# Dimensionless units

Martin Hill mchill at dial.pipex.com
Fri Feb 18 08:58:13 PST 2005

```Anita Richards wrote:
>>I suspect that for Jy/sr in the above example we should be using length^2
>>stradians and arcmins *are* just scaling factors, ie ratios.  In going from
>>Jy/sr to Jy  you are implying that you want the value for a fixed proportion
>>of a sphere (depending on the scaling factor used for stradians) but you
>>haven't yet defined the projection from the sphere. If my geometry is right!
>>ie we are back to defining a description for the value, not its units.
>
> How would length^2 solve anything, since in most cases you will have
> catalogues of objects at different distances, so that Jy/arcsec^2 or Jy/Sr
> cannot be quantified meaningfully in length units without additional
> information about distance to the object, and in some cases e.g. CMB it is
> meaningless?

Quite right.  I was thinking in the general terms of a surface area on a
sphere (eg a cone), and that such surface areas shouldn't really be
thought of as dimensions until we've projected it onto something at a
known distance - which gives us our length dimension. Sometimes we use
polar coordinates for objects on pixel images, and again we project our
coordinates onto that plane and voila, you have length^2 again.

So indeed once you have a distance to an object, you have a projected
width and height on it.  If we had 'Angle' in the dimension equation the
dimension equations wouldn't match.

> So we do need to have something like 'angle' as a
> dimension in order to recognise that Jy/Sr != Jy .

As Ed has just pointed out, we don't use dimension analaysis to
recognise that Jy/sr != Jy; that's a different job.  I was getting a bit
carried away in saying that from a certain point of view, Jy *is* equal
to Jy/sr, as long as you know what you're projecting.  But I think that
might be a bit silly.

--
Martin Hill
AstroGrid Software Engineer @ ROE
07901 55 24 66

```