Introduction

If appropriately excited, a star will oscillate like a giant spherical instrument. In most stars, including the Sun, surface convection provides the excitation mechanism (Goldreich and Keeley, 1977). With turbulent velocities reaching speeds comparable to the local sound speed near the surface of the star, the vigorous convective motions can excite standing acoustic waves. These are known as pressure or p modes because the restoring force arises from the pressure gradient. The broad frequency spectrum of this excitation mechanism gives rise to many oscillation modes, both radial and non-radial, excited simultaneously. These stochastically excited and intrinsically damped oscillations were first detected in the Sun (Leighton et al., 1962), and hence are commonly known as solar-like oscillations.

Oscillation modes can be characterized by the number of nodes, n, in the radial direction, called the radial order, and a non-radial part described by spherical harmonics, each with an angular degree, l, which equals the number of nodal lines on the surface, and an azimuthal order, m, which is the number of those nodal lines crossing the equator of the star. Except for the Sun, we can generally not resolve these oscillations on the surfaces of cool stars and so the surface displacements of modes with high angular degree (l ≳ 4) cannot be observed because regions of opposite phase tend to cancel out.

When stars grow old and the supply of hydrogen fuel is exhausted in the core, their envelopes expand and cool: they become sub giants and eventually red giants. This transition is shown in the so-called HR diagram in Figure 11.1, which indicates the main stages of evolution of a one solar mass star from the “main sequence” where the Sun is currently located (between points 1 and 2) through to the red giant phase discussed in this chapter (beyond point 3).

Introduction

Stars are changing entities in a constant evolution during their lives. At non-secular time scales – from seconds to years – the effect of dynamical processes such as convection, rotation, and magnetic fields can modify the stellar oscillations. Convection excites acoustic modes in solar-like stars, while rotation and magnetic fields can perturb the oscillation frequencies, lifting the degeneracy in the azimuthal component m of the eigenfrequencies (see Chapter 9 for the case in which rotation is slow and first-order perturbative theory can be used). Moreover, the interaction between rotation, convection, and magnetic fields can produce magnetic dynamos, which sometimes yield to regular magnetic activity cycles.

In this chapter we review how stellar dynamics can be studied and explain what long-term seismic observations can bring to the understanding of this field. Thus, we show how we can study some properties of the convective time scales operating in a star like the Sun. We also compare the stratified information we can obtain on the internal (radial) differential rotation from main-sequence solar-like stars to the Sun, and to more evolved subgiants and giants. We complement this information on the internal rotation with the determination of the surface (latitudinal differential) rotation obtained directly from the lightcurves. Indeed, when stars are active there can be spots on their surfaces dimming the light emitted. When the star rotates, the emitted light will be modulated by the presence of these spots with a period corresponding to the rotation rate at the active latitudes (where the spots develop). We finally give a brief summary of stellar magnetic studies based on spectroscopic observations and then we discuss the use of seismology to better understand the stellar magnetism of solar-like stars and the existence of possible magnetic cycles. We conclude this chapter by discussing the seismology of fast rotating stars and, from a theoretical point of view, what are the current challenges to infer properties of the internal structure and dynamics of intermediate-and high-mass stars.

Our Solar System is full of small bodies. The planets occupy an obvious role as the beautiful heavyweights, but smaller bodies are found throughout the Solar System in the form of asteroids, comet nuclei, and moons of Mars and the outer giants. Even Pluto has multiple moons. In spite of their abundance, there is not a large amount of mass in small bodies. An estimate for the mass of all the objects in the main belt puts it at around 4% the mass of the Moon (Krasinsky et al., 2002). Though many of their properties can be understood by looking at the surface and from surface samples, there are many questions that require looking inside the body.

