# Dimensionless units

Martin Hill mch at roe.ac.uk
Fri Feb 18 07:20:50 PST 2005

```On Friday 18 February 2005 1:00 pm, David Berry wrote:
> Martin,
>         It would seem reasonable that a unit conversion scheme should be
> able to determine the conversion from Jy/sr to Jy/arcmin**2 without needing
> UCDs. The conversion is obviously just a simple scaling factor - no need
> for any extra physics or UCDs or anything. But since these two units have
> the same MLT dimensions, how can the conversion be determined on the basis
> of a dimensional analysis alone? Introducing Angle as a dimension along
> with Mass, Length and Time could do it.

Hi David

I understood that a scaling factor was part of the dimension-based unit
converters (we can't do any unit conversions otherwise?)  So converting
from Jy/sr to a dimension equation will give the same result as converting
Jy/arcmin^2 to a dimension equation, as the scaling values used for radians
and arcmins will be different.

However you can, in principal, ask a dimension-based unit converter to
convert from Jy/sr to Jy, and it will not be able to tell you that this is,
at first glance, just silly.

So it is tempting to add Angle to the dimension set, but I think it will
introduce more problems than it solves.  For example, I seem to remember that
a Parsec is defined as a distance (1 AU) multiplied by an angle resulting in
a distance. Angle doesn't exist in dimension analysis *because* it's a ratio.

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.

Let us stick to proper, accepted dimension analysis for doing dimension-based
unit conversions.  I suspect every physics school student is familiar with
the benefits and pitfalls (perhaps we should get one to contribute!) and
therefore it should be straightforward for anyone else to understand and use
them instead of some special-to-IVOA complication.

We will need other conversion algorithms too, just not not dimension-based.

Cheers

Martin

--
Martin Hill
Astrogrid/AVO, ROE
Tel: 07901 55 24 66

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