Journal for General Philosophy of Science

, Volume 43, Issue 2, pp 347–359

Anomalies and Coherence: A Case Study from Astronomy


    • Philosophisches SeminarUniversität Hamburg

DOI: 10.1007/s10838-012-9196-y

Cite this article as:
Gähde, U. J Gen Philos Sci (2012) 43: 347. doi:10.1007/s10838-012-9196-y


In recent decades, the concept of coherence has become one of the key concepts in philosophy. Although there is still no consensus about how to explicate coherence, it is widely accepted that the appearance of anomalies significantly lowers the coherence of a propositional or belief system. In this paper, the relationship between coherence and anomalies is analysed by looking at a specific case study from astronomy. It concerns anomalies that occurred in the first half of the twentieth century during the attempt to develop a cosmic distance scale. These anomalies could not be removed until several decades after their appearance, which required a fundamental change in astronomical theory. During this process, the astronomical distance scale had to be adjusted by a factor of about 2. This paper focuses on the role that explanatory relations played with respect to the elimination of these anomalies. Thereby, special attention is paid to the explanatory work of astronomical theories or models that were not especially designed for this task.


AnomalyCepheidsCoherenceCosmic distance scale

1 Introduction

In the past decades, coherence has become one of the key concepts in philosophy. Regardless of the importance it has gained in theoretical as well as practical philosophy, there is still no consensus at all about how to adequately explicate it. At least two fundamentally different strategies have been employed for this purpose. The first strategy tries to make a quantitative concept of coherence more precise by referring to several support measures. The probabilistic approaches following this strategy differ from one another concerning the measure of confirmation they choose and how it is incorporated in the explication of the concept. Such approaches have been suggested first by Shogenji (1999), then by Olsson (2002), Fitelson (2003), Douven and Mejis (2007), and others. The second strategy primarily tries to explicate coherence as explanatory coherence, i.e. by focusing on explanatory relations. Here, the approach suggested by Thagard (1989) is widely discussed.

Despite these efforts, so far no explication of the concept of coherence is universally agreed on. All the approaches currently available have severe flaws. Probabilistic views have been charged with the claim that according to them, it is at least unclear if coherence is truth-conducive. The impossibility results of Bovens and Hartmann (2003) and Olsson (2005) cast doubts on whether probabilistic approaches can provide a concept of coherence that could serve as the foundation of a coherentist theory of epistemic justification.

But interpreting coherence as explanatory coherence is also problematic in many respects. The concept of explanation itself is notoriously ambiguous and unclear. Also, the question of how the number and strength of inferential relations in general and explanatory relations in particular can be determined, has so far not been answered convincingly. Additionally, the problem remains unsolved of how the various factors which influence the coherence degree of a belief system can be weighed against each other in a non-arbitrary way.

Regardless of the differences between those conceptions of coherence, there is at least one point on which they all agree: the appearance of anomalies in a belief system lowers its coherence degree significantly. We will understand anomalies as severe and stubborn conflicts between theoretical claims (especially predictions) and empirical data.

The aim of the following is to investigate the connection between coherence, anomalies, and explanations in a rather detailed scientific case study. I will focus on anomalies that occurred in the first half of the twentieth century during the attempt to develop a cosmic distance scale designed to determine the shape and extent of the Milky Way and further galaxies, as well as their relative positions to each other. The attempt to overcome these anomalies proved to be a major challenge. Solving the problem—from the first appearance of the anomalies to their successful theoretical solution—took more than four decades. The following case study is supposed to illustrate how important aiming for coherence in scientific research actually is, and what roles explanatory relations in particular play here.

