Introduction

Nearly all the physical processes that determine the structure and evolution of stars occur in their (deep) interiors. The production of nuclear energy that powers stars takes place in their cores for most of their lifetime. The effects of the physical processes that modify the simplest models of stellar evolution, such as mixing and diffusion, also predominantly take place in the inside of stars.

The light that we receive from the stars is the only information that astronomers can use to study the universe. However, the light of the stars originates from their surfaces. Therefore it would seem that there is no way that the analysis of starlight tells us about the physics going on in stellar interiors.

 

Variable stars

However, there are stars that reveal more about their physics than others. For instance, variable stars are objects for which we can measure a time-dependent light output, on a time scale shorter than that of evolutionary changes. There are two major groups of variable stars, extrinsic and intrinsic variables.

Extrinsic variables do not change their light output directly. For example, the light changes of eclipsing binary stars are caused by two stars passing in front of each other, so that light coming from one of them is periodically blocked. The component stars of eclipsing binaries do not need to be variable themselves. By analysing the temporal light
variations and orbital motion of eclipsing binaries, one can determine their fundamental properties, and by assuming that their components are otherwise normal stars, determine fundamental properties of all stars, such as their mass. In this way, stars and stellar systems can be understood better.

Intrinsic variables, on the other hand, change their light output physically. Supernovae can become brighter than their host galaxies because of the ejection of large amounts of material. Even more revealing are stars that can change their physical size: pulsating variables.

 

Pulsating stars

How can a star change its size? Under some special physical conditions, stars can store energy in some particular interior layers for some time. This would for instance be a zone where a certain chemical species becomes
(partly) ionized. Many stars have such ionisation regions in their interiors as a natural consequence of the gradual increase of temperature and pressure from the stellar surface to the core.

If a (partial) ionization region is located in a region where the thermal time scale is of the same order of magnitude as the dynamic time scale, stellar pulsations can develop (Cox 1980). The energy that is stored during a contraction of the star, is released when it tries to reach its equilibrium state by expanding. Therefore, the star actually expands beyond its equilibrium radius. When the material recedes, energy is again stored in the stellar interior, and the whole cycle repeats: a periodic stellar pulsation evolves. This is a rough description so-called kappa mechanism (Baker & Kippenhahn 1962 and references therein) that drives many different groups of pulsators.

There are other mechanisms that can cause a star to pulsate, for instance stochastic excitation by turbulence in surface convection zones of solar-like stars, or excitation by dust formation in red (super)giant stars. As the existence of stellar pulsations is always linked to some excitation mechanism that requires certain physical conditions, several types of pulsating stars exist in certain regions of the HR Diagram. An overview of many types of pulsating variables is shown below.

Theoretical HR diagram with many classes of pulsating star indicated. The dashed line is the theoretical main sequence, the dotted line the white dwarf cooling track and some evolutionary tracks are also indicated.

 

The previous argument can be reversed: if the instability region of some class of pulsating variable is accurately known, its excitation mechanism must be able to account for it. This is already a first constraint on the interior structure of a pulsator. However, some types of oscillators are suitable for much more detailed studies: multiperiodic radial and nonradial pulsators. The research field that deals with such studies is called asteroseismology.

 

Asteroseismology

We start with a definition: asteroseismology is the study of the interior of (nonradially) pulsating stars by means of their normal mode spectrum. This technique is analogous to the determination of the Earth's inner structure using earthquakes: we use stellar pulsations as "starquakes" or, more scientifically, we use stellar pulsation modes as seismic waves.

What is a pulsation mode? It is nothing more than an individual stellar oscillation. The latter come in many different flavours, and we do not intend to give a complete overview of pulsation modes. Instead, we refer the reader to the excellent monograph by Unno et al. (1989) for more detailed information. For the present purpose, it is most important to know that there are two main groups of pulsation modes, the pressure (p) and the gravity (g) modes. These modes are classified after the force that restores the stellar equilibrium shape following the motion caused by pulsation, either pressure or buoyancy.

Pulsation modes are additionally defined by the shape of the distortions they create on the stellar surface and interior. In general, the pulsations separate the stellar surface into expanding and receding as well as heating or cooling areas. The shape of these distortions can be quantified with (combinations of) spherical harmonics; an example of these is shown below.

Schematic description of the surface distortions produced by pulsation modes with a spherical degree l=3. Whilst the brighter areas of the star are moving outward, the darker parts move inward, and vice versa. Graphics courtesy by W. Zima.

