Pointing and Beam Rotation, JSt, V1.6
Pointing and Beam-Rotation
with an Array-Detector System
(Jürgen Stutzki, Version V1.6)
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Pointing and Beam Rotation, JSt, V1.6
Version
Date
Author
Description
1.1
10. Nov. 2008
JSt
initial draft
1.2
15. Nov. 2008
JSt
consistent description of beam rotator optic (thanks to input
from Urs Graf)
and new chapter on implementation in kosma_control
1.3
20. Nov. 2008
JSt
include new pixel offset variables
1.4
23. Nov. 2008
JSt
include SOFIA l.o.s. rotation, Cassegrain and right Nasmyth
port
1.5
29. Nov. 2008
JSt
include systematic definition of coordinate systems,
numbering of sub-headings
1.6
3. Dec. 2008
JSt
clarified distinction between mapping offsets and pixel
~ E , etc.
offset locations, deleted definitions of ¢
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Pointing and Beam Rotation, JSt, V1.6
1 Introduction
The purpose of this document is to summarize the issues related to the different coordinates systems on
the sky, the telescope, the instrument mounting flange and the instrument focal plane, e.g. in the operation
of a Nasmyth-port or Cassegrain-port mounted beam rotator (the former being e.g. used at NANTEN2
with the SMART array receiver, the latter going to be used at SOFIA with the FIFI-LS istrument) and/or
the additional l.o.s.-degree of freedom for a 3-axis teslescpe mount like e.g. realized with the spherical
oil-bearing mount of SOFIA or in a space telescope. These issues include the nominal beam rotator
setting and the related pointing corrections and mapping offsets for indiviudal detector array pixels.
Experience has shown, that a trial and error approach, in particular with regard to the many possible signflips in the transformation between the various coordinate systems and the rotation angles involved, does
not lead to success: there are too many options and the matter is too complex. Thus, only an ab-initio
approach, consistently describing and analyzing these transformations, makes it possible to achieve a
proper implementation at the telescope.
This document supersedes the technical note Tech_Memo_Beam_Rotator.doc , which was a first
attempt to document these issues for the implimentation at NANTEN2. It aims at giving a full
documentation of the relevant issues in particular towards their implementation within the kosma_control
observing software package, which is in use at NANTEN2 and SOFIA.
1.1 Reference Frames and Coordinate Systems
We have to consider 4 reference frames and associated coordinate systems along the line-of-sight from he
instrument focal plane to the celestial source, each of which (may) rotate against the others. These are:
−
−
−
the sky reference frame, specified by the sky coordinate system in which the celestial source is
observed or mapped. Typically this is an offset coordinate systems in either one of Azimuth/Elevation
(horizon system), Right Acsension/Declination (equatorial system), or Galactic Longitude/Galactic
Latitude relative to the sky reference position. In the following we label this system by the variables
conventionally used for RA/Dec, namely (¢®; ¢± ), which in the context of this document therefore
may stand for any of the above sky coordinates.
the telescope reference frame with its associated telescope coordinate system, which is fixed to the
telescope mechanics and optics. In the following, we label these coordinates with (¢¸; ¢¯ ). For a
ground based telescope with a two-axes (Az=El)-mount, like all radio telescopes and most large
modern optical telescopes, a coordinate system attached to (Az=El) is a natural choice (although not
identical to the (¢Az; ¢El) coordinates, which refer to the sky, we call the axes the same in the
following). For space telescope or the SOFIA telescope, which have three degrees of freedom for
rotation, we define the additional angle in the line-of-sight degree of freedom by »l:o:s: in such a way,
that at »l:o:s:=0° the (¢¸; ¢¯ )-coordinates are aligned with (¢Az; ¢El) and »l:o:s: gives the
counterclockwise rotation angle of the telescope when viewed along its optical axis towards the sky.
The origin of the (¢¸; ¢¯ ) coordinate system is chosen to be the optical axis of the telescope.
the instrument mount reference frame, attached to the instrument mount at the telescope, with its
associated instrument mount coordinate system, which we specify by coordinates (u; v ) in the
following. For a Cassegrain mount, i.e. with fixed orientation lateral position to the telescope, one
naturally chooses these coordinates to be aligned with the telescope coordinate system projected into
the instrument mounting plane, i.e. possibly flipped in the ¢¸ resp. ¢Az -direction due to a tertiary
mirror, like on SOFIA. The origin of the (u; v ) system then is chosen to coincide with the optical axis
of the telescope.
