\documentclass[preprint]{aastex} % AASTeXv5.0
\usepackage{carma_memo}
\usepackage{psfig}

\begin{document}

%%
%% CARMA Memo number. Check web http://www.mmarray.org/memos
%% for most recent memo number.
%%
\carmamemo{52}

%%
%% Uncomment these if this is also an ALMA or BIMA memorandum.
%%
%\almamemo{222} % ALMA Memo number.
%\bimamemo{101} % BIMA Memo number.


\title{Aperture Efficiency, Pointing and Primary Beam Calibration.}
\author{Melvyn Wright$^1$, David MacMahon$^1$, Stuartt Corder$^2$ \& G\"oran Sandell$^3$}
\affil{Radio Astronomy Laboratory, U.C. Berkeley$^1$; ALMA$^2$; Universities Space Research Association$^3$.}

\begin{abstract}
In this memo we present an analysis of observations of compact sources in
an interferometer mosaic pattern to determine the aperture efficiency,
pointing, and primary beam width for the CARMA antennas. The data are
self-calibrated to determine the antenna voltage gains. Gaussian fits to
the antenna voltage patterns are used to analyze the CARMA beam patterns
at 100 \& 230 GHz.  For multichannel observations the antenna bandpass
solutions are also obtained.

Using a hexagonal 7-pointing mosaic pattern works very well at 100
GHz. At 230 GHz good atmospheric phase stability is required to obtain
reliable results.  We measure antenna gains close to 65 and 145 Jy/K
corresponding to aperture efficiencies 50\% and  65\%  respectively for
10 and 6m antennas.  At 100 GHz the data are consistent with a common FWHM
for azimuth and elevation axes. The average FWHM for 10m antennas is 1.20
+/- 0.02. The average for 6m antennas is 1.98 +/- 0.03.  These values
are significantly larger than the the current Gaussian primary beam
model FWHM; 1.07 and 1.92 arcmin for 10m and 6m antennas respectively.
At 230 GHz the FWHM on the elevation axis appears to be smaller than
on the azimuth for several of the antennas. This is possibly caused by
gravitational deformation in both 6m and 10m antennas.  The mean values
are, azimuth: 0.55 +/- 0.02, elevation: 0.50 +/- 0.02 for 10m antennas,
and azimuth: 0.86 +/- 0.02, elevation: 0.83 +/- 0.03 for 6m antennas.

In principle, we obtain an absolute gain calibration using planet
observations with a planet model. However, if the planet is resolved then
the derived gains depend on the detailed surface brightness distribution
in the planet model, and may not produce reliable fits for the aperture
efficiency or primary beamwidth.  For use as an absolute flux density
calibration, correction must be made for absorption lines in planetary
atmospheres.


\end{abstract}

%%
%% Change Record/Revision History
%% This goes AFTER the abstract: the abstract is
%% printed on the front of the cover page and the
%% change record on the back of the cover page.
%% Use \addrevision to add additional revisions.
%% Columns are
%% | revision |date| author| sections/pages affected |
%% |          | remarks                              |

\ChangeRecordBegin
\addrevision{1.1}{2009-June-01}{MCHW}{1-14}{Draft memo}
\addrevision{1.2}{2009-June-15}{MCHW}{1-19}{Draft memo}
\addrevision{1.3}{2009-June-17}{MCHW}{1-23}{added observing log}
\addrevision{1.4}{2009-June-17}{MCHW}{1-23}{mean and rms using smongo}
\addrevision{1.5}{2009-June-18}{MCHW}{1-23}{add to abstract and conclusion}
\addrevision{1.5}{2009-June-25}{MCHW}{24-25}{added 09JUN24 carma/ct007.1E\_1003C273.1 data}
\ChangeRecordEnd


\section{INTRODUCTION}

Measurements of the aperture efficiency, antenna pointing and primary
beamwidth are essential for quantitative calibration of astronomical
images.  An aperture synthesis array samples the cross correlation of
the signals from an array of antennas, (or phased array stations in the
case of the SKA). The signal from each antenna station measures the sky
brightness distribution weighted by the antenna voltage pattern.

For a point source at the pointing center, e.g. a quasar calibration
observation, the array response is well described by the forward gain
and aperture efficiency.  Accurate measurements of the forward gain
require good pointing calibration.
For an extended source distribution, the array response also depends on
the primary beam illumination of the sky for each station pair.

The CARMA telescope is a heterogeneous array, with six 10.4~m, and nine
6.1~m antennas.  The different antenna diameters and primary beam sizes
allow regions up to $\sim 32''$ diameter to be observed with a single
pointing center at 230 GHz.  Regions up to $\sim 64''$ diameter can be
observed with a 7-pointing hexagonal mosaic. Because of the different
primary beam patterns, imaging with a heterogeneous array must be treated
as a mosaic observation even when only a single pointing center is used
for all antennas.

Images of the sky brightness are formed by combining the measured
cross correlations between the antennas. The usual assumption for
aperture synthesis imaging is that the illumination of the sky by the
primary beam pattern is invariant, and the same for all antennas, then
an image of the primary beam weighted sky brightness can be formed from
a Fourier transform of the measured cross correlations. 

