This page last updated on 01/26/2019.
Copyright © 2001-2019 by Russ Meyer
This article started out as a college term paper. I was digging around in a box of old stuff from college and stumbled upon my old text copy. I decided it would make a nice addition to the web site, so I rewrote parts of it to adapt it to that use. It can't be too bad because I got a B on it! Anyway, here it is...take it for what it's worth...
Just looking up at the night sky, it's virtually impossible to tell how far away the stars are. Way-far-away is about the best estimate that can be managed with the naked eye. Finding out how far away the stars are has proven to be a very difficult problem in astronomy. This distance scale problem has given rise to an entire field of study in astrophysics. It has a many varied and fascinating history. The discovery and application of Cepheid variable stars as cosmological distance indicators is one of the greatest revelations of modern astronomy. The following is a survey of the classic Cepheid variables and the development of the Period-Luminosity relationship.
Origins of the Period-Luminosity Relation
In 1908, Miss Henrietta Leavitt studied the Small Magellanic Cloud with the twenty-four inch photographic refractor at Arequipa in Peru. She produced a catalog of 1777 variable stars from plates taken between 1893 and 1906. From these plates, she was able to find the periods for sixteen of the stars. She found that the periods ranged from 1.25 days to 127 days and she also observed that the longer the period of the star, the brighter it was. By 1912, she had increased the number of Cepheids, whose periods were known, to twenty-five, and in that same year published the first formulation of the Period-Luminosity relation. She had discovered that the apparent magnitude of these Cepheid stars decreased in an approximately linear fashion with the logarithms of their period. In this 1912 paper where Miss Leavitt first published the Period-Luminosity relation, she notes, "Since the variables are probably at nearly the same distance from the Earth, their periods are apparently associated with their actual emission of light. It is to be hoped, also, that the parallaxes of some variables of this type may be measured." Obviously, Miss Leavitt recognized the potential Cepheids offered as distance indicators, however, she did not pursue the concept further.
In 1913, Ejnar Hertzsprung of Denmark provided the next major step towards making the Period-Luminosity relation useful for finding distances. Via statistical parallax of thirteen Cepheid stars, he was able to provide data by which Miss Leavitt's Period-Luminosity equation could be calibrated. He also pointed out that the variable stars that Miss Leavitt had discovered had light curves similar to that of the star Delta Cephei and consequently referred to them generically as "Cepheids" for the first time.
In 1918, Harlow Shapley of the Harvard College Observatory attempted a further refinement of the Period-Luminosity relation calibration. Shapley, using statistical parallax and the same basic data that Hertzsprung used, recomputed the zero point of the Period-Luminosity relation. Unfortunately, his zero point, as we know today, was set approximately 1.4 magnitudes fainter than it should have been. There are two major factors responsible for this discrepancy that Shapley didn't account for in his calculations: 1) neglect of interstellar absorption and 2) the then unknown affects of galactic rotation on the proper motions and radial velocities of stars. The next and perhaps most significant mistake Shapley made was that of trying to incorporate RR Lyrae stars into the Period-Luminosity curve. This seemed the natural thing to do because his original 1.4 magnitude error almost exactly compensated for the 1.5 magnitude difference between Population I and Population II stars, thereby making the data appear as though it all fit together.
During the next twenty-five years, Shapley's calibration held sway, despite various attempts to rework the zero point. However, evidence began to mount suggestion that something was wrong with the Period-Luminosity calibration. For instance, it was noticed that all other galaxies, as measured by the Cepheid distance-scale, were smaller than our own, a very peculiar result. Also, as nuclear physicists succeeded in calibrating the rate at which uranium and thorium have been decaying to lead in the rocks of the Earth, their "clocks" made the Earth appear considerably more ancient than the universe. There was also difficulty reconciling Eddington's calculation of the mean density of the Cepheids with the density estimates derived from the relationship of the observed luminosity of these stars to their rate of pulsation.
Observations by Baade during World War II, initially with the 100 inch telescope at Mount Wilson and later with the 200 inch telescope at Mount Palomar, finally opened the way to a resolution of these conflicts. Baade found a basis for classifying stars, including the Cepheids, into two populations. He set out to perform a comparison of the two populations side by side, or in other words, at the same distance. Baade was able to measure the Population I Cepheids of the Andromeda Nebula with great accuracy against the brightest globular-cluster stars, for which absolute magnitude had been established with the help of Population II Cepheids in our galaxy. According to the established Period-Luminosity scale, the distance to the Andromeda Nebula as calculated from its Population I Cepheids, predicted that the bright globular-cluster stars should have an apparent magnitude of 20.9. Baade found that these stars were actually magnitude 22.4, or 1.5 magnitudes fainter. This result implied that the Andromeda Nebula was twice as far away as was previously thought! If it really were twice as far away, this would imply that its size was twice that of current estimates. This size revision brought the Andromeda nebula up to the size of our own galaxy and thus resolved one of the suspicious implications of the Shapley calibrated Period-Luminosity relation. Baade's revision of the Period-Luminosity relation implied that the universe was twice as large as was previously thought. Follow-on work with Baade's revision has shown it to be truly superior to the Shapley calibration. This is primarily due to the recognition of the distinction between Population I and Population II stars and the subsequent exclusion of the RR Lyrae stars in the calibration.
