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\begin{document}
% \SweaveOpts{concordance=TRUE}

\title{ {\Huge Blue Flame and Green Comet} \\ {\Large being a spectral analysis of comet vapor, expounding on its relation to a commonly observed phenomenon on Earth}}
\author{Glenn Davis}
\maketitle
% \thispagestyle{fancy}

% Setup stuff.
<<setup, echo=FALSE, results="hide">>=
require("knitr",quietly=TRUE)
opts_chunk$set(fig.path="figs/ag2-", fig.align="center",
  fig.width=7, fig.height=7, comment="")
knit_hooks$set(output = function(x, options) {
  paste('\\begin{Soutput}\n', x, '\\end{Soutput}\n', sep = '')
})
options(width=90)
if(!file.exists("figs")) dir.create("figs")
@

% ----------------------------------------------------------------------------
\section*{Introduction}

In March, Judy sent me an article \cite{Bennet} about Comet 252P/LINEAR - a green
comet passing close to earth.
Green is not a common color in astronomy so I decided to investigate.
The green color comes from diatomic carbon vapor - C\textsubscript{2} - and its \textit{Swan bands}.
These wavelength bands (about 5 major ones) were discovered in a Bunsen burner flame by 
Scottish physicist William Swan \cite{Swan}, who wrote:

\begin{quote}
\begingroup
\fontfamily{pnc}\selectfont
While in the ordinary flames of coal gas and oil, solid carbon is separated, it
is well known that by burning a mixture of gas and air, the separation of carbon
may be entirely prevented, and a smokeless flame obtained. My attention was at
first accidentally directed to the subject of this paper while using a species of
gas lamp in which this object is effected in a very simple manner. 
As this lamp - the invention of Professor B\textsc{unsen} of Heidelberg - has only lately been
introduced into this country, and as I have made extensive use of it in my experiments, 
it may be proper to explain its construction.
\endgroup
\end{quote}

Swan thought that the the source of the bands was hydrocarbon molecules.
After seven decades of controversy, it was definitively determined that the source of these bands was actually 
diatomic carbon \cite{0959-5309-40-1-313}.
The color of a blue butane flame is a mixture of the colors from hydrocarbons and diatomic carbon.
I thought it would be interesting to
separate the light from hydrocarbons and the light from diatomic carbon
and to display both colors accurately.



<<packs, echo=TRUE, message=FALSE>>=
library( colorSpec )
@


%\section*{Lucid output?}

% ----------------------------------------------------------------------------

\section*{The Flame}

I found this spectrum of the blue part of a butane flame in a Wikipedia article \cite{wiki:1} on the Swan bands:

\begin{center}
\includegraphics[width=0.8\textwidth]{1200px-Spectrum_of_blue_flame_-_intensity_corrected}

Figure 1.  The spectrum of a blue flame
\end{center}

Note that there is a dividing point at about 435 nm.
Below this point the spectrum is from CH (carbyne) and above this point the spectrum is from C\textsubscript{2}.
Fortunately user Deglr6328 also posted the numerical data,
which is included in this package as \code{BlueFlame.txt}.
Here is the same data as plotted using \textbf{R}.

\setcounter{figure}{1}  
<<lee1, echo=TRUE, fig.pos="H", fig.height=3.8, out.width='1.0\\linewidth',  fig.cap='The spectrum of a blue flame'  >>=
path = system.file( "extdata/sources/BlueFlame.txt", package="colorSpec" )
blueflame = readSpectra( path, seq(375,650,0.5) )
par( omi=c(0,0,0,0), mai=c(0.6,0.7,0.3,0.2) )
plot( blueflame )
@

The flame color you see here is accurate on a display calibrated for sRGB \cite{wiki:2}.


\section*{Splitting the Flame}


Here is a simple split into low and high parts:

<<lee2, echo=TRUE, fig.pos="H", fig.height=4, out.width='1.0\\linewidth', fig.cap='The sum of the low and high parts is the original spectrum' >>=
flame.split = chop( blueflame, interval=c(432,435), adj=0.8 )
combo = bind(flame.split,blueflame)
par( omi=c(0,0,0,0), mai=c(0.6,0.7,0.3,0.2) )
plot( combo, subset=c(T,T,F) )
@

Note that the Swan bands, in the high part on the right, are a bluish-green.
The CH bands, in the low part on the left, are violet.
Here is a zoomed-in plot at the splitting interval (432 to 435 nm):
<<lee3, echo=TRUE, fig.pos="H", fig.height=4, out.width='1.0\\linewidth', fig.cap='The sum of the low and high parts is the original spectrum' >>=
par( omi=c(0,0,0,0), mai=c(0.6,0.7,0.3,0.2) )
plot( combo, subset=c(T,T,F), xlim=c(430,440) )
@

The 2 colors you see here are fairly accurate on a display calibrated for sRGB,
except that both are too vivid for sRGB and so they have been clippped.
The sum of these 2 colors is (approximately) equal to the color in Figure 2.
Later these will be displayed as large patches.