The effects of amplitude and phase errors in the calibration of the
measured cross correlations are well known. Less well appreciated are
the effects of using the wrong primary beam illumination for an extended
source distribution. The case for a homogeneous array of antennas has
been well studied. The image fidelity for mosaic observations is limited
by pointing and primary beam errors (Cornwell, Holdaway \& Uson, 1993;
Holdaway, 1998). For a heterogeneous array, it is necessary to determine
the primary beam pattern for each antenna, in order to obtain accurate
images.

In this memo we present observations using a 7-pointing hexagonal mosaic
to determine the aperture efficiency, pointing, and primary beam width
of the CARMA antennas. The integrated observing sequence provides antenna
gains, and primary beamwidth corrected for pointing.

\clearpage
\section{OBSERVATIONS AND DATA REDUCTION}

Mosaic observations were obtained using a standard hexagonal 7-pointing
mosaic of a strong compact source at the nominal pointing center and
half power points for the 10m antennas. The data are self-calibrated
to determine the antenna voltage gains. For quasars we used the MIRIAD
task {\bf mfcal} to determine the antenna-based gains versus time and bandpass.
The voltage gains are determined for each pointing separately with respect
to the flux density of the quasar.  The ruby script {\bf gpfit.rb}
was used to make Gaussian fits to the voltage gains to determine the
pointing and primary beam width of each of the the CARMA antennas.

For strong sources, mfcal also determines the bandpass from the channel data.
For weaker sources, e.g. at 230 GHz  on a 4 Jy source, we can use line=wide
, which is currently  6 x 500 MHz channels to beat down the noise in the gains.

The ruby script {\bf gpfit.rb} makes tables of the peak value, pointing offset
and voltage primary beamwidth from the Gaussian fits to the antenna gains for
each antenna and solution interval. Tables and plots at 100 GHz and 230 GHz are shown
below.

\begin{verbatim}

Observation Log:

1) 06DEC20:07:29:12.0     File: ct006.3c111.2006dec20.1.miriad
13-pointing mosaic on 3c111 at 93/97 GHz. 500 MHz bandwidth. Total observing time 2.56 hours. 
Early attempt to measure the effect of mapping out to the 5% point in the primary beam
by observing a strong compact source (the 3C111 nucleus), in an extended mosaic pattern.
The image fidelity at the 5% points was severely degraded by primary beam errors,
and good Gaussian fits to the primary beamwidth were not obtained.
(gpfit.rb vis='*.*.uv' options=verbose,radecgpfit.rb vis='*.*.uv' options=verbose,azel)
See Figure 1.
The panels on several of the antennas were later adjusted.
Reasonable fits were obtained for some antennas using only the 9 innermost pointings:
(gpfit.rb vis='--0.0.uv,-14.09.-24.41.uv,14.09.24.41.uv,-14.09.24.41.uv,14.09.-24.41.uv,
-28.19.0.uv,28.19.0.uv,-60.0.uv,60.0.uv' options=verbose > hex9.azel.fit)
See Figure 2.
For several of the 6m antennas the fitted FWHM are not good.

2) 09MAR01:21:27:25.0   File: 0237+288 
An "accidental" 19-pointing mosaic on the phase
calibrator 0237+288 (thanks to Tony Wong for providing this data!).  
Total observing time is  1.56 hours.
These observations, spanning 3 hours at 62 MHz bandwidth at
109/114 GHz on 0237+288, don't have enough SNR to do a Gaussian fit in a time
short compared with the phase variations with time. Although reasonable images
of the calibrator could be made at each pointing, the Gaussian fits to determine
the primary beamwidth were not reliable.

3) 09MAR31:13:29:45.5   File: ct007.1D_2303C454.1.miriad 
7-pointing mosaic on 3C454.3 at 223/229 GHz with 500 MHz bandwidth.
Unfortunately using Nyquist sample interval for 100 GHz observations ! 
Good fits were not obtained with sample points at the 5% points in the primary beam pattern.
Total observing time 2.43 hours.   3 x 50 min intervals are shown.
See Figure 3.

4) 09APR17:04:06:49.0  File: ct007.1C_1003C273.1.miriad 
7-pointing mosaic at 95/100 GHz with  500 MHz BW.  Total observing time 0.73 hours
See Table 1 and Figure 4.

5) 09APR18:03:23:44.0   File: ct007.1C_1003C273.2.miriad 
7-pointing mosaic at 95/100 GHz with  500 MHz BW.  Total observing time 0.89 hours
See Table 2 and Figure 5.

6) 09APR26:07:00:41.5   File: ct007.1C_2303C273.1.miriad 
7-pointing mosaic at 230 GHz with 500 MHz BW.  Total observing time 0.83 hours
See Table 3 and Figure 6.

7) 09APR29:06:15:14.0   File: ct007.1C_2303C273.2.miriad 
7-pointing mosaic at 223/229 GHz with 500 MHz BW.  Total observing time 0.80 hours
See Table 4 and Figure 7.

8) 09MAY28:11:06:45.0   File: ct001.1C_230Neptune.2.miriad 
7-pointing mosaic at 223/229 GHz with 500 MHz BW. Total observing time 1.23 hours
See Table 5.

9) 09JUN24:23:22:35.5   File: ct007.1E_1003C273.1.mir
7-pointing mosaic at 95/100 GHz with  500 MHz BW.  Total observing time 0.70 hours
See Table 6.