For a Nasmyth mount, a natural choice is horizontal/vertical, with the horizontal direction pointing to
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Pointing and Beam Rotation, JSt, V1.6
−
the left when looking into the telescope's instrument mounting port and the vertical direction pointing
up. This choice insures that at an elevation angle of 0°, the (u; v ) coordinate system coincides with the
(¢Az; ¢El) coordinates projected into the instrument mounting plane through the tertiary mirror.
The origin of the (u; v ) coordinate system is then chosen to be the point where the beam rotator axis
intersects with the instrument mounting plane, i.e. around which the beam rotator rotates the celestial
image resp. the focal plane array footprint.
the instrument focal plane reference frame, attached to the detector system, with its associated
instrument focal plane coordinated system. We designate these coordinates by (x; y ) in the
following. For a fix-mounted receiver/detector system, there is no reason to not have these
coordinates be chosen such that they are aligned with the (u; v ) direction and the origin to coincide
with the optical axis of the telescope; only the offset of the reference detector pixel from the optical
axis, usually called the boresight-offset, is relevant then.
For an instrument/detector array mounted with a beam rotator, we choose the (x; y ) system such, that
it is aligned with (u; v ) at a beam rotator angle of 0°. The origin is taken to coincide with that of the
(x; y ) system.
1.2 Rotation and Reflections: Coordinate Transformations and SignConventions
In the following one carefully has to distinguish between vectors as physical entities (directions and
length in space) and their coefficients, i.e. coordinate values in a given reference system. A fully
consistent detailed nomenclature would be to elaborate and we are hence somewhat sloppy in often using
the same symbols for the same vector even after e.g. the transformation corresponding to the reflection of
the image plane on e.g. the tertiary mirror, whereas we distinguish between them in other cases, like the
rotation introduced by the beam-rotator. The notation should be clear form the context.
We follow the convention that we count angles as positive, which turn counterclockwise when viewed
from the instrument along the telescope optical axis towards the sky. Correspondingly, the rotation of
b
a 2-dimensional vector ~r = (x; y) counterclockwise by an angle Á into a new vector r~0 = D(Á)
~r is done
b
by the rotation matrix D(Á). In any given coordinate system specified by the unit vectors along the
x
~r ¢ ~ex
coordinate axes, ~ei, the coordinates of ~r are ~ri=x;y = (
)=(
), and the components of the
y
~r ¢ ~ey
µ
¶
cos Á ¡ sin Á
b i;j (Á) = ~ei ¢ D(Á)~
b ej =
rotation matrix are D
: Note that this latter expression holds
sin Á
cos Á
only for right-handed coordinate
handed coordinate systems, the signs are inverted:
µ systems; for left ¶
cos Á
sin Á
b
b ej =
D
ei ¢ D(Á)~
: This is not relevant in the following, because, though
i;j;l.h. (Á) = ~
¡ sin Á cos Á
we use both right- and left-handed coordinate systems, we only calculate coefficients in the right-handed
(Az,El) coordinate system. From the above, it also follows that the coordinates of the same vector viewed
b , are related by
in two different coordinate system S and S', where S' is rotated relative to S by D(Á)
b i;j (¡Á)~rj;S , as ~
b
~ri;S 0 = D
r in S' is oriented relative to the coordinate axes in the same way as D(¡Á)~
r
would be relative to the axes of S.
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Pointing and Beam Rotation, JSt, V1.6
2 Positioning and Focal Plane Orientation
2.1 Pointing the Telescope
The telescope is assumed to be mounted in Az/El coordinates, so that the drive and control system allows
the telescope to be driven to a specified Az/El position on the sky. The telescope pointing model takes
care of all telescope internal misalignments, flexure etc. through appropriate pointing constants and
corrections, which include in particular the Az- and El-pointing corrections corresponding to e.g. the
misalignment between the optical pointing telescope axis, on which the pointing model is established, and
the radio optical axis.
These latter two pointing constants, the
Az-,El-pointing corrections , ±Az; ±El,
are used during observing to quickly correct the telescope pointing, typically on a time-scale of every few
hours. All other pointing constants in the elaborate pointing models of modern telescopes are regularly
determined by pointing sessions on many objects around the sky (in the case of large radio telescopes,
these pointing sources are typically compact and sufficiently bright continuum sources; in the case of
small telescopes with limited point source sensitivity, these pointing sessions use optical guide telescopes
and optical stars).
