set whatever the frame is reflected</dd>
</dl>
<h3>Parameters For Fit:</h3>
<dl>
<dt><samp>--fit [=fit]</samp></dt><dd>method used for fitting of star positions:
<ul>
<li>squares (standard least-squares),</li>
<li>robust (by default).</li>
</ul>
</dd>
</dl>
<h3 id="matchpar">Parameters For Matching:</h3>
<dl>
<dt><samp>--sig xxx.y</samp></dt><dd>mean uncertainty in coordinates of objects
on frames in pixels,default is 1 pixel</dd>
<dt><samp>--sigcat ddd.ddd</samp></dt><dd>mean uncertainty in coordinates of objects
in catalogue in degrees, default is 1 arcsec</dd>
<dt><samp>--fsig d.d</samp></dt><dd>flux errors, default is 1
which fits common observation conditions (clouds, wrong filter). Very bad
bad observations may require larger values.
The parameter significantly affects matching speed.
</dd>
<dt><samp>--minmatch n</samp></dt><dd>Sets count of objects in match sequence.
Default is 5. Crowded fields will require increase the value on 7 or more. The extremely
sparse fields with a few stars only will enough 3-5. To use match algorithm minimal
length is 3. Upper limit is given by <samp>--maxmatch</samp>.</dd>
<dt><samp>--maxmatch n</samp></dt><dd>Set maximum count of objects for matching.
The default is 33 or count of objects in catalogue or in frame.
There is no upper limit, but values over hundredth are probably unusable.
The recommended value for crowded field is 30 - 50.</dd>
<dt><samp>--luckymatch n</samp></dt><dd>Set the number which is added to
value given by <samp>--minmatch</samp> and supposed to indicate
a reliable sequence. For instance, <samp>--minmatch 5</samp>
and <samp>--luckymatch 3</samp> indicates good match when result
has at least 8 successful matches. Default values is 3.
It is reliable like full match and fast like use of first
successful match, if the value is greater than 1.
</dd>
<dt><samp>--disable-lucky-match</samp></dt><dd>
Finish at first success match when <samp>--minmatch</samp>
is reached. Alias for <samp>--luckymatch 0</samp>
</dd>
<dt><samp>--enable-full-match</samp></dt><dd>without this option
finish when a first successful lucky match has occurred.
Full matching is performed when presented, eg. the matching
algorithm explores all possible and acceptable combinations of stars.</dd>
<dt><samp>--disable-flux-check</samp></dt><dd>To improve speed and reliability
of matching, fluxes are used as the additional independent quantity
for checking. This switch disables flux check completely. A possibility
of mismatching will increase, especially on crowded fields.
</dd>
<dt><samp>--disable-rms-check</samp></dt><dd>Normally, RMS (in pixels) is smaller that
xsig*dig (in pixels) for successful fit and one is on condition for the calibration. This
option disables the test and must be used very carefully. Useful in cases when
statistical errors are insignificant to systematical ones (due to an improper
projection).</dd>
</dl>
<h3>Miscellaneous:</h3>
<dl>
<dt><samp>--units</samp></dt><dd>output units: deg, arcmin, arcsec, mas, pix
and auto (default)</dd>
<dt><samp>--disable-save</samp></dt><dd>disable save of calibration to header</dd>
<dt><samp>--remove</samp></dt><dd>remove calibration from FITS header
(keywords like CTYPE, CRPIX, CRVAL, .. and the detailed log)</dd>
<dt><samp>--show-defaults</samp></dt><dd>show default values of parameters</dd>
</dl>
<p>See <a href="man_com.html">Common options</a> for
input/output filenames.</p>
<table>
<caption>Following combinations of mode and options are possible</caption>
<tr><th>Mode</th><th>Options</th></tr>
<tr><td><samp>match</samp></td><td><samp>-c, -r, -R,</samp> <a href="#matchpar">Parameters for Matching</a></td></tr>
<tr><td><samp>sequence</samp></td><td><samp>--seq1, --seq2, -c, -r, -R</samp></td></tr>
<tr><td><samp>manual</samp></td><td><samp>--projection, --xcen, --ycen, --rcen,
--dcen, --scale, --angle, --reflex</samp></td></tr>
</table>
<p>
For manual calibration, use <samp>--projection, --xcen, --ycen, --rcen,
--dcen, --scale, --angle</samp> and don't use <samp>-c, -r, -R</samp>.
The parameters are just interpreted in WCS framework and
stored to FITS header.
</p>
<p>
If an astrometry calibration is already presented in processed header,
one is updated.