\end{verbatim}

\clearpage
\begin{verbatim}


Example of data reduction for a 7-pointing mosaic on a 3.9 Jy source.

#!/bin/csh -vf


# 230 GHz  hexagonal mosaic on 4 Jy source.

# Use  line=wide
      i.e. 6 x 500 MHz channels to beat down the noise in the gains.

# The integration times in mfcal must chosen to give the same number of
solution intervals for each pointing.

echo "integration times chosen to give Number of solution intervals: 3,2,1"
foreach INT ( 50 100 10000)
  echo "integration time =  $INT"
  foreach FILE ( --0.0 30.0 -30.0 15.26 -15.26 15.-26 -15.-26 )
    mfcal vis=$FILE.uv flux=3.9 interval=$INT,$INT line=wide
  end
  gpfit.rb vis='*.*.uv' options=verbose > gpfit.$INT
  plotpbfits.rb -r 0.6 -n 4,4 -d /xs gpfit.$INT
  plotpbfits.rb -r 0.6 -n 4,4 -d gpfit.$INT.ps/ps gpfit.$INT
end


\end{verbatim}

\clearpage
\begin{figure}
\psfig{figure=hex13.azel.ps,angle=-90,width=6.0in}
\caption{Pointing and Primary Beam FWHM at 93/97 GHz. 
06DEC20.   13-pointing mosaic on 3c111.
Figure shows the measured pointing (+), and Gaussian FWHM fits for 6 solution intervals over the 3 hour
observation. Using longer solution intervals, the averaged voltage gains were spoiled by
phase variations. }
\end{figure}

\begin{figure}
\psfig{figure=hex9.azel.ps,angle=-90,width=6.0in}
\caption{Pointing and Primary Beam FWHM at 93/97 GHz. 
06DEC20.  9-pointing mosaic on 3c111 using the inner pointings. }
\end{figure}

\begin{figure}
\psfig{figure=wgpfit.50.ps,angle=-90,width=6.0in}
\caption{Pointing and Primary Beam FWHM using 3C454.3 at 230 GHz. 09MAR31.
Total observing time 2.43 hours.  Fits for  3 x 50 min intervals are shown. There are
significant pointing variations between the 3 x 50 min intervals. Pointing variations
within the 50 min measurement at each of the pointing positions degrade the Gaussian fits
to the primary beamwidth.}
\end{figure}


\clearpage
\section{ANALYSIS}

\subsection{CARMA antenna efficiencies}

Tables 1 \& 2 and Figures 4 \& 5 show the results for 2 observations on 3C273 at 100 GHz.
Tables 3 \& 4 and Figures 6 \& 7 show the results for 2 observations on 3C273 at 230 GHz.

The aperture efficiencies can be calculated
from the peak values (pkval) fitted to the hex7 voltage patterns
(and therefore corrected for pointing)

The peak values are  w.r.t. the measured flux density for 3c273
and the nominal jyperk values in the data

The jyperk in the uv data are  65 and 145.337 for 10 and 6m antennas,
corresponding to aperture efficiencies
32.5/65 = 50\% and 94.5/145.337 = 65\%  respectively.

Using a flux for 3c273, 10 Jy at 230 GHz and 22 Jy at 100 GHz gives a
peak value around 1, consistent with the nominal aperture efficiencies
for 10 and 6m antennas. Lower peak values imply lower aperture efficiency.
At 230 GHz, antennas 6, 10, and 15 have lower aperture efficiency.


gpfit.rb - Fit Gaussian primary beam to hex7 observational data.

 aperture efficiency

 $10m (230GHz) = pkval^2  \times  (10/ Flux{3c273})  \times 50$

 $10m (100GHz) = pkval^2  \times  (22/ Flux{3c273})  \times 50$

 $6m  (230GHz)  = pkval^2  \times  (10/ Flux{3c273})  \times 65$

 $6m  (100GHz)  = pkval^2  \times  (22/ Flux{3c273})  \times 65$


\clearpage
\subsection{CARMA primary beamwidths}

The primary beamwidth is fitted in azimuth and elevation.  Significant
ellipticity is observed in some antennas.  If we get consistent results
we should change the nominal FWHM. Since these observations are similar
to mosaic projects we should get better mosaic imaging performance
using the measured primary beam FWHM. In Figure 8, we plot the fitted
FWHM for all the measured antennas. At 100 GHz the data are consistent
with a common FWHM for azimuth and elevation axes. The average FWHM
for 10m antennas is 1.20 +/- 0.02. The average for 6m antennas is 1.98 +/- 0.03.
These values are significantly larger than the the current Gaussian primary beam model
FWHM 1.07 and 1.92 arcmin for 10m and 6m antennas respectively. 

At 230 GHz the FWHM on the elevation axis appears to be smaller than
on the azimuth for several of the antennas. This is possibly caused by
gravitational deformation in both 6m and 10m antennas.  More extensive
observations at several telescope elevations are required to investigate
this.

The mean values from the two 3C273 datasets ( Tables 3 \& 4) are:
azimuth: 0.55 +/- 0.02, elevation: 0.50 +/- 0.02 for 10m antennas,
and
azimuth: 0.86 +/- 0.02, elevation: 0.83 +/- 0.03 for 6m antennas.
The current Gaussian primary beam model has the same FWHM for both axes:
FWHM 0.47 and 0.83 arcmin for 10m and 6m antennas respectively. 


