EOU-TECHNICAL MANUAL

Rodriguez, W.J., Hennig, L.A., & Henry, G.W. (1997). Finding Periods in Variable Star Data: Using Remote FORTRAN and Local Windows Software. Center of Excellence in Information Systems, Tennessee State University. Nashville, TN: Explorers of the Universe Technical Manual 101-97.

The Explorers of the Universe (http://coe2.tsuniv.edu/explorers) is a Scientific/Literacy Interdisciplinary Project that promotes the investigation of self-directed cases using authentic materials in problem-oriented contexts. Teachers and students are encouraged to think together in extending their knowledge with astronomy and related subject disciplines. An emphasis for incorporating rather than compartmentalizing the curriculum is of primary importance. Astronomy is seen as a venue for incorporating mathematics, literature, music, art, history, and other subjects into students' case base research. Teachers and their students communicate with astronomers via e-mail and publish their papers on the World Wide Web. Metacogntive tools such as concept maps and Interactive Vee Diagrams are used for learning.

Finding Periods in Variable Star Data, Technical Manual 101-97, presents teachers and students with examples of how to analyze variable star data. The sample data contained within the manual has been received from automatic photoelectric telescopes located at the Fairborn Observatory in Washington Camp, near Nogales, Arizona. These automatic telescopes are controlled via the Internet by astronomers at the Center of Excellence in Information Systems at Tennessee State University, Nashville, Tennessee. The astronomers at TSU control the world's largest collection of automatic photoelectric telescopes. They also are able to make the most precise measurements of variable stars known to date. Bill Rodriguez and Lee Ann Hennig have compiled sample data and presented challenging situations for students to apply their mathematics and scientific knowledge. Greg Henry, TSU astronomer, served as a consultant and provided the sample data for analyses. The manual is a precursor for enabling students to analyze data archived by TSU astronomers and to eventually meaningfully transform data information from a star they have initiated for observation by these automatic photoelectric telescopes.

This technical manual is the first written by two high school astronomy teachers and has been field tested and revised by their students. The work is a tribute to those teachers and students who have trusting relationships between each other and see knowledge as constructing meaning from events and objects.

The Explorers of the Universe is under the auspices of the Center of Excellence in Information Systems, Tennessee State University. Dr. Michael Busby is the Director of the Center. It is also supported by NASA through the Tennessee Space Grant Consortium NGT 5-40054.

INTRODUCTION ................................................................................................................................................................................................................. 3

General information ...............................................................................................................................................................................................................4

Logging Into The University School of Nashville System ...................................................................................................................................................5

Establishing an FTP Session ...............................................................................................................................................................................................5

Selecting A Set of Observations For Analysis ...................................................................................................................................................................5

Extracting Data From A TSU APT Data File..................................................................................................................................................................... 6

Initial Analysis of Your Observations (using WPlot) ..........................................................................................................................................................8

Using WPlot to Print Julian Date vs. Delta Mag .............................................................................................................................................................10

Period Finding Using Least-String Techniques............................................................................................................................................................. 12

Period Finding Using Least-Squares, Sine-Fit Techniques ........................................................................................................................................14

Understanding True Periods And Aliases .....................................................................................................................................................................17

Examples of Aliases ........................................................................................................................................................................................................19

Determining The Best Period From A Periodogram.................................................................................................................................................. 23

Determining The Uncertainty of The Period .................................................................................................................................................................26

Producing A Phase Plot .................................................................................................................................................................................................29

Appendix A - Example of Journal Entry Check List..................................................................................................................................................... 32

Appendix B - Aliasing Activity Plots ..............................................................................................................................................................................34

Appendix C - References ...............................................................................................................................................................................................37

Appendix D -Further Reading & Questions .................................................................................................................................................................38

Appendix B - Program Listings ....................................................................................................................................................................................39

This manual is designed to accomplish a number of tasks. It is written in a step-by-step fashion so you can proceed at your own pace. However, do not treat the manual as a cookbook--there are important concepts and questions discussed within the context of analyzing the data acquired by Automatic Photoelectric Telescopes (APTs). You need to remember all of the steps as well as the ideas presented during the analysis and not just skim the manual. Think your way through the analysis procedures and answer all the questions. Take notes, fill in the data tables, and keep a journal. A well-kept journal is invaluable to a scientist. Also, some of you will be collaborating with students at distant schools and it is extremely important that you keep track of what you have done so you can communicate your findings to your collaborators. You may want to keep an electronic version of your journal as well. This way you can email your collaborators. We suggest that you create a directory structure which allows you to place your files in a specific location (by star) so that all important files are kept in one location.

