Student Projects Involving Eclipsing Binaries and Short Period Pulsating Stars
Not infrequently we receive inquiries, often from undergraduate students or their teachers, concerning projects that might be done using charge-coupled device (CCD) cameras on small to medium telescopes. Here we attempt to answer those queries, in a by-no-means exhaustive way, by giving examples of several such projects. The projects are divided according to the amount of observing time available for securing observations. We also assume here that readers are already familiar with the basics of obtaining images and photometry with a CCD camera, leaving us to concentrate on the projects themselves. Our manual of procedures for obtaining CCD photometry of variable stars may be useful. We also note that a basic introduction to variable star astronomy can be found at https://www.aavso.org/education/vsa.
I. Projects for one night of data
It may happen that you have only a single night to secure observations for your project. Despite that limitation, there are still several interesting projects that can be done involving eclipsing binaries or pulsating stars. Moreover, the actual observing part of the project can often be expanded by including searches for associated material in the astronomical literature or in data archives.
1. Observe a minimum of an eclipsing binary star with a single filter. To do this, obtain observations of the target star before or during its decline in brightness and keep observing the star until its return to maximum brightness is well underway. Basics of observing eclipsing binary stars are described in the AAVSO Eclipsing Variables Section website. This generally involves obtaining aperture photometry for the eclipsing binary and at least one comparison star and check star on each image. That webpage also includes lists of some eclipsing binary stars of interest to the AAVSO program (https://www.aavso.org/aavso-eclipsing-binary-section). Observed times of minimum light are vital to tracking the period changes of eclipsing binaries revealing changes in those systems over time.
Not all eclipsing binaries will be suitable single-night targets. Some take more than one night to fall to minimum and recover their uneclipsed brightness. However, almost all the eclipsing binaries listed in the AAVSO program (with the possible exception of AQ Peg) have minima that can be defined with a single night of observing. You will want to observe the star frequently enough that the time of minimum light is well defined. How frequently that is depends on how rapidly the star changes brightness, but your exposure times will also be determined by the size of your telescope and the brightness of the target and comparison stars. With modest observing equipment, exposures between 30 seconds and 5 minutes usually suffice for observing the eclipsing variables in the AAVSO program.
If observing time is limited, it is helpful to have a rough idea in advance of when the eclipse is going to occur. Ephemerides predicting when eclipses are expected for a number of eclipsing binary stars can be found in the AAVSO website (legacy stars) . A portion from one page of these tables of predicted minima is shown below. The predicted universal time of minimum is indicated for each day, rounded to the nearest half hour. “DUR” indicates the duration of the fall in brightness and subsequent rise in hours, while “TOT” indicates how many hours a particular star spends at minimum light.
Of course, the predicted times of minimum might may not match what the star is actually doing. If they always did, there would be no reason to keep observing new minima.
Once you have observed the minimum of an eclipsing binary you can determine its observed time of minimum light. Usually, these are finally given not in the hours, minutes, and seconds of the usual calendar date but in the Julian date system. Your observing software may be able to make the conversion to Julian date automatically. You can calculate the time of minimum light by your own methods, but the Eclipsing Variables Section webpage includes links to some software routines that might be helpful for this.
As the Earth makes its annual circuit of the Sun, the distance that light from the eclipsing binary has to travel to reach us will change. Thus, your observed time of minimum should be transformed to a heliocentric time, the time that would have been observed had you been at the location of the center of the sun instead of on the orbiting Earth. For cases where special precision is needed, the time can be calculated instead for the barycenter of the solar system, which is always close to the sun but is not located at the center of the sun. However, the heliocentric Julian date is usually good enough. It is possible that your observing or reduction software will allow you to make this correction, or you can use one of the several heliocentric Julian date calculators on the web.
Once you have determined the observed time of minimum light, try to estimate the uncertainty in your determination. If your eclipsing binary has been observed in the past, you will probably be able to find ephemerides predicting when minima should occur. Such ephemerides allow you to calculate an O-C value for your minimum, which is the difference between the observed and calculated time of minimum. We saw above that the AAVSO posts ephemerides with approximate times of minimum light for a number of variables.
Figure 1. A V-band light curve of the eclipsing binary W UMa from observations in the AAVSO International Database.
