Cepheid variables

Type of physical science: Astronomy; Astrophysics

Field of study: Stars

A star whose brightness changes over time is called a variable star. Among the periodic variables, the most significant are the Cepheid variables, for they have provided a key to the distance scale of the universe.

Overview

For more than three centuries, stars have been known to vary in brightness. In the late 1600s, the bright star Algol (Beta Persei, commonly called “the ghost”) was explained successfully as an eclipsing two-star (binary) system. In the last two hundred years, many thousands of variable stars of different types have been discovered, both in the Milky Way and in other galaxies. Variables that do not have a name already assigned, such as Polaris, Betelgeuse, or Delta Cephei, are labeled starting with R to Z followed by the possessive of the constellation name; for example, R Coronae Borealis. Then, the letters RR, RS, through ZZ are used. Finally, AA through QZ, omitting J, are used. This system allows for 334 variables in each constellation.

After that, V335, V336, and so on are used.

Several categories of variable stars exist. The broadest classification is into extrinsic and intrinsic variables. The extrinsic variables change brightness because of circumstances external to the star, while intrinsic variables change because of internal physical changes. Most extrinsic variables are eclipsing star systems whose orbital plane is close enough to astronomers’ line of sight for the stars to pass in front of one another.

There is a wide variety of intrinsic variables. They may be exploding stars, such as novas and supernovas, whose brightness increases by many magnitudes; for example, Nova Aquilae 1918 increased eight magnitudes, or fifteen thousand times in brightness. Others vary irregularly, some erratically, some irregularly cyclical, and some semiregularly. Still other variables are periodic.

Among these are two main types: those that display multiple periodicities, such as the Beta Canis Majoris stars and the Delta Scuti stars, and those that are regularly periodic, such as the RR Lyrae (periods of less than a day), the Cepheids with periods from about one-and-a-half to more than fifty days, and the long-period variables with periods above fifty days.

The Cepheids, named after Delta Cephei, whose variability was discovered in 1784, are yellow supergiant stars and the most useful of the variable stars. They have attracted extensive astronomical attention because of their contribution to distance measurement. For the more than six hundred Cepheids discovered within the Milky Way, the periods cluster at seven days, with very few having periods shorter than three days or longer than thirty days. The central tendency varies from galaxy to galaxy in a fashion related to the evolutionary history of the galaxy.

When brightness is plotted against time, a “light-curve” is produced. If the period is more than a few hours, the light-curve has to be pieced together from observations over many nights or with observations from different observatories, a difficult and tedious process. The light-curves of all Cepheids are not the same, although most of them brighten more rapidly than they fade. Each has a characteristic shape with bumps in the curve, a sort of unique fingerprint for each star. Further, the shape of the curve is related to the length of the period of variation.

This is true of Cepheids from other galaxies as well as the Milky Way, an indication that they are the same wherever they occur. Although Cepheids vary from one another, each is extremely regular, with the length of the period constant to one part in a million, but over very long periods (such as decades or centuries), their periods can experience subtle changes.

When Cepheids change brightness, they also change in color and in character of their spectrum. As expected, when brightest, they have the highest surface temperature and are blue in color (or spectral type F). As the Cepheid approaches its brightest level, the lines in its spectrum move toward the violet end, indicating that the surface of the star is moving toward Earth; that is, it achieves maximum radial velocity where radial velocity is motion along the line of sight. As the star approaches its minimum brightness, the spectrum is displaced toward the red end of the spectrum, indicating that the surface is moving away from Earth. Hence, at minimum light, they cool down and are generally of spectral type G. The implication of these motions is that the change in brightness is related indirectly to the physical vibration or pulsation of the star. The brightness is related directly to changes that take place in the surface temperature of the star at the same time as the pulsations. An average Cepheid is estimated to change radius by 7 to 8 percent during a pulsation cycle and changes about one magnitude in brightness during a cycle.

Accumulating intensive studies of Cepheids have established that the period of a star that pulsates as a result of gravitational controls will depend in an inversely proportional way upon its density. The period-luminosity relationship, however, flows more from the relationship between mass and luminosity, which holds for all stars, rather than from pulsation theory. Most notable is that Cepheid-type pulsation occurs only with stars in a fairly narrow band of temperatures related to spectral classes F and G. There is, similarly, a limited range in luminosities from about -1.5 to -6.

