<sect1 id="ai-colorandtemp">

<sect1info>

<author>
<firstname>Jasem</firstname>
<surname>Mutlaq</surname>
<affiliation><address>
<email>mutlaqja@ku.edu</email>
</address></affiliation>
</author>
</sect1info>

<title>Star Colors and Temperatures</title>

<para>
Stars appear to be exclusively white at first glance.
But if we look carefully, we can notice a range of colors: blue,
white, red, and even gold. In the winter constellation of Orion, a
beautiful contrast is seen between the red Betelgeuse at Orion's
"armpit" and the blue Bellatrix at the shoulder. What causes stars to
exhibit different color remained a mystery until two centuries ago,
when Physicists gained enough understanding of the nature of light and
the properties of matter at immensely high temperatures.
</para>

<para>
Specifically, it was the physics of
<link linkend="ai-blackbody">blackbody radiation</link> that enabled
us to understand the variation of stellar colors.  Shortly after
blackbody radiation was understood, it was noticed that the spectra of
stars look extremely similar to blackbody radiation curves of
various temperatures, ranging from a few thousand Kelvin to ~50,000
Kelvin.  The obvious conclusion is that stars are similar to
blackbodies, and that the color variation of stars is a direct
consequence of their surface temperatures.
</para>

<para>
Cool stars (i.e., Spectral Type K and M) radiate most
of their energy in the red and infrared region of the
electromagnetic spectrum and thus appear red, while hot stars (i.e.,
Spectral Type O and B) emit mostly at blue and ultra-violet
wavelengths, making them appear blue or white.
</para>

<para>
To estimate the surface temperature of a star, we can use the known 
relationship between the temperature of a blackbody, and the
wavelength of light where its spectrum peaks.  That is, as you
increase the temperature of a blackbody, the peak of its spectrum
moves to shorter (bluer) wavelengths of light.
This is illustrated in Figure 1 where the intensity of three
hypothetical stars is plotted against wavelength.  The "rainbow"
indicates the range of wavelengths that are visible to the human eye.
</para>

<mediaobject>
<imageobject>
  <imagedata fileref="graph3.png" format="PNG"/>
</imageobject>
<textobject><phrase>Figure 1</phrase></textobject>
<caption><para>The spectra of three hypothetical stars of different temperatures</para></caption>
</mediaobject>

<para>
This simple method is conceptually correct, but it cannot be used to
obtain stellar temperatures accurately, because stars are
<emphasis>not</emphasis> perfect blackbodies.  The presence of various
elements in the star's atmosphere will cause certain wavelengths of
light to be absorbed.  Because these <firstterm>absorption lines</firstterm> 
are not uniformly distributed over the spectrum, they can skew the position of
the spectral peak.  Moreover, obtaining a usable spectrum of a star
is a time-intensive process and is prohibitively difficult for large
samples of stars.
</para>

<para>
An alternative method utilizes <firstterm>photometry</firstterm> to measure 
the intensity of light
passing through different filters. Each filter allows
<emphasis>only</emphasis> a specific part of the spectrum
of light to pass through while rejecting all others. A widely used
photometric system is called the <firstterm>Johnson UBV
system</firstterm>.  It employs three bandpass filters: U
("Ultra-violet"), B ("Blue"), and V ("Visible") regions of the
electromagnatic spectrum.
</para>

<para>
The process of UBV photometry involves using light sensitive devices
(such as film or CCD cameras) and aiming a telescope at a star to
measure the intensity of light that passes through each of the
filters individually. This procedure gives three apparent
brightnesses or <firstterm>fluxes</firstterm> (amount of
energy per cm^2 per second) designated by Fu, Fb, and Fv. The ratio of
fluxes Fu/Fb and Fb/Fv is a quantitative measure of the star's
"color", and these ratios can be used to establish a temperature scale
for stars.  Generally speakiing, the larger the Fu/Fb and Fb/Fv ratios
of a star, the hotter is its surface temperature.
</para>

<para>
For example, the star Bellatrix in Orion has Fb/Fv = 1.22, indicating
that it is brighter through the B filter than through the V filter.
furthermore, its Fu/Fb ratio is 2.22, so it is brighest through the U
filter.  This indicates that the star must be very hot indeed, since
the position of its spectral peak must be somewhere in the range of
the U filter, or at an even shorter wavelength.  The surface
temperature of Bellatrix (as determined from comparing its spectrum to
detailed models that account for its absorption lines) is about 25,000
Kelvin.
</para>

<para>
We can repeat this analysis for the star Betelgeuse.  Its Fb/Fv and
Fu/Fb ratios are 0.15 and 0.18, respectively, so it is brightest
in V and dimmest in U.  So, the spectral peak of Betelgeuse must be
somewhere in the range of the V filter, or at an even longer
wavelength.  The surface temperature of Betelgeuse is only 2,400
Kelvin.
</para>

<para>
Astronomers prefer to express star colors in terms of a difference in
<link linkend="ai-magnitude">magnitudes</link>, rather than a ratio of
fluxes.  Therefore, going back to blue
Bellatrix we have a color index equal to
</para>

<para>
   B - V = -2.5 log (Fb/Fv) = -2.5 log (1.22) = -0.22,
</para>

<para>
Similarly, the color index for red Betelgeuse is
</para>

<para>
 B - V = -2.5 log (Fb/Fv) = -2.5 log (0.18) =  1.85
</para>

<para>
The color indices, like the <link linkend="ai-magnitude">magnitude
scale</link>, run backward. <emphasis>Hot and blue</emphasis>
stars have <emphasis>smaller and negative</emphasis> values of B-V
than the cooler and redder stars
as illustrated below.
</para>

<para>
An Astronomer can then use the color indices for a star, after
correcting for
<firstterm>reddening and interstellar extinction</firstterm>, to obtain 
an accurate temperature of that star.  The relationship between B-V and 
temperature is illustrated in Figure 2.
</para>

<mediaobject>
<imageobject>
  <imagedata fileref="graph2.png" />
</imageobject>
<textobject><phrase>Figure 2</phrase></textobject>
<caption><para>The relationship between a stars's B-V color index and its temperature.</para></caption>
</mediaobject>

<para>
The Sun with surface temperature of 5,800 K has a B-V index of 0.62.
</para>
</sect1>
