[fitswcs] Paper III: dispersed spectra function GRA

[Frank Valdes]

> On Tue 2001/11/26 Mark Calabretta wrote in a message to: fitswcs at NRAO.EDU

> Looks like a very useful addition to Paper III.  For those of us who are
> not familiar with optical spectroscopy I have a few suggestions which may
> make it a bit easier.
> 
> Cheers, Mark
>

Hi Mark and Steve,

Thanks for the careful reading and many recommendations.  I have followed
most of your recommendations and deferred some for later consideration
by Eric and me.

You may not have picked up a draft change since my previous announcement.
But Steve commented about what happens when the incident angle to
a prism is not normal to the face.  I spent alot of time trying to see
if there was a more general though still simple formulation but I could
not find one.  But it was clear that the basic form would be the same
(a sin(beta) relation) and that the departures could be made quite small
with suitable adjustment of parameters.  So I added some comments about
this.

Along those same lines, I remembered that the grism in my example
is not a standard grism and does violate the requirments noted in the
paper.  But the function still fits impressively well to the calibration
data illustrating that this function will still be fine in these cases.
So the description of the MARS grism was corrected.

A new draft is on-line at the same address.  Note that if you don't get
a geometry figure as figure 1 then maybe you need to clear your browser
cache.  This confused me for a few minutes.

Yours,
Frank

* Perhaps "GRI" would be a better algorithm code since it suggests grisms
  as well as gratings.
  
I have adopted this.

* Section 1.1, para 2: the concept of "tangent point" is introduced without
  explanation.  I think a schematic diagram here would help greatly, the
  angles alpha, beta and epsilon and the tangent point should be marked.

I was trying to avoid getting too much into the physics and present this
mostly as a mathematical construct.  But clearly a figure is useful.  I
don't have great tools for this but I did make a figure which is in the
latest version.

* Latex typesetting: use \sin, \cos, \tan, \tan^{-1} and \sin^{-1} to get
  proper typefaces for the trig functions.  You also need a roman subscript
  for r since it is not a variable name, e.g. \lambda_{\rm r}.

Thanks for the help.

* The paragraph associated with Eq. 1 says that "For a pure prism the order
  m is zero".   The zero denominator which this produces in Eq. 1, while
  correct, begs clarification.

I had changed Eq. 1 several times.  Normally the Gm term is on the lambda
side and I have restored it there.  In Eq. 3 I comment about when m and
n'_r cannot be zero.

* n'_r in Eq. 2 is not defined.  I first read it as dn/dlambda (which has
  units of inverse length) but closer inspection shows that it has to be
  lambda_r * dn/dlambda.
  
This is now defined.  I have the Taylor expansion as a function of
z=(delta lambda)/lambda_r.  This then required n' to be multipied by lambda_r.
This is good because it makes n' be of order unity.  However, there is
really no need for this so the Taylor expansion is now purely with respect
to wavelength and so n' is the derivative with wavelength.

* A comment on Paper III in general; I don't think it should preempt the
  name of the intermediate variable, i.e. just use w, not w'.  Paper IV
  will introduce whatever notation is required and we don't want to be
  constrained by what appears in the earlier papers - chances are that it
  will not match this.

Still have to decide on this to be compatible with the rest of paper III
as well as IV.

* In Eq. 4, it's not clear how theta may be non-zero.  However, the diagram
  may make this clear.

I hope the figure makes this clear.

* Typo: "as shown in equations 5".  (Note that A&A has a style rule for this
  it should be Eq. 5 (in latex Eq.~\ref{eq:xyz}) unless at the beginning of
  a sentence when it is spelt in full - Equation~\ref{eq:xyz}.  Similarly
  for Fig. and Sect.)

Thank you very much for this.  I changed this (including the clarifications
in your followup message).

* Eq. 6 has dw'/dlambda but PVj_7 mysteriously gives dlambda/dw'.  (Doesn't
  the formal notation have a vertical bar to the right of the differential
  rather than parentheses?)
  
This was a mistake.  We need to chose one or the other.  It is now consistently
dw'/dlambda.  Note that if we avoid using w' in Paper III it would then
be possible to use w'(lambda_r) to denote this quantity in the same way
as with n'(lambda_r).  While I know what you mean about the vertical bar,
looking at a couple of texts shown using a functional form for things like
Taylor expansions so I have changed to that form.


* The intent of Eqs. 1 to 5 is not made clear until the last paragraph of
  Sect. 1.1.  It would clarify matters if that text could be moved much
  earlier in the section.

I have moved this earlier.  Let me know if this makes more sense.

* You provide the equations for computing lambda as a function of w, but
  what about the inverse?

As was noted in some earlier discussion, the FITS WCS conventions are concerned
only with the forward transformation for WCS interchange.  However, since
it is trivial I added a second paragraph outlining how the inverse
transformation would be computed.

* Section 1.2: I really wonder about the practicality of this.  As you say,
  grisms are defined naturally in terms of wavelength and perhaps it should
  just be left at that.

I'm not sure about leaving this out.  It is inherent in Paper III that
you can have FREQ-XXX, VELO-XXX, etc.  So I am inclined to leave this in.
It would be interesting to get other opinions.

* Section 1.3: how about an example of header interpretation as in Paper II,
  i.e. the working for a particular point so that people can check their
  software.  Also perhaps an example or some comments on header
  construction.

I haven't done this but it might be a good idea.

* Typo Fig. 1 caption: Coude has an acute accent.

OK.

* "l/mm" is a bit obscure, "lines/mm" doesn't take much space.

OK.

* Typo: "equvalent".

OK.