Stellar Structure Models Revisited: Evidence and Data in Asteroseismology

Abstract

This paper advances further an ongoing project to understand the history of stellar structure modelling and its inferential practice. It does so by taking a harder look at the data: how it is collected, analysed statistically, and represented in HR diagrams and stellar structure models alike. The focus is ultimately on the sorts of strong observational constraints revealed in the last two decades within the new and expanding field of asteroseismology. It is argued that the typical inferential practices in asteroseismology, while richly loaded with modelling assumptions of their own, do not raise any circularity worries that may compromise the quality or value of the data.

1 Three Aims in the Philosophy of Stellar Astrophysics

There are three aims to this chapter. First of all, I aim to provide a succinct introduction to stellar astrophysics – particularly as regards stellar structure modelling, with a focus on the sorts of observational data and constraints that are nowadays operative within the field. The chapter may thus provide an informative first point of contact with a fascinating scientific field. The hope is that this will be of use to philosophers and other humanists not specialised in the philosophy of astrophysics – or, for that matter, not even specialised in the philosophy of science.

The second aim is most distinctly philosophical. I claim that there is one lesson regarding scientific modelling at large that becomes very apparent in stellar structure modelling. It concerns the nested nature of modelling; the fact that most models operate against a background that incorporates further models, and where data is routinely tested backwards, as it were – by deducing the values of theoretical parameters from the data given some background assumptions, and where those parameters are then fed into new models that can appropriately generate the data. While this method seems amenable to a bootstrapping model of confirmation (Glymour 1980), I do not here consider in detail issues of confirmation. Instead, I look at how, in the nowadays booming field of observational asteroseismology, our models of stars and their evolution are sensitive to further models of layered stellar interiors, to models of the physics of radiative materials, to models of stellar atmospheres in coronal astrophysics, and to models of the vibration modes that yield the astrophysical data that in turn supports those models.

Let me emphasise from the start that none of this ‘nesting’ of models within models can serve to deny that we have by now robust knowledge of the physics of stellar interiors, and that the accepted typologies of stars, and their evolutionary phases, are a very secure part of our stock of scientific knowledge. As the slogan goes: we do know the interior of our local star, the Sun, rather better than we know the interior of our own planet, the Earth. Yet, it is noteworthy (certainly to an epistemologist) that none of this knowledge is supported by any sort of active laboratory intervention on the system that is the object of our knowledge. We certainly don’t experiment on stars the way we experiment on the objects of our laboratory studies. In stellar astrophysics, we can’t actively intervene on the conditions of the production of the phenomena we study, and we certainly cannot have the sort of experimental warrant that is acquired in the actual causal manipulation of the system of interest in most laboratory experiments.¹ In other words, we don’t materially probe into stars’ interiors in the way we can probe materially into (the surface layers of) our Earth. It is then at least paradoxical that it should be common lore that we know the interior of stars. My second aim is to provide some philosophical insight into why this is so.

As a word of warning, the idea that models are ‘nested’ in astrophysics, which I defend, must be distinguished carefully from Harry Collin’s (1992) much discussed ‘experimenter’s regress’. I do not believe that there is a pernicious circularity in these models of the sort Collins denounces for experiments on gravitational waves.² Nonetheless, there is an obvious circularity in the fact that all models contain assumptions that involve or result from further models. However, I shall argue that, at least as regards stellar astrophysics, such ‘nesting’ of models within models is innocuous to the justification of the models to the extent that the supporting models do not themselves employ the same questionable idealisations that appear downstream the modelling chain. This view gains support from a consideration of the ‘inverse’ or ‘forward’ modelling practices typically engaged in by astrophysicists, which is the focus of much of the discussion throughout.