The pulsations are associated with a layer of ionized hydrogen and helium below the surface of the star. As the star pulsates, the gases alternately ionize and recombine, alternately impeding and permitting the outward flow of radiation. The reason for the limited temperature range for the variables is that a higher temperature results in this layer being too near the surface of the star to be effective. If it is too cool, the layer is too deep in the star and is essentially smothered and ineffectual. In stable stars, the outward pressure of radiation matches the inward pressure of gravity. Cepheids oscillate around this equilibrium point. While these features alone would make Cepheids fascinating objects for study, the most striking feature of all is that the longer the period, the brighter or more luminous the star on an absolute scale. This was the discovery that allowed the Cepheids to be used as a measuring rod for the scale of the universe and made them very significant in the development of twentieth-century astronomy.

Applications

Cepheids would have been just another astronomical phenomenon of interest had it not been for their role in establishing the scale of distance in the universe. At the beginning of the twentieth century, the most reliable means of establishing distance was through triangulation, or measurement of parallax. This method used the orbit of Earth as a baseline, and while that distance is great, on the cosmic scale, it is minute. As a consequence, stars could be measured out to only a few hundred years. Therefore, the search was underway for alternative means of establishing the distance of stars. Several statistical methods based on brightness, color, and other measures were attempted with varying degrees of success.

The major breakthrough came in 1912 when Henrietta Swan Leavitt was working at Harvard’s southern astronomical station at Arequipa, Peru. She had been surveying the Magellanic Clouds, two small satellite galaxies to the Milky Way. Leavitt had already discovered thousands of Cepheid variables in these clouds. One evening before beginning her measurements, she drew up a list of twenty-five of the variables according to their brightness.

She noticed immediately that they were also arranged according to period. The brighter the star, the longer the period. By that simple task, she had discovered the period-luminosity relationship.

She recognized the significance of this discovery and realized that if the distance of even one Cepheid could be known reliably, it could provide a means of determining the distance of almost any Cepheid whose light-curve could be traced anywhere in the universe.

The difficulty was establishing such an absolute scale. Ejnar Hertzsprung made the first attempt to calibrate the Cepheids by a study of the brightest ones, which occur primarily in the spiral arms of the Milky Way. By statistical means, he calculated a mean luminosity for a particular period and estimated the luminosity. His result, which was far too dim, was still startling, for it made the Cepheids very bright stars. The most luminous had an absolute magnitude of -4, about four thousand times as bright as the sun. Following Hertzsprung and improvements made by Harlow Shapley, initial estimates of Cepheid distances were made, and systems of stars, including the Magellanic Clouds that contained Cepheids, were measured. The next step was the identification by Shapley and others of Cepheids in open clusters of stars within the Milky Way. The color-magnitude array was matched with the main sequence stars on the Hertzsprung-Russell diagram for clusters, which was then calibrated by the nearby Hyades cluster, whose distance was known geometrically. As a consequence, when Cepheids were discovered in more distant clusters, there was a means of establishing their distance and getting their luminosity from that. The result was that the Cepheids were found to be even brighter than Hertzsprung had estimated.

The actual use of Cepheids for distance measurement is not as simple. They were discovered in more than 90 galaxies  beyond the milky way with many more awaiting discovery. Three significant complications arise from several factors: First, the relationship between period and luminosity is not a neat line. In the real world, the stars are dispersed about that line, most within a half magnitude of the line. This dispersion means that distance to any galaxy is only poorly known if based upon one or even a few Cepheids. The more Cepheids available, the more accurate the statistical estimate of the distance. Second, the dispersion is related to the location of the Cepheid in the evolutionary sequence of stars. As this factor has become more clearly understood, compensation has allowed the refinement of the measurement of the distance of particular galaxies of interest. A third complication stems from the fact that in more distant galaxies, only the very brightest Cepheids can be observed. Adjustments for this distortion have also been made. In 2023, Webb observations showed that light contamination from nearby stars in crowded images does not appear to create a large enough error to explain the disagreement in measurements of the universe’s expansion rate.

By 1925, the cepheid period-luminosity relation (discovered by Leavitt and calibrated by Hertzsprung and Shapley)  allowed Edwin Powell Hubble to estimate the distance of the Magellanic Clouds and the Andromeda Galaxy (M31). By 1929, the calibration of the Cepheids enabled Hubble to discover the redshift distance (actually, velocity of recession) relationship and was also used to determine distances to nearby galaxies and, through them, to brighter indicators.

