Barry F. Madore and Wendy L. Freedman
In the early decades of this century, Henrietta S. Leavitt recogned that there is a statistical relationship between the average brightness of a Cepheid variable star and the period of pulsation. The periods of these stars range from a few days to a few hundred days, and their luminosities (all of wliich are intrinsically much greater than that of the Sun) span factors of several hundred in brightness. In general, the longer the period of a Cepheid, the brighter is its average thtnninc luminosity.
Because the period of pulsation can be measured independently of distance and because the apparent luminosity of a Cepheid depends on the square of the distance, one can use the intnnsic luminosity, as predicted by the period, in combination with the apparent luminosity to derive a distance. As tcols in the extragalactic distance scale, Cepheid variables provide one of the most accurate means of determining distances to the nearby spiral and irregular-tppe galaxies of the Local Group and somewhat beyond.
One very appealing attribute of Cepheid variables as a class of distance indicators is that the physics for the Cepheid period-luminosity relation is well understood. Put simply, all self-luminous objects, including Cepheids, give off light in proportion to both their area and the surface brightness over that area. For spherical bodies of radius R and surface temperature T, the total luminosity L (integrated over all wavelengths) is found using Stefan's law where L = 4 R2 T4 and the Stefan-Boltsmann constant. Furthermore, the fundamental period of oscillation P of any mechanical system depends only on one thing, the mean density . Low-density systems have longer periods of oscillation than high-density systems, with the equation relating these two quantities having the following form, P1/2 = Q, where Q is a constant. Combining Stefan's law, the P1/2 law, and an assumed mass-luminosiry relation Indicates that the largest stars will have the Inghest luminosities and the longest periods. And that is exactly what is observed.
Of course, with a more rigorous (and more complex) application of physics, we can relax these simplifying constraints and redo the calculations for stellar models havIng a reasonable range of mass and with a variety of surface brightnesses. But these are refinements, certainly necessary for detailed calculations, but not absolutely necessary for an understanding of the underlying process leading to a period-luminosity relation itself. The refinements do tell us that the relation between period and luminosity will have scatter, and that the scatter will have a physical origin due to differences in surface temperature between stars of the same density (i.e., same mass and radius), for it is temperature that determines the surface brightness of sell-luminous objects. Ultimately, Cepheids are well described by a PLC (period-luminosity-color) relation.
It is of interest to note here that the physics that applies to the ensemble of Cepheid variables (that is to the periodluminosity relation as a whole) also applies to the individual stars as they each cycle through their oscillations. Radius variations in a single star give rise to changes in area, which result in changes in luminosity; surface temperature variations around the cycle drive changes in the surface brightness which also affect the total luminosity. There are again two parameters to the problem: The area is a geometrical property of the star and therefore has the same effect on the luminosity nearly independent of wavelength, whereas the surface brightness, radiation theory tells us, is very sensitive to where in the spectrum one observes the star. In the infrared, temperature variations make only a slight contribution to luminosiry differences, whereas in the blue and ultraviolet, the same temperature variations can dominate the luminosiry variations. These changing effects with wavelength can be seen in Fig. l where the cyclical luminosity variation for a Cepheid is shown at different wavelengths. In the near ultraviolet the amplitudes are large and driven primarily by surface temperature variations. In the far infrared almost all of the luminosity change is a reflection of the radius variation which is known to be out of phase and distinctly different in shape with respect to the temperature variations.
Figure 1. Typical light variations as a function of phase for a Cepheid variable as observed at wavelengths rangng from the ultraviolet (top light curve, labeled U) through the blue, visual and red parts of the spectrum (labeled B, V, and R, respectively) out to the near infrared (ending with the bottom light curve at K = 2.2 µm). Note the decreasing amplitude of the light variation, as well as the In shape and the shfft In the phase of rnaxirnurn brightness as longer and longer wavelengths are examilned.
The absolute calibration of the Cepheid periodluminosity relation is based on a small number of Cepheids found in galactic star clusters. These clusters have independent distances, obtained from main-sequence fitting techniques. Additionally, the same main-sequence stars can be used to independently estimate the amount of Interstellar dust obscuring and reddening the light of the Cepheids. Unfortunately, the statistiqs are poor and the intrinsic luminosities and colors of many of the cluster Cepheids are still uncertain. Moreover, most field Cepheids are too far away to have direct parallax measurements made with the present technology. However, an independent calibration will be forthcoming when the refurbished Hubble Space Telescope provides improved direct determinations of the distance to the Large and Small Magellanic Clouds in which hundreds of Cepheids, all essentially at the same distance, will enter the calibration.