The final third goal to this chapter involves updating previous work of mine in this area, and in particular a series of papers produced over a decade ago on the fictional nature of the conditional modelling assumptions in stellar structure modelling. Those papers argued that stellar structure inferences are supported by ‘fictional conditionals’.³ The new and exciting observational science of asteroseismology, and the remarkable data thrown out by the CoRot and Kepler missions, certainly force a revision of some of those claims. At the very least, it is necessary to finess some of the claims regarding the fictional nature of the antecedents/background conditions. Many of those fictional assumptions turn in the light of recent evidence to have been false idealisations.⁴ Yet, it all remains of a piece with the idea that stellar structure models involve indicative conditionals with hypothetical assumptions in their antecedents from which the observational consequences that appear in their consequents can be derived. It may be just as well to start at this point, recapitulating some of those claims before launching into a discussion of how the new observational science of asteroseismology changes, corrects, and enhances the picture.

2 A Very Brief History of Stellar Astrophysics

Arthur Eddington’s The Internal Constitution of the Stars (Eddington 1926) is regarded as a milestone in the history of stellar astrophysics, and as having set the foundations of the discipline for years to come. By the time of its publication, Arthur Eddington (1882–1944) had already been a very young Director of the Cambridge Observatory, secretary and then president of the Royal Astronomical Society, and a fellow of the Royal Society for over a decade. He had played a major role in the 1918 expedition (sometimes named the ‘Eddington expedition’ in his honour) that collected the data that served to confirm Einstein’s theory of general relativity. Eddington is widely credited for the discovery that the energy produced in stars is generated by hydrogen fusion into helium. His 1920 paper with the same title (Eddington 1920) already advanced the hypothesis that Einstein’s postulate of the equivalence of mass and energy, encapsulated in the famous E = mc² equation, allowed for the conversion of mass into energy at rates that would explain the range of luminosity outputs observed for most stars, and he consequently predicted that the temperatures in the interior of stars could reach millions of degrees Celsius. These hypotheses were all confirmed in due course, cementing Eddington’s reputation as the first and foremost expert in the theory of stellar evolution, and hence helping to launch the discipline of stellar astrophysics.

Eddington’s other great contribution to stellar evolution is his re-interpretation of the central or most important regularity in observational astrophysics, the so-called Hertzsprung-Russell (HR) law, as a ‘mass-luminosity relation’ (Eddington 1926, chapter VII). The HR law had been independently discovered in the years 1911–13 by the Danish astronomer Ejnar Hertzsprung (1873–1967) and the American Henry Norris Russell (1877–1957). On analysing the spectral lines of the stars surveyed at the Harvard College Observatory for the Henry Draper catalogue, Antonia Maury (1866–1952) had found a way to group them neatly into spectral classes while retaining information regarding their temperature.⁵ This gave rise to the identification of the spectral classes of stars (O, B, A, F, G, K, M) in the Harvard Classification Scheme, with specific ranges of surface temperatures associated to each class. It was not difficult then to plot in a diagram increasing luminosity versus decreasing surface temperature (spectral class). This soon was noted by Hertzsprung, Russell and others to express a clear correlation between a star’s luminosity (L), a measure of the star’s ‘energy power’, on the one hand, and its effective surface temperature (T_eff), the temperature of the outer layer or photosphere, on the other. In a typical Hertzsprung-Russell diagram (such as the one reproduced in Fig. 7.1), this correlation is represented as a descending sequence from the hottest and brightest stars in the top left-hand corner to the coolest and dimmer ones in the bottom right-hand corner. This is the notorious main sequence, which Eddington was amongst the first to understand, via his postulate of a mass-luminosity relation, as the central sequence of temporal evolution in the lifecycle of stars.

Fig. 7.1

A Hertzsprung-Russell diagram displaying the HR law for GAIA data. (© ESA public domain)