</p>
<h2>Caveats</h2>
<p>
Any distortions, including atmospheric refraction,
are not implemented yet. Gnomonic projection is available only.
</p>
<h2>Examples</h2>
<h3>Sources Of Astrometric Catalogues</h3>
<p>
Virtual Observatory (VO) is intended as the main source
of catalogues. The following example uses cone search
capability of VO to list part of
<a href="http://vizier.u-strasbg.fr/viz-bin/VizieR?-source=I%2F340">UCAC5</a>
catalogue
</p>
<pre>
munipack cone -c UCAC5 -o 0716cat.fits -r 0.1 -- 110.25 71.34
</pre>
<p>
UCAC5 is the recommended astrometric catalogue (default). We must provide
coordinates of centre of frame (use coordinates of assumed object)
and the cone radius (else a catalogue default will be used).
Note parameter -s (sort by a column). The sorting requires
knowledge of catalogue structure but strongly affects speed and
success rate of matching.
</p>
<h2>Manual Calibration</h2>
<p>Manual calibration is designed with its properties:</p>
<ul>
<li>The proper calibration is known.</li>
<li>No catalogue and objects detection on images is required.</li>
</ul>
<p>
To save a calibration to FITS header, find the parameters by hand and
adjust the example
</p>
<pre>
$ munipack astrometry -m manual -p GNOMONIC --rcen=110.471 --dcen=71.351
--scale=5.64e-04 --angle=0.0 0716_1R.fits
</pre>
<p>
This is an example for blazar
<a href="http://www.lsw.uni-heidelberg.de/projects/extragalactic/charts/0716+714.html">0716+71</a>
which observation is in FITS file 0716_1R.fits included to
<a href="ftp://munipack.physics.muni.cz/pub/munipack/munipack-data-blazar.tar.gz">munipack-blazar.tar.gz</a>.
</p>
<p>
The calibration parameters are left untouched. There are are no limits and
no any checks.
</p>
<table>
<caption>Parameters for manual calibration<sup><a href="#manual">[a]</a></sup></caption>
<tr><th>Parameter</th><th>Description</th><th>Units</th></tr>
<tr><td>-p</td><td>Type of spherical projection</td><td></td></tr>
<tr><td>--rcen</td><td>Right Ascension of centre of the projection <i>α</i><sub>c</sub> (CRVAL1)<sup><a href="#tw">[b]</a></sup></td><td>degrees</td></tr>
<tr><td>--dcen</td><td>Declination of centre of the projection <i>δ</i><sub>c</sub> (CRVAL2)<sup><a href="#tw">[b]</a></sup></td><td>degrees</td></tr>
<tr><td>--xcen</td><td>Reference point on a chip <i>x</i><sub>c</sub> (CRPIX1)<sup><a href="#tw">[b]</a></sup></td><td>pixels</td></tr>
<tr><td>--ycen</td><td>Reference point on a chip <i>y</i><sub>c</sub> (CRPIX2)<sup><a href="#tw">[b]</a></sup></td><td>pixels</td></tr>
<tr><td>--scale</td><td>Scale of projection <i>c</i></td><td>degrees/pixel</td></tr>
<tr><td>--angle</td><td>Angle of rotation around centre <i>φ</i></td><td>degrees</td></tr>
<tr><td>--reflex</td><td>Reflected frame</td><td>degrees</td></tr>
<tr><td><samp>file</samp></td><td>a file to calibrate<sup><a href="#fl">[c]</a></sup></td><td></td></tr>
</table>
<div class="notes">
<p id="manual">
<sup><a href="#manual">[a]</a></sup>
The manual calibration is invoked when <samp>-m manual</samp> is presented.
</p>
<p id="tw">
<sup><a href="#tw">[b]</a></sup>
Twines of parameters <samp>--xcen, --ycen</samp> and <samp>--rcen, --dcen</samp>
must be specified together.
</p>
<p id="fl">
<sup><a href="#fl">[c]</a></sup>
This manual calibration works only on a single file. Others types of calibrations
on a list of files.
</p>
</div>
<h2>Matching And Sequence Calibration</h2>
<p>Astrometric calibration is designed with its properties:</p>
<ul>
<li>High precision as possible.</li>
<li>Minimum of manually provided parameters</li>
</ul>
<h3>Prerequisites</h3>
<p>
Astrometry calibration requires detected stars
(see <a href="man_find.html">find</a>),
aperture photometry (see <a href="man_aphot.html">aphot</a>) and an astrometry
catalogue (<a href="man_cone.html">cone search</a>).
</p>
<p>
At the start, astrometry calibration itself can be done, a star table (list)
of objects on an image must be prepared.