Sections 2 and 3 will present the scientific basis of the case study. In particular, I will explain how the cosmic distance scale was construed and what role variable stars played in it. Section 4 will present the anomalies that showed up increasingly in the attempt to use the cosmic distance scale, and the respects in which this lowered the coherence of astronomical theory. Section 5 contains the solution suggested by the German-American astronomer Walter Baade in 1951 at a conference of the International Astronomical Union in Rome. His suggestion removed or at least attenuated most of the anomalies that arose in connection with the cosmic distance scale and hence led to an increase in coherence. The solution remained problematic, however, because Baade had to make several assumptions that, at the time, could not be explained within the framework of his conception. Accordingly, attempts started immediately afterwards to support and explain the brute facts that Baade had assumed by embedding them in the available astronomical background knowledge. I will describe how this was done, and what the consequences were for the coherence of astronomical theory, in sects. 6 and 7. Section 8 contains final remarks about the connection between coherence and truth, in which I will point out two aspects that have been neglected in the discussion so far: the importance of simultaneously removing numerous anomalies by a few assumptions, and the relevance of the explanatory power of a theory (with respect to the anomalies in question) which was not designed with this explanatory power in mind.

2 A Case Study: Variable Stars in General and Cepheids in Particular1

In the ancient world and the Middle Ages, there was hardly any doubt that stars change neither their position, nor their brightness. Aristotle, for instance, contrasted the immutability of the stars with the constant change to which the “sublunar” region is subjected. It is due to a contingent fact that this conviction stayed around for so long: none of the bright fixed stars exhibit a strong light variation—at least if observed from Earth. Hence, the view that stars do not change solidified. This theory was still upheld when stars were observed whose changes in brightness could clearly have been detected even with the limited means of observation available in the ancient world and the middle ages—a fact that obviously illustrates the degree to which observations can be influenced by the background beliefs in question.

This situation changed only in 1572, when Tycho de Brahe discovered a supernova in the constellation Cassiopeia. ‘Tycho’s star’ exhibits a change in brightness that could no longer be overlooked. After this discovery, the dogma of the immutability of the stars had to be given up. The discovery itself was a fortunate coincidence, since supernovae are extraordinarily rare astronomical events: during the last millennium, only five were observed in the Milky Way.

Once the dogma of the immutability of the stars had fallen, more variable stars were soon discovered. In 1596, David Fabricius discovered the variable star Mira in the constellation Cetus. The example of Mira showed that stars can exhibit periodic changes in brightness. The star Algol also periodically changes in brightness and was discovered by Geminiano Montanari in the constellation Perseus in 1696. The period of Algol’s light variation was determined in 1783 by the English astronomer John Goodricke. Goodricke—only 18 years old at that time (and deaf-mute)—accurately interpreted Algol as a so-called optically variable star, i.e. a star that changes its brightness not due to a change in its physical state parameters, but rather because a second object with a lower luminosity moves in front of the star. Goodricke assumed that this was a planet. However, today we know that Algol is part of a binary star system, the fainter component of which causes the light variation. In 1785, Goodricke observed the variability of a star in the constellation Cepheus, which he called Delta Cephei, after its location. The discovery of this star proved to be of tremendous importance for astronomical research, since Delta Cephei belongs to a class of variable stars—the Cepheids—which have played a key role in the construction of a cosmic distance scale. This will be explained in more detail in the next section.

3 Cepheids and their Meaning for Astronomy2

Cepheids are variable stars that exhibit a regular, periodic light variation. The periods of this light variation range from 1 to 50 days. Cepheids undergo pulsations which change their diameter by up to 30 %. All Cepheids are so-called giant stars, i.e. very bright objects. Consequently, they can be observed even from great distances. In the Milky Way, for instance, about a dozen Cepheids are visible to the naked eye. In two neighbouring galaxies of the Milky Way, the so-called Magellanic Clouds, more than 3,000 Cepheids are known.

Cepheids have played a key role in the construction of the cosmic distance scale. In order to explain this role in more detail, we first need some more information about the notion of the brightness of a star.

The brightness with which a star appears when observed from Earth depends on the one hand on its luminosity, i.e. the amount of energy it emits per unit of time, and on the other hand on its distance from Earth: a star which is located closer to Earth appears brighter than a star with the same luminosity, which is further removed from Earth. For this reason, there is a distinction between a star’s apparent and absolute magnitude. The apparent magnitude, designated by the symbol “m” (for magnitudo) is the star’s brightness as observed from Earth. This is distinguished from the absolute magnitude, which is a measure of brightness, but with the distance corrected, i.e. the brightness a star would have if it were observed from a certain standardized distance (1 parsec ≈ 3.26 light-years).3 The absolute magnitude is designated with the symbol “M”. The relation between apparent magnitude, absolute magnitude, and distance from Earth is the following:
$$ m - M = 5 \cdot \log r - 5. $$

If not only the apparent magnitude m, but also the absolute magnitude M of a star, were known, its distance from Earth r could be calculated by (1). This is usually not the case, however. The apparent magnitude m of a star can be determined by photometric methods. By contrast, we do not have direct access to its absolute magnitude M as there is no possibility to observe a star from the prescribed standardized distance.

