what must be measured to determine distance by the cepheid variable star method?

Learning Objectives

By the stop of this section, you will be able to:

• Describe how some stars vary their light output and why such stars are important
• Explicate the importance of pulsating variable stars, such as cepheids and RR Lyrae-type stars, to our written report of the universe

Let'south briefly review the cardinal reasons that measuring distances to the stars is such a struggle. As discussed in The Brightness of Stars, our problem is that stars come up in a bewildering variety of intrinsic luminosities. (If stars were lite bulbs, we'd say they come up in a wide range of wattages.) Suppose, instead, that all stars had the same "wattage" or luminosity. In that example, the more than distant ones would always look dimmer, and we could tell how far away a star is simply by how dim it appeared. In the real universe, however, when we look at a star in our heaven (with eye or telescope) and measure its apparent brightness, nosotros cannot know whether it looks dim because it's a depression-wattage bulb or because information technology is far abroad, or perhaps some of each.

Astronomers need to observe something else about the star that allows us to "read off" its intrinsic luminosity—in effect, to know what the star'southward true wattage is. With this data, we tin can and so attribute how dim it looks from Earth to its distance. Call back that the apparent effulgence of an object decreases with the foursquare of the distance to that object. If two objects have the same luminosity but one is three times further than the other, the more afar one will await nine times fainter. Therefore, if we know the luminosity of a star and its apparent brightness, we can calculate how far away information technology is. Astronomers have long searched for techniques that would somehow allow us to determine the luminosity of a star—and it is to these techniques that we turn adjacent.

Variable Stars

The breakthrough in measuring distances to remote parts of our Milky way, and to other galaxies likewise, came from the study of variable stars. Well-nigh stars are constant in their luminosity, at to the lowest degree to within a percent or two. Like the Sun, they generate a steady flow of energy from their interiors. However, some stars are seen to vary in brightness and, for this reason, are called variable stars. Many such stars vary on a regular bicycle, like the flashing bulbs that decorate stores and homes during the winter holidays.

Let'south define some tools to assistance united states keep track of how a star varies. A graph that shows how the effulgence of a variable star changes with time is called a light curve (Figure ane). The maximum is the point of the light curve where the star has its greatest brightness; the minimum is the point where it is faintest. If the light variations repeat themselves periodically, the interval between the ii maxima is called the catamenia of the star. (If this kind of graph looks familiar, it is because nosotros introduced it in Diameters of Stars.)

Figure 1: Cepheid Lite Curve. This light bend shows how the brightness changes with time for a typical cepheid variable, with a menstruum of about six days.

Pulsating Variables

There are two special types of variable stars for which—as we will see—measurements of the light curve give us accurate distances. These are called cepheid and RR Lyrae variables, both of which are pulsating variable stars. Such a star really changes its bore with time—periodically expanding and contracting, equally your chest does when you breathe. We now empathize that these stars are going through a brief unstable stage late in their lives.

The expansion and wrinkle of pulsating variables tin be measured by using the Doppler effect. The lines in the spectrum shift toward the blue every bit the surface of the star moves toward u.s.a. and then shift to the red every bit the surface shrinks back. As the star pulsates, it also changes its overall color, indicating that its temperature is also varying. And, most important for our purposes, the luminosity of the pulsating variable also changes in a regular style every bit it expands and contracts.

Cepheid Variables

Cepheids are large, yellow, pulsating stars named for the first-known star of the group, Delta Cephei. This, past the style, is another example of how confusing naming conventions go far astronomy; here, a whole class of stars in named afterward the constellation in which the showtime one happened to be found. (We textbook authors can just apologize to our students for the whole mess!)

The variability of Delta Cephei was discovered in 1784 by the immature English language astronomer John Goodricke. The star rises rather chop-chop to maximum calorie-free and then falls more slowly to minimum low-cal, taking a total of 5.iv days for one cycle. The bend in Effigy 1 represents a simplified version of the light curve of Delta Cephei.

Several hundred cepheid variables are known in our Galaxy. Most cepheids have periods in the range of 3 to l days and luminosities that are well-nigh 1000 to x,000 times greater than that of the Sunday. Their variations in luminosity range from a few percentage to a cistron of 10.

Polaris, the North Star, is a cepheid variable that, for a long fourth dimension, varied by i tenth of a magnitude, or past nearly 10% in visual luminosity, in a flow of just nether 4 days. Contempo measurements indicate that the amount by which the effulgence of Polaris changes is decreasing and that, quondam in the time to come, this star will no longer be a pulsating variable. This is just one more piece of show that stars really do evolve and change in fundamental ways every bit they historic period, and that beingness a cepheid variable represents a stage in the life of the star.