 

Between the expanding and receding surface areas, no motion takes place. The lines along which this is the case are called the node lines. The number and direction of these node lines are used for the classification
of the pulsation modes. The total number of node lines on the stellar surface is the spherical degree l, which must in all cases be larger than or equal to zero. The number of node lines that are intersected when travelling around the stellar equator is the azimuthal order m. Pulsation modes with m different from zero are travelling waves, and as they can run either with or against the rotation of the star, m can lie in the interval [-l, +l]. Finally, the number of node lines in the stellar interior is called the radial overtone k (or sometimes n).

The radial overtone number k (or n) is the number of nodal lines in the stellar interior. Such as in a musical instrument, standing waves create the "sound" of a star. Graphics courtesy by V. Antoci.

 

In this framework, radial pulsations can just be seen as modes with l=0; all other modes are called nonradial oscillations. A mode with k=0 is called the fundamental mode, a mode with k=1 is named the first overtone, etc. Pulsation modes with periods longer than that of the radial fundamental mode are usually g modes, whereas p modes have periods equal or shorter than that; radial pulsations are always p modes.

Now, asteroseismology takes advantage of the fact that some stars can oscillate in many of these radial and/or nonradial modes simultaneously. This is the key for asteroseismology, as each pulsation mode carries information about the region in which it propagates, its pulsation cavity. Every single pulsation mode has a different cavity, and its oscillation frequency is determined by the physical conditions in its cavity.

Therefore, interior structure models of the stars can be refined by measuring the oscillation frequencies of pulsating stars and by reproducing them with stellar models. The prerequisite for successful seismic modelling, however, is that the observer provides the largest possible number of intrinsic pulsation frequencies with a correct identification of all the underlying modes to the theorist. Should the observer fail to do so, for instance by over-interpreting the observational results or by providing incorrect mode identifications, any seismic model computations will lead to incorrect results.

Asteroseismology can go even beyond an improved understanding of stellar structure. For instance, some of the most massive pulsating stars will become Type II Supernovae in the future. Supernova explosions are mostly
responsible for the enrichment of the interstellar medium. If we know the interior structure of an immediate supernova progenitor, we will be able to constrain the chemical evolution of galaxies. Asteroseismology of pulsating white dwarf stars teaches us about nuclear reaction rates, neutrino physics and the history of the object's evolution on the Asymptotic Giant Branch. It is therefore clear that asteroseismology has a large impact on astrophysics in general.

 

Success stories and challenges

The greatest successes of asteroseismology were obtained for the star closest to us, the Sun. The interior structure of the Sun was modelled to fine detail (e.g., see Christensen-Dalsgaard 2002 for an extensive review) because millions of pulsation modes can be used for seismic analyses. Since the solar surface can be resolved in two dimensions, even local asteroseismology (e.g., dealing with subsurface and meridional flows) can be carried out. However, the recent revisions in the solar elemental abundances (Asplund et al. 2004) suggest that our present solar model needs some modifications.

Comparing the Sun and the distant stars, two major problems for asteroseismology are immediately obvious: asteroseismologists have to work with integrated light as stellar surfaces cannot be (sufficiently) resolved and much less light is available. This means that only a restricted range of modes, depending on their l values, is available for analysis, and that the accuracy of asteroseismic observations will be poorer than helioseismic data.

Nevertheless, asteroseismology of some pulsating white dwarf stars (e.g., see Winget et al. 1991, 1994) was quite successful, resulting in exact determinations of masses, rotation periods, luminosities, and chemical element layer masses, as well as in estimates of magnetic field strengths.

Why was this possible? The frequencies of the pulsation modes of these stars were arranged in clear patterns. Pulsating white dwarfs oscillate in g modes of high radial order. Asymptotic theory then predicts that consecutive overtones are equally spaced in period (Tassoul 1980). Consequently, the pulsation modes of these stars could be identified by just examining their frequency spectra.

Regrettably, this mode identification method does not work for all pulsating stars. Even in cases where a sufficient number of pulsation modes was observed, so that patterns within their frequencies were to become visible, they escaped detection (e.g., see Handler et al. 2000). A second problem that is common in the analysis of asteroseismic data is that only a small percentage of the theoretically predicted number of pulsation modes is observed in reality, and that it is not clear which particular set of modes the star chooses to excite to measurable amplitude.

