1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
|
<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 3.2//EN">
<!--Converted with LaTeX2HTML 97.1 (release) (July 13th, 1997)
by Nikos Drakos (nikos@cbl.leeds.ac.uk), CBLU, University of Leeds
* revised and updated by: Marcus Hennecke, Ross Moore, Herb Swan
* with significant contributions from:
Jens Lippman, Marek Rouchal, Martin Wilck and others -->
<HTML>
<HEAD>
<TITLE>Focal-Plane Astrometry</TITLE>
<META NAME="description" CONTENT="Focal-Plane Astrometry">
<META NAME="keywords" CONTENT="sun67">
<META NAME="resource-type" CONTENT="document">
<META NAME="distribution" CONTENT="global">
<META HTTP-EQUIV="Content-Type" CONTENT="text/html; charset=iso_8859_1">
<LINK REL="STYLESHEET" HREF="sun67.css">
<LINK REL="next" HREF="node227.html">
<LINK REL="previous" HREF="node225.html">
<LINK REL="up" HREF="node197.html">
<LINK REL="next" HREF="node227.html">
</HEAD>
<BODY >
<BR> <HR>
<A NAME="tex2html2713" HREF="node227.html">
<IMG WIDTH="37" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="next" SRC="next_motif.gif"></A>
<A NAME="tex2html2711" HREF="node197.html">
<IMG WIDTH="26" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="up" SRC="up_motif.gif"></A>
<A NAME="tex2html2705" HREF="node225.html">
<IMG WIDTH="63" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="previous" SRC="previous_motif.gif"></A> <A HREF="sun67.html#stardoccontents"><IMG ALIGN="BOTTOM" BORDER="0"
SRC="contents_motif.gif"></A>
<BR>
<B> Next:</B> <A NAME="tex2html2714" HREF="node227.html">Numerical Methods</A>
<BR>
<B>Up:</B> <A NAME="tex2html2712" HREF="node197.html">EXPLANATION AND EXAMPLES</A>
<BR>
<B> Previous:</B> <A NAME="tex2html2706" HREF="node225.html">Radial Velocity and Light-Time Corrections</A>
<BR> <HR> <P>
<P><!--End of Navigation Panel-->
<H2><A NAME="SECTION000520000000000000000">
Focal-Plane Astrometry</A>
</H2>
The relationship between the position of a star image in
the focal plane of a telescope and the star's celestial
coordinates is usually described in terms of the <I>tangent plane</I>
or <I>gnomonic</I> projection. This is the projection produced
by a pin-hole camera and is a good approximation to the projection
geometry of a traditional large <I>f</I>-ratio astrographic refractor.
SLALIB includes a group of routines which transform
star positions between their observed places on the celestial
sphere and their <IMG WIDTH="42" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
SRC="img20.gif"
ALT="$[\,x,y\,]$"> coordinates in the tangent plane. The
spherical coordinate system does not have to be <IMG WIDTH="42" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
SRC="img3.gif"
ALT="$[\,\alpha,\delta\,]$"> but
usually is. The so-called <I>standard coordinates</I> of a star
are the tangent plane <IMG WIDTH="42" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
SRC="img20.gif"
ALT="$[\,x,y\,]$">, in radians, with respect to an origin
at the tangent point, with the <I>y</I>-axis pointing north and
the <I>x</I>-axis pointing east (in the direction of increasing <IMG WIDTH="13" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img24.gif"
ALT="$\alpha$">).
The factor relating the standard coordinates to
the actual <IMG WIDTH="42" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
SRC="img20.gif"
ALT="$[\,x,y\,]$"> coordinates in, say, millimetres is simply
the focal length of the telescope.
<P>
Given the <IMG WIDTH="42" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
SRC="img3.gif"
ALT="$[\,\alpha,\delta\,]$"> of the <I>plate centre</I> (the tangent point)
and the <IMG WIDTH="42" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
SRC="img3.gif"
ALT="$[\,\alpha,\delta\,]$"> of a star within the field, the standard
coordinates can be determined by calling
sla_S2TP
(single precision) or
sla_DS2TP
(double precision). The reverse transformation, where the
<IMG WIDTH="42" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
SRC="img20.gif"
ALT="$[\,x,y\,]$"> is known and we wish to find the <IMG WIDTH="42" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
SRC="img3.gif"
ALT="$[\,\alpha,\delta\,]$">, is carried out by calling
sla_TP2S
or
sla_DTP2S.
Occasionally we know the both the <IMG WIDTH="42" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
SRC="img20.gif"
ALT="$[\,x,y\,]$"> and the <IMG WIDTH="42" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
SRC="img3.gif"
ALT="$[\,\alpha,\delta\,]$"> of a
star and need to deduce the <IMG WIDTH="42" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
SRC="img3.gif"
ALT="$[\,\alpha,\delta\,]$"> of the tangent point;
this can be done by calling
sla_TPS2C
or
sla_DTPS2C.
(All of these transformations apply not just to <IMG WIDTH="42" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
SRC="img3.gif"
ALT="$[\,\alpha,\delta\,]$"> but to
other spherical coordinate systems, of course.)