If you are only interested in the time of minimum light, it does not usually matter which filter you use. However, it is usually advisable to use a standard filter. Unfiltered light curves can be affected by differential atmospheric absorption if the variable and comparison star are not the same color. Only if all of the data are recorded through a minimal air mass or if the star has a high amplitude, can unfiltered data be counted on to produce as accurate a time of minimum as will observations obtained through a standard filter. Often observations are obtained in the Johnson V-band, and standard magnitudes in V and often in other passbands can be found in the photometry tables associated with AAVSO charts.
Figure 2. Minimum of the eclipsing binary U Sge (V-band). This binary stays at minimum brightness for a significant time, as indicated by the flat bottom to the eclipse, indicating that this is a total eclipse as opposed to a partial eclipse where there is no flat bottom.
We saw above that the AAVSO posts ephemerides with approximate times of minimum for a number of eclipsing binaries. However, in calculating your value of O-C, you probably want to use a formula that gives the time of predicted minimum with greater precision than those listed on the AAVSO ephemeris pages. The information necessary to make such calculations can be found for a selection of eclipsing binaries in the RBStarsDetails file at https://sites.google.com/site/aavsoebsection/legacy-stars. The heliocentric Julian date of minimum can be calculated from the equation
HJD(min) = epoch + period x E,
where epoch is the initial epoch of minimum light and E is the number of cycles that have elapsed since that initial epoch. Detailed formulas for calculating times of minimum for your star might also be found in publications in the astronomical literature. Many astronomical papers can be found and accessed using the NASA Astrophysics Data System (http://adsabs.harvard.edu/abstract_service.html). Remember that many eclipsing binary light curves have two minima in each period interval, and that the ephemerides may predict only the time of the main or primary minimum.
Once you have determined your O-C value, you can answer the question: Does the actual time of minimum light match that predicted by the ephemeris to within the expected uncertainty of your observations? You might want to search the astronomical literature for times of earlier minima and make a more extensive O-C diagram, a diagram of the differences between observed and predicted (calculated) times of minimum as a function of time.
Figure 3. O-C diagram for the eclipsing binary SW Lac. The points do not follow a straight line, which indicates that the period of SW Lac has changed over time.
This is discussed more completely in the Analysis (O-C) portion of the AAVSO eclipsing variables website (https://sites.google.com/site/aavsoebsection/analysis-of-times-of-minima). It is worth noting that both period changes and apsidal precession can change the observed times of minima in an eclipsing binary system, though an explanation for how each of these might occur is beyond the scope of this write up. However, the references listed at the end of this writeup can tell the reader more about why astronomers find the studying of period changes of eclipsing variables to be important to answering of a variety of astrophysical questions.
2. Observe a single maximum of a d (delta) Scuti, SX Phoenicis, or RR Lyrae star. The idea here is similar to that in section (1) above, but instead of observing the decline and recovery in brightness of an eclipsing binary star, observations cover the climb to maximum brightness of a short period pulsating star and the beginning of its subsequent decline. How long one has to observe to do that depends upon the period and behavior of the particular star being observed.
d Scuti and SX Phoenicis variables
d Scuti stars (and their population II counterparts, SX Phoenicis variables) have short pulsation periods, usually less than 0.2 day and sometimes shorter than 0.1 day. Thus, one can often observe not just the maximum but the entire light cycle with several hours of observing. Many of these variables have small amplitudes, and change in brightness by less than 0.1 magnitude in V during the pulsation cycle. However, some have larger amplitudes, approaching a full magnitude. Some pulsate with more than a single period and have amplitudes which change over time. Several variables of this type are listed in the webpage of the AAVSO Short Period Pulsator Section (https://sites.google.com/site/aavsosppsection/). A few well-known d Scuti and SX Phoenicis variables, for which AAVSO charts with comparison stars are available, are listed in the table below:
|
Variable |
Period (days) |
V amplitude (mag) |
Comments |
|
CY Aqr |
0.061038408 |
0.8 |
|
|
VZ Cnc |
0.178363704 |
0.7 |
|
|
XX Cyg |
0.134865113 |
1.1 |
|
|
DY Her |
0.148631353 |
0.5 |
|
|
VX Hya |
0.148631353 |
0.7 |
Variable amplitude, multi-periodic |
|
AN Lyn |
0.09827292 |
0.1 |
Variable amplitude |
|
SZ Lyn |
0.12053626 |
0.5 |
Short period pulsator in a binary system (not eclipsing) |
|
AE UMa |
0.086017055 |
0.7 |
Variable amplitude, multi-periodic |
With data from only a single night of observing it is often easier to interpret the light curves of those variables that pulsate mainly with a single period and which do not have strongly variable amplitudes. Since these variables show quick changes of brightness, the cadence of observations should be frequent. With observations every minute or so, the light curve can usually be well defined. The length of your own exposure times will depend both upon the brightness of the target and the size of your telescope.