The significance of Cepheids for astronomy lies in the fact that their absolute magnitude can, nevertheless, be determined quite easily. A discovery made by the American astronomer Henrietta Leavitt in 1912 while she was observing Cepheids led to this possibility. First Leavitt noticed that with these variable stars, apparent magnitudes and periods of light variation correlate: the greater the apparent magnitude is, the longer its period lasts. Assuming that the diameter of the Andromeda Galaxy is low compared to its distance from Earth, all these objects had to be nearly equally far away from Earth. So the relation between apparent magnitudes and periods had to correspond to a relation between absolute (distance corrected) magnitudes and periods. Based on her measurements, Leavitt discovered the following connection between the absolute magnitudes M and the periods P of the light variation4:
$$ M = \alpha + \beta \cdot \log P. $$

In order to use this so-called period-luminosity relation for determining distances, it first had to be calibrated: the constants α and β had to be determined. The first calibration was carried out in 1917 by Harlow Shapley who employed measurements of comparatively close Cepheids in the Milky Way, whose distance from Earth could be determined by different methods.5

The calibrated period-luminosity relation provides the theoretical background for one of the most important methods for measuring distances in astronomy. Say a Cepheid is observed in the Milky Way or a different galaxy. First, the star’s period of light variation P is measured. Inserting P in the period-luminosity relation shows its absolute magnitude M. If now its apparent magnitude m is determined by photometric measurements from Earth, inserting M and m in (1) results in the distance of the object from Earth.

The significance of this method for measuring distances is based on Cepheids being particularly bright objects. Because of their brightness, they can still be observed from great distances. They are cosmic milestones which—after the calibration of the period-luminosity relation—made it possible for the first time to measure almost the whole universe, as far as it was known at the time. This led to fundamental astronomical discoveries. It was now possible to estimate the size and shape of the Milky Way for the first time, as well as the position of the sun in it. This revealed that the planetary system is not after all located in the centre of the Milky Way—as had been assumed until then—but rather at the edge, in a spiral arm. In 1924, Edwin Hubble, after having discovered Cepheids in the Andromeda Galaxy, was able to determine their distances at least approximately. Hubble’s data showed that these objects—and hence the Andromeda Galaxy itself—were far too far away to belong to the Milky Way. This, for the first time, allowed a confirmation of the hypothesis that there are more galaxies than just the Milky Way—a hypothesis that was first formulated by Kant (1755) in his Universal Natural History and Theory of Heavens:

„Wir haben mit Erstaunen Figuren am Himmel erblickt, welche nichts anders, als solche auf einen gemeinschaftlichen Plan beschränkte Fixsternsystemata, solche Milchstraßen, wenn ich mich so ausdrücken darf, sein…“6

At the beginning of the twentieth century, several astronomers had suspected that the hypothesis was correct. But it could only be empirically confirmed in a convincing manner, for the first time, by measuring distances by means of the Cepheids.

4 The Cepheid Anomalies and the Coherence of Astronomical Beliefs

Construing a cosmic distance scale with the help of the Cepheids was a breakthrough in astronomical research. Yet at the same time, the scale had numerous problems—during its use, an increasing amount of anomalies appeared. I will now explain four of these anomalies and how they are connected with the coherence of astronomical theory.