The HR correlation expressed in this diagram is sometimes referred to as the main ‘empirical law’ of stellar astrophysics, but this is a misnomer for a couple of reasons. First, the quantities plotted in an HR diagram are not directly observed in any meaningful sense of the term. They are rather inferred, by means of some simple phenomenological laws and extrapolations, from those quantities that can in fact be observed. The two quantities that are directly observable are the incident radiation flux from a stellar source into a telescope on earth, also known as the star’s apparent brightness (I _obs), and the characteristic set of spectral lines of the radiation received on earth, or distribution of radiation intensity per wavelength, also known as the electromagnetic spectrum (I _λ) of the source. Add a third, the star’s distance from earth (d), which is not strictly speaking observable but can be calculated by geometric means independent of any consideration of stellar structure, namely ‘parallax’, or cluster analysis. Given precise values for these three quantities (I _obs, I _λ, d) for any given stellar source, it is possible to derive the luminosity (L), effective temperature (T _eff), and chemical composition of the star as follows (see Suárez 2013, 239–240; or Tayler 1994, chapter 2). The luminosity (L) of a star is its ‘energy power’ or the amount of energy radiated per unit time – a simple function of distance and observed incident radiation flux: L = 4πd ² I _obs. The effective surface temperature can then be derived from the luminosity under the assumption that the star is a blackbody spectrum as follows: ${T}_{eff}^4=L/4\pi {R}^2\sigma$, where σ is the Steffan-Boltzmann constant and R is the radius of the star.

The second and most important reason why the HR law is not ‘empirical’, though, is that the ‘main sequence’ in an HR diagram is not merely the diagrammatic representation of a bare statistical correlation between the values of the luminosities and spectral types of the stars in the sky, as observed at a given time. As Eddington pointed out, it goes much further in describing the evolution of each star throughout its entire lifecycle. Stars typically begin their lifecycle as coalescing gas in the wake of supernova explosions. They thus start out as high temperature and great luminosity objects, and they reach their maturity as colder and less luminous objects further down the main sequence. All stars eventually meet their end away from the main sequence, as luminous but cool ‘red giants’ if they were initially not very massive (less than eight solar masses); otherwise, the star’s core explodes in a supernova explosion – a very rare occurrence. Eventually a star will radiate away its energy and outer layers, and then end up as a very dense, and hence hotter, but much dimmer ‘white dwarf’ (In Fig. 7.1, red giants correspondingly sit to the upper right-hand corner of the HR diagram, while white dwarfs are in the lower left-hand corner). In other words, for each individual star, its evolution in time will necessarily take it from a higher position at the left side of the main sequence in an HR diagram to a lower position at the right end of this same sequence. Hence, the HR law is not a mere statistical regularity but the expression of the fundamental process of dynamical evolution that applies to the lifecycle of every star. It took the genius of Eddington to turn a statistical empirical regularity into the fundamental dynamical law of stellar astrophysics.

The discovery of this dynamical law of stellar evolution has deeply shaped the way the field of stellar astrophysics has evolved. The HR law of evolution down the main sequence led to the subsequent history of stellar astrophysics as a field overwhelmingly dominated by the effort and need to correctly model the structure of stellar interiors, often as a simple function of only the stellar mass. For if we can reduce the parameters describing the internal physics of the star to the initial mass of its coalescing gas, at the birth of the star, we are then able to predict its entire subsequent lifecycle, as it moves down the HR diagram’s main sequence. So, we are in an optimal position to generate accurate predictions for what are known as the ‘observable quantities’ of stellar astrophysics, namely the luminosity and spectral type (or effective temperature of the photosphere), precisely the quantities that get plotted in an HR diagram. This mode of inference from model parameters to observable quantities is what is known in the field as ‘forward’ or ‘inverse’ modelling. It does not allow us to infer the physical conditions prevalent in the star from its observable conditions at the photosphere, but rather the other way round: One must start hypothesising some values of the parameters (some description of the internal structure of the star) to derive the observable consequences that can then be tested against the empirical data.

This inferential procedure can evidently lead to suitably modify some of these parameters retrospectively, by fine-tuning the initial description within some margins both for the parametrization chosen, and the initial values for some of those parameters. It is not a procedure that can ever settle, or in any way determine, the value of those parameters. On the contrary, underdetermination is rife here since we can produce models with critically different hypothetical descriptions of the (presumably unobservable) properties of the interior of a star yielding nonetheless approximately equally correct values for its observed quantities.⁶ That is, nothing in an HR diagram can determine precisely what goes on in the actual interior of any star. We can merely postulate some description of a star’s age, chemical composition and initial mass at birth, shape and layered structure, energy transfer mechanism and so on, and then use the models to appropriately deduce from these hypothetical physical processes and parameters within the star some of the star’s observed quantities. In other words, the ‘forward’ inferences that take us from the values of the central parameters in stellar structure models describing the (unobservable, hypothetical) stellar interior, to the (observable, actual) photosphere, cannot ever hope to settle what goes on inside a star, for any star, whatever its position in the main sequence. Thus, either the interior of a star remains a useful fiction (as useful as many other fictions that have routinely been employed to great benefit for inferential purposes throughout the history of science); or there is another set of ‘observable’ quantities beyond those plotted in HR diagrams, one that can provide us with information regarding those elusive interiors of stars.