This instruction manual is intended for use by students after they understand how the Tennessee State University Automatic Photoelectric Telescopes (APTs) collect and reduce photometric data. It is also important to understand what the data in the files represents. For a basic information on the APTs operated by TSU, see their web pages at:

and follow the links to the Fairborn Observatory and the Automated Astronomy Group. Also see the references listed in Appendix C.

When reading the instructions in this manual, the font and emphasis, for example bold-face, have special meanings. Here is the key to the conventions used in the instructions.

The names of programs are italicized. If they are also bold-faced then you are entering the program name to run the program.

Special keys such as the ENTER key that you either press or hold are typed in uppercase.

During the data analysis you will use a combination of software run on your PC as well as software which runs on a remote computer. To accomplish this you will run a set of programs simultaneously and swtich between them as you need them. Briefly, here is a quick run-down of the programs you will need to complete your analysis.

Telnet: opens a connection to a remote computer. In this case, a PC running the Linux operating system.

10extract and 16extract: extracts data from the 10 inch and 16 inch APT data files into a more easy to analyze form.

least-str: performs a least string analysis on your data to help you determine the best period..

least-sqr: performs a least squares sine-fit analysis on your data to help you determine the best period.

These remote programs were originally written in Fortran and converted to C enabling them to run on the Linux operating system. Linux is a free version of the UNIX operating system.

Bascially you will use Telnet to run the programs 10extract, 16extract, least-str, least-sqr, and phase on the remote computer. Once you have run the analysis your need to transfer the data files for plotting onto your machine. You use FTP to do these transfers. Once you transfer the data files to your machine, you use WinPlot to plot the various graphs.

First, start the programs we use to analyze the data. You need to simultaneously run three programs¹ on your windows machine: Telnet, FTP, and WPlot². To do this first double click on the WPlot icon to start the first program. Next, hold the ALT key down and touch the TAB key. Notice that a box appears with a program name in it. Keep touching the TAB key until the Program Manager appears and release the ALT key. Once you are back in the Program Manager, double click on the FTP icon. Now, holding the ALT key touch the TAB key until Program Manager appears once again. Now double click on the Telnet icon. You are now ready to begin analyzing your data..

Connect to 206.23.18.2. Your login and password are the same as the telnet session.

Hold the ALT key down and touch the TAB key until Telnet appears in a box, then release the ALT key.

Whenever you are in the Telnet session, you are entering commands and running programs on the remote computer running the Linux operating system. This is a UNIX-type system which means that it is case sensitive, and some of the commands you may be used to are not the same. For example, to list the files in a directory you type ls rather than dir. Notice the ls typed in lowercase, all Linux system commands must be entered in lowercase. Let’s list the files and directories available to you at this time.

The data for each star you will study are stored in separate directories. Once you get more proficient, you can create you own directories and store the data and information in a structure you create.

Change to the directory of the star you wish to analyze, let’s start with star-1.

Type ls and press the ENTER key to see the name of the file. Notice the file has the same name as the directory except it has a .dat extension. This file contains the V, R, and I (visible, red, infrared) differential magnitudes of the variable-comparison and check-comparison stars for the 10 inch APT.

Run the 10extract program to breakout the star-1.dat file into 6 separate files, each containing the data for a single filter on the variable or check star for ease of plotting the individual light curves. The 10extract program removes any lines containing 99.9999 from the file. The APT software performs a good deal of statistical tests on the data the APTs collect. If any of the data falls outside a specified range it is rejected as being inacccurate and the software places a 99.9999 as the data for that set of observations. An example of why this would occur is clouds that blow in during a set of observations. The brightness measurments would vary tremendously during the observation set as a result of the clouds blocking some or all of the light from the star. You would not want to include this data in your analysis so it is makred as bad with the 99.9999. Refer to the web pages for more details. 10extract also truncates the Julian date by removing the left-most two digits in the date. The Julian date is a numerical date such as 2449838.6878. When you run the 10extract program it truncates the date to 49838.6878 to make plotting the data easier. Let’s begin the analysis!

