# Global seismology of the Sun

## Abstract

The seismic study of the Sun and other stars offers a unique window into the interior of these stars. Thanks to helioseismology, we know the structure of the Sun to admirable precision. In fact, our knowledge is good enough to use the Sun as a laboratory. We have also been able to study the dynamics of the Sun in great detail. Helioseismic data also allow us to probe the changes that take place in the Sun as solar activity waxes and wanes. The seismic study of stars other than the Sun is a fairly new endeavour, but we are making great strides in this field. In this review I discuss some of the techniques used in helioseismic analyses and the results obtained using those techniques. I focus on results obtained with global helioseismology, i.e., the study of the Sun using its normal modes of oscillation. I also briefly touch upon asteroseismology, the seismic study of stars other than the Sun, and discuss how seismic data of others stars are interpreted.

### Keywords

Sun: interiorSun: helioseismologySun: rotationsStars: asteroseismology## 1 Introduction

The Sun and other stars oscillate in their normal modes. The analysis and interpretation of the properties of these modes in terms of the underlying structure and dynamics of the stars is referred to as global seismology. Global seismology has given us an unprecedented window into the structure and dynamics of the Sun and stars. Arthur Eddington began his book *The Internal Constitution of the Stars* lamenting the fact that the deep interior of the Sun and stars is more inaccessible than the depths of space since we so not have an “appliance” that can “\(\ldots \)*pierce through the outer layers of a star and test the conditions within”*. Eddington went on to say that perhaps the only way of probing the interiors of the Sun and stars is to use our knowledge of basic physics to determine what the structure of a star should be. While this is still the dominant approach in the field of stellar astrophysics, in global seismology we have the means of piercing the outer layers of a star to probe the structure within.

The type of stellar oscillations that are used in helio- and asteroseismic analyses have very low amplitudes. These oscillations are excited by the convective motions in the outer convection zones of stars. Such oscillations, usually referred to as solar-like oscillations, can for most purposes be described using the theory of linear, adiabatic, oscillations. The behaviour of the modes on the stellar surface is described in terms of spherical harmonics since these functions are a natural description of the normal modes of a sphere. The oscillations are labelled by three numbers, the radial order *n*, the degree \(\ell \) and the azimuthal order *m*. The radial order *n* can be any whole number and is the number of nodes in the radial direction. Positive values of *n* are used to denote acoustic modes, i.e., the so-called p modes (p for pressure, since the dominant restoring force for these modes is provided by the pressure gradient). Negative values of *n* are used to denote modes for which buoyancy provides the main restoring force. These are usually referred to as g modes (g for gravity). Modes with \(n=0\) are the so-called fundamental or f modes. These are essentially surface gravity modes whose frequencies depend predominantly on the wave number and the surface gravity. The degree \(\ell \) denotes the number of nodal planes that intersect the surface of a star and *m* is the number of nodal planes perpendicular to equator.

*n*have the same frequency. Asphericities such as rotation and magnetic fields lift this degeneracy and cause ‘frequency splitting’ making the frequencies

*m*-dependent. It is usual to express the frequency \(\nu _{n\ell m}\) of a mode in terms of ‘splitting coefficients’:

*a*’ coefficients and \(\mathcal {P}\) are suitable polynomials. For slow rotation the central frequency \(\nu _{n\ell }\) depends only on structure, the odd-order

*a*coefficients depend on rotation, and the even-order

*a*coefficients depend on structural asphericities, magnetic fields, and second order effects of rotation.

The focus of this review is global helioseismology—its theoretical underpinnings as well as what it has taught us about the Sun. The last section is devoted to asteroseismology; the theoretical background of asteroseismology is the same as that of helioseismology, but the diverse nature of different stars makes the field quite distinct. This is, of course, not the first review of helioseismology. To get an idea of the changing nature of the field, the reader is referred to earlier reviews by Thompson (1998b), Christensen-Dalsgaard (2002, 2004) and Gough (2013b), which along with descriptions of the then state-of-art of the field, also review the early history of helioseismology. Since this review is limited to *global* seismology only, readers are referred to the *Living Reviews in Solar Physics* contribution of Gizon and Birch (2005) for a review of *local* helioseismology.