</p>
<p>
The object detection (as a side effect is the aperture photometry)
is relative straightforward (detected objects are stored
to another HDU with label FIND):
</p>
<pre>
$ munipack find -f 2 -t 5 0716_1R.fits
$ munipack aphot 0716_1R.fits
</pre>
<p>A catalogue with reference stars can be got with help of
<a href="vobs.html">Virtual Observatory</a>:
</p>
<pre>
$ munipack -o 0716_cat.fits -r 0.2 cone -- 110.25 71.34
</pre>
<p>
Selected astrometric stars in radius 0.2° around centre
α = 110.25° and δ = 71.34° are saved to the FITS table
0716_cat.fits.
</p>
<p>
Note selection of catalogue UCAC5. The objects stored to the output file are
sorted by magnitude designed as 'f.mag'. You must known catalogue structure
before
use of -s (simply get catalogue without -s option, look for structure and
than use -s with right parameter). Sorting importantly increase probability
of successful matching.
</p>
<h3>Invoking of Matching</h3>
<p>
In case of matching, when your are a lucky user, following command would
give you the excellent job:
</p>
<pre>
$ munipack astrometry -c 0716_cat.fits 0716_1R.fits
</pre>
<h3>Invoking of Sequence</h3>
<pre>
$ munipack astrometry -m SEQUENCE \
--seq1 5,6,9,16,17,18,19,21,23,22,24,26,27,29,28,30,32 \
--seq2 1,2,3,5,4,8,11,13,7,10,6,12,9,18,14,19,17 \
-c 0716_cat.fits 0716_1R.fits
</pre>
<table>
<caption>Parameters for astrometric calibration</caption>
<tr><th>Parameter</th><th>Description</th></tr>
<tr><td>-p</td><td>Spherical projection</td></tr>
<tr><td>-c</td><td>Reference astrometry catalogue<sup><a href="#cm">[‡]</a></sup></td></tr>
<tr><td>-r</td><td>Reference already calibrated frame<sup><a href="#cm">[‡]</a></sup></td></tr>
<tr><td>-R</td><td>Reference frame for relative astrometry (no projection)<sup><a href="#cm">[‡]</a></sup></td></tr>
<tr><td>--col-ra</td><td>Label of Right Ascension column</td></tr>
<tr><td>--col-dec</td><td>Label of Declination column</td></tr>
<tr><td>--col-pm-ra</td><td>Label of proper motion of Right Ascension column</td></tr>
<tr><td>--col-pm-dec</td><td>Label of proper motion of Declination column</td></tr>
<tr><td>--minmatch</td><td>match sequence minimal length<sup><a href="#nm">[†]</a></sup></td></tr>
<tr><td>--maxmatch</td><td>match sequence length<sup><a href="#nm">[†]</a></sup></td></tr>
<tr><td><samp>file</samp></td><td>a file to calibrate</td></tr>
</table>
<div class="notes">
<p id="cm">
<sup><a href="#cm">[‡]</a></sup>
Parameters -c, -r and -R are exclusive mutual.
</p>
<p id="nm">
<sup><a href="#nm">[†]</a></sup>
Parameters --minmatch and --maxmatch affects both speed (greater is slower)
and chance of matching (lower is worse). The recommended values
for --minmatch are 5 - 10 (default is 5) and --maxmatch 20 - 200 (default 33).
</p>
</div>
<h2>Tips For Usage</h2>
<h3>Successful Matching</h3>
<p>
Matching needs at least <samp>--minmatch</samp> common stars
on frame and catalogue as selected <samp>--maxmatch</samp>.
When matching is failed, check all centre, radius and proper
reference catalogue.
</p>
<p>
As the best diagnostics tool is the astrometry in xmunipack
(Tools->Astrometry).
</p>
<h3>Fine Tune Of Calibration</h3>
<p>
The matching of the reference catalogue with detected stars is
extremely complicated procedure. Therefore, there are tun-able
parameters for both matching and fitting algorithms.
In doubts, check ones:
</p>
<ul>
<li>The sorting of the table with astrometric stars.
Stars would be sorted from the most brighter to faintness ones
(parameter -s of the cone search,
tables of detected objects are sorted by default)</li>
<li>The search region is overlapping of the field of view of
matched pictures.</li>
<li>Try
increase number of stars used for matching (--maxmatch)
and a match sequence length (--minmatch). Large values of both
parameters slow-downs matching.</li>
</ul>
<h3>Tracing</h3>
It is always possible to get detailed log of processing
invoking of <samp>--verbose</samp> parameter.
The usual log will like
<pre>
$ munipack astrometry --verbose 0716_006V.fits
Debug: Launching `astrometry' ...