Here are the questions you must answer and the responses you should use on the star-1.dat file.

Once the program has created var.v, var.r, var.i, chk.v, chk.r, and chk.i you should grab a few pieces of information from these files for later use in your analysis. You need to know the maximum delta magnitude for any file you submit to the program least-sqr for periodogram analysis. This value is used by the program to change the logarithmic magnitude scale into a linear scale for analysis. So, let’s find the maximum values now by using the Linux sort command. Don’t worry that the output scrolls by you on the screen - you are only interested in the last row of data.

This command tells the sort program to ignore blank spaces (-b) and begin sorting after the first column (+1). The last line in the screen output will contain a truncated Julian Date followed by the largest delta magnitude. Please record the file name, Julian Date, and this delta magnitude in your journal for future reference. Repeat the above process for var.r and chk.v. For example, your journal might contain a table which looks like the one in Figure 1.

Now, FTP the files created by 10extract to your machine for graphing. To do this Hold the ALT key down and touch the TAB key until a box appears with FTP in it. When it does, release the ALT key.

Once you are in the FTP session, mark the files var.v, var.r, var.i, chk.v, chk.r, chk.i, and star-1.dat for download. Place the files in the proper directory on your machine which you can easily locate from within the WPlot program.

Using Write or another editing program you should print the star-1.dat data file. It is good to have a copy of the data so that you can find the dates of the observations for possible use during editing later in the analysis. You should print the file before using the phase program. I suggest using Write or Notepad as both of these programs put small demands on the computer’s resources. Running Microsoft Word might cause your computer to lock up after you have all the software running and have created and printed numerous plots.

Hold the ALT key down and touch the TAB key until a box appears with WPlot in it. We are going to plot the observations as a function of the date they were acquired to see if we can find a period by inspection. To do this, follow these instructions:

Take a look at your graph - is there an apparent period? Don't be too quick to leap at whatever period appears. You will find out later that sometimes what you see on this plot is not the real period but a multiple of the real period.

FIGURE 2 should very closely resemble what is now on your screen. Doesn’t there seem to be a very nice pattern to the data points? Make a note of the pattern you see. Now, print the graph so that you have a hard copy to keep in your journal. You must print all your graphs so that you can quickly refer to them without using the computer. It also makes Dr. Alvarez really happy to see what you have done! Seriously, it is important to have copies of your plots so you can review them visually when you are away from the computer. Notice on the graph that the date is really a truncated form of the full Julian Date. Refer to your printout of star-1.dat for comparison of the Julian Dates.

WPlot does not allow you to print multiple copies so you will need to print one copy and then use a copier to make copies for each member of the group. It is important that each group member have copies of all the data printouts as well as plots.

Instruct WPlot to connect the data points (by making a line-plot) and look at the new plot to see if your opinion of the variation in brightness remains the same.

FIGURE 3 represents the new plot. What happened to the pattern? There still is a pattern in the data - can you see it? A hint, think of tuning two instruments to the same note. What do you try to eliminate?

Now, load, plot, and print the remainder of the data files var.r, var.i, and chk.v. It really isn't necessary to look at all the chk files at this time. Remember to give your plots useful titles and labels. Between plots you should reset WPlot by

You have a choice of two programs which search for periodicity in data. Refer to the web pages for detailed explanations of how these programs analyze your observations. The program least-str uses the least-string method of determining the period while the program least-sqr uses a least-squares method to fit a series of sine curves to the data. To begin let’s use both and compare the results. First we’ll use the least-str program.

Here is the dialog between you and the program. Enter the numbers as they are typed below so that you can make comparisons with the information here to make sure you arrive at the correct results.

A file containing the data for a periodogram is automatically saved under the name lststring.dat. Now, immediately move the file to another name so it will not be overwritten the next time you run the program least-str. To do this,

Now, run least-str once again using the chk.v file. Since this file is a series of differences in magnitudes between two stars that are known (to the best of our knowledge) not to vary, running and plotting a periodogram of this file will allow you to see what a periodogram looks like in which there is no periodicity in the data.