Solar oscillations were first discovered by Leighton et al. (1962) and confirmed by Evans and Michard (1962). Later observations, such that those by Frazier (1968) indicated that the oscillations may not be mere surface phenomena. Subsequently Ulrich (1970) and Leibacher and Stein (1971) proposed that the observations could be interpreted as global oscillation modes and predicted that oscillations would form ridges in a wave-number v/s frequency diagram. The observations of Deubner (1975) indeed showed such ridges. Rhodes et al. (1977) reported similar observations. Neither the observations of Deubner (1975), nor those of Rhodes et al. (1977), could resolve individual modes of solar oscillations. Those had to wait for Claverie et al. (1979) who, using Doppler-velocity observations integrated over the solar disk, were able to resolve the individual modes of oscillations corresponding to the largest horizontal wavelengths, i.e., the truly global modes. They found a series of almost equidistant peaks in the power spectrum, just as was expected from theoretical models.

Helioseismology, the study of the Sun using solar oscillations, as we know it today began when Duvall and Harvey (1983) determined frequencies of a reasonably large number of solar oscillation modes covering a wide range of horizontal wavelengths. Since then, many more sets of solar oscillation frequencies have been published. A large fraction of the early work in the field was based on the frequencies determined by Libbrecht et al. (1990) from observations made at the Big Bear Solar Observatory (BBSO).

Accurate and precise estimates of solar frequencies require long, uninterrupted observations of the Sun, that are possible only with a network of telescopes. The Birmingham Solar Oscillation Network (BiSON; Elsworth et al. 1991; Chaplin et al. 2007) and the International Research on the Interior of the Sun (IRIS; Fossat 1991) were two of the first networks. Both these networks however, did Sun-as-a-star (i.e., one pixel) observations, as a result they could only observe the low-degree modes with \(\ell \) of 0–3. To overcome this limitation, resolved-disc measurements were needed. This resulted in the construction of the Global Oscillation Network Group (GONG: Hill et al. 1996). This ground-based network has been making full-disc observations of the Sun since 1995. There were concurrent developments in space based observations and the Solar and Heliospheric Observatory (*SoHO*: Domingo et al. 1995) was launched in 1995. Among *SoHO*’s observing programme were three helioseismology related ones, the ‘Solar Oscillations Investigation’ (SOI) using the Michelson Doppler Imager(MDI: Scherrer et al. 1995), ‘Variability of solar Irradiance and Gravity Oscillations’ (VIRGO: Lazrek et al. 1997) and ‘Global Oscillations at Low Frequencies’ (GOLF: Gabriel et al. 1997). Of these, MDI was capable of observing intermediate and high degree modes. BiSON and GONG continue to observe the Sun from the ground; MDI stopped acquiring data in April 2011. MDI has been succeeded by the Heliospheric and Magnetic Imager (HMI: Schou et al. 2012; Scherrer et al. 2012) on board the Solar Dynamics Observatory (SDO: Pesnell et al. 2012). In addition to obtaining better data, there have also been improvements in techniques to determine mode frequencies from the data (e.g., Larson and Schou 2008; Korzennik et al. 2013a) which has facilitated detailed helioseismic analyses. Descriptions of some of the early developments in the field can be found in the proceedings of the workshop “Fifty Years of Seismology of the Sun and Stars” (Jain et al. 2013).

