It’s sometimes satisfying in this age of new rockets to the moon, space probes around the satellites of Jupiter and Saturn, and the razzle-dazzle images of JWST, to reflect upon nuts-and-bolts astronomy and how things we now take for granted came to be. I also like to take the opportunity to celebrate women in science whenever I can. And, for that matter, the process of science – how simple science can be done. So I’m posting a quick historical snapshot here of the 1912 ground-breaking discovery of Henrietta Swan Leavitt, one of the group Harvard Observatory women employed by the astronomer Edward Pickering at the turn of the last century to examine the observatory’s vast collection of photographic plates to accurately determine the brightnesses (magnitudes) of all measurable stars. In her study of certain stars whose light was variable, Leavitt discovered a remarkable, encoded message: the first reliable ‘cosmic distance indicator.’ It became game-changer for astronomical science. A few decades later Edwin Hubble would use her method to discover that the universe was expanding.

The story below is simple to grasp and shows how the scientific method can work in discovering laws of nature. Leavitt found a correlation among phenomena in a certain class of stars: the *rate* of their fluctuation was predictably linked to their intrinsic *brightness*. To see how she did it, we’ll follow the procedure I’ve used for years in classes and in science writing: first, we will use Leavitt’s original historical data to *re-create* her discovery. Next, we’ll divide the matter into six logical parts: (1) the problem to be solved; (2) the facts given; (3) the assumptions made; (4) the method we’ll use to solve the problem; (5) the calculations or procedure to solve the problem; and (6) concluding observations interpreting or adding context to what we have found. Again, in brief: *problem, given facts*, *assumptions*, *method*, *calculations*, and *observations*. Binning a problem into categories like this helps avoid common errors in doing or thinking about science. Students appreciate its clarity. By sticking to procedures like this, you too can avoid confused and sloppy thinking! But enough professorializing (not sure that’s a word). On to our story . . .

*Background*

The Magellanic Clouds (or the Clouds of Magellan) are two wispy, irregular, satellite galaxies that hover near our Milky Way, visible only in the Southern hemisphere. They proved to be a trove of discovery for astronomers in the early years of the 20th Century. In them lie a great number of *variable stars*, whose regular light changes over time were apparent on photographic plates taken by Harvard’s 26” Bruce telescope in Aerequipa, Peru. Analysis of these variations required painstaking investigation of hundreds of plates taken over many years. This task was undertaken by Henrietta Swan Leavitt, to whom you were just introduced, a talented, diligent assistant employed by Professor Edward Pickering at Harvard – one of the many women working there whose discoveries were to transform astronomy. She focused on the stars of the Small Magellanic Cloud (SMC), which hovers near the South Celestial Pole in the constellation of Tucana, the Toucan. The SMC, a dwarf companion galaxy to our Milky Way, is about 200,000 light years away from us.

The variability of some stars in our own galactic neighborhood had been known from ancient times, but the camera and spectrograph employed on better telescopes at the end of the 19th Century opened up the possibility of serious systematic study of these objects. And this included variables in the Magellanic Clouds, which were much too distant to detect in an earlier era. Stellar variations show up on photographic plates as larger or smaller points of greater or less density, and those variations, matched with the time of the photograph, yield – over a sequence of photographs – the *period* of the variability. With enough photographs, and careful determination of their apparent magnitudes on each plate, and the known times of the exposures, the *light curves* of any variable can be determined. Leavitt’s big discovery within the stars of the SMC was to show a correspondence between the *brightness* of the SMC variable stars and their *periods of variation*: the brighter the star, the longer the period.

In 1904 – only two years after being appointed to the permanent staff of the observatory – she published the periods and magnitudes of 17 SMC variables in *Harvard Annals* 60, No.4. She had noticed this relationship back then, but thought the number was too small to draw general conclusions. By 1912 she was able to publish a table of the brightness and periods of 25 stars, where her “remarkable relation between the brightness of these variable and their periods” was plainly evident.

*Why the Magellanic Clouds?*

How did Leavitt know that the bright stars were not brighter than others because they might be closer? The brightness of a star is after all a mix of its intrinsic brightness and its distance from us. In our galaxy we learned to adjust apparent magnitudes of stars to what they would be at a standard distance of ten parsecs, which is the definition of the uniform standard of *absolute* magnitude. But – and here is the key – because Leavitt’s variables were all in the SMC, they could be regarded as nearly the same distance away from Earth – in same way that people in various regions of Spain are *about* all the same distance from San Francisco. So, the relative *apparent* magnitudes of stars in the LMC – one against the other – should correspond pretty nearly to their relative *absolute* magnitudes and suffice as the magnitudes against which the periods could be compared.