3 ‘Fictional Conditionals’ in Stellar Structure Modelling

Let us consider the first option first: the modelling descriptions of stellar interiors within stellar structure models are essentially fictions, adequate only for the purpose of expedient inference to the star’s observable quantities, namely its luminosity (L), and the effective temperature of its photosphere (T_eff).⁷ Such fictional assumptions have no further cognitive value beyond the convenience of their expediency. At the most abstract level, stellar structure modellers assume that a star is defined as a cloud of gas uniformly constituted by a mixture of hydrogen and helium, bound together by self-gravitation, and radiating energy from an internal source at its core (Prialnik 2000, 1). It then seems extraordinary that the star would maintain itself in equilibrium for vast periods of its lifetime (for millions of years). This is due to the exquisite way in which the forces balance themselves out in the cloud of gas. The inward gravitational force is perfectly balanced by the outward radiation pressure; otherwise, the star would collapse under its own weight. Conversely, gravity prevents the star’s matter from blowing away under the outwards pressure exerted by the radiation. As Eddington himself put it almost a century ago (1926, 20): “We may think of a star as two bodies superposed, a material body (atoms and electrons) and an aethereal body (radiation). The material body is in dynamical equilibrium but the aethereal body is not; gravitation takes care that there is no outward flow of matter, but there is an outward flow of radiation”. What prevents all radiation from diffusing away instantaneously is the opacity of the material of the star – which we represent by means of an absorption coefficient.

Nevertheless, this abstract description of a star is clearly a convenient fiction. A star’s boundaries are rather imprecise, and the extensive area where the surrounding atmosphere interacts with the inter-stellar gas is the locus of extraordinary physical processes and events which are interesting in themselves, and which affect the radiation passage from emission to reception on earth. In other words, a star is not a closed system, but is in constant contact with its environment, the interstellar medium. The study of this interaction is now the remit of an expanding field known as ‘coronal astrophysics’, which has nonetheless traditionally been ignored for the purposes of modelling stellar structure and stellar evolution. As regards the forces acting on the constitutive gas, although self-gravity dominates it clearly is not the only force. Besides the radiation forces, there are also magnetic forces at play, which can occasionally have dramatic effects on the shape of the star and the ensuing surface temperature distribution over the photosphere. Finally, while young stars tend to be composed mainly of hydrogen and helium, as they move down the main sequence, they will generate elements with heavier atomic numbers such as oxygen, carbon, and nitrogen, which they then eject into the interstellar medium (and which can thereby be present in still younger stars formed in the vicinity, particularly in the wake of supernova explosions). This is the sense in which stars are popularly said to be the ‘kitchen’ of the universe, where the heavy elements that make life possible are ‘cooked up’.

Thus, a real star will typically not look much like the perfectly symmetrical ball of uniformly distributed hydrogen and helium gas mixture in perfect equilibrium described in stellar structure textbooks. A real star like our Sun looks a bit more like the object depicted in Fig. 7.2. We suppress a great deal of the physical detail that we know to be present in a star when we model it in accordance to the four ubiquitous assumptions in stellar structure modelling: (i) isolation (IA) from interstellar medium, (ii) blackbody radiative equilibrium (EA), (iii) uniform composition (UCA) of hydrogen and helium (roughly at 70–30% respectively), (iv) gravity as the only self-bounding force, which yields the assumption of perfect spherical symmetry (SSA) of the star’s layers, including the photosphere. These assumptions together combine to great effect in the building of concrete stellar structure models that take in as initial conditions the description of the internal state of the star at each of its layers, as parametrized solely by the mass of the star (the ability to parametrize singly by mass is a consequence of the UA and SSA assumptions which together entail that the mass of the star grows linearly and monotonically with radial distance from the centre of the sphere).