Asteroseismology, the seismic study of stars other than the Sun, took longer to develop because of inherent difficulties in ground-based observations. Early attempts were focused on trying to observe pulsations of \(\alpha \) Cen. Some early ground-based attempts did not find any convincing evidence for solar-like pulsations (e.g., Brown and Gilliland 1990), though others could place limits (e.g., Pottasch et al. 1992; Edmonds and Cram 1995). It took many more attempts before pulsations of \(\alpha \) Cen were observed and the frequencies measured (Bouchy and Carrier 2001; Bedding et al. 2004; Kjeldsen et al. 2005). Other stars were targeted too (e.g., Kjeldsen et al. 2003; Carrier et al. 2005a, b; Bedding et al. 2007; Arentoft et al. 2008). The field did not grow till space-based missions were available. The story of space-based asteroseismology started with the ill-fated Wide-Field Infrared Explorer (WIRE). The satellite failed because coolants meant to keep the detector cool evaporated, but Buzasi (2000) realised that the star tracker could be used to monitor stellar variability and hence to look for stellar oscillations. This followed the observations of \(\alpha \) UMa and \(\alpha \) Cen A (Buzasi et al. 2000; Schou and Buzasi 2001; Fletcher et al. 2006). The Canadian mission Microvariability and Oscillations of Stars (MOST: Walker et al. 2003) was the first successfully launched mission dedicated to asteroseismic studies. Although it was not very successful in studying solar type stars, it was immensely successful in studying giants, classical pulsators, and even star spots and exo-planets. The next major step was the ESA/French CoRoT mission (Baglin et al. 2006; Auvergne et al. 2009). CoRot observed many giants and showed giants also show non-radial pulsations (De Ridder et al. 2009); the mission observed some subgiants and main sequence stars as well (see e.g., Deheuvels et al. 2010; Ballot et al. 2011; Deheuvels and the CoRoT Team 2014). While CoRoT and MOST showed that the seismic study of other stars was feasible, the field began to flourish after the launch of the *Kepler* mission (Koch et al. 2010) and the demonstration that *Kepler* could indeed observe stellar oscillations (Gilliland et al. 2010) and that the data could be used to derive stellar properties (Chaplin et al. 2010). Asteroseismology is going through a phase in which we are still learning the best ways to analyse and interpret the data, but with two more asteroseismology missions being planned—the Transiting Exoplanet Survey Satellite (TESS; launch 2017), and PLATO (launch 2025)—this field is going to grow even more rapidly. We discuss the basics of asteroseismology in Sect. 11 of this review, however, only a dedicated review can do proper justice to the field.

This review is organised as follows: Since it is not possible to perform helioseismic (or for that matter asteroseismic) analyses without models, we start with the construction of solar and stellar models in Sect. 2. We then proceed to derive the equations of stellar oscillation and describe some the properties of the oscillations in Sect. 3. A brief history of solar models is given in Sect. 4. We show what happens when we try to compare solar models with the Sun by comparing frequencies in Sect. 5. The difficulty in making such comparisons leads us to Sect. 6 where we show how solar oscillation frequencies may be inverted to infer properties of the solar interior. The next three sections are devoted to results. In Sect. 7 we describe what we have learnt about spherically symmetric part of solar structure. Deviations from spherical symmetry—in terms of dynamics, magnetic fields and structural asymmetries—are described in Sect. 8. The solar-cycle related changes in solar frequencies and the deduced changes in the solar interior are discussed in Sect. 9. Most of the results discussed in this review were obtained by analysing the frequencies of solar oscillation. There are other observables though, such as line-width and amplitude, and these carry information on how modes are excited (and damped). The issue of mode-excitation is discussed briefly in Sect. 10. Finally, in Sect. 11, we give a brief introduction to the field of asteroseismology.

## 2 Modelling stars

In order to put results obtained from seismic analyses in a proper context, we first give a short overview of the process of constructing models and of the inputs used to construct them. Traditionally in astronomy, we make inferences by comparing properties of models with data, which in the seismic context means comparing the computed frequencies with the observed ones. Thus, seismic investigations of the Sun and other stars start with the construction models and the calculation of their oscillation frequencies. In this section we briefly cover the field of modelling. We also discuss how solar models are constructed in a manner that is different from the construction of models of other stars. There are many excellent textbooks that describe stellar structure and evolution and hence, we only describe the basic equations and inputs. Readers are referred to books such as Kippenhahn et al. (2012), Huang and Yu (1998), Hansen et al. (2004), Weiss et al. (2004), Maeder (2009), etc. for details.

### 2.1 The equations

The most common assumption involved in making solar and stellar models is that stars are spherically symmetric, i.e., stellar properties are only a function of radius. This is a good approximation for non-rotating and slowly-rotating stars. The other assumption that is usually made is that a star does not change its mass as it evolves; this assumption is valid except for the very early stages of star formation and very evolved stages, such at the tip of the red giant branch. The Sun for instance, loses about \(10^{-14}\) of its mass per year. Thus, in its expected main-sequence lifetime of about 10 Gyr, the Sun will lose only about 0.01 % of its mass. The radius of a star on the other hand, is expected to change significantly. As a result, mass is used as the independent variable when casting the equations governing stellar structure and evolution.

