Dimensionless units

[Ed Shaya]

David and Martin

Again, we are getting the two types of dimensional analyses confused.
Unit Analysis specifies the conversion fully:

Jy/sr *  sr/[(pi/4)deg]^2  * [deg/(60 arcmin)]^2 = [4*60/pi]^2 
Jy/arcmin2 or something like that.

This type of analysis should almost always work and be valid and easily 
automated.  I can think of some oddball occassions, like using redshift 
for distances to planets etc, that would be a mistake.   For conversion 
purposes, we should add radian to the basic SI units: meter, kilogram, 
second, candela, ampere, kelvin, mole.  Since mole is not terribly 
useful in astronomy, we could switch number count for mole.  Then one 
can convert anything to these base units and do comparisons on a common 
basis.  I think this is part of  Pedro's system and it is also the basis 
of what Brian and I wrote a schema for many years ago.

The base units are not the dimensions.   Don't get them confused.  A 
mole is dimensionless, but one can still do unit analysis with moles!

The dimensional analysis where one uses just the dimensions doesn't tell 
you anything about how to
go from Jy/sr to Jy/arcmin**2 other than to tell you that it is not 
dimensionally wrong.  Pedro's system does use real dimensional analysis 
for spectra by taking the ratio of the dimensions of two quantities and 
if the ratio is L/T then it assumes that a velocity can be used to do 
the conversion and since spectra are EM waves, the velocity is probably 
the speed of light.  

I wonder, is a SED only allowed to be spectra?  If I have Fnu/pc (flux 
along a disk) and Fnu/sec (rate of change of the flux), are these 
allowed in a SED?  If so then we will need to rely on UCD or ontology to 
stop Pedro's system from doing a dimensional analysis and converting one 
to the other.  I am sure he is aware of this. 

EdDavid 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.
>>    
>>
>
>But the scaling factor as I understand it is just the ratio of the units
>you want, to the SI units implied by the dimensional analysis. That is,
>The DIMEQ for "Jy" is "MT-2", implying SI units of kg/s/s, and the
>corresponding SCALEQ value of 1.0E-26 means that 1 Jy is 1.0E-26 kg/s/s/.
>But since angle does not enter into the dimensional analysis, neither will
>it enter into the scaling factor. So I would expect Jy/sr and Jy/arcmin**2
>to have the same scale factor.
>
>  
>
>>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.
>>    
>>
>
>Is this progress, given that a straightforward string parsing system could
>tell you this?
>
>David
>
>
>  
>