# Dimensionless units

Ed Shaya edward.j.shaya.1 at gsfc.nasa.gov
Thu Feb 17 12:07:40 PST 2005

```Martin Hill wrote:

>Units and dimensions are not ontologies (thank goodness!)
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I agree, but units analysis should be constrained by ontologies.

>We don't expect the units to fully define what we're measuring, but they are
>an important part of that definition.  So, in this case, we don't expect um
>to tell us we have a wavelength, instead we know 'in some other way' that we
>have a wavelength value, and its units are in um.
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And, that 'some other way' is?

>The equivelence between wavelength and frequency and energy in light, say,
>requires knowing that we are dealing with photons (the equivelence is
>different for waves in beer), so we can make use of the units to simplify the
>ontology; if we know 'in some other way' that we have photons, then the units
>can be length, time-1 or energy.
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The speed of light c is a function of the index of refraction so your
automated converter between frequency and wavelength needs to know the
index of refraction as well.  So, we get into problems of c in air or c
in space.  In the radio the electron density can make a difference but
let's assume a vacuum in space.

>Similarly the difference between surface flux and observed flux need to be
>described, but is a problem for ontologies not units or dimensions.
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Yes.  So long as there is awareness of the ontological class by the
units converter.

>Units and dimensions allow us to convert and compare things that we know 'in
>some other way' are the same. These need not be simple comparisons; Pedro has
>demonstrated that you can build an SED from dimension analysis, because you
>already know which values go on the 'brightness' axis and which go on the
>'wavelength' axis, even if the units are all different.
>
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If I put an rate of change of energy spatial density vs wavelength in
an SED with units w/um^3.   Will it get mixed up with a Flux density of
w/um^2/um?
On the other hand will nuFnu be allowed to convert to Fnu, as it well
can be, even though they are different UCDs.

Also, I now understand that we are talking about dimensional analysis in
terms of
km/THz = 1e3 m * 1e12 s
Not dimensional analysis in terms of
km/THz = L*T.
There are two types of dimensional analyses.  One (the real one) is
useful to determine whether an equation is roughly correct by checking
the dimensions (M, L, T,...) on both sides of the equal sign.  The other
is really a units analysis because it is doing arithmetic with the units
(kg, m, s..) not just the dimensions.  This one needs to know the
conversions between kg and pounds and milligrams, the other does not.

Ed

>Cheers
>
>Martin
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>On Thursday 17 February 2005 6:33 pm, Ed Shaya wrote:
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>>Pedro,
>>
>>    If you are given units of w/cm^2/um then there is no way for a
>>computer to know that the um refers to wavelength.  A human can decide
>>based on complex reasonable judgement and experiences but it is always
>>just a guess.  If I have light emanating from  boxes that are 1 cm in
>>height and width and 1 um in depth, then I have the right to express
>>this as w/cm^2/um.  Note that this is dimensionally watts/volume.  In
>>this case, we would not want a conversion to Janskys with the assumption
>>that the um is a wavelength.  You need a system in which the physical
>>objects within the units are revealed, such as Light Energy
>>Flux/wavelength or Light Energy Flux/Frequency where Flux is already
>>defined to be Quantity Flow/Area/Time.  This is to separate it from
>>Quantity Change/Area/Time because you do not want to mix up quantity
>>sources with quantity motion.   There is also possible confusion between
>>Flux at the observer and Flux at the surface of the source, so area
>>needs to be more specific.
>>   Likewise with the problem of Jy/arcsec^2, this needs to be described
>>as Light Energy Flux/Frequency/angle^2.
>>    Now, maybe the idea is that the UCD or a more precise class of the
>>quantity has this detailed description of what it is and then one
>>proceeds to do conversions between two quantities of the same class.
>>Any  automated tool will need to understand the class as described here
>>to know how to go about a conversion.  After breaking things down to
>>just dimensionality, some of the information needed to do the
>>comparisons and numerical conversions is lost.  Dimensional analysis is
>>what you do when you are ignoring amplitude and want a sanity check that
>>unit agree.
>>   Does anyone know the origin of the expression "Nothing is simple"?
>>
>>Ed
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