Possible
Pointing Types

**Tilings
for each pointing Type**

**Areas
covered
Pointings
for ALFA pulsar survey
**

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.

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

- 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**. T**he 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 "

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.

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.

Last updated 16th of February 2004.
Please send comments and questions to pfreire@naic.edu.