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Lab Manual

Temperature and Luminosity of Stars: Wein's Law and the Stephan-Boltzmann

Law

 

Irina Golub January 30, 2017


Read University Physics Volume 3 Chapter #6: PHOTONS AND MATTER WAVES

Learning Objectives

·        To understand thermal spectra

·        To understand Wien's Law and the Stephan-Boltzmann Law

·        To understand how thermal spectra can be used to evaluate the temperature of a star

·        To understand how temperature and radius of a star determine a star's luminosity


Introduction: In this activity we will learn how light from a star can tell us its temperature and how much energy per second the star is emitting - its luminosity.


Light that is emitted from stars, like light from any hot object made of many closely packed atoms, forms a continuous spectrum called a thermal spectrum. The range of wavelengths emitted and the relative intensity of the light for each wavelength are dependent on the temperature of the object.


In this activity we will use a PhET simulation to study the thermal spectra of stars in order to understand the relationship between these spectra and temperature. We will see how this relationship results in two laws that describe these spectra -- Wein's Law and the Stephan-Boltzmann Law.


Go to PhET Simulation Blackbody- Spectrum


As you work through this activity, please answer the questions

Part 1: Understanding the graph


This simulation plots a graph that describes the spectrum emitted by a hot object. The horizontal axis indicates the wavelength of the emitted photons and the vertical axis indicates the amount of energy per second (power) emitted by one square meter of the object at a particular wavelength.


To investigate this further, set the temperature to 5800 K. Either move the slider or type the numbers in the temperature box. 5800 K is approximately the temperature of the Sun.


Note that the horizontal axis specifies the emitted wavelengths (l) in units of micrometers, µm. One micrometer equals 1000 nanometers. (1µm = 1000 nm.) For example, look at the visible spectrum. The longest wavelength of the visible spectrum is approximately 0.7µm = 0.7x1000nm or 700 nm.


Question 1. Approximately, what is the shortest visible wavelength? Express your answer in µm and nm.


Observe the peak of the curve. The Sun emits the most energy at this wavelength. We will label this wavelength λpeak-- the wavelength where the curve is at its peak.


Question 2: What color corresponds to λpeak?


Question 3. Approximately, what is the value for λpeak? Express your answer in

µm and nm. (Check the box "show ruler" and use the ruler to help approximate the value for λpeak.)


The vertical axis indicates for each wavelength the amount of energy per second emitted by one square meter of the surface of the emitting object, the Sun in this setting. Recall that "energy per second" is "power," measured in watts, W. MW indicates "megawatts." 1MW = 1Million Watts. Thus the vertical units are MW/m2. Note that the only values on the graph are 0 and 100. You can use the ruler to estimate values in between. For example, to measure value for the peak wavelength, rest the bottom of the ruler on the horizontal axis, and note that 30cm corresponds to 100 MW/m2. The peak measures 24 cm. So the value at this wavelength = (24/30) x 100 MW/m2 = 80 MW/m2.


Hit the "save" button!


Enter your values for the Sun in Table 1 on the next page.


Part 2: Other Stars at other Temperatures


A.   Change the temperature to 3800K. Compare this curve to the saved one for the Sun.


Question 4: In what two ways does this peak differ from that of the Sun?


For easier viewing, magnify the curve by pressing the + button on the left. Note that the maximum reading on the vertical axis changes. Make measurements to complete the Table 1 entries for this temperature.


B.   Change the temperature to 6600K. Compare this curve to the saved one for the Sun. For easier viewing, press the - button on the left. Make measurements to complete the Table 1 entries for this temperature.


Question 5: In what two ways does this peak differ from that of the Sun?


C.   Make measurements to complete the Table 1 entries for a temperature of 8300 K, adjusting the scales as necessary.


Table 1: Thermal Spectra Characteristics for Different Temperatures


Temperature

λpeak (nm)

λpeak (color, UV, or IR)

Peak Energy/sec/m2

(MW/m2)

3800 K

5800 K (Sun)

6600 K

8300 K


Part 3: Wien's Law


A. Qualitative discussion


Question 6: Based on your values in Table 1, how does the value of λpeak change as the temperature increases?


Wien's Law states: Hotter objects emit photons with a higher average energy, which means a higher frequency and, therefore, a shorter wavelength.


Question 7: Do the values in Table 1 support this law or not? Explain your choice.


B.   Quantitative discussion


Quantitatively, Wien's law reflects this inverse relationship between temperature and peak wavelength. It states that if wavelength is measured in nm and temperature is measured in Kelvin, then



T = 2,900,000

lpeak


Use the values you measured for λpeak and Wien's law to determine T and compare to the values from the PhET. Enter the values in Table 2.


Table 2: Comparison with Wien's Law

 

Temperature

(From PhET)

λpeak (nm)

Temperature

(Wien's Law)

% error

3800 K

5800 K (Sun)

6600 K

8300 K

Part 4: The Stephan-Boltzmann Law

A.   Qualitative discussion



 

The picture above is a screen shot of the 5800K and 6600K thermal spectra.


Question 8: Which radiates more MW/m2 at infrared wavelengths? Question 9: Which radiates more MW/m2 at visible wavelengths? Question 10: Which radiates more MW/m2 at ultraviolet wavelengths?



The Stephan-Boltzmann Law states: Each square meter of a hotter object's surface emits more light at all wavelengths.


Question 11: Do your observations of this figure support this law or not? Explain your choice.


B.   Quantitative discussion

The Stephan-Boltzmann law can be expressed mathematically as follows:


Use your data calculate the power for each T:

·        The energy emitted per second varies as the fourth power of the temperature so that, for example, doubling the temperature results in 2x2x2x2 = 16 times as much energy per second.