*m*is the mass enclosed in radius

*r*and \(\rho \) the density.

*l*flows through a shell of radius

*r*per unit time as a result of nuclear reactions in the interior. If \(\epsilon \) be the energy released per unit mass per second by nuclear reactions, and \(\epsilon _\nu \) the energy lost by the star because of neutrinos streaming out without depositing their energy, then,

*t*is time, and \(\delta \) is given by the equation of state and defined as

*g*for gravity) because they denote the release of gravitational energy.

The last set of equations deals with chemical composition as a function of position and time. There are three ways that the chemical composition at any point of a star can change: (1) nuclear reactions, (2) the changes in the boundaries of convection zones, and (3) diffusion and gravitational settling (usually simply referred to as diffusion) of helium and heavy elements and other mixing processes as well.

*i*, \(r_{ji}\) is the rate at which isotope

*i*is formed from isotope

*j*, and \(r_{ik}\) is the rate at which isotope

*i*is lost because it turns into a different isotope

*k*. The rates \(r_{ik}\) are external inputs to models.

*i*in the convection zone is:

*i*at \(m_1\) and \(m_2\) respectively.

*D*is the diffusion coefficient, and \(\nabla ^2\) is the Laplacian operator. The diffusion coefficient hides the complexity of the process and includes, in addition to gravitational settling, diffusion due to composition and temperature gradients. All three processes are generally simply called ‘diffusion’. Other mixing process, such as those induced by rotation, are often included the same way by modifying

*D*(e.g., Richard et al. 1996).

*D*depends on the isotope under consideration, however, it is not uncommon for stellar-evolution codes to treat helium separately, and use a single value for all heavier elements.

Equations (2), (3), (5), and (7) together with the equations relating to change in abundances, form the full set of equations that govern stellar structure and evolution. In most codes, Eqs. (2), (3), (5), and (7) are solved for a given \(X_i\) at a given time *t*. Time is then advanced, Eqs. (8), (10), and (11) are solved to give new \(X_i\), and Eqs. (2), (3), (5), and (7) are solved again. Thus, we have two independent variables, mass *m* and time *t*, and we look for solutions in the interval \(0\le m \le M\) (stellar structure) and \( t \ge t_0\) (stellar evolution).

Four boundary conditions are required to solve the stellar structure equations. Two (on radius and luminosity) can be applied quite trivially at the centre. The remaining conditions (on temperature and pressure) need to be applied at the surface. The boundary conditions at the surface are much more complex than the central boundary conditions and are usually determined with the aid of simple stellar-atmosphere models. As we shall see later, atmospheric models plays a large role in determining the frequencies of different modes of oscillation.

The initial conditions needed to start evolving a star depend on where we start the evolution. If the evolution begins at the pre-main sequence phase, i.e., while the star is still collapsing, the initial structure is quite simple. Temperatures are low enough to make the star fully convective and hence chemically homogeneous. If evolution is begun at the Zero Age Main Sequence (ZAMS), which is the point at which hydrogen fusion begins, a ZAMS model must be used.

The situation in the convection zones is tricky. Convection is, by its very nature, a three-dimensional phenomenon, yet our models are one dimensional. Even if we construct three-dimensional models, it is currently impossible include convection and evolve the models at the same time. This is because convection takes place over time scales of minutes to hours, while stars evolve in millions to billions of years. As a result, drastic simplifications are used to model convection. Deep inside a star, the temperature gradient is well approximated by the adiabatic temperature gradient \(\nabla _\mathrm{ad}\equiv (\partial \ln T/\partial \ln P)_s\) (*s* being the specific entropy), which is determined by the equation of state. This approximation cannot be used in the outer layers where convection is not efficient and some of the energy is also carried by radiation. In these layers one has to use an approximate formalism since there is no “theory” of stellar convection as such. One of the most common formulations used to calculate convective flux in stellar models is the so-called “mixing length theory” (MLT). The mixing length theory was first proposed by Prandtl (1925). His model of convection was analogous to heat transfer by particles; the transporting particles are macroscopic eddies and their mean free path is the “mixing length”. This was applied to stars by Biermann (1948), Vitense (1953), and Böhm-Vitense (1958). Different mixing length formalisms have slightly different assumptions about what the mixing length is. The main assumption in the usual mixing length formalism is that convective eddies move an average distance equal to the mixing length \(l_m\) before giving up their energy and losing their identity. The mixing length is usually defined as \(l_m=\alpha H_P,\) where \(\alpha \), a constant, is the so-called ‘mixing length parameter’, and \(H_P\equiv -{\,\mathrm {d}}r/{\,\mathrm {d}}\ln P\) is the pressure scale height. Details of how \(\nabla \) is calculated under these assumption can be found in any stellar structure textbook, such as Kippenhahn et al. (2012). There is no a priori way to determine \(\alpha \), and it is one of the free parameters in stellar models. Variants of MLT are also used, and among these are the formulation of Canuto and Mazzitelli (1991) and Arnett et al. (2010).