To determine the Az-,El-pointing corrections, the observers scans across a pointing source (bright planet
or the Sun) and determines the Az-,El-offsets, or more precisely:
the position offset of the source in Az-,El-coordinates relative to the nominal pointing: ¢Az; ¢El
In order to correct the pointing, i.e. to make the telescope point to this particular object, the pointing
corrections have to be increased by the values of the Az-,El-offsets:
correcting the pointing: ±Az; ±El ! ±Az + ¢Az; ±El + ¢El.
The ¢Az; ¢El-offsets giving the apparent position of the source in Az,El-coordinates, i.e. the mapping
offsets, on the sky, the position offsets of the detector pixel projected on the sky, (±a; ±e), has the
opposite sign: (±a; ±e) = (¡¢Az; ¡¢El).
2.2 Horizon System and Source Coordinate System
The astronomical observations are typically not done in the horizon coordinate system (Azimuth,
Elevation), but in a celestial coordinate system, e.g. equatorial coordinates right ascension (R.A., ®) and
Figure 1: northern sky, non-circumpolar source, setting
declination (Dec, ± ) or Galactic longitude and latitude. The angle between these coordinate systems
(which in the following is needed to specify the proper orientation of the beam rotator of the array
receiver) is defined as the angle from the second axis of the celestial coordinate system to the second axis
of the horizon system (Elevation). We denote this angle by » in the following. The reason for using the
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Pointing and Beam Rotation, JSt, V1.6
rotation angle for the second axes rather then the first is, that the convention has the (R.A., Dec.)-system
to be left-handed, with R.A. pointing opposite to Az at source transit, so that the first axes have an
additional 180° angle between them.
Figure 2: northern sky, circumpolar source, setting
Figure 3: southern sky, non-circumpolar source, rising
Note that in particular for observations in the Horizon-system (Az,El), e.g. for pointing observations on
planets, the angle » =0° by definition.
Figure 1 to Figure 4 illustrate the typical orientation, sign and magnitude of » for non-circumpolar and
circumpolar sources on the northern and southern sky and for the case of (R.A., Dec.)-observations. The
values of » in these different cases are summarized in Table 1. Note the singular behavior for sources
transiting at zenith as well as when viewed from the equator.
Figure 4: southern sky, circumpolar source, rising
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Pointing and Beam Rotation, JSt, V1.6
hemisphere
Northern Sky
object location
non-circumpolar
circumpolar
Southern Sky
non-circumpolar
circumpolar
range of »
rising
-90 to 0°
setting
0 to 90°
rising
0 to -180°
setting
180 to 0°
rising
-90 to -180°
setting
180 to 90°
rising
-180 to 0°
setting
0 to 180°
Table 1: Range and sign of angle between source coordinates ¢®; ¢± and horizon system coordinates
¢Az; ¢El.
2.3 Telescope-Reference Frame: Line-of-Sight Rotation in 3-axes-telescope
mounts
Some telescopes allow rotation around 3 axes. This is the case for SOFIA, where the telescope floats on
its ball bearing, or for space telescopes, which have no mechanical reference frame at all. In this case, the
line-of-sight rotation angle of the telescope, i.e. a rotation around an axis aligned along its optical axis,
specified by the additional degree of freedom.
As defined above, the l.o.s.-angle »l:o:s: is the angle specifying the counterclockwise rotation from the
nominal position, i.e. aligned with the (Az; El)-direction , to the actual orientation of the (¸; ¯ )coordinates. A two-axes (Az; El)-mounted telescope with the natural choice of the ¸; ¯ coordinates being
aligned with (Az; El) as discussed above, thus corresponds to the case of »l:o:s:=0°.
Thus by extending the above definition of » and specifying as »s the angle from the sky reference frame
Figure 5: line-of-sight rotation for a three-axes telescope
coordinate system to the horizon system as formerly named » without an index, and with »l:o:s: being the
angle from the horizon system to the telescope reference frame coordinates, we have the total angle from
the sky reference frame coordinates to the telescope reference frame coordinates given by
» = »s + »l:o:s:.
The resulting geometry is illustrated in Figure 5.
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Pointing and Beam Rotation, JSt, V1.6
2.4 Instrument Mount Reference Frame
Here we have to distinguish between the case of a Nasmyth mount with the corresponding elevation
dependent rotation between the telescope focal plane and the instrument mount, or a Cassegrain-mount
or Cassegrain-like mount (i.e. with an additional tertiary mirror like on SOFIA), where the telescope focal
plane is fixed relative to the instrument mount.