Additional adjustments in the distance scale were necessary for further study of the variable stars. In 1944, Henry Mineur found that the variables with periods shorter than one day (now called the RR Lyrae variables) were about one magnitude fainter than the classical Cepheids, a fact that immediately explained why they were not found as expected in the Magellanic Clouds. They were too dim to be seen by the equipment available at that time.

Another major adjustment in the distance scale came with the discovery by Walter Baade in 1944 of two populations of stars, which resulted in more than doubling the distance of the galaxies.

The significance of the study of Cepheids is that two types of Cepheids were also discovered: the classical Cepheids belong to population I, and type II Cepheids belong to population II. The classical Cepheids are found in the disk and spiral arms, while the type II Cepheids are found primarily in the globular clusters that form a spherical halo around the center of the galaxy. The prototypical Cepheid variable (δ Cephei) was discovered in Cepheus. These pulsating yellow supergiants typically have periods in the range of a day to more than fifty days. Their light-curve, luminosity, and color differed from the classical Cepheids, and they had a luminosity about a magnitude and a half dimmer than the classical Cepheids, which led to erroneous distances for all star clusters and galaxies that were based upon their use. These stars occur both in and out of globular clusters, but they do occur far from the galactic plane, while the classical Cepheids all lie near the galactic plane and tend toward the galactic center. Baade’s adjustment was confirmed with the discovery of the nineteenth magnitude RR Lyrae stars in the Magellanic Clouds, which led to more than doubling of the distance of the Clouds, as well as the rest of the galactic distance scale.

For more remote galaxies where individual variables cannot be observed, the amount of shift toward the red end of the spectrum for the galaxy as a whole is used to estimate the distance.

Nevertheless, even this method depends upon the calibration for the near galaxies given by the Cepheids. Cepheids have, therefore, proven the most valuable tool in the effort to measure galactic distances. In addition to measurements involving distance, they are also useful in any studies aided by the need for bright stars. Thus, they have been used extensively for studies of the size, structure, and galactic rotation of the Milky Way, for their radial velocities do confirm the rotation, although the bright O and B stars actually give somewhat better confirmation. They have also been prominent in studies of stellar evolution. It would be difficult to overestimate the progress that modern astronomy has made as a consequence of improved understanding of the nature and functions of the Cepheid variable stars.

Context

The study of Cepheid variables has been prominent in twentieth- and twenty-first century astronomy.  In the three major divisions of astronomy (planetary, galactic, and extragalactic), the study of Cepheids is primarily within galactic studies. They were prominent, however, in initially establishing the distance of exterior galaxies and remain a necessary link in calibrating the redshift to the velocity of recession relationship.

The foundational period of study of Cepheids was during the first half of the twentieth century. Since then, telescopes and detectors in space have led to dramatic new discoveries that overshadow Earth-based study of Cepheids. Individual research efforts continue, tending to refine what was previously known about the Cepheids. The Cepheids and other variable stars became prominent in later years for their usefulness in studies of stellar evolution and as distance indicators. Cepheids remain a fundamental, high-priority, and active field of research in both stellar astrophysics and cosmology. In 2024, observations from NASA’s James Webb Space Telescope and Hubble extended Cepheid-based distance tests to the galaxy NGC 5468 and showed that the disagreement over the universe’s expansion rate, called the Hubble tension, still remains.

Cepheids are a beacon of brightness that has aided in unlocking the scale of distances in the universe. They have also given humankind a more accurate picture of humanity’s place in space and the grandeur of the universe. The study of variables contributed to the unlocking of the distance scale so that the expanding universe could be discovered. Variable stars helped establish the distance scale that supported later studies of the expanding universe. Credit must be given to the pioneering astronomers who devoted much time to the study of Cepheids and other variable stars in the early part of this century.

Principal terms

LIGHT-CURVE: a graph that indicates how the magnitude of a star changes with time

MAGNITUDE: a scale for measuring brightness, for which each magnitude is approximately two and a half times brighter; the lower the magnitude, the brighter the star

PERIOD: the time interval between two successive maximum points on the light-curve of a variable star

RADIAL VELOCITY: movement in the line of sight

SPECTRAL CLASSES: an arrangement of stars using the letters O, B, A, F, G, K, and M from hot blue stars to dim red ones; the sun is a G star, with a surface temperature of about 5,000 kelvins

SPECTRUM: the array of colors or wavelengths obtained when light is dispersed by means of a prism or grating

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