The first anomaly concerned a far-reaching cosmological consequence of the cosmic distance scale. In a ground-breaking paper Edwin Hubble (1929) reported that the spectral lines of far removed galaxies shift towards longer wavelengths. He discovered that a linear relation holds between the redshift of these spectral lines and their distance from our galaxy. In the light of Einstein’s general theory of relativity, this redshift can be explained as being caused by the expansion of the universe—an explanation which Hubble himself never seems to have accepted.7 From the redshift of the galaxies’ spectral lines, their recession velocities can be obtained and these, once again, are proportional to their distance (Hubble’s law). The factor of proportionality is called Hubble’s constant H. Hubble’s law was discovered by determining the distances of some galaxies with the help of Cepheids that were detected in them. Then the velocities with which these Cepheids—and hence the galaxies they were located in—move away from us were calculated by using the redshift of their spectral lines. Based on the (however problematic) assumption that H indeed possessed the same value all through the whole expansion process, it was possible to trace back this process and calculate when all galaxies were concentrated at one point, thus allowing astronomers to estimate the age of the universe. The estimated age was about 2 billion years. This number clearly contradicted the estimated age of Earth. Based on geological and radioactive dating methods carried out between 1940 and 1950, Earth’s age was supposed to be 3–3.5 billion years.8 So there was a strict logical contradiction within the set of scientific theories: either the estimations of the age of the universe based on the period luminosity relation of Cepheids, or the age estimations based on geological methods and radioactive dating, had to be false.

In the discussion of coherence, almost all authors initially insisted on logical consistency as a necessary condition for coherence. But it soon became obvious that this demand was too strong. The paradox of the preface, for instance, shows that our belief system usually contains inconsistencies. If logical consistency were a necessary condition for coherence, any systems of belief of this kind would equally be incoherent—which contradicts our intuition that—despite individual inconsistencies—we can in fact distinguish between more and less coherent belief systems or sets of propositions. Consequently, the demand for logical consistency has become much weaker in recent suggestions of how to explicate coherence. Usually, it is merely stated that strict logical contradictions in a set of beliefs or propositions significantly lowers its degree of coherence—regardless of how the latter is characterized. This was exactly the case with the anomaly that I have just described. Immediately after its discovery, attempts were made to remove the contradiction.

But instead of a strict logical contradiction, anomalies often only constitute a weaker form of inconsistency, which is called a probabilistic inconsistency. A probabilistic inconsistency occurs if someone claims that something is the case, but the available background knowledge makes this seem highly improbable. The Cepheid anomalies provide several examples of this weaker form of inconsistency, too. For instance, the work of Shapley, Hubble, and others made it possible to compare the size of the Milky Way with the size of other galaxies. The comparison led to a surprising result: the Milky Way seemed to be significantly bigger than all other galaxies whose extent could be determined. Obviously this did not constitute a strict logical contradiction with the background knowledge. Of course one could not definitely rule out that the Milky Way, for contingent reasons, is bigger than all other galaxies. However in the light of the available background knowledge, this result seemed extremely unlikely.

The latter anomaly is also interesting for another reason. Kuhn and others have pointed out the relevance of one’s world-view as background beliefs for the development of empirical theories. One typical example would be the belief that it is highly probable that the Milky Way plays no outstanding role in the universe. One will hardly find an explicit formulation of it in an astronomy textbook. Nevertheless, at the beginning of the twentieth century, it was so deeply entrenched in the background beliefs of astronomers that Shapley’s and Hubble’s conflicting result about the size of the Milky way compared to other galaxies—even though it was just an instance of a probabilistic inconsistency—immediately raised doubts about the correctness of the result.

Another example of probabilistic inconsistency is an anomaly concerning the luminosity of so-called globular clusters. These globular clusters are spherical collections of 104–107 stars with a high stellar density in their centre. A comparison of the absolute magnitudes of globular clusters in the Milky Way with that of globular clusters in the Andromeda Galaxy showed that the objects in the Andromeda Galaxy on average seemed to be much fainter than those in the Milky Way. No plausible explanation could be found.