The Menses-Luminosity Relation

The importance of cepheid variables lies in the fact that their periods and average luminosities turn out to be directly related. The longer the period (the longer the star takes to vary), the greater the luminosity. This menstruation-luminosity relation was a remarkable discovery, one for which astronomers still (pardon the expression) thank their lucky stars. The flow of such a star is easy to measure: a skilful telescope and a good clock are all you need. In one case you lot have the period, the relationship (which tin be put into precise mathematical terms) will give y'all the luminosity of the star.

Let's be clear on what that means. The relation allows you to essentially "read off" how bright the star really is (how much energy it puts out). Astronomers tin then compare this intrinsic brightness with the apparent brightness of the star. As nosotros saw, the deviation between the two allows them to summate the distance.

The relation between period and luminosity was discovered in 1908 by Henrietta Leavitt (Figure ii), a staff fellow member at the Harvard College Observatory (and one of a number of women working for low wages assisting Edward Pickering, the observatory's director; come across Annie Cannon: Classifier of the Stars). Leavitt discovered hundreds of variable stars in the Big Magellanic Cloud and Small Magellanic Cloud, 2 great star systems that are really neighboring galaxies (although they were not known to exist galaxies so). A pocket-size fraction of these variables were cepheids (Figure 3).

Figure 2: Henrietta Swan Leavitt (1868–1921). Leavitt worked equally an astronomer at the Harvard Higher Observatory. While studying photographs of the Magellanic Clouds, she found over 1700 variable stars, including 20 cepheids. Since all the cepheids in these systems were at roughly the same distance, she was able to compare their luminosities and periods of variation. She thus discovered a primal relationship betwixt these characteristics that led to a new and much amend fashion of estimating cosmic distances. (credit: modification of work past AIP)

These systems presented a wonderful opportunity to study the behavior of variable stars independent of their distance. For all applied purposes, the Magellanic Clouds are so far away that astronomers can assume that all the stars in them are at roughly the same distance from us. (In the same manner, all the suburbs of Los Angeles are roughly the same distance from New York City. Of course, if you are in Los Angeles, you will observe annoying distances between the suburbs, but compared to how far away New York City is, the differences seem small-scale.) If all the variable stars in the Magellanic Clouds are at roughly the aforementioned altitude, then whatsoever difference in their apparent brightnesses must be caused past differences in their intrinsic luminosities.

Figure iii: Big Magellanic Cloud. The Large Magellanic Cloud (and so named considering Magellan'south crew were the showtime Europeans to record it) is a small, irregularly shaped milky way near our ain Milky way. It was in this galaxy that Henrietta Leavitt discovered the cepheid period-luminosity relation. (credit: ESO)

Leavitt found that the brighter-appearing cepheids ever have the longer periods of light variation. Thus, she reasoned, the period must be related to the luminosity of the stars. When Leavitt did this work, the distance to the Magellanic Clouds was not known, and so she was only able to show that luminosity was related to period. She could non make up one's mind exactly what the relationship is.

To define the period-luminosity relation with bodily numbers (to calibrate it), astronomers first had to measure the actual distances to a few nearby cepheids in some other mode. (This was accomplished by finding cepheids associated in clusters with other stars whose distances could be estimated from their spectra, as discussed in the side by side section of this chapter.) Just once the relation was thus defined, it could give us the distance to any cepheid, wherever it might be located (Figure 4).

Figure 4: How to Use a Cepheid to Measure Distance. (a) Find a cepheid variable star and mensurate its catamenia. (b) Use the flow-luminosity relation to summate the star's luminosity. (c) Measure the star's apparent brightness. (d) Compare the luminosity with the apparent effulgence to calculate the distance.

Hither at concluding was the technique astronomers had been searching for to break the confines of altitude that parallax imposed on them. Cepheids can exist observed and monitored, it turns out, in many parts of our own Galaxy and in other nearby galaxies as well. Astronomers, including Ejnar Hertzsprung and Harvard'south Harlow Shapley, immediately saw the potential of the new technique; they and many others set up to work exploring more than distant reaches of infinite using cepheids every bit signposts. In the 1920s, Edwin Hubble made i of the nigh significant astronomical discoveries of all time using cepheids, when he observed them in nearby galaxies and discovered the expansion of the universe. As we will see, this work still continues, as the Hubble Infinite Telescope and other modern instruments try to identify and measure individual cepheids in galaxies farther and farther abroad. The near afar known variable stars are all cepheids, with some about 60 million light-years away.

