Henrietta Leavitt revealed the first indication of a period-luminosity relation in variable stars. Following the examination of hundreds of photographic plates obtained between 1893 and 1906 at Harvard College’s observatory in Peru, she produced a catalogue of 1777 variable stars in the Magellanic Clouds. Among these stars, 16 appeared in a sufficient number of plates for their periods to be determined. When tabulated in order of increasing magnitude a pattern emerged. Leavitt succinctly observed, "It is worthy of notice that in Table VI the brighter variables have the longer periods." (Leavitt 1908)

In 1912 Leavitt produced more data on the period-luminosity (PL) relation (Leavitt 1912). She had now managed to obtain the magnitudes and periods for 25 variables in the Small Magellanic Cloud (SMC). She wrote:

Fernie (1969, p708) wrote that Leavitt, "is sometimes unjustly accused of not having appreciated the significance of her discovery". Careful reading of Leavitt’s 1912 paper indeed reveals significant insight, she wrote:

- thereby connecting their light curves with other known variables. She added:

and tantalisingly, a few lines later she wrote:

Of course the term parallax was, and remains, a synonym for distance. But she did not expand further on this point.

It is suggested that if Leavitt was culpable of anything, it was merely a degree of understatement, and a good measure of scientific caution. Perhaps modern astrophysics has tended in the opposite direction and Leavitt’s reputation has suffered as a result of the present fashion for speculation.

Leavitt’s graphs of the PL relation are reproduced in Figures 1 and 2, Fig. 2 being the magnitude – logP plot.

Whilst Fig. 2 does not represent a calibration of the PL relation in the modern sense (there is no zero point since the distance to the SMC was not known), these data were nevertheless used extensively by Hertzsprung and others.

For the time being, we will accept this PL relation as a purely empirical law. The physical justifications underlying it will be dealt with in later chapters.

Danish astronomer Ejnar Hertzsprung realised that if the PL relation could be calibrated, then the absolute magnitudes of members of this group of variable stars might be determined directly from their periods. It should then be a straightforward matter to obtain their distance moduli and hence their distances. This was, and remains, the prime motivation behind Cepheid calibration.

Fig. 3: The light curve for d Cephei, characteristic of Leavitt’s variables
(reproduced in Hoskin 1999 p279)

We now know that the variables identified by the early calibrators actually consisted of stellar populations with different characteristics, and this did not come to light until Baade’s observations in the 1940s and 50s. Nevertheless it is interesting to note that, certainly in what would now be termed extragalactic work, the immense luminosity of the Population 1 Cepheids tended to act as a natural observational filter – the fainter Population II stars were unresolvable at these distances. It was only when astronomers turned their attention to the globular clusters that serious confusion between the populations arose.

In his seminal paper, Hertzsprung (1913) made the first attempt to calibrate a PL relation. In modern notation this takes the form:

<M> = a + blog₁₀P

At first sight calibration of the PL relation might seem a straightforward task. All that is required are accurate values for a (the zero point) and b (the slope). However, great difficulties lie in the technicalities of obtaining these two quantities. The zero point is particularly problematic and, despite technological advance, remains an area of controversy to this day. In the absence of a zero point, the PL relation remains a relative measure of distance. However, with an incorrect zero point, every distance determination using the relation is invalid - regardless of the precision of the slope.

Hertzsprung used statistical parallax to obtain his zero point for the Cepheids. Traditional annual parallax techniques are not capable of determining distances to even the closest Cepheids because the 2AU base line is not long enough. Statistical (and mean secular) parallax methods use the Sun’s motion relative to the local standard of rest (LSR) to run a longer base line. In this respect we are fortunate that the Sun’s peculiar velocity is rather high (about 13 kms^-1 in the direction of Hercules) (Webb 1999 p118). This amounts to about 2.8 AU per year. Clearly the subject stars have their own peculiar velocities relative to the LSR. This is why a statistical treatment is necessary to attempt to remove these effects under the assumption that their average peculiar velocity is zero. Hertzsprung was hampered by the inaccuracies of this technique, particularly given his small sample of only 13 Cepheids and imprecise knowledge of the motion of the Sun relative to the rest of the Galaxy.

For his slope Hertzsprung used Leavitt’s data, after converting her photographic magnitudes to visual ones with a colour correction. In modern notation, Hertzsprung obtained a PL relation of the form:

<M_v> = -0.6 – 2.1LogP

Hertzsprung now used his calibration to estimate the distance to the SMC and arrived at the extraordinary figure of 3000 light years in his 1913 paper. This result is extraordinary - because it is so clearly incorrect! Fernie (1969) suggests it may have been miscalculated or misprinted. This is almost certainly true for the following reasons:

A modern calibration of the PL relation (Feast & Catchpole 1997) gives

<M_v> = -1.43 – 2.81LogP

The two calibrations diverge and so the discrepancy becomes larger with period. But even at a period of 50 days the difference between the two calibrations is only about 2.5 magnitudes, corresponding to a 3-fold underestimate of distance. Current estimates place the SMC at a distance of about 170,000 ly - it seems almost inconceivable that Hertzsprung’s published estimate could have been intentionally out by a factor of 50!

Hertzsprung's paper initially quotes the distance to the SMC as a parallax of 0.0001", which obviously corresponds to a distance of 10 kpc or 30,000 ly. So, given that the published distance is almost certainly a typographical error involving a missing zero; why did Hertzsprung not draw proper attention to it? Perhaps he thought the mistake was obvious; or perhaps he did not mind having such a controversial result unintentionally obscured in this way - we can only speculate.

Fig. 4: The Small Magellanic Cloud occupied centre stage in early Cepheid work. The LMC is now the focus of attention because its front to back dimensions are smaller.