Fig. 7.2

The internal structure of a typical average main sequence star. (© ESA/NASA public domain)

These assumptions yield the four equations of stellar structure: hydrostatic equilibrium, continuity, radiative transfer, and thermal equilibrium (Prialnik 2000, Ch. 5; Tayler 1994, Ch. 3), and it is these four equations jointly that swiftly yield values for the ‘observable’ properties that are plotted in an HR diagram: luminosity (L), effective temperature (T_eff), and mass fraction (I_λ) at the photosphere. In other words, the assumptions are in place not because they are idealised approximations to the nature of stars. Barring uniform chemical composition (UCA), which clearly is an idealisation, the others would seem, at this point, to be working rather as fictional posits. They are employed because they are effective in allowing modellers to derive expedient predictions for the observable quantities in an HR diagram from mere estimates of radius, or overall mass.

Suárez (2013) argued that the inferences that these four equations licence towards the ‘observable’ HR quantities can be formally represented by means of fictional conditionals: Indicative conditionals that operate against the background of the four fictional assumptions (IA, EA, UCA, SSA), or have those assumptions appear in their antecedent.⁸ If so, such fictional conditionals allow for expedient inference without necessarily requiring the truth, or even truth-aptness, of the assumptions – just figuring as presuppositions or as part of the antecedents of the relevant inferences. But are IA, EA and SSA really fictional assumptions required merely for expedient inference? Or are they rather idealisations such that evidence can be provided for their suitable modification at least in some cases? Can we ‘peek into’ the interior of stars directly, that is, without presupposing that these fictional assumptions apply instrumentally in all our models? It turns out that we can, and such evidence is available.

4 Asteroseismology: The Observational Basis of Stellar Astrophysics Revisited

The second option is to find other observational means that allow us to probe deep into the interior of stars directly, without making any fictional assumptions regarding isolation, equilibrium, uniform composition, or spherical symmetry. The nowadays booming field of asteroseismology, which was merely nascent 20 years ago, provides such means to suitably modify the assumptions for a multitude of actual stars. The discovery and thorough analysis of observed seismic oscillations in stars has thus provided us with detailed knowledge of their interior. This is particularly the case for our local star, the Sun, and the claim that we know its interior rather well is nowadays supported by the new science of helioseismology.⁹

There is at any rate now a vast amount of data regarding stellar oscillations, some of it still awaiting full analysis. Most of it was collected in the CoRoT mission of the European Space Agency (ESA), which run between 2006 and 2013; and in NASA’s successive Kepler missions, following on the initial launch of the Kepler space telescope in 2009 all the way to its retirement in 2018.¹⁰ Such observational data has undoubtedly revolutionised our understanding of stellar structure models, imposing strong constraints on the idealising assumptions involved.¹¹ The study of asteroseismology thus provides us with insight into how astrophysical data is collected, analysed statistically, and represented in characteristically expanded HR diagrams. It turns out that the inferences from observed data to the data models – and those from these data models onto the parameter space in theoretical models – are predictably rich with modelling assumptions of their own. These assumptions in turn play critical roles in determining the quality of the data, and how precisely it weighs for and against the different idealisations employed in stellar structure models.

