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Fall 2010
Name: _________________________________
How Close is our Nearest Neighbor?
Description:
From Shapley’s experiment we have learned that our Sun is located about halfway from the center of the Milky Way to its outer edge. Our Milky Way contains hundreds of billions of stars and is one of billions of galaxies in the universe. The universe is vast, and most of the universe is empty – no stars, no dust, no gas, no galaxies. Yet, our Milky Way is located in a group of about 35 galaxies. The galaxies in our group are so far apart that our closest neighbor is still thousands of light years away.
The distances between galaxies are so large that they cannot be measured using parallax. Paradoxically, the galaxies in a group are so close together that the expansion of the universe does not dominate their motion, so the Hubble law cannot be used to measure the distances to galaxies that are members of the Local Group. To determine the distance to our nearest neighboring galaxies requires a clever astronomy experiment.
When Shapley did his experiment, he had to measure the distances to globular clusters. To do this, he used Henrietta Leavitt’s discovery that certain variable stars obeyed a period-luminosity law so that their luminosities could be determined by measuring their periods of variation. These variable stars have been found in other galaxies, including our nearest neighbors.
Introduction:
Cepheid variable stars are simply stars whose brightness varies regularly. They are called Cepheids because the first such star was found in the constellation Cepheus. Henrietta Leavitt’s work studying these stars showed that their period of variation (how long it took them to go from maximum brightness to minimum brightness and back to maximum brightness again) was directly correlated to their luminosity. That is, the more luminous the star, the longer it would take for that star to change its brightness.
What is most important about this discovery is that it makes it possible to calculate the luminosity of a star, without having to measure its distance. Rather, one can measure its periodicity and determine its distance from Earth, using the distance modulus formula.
In this exercise, you will derive the period-luminosity relationship, using modern data. Then, you will apply this law to a given period and apparent magnitude of a Cepheid variable star found in the Large Magellanic Cloud, one of our nearest galaxy neighbors, and calculate the distance to this galaxy using the law you derived.
1. Plot the data from the table below using Excel. Plot the log of the period (Log P) on the x-axis and the absolute magnitude on the y-axis. [Use “Scatter plot.” Nothing to “report” for this question.]
2. Fit a line to the data. Use the fit line function to fit a line to the data and display the equation of the best fit line. [After plotting, right-click on a point and choose “Add Trendline” and pick “line.” In pop-up window, choose “Options” tab and click on “Display equation on chart.” Remember to give your chart a title. Upload your final graph as part of the lab assignment. Otherwise, nothing to “report” for this question.]
3. Calculate the distance to the Large Magellanic Cloud. Using the data below, along with your linear fit to the data from step 2, calculate the distance to the LMC. The data for a star in the LMC is as follows: P = 4.76 days. The apparent magnitude (mV) of this star is 15.56. Use the distance modulus formula to calculate the distance: d = 10(m – M+5)/5. (See “Lab #5: H-R Diagram” for help doing this calculation.)
[Type work & answer here]
4. Compare the distance you calculated using the period-luminosity relationship that you derived to the accepted distance to the LMC. Look up the distance to the LMC on NED and compare the answer you got to the distance listed. Calculate the percent error.
[Type work & answers here]
5. Repeat for the Small Magellanic Cloud (SMC). Make a graph for the SMC using log (Peroid) on the x-axis and the apparent magnitude along the y-axis. Plot a trendline and display the equation of the best fit line. [Use “Scatter plot.” Remember to give your chart a title and axes labels. Upload your final graph as part of the lab assignment. Nothing to “report” for this question.]
6. Compare the slopes of your two graphs. In both of your graphs (one for the LMC and the other for the SMC), you had the equation of the line placed on the graph. It is in the form y = mx + b where both x and y refer to the x and y coordinates, respectively, b is the y-intercept, and m is the slope. Referring to the two equations, answer the following questions:
a.) What is the slope of the line for the LMC?
[Type answer here]
b.) What is the slope of the line for the SMC?
[Type answer here]
c.) Compare the two slopes. (Refer to the Math Review worksheet for assistance.)
[Type work & answer here]
7. Calculate the distance to the SMC. This time, you have the apparent magnitudes for the stars but not their absolute magnitudes. Find their approximate absolute magnitudes by looking-up the log of their period on the LMC graph and writing down the absolute magnitude associated with it based upon the trendline. Record your answers below. Then, using two SMC stars of your choice, calculate the distance to the SMC using the distance modulus formula.
[Type work & answers here]
8. Compare the distance you calculated using the period-luminosity relationship that you derived to the accepted distance to the SMC. Look up the distance to the SMC on NED and compare the average of your two answers you got to the distance listed. Calculate the percent error.
[Type work & answers here]
OPTIONAL STEP 8: Write a paper describing how you derived a period-luminosity law and calculated the distance to the nearest galaxy. Include the comparison you made to the accepted distance to the LMC. Discuss any source of error that you can determine.
LMC Data Table
|
Star Name |
MV |
Period (days) |
Log P (days) |
|
|
RY Cas |
-8.54 |
12.1373 |
1.0841 |
|
|
FM Cas |
-5.83 |
5.8094 |
0.7641 |
|
|
SU Cas |
-2.24 |
1.9493 |
0.2899 |
|
|
RX Cam |
-2.02 |
7.9120 |
0.8983 |
|
|
SZ Tau |
-1.03 |
3.1489 |
0.4982 |
|
|
Alpha UMi |
-3.60 |
3.9698 |
0.5988 |
|
|
delta Cep |
-3.32 |
5.3663 |
0.7297 |
SMC Data Table
|
Star |
App. Mag. |
log(Period) |
Mv |
|
|
HV1871 |
1.2413 |
17.21 |
0.0934 |
|
|
HV1907 |
1.6433 |
16.96 |
0.216 |
|
|
HV11114 |
2.7120 |
16.54 |
0.433 |
|
|
HV2015 |
2.8742 |
16.47 |
0.459 |
|
|
HV1906 |
3.0655 |
16.31 |
0.486 |
|
|
HV11216 |
3.1148 |
0.493 |
||
|
HV11113 |
3.2139 |
16.56 |
0.507 |
|
|
HV212 |
3.9014 |
15.89 |
0.591 |
|
|
HV11112 |
6.6931 |
15.69 |
0.826 |
Percent error = | difference between your value and the accepted value | × 100%
Accepted value 1
Where the vertical bars in the numerator | … | mean “absolute value”
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