After the collapse of Shapley’s enduring calibration there was a flurry of activity to re-calibrate. Fernie’s antelopes were indeed charging off, but not in a new direction: new directions might have been a better description. If section one and two were the Cepheid scale’s Iron and Bronze ages, then what should we make of this period? Perhaps all scientific ideas have a dark age at some stage in their development c.f. the caloric in thermodynamics, the ether in electromagnetism. In the Cepheid story, as in human history, there was no sudden leap from the Bronze age to the Renaissance. This was a confused period – mediaeval in character; because the modern picture appears simpler. The problem was not a lack of ingenuity or imagination in the astronomers of the time – it was primarily observational. One might argue that, whilst the early Cepheid astronomers were limited primarily by telescopic resolution, the astronomers of this period were hampered by photometry. The tortuous twists and turns, revisions and counter-revisions of the time make tracing a path through this period difficult. A semi-theoretical approach will be applied, and what emerge as key papers will be used to illustrate the development of ideas. These papers have not always been selected on the basis of their importance - after all, many of the conclusions drawn were incorrect - some have been chosen simply because they capture the thinking of the time.

The primary aim of the period was to remove the observed scatter in the PL relation. The chief cause was presumed to be the finite width, and therefore temperature, of the instability strip. This is what led to the development of Period-Luminosity-Colour (PLC) relations. However, it was repeatedly noted that the new PLC relations appeared to offer no advantage over ordinary PL ones – which sounds absurd! An attempt to interpret this apparently contradictory situation will be given later.

Fig. 8: The instability strip for pulsating stars on the H-R diagram.
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Allan Sandage (1958) wrote:

(which is essentially the modern view of the cause of the scatter observed at the time).

Sandage continued:

Sandage presented a semi-theoretical approach to the PLC relation along the following lines:

The pulsation equation for Cepheids is , where P is the period, r the density and Q a structural constant, although as its name implies, it is likely to vary with the physical state of the star. The degree of this variation is far from certain, but modern observations indicate that it is a reasonably robust quantity (Madore & Freedman, 1991, seemed content to treat it as a constant in their own linear semi-theoretical approach). We will only consider pulsation in the fundamental mode here.

Using the relations:

, and

Sandage obtained:

(1)

Sandage now proceeded to eliminate variables: he would eliminate the mass with a mass-luminosity relation; convert M_bol to M_v, and express logT_e in terms of a colour relation.

Whilst equation (1) and its underlying assumptions appear reasonable, the principle difficulties lie in these eliminations. The final expression will inevitably be a combination of approximations, and the veracity of any relation so obtained must always be questionable – particularly if it is intended to be a refinement!

To eliminate the mass term Sandage used a rather dated empirical mass-luminosity relation after Kuiper (1938). He then reasoned that if the evolutionary tracks of Cepheids were similar to those of other giants, "the customary assumption of a 1-mag. rise from the age-zero main sequence" ought to be applied. Accordingly he subtracted 1 from the intercept in Kuiper’s relation and gave, "as an approximation to the mass-luminosity law for Cepheids":

Fig. 9: Kuiper’s empirical mass-luminosity relation (1938)
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For the bolometric correction, Sandage used data from Eggen (1956) to obtain:

And for the temperature-colour relation, he applied an approximation based on a temperature-spectral class relation, and a spectral class-colour relation by Johnson & Morgan (1953) to obtain:

The two relations above illustrate the appearance of the ubiquitous colour index whenever conversions of this type were made. The colour index (B-V) of a star is simply its magnitude in a band of wavelengths centred on the blue, minus its magnitude in the visual, or 'yellow', part of the spectrum. Because magnitudes decrease with increasing luminosity, (B-V) is essentially a measure of "redness", and broadly speaking indicates lower temperatures as its value increases. In principle colour indices are capable of measuring the effective temperatures of stars by determining the relative flux at two points on their (assumed) blackbody curves. But in practice colour indices are subject to a number of systematic errors. Arguably the most serious effect is "reddening", caused by interstellar dust scattering short wavelengths out of the line of sight. Although astronomers were aware of this problem, they underestimated it, and with simple two-colour photometry it proved virtually impossible to determine whether the colour index of a Cepheid represented its intrinsic colour or the effects of reddening. Disentangling these two effects proved to be a major difficulty. Attempts were nevertheless made to take reddening into account. At this time reddening was usually termed the colour excess, notated E(B-V) and was defined as the difference between the observed colour and what was believed to be the intrinsic colour. Colour excesses can generally be taken to be positive quantities.

