BIMA MEMO No. 32
Strategies for Multiple Pointing Observations.
W.J.Welch, 30-Nov-93
Multiple pointing observations may be done either to get more
uniform sensitivity over the primary beam for a source that nearly
fills the beam, or it may be used to map a source that is larger than
the primary beam. In either case, it is necessary to carry out a
complete set of all the pointings while still within each uv cell of
the source visibility function, in order that the source visibility be
properly sampled. This may be difficult if the source is large
compared to the beam, requiring many pointings, or if the desired
angular resolution is high. This is because the observation may pass
completely through the basic uv cell before it is possible to complete
all the different pointings. It is always possible to carry out the
observations by using tracks on successive days, but here we discuss
the limitations to obtaining the complete sampling in one single
track.
Note also that "multiple pointings" represents part of one
strategy for obtaining complete uv coverage, including the shortest
spacings. That is, in the most compact array, a few multiple
pointings at half primary beam spacings combined with a map made with
one of the array elements provides all the short spacing data for a
map which is fully sample over the extent of the primary beam. No
further single dish data is required.
If a single pointing is used for an interferometric observation,
the primary antenna beamwidth determines the maximum effective size of
the source, regardless of what the actual source extent may be. It is
close to twice the one half power beamwidth. The visibility function
is then "band limited" by this amount, and it is sufficient to measure
visibilities at sample intervals of:
\Delu,\Delv \leq 1/(2\The|1/2|)
If multiple pointings (mapping) are used the effective source
size may well be larger, with size \The|s|. Then the visibility
sample interval is smaller:
\Delu,\Delv \leq 1/\The|s|
Whatever the source size, the beam spacing interval that must be
used is the usual one half of the primary beamwidth, \The|1/2|.
Strictly, this spacing should be one half of \lam/D, but the
difference is usually small.
Here is the question: Is there time enough to carry out the
multiple pointings in the time available in one visibility sample
cell? If the source size is \The|s| and is approximately square, the
number of pointings at half beamwidth spacings is:
n \app [2\The|s|/\The|1/2|]^2^
Any particular array has a longest baseline which specifies the
outer envelope of the u-v coverage. It defines the angular resolution
of the array. It also is the track that moves most rapidly through the
u-v sample cell. Like all the tracks, the outer one is a section of an
ellipse. Motion through the sample cell is most rapid in the limiting
case of a circle (for a source at the pole), so we will use that case
to discuss the available time for the multiple pointings. Let the
radius of the outer track be \bet|0|. Then the resolution of the
array is approximately 1.4/(2\bet|0|) = \The|r|. This outer track
covers a quarter of a circle of radius \bet|0| in six hours. Its
length is \bet|0|\pi /2. The total time in one cell is approximately:
\Delt' = \Delu/(\pi \bet|0|/2) x 6 hours
Substituting from above, we get for the total time in one cell:
\Delt' = 320 minutes (\The|r|/\The|s|)
If n is the number of pointings required, then dividing by n
gives the amount of time available to each pointing::
\Delt = 80minutes [(\The|1/2|)^2^x\The|r|/(\The|s|)^3^]
It is clear that for a given half beamwidth the maximum source size
that can be mapped in one track will be strongly limited by the cube
of the source size in the denominator of the above expression.
The ultimate limit to the size of the field that can be done at
any given resolution is the overhead, or time lost, in moving from one
pointing to another and beginning an integration at the new position.
For example, if the lost time were only one second, a total time of
only 10 seconds at each pointing would suffer only a 10% loss in
observing time. That 10% loss could be used as the criterion for the
maximum practical field size for the given resolution.
_Some Examples_
1. \The|1/2| = 120", \The|s| = 120", \The|r| = 5".
This corresponds to the HCN observation of IRC+10216 scheduled for the
present BIMA C/B array. From the relations above we find \Delt' = 13.3
minutes, n = 4 pointings, and 3.3 minutes is the time available to
each pointing position. Thus, about 3 60 second integrations may be
done on each position. These 3 may be done in succession, eliminating
some of the time that would be lost in repositioning the antennas
after each integration. Note that in estimating accurately the time
available one must know the times lost in starting up an integration
and the time lost in an antenna move.
2. Suppose this project were pushed on to higher resolution with an
A array giving about 2" resolution. In that case, there would be just
1.3 minutes for each pointing in the outer most uv cell. That would
correspond to about one 60 second integration at each antenna
position.
3. For this case, double the source size to four arc minutes,
keeping the resolution at 5". All other factors being equal, the
available time per position is reduced by a factor of eight to 3.3/8,
or about one half minute. This source size, just twice the beamwidth,
is about the maximum that could be done.
_A few general conclusions can be drawn._
(a) A source diameter of about twice the primary beam half power
width is about all that one can do in a single track. Bigger sources
require more tracks on more days.
(b) In order to minimize time lost in repositioning the antennas,
there needs to be a separate specification of the integration times
and the number of repeated integrations that can be made at the same
pointing position.
(c) In any case, both the integration start-up and the antenna
repositioning times must be kept as short as possible.
WJW 11/28/94