For example, a central question of the bodies' origin and evolution is whether the asteroid or comet is an accretion of small objects into something larger, or is it a small piece of an originally larger object. An important clue to answering this question is the interior structure of the asteroid or comet nuclei. Being able to look inside would allow us to see the structure and whether the object is one solid piece, a small number of large pieces, or a lot of small objects held loosely together (the oft referred to “rubble pile”). In addition, understanding the interior structure is important if we want to move an asteroid or a comet nucleus, either for engineering reasons or to prevent one from striking Earth.

There are two ways of looking inside an object. Electromagnetic radiation or high-energy particles is one approach, which will not be discussed here save to mention that the power requirements are large and penetration depths of things such as “ground-penetrating radar” are not large. The second approach is to use mechanical waves propagating through the body, i.e., seismology.

The many successes of seismology have been described in the chapters in this book.

In this chapter we briefly summarize how angular momentum is being transported and exchanged between convective and radiative zones in stars. We discuss what physical processes influence the internal rotation history of stars on short to long (secular) time scales.

The astrophysical context

Stars are rotating magnetic bodies with complex internal and external dynamics. Observations using helioseismology (e.g., García et al., 2007), asteroseismology (e.g., Deheuvels et al., 2014), and spectropolarimetry (e.g., Donati and Land street, 2009) techniques put more and more constraints on this intricate dynamics. To get a complete and coherent picture of dynamical processes in stars and of the associated transport of angular momentum that goes beyond the “standard” modeling of stellar structure and evolution (Maeder, 2009) one needs to develop new models by introducing an improved physical description of these time-dependent processes. However, to simulate such processes in a star in full detail requires treating spatial and temporal scales spanning about 10 orders of magnitude. This is clearly not yet feasible, even with the most powerful computers available today. Therefore, one can choose to describe what occurs on a dynamical time scale (such as a convective turnover time or stellar magnetic cycles) or on the long-term evolution where the typical characteristic time scale is the dominant nuclear reactions. The same applies for spatial scales. One has to choose which relevant scale one needs to model in order to accurately describe the spatial dependence of the physical processes (convection motions, MHD instabilities, transport and mixing processes, surface dynamics).This is the reason why it is nowadays necessary to use and couple 1D, 2D, and 3D models to get a global picture of macroscopic MHD transport processes in stars over short to secular time scales.

In this chapter, we report on the state of the art of the modeling of the transport of angular momentum in stars both in convection and in radiation zones and we present our main contributions to this field of research.

Introduction

The frequencies of the global resonant modes of oscillation of the Sun depend on its structure and dynamics. Since we are able to observe and determine frequencies of many of the Sun's oscillation modes, the observed frequencies can be “inverted” to determine solar structure and dynamics. Inversions of solar oscillation frequencies have proved to be extremely successful in determining solar internal structure and dynamics as well as in testing solar models and inputs to solar models. We discuss here some of the major results that have been obtained through inversions. Recent reviews of inversions for solar structure and dynamics may be found in Christensen-Dalsgaard (2002), Basu and Antia (2008), and Howe (2009).

There are a number of sources, both ground based and space based, for helioseismic data. The Birmingham Solar Oscillation Network (BiSON; Chaplin et al., 1996) is the predominant ground-based source of data for modes of low spherical-harmonic degree ℓ (0 ≤ ℓ ≤ 3) while the Global Oscillation Network Group (GONG; Hill et al., 1996) is the source of ground-based intermediate degree modes (ℓ up to about 150). Instruments such as Global Oscillations at Low Frequencies (GOLF; Gabriel et al., 1997) and Variability of solar Irradiance and Gravity Oscillations (VIRGO; Lazrek et al., 1997) on the Solar and Heliospheric Observatory provide space-based data on low-degree modes, while MDI was until recently the source of intermediate degree data. The Helioseismic and Magnetic Imager (HMI; Scherrer et al., 2012) on board the Solar Dynamic Observatory (SDO) collects data on modes up to ℓ = 3000, though the frequencies of all visible modes cannot be determined easily.

