Possible Pointing Types
Tilings for each pointing Type
Areas covered
Pointings for ALFA pulsar survey

Possible pointing types.

  Let us suppose we are conducting a perfect pulsar (or galaxy!) survey with non-drifting, single beam pointing, covering a given region of the sky. No holes in that region can be left. The best possible coverage will probably look like the picture below.
   In this picture one has defined that, for each pointing, a circle indicates the points for which the sensitivity is half (or some other fraction) of what it is in the centre. The diameter of each circle is what is normally called the beam diameter.
   In the arrangement showed, all the points in the sky are at least sampled with half (or some other acceptable fraction) of the sensitivity of the center of the beam. No other pattern achieves this so efficiently.

Figure 1: Optimal beam arrangement for a single-beam survey.
 

Now, whith ALFA we can cover seven beams simultaneously. They do not overlap, and six of them are arranjed in an hexagon around the central beam. Again, it is desirable that the whole region being surveyed is covered in such a way that all points of the region are covered with at least half the sensitivity of the outer beams. Therefore, the individual beams have to fit in an overall pattern that looks like that of Figure 1.

There is an infinite number of ways in which this can happen. In the next three figures, we present the three most compact ways of accomplishing this.

Fig. 2 - Nearest non-contiguous circle from center (Type 1 tiling). D is the distance between neighbouring beams, the green circle is the ALFA beam, for comparison.

Fig. 3 - Second nearest non-contiguous circle (Type 2 tiling). D is the distance between neighbouring beams, the green circle is the ALFA beam, for comparison.

Fig. 4 - Third nearest non-contiguous circle (Type 3 tiling), with black and partially filled circles indicating the two possible alternatives. D is the distance between neighbouring beams, the green circle is the ALFA beam, for comparison.

In Figure 2, the separation between beam centers is 1.5 times the background circle diameter (let us say s = 1.5).
In Figure 3, s =1.732... (square root of 3).
In Figure 4, s = 2.291 [(square root of 21)/2].

The ALFA feed separation between the centers of neighbouring feeds has already been selected. It is 25 cm x 26 cm on the focal plane (TE11 horns). The telescope optics make the sky positions of the centers of the six outer beams sit on an ellipse in the sky with minor axis of 658 arcseconds and major axis of 768 arcseconds. Therefore, the separation between neigbouring beams is 329 arcseconds (D') and 384 arcseconds (D) along perpendicular directions. The center of the central beam is 33 arcminutes from the center of this ellipse (see the "Final Feed Selection Study" in the ALFA memo series).

Because the projection in the sky of the positions of the feeds is elliptical, the pointings can only fit in the pointing grid if we stretch it in one direction. In the figures above we chose to stretch the grid in the vertical sense.

The average beam is itself an ellipse with average minor and major axis of 204 and 232 arcseconds, these are generally aligned with the same direction as the ellipse defined by the outer beams. The ratio between separation and beam diameters is therefore 1.61 and 1.66 along the directions of the minor (horizontal) and major (vertical) axis respectively.

Which pointing?

Since for the fixed feed separation of 26 cm we have s = 1.6, we can choose Type 1 tiling (s = 1.5) or Type 2 tiling (s = 1.732...).

The reason is the following: the size of the pointing grid (i.e., the space between two neighboring black circles in Figures 2,3 and 4, and the corresponding scaling of the background of circles) is set by the beam separation in the sky only. And that has been already fixed by the feed dimensions! D is 384 arcseconds , independently of the pointing type we choose.
 

• If we select Type 1 tiling, then covering the plane as indicated in Figure 2 (and 5) will require beams with a major axis that is no smaller than D / 1.5, i.e., 256 arcseconds. The major axis of the ALFA beams is slightly smaller than this (only 232 arcseconds, see Figure 2). This means that if we select this tiling strategy, some parts of the sky will not be sampled at half the sensitivity of the center of the beam, mainly near the points where three "circles" meet. The advantage of this survey is that beam overlap is reduced to a minimum.

• If we select Type 2 tiling, then covering the plane as indicated in Figure 3 (and 6) will require beams with a major axis no smaller than  D / 1.732, or 221 arcseconds. The ALFA beams are slightly larger than this (see Figure 3), which means that there will be significantly more beam overlap.

• If we select Type 3 pointing, then covering the plane as indicated in Figure 4 (and 7) would require beams no smaller than D / 2.291, i.e., 167 arcseconds. The real ALFA beam is MUCH larger than this (see Figure 4). Adopting such a strategy would lead to massive beam overlap, and therefore to a great waste of observing time.

Tilings for each pointing Type

The selection of the type of pointing is important because of the overall sky coverage.

