[fitsbits] [mhvk at astro.utoronto.ca: Question about FITS format for logarithmic units]

[Marten van Kerkwijk]

Dear Lucio,

>> The background to this request actually is not directly related to FITS, 
>> but rather to an attempt to introduce functional units in astropy
>
>    What I call an ambitious goal. See comment above.

Actually, if one thinks of them as quantities with regular physical
units which just happen to have an operation applied to them (very much
like defined by FITS), it turns out not to be that hard to implement.
For a worked-through example of what I'll put forward for inclusion in
astropy shortly, see

http://www.astro.utoronto.ca/~mhvk/functional_units_magnitudes.html

> Well, I won't call them "magnitudes with a unit" but more "magnitudes with 
> a method" (e.g. AB or Vega), and of course there are also "magnitudes 
> with a band" (UVBRI JHK u*g'r'i'z' etc. etc.), and of course relative or 
> absolute magnitudes.

After a very long similar discussion, I was convinced by my astropy
colleagues that there are really two separate types of magnitudes: AB
and ST, which are *defined* in terms of physical units and are similar
to dex(cm/s^2) or dB(mW), and the magnitude systems we usually think of
(Vega, SDSS, ...) which cannot be expressed in physical units (except
through an additional calibration step).

Anyway, while the above discussion is very interesting, I do hope it
will not distract from my actual questions, the first one at least I
would hope is relatively trivial:

(1) How would I indicate a dimensionless but logarithmic quantity such
    as dex? If I understood the standard correctly, log(surface gravity)
    might have the unit "log(cm/s2)", but how about a dimensionless one
    (like metallicity).  Would it be "log()", or, by analogy with
    magnitude, just "log"?

(2) If I wanted to represent decibels, would "10*log(unit)" be
    recommended? Or "10^-1 log(unit)" to be more like the "deci" prefix
    (but which I think would be more confusing).

(3) If I wanted to represent magnitudes *with* a unit, such as AB
    magnitudes, what would be the recommended format?  I only noticed
    "mag" without units, but is "mag(unit)" allowed?  Or would one use
    "-2.5*log(unit)"? (Though this would seem to break the rule that
    scales can only be powers of 10).

(4) As a particularly gruelling example of the above, how would one
    represent AB magnitudes?  In principle, an inelegant but correct way
    might be "-2.5*log(10^(-0.4*48.6) mW/(m2*Hz))".  Would you have a
    recommendation?

With all best wishes,

Marten

p.s.  My colleage Michael Droettboom suggested just to take wcslib as
      effectively defining the standard.  I'll look into what that
      does.  It certainly has the advantage that anything we produce can
      be imported fairly widely.