The basic phenomenon underlying all asteroseismology research is the regular pulsation in a star’s observed brightness due to internal gravitational or acoustic oscillations caused by rotational or convection forces within it (Aerts 2015, 38–39). These investigations enable modellers to estimate both the opacity, or absorption coefficient through the star, and its hydrostatic equation state, while offering opportunities to measure diffusion through slow mixing; overshoot from convective cores of stars that have them (typically large stars, certainly larger in mass than the Sun); progressive mass loss, and near-surface convection in very old red giants (Christensen-Dalsgaard 1999, 1). Asteroseismologists study the oscillations in stars due to both pressure (in the convective layers of the star) and gravity (throughout the star). These are known as the g- and p- modes of oscillation; there are also mixed g-p modes that combine gravity and pressure waves and are most informative regarding the deep structure of the star (Aerts 2014, 155; Aerts et al. 2010, Ch. 7). For each of these modes, there are a range of oscillation nodes going from purely radial (i.e., arising at the core and expanding regularly outwards, as in a regularly pulsating sphere that expands and contracts repeatedly), to very non-radial (for instance when the star is literally deformed at two opposite ends of a quadrant at its surface, alternating north and south of the equator). In fact, stars show an extremely rich oscillatory pattern of nodes for both pressure and gravity modes, and to properly detect all this oscillatory behaviour requires very extensive longitudinal studies over many years. This is the reason why asteroseismology did not really take off until the launch of the CoRoT and Kepler missions. Only when such extensive longitudinal data for a single star is conveniently aggregated by Fourier transform methods, can we obtain an overall oscillatory profile for a star, such as that depicted in Fig. 7.3 for KIC 4726268, which aggregates the Kepler mission data for that star.

Fig. 7.3

The asteroseismical portrait of star KIC 4726268, as observed by the Kepler satellite (Reproduced from Aerts 2015, p. 39 with kind permission from Conny Aerts, as well as AIP publishing)

The existence of rotational and convection forces inside the star is contrary to at least two of the assumptions that run through most stellar structure modelling, namely the equilibrium (EA) and spherical symmetry (SSA) assumptions. If there is convection inside the star, that entails the energy transfer is not entirely radiative, but in some layers of the star at least energy gets transferred by means of convectional plasma movement (in essence: huge flows of parts of the stellar gas from some regions into other, presumably cooler, regions in the star). If this is so, the star cannot in fact be entirely in a state of thermal equilibrium, and the (EA) is not a convenient fiction but a false idealisation in at least some of the regimes within the star.

On the other hand, if there are rotational forces inside the star, it means that different regions of the star rotate at different speeds – and this ought to generate deformations of the layers of the star in different regions. It is nowadays known – precisely out of helioseismology data regarding its pulsating oscillations – that the Sun experiences higher rotational speed of its radiating photosphere in the equator than the poles. The rotation period at the equator is about 25 days while that at the poles is about 35 days (Parker 2000, 27), which deforms the Sun into an oblong at the equatorial axis (i.e., the radius of the Sun is slightly longer to the equator than the pole). Since we now have good evidence in asteroseismology that stars experience similar differences in rotational speed due to divergent rotational forces, we know that the assumption of spherical symmetry (SSA) is not a convenient fiction but a false idealisation for most, if not all, stars.¹²

We have also by now gained – by similar asteroseismical methods – a lot of knowledge regarding the energy transfer mechanisms inside a multitude of stars and star types. The consensus nowadays is that young stars which have not yet burnt their hydrogen exhibit a certain pattern. The very small ones (with masses less than or equal to the solar mass) possess a purely radiative core in thermal equilibrium, but a large convective layer that experiences considerable rotational and tidal forces. This is represented accurately in Fig. 7.2 for our Sun, and already shows that the EA assumption is a very rough approximation to the outer layer of stars like ours. Those stars that are a little more massive (with masses between one and two solar masses), develop a convective inner core, and thus present a three-layered structure, with a small convective core, an extensive radiative layer in equilibrium, and a shallower outer convective layer. Finally, those stars that are very massive (above two solar masses) possess a small convective core and an often very large radiative outer layer only. Older stars that have burnt most of the fuel (i.e., red giants) possess some sort of convective envelope but its extent is not well known – they also often exhibit changing and irregular patches of convective and radiative energy transfer throughout (Aerts 2015, 37; Bedding 2011).¹³ We see then that asteroseismology shows that the equilibrium, spherical symmetry, and uniform composition assumptions no longer operate as fictions, but that they turn out to be false idealisation in most cases. We even now have some good estimates, supported by evidence, of how they differ from the truth in many stars.

