Blackbody Radiation
Temperature Dependence of the Light Output vs. Wavelength Graph
Look at the Blackbody Curve simulation on the course website under the section “Activity 2-2: Investigating Blackbody Radiation.” This shows a graph of light output vs. wavelength for an object. The slider at the right allows you to control the object’s temperature. The number at the top of the slider is the temperature in Kelvin. Move the slider up and down, and observe what happens to the graph of light output vs. wavelength. You can also click on the temperature number and directly type in the temperature you want.
The horizontal axis represents the wavelength of light in units of micrometers (mm). Bands of colors (the rainbow on the graph) show where the wavelengths of visible light are. Wavelengths just shorter than purple are ultraviolet light, and wavelengths just longer than red are infrared light. The smallest wavelength shown on the graph is 0 mm and the largest wavelength displayed is 3 mm which is in the infrared portion of the electromagnetic spectrum.
Above the graph are three colored circles. The three circles give an indication of how much blue(B), green(G), and red(R) light is being emitted by the object. The star shaped object to the right of these circles shows what color the object would actually appear to the human eye given the fact that all of the emitted colors would be mixed together. Play with this a little bit to see how it works.
Set the temperature to be 3500 K. The vertical axis of the graph represents the intensity of the light. Note that the vertical axis currently goes from 0 to 100. Also, note that at this temperature, there is a greater intensity of red light than blue light being given off by the object. You can see this in two ways. First, the graph (red line) is higher in the red light portion of the spectrum than in the blue light part of the spectrum. So, the intensity of red light being emitted is greater than the intensity of blue light being emitted. Also, if you look at the three circles (BGR), you’ll notice that the red circle is brighter than the blue circle. As a result, when all of these colors are mixed together, they would appear slightly reddish to your eye (color of the star to the right of the BGR cirlces).
You can see the graph a little better if you change the vertical scale. At the top of the vertical axis, you should see “zoom in (+)” and “zoom out (-)” magnifying glasses. Click the (+) magnifying glass twice until the vertical axis goes from 0 to 10 instead of 0 to 100. Now, the graph is easier to see since it fills up the screen instead of being compressed near the bottom of the graph. There are also zoom tools for the horizontal axis. You will use these in the activity below when the graph becomes too difficult to see.
1) Set the slider to the temperature of the human body (about 300 Kelvin). On the vertical axis, click the (+) zoom tool till the max value is 0.00003. On the horizontal axis, click the (-) zoom tool till the max value is 24. What wavelengths of light is the object emitting (note that the visible colors of light are far to the left)?
Would you be able to see any of the light this 300 K object emits? (HINT: Look at the BGR circles and the star at the top of the simulation. How much visible light is there?) Would you be able to see the emitted light with an infrared camera?
2) Now increase the object’s temperature to 1500 Kelvin. On the vertical axis, click the (-) zoom tool till the max value is 0.1. On the horizontal axis, click the (+) zoom tool till the max value is 3. What wavelengths of light are emitted by the object now?
How does the amount of infrared light emitted compare to the amount of visible red light emitted?
Note that the star at the top next to the BGR circles is slightly reddish in color. Explain why the object appears red in color.
3) Now increase the object’s temperature to 5000 Kelvin and adjust the vertical and horizontal scales so that you can see the graph well. This is about the temperature of a light bulb filament. What wavelengths of light are emitted by the object now?
Explain why the object appears white in color.
4) Now increase the object’s temperature to 10,000 Kelvin and adjust your vertical and horizontal axes. Explain why the object appears blue in color.
Notice that as the temperature increases, the appearance of the object goes from red to orange, then to yellow, then to white, and finally to blue. You can use your mouse to move the temperature slider up and down smoothly and watch the color the star appears to be to your eye. Even though the object emits green light, we never see it as being green. Explain why this is.
5) As you increase the temperature with the slider, explain how the peak wavelength changes?
6) As you decrease the temperature with the slider, explain how the peak wavelength changes?
Relationship between Peak Wavelength and Temperature
1) Go to the following website: http://astro.unl.edu/classaction/light.html
2) At the bottom left, click on the “Animations” tab, then select the “Blackbody Curves (NAAP)” simulation.
3) Take a few minutes to learn how the features of this simulation work. Learn how to add and remove curves and change their temperatures.
a) Click the “add curve” button one or more times.
b) Change the temperature slider.
c) Select another curve (click on the curve’s information in the table at right) and
change the temperature of this curve.
d) Remove all but two curves.
4) Learn the “vertical scale” options. Have two curves in the window.
a) Check the “auto scale all curves” option and change the temperature of one
curve.
b) Check the “auto scale to selected curve” option and change the temperature of
one curve.
c) Check the “lock scales” option and change the temperature of one curve.