Unfortunately, the effects of interstellar reddening due to intervening dust both in our own galaxy and in the parent spiral galaxies of extragalactic Cepheids can be quite severe and are very much stronger.in the blue as compared to the red and Infrared. A multicolored approach to the application of Cepheids to determining the distance scale is therefore required. Using modern electronic detectors (such as charge-coupled devices) and a variety of filters turning from the blue to the very near infrared, this latest, hybrid approach allows for both the distance and the total reddening to individual external galaxies to be sirnultaneously solved for by a careful application of the periodluniinosity relation, given an a priori knowledge of the interstellar extinction law. Without a determination of the extinction, all other distance estirnates are upper limits, overestimating the true distance. Examples of periodluminosity relations constructed by this technique can be seen in Fig. 2 which consists of a montage of optical wavelength observations of Cepheids in the Large and Small Magellanic Clouds, followed by the dramatically narrower period-luminosity relations found at infrared wavelengths. These latter relations are so well defined and narrow that they have allowed astronomers to calculate not only the distances to the Magellanic Clouds but also the three-dimensional shape and orientation of these galaxies in space.
Figure 2. Multiwavelength period-luminosity relations for Cepheids observed in the Small Magellanic Cloud (open circles) shifted into magnitude registry with data for Cepheids in the Large Magellanic Cloud. Note the decreased scatter as one goes from the blue wavelength data (at the top left of the figure) to the near-infrared data (at the bottom right). The individual PL relations are shifted vertically in magnitude and horiitontally in period for display purposes only.
With the application of modern techniques to the Cepheid periodluminosity relation, it is now widely agreed that the extragalactic distance scale is relatively secure for galaxies within, and slightly beyond, the Local Group. For galaxies with Cepheids identified and observed, the distance estimates, out to approximately 2 Mpc, are now agreed upon at the 10% level. However, a factor of 10 further away at 20 Mpc, where Cepheids have not yet been discovered, the uncertainty, as judged by rival factions, rises to a factor of 2 difference in opinion. Much of this uncertainty is expected to disappear with the optically corrected imaging phase of the Hubble Space Telescope expected to begin in late 1993 or 1994. One of the major missions that MST is committed to working on is the extragalactic distance scale. MST will be capable of discovering Cepheids in galaxies that sample a much larger volume of space than can ever be imaged from the ground (because of atmospheric turbulence). A Cepheid-based distance to one or two galaxies at 20 Mpc (for instance in the Virgo Cluster, which is probably the practical limit even for HST) will not end the controversy about the size and age of the universe; but such observations will certainly go a long way toward narrowing the divergence of opinion. On the other hand, with many more Cepheid-based distances to galaxies Inside that volume lirnit, secondary distance indicators can be calibrated with some statistical certainty, and these altemate techniques can then be pushed deep into the extragalactic space where the pure Hubble flow is expected to be revealed.
But the history of science teaches us that even our most modest expectations are not always met as we venture into new regimes. To be sure, questions of importance, such as those concerned with the age, size, and structure of the universe, will never completely go away; given time they will only become more interesting. Without Cepheid variables to lead the way in the extragalactic distance scale, our view of the universe would certainly be far less secure than we hope it is today.
Freedman, W. L. (1988). Cepheid distances to nearby galaxies. In The Extragalactic Distance Scale, S. van den Bergh and C. Pritchet, eds. Astronomical Society of the Pacific, San Francisco, p. 24.
Madore, B. F., ed. (1985). Cepheids: Theory and Observations. Cambridge University Press, Cambridge.
Madore, B. F. and Tully, R. B., eds. (1986). Galaxy Distances and Deviations from Universal expansian. Reidel, Dordrecht.
Sandage, A. R., Sandage, M., and Hristian, J., eds. (1975). Galaxies and the Universe. Stars and Stellar Systems, Vol. 9, University of Chicago Press, Chicago.
Adapted from The Astronomy and Astophysics Encyclopedia, ed. Stephen P. Maran