Use FTP to download the files var-v.str and chk-v.str to your machine and use WPlot to plot the periodograms. Remember to print the plot - it will come in handy later when you are discussing the best period fit. The suggested labels and titles for the plot of the file var-v.str are:

Another method used to analyze data for periodicity is a least-squares fit - to find a full discussion of this method please consult the web pages.

Here is the dialog between you and the program. Enter the numbers as they are below so that you can make comparisons with the web pages if you have trouble. Recall that immediately after running the 10extract program you sorted the files var.v, var.r, and chk.v and recorded the last line on the display. Well, we are going to use the number in the second column (delta magnitude) in the least-sqr program. By telling the program the greatest amplitude in the file it is able to convert from the magnitude (logarithmic scale) to a linear scale. This number is entered when the program asks for the "MAGNITUDE FOR LIGHT UNITS".

After the program completes its analysis it prints a summary of the analysis which looks like this:

The previous information is very important later in the analysis. You can either copy certain data items down into your journal or use the "cut and paste" ability of Windows to copy this section into a text-editor such as Write. Here are the important items:

N represents the number of data points. Required for calculating the uncertainty in your period which you must do later in the analysis.

EPOCH is the truncated (less the left two numbers) of a standard Julian Date. You can use this date later in the phase program when it asks for an epoch when producing a phase plot file.

MINOMC is the least of the sums of the squares of the residuals at the indicated period. This period might be the best fit for the data.

MAXAMP is the maximum amplitude of the sine wave used to fit the data. When a best fit arises it will produce a maximum amplitude.

A file containing the data for a periodogram is automatically saved under the name pgram.dat. Now, immediately move the file to another name so it will not be overwritten the next time you run the least-sqr program. To do this,

WARNING! You must read the file var-v.sqr into WPlot differently than previous files! var-v.sqr is in a three column form and will produce useless plots unless read properly. Here are the steps to correctly reading the three column data file.

Figure 5 represents what your least-squares periodogram will look like. Notice the spike at slightly less than 2 days. Since this is the least of the sums of the squares of the residuals, it represents a good choice for the period of this data. However, there is a spike at a little more that 2 days - what is this? This spike is an aliasing period caused by the regular manner in which astronomers collect data - roughly once per day. Because this is a very important problem in analyzing phenomena that vary periodically, we need to spend time discussing an thinking about how these "aliases" come about and how we can predict them so we will find the true period and not a false period. Do not jump over this part! It is not an aside, but an important method of data analysis.

When searching for periodicity in natural phenomena such as varaible stars, one must be certain that the repeating pattern one finds is a real pattern and not one caused by the interplay of the phenomena and the data collection method. Most experiments done in the high school laboratory are designed to allow you to see the repeating pattern - for example, the period of a pendulum. You can watch it repeat it’s swing back and forth and have the luxury of timing complete periods to determine its period. However, astronomers (visual ones) may only collect data during the nighttime and during the season in which the star is above the horizon at night. Therefore, our data collection routines force us to use bits and pieces of data to determine the entire picture. Another problem is that most of the objects we study vary over a time period greater than one evenings observations. When is the last time you collected data during one laboratory period then came back each day and collected more data on a phenomena that some varied? Let’s take a simple pendulum for an example.

The table below contains data acquired by measuring the position of a pendulum bob each second for a time period of 13 seconds.

Plot the above data on graph paper and see if you can determine the period visually. Looks pretty good doesn’t it? A nice 9 second period seems to fit rather nicely - let’s all go home and call it a suceessful day at the lab! Well, there might be a little more work to be done here because this data was collected differently than we usually collect photometric data. Notice that this data was collected at EXACTLY 1.0 second intervals - something that might sound harmless, but can cause problems if we are not careful in our analysis. Remember too, that as astronomers, we collect data at one day intervals for much of our work. What we have done is setup a "frequency" of data collection. If the star varies as an integer multiple of this frequency, we might get an "alias period’ rather than the real period. Look at Figures 6 & 7, they are plots of the above data and should look very similar to your plots.

Figure 6 is just a scatter plot of data set #1. Visually it is very easy to see a repeating pattern. Let’s sketch a curve to the points and see what it looks like.

