By the beginning of the 20th century, progress in determining the distances to the stars had slowed to a crawl. The trigonometric parallax method had pretty much pooped out. Evidence was growing that the stars vary greatly in absolute (intrinsic) brightness, precluding any direct application of the inverse square law.
Fortunately, it is always darkest just before the dawn. The combination of two advances, one technological and one scientific, was about to provide astronomers with a new and very powerful distance measurement tool. The technological development was the newly acquired capability to take photographs of telescopic images. Among other benefits, “astrophotography” would take much of the guesswork out of determining the precise position of a star relative to other stars in its immediate vicinity, breathing new life the trigonometric parallax method and revealing the distances to many additional stars. Meanwhile, the scientific discovery in question built on a much earlier discovery, made soon after Galileo first turned his telescope to the heavens, that some stars fluctuate in brightness on a fixed, predictable schedule, like Old Faithful.
Here’s the story. In 1794, a young English astronomer named John Goodricke identified a distinct subclass of these “variable” stars that would eventually be called “Cepheid” variables. Although Goodricke had no way of knowing it at the time, Cepheids are giant yellow stars, much brighter and more massive than the Sun. In addition to being large and luminous, Cepheids display a very distinct fluctuation pattern. The figure below this paragraph shows the pattern for “Delta Cephei”, (or “Delta Cep” for short), one of the few Cepheids actually visible to the naked eye (the North Star, Polaris, is another). Like all Cepheids, Delta Cep brightens quickly, dims more slowly, and then brightens quickly again, yielding a distinctive “shark fin” shape to the plot of the star’s change in brightness over time (if anyone is interested in why Cepheids behave this way, I’ll be happy to explain it in the comment section).
Delta Cep fluctuates on a very regular schedule, completing one full cycle every 5.4 days; this is known as Delta Cep’s period. In common with Delta Cep, all Cepheids fluctuate on very regular schedules. However, different Cepheids have very different fluctuation periods that range from about one day (for the shortest period Cepheids) to over four months (for the longest). In the figure below, I show the fluctuation periods for three actual Cepheids that range from a single day to over thirty days. As you can plainly see, the characteristic “shark fin” shape is present even for the longest period Cepheid, just greatly stretched out in time.
Regardless of their fluctuation period, Cepheids are quite uncommon. By 1900, only about 30 were known, none close enough to measure a parallax angle and determine its distance. Back in 1891, however, Harvard College had deployed a 24-inch telescope on a mountaintop in Peru – the first large aperture telescope to be deployed in the Southern Hemisphere. From 1893 to 1906, this instrument was used to take hundreds of photographs of the Small Magellanic Cloud (hereafter, the “SMC”), a prominent southern sky object named after Ferdinand Magellan, whose crew brought it to the attention of Europeans following their around-the-world voyage in 1519. As shown in the left-hand photograph below, the SMC looks like a small fuzzy patch to the naked eye. The right-hand image, taken through a telescope, reveals the “cloud” for what it really is: a “dwarf galaxy” containing many millions of stars!
The telescope in Peru was big enough to resolve the SMC into individual stars. Exploiting this fact, in 1905 an extremely diligent Harvard astronomer named Henrietta Leavitt began a long-term project to scrutinize the SMC photographs in order to identify variable stars. After several years of painstaking effort, Leavitt found about a thousand variables, 25 of which pulsed with the distinctive shark-tooth shape that marked them as Cepheids. The shortest-period member of the 25 had a fluctuation period of about one day; the longest-period member, about 128 days.
Next, Henrietta proceeded to measure the apparent (observed) brightness of each SMC Cepheid right at the top and the bottom of their fluctuation cycles (i.e., at the brightest and dimmest points of their respective cycles). The range turned out to be quite substantial, with the brightest member of the 25 being thousands of times more luminous than the dimmest. Henrietta surmised, correctly as it turned out, that the distance to the SMC was so great that any further differences in distance between her Cepheids and the Earth must be quite inconsequential; to all intents and purposes, they were equally far away. In turn, this meant that the large differences in apparent brightness among the Cepheids in her sample had to reflect actual differences in their “real” or intrinsic brightness – as opposed to an artifact of their being at different distances from Earth.
Henrietta’s finding that Cepheids differ greatly in intrinsic brightness gave tremendous importance to her next discovery, which came about when she plotted the fluctuation period of each of the 25 Cepheids (X axis) against their apparent brightness at both the top and bottom of their fluctuation cycle (Y axis). I’ve taken the liberty of reproducing the actual plot from Henrietta’s scientific paper on this topic below.
For both the top and bottom plots, the key thing to notice is the very strong relation between each Cepheid’s period and its intrinsic brightness: the longer the Cepheid’s period, the brighter it is. This “period-luminosity” relation showed up even more clearly in the next graph, which shows what happened when Henrietta re-plotted the relation after converting each Cepheid’s period to log-linear coordinates (this just means that she took each Cepheid’s period in days and determined to what exponent the number 10 would have to be raised to equal that number. For example, a Cepheid with a period of exactly 10 days would have a value of “1” in her log coordinate system; a Cepheid with a period of 100 days would have a value of “2”, and so on). Now, the “period-luminosity” relation was revealed to be so strong that if you fit a straight-line function to the dots, as Leavitt did in the figure, all 25 Cepheids fell very close to the line at both the minimum and maximum points of their brightness cycles.
I’m going to end this blog with an explanation for another aspect of these graphs that might be puzzling. Although I’ve explained the log scale along the X-axis, you might still be confused on the other (Y) axis by the fact that the numbers (the brightness scale) go from higher at the origin (the bottom) to lower at the top. This means that the brighter, long-period Cepheids on the right of the period-luminosity function are associated with a lower number on the brightness scale than the dimmer, lower-period Cepheids down and to the left.
Why is the number lower, when I just finished noting that the longer-period Cepheids on the right side of the graph were brighter than the shorter-period Cepheids on the left? The blame belongs to the ancient Greeks, who classified all the stars they could see in the night sky according to their perceived importance. They declared the very brightest stars, like Sirius, of “first” (“most important”) magnitude. Stars of the second magnitude were dimmer, and therefore less important, and so on, all the way down to stars of the sixth magnitude, so dim that they were barely visible (and therefore least important). Later astronomers adopted this magnitude scale, assigning bright stars (like long-period Cepheids) to lower magnitudes than dimmer stars (like short-period Cepheids). And while we’re on the topic of magnitude, it’s worth noting that the ancient Greeks included only stars in their magnitude scale, not brighter objects like Venus, the Moon, and the Sun. When later astronomers began to add these objects to the scale, they had to go to smaller (negative) numbers to convey their apparent brightness (that is, the brightness as measured from here on Earth). For example, when at its most dazzling, Venus shines with an apparent magnitude of -4.5; The Sun shines with an apparent magnitude of about -27. On the other end of the scale, stars that are way too dim to be seen in anything but telescopes have apparent magnitudes that extend all the way “up” to +30. As you can see from Henrietta Leavitt’s plots, the Cepheids in the SMC had apparent magnitudes all the way up in the mid teens, though they were far too faint to be seen with the naked eye. So the rule of thumb is: The lower the magnitude, the brighter the star.
In the next blog, we’ll continue the discussion of Leavitt’s period-luminosity function and what astronomers could do with it.