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   pdfauthor = {Jan\ Hollan},
   pdftitle  = {Magnitude?...},
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\title{\vspace*{-2cm}\bfseries Magnitude\dots?}
\author{Jan Hollan}
\date{1998--09--01}

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\noindent
 Among several ``obtrusive medieval features'' in astronomy there is one,
 which is simple to solve, but difficult to obey the chosen solution. I mean
 the ridiculous way of indicating, how bright the stars are. For
 two thousand years the stars had been just sorted into several classes.
 Only recently, in the 19th century, this procedure was somehow defined
 and refined. So, we have not just stars of \emph{first brightness class},
 \emph{second brightness class}, etc., but we can use
 decimal numbers as well. However, we do it
 rather silly.

 The problem is not that we would not get accurate results. Similarly,
 as in case of positions of celestial bodies, we only keep secret, what
 we really mean. What do our numbers mean. This secrecy is useless, and
 I am sure that it is harmful. No wonder that we are misidentified with
 astrologists. What we miss to be taken for physicists?

 In the astronomical photometry just a little. The information, how bright
 a star is must be expressed by a value of a physical quantity. And to this
 quantity, some unit of measurement should belong.
 %We call that quantity,
 %respecting the ancient sorting of stars, \emph{faintness}, and its unit
 %\emph{magnitude}.
 For historic reasons, this unit is called \emph{magnitude}. But to prevent
 confusion, the quantity itself must have another name. We propose to use
 the term \emph{faintness} when referring to this physical quantity.
 (English linguists may replace it by a more suitable word.)
 To be able to define the quantity easily, we need one
 more quantity, \emph{brightness} (its unit being lumen per square meter).
 Then it is sufficient to give brightness corresponding to the
 faintness of zero magnitudes (approximately to that of the star Vega), and
 to define that
 stars which are five magnitudes fainter (i.e., with faintness of five
 magnitudes) are hundred times fainter (i.e., their brightness is just
 0.01 of the brightness of Vega). Difference of 2.5\,mag means the ratio of
 brightnesses of 10, stars 7.5\,mag faint illuminate us thousand times less
 than Vega. Faintness is a dimensionless and a logarithmic quantity. Some
 quantities in acoustics are of similar type, what is not strange:
 they also try to match the rules of human perception.

 Nowadays photometry is seldom based on human vision. Therefore many
 similar quantities have been used,
 each applying to some filter used for observation of
 visible and unvisible radiation from the universe. Transmissivity curve of
 one of them (of the V filter) is similar to the sensitivity curve of
 human eye (hence V for Visual). Almost all faintnesses are defined in such a
 way that Vega is approximately zero magnitudes faint (for setting the
 base level more accurately, a set of stars is used). There is just one
 notable exception, a ``filter-less'' 
 \emph{bolometric faintness}\footnote{Bolometric faintness is the first
quantity in astronomical "photometry" (radiometry, in fact),
which can be attached to SI without problems. A star 
which is 10 parsecs distant has bolometric faintness 0 magnitudes, when it
is an isotropic source of radiant flux of 3.055$\cdot$10$^{28}$\,W. This
value proposed by Roger Cayrel has been adopted at the assembly of
the International Astronomical Union in 1997\cite{Kyoto}. However, the
quantity is called magnitude, and it has no unit in the resolution from
1997.}
 (encompassing the whole electromagnetic spectrum), where Vega has some
 $-0.3$\,mag. Thanks to this convention one needs not much care of
 which filter has been used indeed when reading popular texts. If e.g.
 the text reads that ``the minor
 planet was of \emph{fifteenth brightness class}'', i.e., that its faintness
 was approximately fifteen magnitudes, one understands
 that it was some one million
 times fainter than Vega, or than a fixed star of zero-th brightness class
 (fifteen is three times five, and the third power of one hundred is
 one million).

 It is a bit queer language, but it is not vague. Unfortunately, it becomes
 vague in common texts. Words \emph{brightness class, brightness,
 faintness} and \emph{magnitude} have been arbitrarily interchanged.
 The last one is
 sometimes used not for a unit, but for a quantity. Moreover, it has been
 often combined with an ordinal numeral.
 However, an ordinal numeral suits just to the ancient way
 of sorting into brightness classes.

 The most substantial terminological problems of this kind occur naturally
 in some Romanic languages. Those brightness classes have
 been traditionally called ``magnitudes'', i.e., sizes,
 and it has been well understood in these languages.
 Unfortunately, the same situation has developed in English.
 Other languages can avoid
 confusing name of the unit for the name of the class simply by using their own
 word for ``size''. ``Magnitude'' is foreign word for them, and is not
 felt as a synonym for size. E.g., in German, the brightness class is
 called \emph{Gr\"o\ss{}e}, in Czech it is \emph{velikost}. Both these
 languages have one more advantage over English: they have a standard
 (maybe felt a bit obsolete, but all the more traditional) name for the
 faintness.
 In German it is \emph{Sterngr\"o\ss{}e}, in Czech \emph{hv\v{e}zdn\'a
 velikost}. Or is it a disadvantage? Faintness rises, when the star gets
 fainter, and everybody would expect it to do so. It is far from being the
 case for Sterngr\"o\ss{}e.

 I insist that \emph{magnitude} should never be used as a name of the
 quantity. Using it as a synonym for a brightness class should be mostly
 avoided. If one does not like saying ``a star of the third brightness
 class'', one can say ``the star has three magnitudes'' or ``the star is some 3
 mag'' instead.

