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