5 From Experimenter’s Regress to Modeller’s Nest

The vast amount of asteroseismology data that we now possess is deeply affecting our understanding of stellar interiors. The data both impose stringent constraints on some of the parameter values in stellar structure models, and they force us to modify some of their central assumptions. These data demonstrate that some of the assumptions that have characterised stellar astrophysics from its historical origin are demonstrably very far from the truth in many cases. At least for those stars for which we have recorded enough longitudinal data, over a long enough period, their oscillations are hardly compatible with the spherical symmetry and equilibrium assumptions. Therefore, asteroseismology rather dramatically expands our understanding of the evolutionary phases of stars too. This is true to the extent that the HR diagrams that we are most likely to see nowadays include shady areas representing the oscillations in luminosity and effective temperature for most stars (see Fig. 7.4).

Fig. 7.4

A portion of the HR diagram accounting for characteristic oscillations revealed in asteroseismology (Reproduced from Aerts 2015, p. 37 with kind permission from Conny Aerts, as well as AIP publishing)

Nevertheless, some legitimate worries concern circularity of inference: To what extent are we assuming the very models of stellar structure, including some of their formidable assumptions, in the study and statistical analysis of the asteroseismology data that we use to adjust the parameters and the assumptions in those models? A preliminary version of this worry is familiar to anybody who has been exposed to the heated debates surrounding Harry Collins’ (1992) “experimenters’ regress”. However, there is no threat of such a regress in the CoRoT and Kepler missions. The recording of the receiving star radiation by the space telescopes does not assume any specifics about the interiors of the objects that produce them. The oscillations in intensity recorded at the telescopes are insensitive to any of the features of the stars as modelled. It would make no difference, for instance whether we model the stars as having a convective or radiative core. It’s rather the other way round – the recorded data set out constraints on how we can possibly model those stellar sources.

Furthermore, one advantage of a purely observational science like astrophysics is precisely that there is no material interaction with the object that is causing the recorded data, so there can be no experimental ‘infection’ of the source. Similarly, though, for the recording instruments. Unlike, say, Collins’ story with gravitational wave detection, the instruments themselves are not subject to the laws of the models under test. Rather, the assumptions and models that are under test in asteroseismology research into stellar interiors are themselves playing no role in the production or recording of the data. They are not cosmological assumptions, and they do not apply to any of the interstellar medium the radiation must travel through to reach the telescope nor to the space (space-time) that these instruments operate in.¹⁴

However, there are obvious modelling circularities at play. The data must be statistically analysed and modelled appropriately to generate informative diagrams such as those in Figs. 7.3 and 7.4. Most importantly, asteroseismology requires what is known as a forward seismic model: “a model that takes the physical properties of a star as input parameters and predicts the star’s oscillations” (Aerts 2015, 38). That is, one must start with some description of the internal workings of the star and work one’s way up to the photosphere luminosity and spectral class. This generates a sort of nested modelling practice, in which we first model the star, then deduce the oscillatory frequencies, then compare such oscillations to the actual observations, then correct the parameters and assumptions backwards as needed. While this is certainly worthy of further study, there is no prima facie offence to either logic or evidence in such a method, which is typical of a large range of our observational sciences.

Here the comparison with models of the interior of our planet in Earth science and geophysics is certainly instructive. Miyake (2015) describes how Preliminary Reference Earth Models (PREMS) are used to turn geological data into evidence for or against different values of parameters and assumptions in more sophisticated Earth models. There is some sense of circularity here: “Suppose I create an initial model, and then I study the deviations from this model. These deviations are then taken to be evidence for, say, casual factors that must be taken into account in the model. I then add these causal factors and improve the model” (Miyake 2015, 826). Similarly, in stellar astrophysics, we take the original HR diagram (Fig. 7.1), and the sort of preliminary stellar structure models that support it, as discussed in Sect. 7.3, and then study deviations from these models’ predictions in asteroseismological data. Thus, it turns out, for example, that some stars’ layers rotate at speeds different to and incompatible with some of the assumptions in the preliminary models. This leads us to reject such assumptions, to measurable or quantifiable degrees, and to correspondingly modify the HR diagram. The more complex version of the HR diagram (Fig. 7.4) is then fed back into more sophisticated models of stellar structure and interiors that answer to such asteroseismological data. While this ‘bootstrapping’ procedure may be surprising from a naïve hypothetico-deductive point of view, it does not impugn the empirical status of the resulting scientific knowledge in any way.¹⁵