Once you have observed the rise and fall of one of these stars sufficiently well to determine the time of maximum light, you can convert the observed time to a heliocentric Julian date (as suggested for eclipsing binary targets). As with the eclipsing binaries, previous observations have often allowed astronomers to construct ephemerides for these stars, predicting when times of maximum brightness occur.
Figure 4. V-band observations of the d Scuti variable CY Aqr from data in the AAVSO International Database.
You can test whether your observed time of maximum light is consistent, within the uncertainties, with the predictions of a selected ephemeris. Obviously, if your variable is known to pulsate simultaneously with several periods (VX Hya for example, has two main periods), interpreting your data will be harder. However, at least in the case of stars believed to have a single main period you can search the literature (and the AAVSO International Database) for earlier observations from which times of maximum brightness can be determined. You may need to convert those times to heliocentric Julian dates if they are not already given in that format.
If you have an ephemeris giving predicted times of maximum light, and you have observed times of several maxima, you can construct an O-C diagram. As with the eclipsing binaries, determining the correct number of pulsation cycles that has elapsed since the last prior observation is important to constructing a correct O-C diagram. This is harder to do if the period of a variable is changing but is nonetheless often possible if a star has been frequently observed. If you can construct a reliable O-C diagram, see whether the pulsating star has a changing period and think about what that might mean. It helps to remember that the pulsation period of a star usually obeys some form of the pulsation equation: P√r= Q, where P is the pulsation period, r is the mean density of the star, and Q is the pulsation constant for that type of variable. Thus, an increasing period can mean a decreasing density and a decreasing period an increasing density, possibly revealing the evolution of the star through the Hertzsprung-Russell diagram.
If you observe with more than one filter, you can determine whether the color of the star changes during your observations by making a color index – for example, B-V or V-I. A color change might be indicative of a change in surface temperature during the pulsation cycle, with a bluer color generally meaning a higher surface temperature.
If you are able to observe more than one maximum of a short period d Scuti or SX Phoenicis variable on a single night, you can use those observations to obtain a rough value of the pulsation period just from that one night of observing. With more than one maximum you can also investigate whether the light curve repeats exactly from one cycle to the next. If not, something else is going on besides the repetition of a single pulsation cycle.
RR Lyrae Stars
At 0.2 to 1.0 day, the periods of RR Lyrae stars are almost always longer than those of d Scuti or SX Phe variables. This usually makes it impossible to observe a complete light cycle during a single night of observing, except in the cases of the shortest period RR Lyrae variables. However, with a few hours of observing, the rise, peak, and start of the decline can be observed, allowing the determination of the time of maximum light. If observations can be obtained every minute or so, the light curve of an RR Lyrae variable can usually be very well defined. As with the shorter period variables, the time of maximum can be compared with the predicted time (if a prediction is available) and it might be possible to construct an O-C diagram. RR Lyrae variables in the AAVSO Short Period Pulsator program are listed in that section’s website, which also includes ephemerides predicting times of maximum light. The periods of some RR Lyrae variables have not changed in decades whereas others, such as XZ Cygni, have exhibited multiple period increases and decreases.
Figure 5. V-band observations of a maximum of the RR Lyrae star XZ Cygni, based upon data from the AAVSO International Database.
Figure 6. This O-C diagram, calculated assuming a constant period for XZ Cyg, shows that the period of that RR Lyrae star has undergone both increases and decreases during the past century.