Whilst reddening was undoubtedly the biggest obstacle to successful PLC calibrations, there were other difficulties. With increasing metallicity the spectra of stellar atmospheres deviate further and further from those of blackbodies. A major cause of this is line blanketing, produced by frequency selective absorption by metals, predominantly at shorter wavelengths (Figure 10).

Unaware of these effects, which with hindsight outweighed all the other subtleties introduced into his PLC calibration, Sandage arrived at the following result:

which was the first statement of a PLC relation, by virtue of the fact that it includes an explicit colour term. In the literature this is all that distinguishes a PLC from a PL relation: a PL relation lacks an explicit colour term, but this does not preclude it from having some form of colour or temperature correction included in its derivation, based, for example, on some average result.

Fig. 10: The effects of line blanketing on the Solar spectrum, the dashed line is the theoretical blackbody curve.
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Sandage went on to investigate the nature of Q, the structural "constant". Eggen (1951), in studying the maxima and minima of numerous light curves, believed that he had discovered a period - colour amplitude - luminosity amplitude relation in the classical Cepheids which pointed to three distinct types: A, B and C. Sandage deduced two values for Q, one for Eggen types A and B, the other for C. In a short and stinging criticism, Reddish (1959) showed that Sandage had made a basic error in his assumptions, which rendered his conclusions regarding Q baseless. Later work, particularly by Kraft, also showed that this division was unnecessary.

In the early 1960s Kraft published a series of detailed papers (1960a,b, 1961a,b,c) which drew together all that was known about Cepheid calibration at the time. In them he was able to make use of the handful of Cepheids recently discovered in Galactic (open) clusters. Whilst these Cepheids were insufficient in number to re-calibrate the slope of the relation, they were useful in distance calibration and in attempts to estimate colour excesses (reddening). It was now possible to directly compare the colours of these Cepheids with those in external galaxies such as the SMC and LMC. A second colour excess determination compared the colours of the galactic Cepheids with those of nearby, assumed unreddened, supergiants of the same spectral type (Kraft 1960a). Colour excesses were also estimated with empirical spectral type – intrinsic colour relations (Kraft 1960a,b) (Fig. 11). Using these data Kraft formulated a period - mean colour relation for Cepheids for use in a new PLC calibration (Kraft 1961c). Clearly a range of techniques were available to Kraft, but systematic errors were unavoidable using two-colour photometry, and even in the absence of these errors the results would remain rather crude estimates. The scatter in Kraft’s period-mean colour relation is a clear indication of this (Figure 12).

Fig. 11: Spectral type-colour relation (Kraft 1961c)
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Fig. 12: Period-mean colour relation (Kraft 1961c)
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Given a colour determination system based upon so many approximations and assumptions, it might seem ambitious to be concerned with relative details like the nature of the structural 'constant' Q; but this was precisely what pre-occupied Kraft and others. To the astronomers of the time it appears to have become something of an obsession. Kraft (1961c) was concerned with the possible variability of Q with period and proceeded to investigate this as follows:

The basic starting relation was:

(2)

Into this the following four functions were substituted:

Giving:

. (3)

Along the centre of the stability strip (ridge line):

Substituting these into (3) gave:

(4)

Kraft reasoned that if Q was independent of P, then the coefficient of logP in (4) must be zero. However, when he inserted his values for the coefficients he found that the slope (b₆) would be forced to differ from that previously determined by Arp (1960) for the SMC. Kraft decided that there were only two possibilities: either the slope of the PL relation along the ridge line was not a universal number or Q is indeed a function of P. He opted for the latter.

