Introduction

Helioseismology, the study of the Sun using solar oscillations, has provided us with the means to probe the solar interior. Since the discovery of the oscillations in 1962 (Leighton et al., 1962) and their interpretation as global oscillation modes by Ulrich (1970) and Leibacher and Stein (1971), helioseismology has been used extensively to study the interior of the Sun, mainly through inversions of solar frequencies. With space missions such as CoRoT (Baglin et al., 2006) and Kepler (Borucki et al., 2010) now observing oscillations of other stars, inversions of stellar frequencies may soon be feasible. There are two ways by which we could use seismic data to make inferences about the stars. The first way involves trying to find models whose frequencies match the observed frequencies, usually referred to as “forward modeling.” This is essentially what is done in most fields of astronomy. The end result of the process is a model that is the best match to the observations. The second way is to invert the data. Inversions use the data directly to make inferences about the star. In the case of inversions, we can make a distinction between the structure of the star and the structure of the best-fit model. These days inversions are used to study the solar interior, while forward modeling is used to study other stars.

It is not possible to do an inverse analysis unless we can do the forward analysis.

Introduction

My brief for the IAC Winter School was to cover observational results on helioseismology, flagging where possible implications of those results for the asteroseismic study of solar-type stars. My desire to make such links meant that I concentrated largely on results for low angular-degree (low-l) solar p modes, in particular results derived from “Sun-as-a-star” observations (which are, of course, most instructive for the transfer of experience from helioseismology to asteroseismology). The lectures covered many aspects of helioseismology – modern helioseismology is a diverse field. In these notes, rather than discuss each aspect to a moderate level of detail, I have instead made the decision to concentrate on one theme, that of “sounding” the solar activity cycle with helioseismology. I cover the topics from the lectures and I also include some new material, relating both to the lecture topics and to other aspects I did not have time to cover. Implications for asteroseismology are developed and discussed throughout.

The availability of long time series data on solar-type stars, courtesy of the NASA Kepler Mission (Chaplin et al., 2010; Gilliland et al., 2010) and the French-led CoRoT satellite (Appourchaux et al., 2008), is now making it possible to “sound” stellar cycles with asteroseismology. The prospects for such studies have been considered in some depth (Chaplin et al., 2007a, 2008a; Metcalfe et al., 2007; Karoff et al., 2009, e.g.), and in the last year the first convincing results on stellar-cycle variations of the p-mode frequencies of a solar-type star (the F-type star HD49933) were reported by García et al. (2010).

Background

The XXII Canary Islands Winter School of Astrophysics, organized by the Instituto de Astrofísica de Canarias (IAC), focuses on the new advances and challenges that asteroseismology provides in the domains of stellar structure, dynamics, and evolution. Every year the Winter School welcomes around 60 Ph.D. students and young postdocs and provides a unique opportunity for them to broaden their knowledge in a key field of astronomy.

Scientific rationale

When oscillations of the Sun were first discovered, a new era of science began. The observed frequencies could be used to probe deep into the stellar interior, the only measurements that could possibly pierce the stellar surface. Today, “helioseismology” has been responsible for some of our deepest understanding of the Sun: we know the radial and longitudinal rotation profile of the interior, we have measured the depth of the outer convection zone, and it has helped solve the so-called neutrino problem when the observations and theory predicted a much hotter central temperature than the observed neutrinos predicted. Today, these seismic observations are not only available in much higher quality, but they are also available for hundreds of other stars. In the last few years, many space missions (CoRoT and Kepler) have produced these data of exquisite quality, and for the first time we are in a position to study the Sun in the context of other stars, measure the fundamental parameters of single field stars to within 2 percent, learn about diffusion processes and the effects of rotation on the stellar structure, and test opacities and equations of state in extreme conditions.

What are solar-like oscillations?

Oscillations in the Sun are excited stochastically by convection. We use the term “solarlike” to refer to oscillations in other stars that are excited by the same mechanism, even though some of these stars may be very different from the Sun. The stochastic nature of the excitation produces oscillations over a broad range of frequencies, which in the Sun is about 1 to 4mHz (the well-known 5-minute oscillations). Stellar oscillations can also be excited via opacity variations (the heat-engine mechanism, also called the k mechanism), as seen in various types of classical pulsating stars (Cepheids, RR Lyraes, Miras, white dwarfs, δ Scuti stars, etc.).

For a star to show solar-like oscillations, it must be cool enough to have a surface convective zone. In practice, this means being cooler than the red edge of the classical instability strip, which includes the lower main sequence, as well as cool subgiants and even red giants. Indeed, solar-like oscillations with periods of hours (and longer) have now been observed in thousands of G and K giants (see Section 3.9). There is also good evidence that the pulsations in semiregular variables (M giants) are solar-like (Christensen-Dalsgaard et al., 2001; Bedding, 2003; Tabur et al., 2010), as perhaps are those in M supergiants such as Betelgeuse (Kiss et al., 2006).

What about hotter stars? By definition, solar-like oscillations are excited stochastically in the outer convection zone.

Basic Model Ingredients

From Chapters 1 and 2 we know that low- to intermediate-mass stars (1 to 8 M⊙) are found to evolve along the Asymptotic Giant Branch (AGB). These stars are surrounded by large, extended dust shells and are characterized by pronounced time variations. This is particularly true for the main constituents of the AGB – Miras and Long-Period Variables (LPVs) – the light curves of which exhibit a more or less well-defined periodicity, in this way showing a kind of an oscillating behavior. This is assumed to be caused by pulsations of the deeper layers driven by kind of a κ mechanism (see e.g., Section 1.4.4). By these internal pulsations, hydrodynamic waves are generated that travel outward into a medium with decreasing density and temperature, causing the waves to increase in amplitude and finally grow and steepen to shock waves.

These shock fronts moving outward through the atmosphere have a significant bearing on the actual local thermal and chemical state of the shell, producing either favorable or unfavourable conditions for grain nucleation and growth, respectively. By means of these processes, a complex interplay between the internal pulsation, the dynamics of the circumstellar shell, and dust formation and growth is induced, the nonlinear treatment of which allows a reliable understanding not only of the detailed shell dynamics and its particular wind characteristics but also of its detailed spectral appearance, as illustrated in Figure 16.1, where the causal interplay of the various processes that govern the local and global dynamic shell structure is sketched.

The absorption and scattering properties of dust grains strongly depend on the ratio of the particle size to the wavelength of the wave interacting with a grain. Generally it is assumed that most of the grains in circumstellar dust shells are much smaller in size than 1 μ m but that some fraction of the grains has sizes up to a few microns (cf., e.g., Jura 1996). This assumption is based on (1) the observation that circumstellar grains seem not to strongly scatter radiation in the visual and infrared wavelength regions, but that scattering becomes important in the ultraviolet (UV) region (cf. Kruszewski et al. 1968; Serkowski and Shawl 2001) and (2) the theory of scattering of electromagnetic radiation by small particles that shows particles to strongly scatter radiation only if their size is comparable with or larger than the wavelength of radiation. The typical wavelengths of the radiation emitted by stars and their circumstellar dust shells considerably exceed the size of most of the grains, and one can determine the extinction properties of the dust in the limit case of small particles. Only if one is interested for some reason in the UV part of the spectrum has one to consider the case of grains bigger than the wavelength, but even then the particles are not very big compared with the wavelengths. In the theory of circumstellar dust shells there is fortunately no need to consider really big grains, and one avoids the problems encountered by calculating their extinction properties.