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

[Tim Pearson]

I think it is stretching things too far to use the units field to indicate that the quantity is logarithmic. This information belongs in the name of the quantity (TTYPE), not the name of the unit (TUNIT). Note that what matters is the quotient TTYPE/TUNIT. You can move things between TTYPE and TUNIT so long as the quotient is unchanged. Conventionally, the units go in TUNIT, but this doesn't work for logarithmic quantities.

The following guidelines are from the Royal Society's report on "Quantities, Units, and Symbols" (1975):
"The value of a physical quantity is equal to the product of a numerical value and a unit: physical quantity = numerical value x unit. Neither any physical quantity nor the symbol used to denote it should imply a particular choice of unit. Operations on equations involving physical quantities, units, and numerical values should follow the ordinary rules of algebra. ... When numerical values of a physical quantity are tabulated, the expression to be placed at the head of a column should be a pure number, such as the quotient of the symbol expressing the physical quantity and the symbol for the units used."

In this light, a unit like "log(cm/s2)" is incorrect and meaningless: you cannot take the logarithm of a dimensional quantity (at least, it doesn't help you much to do so). 

If you are presenting a quantity like surface gravity ("g", say) in a FITS file (or indeed any sort of table) the values in the table are pure numbers: numerical value = quantity/unit, i.e., they are values of TTYPEn/TUNITn in the FITS convention. You have two options:

   Quantity			Unit	
   TTYPEn			TUNITn			
1. g				m s^{-2}
2. log[g/(m s^{-2}]		...

The second case is a dimensionless quantity, and has no units, although sometime you might want to name the units as "dex".  

Of course, FITS keywords are inconvenient for representing units with exponents, but that is another issue. [Note also that FITS discourages use of non-SI units like cm.]

The different types of magnitudes are not different units. They are different measures of flux (weighted by different bandpasses), albeit all expressed in the dimensionless logarithmic magnitude scale. So don't use TUNIT to indicate that the magnitudes are on the AB scale; use TTYPE or some other mechanism.

Similar issues arise, e.g., in labeling graphs. The axis label should be e.g., "ln(p/MPa)", not "ln(p) [MPa]".

- Tim

On Dec 9, 2013, at 6:35 AM, Marten van Kerkwijk wrote:

> (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?