### 2.2 Inputs to stellar models

The equations of stellar structure look quite simple. Most of the complexity is hidden in the external inputs needed to solve the equations. There are four important inputs that are needed: the equation of state, radiative opacities, nuclear reaction rates and coefficients to derive the rates of diffusion and gravitational settling. These are often referred to as the *microphysics* of stars.

#### 2.2.1 The equation of state

The equation of state specifies the relationship between density, pressure, temperature and composition. The stellar structure equations are a set of five equations in six unknowns, *r*, *P*, *l*, *T*, \(X_i\), and \(\rho \). None of the equations directly solves for the behaviour of \(\rho \) as a function of mass and time. We determine the density using the equation of state.

The ideal gas equation is good enough to make simple models. However, the ideal gas law does not apply to all layers of stars since it does not include the effects of ionisation, radiation pressure, pressure ionisation, degeneracy, etc. Among the early published equations of state valid under stellar condition is that of Eggleton, Faulkner and Flannery (EFF; Eggleton et al. 1973). This equation of state suffered from the fact that it did not include corrections to pressure due to Coulomb interactions, and this led to the development of the so-called “Coulomb Corrected” EFF, or CEFF equation of state (Christensen-Dalsgaard and Däppen 1992; Guenther et al. 1992). However, the CEFF equation of state is not fully thermodynamically consistent.

Modern equations of state are usually given in a tabular form with important thermodynamic quantities such as \(\nabla _\mathrm{ad}\), \(C_p\) listed as a function of *T*, *P* (or \(\rho \)) and composition. These include the OPAL equation of state (Rogers et al. 1996; Rogers and Nayfonov 2002) and the so-called MHD (i.e., Mihalas, Hummer and Däppen) equation of state (Däppen et al. 1988; Mihalas et al. 1988; Hummer and Mihalas 1988; Gong et al. 2001). Both OPAL and MHD equations of state suffer from the limitation that unlike the EFF and CEFF equations of state, the heavy-element mixture used to calculate them cannot be changed by the user. This has lead to the development of thermodynamically consistent extensions of the EFF equation of state that allow users to change the equation of state easily. The SIREFF equation of state is an example of this (Guzik and Swenson 1997; Guzik et al. 2005).

#### 2.2.2 Opacities

We need to know the opacity \(\kappa \) of the stellar material in order to calculate \(\nabla _\mathrm{rad}\) [Eq. (12)]. Opacity is a measure of how opaque a material is to photons. Like modern equations of state, opacities are usually available in tabular form as a function of density, temperature and composition. Among the widely used opacity tables are the OPAL (Iglesias and Rogers 1996) and OP (Badnell et al. 2005; Mendoza et al. 2007) tables. The OPAL opacity tables include contributions from 19 heavy elements whose relative abundances (by numbers) with respected to hydrogen are larger than about \(10^{-7}\). The OP opacity calculations include 15 elements. Neither table is very good at low temperature where molecules become important. As a results these tables are usually supplemented by specialised low-temperature opacity tables such that those of Kurucz (1991) and Ferguson et al. (2005).