Theory of the Period-Luminosity Relation
The reason that Cepheid stars vary in brightness in a regular periodic fashion is that the star itself is actually pulsating, that is, the star expands and contracts in a regular repetitive motion. The point of concern is the mechanism that keeps the star pulsating. The mechanism has to do with the outer layers of the star and the way they allow light to be transmitted through them. As a Cepheid contracts, the density of its atmosphere increases, and helium atoms are formed by the combination of helium ions with electrons. These helium atoms absorb light quite effectively and, as a consequence, radiant energy from the star is kept inside. The star heats up because of this and begins to expand. As the star expands, the density of the stellar atmosphere decreases, allowing the helium atoms to become ionized by the emerging radiation. The atmosphere is then able to allow light to pass through, thereby causing the expansion to diminish. As the pressure driving the expansion of the stellar atmosphere decreases, the outer layers of the atmosphere begin to fall back in, and the star enters a new contraction cycle.
The period of pulsation is determined by the average density of the star. The denser it is, the more rapidly it vibrates. Thus, the range of periods of Cepheid variables represents a range of densities.
This theory of pulsation is derived from the observations of the regularly varying apparent magnitude and radial velocity of Cepheid variable stars.
Application of the Period-Luminosity Relation
Given the two observable characteristics of period of oscillation and apparent magnitude, the Period-Luminosity relation provides a means for finding the luminosity of a Cepheid star. From this, it is possible to derive absolute magnitude and therefore the distance of the star. The period of oscillation is derived from a plot of the light curve of the star, as is its average apparent magnitude.
As an example, lets say we are looking through our telescope one night and discover a really cool looking Cepheid variable star. We watch it for several nights and discover that it is pulsating with a period of 6.25 days. Fascinated, we dig out or handy-dandy photometer and measure the magnitude of the star. Whaddayaknow...it has an apparent magnitude of +7. Now we're really cookin'...we're just itchin' to find out how far away this beast is. We look up the Period-Luminosity relation given in absolute magnitude in our handy astrophysics book and find it is:
Cool! Let's plug in our numbers...
So, the absolute magnitude of this newly discovered star is -0.8. Armed with this information, we can calculate the distance to the star. We again reference our astrophysics book and find the equation relating apparent magnitude, absolute magnitude, and distance. It is:
With pounding heart and trembling hands, we plug in our numbers...
In actual calculations, the apparent magnitude should be adjusted to take into account the absorption of starlight by interstellar dust, otherwise the distance to the star would come out further than it actually is. In other words, 364 parsecs is the maximum possible distance to the star in the example above. In reality it is probably somewhat closer. The error in the calculation comes from the fact that interstellar dust is absorbing some of the starlight, making the star appear dimmer than it would otherwise. A dimmer star, as measured by our photometer, implies a greater distance. So, the interstellar dust is fooling us into thinking the star is more distant. It is possible to compensate for the dimming effect of interstellar dust in our computations, but that is beyond the scope of this article.
Cepheids and the Distance Scale
Just as trigonometric parallax has limits to the distances it is capable of spanning, so Cepheids have their limits. When the distances become so great that even the most powerful telescopes on Earth cannot pick out the brightest Cepheid variable, other techniques must be used.
Heliocentric parallax can be used out to the nearest stars. Beyond the closest stars, main sequence fitting, statistical parallaxes, spectrographic parallaxes, and Cepheid Period-Luminosity relation techniques are employed. With these latter techniques, distances to galactic and globular clusters can be determined.
To move beyond our own local galaxy requires that we find objects in other galaxies that we can identify. Because Cepheid stars are inherently bright objects, they provide a bridge to the nearby galaxies. We can apply the Period-Luminosity relation to those Cepheids that can be seen in other galaxies to determine the nature of intergalactic distances.
As bright as Cepheids are, the distances involved quickly become so great that Cepheids cannot be identified. From here on out, Cepheids in nearby galaxies can be used to calibrate distance indicators capable of spanning even greater distances. (Such as H II regions, the brightest stars, absolute magnitudes of galaxies, etc.)
Cepheids are useful only out to approximately four mega-parsecs, but that's still useful for finding distances to the local galaxies and for calibrating other longer range distance indicators.
