The HARP K-mirror (field rotator) hardware is described here. Implementation of a K-mirror field derotator has been successfully accomplished elsewhere: in front of the HERA array at IRAM; see here, Fig.4.

Field rotation, M, is twice the K-mirror angle, K, :

          M   a.k.a.   2K = RPA + E - PA where

• RPA is the Rotator Position Angle, and takes values of 0 or +90 or +180
• E is the elevation
• PA is the parallactic angle for the target, given by

            sin(PA) = cos(observer latitude) * sin(HourAngle) / cos(elevation)     --- (1)

        or

            sin(PA) = cos(observer latitude) * sin(Azimuth) / cos(declination)     --- (2)

  HourAngle is LST-RA, is +ve in the West and -ve in the East, and PA takes the same sign as HA.

The values taken by the parallactic angle, PA, and the (raw) Image rotation angle, M, are shown below for the latitude of the JCMT :

The plots have the zenith (Elevation=90) at the centre and the horizon (Elevation=0) as the largest of the concentric circles. North is right, East down. Notice how the calculated values are discontinuous across the meridian line between the zenith (centre node) and the pole (right node), although the jump is 2pi radians and - as may be deduced from the plot below - has no practical/observational consequences for tracking across the meridian.

The raw values plotted above may be corrected by the first two of the three 'rules' below. The third rule is not applicable in vacuo.

• The primary limitations on the actual position of the K-mirror are the physical limits/hardstops at +55degs : M limits of +110degs. The RPA value is selected from the list of values . . . 0, +90, +180 deg to bring the K-mirror into its operational range.
• Secondarily, a value is selected that will allow the K-mirror to have longest tracking time before hitting one of its physical limits.
• Thirdly, If there are two or more angles with the same tracking time, then the selected angle is the one that will result in the least amount of K-mirror rotation relative to the current K-mirror position.

Applying the rules as much as possible - and resolving cases with no unambiguous choice by adopting the raw angle above - gives the following plot that mostly agrees (in 21 of 28 cases = 75%): with the data recorded to date:

The gaps in some of the tracks, and changes in colour along tracks, indicate locations where there is considerable ambiguity. Experience so far reveals three regions where the predicted Image angle is not always the same as that observed:

1. above elevation=80degs : 40% incorrect
2. the southwest : probably because of the short track times irrespective of the mirror angle adopted: it is here that rule 3 might kick in, for instance.
3. easterly locations : perhaps depending upon whether transit occurs in the north or south - see the colour change along the tracks.

Please use the plot judiciously.

Values of the Parallactic Angle and predictions for the Image Rotation angle ( = 2 x K-mirror angle) - or some relevant information in cases that lead to ambiguous results - may be obtained at the command line by typing

      /home/imc/pointing/progs/getPA.com   azimuth(degs)  elevation(degs)

Misalignments
It is misalignments, zero-point errors and non-orthogonalities in the construction of the antenna that lead to the need for the current 7 parameters of the pointing model. For similar reasons errors or deficiencies in the construction or installation of the K-mirror will require similar modelling in order to optimize the pointing accuracy when HARP is used.

The mirror system may be misaligned in two ways:

1. its axis of rotation is not parallel with the elevation axis
2. its axis does not intercept the centre of HARP array

Per Friberg (JCMT) described the relevant coordinate systems in which to describe the misalignments and their consequences on telescope pointing:

----------------- email of 13 Dec 2005 ----------------------------------
Date: Tue, 13 Dec 2005
From: Per Friberg 
To: Richard Hills 
Cc: {many others}
Subject: HARP pointing offsets

I am in the following always looking out towards the sky:

i) Define a coordinate system fixed in the cabin. With the antenna pointing at
   Zenith the y-axis vertical (+up) and the x-axis horizontal (+ from 
   elevation axis towards RxB) and the z-axis along the elevation axis.

A x,y offset in this coordinate system defines a specific point on the TMU 
and has a fixed Az,El offset (we are not considering telescope axis tilts 
and deflections here).

We have dAz = y
        dEl = x

ii) Define the x',y' system fixed in the HARP Nasmyth frame with 
    y vertical (+up), x-axis horizontal (+ to the right) and the z-axis 
    along the telescope elevation axis. This coordinate system rotates 
    with respect to the cabin fixed system according to:

x = sin(E)x' - cos(E)y'
y = cos(E)x' + sin(E)y'


The HARP pixels have fixed x',y' coordinates but might not be centered 
on (0,0). We will rotate the x',y' offsets to the x,y system and then pass 
it through an offset K-mirror.


Considering the K-mirror - it inverts the offsets around the "fast" axis. 
The K-mirror is fixed in the x,y system but we can also rotated it around 
it's internal axis the angle K. K=0 corresponds to the fast "axis" being 
along the x-axis. Let the K-mirror internal axis be located at x0,y0 in 
the x,y system. After some manipulation we find that -

If the input coordinate are x1,y1 this is converted to

x2 - x0 =  cos(2K)*(x1-x0) - sin(2k)*(y1-y0)
y2 - y0 = -sin(2K)*(x1-x0) - cos(2K)*(y1-y0)

where x2,y2 is the output coordinates. Putting it together we have

x2 = x0 + cos(2K)*[sin(E)x'-cos(E)y'-x0]
        - sin(2K)*[cos(E)x'+sin(E)y'-y0]
y2 = y0 - sin(2K)*[sin(E)x'-cos(E)y'-x0]
        - cos(2K)*[cos(E)x'+sin(E)y'-y0]

x2 = x0 + x'*[ sin(E)cos(2K)-cos(E)sin(2K)]
        + y'*[-cos(E)cos(2K)-sin(E)sin(2K)] - x0*cos(2K) + y0*sin(2K)
y2 = y0 + x'*[-sin(E)sin(2K)-cos(E)cos(2K)]
        + y'*[ cos(E)sin(2K)-sin(E)cos(2K)] + x0*sin(2K) + y0*cos(2K)

x2 = x0*(1-cos(2K)) + y0*sin(2K)     + x'*sin(E-2K) - y'*cos(E-2K)
y2 = x0*sin(2K)     + y0*(1+cos(2K)) - x'*cos(E-2K) - y'*sin(E-2K)


Finally

dAz = x0*sin(2K)     + y0*(1+cos(2K)) - x'*cos(E-2K) - y'*sin(E-2K)
dEl = x0*(1-cos(2K)) + y0*sin(2K)     + x'*sin(E-2K) - y'*cos(E-2K)


Encouraging(ly) it does agree with the previous email!! So hopefully it is 
correct.