\clearpage


\begin{verbatim}
---------------------------------------
Table 1. ct007.1C_1003C273.1.gpfit.0.2
---------------------------------------

#ant pol fit iters  :  pkval     az_off   el_off  az_fwhm  el_fwhm  |residual|
###############################################################################
01 I fit 1 iters 7  : 9.96e-01   -0.072   -0.034    1.162    1.224  |X|= 0.0151
02 I fit 1 iters 7  : 1.00e+00   -0.016    0.020    1.198    1.167  |X|= 0.0198
03 I fit 1 iters 7  : 1.04e+00   -0.041   -0.065    1.148    1.183  |X|= 0.0076
04 I fit 1 iters 7  : 9.77e-01   -0.036   -0.030    1.227    1.221  |X|= 0.0128
05 I fit 1 iters 7  : 9.63e-01   -0.051   -0.088    1.216    1.203  |X|= 0.0180
06 I fit 1 iters 7  : 9.80e-01   -0.006   -0.007    1.216    1.195  |X|= 0.0203
07 I fit 1 iters 8  : 9.75e-01    0.005   -0.034    2.044    1.948  |X|= 0.0032
08 I fit 1 iters 9  : 9.61e-01    0.007   -0.075    2.055    2.160  |X|= 0.0083
09 I fit 1 iters 9  : 9.48e-01    0.032   -0.007    1.841    1.950  |X|= 0.0142
10 I fit 1 iters 8  : 9.55e-01    0.019   -0.142    1.920    1.991  |X|= 0.0036
11 I fit 1 iters 8  : 9.98e-01    0.053   -0.030    1.944    1.989  |X|= 0.0087
12 I fit 1 iters 8  : 9.51e-01   -0.026   -0.105    1.972    2.022  |X|= 0.0043
13 I fit 1 iters 8  : 9.77e-01    0.043   -0.083    1.911    1.915  |X|= 0.0050
14 I fit 1 iters 8  : 9.57e-01   -0.021    0.023    1.906    1.945  |X|= 0.0139
15 I fit 1 iters 8  : 9.71e-01    0.045   -0.187    2.009    2.035  |X|= 0.0064

                         az_fwhm            el_fwhm
        10m antennas:  1.20 +/- 0.03      1.20 +/- 0.02
         6m antennas:  1.95 +/- 0.07      1.99 +/- 0.07
\end{verbatim}

\clearpage
\begin{figure}
\psfig{figure=ct007.1C_1003C273.1.gpfit.0.2.ps,angle=-90,width=6.0in}
\caption{Pointing and Primary Beam FWHM at 100 GHz. File ct007.1C\_1003C273.1.gpfit.0.2.ps}
\end{figure}

\clearpage

\begin{verbatim}
---------------------------------------
Table 2. ct007.1C_1003C273.2.gpfit.0.2
---------------------------------------

#ant pol fit iters  :  pkval     az_off   el_off  az_fwhm  el_fwhm  |residual|
###############################################################################
01 I fit 1 iters 7  : 9.97e-01    0.001   -0.002    1.182    1.213  |X|= 0.0025
02 I fit 1 iters 7  : 1.01e+00   -0.015    0.013    1.177    1.155  |X|= 0.0177
03 I fit 1 iters 7  : 1.04e+00   -0.028   -0.015    1.183    1.163  |X|= 0.0033
04 I fit 1 iters 7  : 9.83e-01   -0.039   -0.042    1.211    1.161  |X|= 0.0051
05 I fit 1 iters 7  : 9.66e-01   -0.034   -0.004    1.227    1.188  |X|= 0.0125
06 I fit 1 iters 7  : 9.88e-01   -0.005    0.024    1.199    1.185  |X|= 0.0147
07 I fit 1 iters 8  : 9.89e-01    0.033   -0.073    1.984    1.997  |X|= 0.0013
08 I fit 1 iters 8  : 9.67e-01    0.002   -0.061    2.033    1.989  |X|= 0.0030
09 I fit 1 iters 8  : 9.50e-01    0.018   -0.105    1.974    1.962  |X|= 0.0022
10 I fit 1 iters 8  : 9.65e-01    0.042   -0.067    1.975    1.952  |X|= 0.0025
11 I fit 1 iters 8  : 9.97e-01    0.018   -0.100    1.992    1.965  |X|= 0.0012
12 I fit 1 iters 8  : 9.51e-01   -0.006   -0.090    1.971    2.038  |X|= 0.0034
13 I fit 1 iters 8  : 9.70e-01    0.059   -0.044    2.014    2.018  |X|= 0.0045
14 I fit 1 iters 8  : 9.59e-01    0.006   -0.015    2.071    1.984  |X|= 0.0014
15 I fit 1 iters 8  : 9.85e-01    0.020   -0.074    2.008    1.993  |X|= 0.0021

                         az_fwhm            el_fwhm
        10m antennas:  1.20 +/- 0.02      1.18 +/- 0.02
         6m antennas:  2.00 +/- 0.03      1.99 +/- 0.03
\end{verbatim}

