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

David Berry dsb at ast.man.ac.uk
Sat Feb 19 02:26:34 PST 2005

```Ed,
So you are suggesting that the IVOA specifies that no data should have
logarithmic units? Anyone else got any views on this?

Obviously my example of log(Jy)/sr was a bit contrived, but what about
spectra in which the spectral position is given in units of log(Hz)?

David

On Fri, 18 Feb 2005, Ed Shaya wrote:

> David Berry wrote:
>
> >AST can determine and evaluate mappings between any two equivalent
> >units strings based on FITS-WCS paper I, with a range of tolerated
> >variations (cm2->cm^2, micron->um, etc). This includes the log, ln, and
> >exp functions which FITS-WCS paper I specified. But I cannot currently
> >see how to do a dimensional analysis of say "log(Jy)/sr". Anyone got any
> >ideas?
> >
> >
>
> I don't think you should try to handle log(Jy)/sr.  Nor should you
> handle [Jy + 7]/sr.  These both break the rules of a physical formula.
>    During scientific manipulations you should only take log, exponents,
> or trigonometric functions of unitless variables.  If you have exp(kt),
> k needs to have dimension 1/T.     One sees the seriousness of taking
> the logarithm of a unit when you look at the expansion:
> ln(x) = (x-1)/x + 1/2[(x-1)/x]^2 + 1/3[(x-1)/x]^3 ...
> There is just no defineable resulting dimension to this.
>
> Now, log(Jy)/sr is particularly insidious, because you can not recover
> back to Jy/sr unless you know what number of sr was used to derive this
> value.  If you took 10 Jy, and take the log(Jy)=1 and divided by 1 sr,
> then to undo you just exponentiate.  But you took 15 Jy and tak
> log(Jy)=1.17, and divide by 1.5 sr you get 0.784.  So log(15Jy)/1.5 sr
> is not the same as log(10Jy)/1sr.  So what does it mean physically, nothing.
>
> So, you ask, what about  the thousands of tables and plots with
> log(whatever)  in them?  It is fine for display or as a compact form to
> store or show anything after you take its log.  But, if you are going to
> do any further processing of it, you need to undo it (exponentiate).
>
> Think about FITS integer format, it takes the values and adds a constant
> and then multiplies by a constant.  You can't do dimensional analysis
> with these numbers because, what are the units of (Jy + 7)?
>
> Bottom line, <units name="Jy/sr"  factor="1e7"
> storageAlgorighm="log(\$v)+17">
>
> I hope that helps,
> Ed
>
>
>

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