Kraft therefore determined Q in terms of P using equation (4). When this was substituted into (3) he obtained his PLC relation, of the form:

In 1967, when Fernie recalibrated the PL and PLC relations, he criticised Kraft's analysis on several counts (Fernie 1967a). He argued that it was possible for the slope to be a non-universal number and for Q to be a function of P. It was also suggested that the mass-luminosity relation adopted by Kraft was unreliable (like Sandage before him, Kraft had been forced to make assumptions about the evolutionary tracks of the Cepheids; Fernie believed that these were too simplistic).

Fernie proposed an elegant way to deal with these difficulties - essentially by avoiding them altogether. Using the Baade-Wesselink method he obtained an empirical period-radius relation for Cepheids. This meant that instead of using equation (1) or (2) he was able to use a basic Stefan’s Law equation as his starting point:

Into this he substituted his period-radius relation, along with a temperature-colour and luminosity-colour relation, to obtain a PLC relation of the same form as Kraft’s. The difference being that a mass-luminosity function, with its problematic dependence on assumed evolutionary tracks, had not been required. Additionally, considerations of Q had been avoided altogether. This was a novel and elegant approach which, unfortunately, would ultimately be let down by systematic errors in the colour relations.

When Fernie inserted his values he obtained the following:

(5)

The interesting feature of this relation is the lack of mean values – a consequence of the dimensional simplicity of the period-radius relation. As such, approximations aside, there is no a priori reason why M_v and (B-V) cannot be considered instantaneous quantities. This was the logical justification for Fernie’s assertion that the relation could be applied to individual stars – throughout their cycles. What Fernie argued was that if (5) were applied to a single star, then in the absence of phase-lag effects it would produce a line on a colour-magnitude diagram with slope .

Fernie had earlier observed:

Clearly equation (5) suggests that = 2.0, but Fernie gathered data to investigate a possible refinement:

Fig. 13: Plot used for the refinement of the (B-V) coefficient. (Fernie 1967a)
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Hence equation (5) became:

(6)

Fernie’s paper is full of interesting features like this: non-linear PLC and PL relations cropped up quite frequently during the period, but equation (6) appears to be the earliest example. Fernie believed that this might account for the apparent differences in slope observed in different systems. Clearly if he was right about the quadratic nature of the PLC relation, all previous calibrations, including Kraft’s, had been off the mark. It is worth noting that values of (B-V) typically range from about 0.5 at 3 days to 0.9 at 20, only exceeded unity at around 30 days. The quadratic term would therefore have a relatively small effect, particularly for small periods.

Fernie now proceeded to obtain a PL relation from his PLC. He had previously determined a period-colour relation (Fernie 1967b) (Figure 14):

The scatter in Figure 14 clearly indicates this was an estimate, nevertheless it was substituted into the PLC relation to obtain the non-linear PL relation:

What followed was very strange indeed, and the history of the period is punctuated with observations like this: when Fernie compared his PL relation to his PLC relation, he concluded that:

Fig. 14: Empirical period-colour relation (Fernie 1967b)
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Fig. 15: PLC versus PL plot (Fernie 1967a)
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Figure 15 shows a plot of his PLC relation (black dots) overlaid on the calculated PL relation (solid line). He determined that even in the most extreme cases the difference could be only +/- 0.2 magnitudes.

The logical difficulty is this: leaving aside the quadratic term (which only serves to bend the curve slightly), why should the scatter suddenly disappear in this way? A PL relation consists essentially of a slope and an intercept. However subtle its derivation, how could simply finding new values for these two parameters suddenly remove the scatter? A comparison of Leavitt’s results (Fig. 2) and Fernie’s (Fig. 15) clearly illustrate this contradiction.

Fernie observed that the term in (B-V) and the term in (B-V)² had opposite signs. By partially differentiating equation (6) w.r.t. (B-V) and setting it to zero, he showed that the PLC relation coincided exactly with the PL relation at a value of (B-V) = 0.85, corresponding to a period of about 18 days. Thus the two relations pivoted about this common, fairly central point. But Fernie's analysis could not explain the removal of the observed scatter – the original aim of the PLC relation.