Let me point out one more anomaly, which particularly raised the suspicion of some astronomers that the cosmic distance scale based on Cepheids was flawed. This anomaly concerned RR Lyrae variables (so called after the constellation Lyra, in which they were first found). RR Lyrae are periodic variable stars, too, whose light variation follows a period-luminosity relation. Their absolute magnitude is slightly lower than that of Cepheids. The interval in which their luminosities and absolute magnitudes lie was known through measurements with relatively close RR Lyrae variables. Based on these data and Hubble’s estimation of the distance of the Andromeda Galaxy from the Milky Way, one expected to observe RR Lyrae variables in the Andromeda Galaxy. However, not a single RR Lyrae variable was found there—in contrast to the at least prima facie plausible assumption that all galaxies of the same type (spiral galaxies etc.) contain a similar percentage distribution of the various types of stars. This contradiction, too, significantly lowered the coherence of astronomical theory and supported the belief that something was wrong with the cosmic distance scale.

At the same time, determining distances with the help of the period-luminosity relation of the Cepheids proved to be crucial for learning about the shape, relative position and distance of galaxies. It thus became an indispensable tool in astronomical research. From a coherentist point of view, it allowed for various new inferential relations within astronomical theory, significantly changing our view of the Milky Way and the universe. It is no surprise, then, that serious efforts were made to remove the anomalies.

How these anomalies were interrelated remained unclear. Only in 1952—40 years after the discovery of the period-luminosity relation by Leavitt and 20 years after its calibration by Shapley—the German-American astronomer Walter Baade presented a solution at a conference of the International Astronomical Union in Rome. It removed or at least weakened most of the anomalies. I will describe Baade’s solution and its consequences for the coherence of astronomical theory in the next section.

5 Baade’s Solution of the Anomalies: A Gain in Coherence for Astronomical Theory9

As early as 1944, Baade claimed that all stars can be categorized into two different classes, the so-called stellar populations.10 In the Milky Way, stars of population I are mostly located in the galactic disk. These are ‘young stars from old matter’. This means that the stars consist of matter that has already been part of the development of a star before. Due to this fact, heavier elements had accumulated in the star matter, i.e. elements with a higher atomic mass than helium.11 Because stars of population I are mostly located in the galactic disk, this population is also called disk population. In contrast, stars of population II are mostly found in the halo surrounding the galactic disk, especially in globular clusters. For this reason, population II is also called the halo population. The stars belonging to it are ‘old stars from young matter’: they consist of matter in which no elements heavier than helium have so far accumulated.

Baade’s solution to the anomalies started with the assumption that the borderline between stars of population I and II also leads through the class of Cepheids. He distinguished between Cepheids of population I (the so-called classical Cepheids) and Cepheids of population II (the W Virginis stars). Both kinds of Cepheids fulfil the period-luminosity relation, but with different values for α and β: population I Cepheids are much brighter than population II Cepheids, i.e. they possess a higher absolute magnitude as compared to W Virginis stars.

Baade was able to show that, with this distinction, most of the anomalies disappeared or were at least attenuated. He argued as follows: the period-luminosity relation was initially calibrated with globular clusters from the Milky Way. These globular clusters belong to population II, i.e. to the fainter Cepheids. When measuring the extent of the Milky Way, one had referred to same type of star. Hence the results were mostly correct. The assumptions about the shape and size of the Milky Way did not need to be revised. The mistake only happened when astronomers tried to use this relation in order to determine the distance of the Andromeda Galaxy. In this attempt, they had referred to Cepheids belonging to the brighter population I class. This meant that the absolute magnitude of these objects was much higher than had initially been assumed. In order to have the apparent magnitude with which they are actually observed from Earth, they had to be far more removed than previously assumed. What this meant exactly was that the whole cosmic distance scale had to be adjusted by the factor 2. The Andromeda Galaxy now was about twice as far removed than previously assumed.12

This also had consequences for the estimated extent of this spiral galaxy. In order to appear the size which we observe from Earth, it had to be much bigger than astronomers had thought, because it is further away. The same was true for any other galaxy. The first anomaly disappeared: the estimated size of the Milky Way remained the same. But all the other galaxies were much larger than previously assumed. For the Milky Way, this meant that there was no longer any need to assume that it was more extensive than all the other star systems.

The second anomaly, too, could now be removed. According to Baade’s ideas, the distance of the Andromeda Galaxy from the Milky Way was about twice as large then previously thought. Hence the globular clusters in it must also have a higher absolute magnitude than one had assumed in order to have the apparent magnitude observed from Earth. After this adjustment, the mean value for their absolute magnitude was no longer any lower than the corresponding mean value for globular clusters in the Milky Way.

