Light Curves of Variable Stars

Variable stars are stars whose brightness changes over time. We will observe two naked-eye variable stars (Delta Cephei and Beta Lyrae) over a period of several weeks, then graph their light curves to verify their periods and ranges of brightness.

Background Reading: Stars & Planets, p. 279-281 (Variable Stars) ; p. 9-10 (Star brightness); p. 263-277 (Life cycle of Stars); p. 278 (Double and Multiple Stars)


To the naked eye, the vast majority of stars appear to be constant both in location (relative to other stars) and in brightness. It turns out that neither of these is true; stars do change in position and brightness, but more slowly than can be easily noticed during one human lifetime. A few special stars, however, vary their brightness with a rather short, regular period. These variable stars are now known (with modern observational methods) to number well over 30,000 in our region of the Milky Way alone (there exist also non-periodic variables, which we do not consider here). As measurements of stellar brightnesses become more and more precise, the number of stars known to be variable in both our own Galaxy and other nearby galaxies continues to grow.

WHAT CAUSES VARIABILITY?

Variable stars are classified based on the actual physical mechanism that causes their brightness to vary. Pulsating variables are stars which brighten and dim due to a physical change in their size and surface temperature. We will observe Delta Cephei, which belongs to this class of variable stars.

Eclipsing binaries appear to brighten and dim because the light from one star is blocked from view (occulted or eclipsed) by an orbiting companion star. As a result, the light from an eclipsing binary is generally constant except for those brief periods when one star is actually in front of the other. While Algol (the "demon star" in Perseus) is the most famous and easily observed example, Perseus will not be easily visible for us until the end of the Fall semester. Instead, we will observe Beta Lyrae, another naked-eye eclipsing binary.

HOW PULSATING VARIABLES WORK

Most stars, like our own Sun, are in a tug-of-war-like state of hydrostatic equilibrium: gravity tries to make the star smaller, but the heat released by nuclear reactions at its center tries to make the star expand. These two forces balance when the star is at its equilibrium size, and that gives the constant radius that we see for any non-variable star, like our Sun. In the elegant words of Frank Shu (The Physical Universe, page 94), "The present radius of the Sun has just the right value to maintain a central temperature which provides a rate of nuclear energy generation that exactly balances the leakage rate due to the random walk of the photons from center to surface". The leakage rate is controlled by the opacity of the gas composing the star.

As stars get older, however, there are changes in the star's internal structure, and sometimes the star goes through an unstable stage, during which it alternatively expands and contracts. To fix ideas, we will consider Delta Cephei, the prototype of the "cepheid" pulsating variables. In order to understand what happens, let us consider first a stable, non-variable star. Suppose that, at first, the star's radius R decreases in response to some very slight internal perturbation. Having been disturbed, the star is now slightly out of equilibrium, and a damped oscillation starts which will bring the star back into hydrostatic equilibrium. Note that nothing happens to the most internal layers of the star; the nuclear sources of energy are not affected, and we are concerned only with the behavior of the outer layers of the star. Let us follow the consequences of the perturbation. The star becomes overcompressed (smaller radius) which increases the temperature. Higher temperature normally translates into lower opacity. Then more photons leak out from the overcompressed region. You can think of this as a kind of "valve" that opens and releases some of the overpressure that drives the gas outward. Thus the star does not expand as much as it would in the absence of the "valve effect". When the star is maximally expanded the temperature drops, which means more opacity. Now more photons are trapped, and pressure builds up faster, helping to stop the subsequent contraction, so that the star quickly regains hydrostatic equilibrium and the oscillation is damped.

Now we can try to understand what happens to the cepheid variable. In this special case, at a certain depth within the star, not far from the surface, the opacity increases with temperature (because Helium is being ionized). Now, as a result of the initial perturbation, the "valve" is acting in the opposite way, and sets up forces that more than suffice to restore equilibrium; the overreaction of the system causes the star to overshoot. The following cycle is operating:

As the star's radius R grows and shrinks, the star's size physically pulsates, giving these stars their name "pulsating variables." R swings back and forth around some average radius, on each pulsation first overshooting one way, then overcorrecting the other way. The surface temperature of the star also varies with the same period, as a consequence of the changes happening inside. The luminosity of the star is a function of its radius and surface temperature. In the case of the cepheid, maximum luminosity happens when the star is close to maximum surface temperature, with the radius at intermediate values, but increasing at the maximum rate; and minimum luminosity happens when the star is close to minimum surface temperature, with the radius approximately the same as at maximum, but now decreasing at the maximum rate (do not worry, you do not need to remember these details). The periodic increase and decrease of the star's luminosity is what we see with our eyes (or telescopes) as varying brightness. This cycle can repeat with amazing regularity for thousands of years! Eventually, the pulsations will stop because of further changes in the internal structure of the star, as it evolves into older age and prepares to run out of nuclear fuel and end its nuclear-energy-generating life by becoming a white dwarf.