\clearpage
\begin{figure}
\psfig{figure=ct007.1C_1003C273.2.gpfit.0.2.ps,angle=-90,width=6.0in}
\caption{Pointing and Primary Beam FWHM at 100 GHz. File ct007.1C\_1003C273.2.gpfit.0.2.ps}
\end{figure}

\clearpage

\begin{verbatim}
--------------------------------------
Table 3. ct007.1C_2303C273.1.gpfit.60
--------------------------------------

#ant pol fit iters  :  pkval     az_off   el_off  az_fwhm  el_fwhm  |residual|
###############################################################################
01 I fit 1 iters 7  : 1.01e+00    0.008    0.009    0.545    0.521  |X|= 0.0369
02 I fit 1 iters 7  : 1.05e+00   -0.009   -0.001    0.544    0.507  |X|= 0.0168
03 I fit 1 iters 7  : 1.01e+00   -0.013    0.023    0.519    0.519  |X|= 0.0095
04 I fit 1 iters 7  : 9.05e-01    0.025    0.020    0.554    0.524  |X|= 0.0112
05 I fit 1 iters 7  : 1.01e+00    0.016    0.019    0.541    0.508  |X|= 0.0062
06 I fit 1 iters 7  : 7.96e-01    0.035   -0.032    0.576    0.530  |X|= 0.0162
07 I fit 1 iters 50 : 7.71e-01   10.840   -9.220    9.981   11.584  |X|= 0.0080
08 I fit 1 iters 8  : 9.03e-01   -0.004    0.083    0.887    0.918  |X|= 0.0059
09 I fit 1 iters 8  : 9.47e-01   -0.045    0.021    0.837    0.855  |X|= 0.0052
10 I fit 1 iters 8  : 9.11e-01   -0.036   -0.013    0.843    0.868  |X|= 0.0011
11 I fit 1 iters 8  : 1.01e+00   -0.007   -0.002    0.834    0.894  |X|= 0.0018
12 I fit 1 iters 8  : 9.61e-01   -0.006   -0.030    0.846    0.834  |X|= 0.0126
13 I fit 1 iters 8  : 9.94e-01   -0.025   -0.052    0.849    0.805  |X|= 0.0048
14 I fit 1 iters 8  : 9.93e-01    0.013    0.023    0.930    0.890  |X|= 0.0047
15 I fit 1 iters 8  : 8.92e-01   -0.021    0.035    0.852    0.851  |X|= 0.0068

Notes:
. The Gaussian fits for ant 7 failed ( iters=50)
. Ant 6, 10, and 15 have lower aperture efficiency ( lower pkval)

                         az_fwhm            el_fwhm
        10m antennas:  0.55 +/- 0.02      0.52 +/- 0.02
         6m antennas:  0.86 +/- 0.03      0.86 +/- 0.04
\end{verbatim}


\clearpage
\begin{figure}
\psfig{figure=ct007.1C_2303C273.1.gpfit.60.ps,angle=-90,width=6.0in}
\caption{Pointing and Primary Beam FWHM at 230 GHz. File ct007.1C\_2303C273.1.gpfit.60.ps}
\end{figure}

\clearpage

\begin{verbatim}
--------------------------------------
Table 4. ct007.1C_2303C273.2.gpfit.60
--------------------------------------

#ant pol fit iters  :  pkval     az_off   el_off  az_fwhm  el_fwhm  |residual|
###############################################################################
01 I fit 1 iters 8  : 1.07e+00   -0.013   -0.037    0.510    0.485  |X|= 0.0391
02 I fit 1 iters 50 : 8.36e-02   -4.176   -9.600    5.218    8.550  |X|= 0.0013
03 I fit 1 iters 7  : 1.06e+00    0.062   -0.029    0.522    0.479  |X|= 0.0344
04 I fit 1 iters 7  : 9.50e-01    0.055   -0.050    0.554    0.468  |X|= 0.0204
05 I fit 1 iters 7  : 1.01e+00    0.055   -0.034    0.548    0.485  |X|= 0.0107
06 I fit 1 iters 7  : 8.69e-01    0.028   -0.050    0.553    0.483  |X|= 0.0065
07 I fit 1 iters 50 : 4.98e-01   -2.862   10.928   13.647    8.472  |X|= 0.0033
08 I fit 1 iters 8  : 9.43e-01   -0.077    0.006    0.869    0.818  |X|= 0.0050
09 I fit 1 iters 8  : 9.73e-01   -0.098   -0.061    0.877    0.792  |X|= 0.0023
10 I fit 1 iters 8  : 9.37e-01   -0.026   -0.097    0.845    0.777  |X|= 0.0043
11 I fit 1 iters 8  : 1.05e+00   -0.085   -0.015    0.844    0.823  |X|= 0.0048
12 I fit 1 iters 8  : 9.89e-01   -0.034   -0.068    0.850    0.836  |X|= 0.0038
13 I fit 1 iters 8  : 1.02e+00   -0.005   -0.106    0.862    0.804  |X|= 0.0054
14 I fit 1 iters 8  : 1.03e+00   -0.048   -0.053    0.864    0.841  |X|= 0.0067
15 I fit 1 iters 8  : 9.33e-01   -0.068   -0.022    0.879    0.818  |X|= 0.0070

Notes:
. The Gaussian fits for ant 2 and 7 failed ( iters=50)
. Ant 6, 10, and 15 have lower aperture efficiency ( lower pkval)