Solar structure

As described in Chapter 9, inversions to determine solar structure proceed through the linearization of the oscillation equations around a known solar model. What is obtained from the inversions is the relative differences in the structure – in particular between the sound speed, density, or adiabatic index Γ1 profiles – of the Sun and the model.

Introduction

The theory of hydrodynamics or, where necessary, magneto hydrodynamics, provides an appropriate mathematical framework to study solar and stellar oscillations. Well-developed hydrodynamic descriptions of global oscillations of gravitationally bound gaseous spheres, aimed primarily at understanding stars such as Cepheid, delta-Scuti, and RR Lyrae variables, resulted from studies over half a century following that of pioneers such as Pekeris, Rosseland, and Chandrasekhar (see Ledoux and Walraven, 1958). The discovery of global non-radial oscillations on the solar surface (Leighton et al., 1962) led eventually, though not immediately, to a successful utilization of such theoretical machinery for helioseismology. It was also realized early on (Goldreich and Keeley, 1977) that the excitation of small amplitude non-radial oscillations on the Sun is due to a mechanism different from that responsible for the typically large amplitude oscillations exhibited by the above variable stars (known as the classical variables), even though both these types of oscillations are acoustic in nature. Observed solar oscillations are now recognized as mainly due to acoustic waves generated by turbulent convection near the surface layers of the outer convection zone, intrinsically damped and stable, hence limited to small amplitudes and linear evolution (Goldreich et al., 1994). These properties define the solar-like oscillations. There are also internal gravity waves, which, with buoyancy supplied by perturbations in gravity as the restoring force, propagate only in the sub-adiabatic (i.e., temperature gradients smaller than that required for convection) radiative interior beneath the convection zone, where they are evanescent. The surface gravity wave lives on the solar surface propagating horizontally.

Global helioseismology

Helioseismology consists of modeling the interior propagating acoustic and gravity waves, and the surface gravity wave, as small-amplitude linearly evolving perturbations to a background model of the Sun, and determining observable quantities to accuracies better than they are measured with. This enables updating the model iteratively for the best match with observations, thereby determining the internal structure of the Sun (Christensen-Dalsgaard, 2002).

The complex dynamics of highly turbulent solar magneto convection is a source of acoustic waves, and also various self-organization processes, observed on different scales on the Sun: from tiny vortex tubes to large-scale convection and active regions. Recent advances in computational capabilities have made it possible to create realistic numerical models, based on first principles, including radiative energy transfer, ionization, and magnetic fields. The simulations have provided important insight into the mechanism of acoustic emission on the Sun, and the physics of wave interactions with flows and magnetic fields. Observations of subsurface dynamics of the Sun by helioseismology are obtained by measuring acoustic travel-time anomalies, and reconstructing variations of the sound speed and behavior of convective flows in the solar interior. However, strong stratification of the turbulent convective medium, in homogeneity of flow, and complicated topology of the magnetic field create uncertainty in interpretation of helioseismology observations. Because there are no direct subsurface observations, a synergy of the magneto hydrodynamic (MHD) numerical simulations and observations provides a basis for verification and improvement of helioseismology methods.

Introduction

Understanding and characterization of solar magneto convection is a key problem of heliophysics and astrophysics. Solar turbulence driven by convective energy transport determines the dynamical state of the plasma, and leads to excitation of acoustic waves. Observed oscillations contain information about hidden dynamics of the subsurface layers, and are a powerful tool to probe the solar and stellar interior conditions.

Large parallel supercomputers, especially those developed in the last few years, allow us to perform large-scale realistic simulations of MHD processes on the Sun. These simulations have greatly improved our understanding of the multi-scale structure and dynamics of solar convection. A characteristic feature of simulations of this type is that they include all essential physics from first principles, and take into account the real-plasma equation of state, radiative transfer, chemical composition, ionization effects, and the effects of magnetic fields, but they do rely on subgrid-scale models of turbulence.