Baade’s suggestion also led to a redetermination of Hubble’s constant H. The value of H obtained thereby was considerably smaller than what Hubble had estimated. This, in turn, led to an estimation of the age of the universe that was about twice as high as previously assumed on the basis of Hubble’s law. This at least softened the conflict with other age estimations available at the time.13

Baade’s suggestion thus increased the coherence of astronomical theory by helping to remove or at least weaken the aforementioned anomalies. This success came at a price, however: various brute facts had to be accepted for which no plausible physical explanations were available at the time. The theoretical background of the pulsation process of the Cepheids and the period-luminosity relation in particular remained unclear. Why population I Cepheids had other values for α and β in the period-luminosity relation than population II Cepheids remained equally unclear.

The majority of approaches to explicating coherence attribute a crucial role to explanatory relations for the coherence of a belief or proposition system. From this perspective, one would have to regard the explanatory gaps of Baade’s solution as a severe weakening of the coherence of astronomical theory. Hence it is unsurprising that, immediately after the publication of Baade’s suggestion, attempts were made to supply the missing explanations. The first step was to understand the mechanism behind the Cepheids’ pulsation process. In order to solve the problem, it was possible to go back to an old idea that Arthur Eddington had already published in 1926. He himself, however, had discarded it again because it did not fit coherently into the available astronomical background knowledge. This will be explained in the next section.

6 The Kappa Mechanism: Removing an Explanatory Gap

For most of its time, a star is in balance. The gas pressure in particular—which, on its own, would lead to an expansion of the star—is compensated by gravitational force. Say a star becomes compressed. Gas pressure would expand it again, and even beyond its original radius. After that, the gravitational force would overbalance and compress the gas quantity. The star would begin to carry out an oscillatory motion. But damping mechanisms—which I cannot explain in detail here—would lead to an end of the pulsation after about 100 years. However, we know that Delta Cephei—at least since its observation by John Goodricke—pulses with almost the same period. Astronomers were thus faced with the task of explaining which mechanism keeps the pulsation going.

The, at least in principle, correct answer can already be found in Eddington (1926). Eddington based his theory on the assumption that the opacity of the star’s matter increases when being compressed. This would make it heat up, the radiation pressure would increase, and the star would expand. The expansion would then decrease the density of the star’s matter—and consequently, its opacity. The star’s matter would become more penetrable for the radiation. The radiation pressure would decrease, and the star would collapse again. This mechanism, which is also called kappa mechanism after the coefficient of absorption κ, might counter the damping of the pulsation and sustain it over longer periods of time.

The problem with this explanation, however, was that it employed an assumption that could not be coherently fitted with the astronomical background knowledge of the time. According to most astronomers—including Eddington himself—the opacity of a star’s matter should decrease under compression, not increase. So opacity’s dependence on density would dampen the pulsation process instead of supporting it. This is why Eddington discarded his own explanation and searched for alternatives.

The situation did not change until 1953, when S. A. Zhevakin was able to show that the opacity of star matter does indeed decrease under compression, but that this proposition is not true for matter close to the surface of the star: within a certain temperature range, the opacity can increase here under compression. Zhevakin could also show that this mechanism is sufficient to compensate the damping of the pulsation process due to the effects on the inside of the star, and to maintain the oscillatory motion.14

What is the impact of Zhevakin’s work regarding the coherence of astronomical theory? Baade was able to increase its coherence significantly with his solution, by showing how to remove the anomalies. However, his theory was based on several assumptions for which no accepted physical explanations were available. Zhevakin was able to fill this explanatory gap at least to some extent. By referring to Eddington’s initial idea, he could provide an explanation for the oscillatory mechanism of pulsating stars—and the duration of the pulsation process in particular—which could be smoothly embedded in the physical background knowledge. It also made use of premises belonging to the established standard knowledge from thermodynamics.

