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0001 <sect1 id="ai-colorandtemp">
0003 <sect1info>
0005 <author>
0006 <firstname>Jasem</firstname>
0007 <surname>Mutlaq</surname>
0008 <affiliation><address>
0009 </address></affiliation>
0010 </author>
0011 </sect1info>
0013 <title>Star Colors and Temperatures</title>
0014 <indexterm><primary>Star Colors and Temperatures</primary>
0015 <seealso>Blackbody Radiation</seealso>
0016 <seealso>Magnitude Scale</seealso>
0017 </indexterm>
0019 <para>
0020 Stars appear to be exclusively white at first glance.
0021 But if we look carefully, we can notice a range of colors: blue,
0022 white, red, and even gold. In the winter constellation of Orion, a
0023 beautiful contrast is seen between the red Betelgeuse at Orion's
0024 "armpit" and the blue Bellatrix at the shoulder. What causes stars to
0025 exhibit different colors remained a mystery until two centuries ago,
0026 when Physicists gained enough understanding of the nature of light and
0027 the properties of matter at immensely high temperatures.
0028 </para>
0030 <para>
0031 Specifically, it was the physics of
0032 <link linkend="ai-blackbody">blackbody radiation</link> that enabled
0033 us to understand the variation of stellar colors.  Shortly after
0034 blackbody radiation was understood, it was noticed that the spectra of
0035 stars look extremely similar to blackbody radiation curves of
0036 various temperatures, ranging from a few thousand Kelvin to ~50,000
0037 Kelvin.  The obvious conclusion is that stars are similar to
0038 blackbodies, and that the color variation of stars is a direct
0039 consequence of their surface temperatures.
0040 </para>
0042 <para>
0043 Cool stars (i.e., Spectral Type K and M) radiate most
0044 of their energy in the red and infrared region of the
0045 electromagnetic spectrum and thus appear red, while hot stars (i.e.,
0046 Spectral Type O and B) emit mostly at blue and ultra-violet
0047 wavelengths, making them appear blue or white.
0048 </para>
0050 <para>
0051 To estimate the surface temperature of a star, we can use the known
0052 relationship between the temperature of a blackbody, and the
0053 wavelength of light where its spectrum peaks.  That is, as you
0054 increase the temperature of a blackbody, the peak of its spectrum
0055 moves to shorter (bluer) wavelengths of light.
0056 This is illustrated in Figure 1 where the intensity of three
0057 hypothetical stars is plotted against wavelength.  The "rainbow"
0058 indicates the range of wavelengths that are visible to the human eye.
0059 </para>
0061 <para>
0062 <mediaobject>
0063 <imageobject>
0064   <imagedata fileref="star_colors.png" format="PNG"/>
0065 </imageobject>
0066 <caption><para><phrase>Figure 1</phrase></para></caption>
0067 </mediaobject>
0068 </para>
0070 <para>
0071 This simple method is conceptually correct, but it cannot be used to
0072 obtain stellar temperatures accurately, because stars are
0073 <emphasis>not</emphasis> perfect blackbodies.  The presence of various
0074 elements in the star's atmosphere will cause certain wavelengths of
0075 light to be absorbed.  Because these absorption lines are not uniformly
0076 distributed over the spectrum, they can skew the position of
0077 the spectral peak.
0078 Moreover, obtaining a usable spectrum of a star
0079 is a time-intensive process and is prohibitively inefficient for large
0080 samples of stars.
0081 </para>
0083 <para>
0084 An alternative method utilizes photometry to measure the intensity of
0085 light
0086 passing through different filters. Each filter allows
0087 <emphasis>only</emphasis> a specific part of the spectrum
0088 of light to pass through while rejecting all others. A widely used
0089 photometric system is called the <firstterm>Johnson UBV
0090 system</firstterm>.  It employs three bandpass filters: U
0091 ("Ultra-violet"), B ("Blue"), and V ("Visible"); each occupying different regions of the
0092 electromagnetic spectrum.
0093 </para>
0095 <para>
0096 The process of UBV photometry involves using light sensitive devices
0097 (such as film or CCD cameras) and aiming a telescope at a star to
0098 measure the intensity of light that passes through each of the
0099 filters individually. This procedure gives three apparent
0100 brightnesses or <link linkend="ai-flux">fluxes</link> (amount of
0101 energy per cm<superscript>2</superscript> per second) designated by Fu, Fb, and Fv. The ratio of
0102 fluxes Fu/Fb and Fb/Fv is a quantitative measure of the star's
0103 "color", and these ratios can be used to establish a temperature scale
0104 for stars.  Generally speaking, the larger the Fu/Fb and Fb/Fv ratios
0105 of a star, the hotter its surface temperature.
0106 </para>
0108 <para>
0109 For example, the star Bellatrix in Orion has Fb/Fv = 1.22, indicating
0110 that it is brighter through the B filter than through the V filter.
0111 furthermore, its Fu/Fb ratio is 2.22, so it is brightest through the U
0112 filter.  This indicates that the star must be very hot indeed, since
0113 the position of its spectral peak must be somewhere in the range of
0114 the U filter, or at an even shorter wavelength.  The surface
0115 temperature of Bellatrix (as determined from comparing its spectrum to
0116 detailed models that account for its absorption lines) is about 25,000
0117 Kelvin.
0118 </para>
0120 <para>
0121 We can repeat this analysis for the star Betelgeuse.  Its Fb/Fv and
0122 Fu/Fb ratios are 0.15 and 0.18, respectively, so it is brightest
0123 in V and dimmest in U.  So, the spectral peak of Betelgeuse must be
0124 somewhere in the range of the V filter, or at an even longer
0125 wavelength.  The surface temperature of Betelgeuse is only 2,400
0126 Kelvin.
0127 </para>
0129 <para>
0130 Astronomers prefer to express star colors in terms of a difference in
0131 <link linkend="ai-magnitude">magnitudes</link>, rather than a ratio of
0132 <link linkend="ai-flux">fluxes</link>.  Therefore, going back to blue
0133 Bellatrix we have a color index equal to
0134 </para>
0136 <para>
0137    B - V = -2.5 log (Fb/Fv) = -2.5 log (1.22) = -0.22,
0138 </para>
0140 <para>
0141 Similarly, the color index for red Betelgeuse is
0142 </para>
0144 <para>
0145  B - V = -2.5 log (Fb/Fv) = -2.5 log (0.18) =  1.85
0146 </para>
0148 <para>
0149 The color indices, like the <link linkend="ai-magnitude">magnitude
0150 scale</link>, run backward. <emphasis>Hot and blue</emphasis>
0151 stars have <emphasis>smaller and negative</emphasis> values of B-V
0152 than the cooler and redder stars.
0153 </para>
0155 <para>
0156 An Astronomer can then use the color indices for a star, after
0157 correcting for reddening and interstellar extinction, to obtain an accurate temperature of that star.
0158 The relationship between B-V and temperature is illustrated in Figure
0159 2.
0160 </para>
0162 <para>
0163 <mediaobject>
0164 <imageobject>
0165   <imagedata fileref="color_indices.png" />
0166 </imageobject>
0167 <caption><para><phrase>Figure 2</phrase></para></caption>
0168 </mediaobject>
0169 </para>
0171 <para>
0172 The Sun with surface temperature of 5,800 K has a B-V index of 0.62.
0173 </para>
0174 </sect1>