NOT ALL STARS SHINE AS steadily as the sun. A small percentage visibly fluctuate in brightness. But far from being mere curiosities, variable stars offer much information about themselves and the universe as a whole. Among the most useful are the so-called type I Cepheids, because they are good indicators of distance. This property comes about because a definite relation exists between a Cepheid's period of variability and its luminosity: the longer the period, the intrinsically brighter the star. Finding a Cepheid's period and measuring the apparent brightness enable astronomers to deduce its distance from the earth [see "The Expansion Rate and Size of the Universe," by Wendy L. Freedman, page 54].

Getting accurate distance figures is usually not straightforward. Because of the range of magnitudes in the period-luminosity relation, distance calculations can be off by a factor of two or three. To correct for this uncertainty, astronomers generally rely on observations of several Cepheids in proximity. For example, I discovered 29 Cepheids in the galaxy NGC 3109. I fit their apparent magnitudes at maximum light into the graph of the period-luminosity relation [see Figure 3]. The fit yielded a distance of 1.9 megaparsecs. Even this figure is only approximate, since it is not corrected for absorption by the intervening interstellar matter. Accurate distance measurements usually demand large telescopes, filters and complicated techniques to extrapolate measurements from one celestial body to another. As such, determining distances is a task best left to the professional.

Yet the amateur is not left completely out in the cold. You can readily determine a Cepheid's brightness over time and display the information as a light curve. One of the most useful resources in making observations of variable stars is the Astronomical Almanac. This reference (available from the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C., 20402) lists the variable stars visible to the naked eye. The information includes the names of the stars, their type and . location and their period of variability. The almanac also lists a recent time of maximum light, which can be used to predict future maximum brightness.

Stellar brightness is expressed in terms of the magnitude scale. Each unit on the scale corresponds to a luminosity difference of 2.5 times. Fainter stars have larger magnitude numbers; the brightest objects in the sky have negative numbers. For instance, the sun shines at magnitude -26. The unaided eye can detect objects up to a magnitude of about six.

The times given in the almanac are in Julian days, which is a running decimal number related to the dates and times of events. Julian dates are much more convenient than ordinary calendar dates. To obtain the interval between events, you simply subtract the dates. Section B of the almanac lists the equivalent calendar dates.

The general strategy for observing Cepheid variables is to measure the brightness of a Cepheid at many times over many cycles. There are two ways to collect the data on light variation. The first is simply to use the unaided eye and make informed estimates. The second is to photograph the star and determine the brightness by measuring the size of the image.

In either case, begin by making the first set of measurements close together over one or two cycles, so you can get a rough idea of the period. Then wait for a few cycles and take a second set of measurements. But do not wait too long: when the number of cycles multiplied by the uncertainty in the period equals one period, you would be off in the cycle count by one. Pause for several more cycles before taking a third set of observations. You can repeat this process indefinitely. Knowing the number of cycles between the observations, you can "phase" together your observations to produce an accurate value for the period. I use a computer program to perform the task; the section "Finding the Period" below describes the algorithm.

The procedure to observe Cepheids with the naked eye is simple in principle and is attributed to the German astronomer Friedrich W. A. Argelander You select ordinary (nonvariable) stars to compare with the Cepheids. The apparent magnitudes of the comparison stars should cover the full range of the variable's brightness. The comparison stars should be located in the sky as closely as possible to the variable, so that each comparison star and the variable are in the same field of view. Ideally, the comparison stars should differ in steps of 0.3 to 0.5 in magnitude. I label each comparison star in an alphabetical order based on brightness, with "a" being the brightest.

At the time of observation, decide which two comparison stars have magnitudes that bracket those of the variable. Then try to estimate to the nearest tenth the relative brightness of the Cepheid in the bracketed interval. Thus, if you estimate that the magnitude of a variable is 0.4 between those of comparison stars b and c, write b4c as the magnitude (the Cepheid observer's shorthand). You can convert to the actual value by interpolating between the known magnitudes of b and c.

Of course, suitable comparison stars may not lie close enough to the variable. In that case, you would need to shift your line of sight back and forth between the stars to estimate relative brightness. Visual memory is very short; when you concentrate on the seconcl star, your eye has "forgotten" how bright the first one looked. In addition the twinkling of stars makes it more difficult to judge their brightness. Even so, with practice, your ability to estimate brightness can improve. In fact, with only three levels of brightness, the period of a variable can be found with reasonable accuracy.