But still Zhevakin was not able to explain all of the brute facts in Baade’s solution. He could not explain why a period-luminosity relation holds for Cepheids. Neither could he explain why population I Cepheids had other values for the constants α and β, respectively, than population II Cepheids. An explanation of these phenomena only became available when the pulsation process of Cepheids could be fitted into a general theory of stellar structure and evolution.

7 Further Progress in Explanation: Fitting Baade’s solution into a Theory of Stellar Structure and Evolution

In his pioneering work from 1926, Eddington was one of the first to articulate the assumption that the energy emitted from the stars stems from thermonuclear processes. In order to understand these processes in more detail, astronomers initially focused on the phase of stellar evolution in which stars cover their energy requirement by means of central hydrogen burning, during which hydrogen nuclei fuse to helium nuclei. This phase is comparatively easy to describe because, in it, the stars are at least close to an equilibrium state. Their physical parameters (especially size, temperature, and pressure) only change slowly during this phase. Such objects are called main sequence stars. The name refers to a fundamental diagram in astronomy, the Hertzsprung-Russell diagram, which shows the absolute magnitude of stars versus their spectral types.15 In this diagram, stars with central hydrogen burning are on a long band called main sequence or dwarf band. The calculations showed, amongst other things, that the stars’ evolution is crucially dependent on their mass.

When the hydrogen reserves in a star’s core are used up, shell burning begins—the fusion of hydrogen to helium in shells surrounding the centre of the star. During this process, the star expands significantly: it becomes a red giant.16 In the middle of the twentieth century an increasing number of attempts were made to quantitatively capture the transition from main sequence stars to giant stars within the framework of a theory of stellar structure and evolution. In one of these model calculations, Kippenhahn and his colleagues noticed that, during their evolution, stars with seven times the mass of the Sun cross—due to their absolute magnitude and spectral type—the Cepheid area on the Hertzsprung-Russel diagram several times. By using work that Kippenhahn had carried out with Baker in 1960, they examined whether these stars oscillate while crossing the Cepheid area. The model calculations showed that the objects pulsated each time they crossed the Cepheid area of the Hertzsprung-Russell diagram, and that the observed oscillation frequencies of Cepheids could be easily fitted into these model calculations.17 Within the framework of the rapidly developing theory of stellar structure and evolution, astronomers also succeeded in finding a convincing physical explanation for the period-luminosity relation, as well as for the fact that population I Cepheids have other values for α and β than population II Cepheids.

This removed the crucial explanatory gaps from which Baade’s solution for the Cepheid anomalies had initially suffered. Zhevakin managed to give a plausible physical explanation—based on the work of Eddington—for why Cepheids undergo an oscillating process that does not stop after about 100 years. The achievement of Kippenhahn and his colleagues consisted in fitting a description of the Cepheids’ oscillating process within a general theory of stellar structure and evolution, and in proving that Cepheids can be interpreted as stars that are in a kind of transitional state between main sequence and giant stars.

In almost all attempts to define coherence, decisive importance is assigned to the number and strength of the inferential relations for the coherence of a propositional system. The model calculations of Kippenhahn created numerous new inferential relations, including surprising explanatory relations. They combined observational propositions about Cepheids with the theory of stellar structure and evolution. They also led to an increasingly close merging of the theory itself with the established physical background theories.

8 Two Remarks About the Relation Between Coherence and Truth

At the beginning, I mentioned that there are strong doubts, especially concerning probabilistic conceptions of coherence, as to whether a relation between the coherence of a propositional or belief system—understood this way—and its truth can be established. But also when coherence is understood as explanatory coherence, the connection between coherence and truth is controversial.

Against this background, it should be astonishing how confident astronomers were that Baade’s solution for the Cepheid anomalies was adequate, as well as Kippenhahn’s explanation of the Cepheid state as a transitional state in the evolution of main sequence stars into giant stars. They thought that these were not only possible explanations, but by and large the correct explanations for the relevant phenomena. Two aspects of the case study I described here are probably responsible for this certainty, and I would like to point them out as a final remark.