With a Type 1 pointing, there are two possible pointing patterns, as indicated in Figure 5. Keep in mind that the size of the "circles" in Figure 5 is 256 arcseconds in the vertical axis and 219 arcseconds in the horizontal axis, which is slightly larger than the half-power axis of ALFA's beams (232 and 204 arcseconds respectively). These "circles" are the effective "survey beams" for Type 1 tiling.

Figure 5: The two possible pointing strategies that could be chosen if the ALFA survey pointing is to be of Type 1. Again, the red lines define the global pointing pattern. Whenever two such lines meet, three pointings (with the central beams indicated by the numbers) must be made to cover the survey region completely. All the beams in the same pointing have the same colour. In the left, the three pointings are made along a line, on the right they are made in a triangle.

With a Type 2 pointing, one can completely cover the sky with the pattern depicted in Figure 6. Keep in mind that the size of the "circles" in Figure 6 is 221 arcseconds in the vertical axis and 190 arcseconds in the horizontal axis, which is slightly smaller than the half-power axis of ALFA's beams. These "circles" are the effective "survey beams" for Type 2 tiling.

Figure 6: Pointing strategy implied by choosing a separation of 26 cm for the ALFA feed horns. The red lines define the global pointing pattern. Whenever two such lines meet, four pointings (with the central beams indicated by the numbers) must be made to cover the survey region completely. All the beams in a given pointing have the same colour. The arrows with D' define the scale of the drawing, their length is 329 arcseconds (5.483 arcminutes).

Finally, if the pointing was of Type 3 (very inneficient), one would have to cover the sky with the rather complex pattern depicted in Figure 7.

Figure 7: The purple lines define the global pointing pattern. Whenever two such lines meet, seven pointings (with the central beams indicated by the numbers) must be made to cover the survey region completely. All the beams in a given pointing have the same colour. The red and green lines indicate hexagonal regions covered with the same sensitivity.
 

Sky Areas Covered
 

To calculate the area covered by a single "survey beam", and take into account beam superposition, we will now assume for each beam an hexagonal shape (the only reason for this is that the hexagons tile the plane without superposition).

We now calculate the areas covered for two tiling strategies: Types 1 and 2.

Type 1 tiling. In Figure 5, on the left (and in more detail in Figure 8), we can see that 3 times the radius r of each "survey beam" is equal to D.
Therefore, r = 128.0 arcseconds, as mentioned here. The apothema ap of the corresponding hexagon can be calculated by ap = r cos 30 = 110.85 arcseconds.

The area of an hexagon, assuming no deformation in the underlying pattern, is 3 ap r =  45566.88 square arcseconds.
However, we should not forget that the whole pattern is deformed along the perpendicular axis, is is shrunk horizontally by the ratio D'/D= 0.857. Therefore, the area covered by a single beam is also shrunk in the "horizontal" axis by the same ratio: it is 39040.37 square arcseconds, or 10.844 square arcminutes, or 0.00301 square degrees.
The area covered by a single ALFA pointing is seven times this: 0.0210866 square degrees.
It takes 47 pointings to cover a square degree.
 
 

Figure 8: closeup of Figure 5, assuming no deformation in the underlying circle pattern (i.e., a tiling with regular hexagons with no deformation).

Type 2 tiling. In Figure 6 and, in more detail in Figure 9, we can see that 4 times the apothema (ap') of the hexagon corresponding to the "survey beam" is D', which has a length of 329 arcseconds, therefore the apothema is 82.25 arcseconds. The radius from center to vertex of the hexagon (r') is given by

r' = ap' / cos 30 = 94.97 arcsec

Figure 9: closeup of Figure 6, assuming no deformation in the underlying circle pattern (i.e., a tiling with regular hexagons with no deformation).

The area of an hexagon, assuming no deformation in the underlying pattern, is 3 ap' r '= 23433.85 square arcseconds .
However, we should not forget that the whole pattern is deformed along the perpendicular axis, is is streched vertically by D/D' = 1.17.
Therefore, the area covered by a single beam is also stretched in the "vertical" axis by the same ratio: it is 27339.89 square arcseconds, or 7.594 square arcminutes, or 0.002109 square degrees.
The area covered by a single ALFA pointing is seven times this: 0.01477 square degrees.
It takes 68 pointings to cover a square degree.

Type 1 tiling, with realistic beam patterns (see here for .eps version):

Figure 10: same as Figure 5, but this time with the real sky footprint of ALFA, as tabulated for the TE11 horns in page 19 of the "Final Feed Selection Study" in the ALFA memo series. Each ellipse depicts a sensitivity level of -3 dB compared to the center of the beam. Beams with the same number belong to the same ALFA pointing.