By inverting these offsets we have the pointing terms. We can either 
select the HARP pixel coordinates around the true Nasmyth (x',y') = (0,0) 
in which case we have the pointing corrections

-dAz = x0*sin(2K)     + y0*(1+cos(2K))
-dEl = x0*(1-cos(2K)) + y0*sin(2K)

Assuming any offset is small this is the simplest choice. The other 
possibility is to select xH = x'-x0', yH = y'-y0' in which case we also 
will have E-2K terms.

-dAz = x0*sin(2K)     + y0*(1+cos(2K)) - x0'*cos(E-2K) - y0'*sin(E-2K)
-dEl = x0*(1-cos(2K)) + y0*sin(2K)     + x0'*sin(E-2K) - y0'*cos(E-2K)

----------------- end of email of 13 Dec 2005 ----------------------------------

He later speculated on the need for additional terms:

----------------- email of 04 Dec 2006 ----------------------------------

Date: Mon, 4 Dec 2006
From: Per Friberg 
Subject: Harp/K-mirror pointing terms

I think there is a potential for more pointing terms. We have the once we 
have discussed before

i)  Image rotation which not is a pointing term  - used to figure out the
    position of a pixel relative to the telescope nominal direction (for
    detector tracking, offsetting in Fplane and by the beampos task to
    tell ACSIS where to grid the pixel).

    dAz = -xn*cos(E-M) - yb*sin(E-M)
    dEl =  xn*sin(E-M) - yb*cos(E-M)

ii) A pointing shift common to all pixels due to the elevation and 
    K-mirror axes not are coincident. With x0 & y0 denoting the 
    displacement I get.

    -dAz = -x0*sin(M) - y0*cos(M)
    -dEl =  x0*cos(M) - y0*sin(M)

    I have changed sign of the offsets since the pointing term should
    remove the offset.

E is the elevation, M is the image rotation i.e. 2*K where K is the 
K-mirror angle. The x and y are measured in coordinate system I have 
defined. If these are the same as the telescope uses is another question. 
Further the telescope I think used M-E which changes sign of the sin 
terms.

In addition any tilt of the K-mirror axis or image relative the elevation 
axis will give rise to additional effects. However, I believe these will 
have the same functional form as the term above since the angle offset is 
reversed along in the plan of the K while not affected along the 
perpendicular axis. Which is just what happens for positional offsets.

There is also the possibility of the K-mirror having internal alignment 
errors. One example would be a translation or angle offset of a mirror. If 
I get this right this should give rise to a pointing offset depending only 
on the K mirror angle. This since the effect is an translation of the 
image on the TMU. i.e you expect terms of the form

  -dAz = x1*sin(K) + y1*cos(K)
  -dEl = x1*cos(K) - y1*sin(K)

Again with my definition of x and y axes.

In short we need to look for coefficients for terms promotional to the sin 
and cosine of M-E, M and M/2.

----------------- end of email of 04 Dec 2006 ----------------------------------

I have since adopted the 'full-6' version:

   -dAz =  + xo*sin(2K)         -dEl = + xo*(1-cos(2K)) 
           + yo*(1+cos(2K))            + yo*sin(2K)     
           - x'*cos(E-2K)              + x'*sin(E-2K) 
           - y'*sin(E-2K)              - y'*cos(E-2K) 
           - x2*sin(K)                 + x2*cos(K)
           - y2*cos(K)                 - y2*sin(K)

i.e.
    dAz =  - xo*sin(2K)          dEl = + xo*cos(2K)
           - yo*cos(2K)                - yo*sin(2K)
           + x'*cos(E-2K)              - x'*sin(E-2K)
           + y'*sin(E-2K)              + y'*cos(E-2K)
           + x2*sin(K)                 - x2*cos(K)
           + y2*cos(K)                 + y2*sin(K)

or (later) the 4-parameter version

    dAz =  - xo*sin(2K)          dEl = + xo*cos(2K)
           - yo*cos(2K)                - yo*sin(2K)
           + x'*cos(E-2K)              - x'*sin(E-2K)
           + y'*sin(E-2K)              + y'*cos(E-2K)

where E=Elevation, and K is the K-mirror angle, and there has been a change in the names used for the constant terms.

My own program to determine values for these 6(4) parameters from pointing or 'arc' datasets requests an initial value set - for which (0,0,0,0, 0,0) has been useful, so far. It then searches 6(4)-space, centred on the initial guess, up to parameter value limits requested by the program and at a resolution also requested by the program. The solution sits in a depression in 6(4)-space (see the analysis of 20070103) with slopes in the yo direction that are about 5x steeper than in the other directions. The search algorithm takes this into account and modifies both the yo-search space and the yo-resolution accordingly.

In the case (extant until at least this date, UT20070508) where POINTINGs and FIVEPOINTs are done in AZ-EL coordinates, and hence where (twice) the k-mirror angle is exactly equal to the elevation, the x' and y' terms collapse to constants - and should be removed from the analysis.