Try the technique on Delta Cephei, which has an apparent visual magnitude of 3.48 at maximum and 4.37 at minimum. Locate Beta Cephei (magnitude 3.23) from a star chart and use it as comparison star a. Use Epsilon Cephei, at a magnitude of 4.19, as comparison star b. Observe Delta Cephei once a night for a week or two, and write down its time and brightness as compared with a and b. If the brightness of Delta Cephei is in between those of a and b, simply write a5b. These three values can provide a crude estimate of the period. With a little practice and some luck with the weather, you should be able to refine your estimates. The almanac lists other Cepheids and suitable comparison stars if Delta Cephei is not easily seen from your location.

A more accurate means to record Cepheid variability is through photography. Because film can detect objects K' too faint for the naked eye, you can conduct a photographic study of Cepheids not even listed in the almanac. You will need a camera mounted to your telescope and a clock-drive unit (a tracking mechanism that follows the apparent motion of a celestial body).

For a variable such as Delta Cephei, a typical exposure through a small telescope would be about 10 seconds. You should, however, take several photographs with different exposure times and select the best time. I recommend doing your own developing so that you can keep the processing consistent from one roll of film to the next. Film records the light intensity of stars as size: the brighter the star, the larger the image. Thus, you will need to correlate the diameter of a stellar image with a star whose magnitude is known (magnitudes are printed in the almanac and other handbooks). In my astronomy course, I use the open cluster NGC 6940 to teach my students how to perform the task. I use a 12-power comparator (essentially a loupe) to measure the images of the 15 stars in the cluster on an enlarged print. The comparator contains a reticle that has a linear scale with 0.1-millimeter divisions (available from scientific-equipment supply companies). You should be able to estimate sizes to within a few hundredths of a millimeter.

A plot of the magnitudes of these stars against the logarithm of the square of the diameters yields a straight line [see illustration below right]. Once calibrated, you can find the magnitude of any star (including possible Cepheids) in the field of view. Just measure the star's diameter and use the calibration line to convert. Each photograph will have to be calibrated in this way.

Indeed, it may be that the magnitudes of the stars you need for calibration are not listed in any handbook. In that case, you must calibrate those stars against stars whose magnitudes are listed. This secondary calibration needs to be done only once. You can then use these secondary standards against which to measure the variable.

To ensure accuracy in secondary calibrations, you should take certain precautions. First, use the same exposure time for all the photographs. Second, shoot everything on the same roll of film, so that the images experience the same developer concentration and processing time. Third, work from negative prints; it is easier to measure dark images on a light background rather than vice versa. (I make such prints from contact negatives of the original negatives.) Finally, correct for the fact that you are looking along different optical paths through the atmosphere.

The most systematic method is to convert your magnitudes to what they would be if you were looking straight up. For visual wavelengths, the amount of correction is equal to about 0.14 (secant z - 1), where z is the angle between the zenith and photographed field. Once you have calibrated your photographs, you can find the period by using the algorithm described in "Finding the Period" below.

You can construct light curves by plotting magnitude versus phase. Remember to plot the smaller magnitudes (brighter values) toward the top of the ordinate. The conventional way to display light curves is to show two complete cycles.

If you decide to pursue more advanced kinds of observations, you can obtain instructions and software for data analysis from many sources. One of the most useful is the American Association of Variable Star Observers (AAVSO), which can supply finder charts for thousands of variable stars as well as their corresponding comparison stars. The association can recommend projects suitable to each observer's geographic location, equipment, observing conditions and schedule. Interested readers can contact the AAVSO at 21 Birch Street, Cambridge, MA 02138. Amateur astronomy magazines also provide information on Cepheids.

Finding the Period

For a sample data set and a copy of the program that finds the period (written in FORTRAN), send a self-addressed, stamped business envelope to the Amateur Scientist, Sighting Cepheid Variables, Scientific American, 415 Madison Avenue, New York, NY 10017-1111.

Bibliography

THE REALM OF THE NEBULAE. Edwin P. Hubble. Yale University Press, 1936 (reprint 1982).

THE NATURE OF VARIABLE STARS. Paul W. Merrill. Macmillan Company, 1938.

CEPHEIDS: THEORY AND OBSERVATIONS. Edited by Barry F. Madore. Proceedings of the IAU Colloquium No. 82. Cambridge University Press, 1985.

THE EXTRAGALACTIC DISTANCE SCALE. Edited by Sidney van den Bergh and Christopher J. Pritchet. Astronomical Society of the Pacific Conference Series, Vol. 4; 1988.

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