Nevertheless, as a methodological issue, the ‘nested’ nature of stellar structure models is remarkable. These models derive from earlier, simpler models, which they often subsume as limiting cases. And each is nowadays underpinned and severely constrained by asteroseismological data. The data itself has been modelled to render it intelligible; and to yield significant observational constraints on the parameters in the models. Thus, we find that stellar structure models are supported by further models, both in their historical development, and in their relation to the data that they account for. And, again, while there is no epistemic circularity involved here, the consequences for our understanding of the nature of those models and their relationship to their target phenomena are startling and deserve attention.¹⁶ Let me here outline briefly three of these consequences that await further study.

First, Sibylle Anderl (2018) has defended the use of simple but highly idealised models in astrophysics, and this is a lesson that stellar structure models also bear out. In Sect. 7.2 I argued that the entire field of stellar astrophysics modelling emerged from some very straightforward interpretations of correlation data in HR diagrams as dynamical laws, together with highly simplifying fictional assumptions regarding stellar structure and evolution (the IA, UCA, EA, and SSA assumptions). Together these jointly yield a template for powerful models of stellar interiors accounting for such data. It is hard to imagine how stellar astrophysics could possibly have developed historically as a discipline without such straightforward, relatively simple, yet inferentially robust early models. Anderl (2018, 828) connects the intuitions lying behind such models with the pursuit of understanding – and this seems fitting here too. While the early models may have been convenient fictions, they provided richly layered understandings of the physics of stellar interiors. Astroseismology does not fundamentally alter that picture – we still very much understand the physical processes operating in stellar interiors in the terms laid out by the four equations described in Sect. 7.3 in this chapter. We are just adding significant detail deriving from the specific asteroseismological profile for each star.

Second, Daniela Bailer-Jones – a pioneer in the philosophy of astrophysics over 20 years ago – used to emphasise¹⁷ that most models in astrophysics are composed of more specific sub-models aiming to capture different parts of the overall causal mechanism putatively responsible for their complex target phenomena. While Bailer-Jones’ favourite cases of this sort of composition were cosmological models of galaxy formation, similar lessons apply to models of stellar structure too. For instance, asteroseismological models must consider both acoustic and gravity pressure modes in a typical star, as well as the range of mixed modes they generate. Each mode is responsive to a different mechanism, and the ‘art of stellar modelling’ requires a judicious choice of nodes in each mode for each star. In other words, it calls for judicious combinations of the underlying mechanisms.

Finally, and most tentatively, there is an issue concerning nested modalities within nested models in astrophysics, which Elena Castellani and Giulia Schettino explore in recent work (Castellani and Schettino, 2023). Inasmuch as a model describes a possible mechanism partly or fully responsible for a phenomenon, it lays down a possibility space. We may then wonder what sort of possibility is explored in nested models, where the possibility operator in the overall or ulterior model arguably ranges over the more primary possibility spaces of the simpler or antecedent models. One conjecture is that such possibility spaces obey a simple multiplication rule, and that degrees of possibility thus behave as classical probabilities. But how and on what grounds do we impose a data-sensitive probability measure on, say, the arrays of distinct models of convective flows in the convective envelope inside a star? Undoubtedly, these are issues that deserve further study.

6 Conclusions

In this chapter, I have aimed to (i) provide a historical introduction to the exciting new field of asteroseismology, and how the observational data coming from asteroseismological research significantly constrains stellar structure modelling; (ii) update my 15-years old account of fictions in structural modelling in the light of such new data, and (iii) outline some of the new avenues of philosophical research that open up in the wake of the ‘nested’ nature of the models involved. While stellar structure modelling throws out many interesting puzzles and distinctions, I have argued that there is no ‘vicious circularity’ arising from the nested nature of the models involved that could impugn the empirical knowledge that asteroseismology affords us. We do indeed know the interior of our Sun and the stars to a greater extent than we know many other systems in the observational or even experimental sciences.