The Interaction of Light and Matter130134In order to understand how starlight is produced we also need a number of basic ideas about the motion and energy of atoms. Perhaps the most basic of these ideas is that of temperature. . TemperatureTemperature is a measure of the average (kinetic) energy of a gas. A gas in which the atoms or molecules are moving rapidly has a higher temperature than the same kind of gas in which the atoms move sluggishly. The video clips in Table 6.2 illustrate this. Also note that the temperature is measured in units of Kelvin degrees (named after Lord Kelvin). The Kelvin temperature scale is the one most commonly used in astronomy and differs only a little from the Celsius (or centigrade) scale that you are familiar with. The Kelvin scale begins at absolute zero which is about 273 C. Room temperature is about 300 K, water freezes at 273 K and boils at 373 K.
. Temperature vs HeatThese are two easily confused ideas that can be separated by a simple question ..."Which would you rather have happen: A tiny drop of molten steel drops on your hand or an entire kettle of boiling water?" The water is "only" at 100 C while the steel may be 1500 C or higher yet the answer is simple. Neither would be pleasant but the kettle would be far more dangerous. The kettle has more heat which is a measure of the total energy of all the atoms in the system. In future discussions we will encounter situations in which we will talk about very high temperatures which may or may not involve a correspondingly high heat content.
Example 6.1 If temperature is related to average energy of motion what does this suggest about the motion of a gas at a temperature of 0 K? Solution:In classical physics the concept of temperature was directly related to the idea of atomic motion. At absolute zero it was believed all motion would cease. In quantum physics there will still be subtle "vibratory" motion but motion would be at its minimum possible. . Incandescence  The Black Body Spectrum Why do hot objects glow? This is a simple question that has a surprisingly subtle and complex answer that literally transformed physics. The answer embraces two important ideas:

Figure 6.8 The Black Body Explorer. You can use this applet to investigate the black body curve as well the idea of bandpass filters. 
Why is the Black Body Curve Important?
Our main purpose in studying the black body curve is because  to a first approximation  stars behave like black body radiators. That means that the physics of black bodies can be used to tell us a great deal about the stars. There are two very important properties of black bodies that are directly applicable here:
Wien's Law
. In this formula if T is expressed in degrees K then the wavelength will be in nanometers. The applets in both Figure 6.8 and 6.9 can be used to illustrate this idea.
Example 6.2 Use the applet in Figure 6.8 to determine the wavelength of maximum emission for a star that has a surface temperature of 10 000 K. Confirm this by using the formula and then use the applet in 6.9 to determine the colour that this star would be expected to have.
Solution The applet shows that the peak emission occurs at 290 nm. You can verify this quite easily by using the formula . Note that the values differ by a few percent. The applets use the more accurate version of Wien's Law. Using the applet in Figure 6.9 you can see that this star would have a distinctly bluish color. 

Figure 6.9 StarColor  Applet showing how the colour of a star changes with temperature. 
Example 6.3 Suppose we determine that a star emits the maximum amount of light at 700 nm. What is a good estimate for the surface temperature of the star?
Solution: Just rearrange Wien's Law to read The star is cooler than our sun.
Another useful idea relates the temperature of an object to the amount of energy it will emit. This is called Stefan's Law. In words... the intensity of light emitted by an object is proportional to the fourth power of the Temperature of that object, in symbols...
The symbol "E" denotes the energy emitted each second per square meter. The symbol 's' is called the StefanBoltzmann Constant and has a numeric value of 5.67 X 10^{8} J/m^{s}K^{4}.
Another way to express this is to use the concept of Flux which is a more specific term that combines
energy per second" as power which is measured in Watts (W) emitted per square meter. The black body applet uses Flux.
You should notice that the energy emitted by an object depends very strongly on its temperature. A hot star emits much more energy per unit area than a cool one. We can illustrate this in the next example:
Example 6.4 Stars A and B are shown in Figure 6.10. How much more energy does star B emit than star A? (Assume that the stars are identical in size).
Solution: The easiest way to approach this is to realize that star B is "3 times hotter" than star A. Since the energy emitted increases with the 4^{th} power of temperature this means that star B will emit (3 x 3 x 3 x 3) = 81 times as much energy! You can also do this with the equations if you set up ratios: Notice how big a difference there is in energy emitted. As we will see in a future chapter, big hot stars emit much, much more energy each second than smaller stars and as such, have much shorter life spans. 

Figure 6.10 
The applet Black Body Curve (provided in Figure 6.8) illustrates Stefan's Law by enabling you to shade in the area under the black body curve. This area represents the total energy emitted per square meter and calculated by Stefan's Law. To see how you could use this compare the energy emitted by the star Vega (T_{surface} = 9600 K) and our Sun (T_{surface} = 5800 K).
The red shaded region represents the light energy emitted by Vega. The green curve (much lower on the scale) shows how much less energy our Sun emits. Also  notice that the hotter star outshines the cooler star at all wavelengths. To compare the amount of energy emitted per square meter look carefully at the right hand panel of the applet. Vega emits 4.81 x 10^{8} W/m^{2} while the Sun emits 6.42 x 10^{7} W/m^{2} . If you express these as a ratio then you will see that Vega emits about 7.5 times as much energy per square meter as does our Sun.  
Figure 6.11 Area under the Black Body curve indicates the total energy emitted. This is what the StefanBoltzmann formula calculates. 
Taking a Star's Temperature  Color Indices
Figure 6.13 Black Body Explorer in "filter mode"  click on image to launch applet in separate window and then select the filters tab in the upper right corner. 
If you run the applet Black Body Explorer and select the filters tab (upper right corner) and then adjust the temperature to 3500 K you will see a series of coloured regions appear under the black body curve. These curves represent the amount of light produced by Aldebaran in each of the regions or passbands. By measuring the brightness of a star in each filter and then comparing these magnitudebrightness measures a very good approximation of the star's temperature can be made. If you measured the brightness of Aldebaran through U, B and V filters you would get the following results: 


Table 6.2 Broadband filter wavelengths 
V magnitude  0.85  Remember that the magnitude scale goes backwards! So this tells us that Aldebaran emits a lot more light in the longer wavelength part of the visible spectrum. This tells us that Aldebaran is a cool star. 
B magnitude  2.39  
U magnitude  4.29 
Solution: At 12 000 K peak emission will be around 250 nm  in the ultraviolet so you should expect that this star will be quite different than Aldebaran. The applet Black Body Explorer produces the following graph which shows clearly that the amount of light in the U region is greater than the B region which is greater than the V region. This means that the U magnitude will be smallest and the V magnitude greatest for this star.  
Color Index  An Approximate Stellar Temperature Indicator
Example 6.xx and the example of Aldebaran provide examples of how one can quickly estimate stellar temperature. Obtain the magnitude of the star in different spectral bands and compare the magnitudes by subtracting them. The difference in magnitudes is called the colour index for the star and this turns out to be a good temperature indicator. Table 6.3 summarizes this.
Colour Index 
Temperature Interpretation 
Example 
UB < 0 
Hot star (T > 10 000 K) 
Alnilam is one of the belt stars in Orion. It is very hot (25 000 K). UB = 1.03 
BV <0 
Hot star (T > 10 000 K) 
For Alnilam B  V = 0.19 
BV > 0 
Cooler stars ( T < 10 000 K) 
Arcturus is a cool and bright red giant star in the constellation Bootes. It has a BV of about 1.23 and a surface temperature of about 4300 K 
Table 6.3 Colour indices as temperature estimators 
Practice
To understand the interaction between light and matter
Chp7.2