2.4.1 Nasmyth Mount and Elevation correction
We now consider the sky image in a focal plane attached to the Nasmyth port, i.e. a port which is
stationary in its horizontal and vertical axis, but rotates with the telescope in azimuth. The tertiary mirror,
mounted so that it moves with elevation, rotates the direction of the (Az,El) coordinates with elevation
angle. In addition, the reflection on the tertiary inverts the orientation of the (Az/El) coordinates. This
nd
reflection also changes the sign of the rotation angle between the 2 axis of the source coordinate system
(e.g. Dec.) and the direction of the Elevation axis.
We choose the Nasmyth focal plane coordinate axes (u,v) in such a way, that their direction coincides
with the imaged (Az/El) coordinates at an elevation of 0°; due to the tertiary reflection, the (u,v)
coordinate system thus is left-handed. The resulting geometry is sketched in Figure 6 for the left Nasmyth
port. For the right Nasmyth port, the orientation of the rotation with elevation is opposite.
Figure 6: Nasmyth port view (left Nasmyth port)
In the following, we denote with ~r the position vectors relative to the Nasmyth focal plane, i.e. (u,v)coordinates; with ~a = (±a; ±e) we denote offset position vectors in the horizon system, i.e. (¢Az; ¢El)u
coordinates. In Figure 6, we have denoted with ~rE = ( E ) the position, where the elevation axis hits
vE
uR
this plane, and with ~rR = (
) the position of a reference point in the (u,v) plane, which for the
vR
moment can be taken to be the position of a single pixel detector element in a receiver mounted fixed to
the Nasmyth port. d~E is the vector from the (Az,El) reference point, i.e. nominal pointing position, which
here is assumed to be at (¢Az; ¢El) = (0; 0), to the elevation axis. d~R;0 is the vector from there to the
reference point R in the Nasmyth focal plane, which is a fixed vector in this coordinate system. Figure 6
shows the situation for two different elevations, El (black) and El' (gray), and hence with the
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Pointing and Beam Rotation, JSt, V1.6
correspondingly different rotation of the (¢Az; ¢El) axes viewed from the Nasmyth port.
If viewed in the (¢Az; ¢El)-coordinate system, the situation is as shown in Figure 7. Note that the
reflection on the tertiary mirror changes the sign of the rotation angles. Across the elevation range, the
reference position R, i.e. the detector, is located on a circle arc centered at E, which is given by
~aR (El)
=
=
=
±aR
)
±eR
d~E + d~R
b El )d~R;0 ;
d~E + D(®
(
where d~R;0 is the vector from the elevation axis to the Nasmyth focal plane reference point R at elevation
0° and we have defined the elevation correction angle ®El = ¡El. Note the negative sign of the angle in
the case of the left Nasmyth port, reflecting the fact that the elevation correction results in a clockwise
rotation of d~R;0 with increasing elevation. For the right Nasmyth port of the telescope, the angle would be
the opposite:
½
+El; right Nasmyth
.
®El =
¡El; left Nasmyth
With these detector positions in the (ΔAz,ΔEl) plane, the source appears at opposite offsets in an Az/El
map, so that, following the above definition of (Az/El)- offsets, we have
¢Az
¡±aR
b El )d~R;0 .
(
)=(
) = ¡d~E ¡ D(®
¢El
¡±eR
The parameters d~E and d~R;0 are measured by fitting a circle to the observed and inverted, i.e. (
¡¢Az; ¡¢El)-offsets of a pointing source, which trace a circle with elevation.
Figure 7: Az,El offsets of Nasmyth focal plane
reference position at different elevations
Note that in the above we have chosen an arbitrary origin of the Nasmyth focal plane coordinate-system
(u,v). The resulting equations for the Nasmyth-Elevation pointing correction are, of course, independent
of this choice. A convenient choice in practice would be to take the fixed position of the elevation axis as
the origin of the (u,v)-coordinate system.
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Pointing and Beam Rotation, JSt, V1.6
2.4.2 Cassegrain and Cassegrain-like mounts
In this case, the instrument mount reference frame is fixed relative to the telescope and can thus be chosen
to be identical to the telescope reference fram coordinates.
Thus, there is no elevation dependent rotation of the field-of-view, and correspondingly the elevation
correction angle ®El introduced above, is effectively constant =0°. The offset from the elevation axis to
the reference point R in the focal plane is then an elevation independent, additive offset:
±aR
b El = 0)d~R;0 = d~E + d~R;0.
~aR = (
) = d~E + D(®
±eR
It nevertheless makes sense to keep both parameters, d~E and d~R;0 , as one is the telescope port-dependent
offset of the elevation axis relative to the nominal pointing, the other one the instrument dependent offset
from there to the reference point in the ports focal plane. The latter may be either the offset of the
(reference) detector pixel for a fix-mounted instrument, or the position of the beam rotator axis for an
instrument mounted with a beam rotator.