As a result, additional observational methods that allow an identification of the stellar pulsation modes are required. The interplay of the different surface distortions and the changes in gravity and temperature caused by a given mode combined with limb darkening effects give us this possibility. Therefore, amplitude ratios or phase shifts between different photometric passbands, also in combination with radial velocity data, can reveal the pulsation mode (Dziembowski 1977). The line profiles of an oscillating star will also reflect the pulsations (Ledoux 1951) and can thus be used to identify the mode. The shape of the mode can be reconstructed by Doppler Imaging (Hatzes 1998). We conclude that one can derive, or at least constrain, the l and m values for at least the strongest pulsation modes of a given star.

However, some words of caution are necessary here: unless the identifications from any individual technique are unambiguous, extreme care must be taken when applying such methods. If possible, cross-checks between different techniques must be made, but it must also be checked whether they are indeed (mostly) independent. We refer to the discussion by Balona (2000) who showed that the agreement of the mode identifications derived from one photometric and one spectroscopic technique were not due to their reliability, but resulted from the two methods being basically sensitive to the same quantities.

In planning an observational project one should therefore take care that mode identification methods can be applied. However, their limitations must be kept in mind and must be critically examined.

Another observational challenge for asteroseismology is that pulsational signals of extremely low amplitude (less than 1 millimagnitude or a few centimetres/second) need to be detected reliably, requiring the acquisition of large amounts of high-quality measurements. Finally, the observer should be careful to measure stars not only have identifiable pulsation modes, but also can be treated by theory within its current limitations. Close interaction between observers and theorists is therefore required. In fact, the subfields of asteroseismology where the first successes were obtained, benefited from just this interaction.

 

Multisite observing campaigns

To detect the required number of pulsation frequencies of a star with the necessary reliability, measurements from a single site are usually not sufficient. The regular daytime breaks caused by the Earth's rotation leave too many gaps in the data. As a consequence, ambiguities in the identification of the intrinsic frequencies arise. These become worse the closer one comes to the observational noise level, because the relative effect of the interference of intrinsic variations with noise becomes stronger.

Therefore, an asteroseismic data set should have gaps as few and as small as possible. There are two basic strategies to reach this goal, observations from space satellites, or multisite campaigns. The latter involve collaboration of colleagues interested in the matter over the whole globe. Each group applies for observing time at their home observatory for the same time slot, observers are sent to sites where no local collaborators are available, and then the whole network measures the same target. In this way, daytime gaps can often be minimised, and nighttime gaps only occur at geographical longitudes covered by few sites and suffering from unfortunate weather conditions.

A light curve gathered during a Delta Scuti Network observing campaign. The measurements are colour-coded in the same way as the observatory where they were taken. Graphics courtesy by W. Zima.

 

This type of observing has been done more than 50 years ago for the first time, when communication between the different collaborators was much less efficient than nowadays (de Jager 1963). However, worldwide campaigns were carried out regularly within the last 20 years with collaborations such as the Delta Scuti Network, founded by Michel Breger in 1983 (e.g., see Breger et al. 1987) or the Whole Earth Telescope (Nather et al. 1990). The latter network is particularly well known not only for its scientific success but also for its real-time communication between a headquarters staffed 24 hours per day and the individual observers at the remote sites.

 

Space asteroseismology

The precision of measurement of stellar light variations is compromised by the Earth's atmosphere. In particular, scintillation and variations in sky transparency preclude accuracies considerably below one thousandth of a magnitude per single brightness measurement. Going to space resolves this problem. The first mission that achieved a precision sufficient for asteroseismology was WIRE, using its star trackers after the failure of the main mission (Buzasi et al. 2000). The first dedicated satellite that had asteroseismology as its main purpose is MOST (Walker et al. 2003), operating since 2003. CoRoT (Baglin 2003) was launched in December 2006, and Kepler (Christensen-Dalsgaard et al. 2008) followed in March 2009. Both of the latter satellites also have the capability and purpose to find extrasolar planets. The Austrian-Canadian-Polish BRITE mission (Zwintz & Kaiser 2008) is supposed to be launched in early 2011.

 

Conclusions and outlook

We believe that the future of asteroseismology is a bright one. Some classes of pulsating stars recently became available for seismic modelling. Mode identification methods are maturing and will become more and more reliable. Several interesting individual objects were discovered recently ("hybrid" pulsators showing both low-order p and g modes as well as high-order g modes). Because of the construction of giant mirrors, telescopes of the 2 - 4 metre class are becoming more easily available for stellar astronomy, which will be particularly helpful for the application of spectroscopic mode identification methods. Asteroseismic satellite missions are already delivering data of unprecedented quality, and it is hoped that this will continue over many years to come.

 

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