 Most astronomers are vulnerable to use confusing language when speaking
 about stellar photometry, as they have been educated that way. It takes
 some time to get rid of it, even if one tries hard. Even me, an author of
 this proposal of transparent terminology, lasted years, to adhere to it
 completely and to avoid most mistakes.


\section*{Why?}

 Maybe you have your own system for speaking about how bright the
 stars are, and you believe it to be coherent. I doubt you are able to keep it
 coherent indeed, if it differs substantially from the above-mentioned one.
 It is really
 difficult to do without four terms: the one for the linear quantity (which we
 call brightness), the others for the dimensionless logarithmic quantity
 (we call it
 faintness), for its unit (we call it magnitude) and finally for the
 ancient way of speaking about stars (if its using cannot be avoided, we
 call it brightness class).

 Even four distinct terms are not entirely satisfactory, as there are more
 linear quantities differing in their dimensions: the old one is
 photometric (based on \emph{luminous intensity} as a basic SI\cite{SI}
 quantity and on \emph{candela} as its unit)\footnote{It may be not quite
 clear whether
 photopic (daytime) or scotopic (dark-adapted) vision and units are more
 appropriate.}, and all the new ones are energetic (having nothing to do
 with candelas and lumens, but based just on length, mass and time, and
 expressed in \emph{watts per square metre})\footnote{%
 Astronomers often misuse word \emph{luminosity} for (radiant) output of a
 star. However, the word \emph{luminosity} suggests to be a short for
 \emph{luminous intensity} (in Czech, e.g., its translation
 \emph{sv\'{\i}tivost} is indeed the only name for luminous intensity).
 Confusion with a basic quantity of the SI is then easily possible. All the
 worse, electromagnetic radiation of stars is not just light, what
 \emph{luminosity} suggests. It is so easy and clear to say simply
 \emph{output} when one means radiant flux from a given astronomical object.

 The worst term used by astronomers is, however, ``surface brightness''.
 The quantity that they understand by it is
 \emph{brightness per solid angle}, which
 is the same as \emph{luminance} or rather \emph{radiance} through a given
 filter. It should be measured in candelas per square metre, or in watts
 per steradian and square metre. If one wishes to express it in a
 logarithmic way, it is quite correct to say, e.g., that ``one square
 second of this nebula is 20 mag faint''. One can table faintness of a
 given solid angle in a list of astronomical objects
 to express their luminances
 indirectly. But no sound quantity can possibly have a unit of ``magnitude
 \emph{per} square second''.

 Two simple programmes, which convert logarithmic and linear photometric and
 radiometric quantities, are available at
 \wwwo{http://astro.sci.muni.cz/pub/hollan/programmes/photometry}.
 }

 The most common habit is that there is no unit at all. Magnitude is used
 then as a denomination of the quantity. Having no unit poses large stylistic
 problems, and it is difficult to express small differences of the
 discussed quantity. In my proposal, the star can brighten by three
 millimagnitudes, 3~mmag. Try to examine your texts, whether the word
 \emph{magnitude} does not mean sometimes a unit, in spite the author trying to
 avoid that. If you find sush phrases (what I consider probable),
 it will be the same situation as measuring length in
 ``lengths'' (not horse lengths, just in lengths). Let's have an example from
 a magnificient textbook \emph{An Introduction to Modern
 Astrophysics}\cite{MA}:
 \begin{quote}
  Using sensitive instruments called \textbf{photometers},
  astronomers can measure the apparent magnitude of an object with an
  accuracy of $\pm0.01$ magnitude, and \emph{differences} in magnitudes with
  an accuracy of $\pm0.002$ magnitude.
 \end{quote}
 In spite of the book being quite clear and rigorous as far as pure physics
 of the universe is concerned, here the vague astronomical heritage took
 over. Isn't magnitude a unit in the quoted sentence? If not, do the
 authors really mean that accuracy is one per cent of ``the quantity named
 magnitude''? For a star ``of magnitude 1'' the uncertainty would be then
 0.01 (of no unit), for a star ``of magnitude 10'' it would be 0.1, and for
 a star ``of magnitude 0'' it would be none at all! No, they use the same
 word both for the quantity and for its unit (and for the ancient
 brightness class as well, elsewhere).

 Another system is using \emph{brightness} as a name for the quantity that I
 call \emph{faintness}. Definition of that quantity becomes then more
 difficult. But the main defect of such a system is the silly behaviour of
 the quantity: when the star grows brighter, its ``brightness'' goes smaller.

 If you know some other terminology which remains always coherent and is
 not confusing, I would be obliged to you if you get in touch with me.

\begin{thebibliography}{9}

 \bibitem{JH}
 More information from the author
 is currently available only in Czech. See \\
 \www{http://astro.sci.muni.cz/pub/hollan/a\_papers/}
     {http://astro.sci.muni.cz/pub/hollan/a_papers/}

 \bibitem{MA}
 Carrol, B. W., Ostlie, D. A.: An Introduction to Modern Astrohysics.
 Adison-Wesley Publ. Comp, Inc., 1996. ISBN 0-201-54730-9. \\
 \wwwo{http://astrophysics.weber.edu}

 \bibitem{SI}
See, e.g.,
\wwwo{http://physics.nist.gov/cuu/} \\
 or \wwwo{http://www.bipm.fr}.

\bibitem{Kyoto}
Andersen, J. (ed.): Transactions of the IAU, vol. XXIIIB,
\emph{Proceedings of the 23rd General Assembly}, 
p. 141 and 181.
Kluwer Academic Press, 1999.
ISBN 0-7923-5588-1.

\end{thebibliography}


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 The author thanks to his colleague Jind\v{r}ich \v{S}ilh\'an for much
 improving the language of the paper.

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