John Goodricke

The cursory life of John Goodricke is a testament to the human spirit nether arduousness. Born deaf and unable to speak, Goodricke nevertheless made a number of pioneering discoveries in astronomy through patient and careful observations of the heavens.

Figure five: John Goodricke (1764–1786). This portrait of Goodricke by creative person J. Scouler hangs in the Royal Astronomical Club in London. At that place is some controversy about whether this is actually what Goodricke looked like or whether the painting was much retouched to please his family. (credit: James Scouler)

Built-in in Kingdom of the netherlands, where his father was on a diplomatic mission, Goodricke was sent back to England at age eight to written report at a special schoolhouse for the deaf. He did sufficiently well to enter Warrington Academy, a secondary school that offered no special assistance for students with handicaps. His mathematics instructor there inspired an interest in astronomy, and in 1781, at historic period 17, Goodricke began observing the sky at his family home in York, England. Within a twelvemonth, he had discovered the effulgence variations of the star Algol (discussed in The Stars: A Celestial Census) and suggested that an unseen companion star was causing the changes, a theory that waited over 100 years for proof. His newspaper on the subject area was read earlier the Royal Society (the master British grouping of scientists) in 1783 and won him a medal from that distinguished group.

In the concurrently, Goodricke had discovered two other stars that varied regularly, Beta Lyrae and Delta Cephei, both of which continued to interest astronomers for years to come. Goodricke shared his interest in observing with his older cousin, Edward Pigott, who went on to find other variable stars during his much longer life. But Goodricke's fourth dimension was quickly drawing to a shut; at age 21, simply 2 weeks after he was elected to the Regal Guild, he caught a cold while making astronomical observations and never recovered.

Today, the Academy of York has a building named Goodricke Hall and a plaque that honors his contributions to science. However if you lot get to the churchyard cemetery where he is buried, an overgrown tombstone has just the initials "J. K." to show where he lies. Astronomer Zdenek Kopal, who looked advisedly into Goodricke'due south life, speculated on why the marker is so small-scale: perhaps the rather staid Goodricke relatives were ashamed of having a "deaf-mute" in the family and could not sufficiently appreciate how much a human who could not hear could all the same see.

RR Lyrae Stars

A related group of stars, whose nature was understood somewhat afterwards than that of the cepheids, are chosen RR Lyrae variables, named for the star RR Lyrae, the best-known fellow member of the group. More common than the cepheids, but less luminous, thousands of these pulsating variables are known in our Galaxy. The periods of RR Lyrae stars are always less than 1 twenty-four hour period, and their changes in brightness are typically less than well-nigh a factor of two.

Astronomers have observed that the RR Lyrae stars occurring in any particular cluster all take about the same apparent brightness. Since stars in a cluster are all at approximately the aforementioned distance, it follows that RR Lyrae variables must all take most the same intrinsic luminosity, which turns out to be about 50 L _{Lord's day}. In this sense, RR Lyrae stars are a piffling bit like standard light bulbs and can also be used to obtain distances, particularly within our Galaxy. Figure 6 displays the ranges of periods and luminosities for both the cepheids and the RR Lyrae stars.

Figure half-dozen: Catamenia-Luminosity Relation for Cepheid Variables. In this class of variable stars, the time the star takes to become through a wheel of luminosity changes is related to the average luminosity of the star. Besides shown are the period and luminosity for RR Lyrae stars.

Key concepts and summary

Cepheids and RR Lyrae stars are ii types of pulsating variable stars. Light curves of these stars bear witness that their luminosities vary with a regularly repeating period. RR Lyrae stars can be used as standard bulbs, and cepheid variables obey a period-luminosity relation, then measuring their periods tin tell u.s.a. their luminosities. Then, we tin calculate their distances past comparing their luminosities with their apparent brightnesses, and this tin can allow u.s. to measure distances to these stars out to over sixty one thousand thousand low-cal-years.

Glossary

cepheid: a star that belongs to a course of yellowish supergiant pulsating stars; these stars vary periodically in brightness, and the relationship betwixt their periods and luminosities is useful in deriving distances to them

calorie-free curve: a graph that displays the fourth dimension variation of the light from a variable or eclipsing binary star or, more mostly, from any other object whose radiations output changes with fourth dimension

catamenia-luminosity relation: an empirical relation between the periods and luminosities of certain variable stars

pulsating variable star: a variable star that pulsates in size and luminosity

RR Lyrae: 1 of a course of giant pulsating stars with periods shorter than ane 24-hour interval, useful for finding distances

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Source: https://courses.lumenlearning.com/astronomy/chapter/variable-stars-one-key-to-cosmic-distances/

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