Such variables are found in our own galaxy as well. Their brightness varies regularly and noticeably. A small percentage of variables are not true variables but are eclipsing double (or multiple) stars (which we may see if the planes of their orbits are in our line of sight). Algol in Perseus is a famous example. True variable stars pulsate: their brightness varies due to the star’s internal dynamics – they swell and shrink over periods from days to years. Among the most common, and important, are the Cepheids. They are named after their prototype, Delta Cephei, whose light variations were observed in 1784 by English astronomer John Goodricke. Now let’s see what Leavitt did.

*Reenacting Leavitt’s Discovery*

*Problem***: **Given Leavitt’s SMC data below, (1) plot the relationship between period and magnitude of the stars in the given sample. See if any relationship can be discerned. (2) Plot the magnitudes against the *logarithm* of the periods to see if any clearer relationship between period and magnitudes of the SMC Cepheids may become apparent.

*Given***: ** Here from Henrietta Leavitt’s 1912 paper are the data she presented on 25 variables in the SMC. H.V. stands for Harvard Variable number. Periods are in days:

*Assumptions***: ** We are assuming that the distance to all the stars in the SMC is about the same. Hence the relationships among their apparent magnitudes will represent the relationships among their absolute magnitudes. We are also ignoring the possible effects of extinction due to interstellar dust. This is an important factor *not* to ignore in systematic study, as we will see below.

__Problem (1)__

*Method***:** We must plot the relationship between magnitude and period for both the minimum and maximum magnitudes. You can do it by hand as Henrietta did, or (as I have done) by a simple Excel plot, or other software.

*Calculations***:**

*Observation***: **This looks just like the curve Leavitt plotted in her paper, “Periods of 25 Variable Stars in the Small Magellanic Cloud”, *Harvard Circular*, No. 173 (1912). She noticed a smooth trend upward, “of remarkable form.” But there was something she still could do to make the relationship clearer. The curve looks like a power relationship. That is, it appears that the *x* value grows exponentially in relation to the *y* value. Such a curve, for example, could look something like this, with a shape similar to Leavitt’s graph:

__Problem (2)__

Plot the magnitudes against the *logarithm* of the periods, as Leavitt did, to see if there will be revealed any clearer relationship between period and magnitude among the SMC variables.

*Given***:** Leavitt’s data above, but including the log of the periods, which are listed below:

*Assumptions***: ** Since we’re just changing the form of the mathematical representation, the assumptions will be the same as above.

*Method***:** We must plot the relationship between magnitude and log period for both the minimum and maximum magnitudes.

*Calculations***:**

*Observations***: **1.Now we have a clearer relationship emerging. Adding Excel trend lines makes it quite noticeable:

The trend is remarkable: There is a direct, linear relationship between the logs of the periods and the apparent magnitudes of the Cepheids in the SMC, at both maximum and minimum brightness. In Leavitt’s words from her historic paper:

A straight line can readily be drawn among each of the two series of points corresponding to the maxima and minima, thus showing that there is a simple relation between the brightness of the variables and their periods. The logarithm of the period increases by about .48 for each increase of one magnitude in brightness…

2. This relationship between period and brightness was to have a profound effect on astronomy. Knowing the period of a Cepheid would allow determination of its brightness (or absolute magnitude) and comparing that with how bright it appeared (apparent magnitude) will tell how far away it is. Harlow Shapley, the astronomer who more than anyone helped develop the consequences of the Cepheid period luminosity relation, expressed the formula for it in his 1943 book *Galaxies* (reprinted many times since)* *in this way (pardon the math): *M* = –1.78 – 1.74 log *P*, where *M* is the absolute magnitude and *P* is the period of variability in days. So, for example, a Cepheid with a 40-day period has an absolute magnitude *M* of – 4.57. This is an exceptionally bright magnitude and confirms that Cepheids are brilliant beacons. This is an important fact: Cepheids are so bright *they can be seen in other galaxies*, allowing us to find the distance to those galaxies.