Debug: VERBOSE = T
Debug: COL_RA = 'RAJ2000'
Debug: COL_DEC = 'DEJ2000'
Debug: COL_PMRA = 'pmRA'
Debug: COL_PMDEC = 'pmDE'
Debug: COL_MAG = 'f.mag'
Debug: CAT = 'cone.fits'
Debug: FILE = '0716_006V.fits' '0716_006V.fits~' ''
Selecting catalogue stars in rank: 1 33
Matching... #: {seq1} -> {seq2} | scale,r: sq.(angle, scale, flux) < Xi2(0.95)
Astrometry calibration of `0716_006V.fits'.
17: 5 6 9 16 17 18 19 21 23 22 24 26 27 29 28 30 32 -> 1 2 3 5 6 4 13 10 7 11 12 9 8 18 14 21 16 | 1767.5 4.53: 1.36 .669E-001 15.7 < 26.0
17: 28 29 5 6 9 16 17 18 19 21 23 22 24 26 27 30 32 -> 14 18 1 2 3 5 6 4 13 10 7 11 12 9 8 21 16 | 1767.5 4.53: .553 .710E-001 15.7 < 26.0
=== Parameters estimation ===
# estim init scale [pix/deg]: 1767.6 0.1
# estim init reflexion: 1.
# estim init angle [deg]: 0.549 0.011
# estim init offset [deg]: 0.01313 0.10040 0.00001 0.00001
=== Absolute deviations fitting ===
# ifault mad acen dcen offset[pix] s[pix/deg] rot[deg]
Debug: MuniProcess elapsed time: 00h 00m 08.945s
Debug: MuniProcess::OnFinish: 24466 0.
1 2 3.16E-05 110.50989 71.29578 24. 92. 1767.4 0.6
2 2 3.17E-05 110.46739 71.34808 0.61E-03 -0.11E-03 1767.4 0.5
3 0 3.13E-05 110.46739 71.34808 -0.90E-02 -0.11E-02 1767.6 0.5
4 0 3.13E-05 110.46740 71.34807 -0.18E-05 -0.83E-05 1767.6 0.5
5 0 3.13E-05 110.46740 71.34807 0.0 0.0 1767.6 0.5
# absfit sign test (total, positive ra,dec): 17 9 8
# absfit final: 31.32E-06 1.768E+03 0.5 110.46740 71.34807 1.0
# astrofit mad=.313E-004 deg .113 arcsec
# astrofit stars to fit: 43
=== Robust fitting ===
Likelihood solution: status=0 (evaluations=2383), log L = -819.
# info s acen dcen offset[pix] [pix/deg] rot[deg]
0 0 3.98E-05 110.46740 71.34807 0.267E-01 0.927E-02 1767.4 0.5
1 2 3.98E-05 110.46740 71.34807 0.253E-01 0.704E-02 1767.4 0.5
2 2 3.97E-05 110.46736 71.34808 -0.355E-05 -0.311E-05 1767.4 0.5
3 2 3.97E-05 110.46736 71.34808 0.00 0.00 1767.4 0.5
# Hessian at minimum:
# 36.000 0.0000 -311.66 -0.35596
# 0.0000 31.000 572.02 0.43994
# -311.66 572.02 0.10717E+07 90.344
# -0.35596 0.43994 90.344 0.29904
# Covariance matrix (no regularisation):
# 0.28150E-01 -0.56872E-03 0.57410E-05 0.32610E-01
# -0.56872E-03 0.33159E-01 -0.14053E-04 -0.45214E-01
# 0.57410E-05 -0.14053E-04 0.96456E-06 -0.26390E-03
# 0.32610E-01 -0.45214E-01 -0.26390E-03 3.5291
# Correlation matrix:
# 1.000 -0.019 0.035 0.103
# -0.019 1.000 -0.079 -0.132
# 0.035 -0.079 1.000 -0.143
# 0.103 -0.132 -0.143 1.000
# solution: 0.000E+00 0.000E+00 5.658E-04 8.862E-03 3.975E-05
# deviations: 5.65E-06 6.13E-06 3.31E-08 6.33E-05 -1.00E+00
# s0,rms: 4.42E-08 3.41E-05 [deg] .123 [arcsec]
# s: 3.97E-05 [deg] .143 [arcsec]
# sign test (total, RA+, Dec+): 43/21.5 17+-2.1 21+-2.3
....
</pre>
<h2>See Also</h2>
<p><a href="astoverview.html">Astrometry Overview</a>,
<a href="dataform_astrometry.html">WCS Header</a>
</p>
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