Historical Application of Cepheids
One way in which Cepheids have played an important role in past efforts in astronomy has been the determination of the placement of the sun in our own galaxy. In 1915, Shapley observed that globular clusters were oddly concentrated in the direction of Sagittarius rather than following the distribution of stars along the Milky Way. Upon closer examination of the clusters, Shapley found that he could make out what he thought were a few Cepheid variable stars. (Actually, they eventually turned out to be RR Lyrae stars.) With his own calibration of the Period-Luminosity relation, Shapley found these closer clusters to be at a distance of about 12,000 parsecs from the sun. Later he estimated M13 to be 30,000 parsecs distant. These distances were far greater than the then accepted size of the galaxy. If these globular clusters proved to be associated with our own galaxy, then our galaxy was far larger than previously thought. What's more, the non-uniform distribution of these clusters would put our own sun way off to one side of the galactic disk, rather than at the galactic center, if indeed these clusters were bonafide parts of our own galaxy. Shapley assumed that the clusters were indeed part of our own galaxy and that they were distributed about the center of the galaxy. He know the distances and directions to the clusters, so he made a three dimensional model of the galaxy and found that his observed clusters did actually center themselves about a point about 16,000 parsecs from the sun in the direction of Sagittarius.
Another problem in astronomy that Cepheids helped to resolve was the question of whether the "spiral nebula," as Herschel called them, were really galaxies in their own right, or were they part of our own galaxy like the globular clusters. Shapley and Herber Curtis were the key figures in this debate. Curtis upheld the older view that the nebula were galaxies of approximately the same size and shape as our own. Shapley pictured the galaxy as outlined by globular clusters and about 300,000 light years in diameter. This ruled out the spiral nebulas as galaxies, because their relatively close distances, as estimated by Shapley, were not consistent with the large diameters required if the spirals were the same size as the galaxy. The problem here is that the distances to the nebula were not known. Even after telescopes had resolved the nebula into stars, the debate continued.
The debate was finally settled in 1924 by Hubble. Edwin Hubble had succeeded in discovering Cepheid variable stars in the M31 nebula. Although variables had been suspected as early as 1922, Hubble confirmed their existence in the outer arms of M31. He derived a distance of 490,000 light years for M31, far beyond the farthest globular clusters that Shapley contended marked the outer limits of the Milky Way.
A third application of Cepheids and the Period-Luminosity relation was made by Sandage and Tammann in their attempt to find the value of the Hubble constant. Sandage and Tammann needed a distance indicator to find the distances to galaxies so distant that individual stars could not be seen. In these galaxies, they were able to make out H II regions and consequently adopted these as one of their long range indicators. Now they had the problem of attempting to calibrate these H II regions for their absolute magnitudes. The distance to nearby galaxies as determined by observation of Cepheid variables provided the necessary link. Because of the high confidence in the Period-Luminosity relation, Sandage and Tammann were assured of the accurate distances to these nearby galaxies, and were able to calibrate the luminosities of H II regions within these galaxies. Thus, the Cepheid variables, via the Period-Luminosity relation, are an important link to the longer range distance indicators.
Advantages of Using Cepheids as Distance Indicators
It is advantageous to use Cepheid variable stars as distance indicators for a number of reasons. For one, the Period-Luminosity relation is fairly well established. That is, the calibration of the relation has been worked out to a high degree of confidence. Another advantage is that the Period-Luminosity relation is relatively easy to use. All one need do is observe the average apparent magnitude and period of oscillation of a star in order to accurately find its distance. No other information need be known about the star. A third advantage is that Cepheid stars are bright enough to be seen at distance much farther than most. This increases the flexibility of using Cepheids as distance indicators, because their usefulness extends over a great range of distances. A fourth possible advantage is that Cepheids occur often enough among the various types of starts, so that a least a few can be usually be found in the star clusters of our own galaxy, and most certainly can be found in other galaxies near our own.
Refinements of the Period-Luminosity Relation
Most of the current "updated" Period-Luminosity relationships incorporate a color term into the equation, making it a Period-Luminosity-Color (PLC) relation.
Along the mean line of the Period-Luminosity relation, it has long been observed that there is a scatter of about one magnitude in the luminosity of the stars. for many years, this was attributed to observational error, or as in the case of the Magellanic Clouds, internal absorption. However, this observed scatter of the data points could actually be a real departure of the stars from the mean line. Highly accurate photoelectric measurements of Cepheids in the Small Magellanic Cloud by Halton Arp of the Mount Wilson and Palomar Observatories have established that the scatter is very mush larger than the observational errors. Sandage suggested that a third variable be introduced to the Period-Luminosity relation to take into account the surface temperature of the star. This third variable is the color of the star, thus the PLC relation. These PLC relations take on the form: M - Alog(P) + b(B-V) + C
There has been some controversy about the relevance of the color term. However, it has been suggested that this controversy can be gotten around by observing Cepheids in the infrared portion of the spectrum.
Cepheids as distance indicators are immensely important. Without them, distances in the universe would be far more difficult to obtain. Cepheids have revolutionized our concept of the universe and our place in it.