The first aspect concerns the simultaneous removal of a greater number of anomalies by a limited amount of the same assumptions. Baade’s solution employed four such assumptions: 1. We have to distinguish between population I stars (“young stars from old matter”) and population II stars (“old stars from young matter”). 2. The borderline between these two populations runs through the class of Cepheids. 3. Population I Cepheids have other values for the parameters α and β in the period-luminosity relation than population II Cepheids. 4. Population I Cepheids have a higher absolute magnitude (luminosity) than population II Cepheids with about the same period of light change. The fact that Baade managed to remove or at least attenuate most of the anomalies mentioned above at once was seen by many astronomers as an indication that these assumptions are at least close to the truth. It was a crucial factor for the acceptance of Baade’s solution.18

The second aspect concerns the explanation of phenomena—which initially constituted anomalies—by theories that were not developed to do this explanatory work. This aspect can be illustrated by Kippenhahn’s model calculations for the pulsation of Cepheids. The background of these model calculations was the developing theory of stellar structure and evolution. To the surprise of the scientists working on this project, model calculations carried out in the framework of this theory with boundary conditions that were typical for Cepheids (mass, luminosity etc.) led to star models undergoing oscillations. The measured periods of the Cepheids’ light changes could be easily fitted into these models. It was precisely the fact that the background theory was not specially designed to do this explanatory work (and thus could not be suitably amended), but nevertheless provided the desired explanations, that increased the astronomers’ confidence that they were on the right track with this theoretical interpretation of the Cepheids.19

Both aspects—the removal of a greater number of anomalies by a limited amount of assumptions and the explanation of phenomena previously regarded as anomalies by a theory not designed for this specific task—are hardly discussed at all in the relevant literature. The case study, however, shows that they might be very helpful for investigating the relation between anomalies, coherence, and truth.


For more details cf. Hoskin (1999), chapters 7 and 8.


For a detailed explanation of Cepheids and their importance for the construction of the cosmic distance scale, cf. Gautschy (2003) and Webb (1999), in particular pp. 137–146.


1 parsec (pc) is the distance from which the astronomical unit (AU)—i.e. the semi-major axis of the ecliptic—is seen under the angle of 1’’.


See Karttunen et al. (2000, 319).


Shapley first determined the distance r of these objects by using problematic statistical methods, on which I cannot elaborate more here (cf. Hoskin (1999, 280). From knowing these distances and the photometrically determined apparent magnitudes m, he calculated the absolute magnitudes M with Eq. (1). After measuring the corresponding periods, (2) allowed him to determine α and β.


“With astonishment we have seen figures in the heavens which are nothing other than such systems of fixed stars restricted to a common plane, such Milky Ways, if I may express myself in this way…“ (Translation by Palmquist 2000).


Cf. Sandage (1988).


Cf. Osterbrock (2001, 162).


For more details on Baade’s solution of the anomalies cf. Osterbrock (2001).


See Hoskin (1999, 298).


In astronomy, these heavier elements are—slightly confusingly—called “metals”.


Alfred Behr had already claimed a year earlier (1951) that the distances between galaxies estimated at the time had to be multiplied by approximately 2. Gamov had pointed out this publication to Baade; cf. Behr (1951), Osterbrock (2001). I would like to thank one of the anonymous referees for the reference to Behr’s publication.


The age estimated for the universe after Baade’s recalibration of the cosmic distance scale was, however, still much lower than today’s estimation of about 13.75 billion years.


Zhevakin’s results, however, did not receive much attention until they were published in English; see Zhevakin (1963), Gautschy (2003).


See Karttunen et al. (2000, 242 ff.).


In the later phase of red giants—depending on the temperature in the star’s core—the fusion of helium into carbon and oxygen may start. For details see Prialnik (2010, 165 f.).


See Kippenhahn and Weigert (1990).


The fact that Baade’s distinction between population I and II stars was later replaced by a more fine-grained distinction shows that his assumptions were indeed only approximately correct.


In a popular science monograph, Kippenhahn (1980, p. 122) wrote: “Thus we conclude from the fact that one can very easily fit Delta Cephei stars with their oscillation properties into the schema for stellar evolution, that everything is more or less right.” (my translation).



The author would like to thank the anonymous referees for their helpful comments on the manuscript.

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© Springer Science+Business Media Dordrecht 2012