The period of pulsating variables can be anywhere from hundreds of days down to only a few hours. The longer the period, the less dense, larger and more luminous the pulsating star is.

CLASSES OF PULSATING VARIABLES

Pulsating variables are divided into classes depending on their period and light curve shapes. We will mention just a few of the most important classes. We have already mentioned the cepheid variables. They have periods between 5 and 60 days. Cepheids are named after the first such star known and studied, Delta Cephei, a naked-eye variable in the constellation Cepheus. In 1908, Henrietta Leavitt discovered that all Cepheid variables have a simple relationship between their period of variability and their average luminosity. Since astronomers have precious few tools to use for determining the distance to objects in the universe, this was a discovery of tremendous importance. Today, this Period-Luminosity Relationship remains one of the most reliable ways to find distances to nearby galaxies.

RR Lyrae variables are similar to Cepheids, but have periods of only 0.5 to 1 day, and can be observed to go from their brightest to faintest magnitude within a single night. Named after the star RR in the constellation Lyra, they are smaller and less luminous than Cepheids. Finally, long-period or Mira variables have periods longer than 100 days, and are named after the star Mira in the constellation Cetus, whose brightness varies from roughly 3rd to 9th magnitude every 332 (or so) days. These stars' variations are not always as regular or as predictable as those of Cepheids and RR Lyraes.

If we make many measurements of the brightness of a variable star and graph them over time, we create a graph known as a light curve. Notice how light curves are "folded" by the period of variability of the star. For example, if the star is known to repeat its variability every 7.0 days, then the horizontal time axis wraps around to the origin again every 7.0 days. This measurement of time "folded" using the period is called phase, and can be expressed either in days, or as the fraction of the star's cycle from start (0.0) to finish (1.0). Here are two such light curves of Cepheid variable stars:

   

You may notice that both Cepheid variables above brighten to their maximum faster than they dim to their minimum. They are "classical Cepheids," as is our star, Delta Cephei. There are other classes of pulsating variables (different age, chemical composition, etc.) whose light curves lack this rapid brightening:

...but they all share a similar overall shape that identifies them clearly as pulsating variables and not eclipsing binaries.

PHASE CALCULATION

If we know the period P of the variable star, and designate some arbitrarily selected time t0 as "phase 0", then we can calculate the phase for any other time t using the following formula: phase = decimal part of [(t-t0)/P]. Of course this phase is a dimensionless number going from 0 to 1. It allows us to better define the light curve by combining observations obtained along a time interval much longer than the period of the variable star.

ECLIPSING BINARIES

The universe is full of binary stars: pairs of stars which are in close orbit about each other. Observational surveys reveal that upwards of 80% of all stars are binary or even multiple; solo stars like our own Sun appear to be the exception.

If binary stars are so common, why don't we see a sky full of eclipsing binary variables? Because a binary system only looks like an eclipsing system if the binary orbit happens to be oriented exactly "edge-on" when viewed from Earth. In that case, one of the binary stars passes exactly through the line of sight from Earth to the other binary star, thereby blocking our view of (occulting) the light from its partner. One half-orbit later, the same thing happens with the two stars reversed: the partner now occults the light from the first star.

Imagine two stars of exactly equal size and brightness, whose orbit is aligned perfectly edge-on to the line of sight from Earth. Most of the time we would have a clear view of both stars side-by-side, and we would measure a combined full "100%" brightness. However, there will be two times during each orbit when one star will pass in front of the other. As one star begins to block our view of the other, we would see the combined brightness of the binary decrease; at the moment they are perfectly aligned, one in front of the other, we would measure only 50% brightness, as the light from the far star is blocked completely by the near star. This occultation, and its corresponding dip to 50% brightness, takes only a small fraction of the time of the full orbit. By examining the shape and spacing of these dips in the light curve, we can deduce some properties of the binary pair:

Here is a computer-model light curve of an eclipsing binary with stars of unequal diameters/brightnesses:

Real-life light curves possess this same overall shape, but each has its own distinctive, non-ideal features:

   

Light curves of eclipsing binaries look substantially different from those of pulsating variables. The binary pair spends much of its time unocculted, so its light curve will spend most of its time near maximum (100%) brightness, punctuated by two relatively sharp dips. By contrast, a pulsating variable is always either growing or shrinking, so its brightness never looks steady, but varies continuously between its maximum and minimum values.

