Introduction

Not all stars shine steadily like our Sun.  Any star that significantly varies in brightness with time is called a variable star.  A variable star that has the peculiar problem with achieving the proper balance between the power welling up from the stellar core and the power being radiated from the stellar surface.  If the energy and pressure build up in the star, it will expand in size.  As this expansion puffs up outer layers of the star outward the star can release some of the pressure built up.  The pressure then drops, and the star contracts again.  In a futile quest for a steady equilibrium, the atmospheres of these pulsating variable stars alternately expand and contract, causing the star's luminosity to rise and fall.  The following figure from Sky and Telescope shows example light curves of four different stars:
 

Any pulsating variable star has its own particular period between peaks in luminosity which we can discover easily from its light curve.  These periods range from as short as several hours to as long as several months.

Most pulsating variable stars inhabit a section of the H-R diagram called the instability strip.  A special category of very luminous pulsating variables lies at the top of this strip: the Cepheid Variables, or Cepheids for short.  These stars fluctuate in luminosity with periods of a few days to a few months.  In 1912, Henrietta Leavitt, an astronomer at Harvard, discovered that the periods of these stars are very closely related to their luminosities.  The longer the period, the more luminous the star.  This Period-Luminosity Relation is seen to be true because larger (and hence more luminous) Cepheids take longer to pulsate back and forth in size.

Once we measure the period of a Cepheid variable, we can use the period-luminosity relation to determine its luminosity.  We can then calculate its distance with the luminosity-distance formula that we have seen before.  In fact, Cepheids provide our primary means of measuring distances to other galaxies and thus teach us the true scale of the cosmos.  One particularly special Cepheid is the North Star, Polaris, which has guided generations of navigators in the Northern Hemisphere.

When Ferdinand Magellan and company returned from their voyage around the world in 1522, they told of two bright patches of light deep in the Southern Hemisphere that appear to the naked eye like displaced portions of the Milky Way.  These Magellanic Clouds are small gravitational companions of our own Galaxy.  The Large Magellanic Cloud (LMC) appears about 8 degrees across, while the Small Magellanic Cloud (SMC) appears about half that size.  The Magellanic Clouds are sufficiently close for use to see that they contain all the different kinds of stars that inhabit our own Galaxy.  Because all of the stars within each cloud have about the same distance, the LMC and SMC make superb natural laboratories that allow astronomers to compare relative stellar properties.  It was the Cepheids in the SMC that Henrietta Leavitt studied when she plotted the average apparent magnitude against the logarithm of the period, like the one shown below.
 
 

There are a few Cepheids in open clusters in our own Galaxy.  From main sequence fitting (see Ex. 1.3), we can find the clusters' distances and the Cepheids' absolute magnitudes.  We can then find the distance of any Cepheid by measuring its period and average apparent magnitude, reading the absolute magnitude from the period-luminosity relation, and applying the magnitude equation.  The supergiant Cepheids are so bright they can be seen immensely far away in other galaxies, not only in our Galaxy and the Magellanic Clouds.  The galaxy M100 has a Cepheid, (here's another picture) which helped determine the galaxy's distance.

We will now make our own Period-Luminosity relation.  Go to the David Dunlap Observatory's Database of Galactic Cepheids website.  Read some of the introduction and description of individual files and then proceed to View, query, or retrieve the tables of the database.   At the top of the page under Display Database Tables, check the Physical Data box, and press the Apply button.  The data page that come up is a very big list so it may take a moment.  When it is finished, do the following:

1. Under the File option, choose the "Save As..." option to save this information.
2. Save the data as a Text File (*.txt).
3. Open the Excel Program.
4. Open the file in Excel; it will ask you what to do with it.  When you try to open this file a Text Import Wizard (1 of 3) box will come up.  It will ask if you want it Delimited or Fixed Width, so check the Fixed Width box and click Next.  The Text Import Wizard 2 of 3 page comes up to let you preview the data.  Click Next again.  Page 3 of 3 follows; click Finish.  This will put the data into the columns of the Excel spreadsheet.
5. The columns that we want are titled PERIOD and MV and represent the Cepheid's pulsation period (in days) and absolute magnitude, respectively. Select the columns that do not say 'Peroid' or 'MV'. Under Edit select Delete. repeat for all columns except 'Period' and 'MV'.
6. You are now ready to create your plot of the Period-Luminosity relation. Highlight the two columns labeled Period and MV. Click on the Chart Wizard (red, yellow, and blue bar graph icon).
7. Choose XY (Scatter).
8. The next window (Chart Wizard - Step 2 of 4) click Next.
9. Under Titles, label the Chart title, Value (X) axis, and Value (Y) axis appropriately. Under Gridline: choose Minor gridlines for X, and Major gridlines for Y.
10. Click Next, then choose New Chart, then Finish.
11. Look at the graph now and compare it to the Period-Luminosity shown above. How is it different? If you want your plot to look like that one you will need to change the axes. First, click on the X-axis. Under Scale, click on Logarithmic scale. This will display the X values as logarithms. Look at the Y-axis. Recall that the more negative the number the brighter the object. Click on the Y-axis and reverse the order of the values.
12. Print your new Period-Luminosity relation.
    

Now, let’s put that Period-Luminosity relation to some use. This sequence of images taken with NASA's Hubble Space Telescope chronicles the rhythmic changes in a Cepheid Variable star (located in the center of each image) in the spiral galaxy M100. The images were taken in visible light in 1994. The Cepheid in this Hubble picture doubles in brightness (24.5 to 25.3 apparent magnitude) over a period of 51.3 days. Use this information to find the distance to the star and therefore, to the host galaxy, M100.

Questions:

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