It really looks as if a 9 second period is an excellent fit. However, take a look at Figure 8 which plots a curve of the REAL preiod of this pendulum! What period do you see on this plot?

It seems that the pendulum varies on a 0.90 second period rather than a 9 second period. To better understand what happened, let’s consider another sytem in which two objects which vary over time seem to cause a new pattern to arise.

When two tuning forks differ slightly in their frequencies, they set up a "beat frequency" in which the volume of the note heard rises and falls in a regular pattern called a beat frequency. Musicians use beat frequencies to tune their instruments. In fact, musicians listen and tune their instruments to remove any beat frequencies Calculating the beat frequency is very easy - it is just the difference in pitch of the two notes played.

The absolute value sign is used because all that we are concerned with the difference - it does not matter if the difference is positive or negative.

By collecting data at a regular interval, we set up a "frequency" - in this case one which was an integer multiple of the real frequency and caused us to see the analog of a beat frequency called an alias period. Remember, in astronomy, we are looking for the period over which the data repeats, not the frequency. All is not lost - we have two choices, collect data at non-regular intervals, or find a mathematical method to determine the real period once we have the alias period and the time interval of data collection. Once such method exists and here is the solution:

So, if we suspected an alias period we couold solve for the true period P_true very easily. Using the numbers from Data Set 1 we have an apparent period (the alias period) of 9 seconds and a data collection (DT) period of 1 second.

Now look at the next two data sets. There may or may not be an alias period. However, you should look at the times at which the data was collected, plot the data and see what turns up. Look in Appendix A for full sized plots of all the data sets with any (if they exist) alias periods as well as the true periods.

After completing the above exercises, refer back to Figure 5 - it looks like the period is just under 2.00d. using the information given by the least-sqr program, you probably know that a minima occured at 1.95d. Let’s see if we can see if the spike at a little more that 2.00d might be an alias period. Using our formula:

P_alias = 2.053d, which agrees very nicely with the plot! There is strong evidence that we can rule out the spike just above 2 days as an alias period and proceed with our analysis.

As you have seen, determining the best period is more than finding the lowest point on a periodogram. If the spike or drop is symmetrical then the lowest point on the graph is probalby the best period. However, if the drop is rough edged (noise in the data) or asymmetrical then you need to resort to other means of determining the best period and you must always take aliasing into account when citing the period of variable star.

Once you feel that you have a reasonable range for the period one method to narrow the range is to re-run the least-sqr program and narrow the range of days around the spike and then using another piece of software to fit a parabola to the data. This will smooth out the noise in the data and probably give a better period. Just running either of the programs with a narrower range might allow you to better choose a minimum point on the periodogram. You may even need to try a few different phase plots based on a series of periodograms to really determine a best fit.

Another option - one that is useful in determining the uncertainty in the data as well - is to expand the scale of the plot of var-v.sqr in WPlot. For example, let’s do that now using the var-v.sqr data file.

By plotting the data as a scatter plot you can manually fit a curve and read the period off the graph. You will also use this plot in the next step - that of determining the uncertainty in the data. Sketch a best fit curve --what do you think the best period is for your data?

It is time to determine the uncertainty (D P) in your data. To do this you need to refer to your journal and read the information generated by the least-sqr program. Remember when it printed some information about the data such as MINOMC, N, and other useful items? Well, you need that information now. Look at your journal and find the minimum sum (MINOMC) of the squares of the residuals (S ) and the number of observations (N). Using those data, solve the equation below:

The number 3 represents the number of degrees of freedom in the analytical process. The three degrees of freedom in our data are the phase, the amplitude, and the mean of the sine curve. The number found above - call it z is used graphically to find the uncertainty in the period.

Now, find your zoomed periodogram plot and construct a horizontal line from the y-axis where the above value lies on the axis. Figure 12 on the next page shows the line drawn on the periodogram.

Now, construct two vertical lines from where the horizontal line intersects the horizontal lines previously drawn as shown in Figure 13.

Read the x-axis where the vertical lines intersect. In the above example the lines intersect at 1.947 days and 1.9510 days. If I choose the period as 1.949 days (the midpoint in the range) the correctly stated period with its uncertainty is:

The variable used for uncertainty is D P, so your D P is 0.003 days. However, you are citing a precision (.001 d) greater than you conducted your analysis (.01d). To make sure of your uncertaintity and period, run the least-sqr program once again using the same start and end days, but step in .001 day increments.