\section*{The Atmosphere}

But wait a minute !  
This blue-green color is what would be viewed from the neighborhood of the comet in space.
We are viewing it through the atmosphere, and we know that the molecules of air scatter blue light
(short wavelength) more than green light (middle wavelength), through Rayleigh scattering. 
So the comet should appear a little greener from the surface of the earth than from space.
Fortunately there is a standard, \cite{ASTM:1}, for the typical transmittance of the atmosphere:

<<lee4, echo=TRUE, fig.pos="H", fig.height=4, out.width='1.0\\linewidth', fig.cap='The transmittance of the atmosphere, for an Air Mass of 1.5' >>=
par( omi=c(0,0,0,0), mai=c(0.6,0.7,0.3,0.2) )
plot( atmosphere2003, xlim=c(375,650), ylim=c(0,1), col='black' )
@

So the blue region is attenuated more than the green region.
An \textit{Air Mass} of 1.5 means that the optical path length of the light is 1.5 times what it is at the zenith.
This means that the comet is being viewed at about 48{\textdegree} from the zenith.
I could not find any data on the transmittance at the zenith itself.
The insignificant dips in the transmittance spectrum are due to molecular absorption (probably water vapor) and not scattering.

And here is the effect of the atmosphere on the Swan bands:

<<lee5, echo=TRUE, fig.pos="H", fig.height=4, out.width='1.0\\linewidth', fig.cap='The transmittance of the atmosphere, for an Air Mass of 1.5' >>=
atmosphere = resample(atmosphere2003,wavelength(blueflame))
swan.atmo  = product( subset(flame.split,2), atmosphere )
specnames(swan.atmo) = "Swan bands, as viewed through the atmosphere"
combo = bind( blueflame, flame.split, swan.atmo )
specnames(combo) = c( "blue flame", 
                      "CH part of blue flame", "Swan bands, as viewed in space", 
                      "Swan bands, as viewed through the atmosphere")
par( omi=c(0,0,0,0), mai=c(0.6,0.7,0.3,0.2) )
plot( combo, xlim=c(375,650), ylim=c(0,0.7), subset=3:4 )
@

The second color is definitely greener than the first, and the next section will show this even better.


\section*{Colored Patches for an sRGB Display}

Here we collect the colors discussed above and display them as large rectangular patches.

<<lee6, echo=T, fig.pos="H", fig.height=4, out.width='1.0\\linewidth', fig.cap='Colors as patches - computed for an sRGB display' >>=
mat = product( combo, BT.709.RGB, wavelength='auto' )
par( omi=c(0,0,0,0), mai=c(0.2,0.2,0.2,0.2) )
plotPatchesRGB( mat, norm=T )
@

In this figure:
\begin{center}
color(blue flame) = color(CH part of blue flame) + color(Swan bands, as viewed in space)
\end{center}
The object \code{BT.709.RGB} is a built-in hypothetical camera.
Except for the blue flame itself, the colors are slightly outside the sRGB gamut.
For this reason they are slightly inaccurate when viewed
on a display calibrated to sRGB, but see the next section.


\section*{Colored Patches for an Adobe RGB Display}

Wide-gamut displays are not that common, but are becoming more so.
Most of them can be calibrated to the Adobe RGB color space  \cite{wiki:3}, 
which is probably the most popular wide-gamut RGB space.
Here are the same large patches as in the previous section, but computed for Adobe RGB.

<<lee7, echo=T, fig.pos="H", fig.height=4, out.width='1.0\\linewidth', fig.cap='Colors as patches - computed for an Adobe RGB display' >>=
mat = product( combo, Adobe.RGB, wavelength='auto' )
par( omi=c(0,0,0,0), mai=c(0.2,0.2,0.2,0.2) )
plotPatchesRGB( mat, gamma=2.2, norm=T )
@

For Adobe RGB, the "CH part of blue flame" is still too vivid (the green is negative)
but the other 3 colors are accurate.
Here is a diagram comparing the 2 color spaces, taken from \cite{wiki:3}

\begin{center}
\includegraphics[width=0.8\textwidth]{CIExy1931_AdobeRGB_vs_sRGB}

Figure 9. A comparison of the Adobe RGB (1998) and sRGB chromaticity gamuts within the CIE 1931 xy chromaticity diagram. 
The sRGB gamut is lacking in cyan-green hues.
\end{center}


\section*{Further Analysis}

Of course, real comet vapor has many more molecules than diatomic carbon.
A better analysis of comet color would include those chemical spectra in the mix.
Also notice that the blue flame spectrum has a positive baseline,
so a better analysis would find the source(s) of this baseline and possibly subtract it.




\pagebreak

\bibliographystyle{apalike}
\bibliography{blueflame}


% ----------------------------------------------------------------------------


\section*{Appendix}
This document was prepared \today \quad with the following configuration:
<<finish, echo=FALSE, results="asis">>=
knit_hooks$set(output = function(x, options) { x })
toLatex(sessionInfo(), locale=FALSE)
@


\end{document}