5) Learn the “horizontal scale” options.
a) Select the “Horizontal Scale” tab. Note how changing the rightmost limit change
the view. Note also that each tick mark represents 100 nm. The other unit you
see on the scale is micrometers (mm). 1000 nm = 1 mm
6) Now create four curves that all show at the same time. Adjust their temperatures to the following values: 3000 K, 3500 K, 4000 K, 4500 K.
a) Click the “Indicate Peak Wavelength” option at the right of the simulation.
Select the 3000 K curve. At what wavelength does the peak of this spectrum
occur? Note that this wavelength is recorded in the table below.
b) Repeat for the other curves and fill in the peak wavelength values in the table.
Formulate a general rule (one or two sentences) that explains how the peak
wavelength changes as the temperatures changes.
c) Now, multiply the temperature by the peak wavelength and write this number
(in scientific notation to three decimal places) in the third column of the table.
The first one has been done for you. What do you notice about these numbers?
3000 / 965.9 / 2.898×106
3500
4000
4500
d) Note that the temperature times the peak wavelength always gives the same
number. How could you use this fact to predict the peak wavelength of a
blackbody spectrum for any temperature?
e) Use your method to predict the peak wavelength for a blackbody at a
temperature of 10,000 K. Show your calculation below.
f) Test your prediction with the simulator. Was it close? What you have
discovered is known as Wien’s Law. It says that the product of the temperature
of a blackbody and the peak wavelength of the spectrum is always the same
number. How could this possibly be useful? Read on!
7) When we look at the spectrum of light coming from a star, we can easily identify the peak wavelength. The peak wavelengths for the spectra of a few stars are listed in the table below.
a) Rank these stars from hottest to coldest. You do not have to calculate anything
to do this. If you need a hint, look back at the rule you wrote in part 6(b) of this
activity.
b) Now, Use what you have learned so far to calculate the surface temperatures of
these stars. Record this in the table below.
c) Go back to the first simulation you used on the course website and figure out
what color each of these stars would appear by moving the slider to the correct
temperature and looking at the star at the top. Put the color in the last column.
Betelgeuse / 828.0
Rigel / 263.5
Sirius B / 118.6
Luminosity and the Light Output vs. Wavelength Graph
The total amount of light energy emitted by any object per unit time is called the object’s luminosity. We can determine something about the luminosity of an object by looking at its light output vs. wavelength graph. Consider the graphs shown below for the objects D, E, and F.
1) Which of these objects (D,E, or F) is putting out the greatest total amount of light (i.e. which object has the greatest luminosity)? Explain how you reach your conclusion.
2) Rank the three objects (D,E,F) from lowest luminosity to highest luminosity. Explain your reasoning.
3) A good way to think of the above activity is that the luminosity is related to the total area under the curve on the graph of light output vs. wavelength. With your pen or pencil, shade in the area under the curves for objects D, E, and F. The greater the area under the curve, the greater the overall light output.
4) Bring up the “Blackbody Curves (NAAP)” simulation again and remove all but one curve. Set the temperature of the curve to 5000 K. In the small table at the right of the simulation you can read off the “area under curve.” Record the area under the 5000 K curve in the table below.
Change the temperature to 10,000 K and record the area under the curve.
Change the temperature to 15,000 K and record the area under the curve.
5000
10,000
15,000
5) Notice that 10,000 K is double 5000 K. When the temperature is doubled, by what factor does the area under the curve increase? To answer this question, take the area under the 10,000 K curve and divide it by the area under the 5000 K curve.
6) Notice that 15,000 K is triple 5000 K. When the temperature is tripled, by what factor does the area under the curve increase?
It is well known that the luminosity of an object, and therefore the area under its blackbody curve, is proportional to the temperature of the object raised to the fourth power. More specifically
L = s×A×T4
where L is the luminosity of the object, s is a constant, A is the surface area of the object, and T is the object’s temperature.
7) According to the above formula, if you double the temperature, by what factor should the luminosity increase?
Is this what you observed when the temperature doubled from 5000 K to 10,000 K?
8) According to the above formula, if you triple the temperature, by what factor should the luminosity increase?
Is this what you observed when the temperature tripled from 5000 K to 15,000 K?
Exercises
1) Below are light output vs. wavelength graphs for two objects, G and H.
a) Compare temperatures of objects G & H. Explain your reasoning.
b) Compare the sizes of objects G & H. Explain your reasoning.
2) Below are light output vs. wavelength graphs for two stars, J and K.
a) Compare temperatures of stars J & K. Explain your reasoning.
b) Compare the sizes of stars J & K. Explain your reasoning.