# Log units (was Re: SED Data Model: Questions and Comments)

Ed Shaya edward.j.shaya.1 at gsfc.nasa.gov
Tue Feb 22 11:29:35 PST 2005

```David Berry wrote:

>Igor,
>
>
>
>>>   So you are suggesting that the IVOA specifies that no data should have
>>>logarithmic units? Anyone else got any views on this?
>>>
>>>
>>Does it mean, that stellar magnitudes will not be allowed, or I'm missing
>>something? It is impossible, astronomers probably will not agree on this!
>>
>>
>
>Magnitude is a bit different because the unit you would use is "mag", not
>log(W/m**2/Hz) (or equivalent). That is, you would not expect a simple
>unit conversion utility to be able to convert "mag" to "W/m**s/Hz". This
>would need extra metadata defining the zero point etc. So I presume Ed
>would accept "mag" as OK.
>
>
>
Well, actually no.  If you want a self-consistent system for
doing automated dimensional analysis,then magnitude should be handled as
what it really is; the logarithm of a ratio, i.e., quite unitless. It
is -2.5*logarithm of the flux compared to Vega's flux in the same band
(except for the new AB system which is already directly related to
Janskys).   Its unit is  "unitless ratio",  but stored as -2.5*log(X).
When we say something is 5 magnitudes brighter than something else, we
are saying that the ratio of brightnesses is 100.  But is magnitude here
a physical unit, or is it just an alternate expression for unitless ratios?

Now, if one considers it to be a ratio stored in logarithmic form then,
like all unitless ratios, it can be given units.  One could consider the
Vega flux in standard bands to be a unit (much like one can consider a
solar mass to be a unit).  Then one would have brightness in, say,
B-band-Zero-Flux units etc.  This unit could be in watts/cm2/Hz or
watss/cm2/um using Vega's absolute brightness and the effective width of
the band. The bolometric magnitudes would be in Bolometric-Zero-Flux (or
Open-band-Zero-Flux) units and convertable to watts/cm2.

Distance magnitudes are 5*log(D/pc)-5 which should be considered as
distance in parsecs, but stored in this weird way.  Conversions and
dimensional analysis are then simplified.

Color is, of course, a dimensional ratio of fluxes in two bands, X=F1/F2
but stored as 2.5*log(X).  Specifying that this is units of magnitude
is a tradition, yes, "but the world is changing, Poppa" (Fiddler).

The UCDs provide the info that these strange storage specification
result in that which we call a magnitude, specifically photo.B.mag or
distance.modulus or photo.color.mag, but the DM of unit should be split
into two parts, one giving the true physical units before any function
gets applied and another part describing the function to go from
physical units to the familiar form.  I suggest that this system is more
easily extensible, transparent, and universal.

Ed

```