We thus expand the above definition of the elevation correction angle ®E to include the case of a
Cassegrain (or Cassegrain-like) port as follows:
8
< +El; right Nasmyth
¡El; left Nasmyth
®El =
:
0
Cassegrain like
2.5 Instrument focal plane and Beam Rotator
Figure 8: Instrument focal plane coordinates
Without a beam rotator, i.e. an instrument mounted fixed to the Nasmyth port, the Nasmyth focal plane is
identical to the instrument focal plane. With a beam rotator, we have to distinguish the two. In the
following, we use (x,y) as the instrument focal plane (IFP) coordinates. For convenience, the intersection
of the beam-rotator axis with this plane is chosen as the origin of this coordinate system. The orientation
is chosen such that at a beam rotator angle ½ =0° the instrument focal plane coordinate system coincides
with the Nasmyth focal plane coordinate system, and hence with the (Az; El) coordinate system
projected into the Nasmyth focal plane at an elevation of 0°.
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Pointing and Beam Rotation, JSt, V1.6
Figure 9: rotated pixel offsets obtained from intrinsic pixel offsets
The purpose of the beam rotator is to rotate the instrument focal plane, so that the orientation of a focal
plane array matches the sky/source coordinate system in an appropriate way, defined by the astronomical
observations.
The array has an internal symmetry, defining an array internal coordinate system (³; ´) which is tilted by
®p and an offset ~c0 relative to the instrument focal plane system (x,y). An alternative instrument focal
plane system (IFP') is the one which is tilted relative to (x,y) by ®p and is labeled by (x',y') in the
following. The N pixels of the focal plane array, labeled by index i = 0; : : : ; N ¡ 1, then have position
vectors, which are the sum of the vector to the array center, ~c0 plus the pixel offsets vector from the array
~i . The resulting geometry is as shown in Figure 8.
center, p
~0
As shown in Figure 9, one can regard the vector ~c0 and p
~0;i as being obtained by rotating the vectors C
~0;i through an angle of ®p , so that
and P
b p ) (C
~0 + P
~0;i )
~c0 + p
~0;i = D(®
~ 0 and P
~0;i are the (back-) rotated offset of the array center and the (symmetric) array-intrinsic
where C
pixel offsets. These, together with the tilt-angle ®p are the array geometry parameters actually fitted to the
array-offsets measured by beam-rotator scans on a pointing source (ideally the Sun), as discussed below.
We now consider the beam-rotator. As illustrated in Figure 10, the image rotation is achieved through the
three reflections on the K-mirror arrangement which effectively results in mirroring the entrance plane
along a line through the origin, which is oriented parallel to the K-mirror arrangement and hence rotates
with the physical angle of the beam rotator, ½=2. The angular offset from this line of §½=2 for the
entrance and exit plane sums up to a relative rotation between the two of an angle ½ .
In essence, a physical rotation of the K-mirror by ½=2 results in a rotation of the instrument focal plane
image by an angle ½ in the Nasmyth focal plane in the same direction. The nominal rotator angle of
½ = ®El = ¡El for pointing observations in the (Az/El) system (see below) at the left Nasmyth port, i.e.
a negative value of the commanded beam rotator angle, corresponds to a clockwise physical rotation of
the beam rotator by an angle El=2.
Figure 11 shows the same situation, but now leaving out the instrument focal plane and including the
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Pointing and Beam Rotation, JSt, V1.6
Figure 10: Beam-rotator rotation: the K-mirror arrangement effectively mirrors the
entrance plane along a line through the origin oriented along the K-mirror plane
and rotating with it. Shown are the mirror images for a rotator angle of 0° (bottom
right), and the mirrored image obtained at an angle of ½=2, effectively resulting in a
rotation by ½ relative to the ½ =0° image.
astronomically relevant source/sky coordinates projected into the Nasmyth focal plane.