II. Projects for several nights of data
It may be that you have several nights in which to secure observations for your target variable. In that case, the range of projects that can be attempted expands.
You might observe several minima of a single eclipsing binary star (or pulsating star, replacing minima with maxima). Do the light curves around minimum light look the same every time? Can you use your multiple minima to determine the period of the star, based only upon your own observations? You also will have more data with which to examine the O-C diagram of the variable, which you can use to determine whether your eclipsing binary (or short period pulsator) has a changing period. You may have enough data to make your own updated ephemeris for predicting times of minimum light. If you observe your eclipsing binary (or pulsating star) at times other than just near minimum (or maximum) brightness, you may be able to construct the entire light curve of the variable, showing its complete change in brightness through one cycle. Obviously, this is easier to do for a short period variable star than for one with a very long period and, as we have remarked, the complete light cycle of a d Scuti or SX Phoenicis star can sometimes be observed in a single night.
As we have already noted, some short period pulsators appear to pulsate with more than a single period, indicating the presence of more than a single pulsation mode. You might want to observe a known or suspected multimode d Scuti or SX Phoenicis star to see whether you find evidence of more than a single period.
Figure 7. The variable maxima of the SX Phoenicis variable AE UMa (V-band).
There are also RR Lyrae variables known to pulsate simultaneously in two (in rare cases even more) pulsation modes. These double-mode RR Lyrae stars typically pulsate in a fundamental mode which has a period longer than 0.5 day and a first overtone mode which has a shorter period, about 0.745 times that of the fundamental mode period. What you see is a combination of these two pulsations. The analysis of the light curves of multiply periodic variables is more complicated than is the case for stars which exhibit only a single periodicity. Nonetheless, software exists which can help you investigate such cases. Examples of this type of analysis can be found in some of the references given at the end of this discussion.
Alternatively, instead of observing a single eclipsing binary (or pulsating star) multiple times, you can observe minima of two or more eclipsing binary stars (or maxima of two or more short period pulsating stars). You can then compare the shapes of the light curves of the different stars. How are they similar? How different?
III. Projects for a semester’s worth of data
It may be that you are able to observe your targets on many nights over the course of a semester or even longer. Once again the range of projects that can be attempted expands in that case. A few projects that can be done but which require extensive data are discussed below.
If the period is not changing rapidly, it may require only one or two times of minimum per year to keep track of the periods of eclipsing binary stars. One exception is cataclysmic stars that eclipse. As you can see from the light curve of UX UMa shown below, the shape of each eclipse does not repeat because of the hot spot on the accretion disk. There is also variation outside of the eclipse. Stars of this type can sometimes be a challenge to observe, since they are fainter than many other eclipsing binaries, but they do have the advantage that the periods are short. However, although that can be an advantage for observing full cycles, it does mean that keeping track of the number of elapsed cycles can be harder than for longer period eclipsing binaries.
Figure 8. This V-band light curve shows the different shape of two different eclipses of the cataclysmic binary variable UX UMa.
You might want to observe multiple minima of eclipsing cataclysmic binaries, not only to keep track of period changes but to note the range of variability from eclipse to eclipse. In addition to UX UMa, U Gem and DQ Her are well-known eclipsing binaries of this type.
RR Lyrae Stars
With more data a more detailed investigation can be attempted for RR Lyrae variables that exhibit more than a single period. A more comprehensive study can be attempted for double-mode RR Lyrae stars, determining the period and amplitude of each pulsation mode . Alternatively, more extensive observations can be used to study the Blazhko period of one of the RR Lyrae variables that exhibits that phenomenon. RR Lyrae stars that show the Blazhko effect (named after Russian astronomer Sergei Blazhko) have a long secondary period in addition to their main pulsation period. This secondary period is often tens of days long, and the shape of the primary light cycle changes throughout this longer Blazhko cycle. Moreover, the behavior of the Blazhko effect does not always repeat precisely from one of these long cycles to the next. RR Lyrae itself, namesake of its class, is known to show the Blazhko effect, with a secondary period that has sometimes been as short as 39 days and at other times as long as 41 days. XZ Cygni, the RR Lyrae star whose O-C diagram we saw above, is also known to show the Blazhko effect. Because the Blazhko periods are usually much longer than the primary pulsation cycle, it can take many nights of observing to figure out what is going on. Nonetheless, if you have sufficient data, it can be fun to try to determine the Blazhko period of such a variable. As with our other variable star projects, you may be able to supplement your observations with data from the literature or the AAVSO International Database.