Once you have determined a best period for your observations you use the phase program to produce a file for plotting. This is the light curve of the variable plotted modulo its period. First, refer to your least-string and least-squares periodograms and select what you think is the best period for your data.

When you produce a data file for a phase plot, the phase program needs to know a date called an epoch. The epoch is any of non-truncated the Julian dates found in the data files. This date (epoch) is used as a 0 point for the phase plot. If you think of plotting a sine wave, usually you see plots which begin at 0 degrees (or radians) and go through a full period. However, there is no reason why you can’t start your plot at 45 degrees and continue the plot for a full period. The plot will just be phase-shifted 45 degrees. This is exactly what is meant by the 0 point of your phase plot. By using the Julian Date you recorded at the very beginning (remember to put on the left-most digits) your phase plot will start at a maximum delta magnitude. You may actually choose any Julian date, but many times it is easier to see the pattern is you atart at a maximium or minimum delta magnitude.

Now use FTP to download the file var-v.pha to your machine and use WPlot to plot the phase plot. Remember to print the plot - it will come in handy later when you are discussing the best period fit. The suggested labels and titles are:

Figure 14 is the phase plot of the photometric data acquired by the 10" Automatic Photoelectric Telescope while observing star-1.dat. The name of this star is SU Cas.

Now that you have successfully completed the analysis of star-1, you should try star-2. This is a good time to review your journal and make sure that it is complete with all plots and notes.

Keeping complete and accurate records is extremely important. You must be able to support your interpretations to converse knowledgeably with others about your analysis. A complete journal provides the information you need to do this as well as allow you and others to follow your progress. A complete journal should contain the information presented on these two sheets as well as any observations you make, conversations you have with others, and email that relates to the analysis. In other words you are collecting information to build a case relating to the period of your star. For all items you must note the date you acquire or analyze the data.

Copies of the Julian Date vs. Delta V, Julian Date vs. Delta R, and Julian Date vs.

Periodogram plots of the Period vs. String Length for the data files listed in the "Saved As" column in the data table above.

Based on these plots, what do you feel is the best period (if any) which fits the data? Please discuss why you have chosen this period, especially if the plot is asymmetrical about the shortest string length.

Periodogram plots of the Period vs. Sum of Squares of Residuals for the data files listed in the "Saved As" column in the above data table.

Based on these periodogram plots, what do you feel is the best period (if any) which fits the data? Please discuss why you have chosen this period - compare it to the period you chose using the least-string method. Do they agree? Which seems to produce an easier-to-read periodogram?

Show all your calculations for computing the uncertainty in the data. Please be sure to work through this part carefully.

What is the accepted period of the star? How well does your period agree with the accepted period? Calculate a percent error based on the accepted period. What sources did you use to find the accepted period of the variable star? Did all the sources agree on the period? What was the error claimed in the accepted period and how does it compare to your uncertainty?

Photoelectric Photometry of variable Stars, Douglas S. Hall and Russell M. Genet, Wilmann-Bell, Inc., 1988, ISBN 0-943396-19-0

Observing Variable Stars, David H. Levy, Cambridge Unbiversity Press, 1989, ISBN 0 521 321131

Robotic Telescopes: Current Capabilites, Present Developments, and Future Prospoects For Automated Astronomy, Gregory W. Henry and Joel A. Eaton, Astronomical Society of The Pacific, 1995, ISBN 0-9377707-98-8

The Study of Variable Stars Using Small Telescopes, edited by John R. Percy, Cambridge University Press, 1986, ISBN 0 521 33300 8

For those of you who now know it all! Here is a few questions for your consideration.

What does the period of a spotted star combined with a spectroscopically measured rsin(i) give you?

If you were to do a periodogram analysis of historical sunspot data (available on the web), what would it tell you? Do you think you might find multiple periods?

What would the periodogram look like if the comparison star were also a variable star?

¹ All Windows programs are available on our anonymous FTP server. FTP to 206.23.16.2 and login as anonymous. Send you email address as the password. Next change to /pub/DOS and get the programs you need.

² WPlot is a shareware program. Please register the program when using it to analyze variable star data.

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