Viewed on the sky, i.e. in the (¢Az; ¢El) plane, the tertiary mirror reflection flips the orientation of the
Az-axis and changes the signs of the rotation angles ½ and ®p , so that the geometry looks like shown in
Figure 12. In order to calculate the location of a particular pixel of the array receiver (no. 3 in the Figure)
in the (¢Az; ¢El)-plane, we now have to take for d~R;0 , i.e. the offset from the elevation axis to the
reference point, the sum of the offsets from the elevation axis to the beam rotator axis, and from thereon
to the array pixel, the latter being the offset at rotator angle 0° rotated by the angle ¡½ (in the
(¢Az; ¢El) plane), i.e.
d~R;0 ! d~R;0 + ~c + ~
pi
=
=
=
b
d~R;0 + D(¡½)(~
c0 + ~
p0;i )
b
b
~
~ .
d~R;0 + D(¡½)
D(¡®
p )(C0 + P0;i )
b
~0 + P
~0;i )
d~R;0 + D(¡½
¡ ®p )(C
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Pointing and Beam Rotation, JSt, V1.6
The location of the reference pixel i in the (¢Az; ¢El)-plane is thus, as a function of Elevation and
rotator-angle, given by
h
i
±a(El; ½)
b El ) d~R;0 + D(¡½
b
~0 + P
~0;i )
~ai (El; ½) = (
) = d~E + D(®
¡ ®p )(C
±e(El; ½)
b El )d~R;0 + D(®
b El )D(¡½
b
~0 + P
~0;i )
= d~E + D(®
¡ ®p )(C
=
b El )d~R;0 + D(®
b El ¡ ½ ¡ ®p )(C
~0 + P
~0;i )
d~E + D(®
Figure 11: Nasmyth focal plane geometry with beam rotator and array receiver
Thus, we get for the mapping offsets of pixel i
¢Azi
±ai
b El )d~R;0 ¡ D(®
b El ¡ ½ ¡ ®p )(C
~0 + P
~0 )
(
) = ¡(
) = ¡d~E ¡ D(®
¢Eli
±ei
~ 0 and P
~0;i , as well as the angle ®p are determined by mapping of a pointing source
The parameters C
(favorably the Sun) repeatedly at a fixed elevation (i.e. fast in time and/or close to transit) with different
~ 0 and ,
rotator angles and fitting the resulting circles against a model of the array geometry (the offsets C
~0;ias well as ®p , are set to zero for this measurement). With the rotator angle ½ tracking with elevation,
P
i.e. setting the rotator angle to its nominal value (see below) for mapping in the horizon system, namely
~ 0 and P
~0;i now activated, the beam rotator axis
½ = ®El ¡ ®p , and the above determined constants C
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Pointing and Beam Rotation, JSt, V1.6
offset from the elevation axis, d~R;0 , and the elevation axis offset d~E can then be determined by following
a pointing source over a large range in elevation, similar to what has been already discussed above for the
case of a single pixel receiver without beam rotator.
Alternatively, a full half day pointing session with the sun elevation covering the range from close to
horizon to culmination (morning) or vice-versa (afternoon), all corrections set to 0, and cycling the beam
rotator angle through a set of e.g. five fixed settings, gives a full set of measurements against which all
corrections can be fitted simultaneously.
Figure 12: Array setting projected back into Az,El coordinates
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Pointing and Beam Rotation, JSt, V1.6
Figure 13: Nominal beam rotator setting with the array aligned with Az,El
2.6 Nominal beam rotator setting
In order to rotate the array so that its pixels are oriented along the axes of a given source coordinate
system (or have a specified rotation against these axes ½source ), one can read from Figure 13, that the
angle has to be set such that ¡½ = ¡(» ¡ El) + ½source + ®p for the left Nasmyth mount, where
¡El = ®El, i.e. the nominal rotator angle setting is
½nom
=
=
®El + » ¡ ½source ¡ ®p
.
®El + »s + »l:o:s: ¡ ½source ¡ ®p
If the actual setting is ½ = ½nom + ¢½ (i.e. ¢½ specifies the offset between the actual and nominal
setting, ¢½ = ½ ¡ ½nom) one has
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Pointing and Beam Rotation, JSt, V1.6
®El ¡ ½ ¡ ®p
=
=
=
®El ¡ (½nom + ¢½) ¡ ®p
®El ¡ (®El + » ¡ ½source ¡ ®p + ¢½) ¡ ®p
¡» + ½source ¡ ¢½
and the offsets between pixel j and i in (¢Az,¢El)-coordinates are
±aj
±ai
b
~O;j ¡ P
~0;i ).