Figure 9. V-band observations of XZ Cygni, obtained over several months in 2015, show that the light curve does not repeat perfectly from cycle to cycle. This happens because XZ Cyg has a long Blazhko period in addition to its primary 0.4666-day period.
Some RR Lyrae stars with Blazhko periods short enough to be studied in a single semester are listed in the table below. Highlighted stars have been subjected to more thorough recent AAVSO observations than some of the others, but all need continued observing.
|
Variable |
Type |
Period |
Blazhko |
|
|
|
(days) |
(days) |
|
SW And |
RRab |
0.4423 |
36.8 |
|
SW Boo |
RRab |
0.5136 |
13 |
|
TV Boo |
RRc |
0.3126 |
33.5 |
|
RW Cnc |
RRab |
0.5472 |
87 |
|
TT Cnc |
RRab |
0.5634 |
89 |
|
XZ Cyg |
RRab |
0.4666 |
57.3 |
|
DM Cyg |
RRab |
0.4199 |
26 |
|
RW Dra |
RRab |
0.4429 |
41.6 |
|
XZ Dra |
RRab |
0.4765 |
76 |
|
RR Gem |
RRab |
0.3973 |
37 |
|
AR Her |
RRab |
0.4700 |
31.6 |
|
DL Her |
RRab |
0.5916 |
33.6 |
|
SZ Hya |
RRd |
0.5372 |
25.8 |
|
RR Lyr |
RRab |
0.5668 |
39 |
|
RZ Lyr |
RRab |
0.5112 |
116.7 |
|
RV UMa |
RRab |
0.4681 |
90.1 |
Cepheids
There may be instances where you can observe for many nights during a semester, but only for an hour or less per night. Cepheid variable stars with periods of days to tens of days might be good targets in that case. Their periods are sufficiently long that they can be difficult to observe adequately with only a few nights of observing. However, you might be able to get one observation a night over many nights. These can be combined to produce a complete light curve showing how the Cepheid changes in a pulsation cycle. Period changes of Cepheids can be studied using this composite light curve. Observing such stars might also provide a good opportunity for making observations with multiple filters.
Some well-known Cepheids, such as d Cephei or h (eta) Aquilae, are so bright as to be visible to the naked eye. Such stars are difficult to observe with the usual telescope-CCD camera setup. Images of these bright Cepheids saturate quickly and it may be impossible to find a comparison star of comparable brightness that fits within your CCD field of view. However, other modes of observing may alleviate these difficulties. The AAVSO DSLR manual describes how one can use a digital single lens reflex camera to obtain photometry of bright variable stars. With a reasonably dark sky, d Cephei and h Aqulae can also be seen with the naked eye. If you observe them digitally, you may be able to compare your digital magnitude measurements to those that you make with your own eye.
Figure 10. B, V, R, and I light curves of d Cephei, namesake of the class of Cepheid variable stars. Observations obtained over many nights have been folded into a single phased light curve.
Fainter Cepheids can be more readily observed with the usual telescope-CCD camera combination. A few interesting Cepheids can be found in Table 3 in the AAVSO Short Period Pulsators section.
If you do take up the observing of variables for any of the suggested projects, we strongly encourage you to report your observations to the AAVSO, where they will be of use to future researchers. The AAVSO webpage describes how to get your observer code and how to report observations to the AAVSO International Database: https://www.aavso.org/.
Useful Books, Papers, and Websites
The AAVSO website and its various subsections are always a good starting point. There you can find a wealth of information about possible observing targets, how to observe, and how to report your observations. Software is also provided for some types of data analysis.
In addition, we list below several books and papers that are illustrative of the results that can be obtained from photometric observations of eclipsing binaries and pulsating variable stars and which may be helpful exemplars in planning and analyzing your own observations. We also list some websites that might provide useful information or software.
