dimensionless units

[Jonathan McDowell]

I think that Martin and Ed have made some very good points
about the subtleties of unit and dimensional analyses, and I'm
going to repeat an old suggestion of mine in the context of this
discussion.

Part of the problem is that Pedro's approach makes total sense within
the context of photon spectra of sources, where things can be
interconverted using some combination of (1) unit analysis and (2) the
dispersion relation nu * lambda = c. But it doesn't extend so easily to
general astronomical data where, as has been pointed out, the UCD and
possibly other metadata may be needed to do a conversion or to determine
whether a conversion makes scientific sense.

I think in the DM context it's important to distinguish between
conversions which are simple scaling  (pc to AU to km) and conversions
which in principle require a UCD (even though you can get away without
one if you are in the spectral context and using nu * lambda = c).
The UCD based conversions are domain and context specific and 
should be in software associated with particular contexts (e.g. a spectral
packag) while the basic units scaling conversion is general and should be in
general software.

So to me the question is: how can we support the very useful functionality
of Pedro's tool in the spectral domain without boxing ourselves into
a corner that won't generalize well?

I've argued with Pedro in the past about this distinction between
dimensional analysis as - I think it was Ed said this? - the answer to
the question 'are these things compatible' with no numeric factor, and
unit analysis where the numerical factors are included. 

I would restate
some of the recent discussion as alleging that Pedro's scheme is
really unit analysis dressed up in the clothes of dimensional analysis,
and I assert that if you restate that scheme as "reduce all units to
their unprefixed SI base unit representation", so that

    mJy ->  10^-23 kg s^-2
    10^-11 erg cm^-2 s^-1 keV^-1   -> 6.215 10^1 m-2 s-1 

and then rename  kg = "M", s = "T", m = "L" so that it's  1.E-23 MT-2 and 6.215E+01 L-2T-1
you have exactly Pedro's scheme. Given that, why not skip the
renaming. If we require (possibly as additional metadata) that
an SI base representation of the units be provided

  ---  Here we have the units as they appear in the relevant ApJ paper ---
    UNITSCAL  = 1.0E-11
    UNIT      = 'erg cm^-2 s^-1 keV^-1'
  ---  Here we have the standardized units for simple software ---
    SISCAL = 62.15 
    SIUNIT ='m^-2 s^-1'

that's really trivial for Pedro's software to parse into his L-2T-1 format, but
it can also be used by unit analysis software. I'm just saying, even if we
have redundant metadata (unit and sifac/siunit) we don't need redundant
DIFFERENT string syntaxes. If we're going to have a string syntax for
the "dimensional" analysis, why not use the SAME syntax that we use
for general unit analysis, since then other people's clever unit software can handle
either UNITSCAL/UNIT or SISCAL/SIUNIT with the same parser.

If clever unit software becomes widespread and standardized we maybe eventually
can drop the SI.. version. 

 - Jonathan