 

OBSERVATIONS

Beta Lyrae (Fall semester)

The most useful nearby stars for comparing its brightness are two of the other stars in the "parallelogram" part of Lyra:

Here is a GIF-format star chart for Beta Lyrae and its comparison stars.

Delta Cephei (Fall semester)

The two most useful nearby comparison stars form the short side of a wedge-shaped triangle with Delta, where Delta is at the point of the wedge. The entire triangle all fits nicely within one binocular field of view:

Here is a GIF-format star chart for Delta Cephei and its comparison stars.

 

Making your observations

Compare a variable star with one of its comparison stars, decide whether the variable star is brighter or fainter, then choose another comparison star and repeat. Your goal is to "bracket" the brightness of your variable star between that of two of its comparison stars, and then make a judgment on the variable's magnitude to the nearest one-tenth of a mag. Don't forget: smaller magnitudes are BRIGHTER; larger magnitudes are FAINTER. For example, suppose that you observe Delta Cephei to be much fainter than its magnitude 3.6 comparison star, but only just slightly brighter than its magnitude 4.2 comparison star. You should record your observation of Delta Cephei as having magnitude 4.1 or 4.0, depending on how dramatic the difference is. If you really can't decide, then record your observation as a range: "4.0-4.1 mag." Don't be discouraged if at first you can't distinguish differences smaller than half a magnitude or so -- with practice, your ability to distinguish more subtle differences in brightness will improve.

Do your best to ignore your expectations for the magnitude of each variable star, and approach each of your observations with an open mind. If the variable star looks "too bright" or "too faint" for what you expect from the night before, you will be tempted to revise your measurement to be more in line with your expectations. Do NOT do this! Look carefully, but record what you see, not what you expect. You can seek explanations for why your measurement might be "off" (using the wrong star for comparison, very bad viewing conditions, etc.), but if you can't find anything wrong with your method, then you should ultimately record what your senses tell you. Perhaps it is just a "random error" (which is entirely expected in science), or perhaps the star really is doing something unexpected! (If you suspect one of your measurements is waaaaaay "off," then discuss it with your instructor. A part of learning to do science is developing judgment about which "off" measurements are the effect of random error fluctuations vs. which might actually be "wrong" and can be thrown out.)

Observing tips and techniques

Recording your observations

Each time you make an observation, be sure to record:
1. Date AND time of observation
2. Magnitude(s) of your target variable star(s)
(to the nearest 0.1 mag)
3. Any relevant comments or problems: were binoculars used, or just naked eye? thin clouds? bright moonlight? twilight? etc. (I recommend a ranking system for how good you think each observation is. For example, I use: "1" for excellent; "2" for average; "3" for questionable.)

Observe the brightness of each variable star at least 12 times over the several weeks of this lab experiment. More than 12 observations is even better, but at least 12 are required for full credit. Do NOT limit yourself to our lab sessions only; you will need to make measurements on your own at least one additional time each week. After you do it a few times, you will probably memorize the important comparison stars, and it will be quick and easy to make an observation. Since our variable stars take several days to vary from brightest to faintest magnitude, observations separated by only a few hours are not useful, so you should make no more than ONE measurement per night.

 

REVIEW QUESTIONS

 

WEB RESOURCES

 

REPORT: Light Curves of Variable Stars

Make the observations described in the OBSERVATIONS section above, and write a report on your work. Include the following:

1. Brief introduction explaining the two main variable star types, and the purpose of your observations
2. Brief procedure section explaining how your observations were made
3. For each variable star, a 4-column table compiling all your observations:
       a. date & time
       b. phase -- your instructor will explain how to calculate this
       c. magnitude
       d. comments
4. For each variable star, a graph of your observations:
        Light curve of the variable star's brightness versus phase
5. Brief discussion of your conclusions from your data (do they agree with what you would expect for a pulsating variable or an eclipsing binary?) and any sources of error or other problems

If you have fewer than 12 observations of each variable star, you will need to borrow a few observations from classmates to bring your total to 12. If you do so, you must: (1) choose observations for nights other than the dates you have already observed; (2) note in your data table's "comments" column the name of the person from whom you received the data.

You do not need to include sketches of the comparison stars or surrounding constellations in your report, unless you want to. You can refer to the names of stars assuming that your reader/grader has ready access to star charts of the region.


Roberto H. Méndez (mendez@ifa.hawaii.edu)

Last modified: October 10, 2005
http://www.ifa.hawaii.edu/~mendez/ASTRO110LAB05/variables.html