lband multibeam system.
(last modified 18feb04)
The lband multi beam system is envisioned to consist of 7 feeds.
The feed system would have to rotate if the outer 6 feeds were to track
the same spot on the sky for extended periods of time.
Do we really need to rotate the feed system and if so, how fast..
The Figure shows
the parallactic angle in relation to the source position at (RA,DEC), the
zenith of the telescope at (LST,AoLatitude), the hour Angle HA, and the
azimuth direction AZ (in the picture the source has already transitted).
When the feed is pointing at the source, the azimuth angle points at the
(RA,DEC) position and the zenith angle moves along this arc till it is
at the (RA,DEC) location (on the sky). As the LST approaches the
RA of the source (ie the source gets closer to transit) the source azimuth
angle goes to zero. Now suppose that a second feed was placed 1 beam width
away from the central horn parallel to the azimuth arm. At time LST, it
would point 1 beam beyond RA,DEC in the az direction. At delta T seconds
later the offset horn is still pointing on the the azimuth direction but
this direction has now rotated about the point (RA,DEC) by the change in
the parallactic angle. The important thing to remember is that if
the central horn points at (RA,DEC), then it is pointing at the location
(RA,DEC) on the celestial sphere. Any rotation about the central horn will
be a "parallactic angle" rotation. The equation for the parallactic
sin(parallactic angle)=sin(azimuth) * cos(latitude)/ cos(declination)
(see Spherical astronomy, Small pg. 49).
Rotating the feed system by the parallactic angle rate will then cause
a horn on the outer ring to stay fixed on the sky. Suppose we don't
rotate the feeds. How long will it take before the outer horn moves by
1/2 of a beam? A linear motion in the focal plane has a plate scale (magnification
or demagnification) that is different in the 2 directions:
14 asecs on the sky per cm perpendicular to the azimuth arm
17 asecs on the sky per cm parallel to the azimuth arm.
The asymmetry comes from the secondary and tertiary mirrors. The
computation is then:
The FIGURES are:
Decide on the largest angle error you will accept (probably 1/2 a beam
of 3.2 arc minutes).
Compute how many centimeters this is in the focal plane using the worst
case plate scale (17"/cm).
Given the radius to the center of the outer horn (30 cm) and the arc Length
from 2., compute the angle MaxDeltaPa (10.78 degrees). This is the largest
parallactic angle change allowed.
Compute the parallactic angle versus hour angle PA(ha). For each PA(ha)
search forward to PA(ha+dt)=PA(ha)+MaxDeltaPa. dt will be the maximum integration
time starting at ha. This needs to be done for each declination.
The integration time plots show that 10 minute integrations (with up to
half a beam motion) can be done at any declination provided the hour angle
keeps you above the horizontal line in figure 4. Long integrations are
primarily needed for the pulsar search program (since they must be coherent).
In this mode you would track all your sources within the hour angle range
that gives you the needed integration time.
Figure 1. shows the parallactic angle rate versus azimuth angle for declinations
8 through 28 (spaced 1 degree steps from Arecibo's latitude). The maximum
rotation rate is .225 degrees per second. The left set (northern sources)
and right set (southern sources) each have declinations stepping 1 degrees
from the AO latitude.
Figure 2. shows the same rate versus hour angle for declinations 0 through
17 degrees (the scale is blown up a bit here).
Figure 3 shows how long you can integrate before the motion equals 1/2
of a beam. I used a beam width of 3.2 arc minutes, an outer horn location
of 30 cm, and a plate scale of 17asecs/cm. I did not take into account
the differential motion caused by the different plate scales. Using
these values, the maximum allowable rotation angle is: maxAngle=bmWd/2*60/plateScale/radius*radToDeg
or 10.87 degrees. The slanted lines on the right are because
the source will set before a longer integration can complete.
Figure 4 is a blowup of Figure 3. For sources 12 degrees from our declination
(5. deg dec and below) a 10 minute integration can be done anywhere on
the dish (as long as you don't set). For a 172114 dec source you can do
a 10 minute integration until 20 minutes before transit and any time 10
minutes after transit.
If you wanted to do 2 ten minute integrations at
the same 7 positions on the sky, you would integrate, move the telescope
by 60 degrees in parallactic angle to rotate the feeds by 1, and then do
the second integration.