Equivalent (and faster)
routines are provided which work directly in <IMG WIDTH="58" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
SRC="img50.gif"
ALT="$[\,x,y,z\,]$"> instead of
spherical coordinates:
sla_V2TP and
sla_DV2TP,
sla_TP2V and
sla_DTP2V,
sla_TPV2C and
sla_DTPV2C.
<P>
Even at the best of times, the tangent plane projection is merely an
approximation. Some telescopes and cameras exhibit considerable pincushion
or barrel distortion and some have a curved focal surface.
For example, neither Schmidt cameras nor (especially)
large reflecting telescopes with wide-field corrector lenses
are adequately modelled by tangent-plane geometry. In such
cases, however, it is still possible to do most of the work
using the (mathematically convenient) tangent-plane
projection by inserting an extra step which applies or
removes the distortion peculiar to the system concerned.
A simple <I>r<SUB>1</SUB></I>=<I>r<SUB>0</SUB></I>(1+<I>Kr<SUB>0</SUB></I><SUP>2</SUP>) law works well in the
majority of cases; <I>r<SUB>0</SUB></I> is the radial distance in the
tangent plane, <I>r<SUB>1</SUB></I> is the radial distance after adding
the distortion, and <I>K</I> is a constant which depends on the
telescope (<IMG WIDTH="10" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
SRC="img298.gif"
ALT="$\theta$"> is unaffected). The routine
sla_PCD
applies the distortion to an <IMG WIDTH="42" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
SRC="img20.gif"
ALT="$[\,x,y\,]$"> and
sla_UNPCD
removes it. For <IMG WIDTH="42" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
SRC="img20.gif"
ALT="$[\,x,y\,]$"> in radians, <I>K</I> values range from -1/3 for the
tiny amount of barrel distortion in Schmidt geometry to several
hundred for the serious pincushion distortion
produced by wide-field correctors in big reflecting telescopes
(the AAT prime focus triplet corrector is about <I>K</I>=+178.6).
<P>
SLALIB includes a group of routines which can be put together
to build a simple plate-reduction program. The heart of the group is
sla_FITXY,
which fits a linear model to relate two sets of <IMG WIDTH="42" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
SRC="img20.gif"
ALT="$[\,x,y\,]$"> coordinates,
in the case of a plate reduction the measured positions of the
images of a set of
reference stars and the standard
coordinates derived from their catalogue positions. The
model is of the form:
<P ALIGN="CENTER">
<I>x</I><SUB><I>p</I></SUB> = <I>a</I> + <I>bx</I><SUB><I>m</I></SUB> + <I>cy</I><SUB><I>m</I></SUB>
</P>
<P ALIGN="CENTER">
<I>y</I><SUB><I>p</I></SUB> = <I>d</I> + <I>ex</I><SUB><I>m</I></SUB> + <I>fy</I><SUB><I>m</I></SUB>
</P>
<P>
where the <I>p</I> subscript indicates ``predicted'' coordinates
(the model's approximation to the ideal ``expected'' coordinates) and the
<I>m</I> subscript indicates ``measured coordinates''. The
six coefficients <I>a-f</I> can optionally be
constrained to represent a ``solid body rotation'' free of
any squash or shear distortions. Without this constraint
the model can, to some extent, accommodate effects like refraction,
allowing mean places to be used directly and
avoiding the extra complications of a
full mean-apparent-observed transformation for each star.
Having obtained the linear model,
sla_PXY
can be used to process the set of measured and expected
coordinates, giving the predicted coordinates and determining
the RMS residuals in <I>x</I> and <I>y</I>.
The routine
sla_XY2XY
transforms one <IMG WIDTH="42" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
SRC="img20.gif"
ALT="$[\,x,y\,]$"> into another using the linear model. A model
can be inverted by calling
sla_INVF,
and decomposed into zero points, scales, <I>x</I>/<I>y</I> nonperpendicularity
and orientation by calling
sla_DCMPF.
<P>
<BR> <HR>
<A NAME="tex2html2713" HREF="node227.html">
<IMG WIDTH="37" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="next" SRC="next_motif.gif"></A>
<A NAME="tex2html2711" HREF="node197.html">
<IMG WIDTH="26" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="up" SRC="up_motif.gif"></A>
<A NAME="tex2html2705" HREF="node225.html">
<IMG WIDTH="63" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="previous" SRC="previous_motif.gif"></A> <A HREF="sun67.html#stardoccontents"><IMG ALIGN="BOTTOM" BORDER="0"
SRC="contents_motif.gif"></A>
<BR>
<B> Next:</B> <A NAME="tex2html2714" HREF="node227.html">Numerical Methods</A>
<BR>
<B>Up:</B> <A NAME="tex2html2712" HREF="node197.html">EXPLANATION AND EXAMPLES</A>
<BR>
<B> Previous:</B> <A NAME="tex2html2706" HREF="node225.html">Radial Velocity and Light-Time Corrections</A>
<BR> <HR> <P>
<P><!--End of Navigation Panel-->
<ADDRESS>
<I>SLALIB --- Positional Astronomy Library<BR>Starlink User Note 67<BR>P. T. Wallace<BR>12 October 1999<BR>E-mail:ptw@star.rl.ac.uk</I>
</ADDRESS>
</BODY>
</HTML>
|