3. How can we use this to find distance? Let us assume there is a Cepheid with a 40-day period at an apparent median magnitude *m* of 21. How far away is it? Remembering that distance in parsecs can be computed from the difference in absolute and apparent magnitudes, and with the formula log *d* (in parsecs) = .2 (*m* – *M*) + 1, we can determine the distance *d *to a Cepheid. (Just raise 10 to the (.2 (*m* – *M*) + 1) power on your calculator.) One parsec is equivalent to 3.26156 light years; it is a more shorthand unit of measurement [1]. Where in our example the Cepheid has a 40-day period and an apparent mean magnitude of 21, its distance *d* by this equation is about 1.3 million parsecs (or 1.3 *mega*parsecs, about 4.2 million light years). (Many textbooks and catalogs provide the *distance modulus* of a galaxy: that is the* m* – *M *value noted in the above equation.)

4. This technique *greatly* extended the range of distances we could measure. In fact, with the power of modern telescopes and detectors, we can measure distances to 20 to 25 million parsecs, all the way to the nearest group of galaxies, and then some. By contrast, the trigonometric parallax method, though improved with Hipparcos and later Gaia satellite measurements, is good only within our own galactic environs.

5. Where the Cepheid is lying near the plane of the Milky Way, extinction of light by dust makes these stars appear dimmer and a correction needs to be made. The formula has a correction factor, so the equation becomes log *d* (in parsecs) = .2 (*m* – *M – δm*) + 1. If *δm* is 1.5 magnitudes, the star is only half as distant. The accurate determination of *δm* in various directions understandably became the focus of much study. Where the star is well away from the plane of the Milky Way, we can assume that *δm* is zero, or close to it. But often the extinction coefficient will be hard accurately to determine.

6. The Cepheids in the Magellanic Clouds are exceptionally bright. They were, and still are, called *classical Cepheids*. Astronomers also discovered what they took to be classical Cepheids in globular clusters that surround the core of our own galaxy. They based the distance calibration upon them since they were easy to study. Harlow Shapley used them to locate the distances to globular clusters surrounding the core of our Milky Way, and thus famously found where our sun was in relation to the center of the galaxy. Edwin Hubble used that same calibration to determine the distance to the Andromeda galaxy, M 31. Only later, in the 1950’s with Water Baade’s research, was it realized that the calibrated Cepheids in globulars are of a slightly different type, with longer periods. They are crucially about 1.5 magnitudes fainter overall than Leavitt’s original SMC classical Cepheids. The fainter variables found in globular clusters and in other older (population II) systems became known as *Type II Cepheids* or *WW Virginis Stars* (after the prototype).* *The Cepheids Hubble saw in Andromeda and other galaxies were the brighter *classical *Cepheids, found in spiral arms and other (population I) star forming regions. They needed their own calibration, since they are brighter by 1.5 magnitudes – so bright ones once thought to be close but dim (under the original Type II Cepheid calibration) were actually dim because they were farther away, by a factor of 2. Hence the original distance to that galaxy had to be doubled!

7. There’s been an effort to refine the relation between classical Cepheid periods and luminosity in modern times, based on Hipparchos and Hubble data. A modern value for the classical Cepheid period luminosity relationship from a series of Hubble Space Telescope measurements would put the absolute magnitude of the star in paragraph 2 above at – 5.51, somewhat brighter than the value using the Shapley formula. Its distance (assuming again a median apparent magnitude of 20) would be found again by the above equation log *d* (in parsecs) = 2 (*m* – *M – δm*) + 1. Assuming zero extinction (for the purposes of discussion), then distance in parsecs is (10 raised to the (.2 (*m* – *M – δm*) + 1) power or) about 2 megaparsecs, or about 6.55 million light years away. This is substantially farther away than the 1.3 megaparsec/4.2 mly distance calculated for that star in paragraph 3 above. So again, the re-calibration of the Cepheid period-luminosity relation, as more and better data comes in, has had dramatic effect on the distance scale.

NOTE

[1] A *parsec* is the distance at which a star must lie for its measured parallax to be exactly one arc second. 1 parsec equals 206,000 astronomical units. The parallax of a relatively close object is its apparent motion with respect to a more distant background as the location of the observer changes. (For example, hold one finger up at arm’s length and compare its apparent motion against the background as you look at it first with one eye then the other, blinking them alternately.)

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