Introduction

Until recently our knowledge of stellar structure and evolution was based almost exclusively on observations of the superficial characteristics of stars, i.e., their surface temperature, gravity, and composition, in some cases combined with the surface luminosity, and in relatively rare cases observational determinations of radii and masses, with observations of eclipsing binaries providing accurate information (e.g., Andersen, 1991). This was combined with modeling, allowing us to infer from these surface properties the structure of the interior of a star, including its evolutionary state and hence age. In a few binary systems it is possible to determine the apsidal motion, which provides a measure of the internal mass distribution (e.g., Claret and Giménez, 2010). Also, in the unique case, amongst non-exploding stars, of the Sun, observations of neutrinos provide information about the nuclear reactions in the stellar interior (for a recent review, see Turck-Chièze and Couvidat, 2011), although these were for a long time at variance with the observations.

Thus, observations only provide somewhat indirect information on which to base a test of our modeling and understanding of stellar evolution. This is a serious concern. Stars are evidently very complex systems, controlled by atomic processes and interactions in the thermodynamic properties of stellar matter, interactions between matter and radiation, and nuclear energy generation that are far from understood or yet reproduced in a controlled manner in the laboratory. Also, they have hydrodynamic instabilities, including those potentially associated with rotation, which are at best treated in a rudimentary fashion in the modeling, but which probably have substantial effects on the evolution of stars. Magnetic fields, which are present on large scales in some stars and which are evident in solar and stellar activity, add another layer of complexity in the physics of stars that is rarely even considered in the modeling.

Stars and planetary bodies have a special place in our study of the universe. Identifying new planets and stars has a long history in astronomy, and studying their complex surface phenomena continues to be an important scientific endeavor in planetary science and astrophysics. With the advent of modern observational techniques and approaches to data analysis, it is now possible to probe their internal structure at increasingly high resolution and to monitor subtle changes in the deep interiors.

Indeed, having more accurate and detailed information about the stellar and planetary interiors helps us understand the evolution of stars and planetary bodies. Such information is invaluable for studying the links between subsurface structures and surface processes, too. Seismology is therefore central to our quests for understanding their interiors and surfaces. It provides a set of powerful and versatile tools capable of revealing the hidden phenomena inside the Earth, the Moon, and the Sun, amongst other stars, asteroids, planets, and their satellites.

Why extraterrestrial seismology?

In seismic studies, it is critically important to be able to relate any measurable or scientifically significant effects associated with seismic waves to the spatially varying physical properties in the subsurface. Research into the modeling of seismic sources and wave propagation in different stellar and planetary media has allowed scientists to build this foundation for imaging the interiors of stars and planetary bodies. Advances in the acquisition, processing, and modeling of seismic data as a whole have drastically improved the quality of the seismic images and information we have about the subsurface.

Encompassing both the theoretical studies of seismic phenomena and technical applications of seismic imaging developed over the past few decades, seismology is both a pure and an applied science. The pure and applied nature of seismology underpins many major breakthroughs across modern solid-earth planetary, and astrophysical sciences: the discovery of the Earth's and lunar cores, the presence of complex time-dependent differential rotation patterns of the solar interior, and the characterization of the inner radiative cores of red giants, to name but a few.

Introduction

Of the many geophysical means that can be used to probe a planet's interior, seismology remains the most direct. Given that the seismic data gathered on the Moon over 40 years ago revolutionized our understanding of the Moon and are still being used today to produce new insight into the state of the lunar interior, it is no wonder that many future missions, both real and conceptual, plan to take seismometers to other planets.

To best facilitate the return of high-quality data from these instruments, as well as to further our understanding of the dynamic processes that modify a planet's interior, various modeling approaches are used to quantify parameters such as the amount and distribution of seismicity, tidal deformation, and seismic structure on and of the terrestrial planets. In addition, recent advances in wavefield modeling have permitted a renewed look at seismic energy transmission and the effects of attenuation and scattering, as well as the presence and effect of a core, on recorded seismograms. In this chapter, we will review these approaches.