                         az_fwhm            el_fwhm
        10m antennas:  0.54 +/- 0.02      0.48 +/- 0.02
         6m antennas:  0.86 +/- 0.01      0.81 +/- 0.02
\end{verbatim}

\clearpage
\begin{figure}
\psfig{figure=ct007.1C_2303C273.2.gpfit.60.ps,angle=-90,width=6.0in}
\caption{Pointing and Primary Beam FWHM at 230 GHz. File ct007.1C\_2303C273.2.gpfit.60.ps}
\end{figure}

\clearpage
\begin{figure}
\psfig{figure=fwhm.ps,angle=-90,width=6.0in}
\caption{Primary Beam FWHM fitted width.}
\end{figure}

\subsection{Absolute gain calibration using planet observations}

In principle, we obtain an absolute gain calibration using planet
observations with a planet model in selfcal. See Table 5. However,
as illustrated in Figure 5, if the planet is resolved then the derived
gains depend on the detailed surface brightness distribution in the planet
model, and may not produce reliable fits for the aperture efficiency or
primary beamwidth.

Figure 9 plots the visibility data points for the USB, and the Neptune model for the LSB at 
223.0 and USB at 229.65 GHz, using a model uniform brightness disk with parameters:
plangle : -23.7280,       plmaj   :  2.28992,       plmin   :  2.25076,
pltb    :  97.5229

For use as an absolute flux density calibration, correction, must be made for absorption lines in
planetary atmospheres.  We observed Neptune at 230 GHz on the CO 2-1 line.
See Figure 10.

\begin{verbatim}
---------------------------------------
Table 5. ct001.1C_230Neptune.2009may28
---------------------------------------

#ant pol fit iters  :  pkval     az_off   el_off  az_fwhm  el_fwhm  |residual|
###############################################################################
01 I fit 1 iters 8  : 1.03e+00   -0.022    0.009    0.474    0.556  |X|= 0.0628
02 I fit 1 iters 7  : 1.04e+00    0.015   -0.008    0.550    0.514  |X|= 0.0139
03 I fit 1 iters 7  : 9.63e-01   -0.021   -0.010    0.503    0.498  |X|= 0.0254
04 I fit 1 iters 10 : 9.56e-01    0.072    0.020    0.737    0.555  |X|= 0.2987
05 I fit 1 iters 7  : 1.05e+00    0.042   -0.033    0.521    0.486  |X|= 0.0070
06 I fit 1 iters 7  : 9.60e-01    0.024   -0.034    0.506    0.468  |X|= 0.0274
07 I fit 1 iters 10 : 9.60e-01   -0.123   -0.043    1.283    0.790  |X|= 0.0448
08 I fit 1 iters 8  : 8.79e-01   -0.150    0.167    0.764    0.761  |X|= 0.0084
09 I fit 1 iters 9  : 9.53e-01   -0.022   -0.057    0.556    0.685  |X|= 0.0729
10 I fit 1 iters 8  : 8.58e-01   -0.026   -0.155    0.751    0.703  |X|= 0.0328
11 I fit 1 iters 9  : 9.97e-01   -0.083   -0.012    0.835    0.872  |X|= 0.0249
12 I fit 1 iters 9  : 1.02e+00    0.055   -0.095    0.934    0.759  |X|= 0.0145
13 I fit 1 iters 9  : 8.99e-01    0.050   -0.109    0.667    0.811  |X|= 0.0497
14 I fit 1 iters 9  : 9.67e-01   -0.101    0.049    0.754    0.859  |X|= 0.0257
15 I fit 1 iters 9  : 8.60e-01   -0.121    0.016    0.832    0.804  |X|= 0.0373
\end{verbatim}
\clearpage
\begin{figure}
\psfig{figure=ct001.1C_230Neptune.2009may28.2.c.0.uvmodel.ps,angle=-90,width=6.0in}
\caption{Visibility for Neptune observations at 230 Ghz in the CARMA C-configuration.
The solid lines show the model visibility for USB and LSB frequency.}
\end{figure}

\clearpage
\begin{figure}
\psfig{figure=showmmnepspect.ps,angle=90,width=6.0in}
\caption{Brightness temperature of Neptune showing strong atmospheric absorption features.
For use as an absolute flux density calibration, correction, must be made for absorption lines in
planetary atmospheres.  We often observe Neptune at 230 GHz on the CO 2-1 line!}
\end{figure}

\clearpage
\begin{verbatim}
---------------------------------------------------------
Table 6. 09JUN24:23:22:35.5   ct007.1E_1003C273.1.gpfit.60
---------------------------------------------------------