To resolve this contradiction, it is proposed that while there is only one type of PLC relation in the literature, there are several types of PL relation. What is meant by this is that a PLC relation is defined as a PL relation with an explicit colour term, but the description "PL relation" can refer to a number of different cases. There are PL relations which are purely observational, and they are full of scatter. If an astronomer wanted to demonstrate the superiority of his PLC relation, he could put these two curves side by side and the improvement would be obvious - it would be attributed to colour corrections that had narrowed the width of the instability strip! Fernie’s comparison, however, was of a different kind. The line he drew in Fig. 15 was that of a theoretical PL relation - it was not plotted from observational data. It was therefore implicit in Fernie’s treatment that the magnitudes of his PL would be properly corrected for extinction. What he had uncovered, which is of some importance, was that if a PLC is compared with a PL relation whose visual magnitudes have also been corrected for reddening, there is virtually no improvement.

It is likely that the colour term (B-V) of the time did not represent a temperature/colour correction as intended, it was probably a reddening correction in disguise. This is perfectly consistent with the PL relations which, like Fernie’s, were obtained from PLC relations by the substitution of some mean colour function; this similarly can be interpreted as merely a mean reddening term. Hence the observed similarities of the relations. The thrust of modern work, which is toward reddening correction rather than PLC relations, would appear to support this and will be considered in more detail later.

The difficulty with reddening corrections, and corrections intended to account for effective temperature, is that they both work in the same direction. This can be illustrated with a very simple example. We can imagine two stars, of the same type, and with the same period, which are judged by simple photometry to have different colours, say red and blue. If the red star is intrinsically red, perhaps lying toward the theoretical red-edge of the instability strip, then its luminosity can reasonably be assumed to be less that that of the blue star. Accordingly, a PLC relation will need to brighten this star, so to speak, in order to correct for this and bring the two stars onto the same line. However, if this same star is intrinsically blue, and merely appears red due to extinction, it will similarly need to be brightened - brought forward - in order to obtain its true magnitude. Decoupling these effects is problematic, and was arguably impossible with two-colour photometry. Modern observations suggest that the reddening effect is by far the most pronounced of the two.

Attempts were also made to correct for the effects of metallicity during this period, but the results have largely been superseded. An early theoretical attempt was made by Reddish (1956) using classical stellar models. Iben & Tuggle (1975) concluded that colour-temperature relations were very sensitive to metallicity, and both Gascoigne (1974) and Stothers (1988) concluded that PLC relations were far more sensitive to metallicity than PL relations. This, once again, was an odd situation. If the aim was to correct for metallicity with a 3 parameter relation, it is hard to see how the simple 2 parameter relation could give better results – at least in the absence of systematic errors or a flaw in the reasoning. In Stothers’ largely theoretical analysis it once again comes down to a matter of terminology and definition. The language of modern astrophysics is characterised by prosaic synonyms for otherwise well defined terms. Accordingly when one reads, period – luminosity - colour, it is tempting to read this as: period - luminosity – effective temperature. But this is not the meaning, PLC relations are easily defined – they contain a colour term.

Thus, whilst Stothers (1988) used a period-luminosity-temperature relation as his starting point, he substituted different functions into it to obtain his PLC and PL relations.

For the former he used a function of the form:

and for the latter:

(calculated along the instability strip’s ridge line: c₁, t₁ are metallicity terms)

The temperature-colour relation above is both more complex in nature, and more difficult to obtain, than the temperature-luminosity relation. Accordingly, when Stothers completed his theoretical analysis, he found that the PLC would be between 5 and 12 times more sensitive to changes in Z than its PL counterpart. He wrote:

Given that both relations cannot be correct if they describe the same physical situation, this seems rather a bold admission. Occam’s razor would immediately suggest a problem with the temperature-colour conversion, but for some reason the accuracy of colour relations was rarely questioned during this period.

Before we leave this period there are two more calibrations that deserve some attention: the first influential; the second controversial. The calibrations of Kraft and Fernie have already been described at some length, principally because they were so characteristic of the time. But a year later Sandage & Tammann (1968) produced a calibration that has stood the test of time. Their approach to the problem was straightforward and pragmatic. They gathered together all the reliable data they could find for Cepheids - not from a single system as was the norm, but from throughout the local group of galaxies. All of these data, from more than 100 stars, were then combined into a single, composite PL relation after simple corrections for reddening. This had not been attempted since Shapley (1918) because it was widely believed that the slope of the PL relation differed between galaxies. With regard to this, Sandage & Tammann (1968) wrote, "The present paper suggests that this view is too pessimistic, and that the apparent differences in slope may not in fact be real, but result from small-sample statistics in the presence of the intrinsic dispersion of the PL relation." To fix their zero point, Sandage & Tammann used the Galactic Cepheids found in open clusters, a small but important set of stars whose absolute magnitudes were known with some confidence.

