 ### The complications of precession for ALFALFA

Here's a quick calculation that might help to demonstrate why it is necessary to track in J2000 instead of leaving the telescope completely fixed.

Approximate values for precessed coordinates are given by the equations:
RA = RA(J2000) + (3.075 + 1.336 * sin(RA) * tan(Dec)) * T
Dec = Dec(J2000) + 20.04 * cos(RA) * T
where T is the time since the time from January 1, 2000 in fractional years, and the offsets in RA and Dec are in seconds of time and arcseconds, respectively. Because we are drifting, we don't care about the precession in RA. We are, of course, interested in delta(Dec) over periods of hours (during a single observing session) and from session to session, possibly spanning a period of several years. Note that it is dependent on both the RA and on the observing epoch. Tracks in Dec(Current Epoch) will not be parallel to tracks in Dec (J2000) because of the RA term.

See for yourself the convenient Coordinate Transformation Calculator at the NASA Extragalactic Database NED

For the spring observing time, we want to map the whole AO sky between 07h30 and 16h30 RA(J2000). Sometimes we might get assigned the full block, but more likely, we will end up with many partial blocks which then have to be stitched together. Also observations will be conducted over a 4-5 year period.

Let's say that at epoch J2005.0, we are assigned a block of time that covers part of this RA range. We can consider two examples (a) a 5 hour block and (2) one that covers the whole 9 hours. The Fixed Az mode takes the input Az (0 or 180d for transit observations), and uses the input Dec(J2000) to calculate an associated ZA. The drift begins when a source of input RA, Dec. (J2000) crosses Beam 0 at that Az,ZA. Currently the telescope then remains at that fixed Az, ZA until commanded otherwise. For ALFALFA, the data taking is continuous (virtually 100% efficient) for many loops of scans of 600 sec each for the entire observing session.

Consider first a drift lasting 5 hours = 18K secs. The track starts at a commanded Az, Dec, and begins taking data when the commanded RA crosses Beam 0. In this example:
-- Track starts at 073000.0 +100000 (J2000) = 073016.5 +095921 (J2005.0)
5 hours later, data taking stops:
-- Track ends at 123016.5 +095921 (J2005.0) = 123001.3 +100100 (J2000)
The ending Dec. is 1' off the starting one in J2000 coords.

[Note: We are ignored the pointing model (which accounts for known offsets of the telescope point due to gravity, etc) but they can be accounted for and are small.]

On the other hand, if we actually observe the full track, then the
-- Track ends at 163016.5 +095921 (J2005) == 1630022 +095959(J2000)
We are lucky to end up at the same place! But the track has not been parallel in Dec (J2000).

ALFALFA will take 6-7 years to complete, and it may be that we end up with gaps in the map that need to be filled in at a later time. Now consider that we come back at J2008.0 to finish the second part of the scan. Unless we keep track of exactly where we left off, we might simply plan to pick up where we "left off" in J2000 coords.
-- Track starts at 123000.0 +100000 (J2000) = 123024.3 +095721 (J2008.0)
and, at the end of that drift series lasting 4 hours = 14.4K sec we have
-- Track stops at 163024.0 +095721 (J2008.0) = 163001.1 +095822 (J2000)
We are off where we want to be in Dec (J2000) by 1'38".

Of course, we could have kept track of where we left off, and thus positioned the telescope at the start of the scan at
-- Track starts at 123001.3 +100100 (J2000) = 123025.6 +095821 (J2008.0)
4 hours later, we stop at
-- Track ends at 163025.6 +095821 (J2008.0) = 1630027 +095922 (J2000)
We are still off by 38".

Hence, unless we apply the small correction which takes us from "truly" fixed azimuth in current epoch coordinates back to J2000, we end up with a mess. Note that even 7 years makes a noticeable difference.

It is for this reason that Mikael Lerner has introduced into CIMA the option of the "almost" fixed azimuth drift. (Thanks, Mikael!)

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