~aj ¡ ~ai = (
)¡(
) = D(¡»
+ ½source ¡ ¢½)(P
±ej
±ei
b
The (¢AZ; ¢El) coordinate system is rotated by D(»)
against the (¢®; ¢±) coordinate system, and
b
vice-versa the (¢®; ¢±)-coordinate system is rotated by D(¡»)
against (¢Az; ¢El). In addition, the
sign of the R.A.-coordinate is flipped. Thus the pixel offset coordinates (±p® ; ±p± ) in the (¢®; ¢± )
-coordinate system are obtained by the inverse rotation from the (¢Az; ¢El)-offsets, namely
µ
¶ µ
¶
µ
¶ µ
¶
¡±p®;j
¡±p®;i
±aj
±ai
b
(
¡
) = D(»)(
¡
)
±p±;j
±p±;i
±ej
±ei
,
b D(¡»
b
~0;j ¡ P
~0;i )
= D(»)
+ ½source ¡ ¢½)(P
b source ¡ ¢½)(P
~0;j ¡ P
~0;i )
= D(½
i.e. they are the pixel offsets rotated by the angle ½source ¡ ¢½ , and flipped for the R.A.-coordinate. In
particular, if ½source = 0 as well as ¢½ = 0 , the (¢®; ¢±) offsets are aligned with the array internal pixel
offsets.
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3 Implementation in the kosma_control environment
Within the kosma_control-environment all parameters are stored and exchanged as KOSMA_file_iovariables in the corresponding KOSMA_file_io-files. The following table summarized, which of the
parameters used above correspond to which of the KOSMA_file_io-variables.
parameter
KOSMA_file_io-variable
KOSMA_file_io-file
comments
general control-parameters
obs_coord_sys_on
KOSMA_obs2tel
coordinate system for source
reference position.
Possible values are:
B1950, B2000, GALACTIC,
HORIZON
obs_coord_sys_del
KOSMA_obs2tel
source mapping coordinate
system,
possible values: see above
obs_coord_sys_focal_plane
KOSMA_obs2tel
type of instrument mounting
flange coordinate system;
possible values are:
HORIZON, i.e.
horizontal/vertical Nasmyth
mount coordinate system
TELESCOPE, i.e. telescope
fixed (e.g. SOFIA)
instr_beam_rotator[ninstr]
Rx_hardware.set
instrument has beam rotator
[Y/N]
instr_tertiary_mirror[ninstr] TEL_hardware.set
instrument port has tertiary
mirror [Y/N]
rx_instr_name
master_parameters
instrument name (for info and
identification)
rx_port
port.set
left or right Nasmyth port
[LEFT/RIGHT]
ninstr
total number of instruments
available
parameters for the rotator angle
»s
tel_angle_focal_plane
KOSMA_tel2obs.set
angle from 2 coordinate axis of
source mapping coordinate
system to El-axis,
calculated by the telescope
system
»l:o:s:
KOSMA_tel2obs.set
actual l.o.s. angle of telescope
tel_los_act
nd
nd
» = »s + »l:o:s:
rotation angle from 2
coordinate axis of source
nd
mapping coordinates to 2
coordinate of telescope focal
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Pointing and Beam Rotation, JSt, V1.6
parameter
KOSMA_file_io-variable
KOSMA_file_io-file
comments
plane system )
El
tel_elv_act
KOSMA_tel2obs.set
actual elevation of the telescope
(not encoder readings)
½source
rot_source_angle
KOSMA_angle_fp.set
additional beam rotator angle
counterclockwise from mapping
coordinates
(user specified)
®p
instr_focal_plane_rotation
[instr_nr]
TEL_hardware.status
value of the array tilt angle,
transferred to
TEL_hardware.status from the
Rx_hardware.status file for the
reference pixel by 'setpoint -p i'
Rx_tilt_1, Rx_tilt_2
Rx_hardware.status
angle to compensate for tilt of
array footprint, determined from
beam-rotator pointing
measurement session
½nom
fp_angle
KOSMA_track_fp.set
commanded rotator angle
½
rotator_angle
KOSMA_rotator.status
actual rotator angle
¢½
- fp_angle_diff
KOSMA_track_fp.set
difference between commanded
and actual rotator angle (minus
sign kept for consistency with
old data sets).
parameters relevant for pixel offsets
~0 , P
~0;i
instr_reference_pos_x,y[nins TEL_hardware.status
C
~ 0+P
~0;ref pixel
tr]=C
Rx_cx_1, Rx_cy_1,
Rx_cx_2, Rx_cy_2,
Rx_gridsize_1,
Rx_gridsize_2,
Rx_px[i], Rx_px[i],
Rx_last_pix_1,
Rx_last_pix_2
d~R;0
Rx_hardware.status
values calculated from variables
Rx_cx,y[i] and Rx_px,y[i] for
reference pixel, transferred to
TEL_hardware.status by
'setpoint -p i'
parameters to calculate pixel
offset in [mm] in focal plane for
first and second sub-array,
determined from beam-rotator
pointing measurement session.
