[daip] Absolute to offset coordinates

[Eric Greisen]

On 10/23/2014 04:44 AM, Andy Biggs wrote:
> Hi Eric. I didn't explain very well I think. My main question concerns
> having 2 source positions for which you want to measure the delta_ra and
> delta_dec. The latter is easy, but for the former you multiply by
> cos(dec). The question concerns which declination to use. I have always
> done what AIPS does, but others seem to take an average.
>
> More specifically, in the OT we can allow a user to enter telescope
> pointing positions as absolute coordinates, but these can then be
> converted into their delta_ra and _dec equivalents, these being
> referenced to the source position. I have always done this like AIPS i.e.
>
> delta_ra = (ra2 - ra1) * cos(dec1)
>
> where ra1, dec1 is the source position. Others seem to use
>
> delta_ra = (ra2 - ra1) * cos((dec1 + dec2) / 2).
>
> There is, for example, an offset calculator at Keck that uses the latter:
>
> http://www2.keck.hawaii.edu/inst/common/offset.php
>
> Presumably AIPS is doing the right thing...
>

By delta_ra and delta_dec you really mean the direction cosines L and M.
Reviewing AIPS Memo 27, these are given by

In -SIN projection:
         L = cos(Dec) sin(RA-RA_0)
         M = sin(Dec)Cos(Dec_0) - cos(Dec)sin(Dec_0)cos(RA-RA_0)
where RA and Dec are the offset point and RA_0,Dec_0 are the initial point.

In -TAN projection it is much more complicated:
The 2 formulae above are divided by
     sin(Dec)sin(Dec_0) + cos(Dec)cos(Dec_0)cos(RA-RA_0)

I do not know what Keck does if you give it a coordinate and then ask 
for an offset.  I would hope that the calculator and the on-line system 
do the same so that the telescope ends up pointed correctly.  ALMA 
should use the -SIN projection (it is a radio interferometer) and so it 
should not average Dec and Dec_0 but use solely Dec.

See AIPS Memo 27 available on the web for derivations.

Eric Greisen