#ant pol fit iters  :  pkval     az_off   el_off  az_fwhm  el_fwhm  |residual|
###############################################################################
01 I fit 1 iters 8  : 9.39e-01   -0.049    0.052    1.162    1.265  |X|= 0.0176
02 I fit 1 iters 8  : 9.06e-01   -0.050    0.113    1.197    1.235  |X|= 0.0311
03 I fit 1 iters 7  : 9.36e-01   -0.077    0.066    1.192    1.180  |X|= 0.0161
04 I fit 1 iters 8  : 9.37e-01   -0.024    0.066    1.230    1.265  |X|= 0.0241
05 I fit 1 iters 8  : 8.75e-01   -0.026    0.124    1.263    1.291  |X|= 0.0280
06 I fit 1 iters 8  : 9.12e-01   -0.055    0.049    1.241    1.254  |X|= 0.0296
07 I fit 1 iters 8  : 8.86e-01    0.039    0.243    2.041    2.210  |X|= 0.0016
08 I fit 1 iters 8  : 8.78e-01    0.079    0.063    2.007    2.086  |X|= 0.0048
09 I fit 1 iters 8  : 8.83e-01    0.034    0.189    2.012    2.180  |X|= 0.0040
10 I fit 1 iters 8  : 8.69e-01   -0.053    0.087    2.059    2.166  |X|= 0.0013
11 I fit 1 iters 8  : 8.99e-01    0.028    0.110    1.973    2.048  |X|= 0.0066
12 I fit 1 iters 8  : 8.93e-01    0.043    0.112    2.030    2.050  |X|= 0.0064
13 I fit 1 iters 9  : 8.89e-01   -0.036    0.115    2.064    2.091  |X|= 0.0090
14 I fit 1 iters 8  : 8.57e-01    0.048    0.124    2.070    2.128  |X|= 0.0051
15 I fit 1 iters 9  : 8.90e-01    0.055    0.176    2.013    2.212  |X|= 0.0065

                         az_fwhm            el_fwhm
        10m antennas:  1.21 +/- 0.04      1.21 +/- 0.04
         6m antennas:  2.03 +/- 0.03      2.13 +/- 0.06
\end{verbatim}

Aperture Efficiency

Using Flux of: 3C273     09JUN02.00 at  92.3 GHz: 17.68 Jy; rms: 2.65 Jy

 $10m (100GHz) = pkval^2  \times  (17.68\pm2.65 / Flux{3c273})  \times 50\%$

 $6m  (100GHz)  = pkval^2  \times  (17.68\pm2.65 / Flux{3c273})  \times 65\%$



\clearpage
\begin{figure}
\psfig{figure=ct007.1E_1003C273.1.gpfit.60.ps,angle=-90,width=6.0in}
\caption{Pointing and Primary Beam FWHM at 100 GHz. File ct007.1E\_1003C273.1.gpfit.60.ps}
\end{figure}


\clearpage
\section{CONCLUSION}

We used an interferometer mosaic pattern to determine the aperture efficiency,
pointing, and primary beam width for the CARMA antennas at 100 and 230 GHz.

We measure antenna gains close to 65 and 145 Jy/K corresponding to
aperture efficiencies 50\% and  65\%  respectively for 10 and 6m antennas.
At 230 GHz, antennas 6, 10, and 15 have lower aperture efficiency.

At 100 GHz the data are consistent with a common FWHM for azimuth and
elevation axes. The average FWHM for 10m antennas is 1.20 +/- 0.02. The
average for 6m antennas is 1.98 +/- 0.03.  These values are significantly
larger than the the current Gaussian primary beam model FWHM 1.07 and
1.92 arcmin for 10m and 6m antennas respectively.  At 230 GHz the FWHM on
the elevation axis appears to be smaller than on the azimuth for several
of the antennas. This is possibly caused by gravitational deformation
in both 6m and 10m antennas.  More extensive observations at several
telescope elevations are required to investigate this.  The mean values
are: azimuth: 0.55 +/- 0.02, elevation: 0.50 +/- 0.02 for 10m antennas,
and azimuth: 0.86 +/- 0.02, elevation: 0.83 +/- 0.03 for 6m antennas.
The current Gaussian primary beam model has the same FWHM for both axes:
FWHM 0.47 and 0.83 arcmin for 10m and 6m antennas respectively.

We obtain an absolute gain calibration using planet observations with
a planet model in selfcal. However, if the planet is resolved then the
derived gains depend on the detailed surface brightness distribution in
the planet model, and may not produce reliable fits for the aperture
efficiency or primary beamwidth.For use as an absolute flux density
calibration, correction, must also be made for absorption lines in
planetary atmospheres.

\clearpage
\section{ References }

Cornwell, T.J., Holdaway, M.A. \& Uson, J.M., 1993, A\&A 271, 697,
{\it Radio-interferometric imaging of very large objects: implications for array design}

Holdaway, M. A., 1998, {\it Mosaicing with Interferometer Arrays}, in
Synthesis Imaging in Radio Astronomy II,
ASP Conference Series, G. B. Taylor, C. L. Carilli and R. A. Perley (eds)

Corder, S., 2008, {}, PhD thesis, Caltech.

Wright, M,. 2002.
CARMA memo 5, {\it Compact Configuration Evaluation for CARMA}

Wright, M., 2007, CARMA memo 38
{\it Image Fidelity as a Function of Source Size and Calibration Errors} 


\end{document}
\subsection{Data Sampling}

The source brightness distribution is illuminated by the antenna reception
pattern for each pair of antennas. The effective primary beam is the
product of the voltage pattern of one antenna and the complex conjugate
of the other. In order to image an extended source we need to correct
the data for this illumination pattern.