A few possibly useful books
Introduction to Astronomical Photometry by E. Budding and O. Demircan (Cambridge Univ. Press, 2007) includes a great deal of information about how to understand the stars from photometric data. It includes a section about period changes in binary systems, and how to study this using the O-C diagram
Understanding Variable Stars by John Percy (Cambridge Univ. Press, 2007). An introduction to both eclipsing and pulsating variable stars.
Pulsating Stars by Marcio Catelan and Horace Smith (Wiley-VCH, 2015). Includes chapters on the main types of pulsating variables.
RR Lyrae Stars by Horace Smith (Cambridge University Press, 1995). Getting a bit old but till useful for basics.
Some possibly useful papers:
Analysis of an eclipsing binary star
The Algol-Type Eclipsing Binary GSC 3002-0454, 2001, Baldwin, M. E., Guilbault, P.R., Henden, A. A., Kaiser, D. H., Lubcke, G. C., Samolyk, G., and Williams, D.B., Journal of the American Association of Variable Star Observers (JAAVSO), 29, 89
Period Analysis, Photometry, and Astrophysical Models of the Eclipsing Binary TW Crucis, 2015, Moriarty, D.,
Journal of the American Association of Variable Star Observers (JAAVSO), 43, Nr. 2.
Double-mode RR Lyrae stars
Secular Variation of the Mode Amplitude-Ratio of the Double-Mode RR Lyrae Star NSVS 5222076, 2010, Hurdis, D., and Krajci, T., Journal of the AAVSO, 38, 1
Secular Variation of the Mode Amplitude-Ratio of the Double-Mode RR Lyrae Star NSVS 5222076, Part II, 2012, Hurdis, D. and Krajci, T., Journal of the AAVSO, 40, 268
Period changes of an RR Lyrae star
XZ Cygni 1965-2002, Baldwin, M.E., and Samolyk, G., 2003, Observed Maxima Timings of RR Lyrae Stars, AAVSO, No. 1.
Blazhko effect of an RR Lyrae star
The Changing Blazhko Effect of XZ Cygni, by LaCluyze et al., 2004, Astronomical Journal, 127, 1653
V1820 Orionis: an RR Lyrae Star With Strong and Irregular Blazhko Effect, de Ponthiere, P., Hambsch, F.-J., Krajci, T., Menzies, K., & Wils, P., 2013, , Journal of the American Association of Variable Star Observers (JAAVSO), 41, 58
Photometry of d Scuti variables
Monitoring Three Less-Studied d Scuti Variables: GW Ursae Majoris, BO Lyncis, and AN Lyncis, 2005, Hintz et al., Astronomical Journal, 130, 2876
A multiple period SX Phoenicis star:
VX Hydrae: Observation of a Sudden Change in a Pulsating Star, 2011, Bonnardeau, M., Dvorak, S., Poklar, R., and Samolyk, G., The Journal of the American Association of Variable Star Observers, 39, 10
Period changes of Cepheid variables
Monitoring Cepheid Period Changes From Saint Mary's University, 1999, Turner, D. G., Horsford, A. J., & MacMillan, J. D., Journal of the American Association of Variable Star Observers (JAAVSO), 27, 5
Some possibly useful websites:
AAVSO website
Peranso software (not freeware) – Can be used to determine the time of minimum for eclipsing binary data or time of maximum for RR Lyr and d Sct data. Also can be used for period searches.
Period04 software -- Another software routine(free) useful for period determinations, including the analysis of multiple periods.
Geos – Published times of maxima for RR Lyr stars.
http://rr-lyr.irap.omp.eu/dbrr/
The Blazhko Project – a little dated but provides useful information about the Blazhko effect.
http://www.univie.ac.at/tops/blazhko/
Lichtenknecker-Database of the BAV – Published times of minima and O-C diagrams for eclipsing binary stars
http://www.bav-astro.de/LkDB/index.php?lang=en&sprache_dial=en
O-C Gateway – Published times of minima and O-C diagrams for eclipsing binary stars
Rolling Hills Observatory ephemeris generator for RR Lyrae stars
http://www.rollinghillsobs.org/perl/calcRRephem.pl
With thanks to our volunteers Horace Smith, Gerry Samolyk and Neil Simmons who contributed to the projects.