·        This only tells us the energy per second emitted by one square meter of the star's surface.

Part 5: Luminosity


Luminosity, L, is the total power (energy per second) that a star radiates into space.


The Stephan-Boltzmann law tells us how much energy per second one square meter of the star radiates into space, so to obtain the total energy radiated we must multiply by the surface area of the star, 4πR2.



L =   T4 x 4πR2


where T is in Kelvin, R, the radius of the star is in m (meters) and L is in watts.

Chose 3 different star from the internet and calculate L for T=: 5700K, 3500K and 7000K

Find the L of the Sun and calculate T


Question 12. What two properties of a star determine its luminosity? Question 13. Can two stars have the same temperature and different luminosities? Explain.

Question 14. Can two stars have the same luminosity and different temperatures? Explain.


Part 6: The blackbody energy distribution observes a Gaussian distribution. Write two (2) hypotheses, each answering the following questions:

 

o   What is the relationship between intensity of the energy radiated by the blackbody and the temperature in degrees Kelvin?

Hypothesis #1 -


o   What is the relationship between the wavelength of the peak (maximum) of the Gaussian curve and the temperature in degrees Kelvin? Hypothesis #2 -



·        Use tracing paper cutouts of the Gaussian curve as a measure of the total radiation intensity [The tracing paper is assumed to be of uniform density. Therefore, the area under each curve is directly proportional to the mass of the paper cutout.]. Each cutout will be massed separately in a balance to the nearest 1,000th of a gram, and the mass will be recorded for each temperature in degrees Kelvin.


·        Start Point: Increase the Intensity axis scale from 100 to 0.1 by toggling the plus (+) circle on the upper left hand side of the screen. Using the Zoom Out Toggle, minus sign (-), on the lower right hand side, scale the Wavelength axis to

24. Lower the temperature of the blackbody to 300 K by lowering the index on the temperature scale. Notice that the curve cannot be distinguished from the wavelength axis.


·        In the table, below, record the temperature, a dash under wavelength maximum (no distinguishable experimental wavelength maximum), and the area underneath the curve as 0.000 g for trial 1.


Trial

Temperature (K)

Wavelength of the Peak Maximum (µm)

Tracing Paper Mass

(g) (as a function of Area under the Curve)

1

---------------

2

3

4

5

6

7

 

·        With the sliding arrow, increase the temperature to slightly over 1000 C. Place tracing paper on the screen and trace the area underneath the Gaussian Curve. Write the corresponding temperature in degrees Kelvin under the curve. The area under the curve is a measure of the intensity of the energy radiated at all wavelengths. Save the trace for the laboratory phase of the experiment.


·        Record the wavelength, in µm, corresponding to the peak (maximum) of the Gaussian Curve and record it in a table of wavelength and temperature in degrees Kelvin.


·        Repeat the procedure for several temperature until the maximum of a curve is at the top of the Intensity chart. This will be the last data point of this experiment.


1) In the laboratory: Mass all traces, and record the data in the table provided above.

 

Post-Phase:


1) With your graphing calculator (or alternately, in an Excel spreadsheet), list the Temperature as L1, the wavelength of the peak maxima in L2, and the area under the curve (energy intensity related by mass, in grams) as L3.

 

2) In Plot 1, graph the Temperature (X-axis: L1) versus the wavelength of the peak maxima (Y-axis: L2). [If you are using Excel, make a graph of these two parameters].


a.              Based on the plot, what relationship do these two parameters appear to hold?       _


b.              List in L4 the reciprocal of L1. Turn off Plot 1, and create a Plot 2 with 1/T (X-axis: L4), keeping Y-axis: L2. Adjust your window to accommodate the new X-axis values. The graph of Plot 2 should give you a linear plot.

 

c.               Do a Linear Regression for L4, L2. Record the slope of the line (include units):         


d.            The Wien Law gives the wavelength of the peak of the radiation


distribution, λmax = 2.90 x 103 µm K .


T

e.               Compare your experimental slope, recorded in 2) c., above, with the accepted coefficient of the Wien Law, 2.90 x 103 µm K. What is the percent error of your determination?


3) Turn off Plot 2, and create a Plot 3 with the Temperature (X-axis: L1) versus relative total energy intensity (Y-axis: L3). [If you are using Excel, make a graph of these two parameters].

 

a.              Based on the plot, what relationship do these two parameters appear to hold?       


b.              According to Maxwell Planck, the total energy intensity of a black body is a function of T4: a quartic (4th power) function. Test this hypothesis by creating a new list (L4 = L14).


i.      Change Plot 3 with the Temperature to the fourth power (X- axis: L4), keeping the total energy intensity (Y-axis: L3). Adjust your Window to show all the values on the X-axis.

 

ii.      The graph should be a linear plot. Do a Linear Regression L4, L3.

 

iii.      Write the slope of the line. Do not forget to write the expected units:            . This value is related to the Stefan- Boltzmann Law, derived from experiments by Stephan (1879) and Boltzmann (1884).



Analyst and Conclusion:

Summarize what you learned from this activity by:

a.      listing the key scientific terms encountered in the activity.

b.      constructing a bulleted list that utilizes these key words (and, as needed, words from previous activities) in summarizing what you learned.


c.      State whether your hypotheses were confirmed or denied. Modify or restate the predictions so that they match the observed laws.


Hypothesis #1 -

 

Hypothesis #2 -


In a brief statement, comment on the use of the graphing calculator as a tool to find patterns between data sets:


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