In CARMA memo 5, we argue that errors in the voltage beam pattern of
the larger antennas which lie within the main lobe of the voltage beam
pattern of the smaller antennas will corrupt mosaic images. If we can
measure the primary beam patterns well enough, then including 10.4 versus
3.5~m correlations provides additional $uv$ data and primary beams to
deconvolve the synthesized and primary beam responses from the image. If
we can not determine the primary beam patterns well enough, the errors
will degrade the image fidelity.  Similarly, if the pointing centers
of the antennas are offset, there will be large uncertainties in the
resulting product of voltage patterns, and the asymmetric primary beam
patterns rotate on the sky.  For mosaic observations we would need to
calculate a primary beam for each pointing, baseline, and integration.
Whilst this is possible in principal, the increased computing burden is
substantial, and the errors in the visibility data are hard to measure.

In order to keep the number and parameterization of the primary beam
models manageable, the most convenient way to make mosaicing observations
is to use the same pointing pattern for all antennas at the sample
interval required for the largest antenna in the sub-array being used.
In view of the discussion above, this mode is also likely to produce the
best image fidelity in practice.
Sampling rates are set by both the largest and smallest antenna diameter.
The Nyquist sample interval for the $uv$ data, $\delta uv = D_{min}/2\lambda$.
The Nyquist sample interval for the pointings, $\delta\theta = \lambda/2D_{max}$.
The $uv$ data for each pointing are oversampled by the larger antennas,
and the pointing is oversampled by the smaller antennas.  There is no
loss in sensitivity since the oversampled data are properly accounted
for in the imaging algorithms.  Using the larger antennas on the
longest interferometer baselines provides a more uniform sensitivity in
the $uv$ data and reduces the required $uv$ data sample rate.

The 23-antenna heterogeneous array can thus be considered as a set of
sub-arrays with a primary beam pattern for each pair of antenna diameters.
Mosaic observations with the 23-antenna telescope (CARMA-23) produce
6 primary beam patterns for each pointing center; the products of the
voltage patterns in cross correlations between the 3 antenna sizes. The
different primary beam patterns illuminate the source structure providing
independent information about the source brightness at each pointing
center.
The primary beam pattern depends on the aperture illumination and pointing
of the antennas.
In practice the mosaicing algorithms usually clip the primary beam
model at the $\sim$ 5\% level, thus avoiding the uncertainties and variations
in the primary beam response at low levels.  Within the 5\% level,
the primary beam patterns are well approximated by Gaussian beam
patterns. Table 1 lists the equivalent antenna diameter, primary beam
FWHM and Nyquist sample interval at 230 GHz.
Note that the 10.4 - 3.5~m correlations have almost the same effective primary
beam pattern as the 6.1~m antennas, providing a method to measure and cross
calibrate the primary beam patterns by comparing images from each sub-array.


Although 10m
antennas are sampled close to the half power
point, the 6m antennas are not. It would be better to sample each antenna
at the half power points 

Mosaic observations

hex7 mosaic to derive pointing, PBFWHM,  jyperk and bandpass

-- pointings at center and FWHM points are best - for the
heterogeneous CARMA telescope this means best using az/el offsets
rather than ra/dec. delay/phase tracking
center on strong compact source.
These are 7 discrete pointings, and Gaussian fitting.
could be 5 pointings with no PA fit - which hex7 can determine,
or 13- or more to fit non Gaussian part of PB of 10m which is within 6m PB.
The frequency and number of pointings needed depends on image fidelity required.

Sounds like telescope software can handle this as is, but may need some
observing script writing ?

2. OTF mosaicing.  This is more ambitious, but from your email, it sounds
like this is possible by incrementing az and el offsets.  The advantage over
the standard discrete pointing mosaic is observing efficiency for large FOV.
I've argued that with OTF it would be difficult to control the synchronous slew
needed to maintain the x-corr PB stability illuminating the target FOV,
but now thinking it maybe possible to finesse this by incrementing az/el offsets
 in a more controlled way at CARMA.
Would need something like synchronized integrations and incrementing az/el
offsets every few seconds.  Does this sound possible ?

3. Array Rx response. This was my initial question.
An array Rx will give different pointings in 6 and 10m antennas. We need to
measure the cross correlation PB response for offset pointings.
We could try this with existing Rxs by using  different Az,El offsets in 6
and 10m antennas. Observations required are same as (1).

 hex7 - antenna voltage patterns
are a great way to analyze beam patterns and feed characteristics/problems -
squint, focus,  pbfwhm, alignment etc.,  Jy/K gains and antenna based bandpass come for free from mfcal on a hex7 mosaic on a calibrator.

using hex7 mosaic on strong source
to determine jyperk, bandpass, pointing, and PB FWHM
from antenna gains.  works very well at 3mm.
Results  on carma at 1mm do not look very stable.

The elliptical PB pattern is probably real and results from
the astigmatism which we saw in the holography at that time,
and later corrected by adjusting the panels
 agree.
This hex7 measurement was at 30" radius  . Gaussian fitting of the pointing and voltage patterns are far from optimum. Also, I had to average for 50min, and there are some pointing changes within the 50min measurement cycle which mess up
the fits.


Suggest we try the standard hex7 script again in a stronger source
at the nominal 10m FWHM points, and see how robust and repeatable the results are.

Better would be to observe with both 6 and 10m at their nominal
FWHM points. This is possible as I understand it from Marc and
Steve, but is not a standard script. We can try that later.