Fig. 16: The composite PL relation (Sandage & Tammann 1968)
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Figure 16 shows their calibration, the envelope lines representing a 0.5 magnitude spread either side of the mean line. Despite having produced an empirical PL relation, Sandage & Tammann had not abandoned the concept of a PLC relation; they identified this 0.5 magnitude scatter with the width of the instability strip. It is also noteworthy that their relation was not linear, but this was because they had incorporated a few variables with periods of 100 days or more. Classical Cepheids are usually taken to lie in the region logP < 1.7 (about 50 days) and the line is straight up to this point.

In this important calibration, Sandage & Tammann had largely laid aside theoretical concerns regarding colour, the pulsation equation and metallicity. What they had done was apply an essentially statistical approach to the problem – but did it pay off? It is a remarkable fact that the PL relation has remained virtually unchanged since this calibration over 30 years ago. At the present time it is in agreement to within a few hundredths of a magnitude.

Sandage & Tammann’s calibration did not signal the end of PLC relations, that of Martin, Warren & Feast (hereafter MWF) (1979) is an excellent example of a PLC calibration from near the close of this, admittedly rather loosely defined, "period". Their use of three wavelength bands (BVI) signified a move to multi-wavelength photometry that would radically alter the character of subsequent work.

Because the degree of reddening varies with wavelength, observations in several bands, with the application of an appropriate law, can enable better de-coupling of the effects of reddening and intrinsic colour. This discrimination increases with the number of bands, and although MWF had only three, they had a high degree of confidence in their intrinsic colour determinations. It was assumed that the PLC relation would be of the form:

Which is familiar enough except that <V> here is the apparent visual magnitude, and the B and V measurements have been separately averaged, rather than the whole colour index. Rather than attempt to infer values for the coefficients using a semi-theoretical approach, MWF solved for them iteratively until the standard deviation of the scatter about some mean relation was minimised. Figures 17 and 18 show their initial results.

Fig. 17: PLC relation (MWF 1979)
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Fig. 18: PL relation (MWF 1979)
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The PLC plot is one of against LogP, and the PL simply <V> against logP. Clearly the graph with the colour correction exhibits far less scatter than the PL relation. Note that neither of these graphs contains any intentional correction for reddening. MWF wrote:

In other words, and it is by now a familiar question, was the reduction in scatter with the inclusion of the -2.70(B-V) term due to intrinsic colour, or did it merely represent an accidental reddening correction?

The normal reddening law is given by:

which relates A_v (the extinction in M_v),  to E(B-V) the colour excess, or degree of reddening. MWF appreciated that the constant in this relation (R»3.3) was uncomfortably close to their value for b (2.70). It would therefore be necessary to attempt to separate the two possibilities. Using BVI band reddening estimates developed by Dean et al. (1978) they determined that the differential reddenings were small, with a standard deviation for a single Cepheid of about 0.05 magnitudes. In short, their analysis suggested that the spread in observed colours represented an actual spread in intrinsic colours: the differential reddening not being sufficient to account for the scatter in the basic PL relation. Figure 19 shows their comparison plots of: <V> (top), reddening corrected <V> (middle) and reddening corrected (bottom). As predicted, the reduction in scatter of the PL relation was small, whilst the PLC relation, as compared to Fig. 17, remained largely unchanged.

MWF therefore asserted that their PLC had been validated, and that a colour term was not a refinement, but a necessity in determining accurate period-luminosity relations. But these findings remain controversial, and later work by Madore and others (see Section 4.1) appears to contradict them. In the next section this issue will be explored further.

Fig. 19: Comparisons of PL and PLC scatter when corrected for reddening (MWF 1979)
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