instr_rotator_axis_x,y[ninstr TEL_hardware.status
],
instr_boresight_offset_x,y[ni
nstr]
offset from elevation axis to
rotator axis, resp. single pixel
boresight (additive) determined
from Nasmyth pointing
measurement session
tel_plate_scale
used to convert mm-offsets in
focal-plane to offsets in arc secs
KOSMA_tel2obs.set
instr_focal_length_correctio
rx-specific correction to focal
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Pointing and Beam Rotation, JSt, V1.6
parameter
KOSMA_file_io-variable
KOSMA_file_io-file
comments
n[ninstr]
length
d~E
instr_elevation_axis_x,y[nin TEL_hardware.status
str]
offsets of elevation axis from
nominal pointing (may be
different for instruments on
different telescope ports)
±a; ±e
fp_offset_x,y
obs_x,y_focal_plane
positional offsets of reference
pixel projected onto the sky in
Az, El, transferred to control_tel
via KOSMA_track_fp.set, and
from thereon to the telescope via
the KOSMA_obs2tel.set-file.
The resulting pointing correction
(in true angles), to be applied by
the telescope task, is the
negative of these values
KOSMA_track_fp.set
KOSMA_obs2tel.set
pointing corrections
±Az , ±El
track_azpoi, track_elpoi
KOSMA_track_point.set Az, El. pointing correction (true
angles)
3.1 Calculating and setting the beam-rotator angle
The task focalplane handles all issues related to the beam-rotator. This includes
●
the calculation of the nominal rotator-angle,
●
setting the beam-rotator to this angle, if a beam-rotator is available,
●
getting the actual beam-rotator angle and calculating the difference between nominal and actual.
focalplane relies on the angle » to be calculated by the telescope astronomical drive software.
3.2 Calculating and setting the focalplane offsets for the reference pixel
The second issue for focalplane is to derive the proper ¢Az; ¢El-offsets and feed them to the telescope
for proper pointing correction. The latter is done through the KOSMA_file_io-file KOSMA_track_fp.set,
which the kosma-control telescope control interface routine control_tel uses to convert to the proper
telescope-drive control parameters, handled to the telescope through KOSMA_obs2tel.set. This includes
the Elevation-dependent Nasmyth-pointing correction as well as the beam-rotator rotated pixel offsets
3.3 Calculating the map offsets from the raw-data during conversion to
CLASS-files
The parameters derived by focalplane and used by control_tel to command the telescope are also written
to the raw-data headers in FITS format. The calibration task kalibrate then uses them to figure out the
appropriate map -offsets for the calibrated data in CLASS format; the code for this in buffers.c.
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Pointing and Beam Rotation, JSt, V1.6
3.4 Fitting the Nasmyth-, rotator- and array-pixel-parameters from pointing
observations
One important step in commissioning the SMART array receiver is the determination of the pixel offsets
and the rotator-/elevation-axis offsets. Traditionally, these are done as two separate steps. The pixel
offsets form the beam-rotator axis are measured by fast sun-scans with different rotator angles at an
almost constant elevation near transit. These measurements are done with the pixel offsets and the tiltangle set to 0.
The analysis (fitting of edge-spikes to the differentiated sun scans) gives the (¢Az , ¢El)-offsets of the
pixels as a function of beam rotator angle. The task rotfit_smart fits the pixel offsets and the tilt angle to
the set of (¡¢Az; ¡¢El) as a function of rotator angle (at approximately constant elevation). The offsets
~ 0 plus the pixel offsets P
~0:i . These
are actually derived by determining the offset to the array center C
latter ones are determined by fitting the grid spacing, taken the array geometry as given. As the grid
spacing is within 2-3 arc secs within 80 , the grid spacing is typically set fixed to this 80 , making easier
to handle map offsets for the observing. rotfit_smart also fits the tilt angle of the array.
In a second step, the Nasmyth correction constants are determined by following a source (the Sun) across
a large elevation range, with the Nasmyth correction constants set to 0, the pixel offsets and tilt angle
activated and the rotator tracking at the nominal angle so that the pixel offsets are rotated into elevationindependent pointing offsets. A fit to the resulting (±a; ±e) = (¡¢Az; ¡¢El) offsets versus elevation
then allows to determine the offset of the rotator axis from the elevation axis
The new approach (still to be fully tested) is to combine these two steps and follow the sun across a large
range in elevation with different rotator setting, and then fit both the pixel offsets and array tilt as well as
the elevation-/rotator-axis offset in a single fit process.
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