Site selection for future planetary seismology missions

The ability of a seismic network to accurately locate an event improves as the number of seismometers increases. On Earth we take for granted that any given event will be relatively well located, due to the comparative ease of installation of seismometers. For planetary applications, we cannot count on a large distribution of stations. Various factors including cost, difficulty of installation, instrumentation longevity, and data transmission severely limit the number of instruments that have been or will be deployed on other planetary bodies. In this section, we will review various methods that can be employed to help determine the best landing sites for future planetary seismology missions, in order to maximize their scientific return. We focus here on the Moon and Mars, although many of these methods are adaptable to other planetary bodies.

Introduction

One of the greatest remaining mysteries surrounding the Sun is the mechanism of its magnetic activity cycle. Most prominently, the number of sunspots repeatedly increases and decreases with an approximate period of 11 years. This must be a result of interactions between various flows and magnetic fields in the Sun, broadly called a dynamo mechanism. Pinning down the mechanism of the solar dynamo would require concerted efforts between theory and observation. Therefore it is of paramount importance, as an observational approach in our attempt to understand the solar dynamo, that we measure these internal flows and, if possible, magnetic fields in the Sun.

Among the various flows, of particular interests are the two components of large-scale flows, i.e., differential rotation and meridional circulation. Convective flows, in particular supergranules, the flows associated with super granulation, are also of great interest. The origin of super granules is still debated, but they are normally considered as convective cells, typically with a spatial scale of 3 × 104 km and a lifetime (∼ the turnover time) of about a day.

The Sun rotates with a period of about a month, but the equator rotates faster than the poles – the Sun rotates differentially. The solar differential rotation is directly responsible for the so-called ω effect, in which the toroidal component (the ϕ component in spherical coordinates) of the magnetic field is produced by shearing the poloidal component (the component in the r–θ plane). It also provides the Coriolis force, which is important in the less understood α effect, in which the poloidal component is produced from twisting the toroidal magnetic field. Meridional flows transport magnetic flux latitudinally by advection. Supergranulation is believed to be important in diffusive transport of the solar magnetic fields. For general reviews of large-scale flows in the Sun and the solar dynamo, readers are referred to Miesch (2005) and Charbonneau (2010), respectively.

As explained Chapters 4 and 5, even basic asteroseismic diagnostics, such as the frequency of maximum power combined with the large frequency separation for solar-like pulsators (e.g., Chaplin et al., 2011a), or the period spacing for high-order gravity-mode pulsators (e.g., Degroote et al., 2010), already lead to values of the global stellar parameters, such as the mass and radius, with a precision far better than that deduced from photometric color indices, spectral lines, or interferometric data. The combination of frequency and period spacings of the dipole mixed modes detected in evolved low-mass stars, allows us to distinguish between red giants with only hydrogen-shell burning or with core-helium burning in addition (Bedding et al., 2011), a property that classical data cannot reveal.

The real benefit of asteroseismic versus classical data becomes even more obvious when modeling the individual oscillation frequencies νnℓm, since they are dependent on, and thus determined by, the details of the interior physics of the stars. Such seismic modeling of the solar oscillation frequencies led to a large step forward in the understanding of the structure of the Sun (Chapter 12), including its interior rotation profile (e.g., Kornilov et al., 2012; Thompson et al., 2003). Likewise, seismic modeling of the oscillation frequencies of other stars offers a unique opportunity to probe the internal physical processes of stars of various birth mass, initial chemical composition, and evolutionary stage. As an example, we shown in Figure 8.1 the observed frequency spectra from three months of Kepler photometry of the binary components and solar analogues 16 Cyg A and B. Finding the correspondence between the observed frequency peaks and those predicted by theory is not only an immensely powerful method to determine the basic fundamental properties of stars, such as their mass, radius, and age, but it also allows us to evaluate standard stellar structure models and improve their input physics.