#### 2.2.3 Nuclear reaction rates

#### 2.2.4 Diffusion coefficients

#### 2.2.5 Atmospheres

While not a microphysics input, stellar atmospheric models are equally crucial: stellar models do not stop at \(r=R\), but generally extend into an atmosphere, and hence the need for these models. These models are also used to calculate the outer boundary condition. The atmospheric models are often quite simple and provide a *T*–\(\tau \) relation, i.e., a relation between temperature *T* and optical depth \(\tau \). The Eddington *T*–\(\tau \) relation is quite popular. Semi-empirical relations such as the Krishna Swamy *T*–\(\tau \) relation (Krishna Swamy 1966), the Vernazza, Avrett and Loeser (VAL) relation (Vernazza et al. 1981) though applicable to the Sun are also used frequently to model other stars. A relatively recent development in the field is the use of *T*–\(\tau \) relations obtained from simulations of convection in the outer layers of stars.

### 2.3 The concept of “standard” solar models

The mass of a star is the most fundamental quantity needed to model a star. Other input quantities include the initial heavy-element abundance \(Z_0\) and the initial helium abundance \(Y_0\). These quantities affect both the structure and the evolution of star through their influence on the equation of state and opacities. Also required is the mixing-length parameter \(\alpha \). Once these quantities are known, or chosen, models are evolved in time, until they reach the observed temperature and luminosity. The initial guess of the mass may not result in a models with the required characteristics, and a different mass needs to be chosen and the process repeated. Once a model that satisfies observational constraints is constructed, the model becomes a proxy for the star; the age of the star is assumed to be the age of the model, and the radius of the star is assumed to be the radius of the model. The most important source of uncertainty in stellar models is our inability to model convection properly. MLT requires a free parameter \(\alpha \) that is essentially unconstrained and introduces uncertainties in the radius of a star of a given mass and heavy-element abundance. And even if we were able to constrain \(\alpha \), it would not account for all the properties of convective heat transport and thus would therefore, introduce errors in the results.

Global parameters of the Sun

Quantity | Estimate | Reference |
---|---|---|

Mass (\(M_\odot \))\(^\mathrm{a}\) | \(1.98892\,(1\pm 0.00013)\times 10^{33}\,\hbox {g}\) | Cohen and Taylor (1987) |

Radius (\(R_\odot \))\(^\mathrm{b}\) | \(6.9599\,(1\pm 0.0001)\times 10^{10}\,\hbox {cm}\) | Allen (1973) |

Luminosity (\(L_\odot \)) | \(3.8418\,(1\pm 0.004)\times 10^{33}\,\hbox {ergs s}^{-1}\) | Fröhlich and Lean (1998) |

Bahcall et al. (1995) | ||

Age | \(4.57\,(1\pm 0.0044) \times 10^{9}\,\hbox {years}\) | Bahcall et al. (1995) |

Nominal solar conversion constants as per IAU resolution B3

Parameter | value |
---|---|

\(1{\mathcal {R}}^\mathrm{N}_\odot \) | \(6.957\times 10^8\,\hbox {m}\) |

\(1{\mathcal {S}}^\mathrm{N}_\odot \) | \(1361\,\hbox {W m}^{-2}\) |

\(1{\mathcal {L}}^\mathrm{N}_\odot \) | \(3.828\times 10^{26}\,\hbox {W}\) |

\(1{\mathcal {T}}^\mathrm{N}_{\mathrm{eff}\odot }\) | \(5772\,\hbox {K}\) |

\(1{\mathcal {GM}}^\mathrm{N}_\odot \) | \(1.327\;124\;4\times 10^{20}\,\hbox {m}^3\,\hbox {s}^{-2}\) |

To be called a solar model, a \(1M_{\odot }\) model must have a luminosity of \(1L_{\odot }\) and a radius of \(1R_{\odot }\) at the solar age of 4.57 Gyr. The way this is done is by recognising that there are two constraints that we need to satisfy at 4.57 Gyr, the solar radius and luminosity, and that the set of equations has two free parameters, the initial helium abundance \(Y_0\) and the mixing length parameter \(\alpha \). Thus, we basically have a situation of two constraints and two unknowns. However, since the equations are non-linear, we need an iterative method to determine \(\alpha \) and \(Y_0\). The value of \(\alpha \) obtained in this manner for the Sun is often called the “solar calibrated” value of \(\alpha \) and used to model other stars. In addition to \(\alpha \) and \(Y_0\), and very often initial *Z* is adjusted to get the observed *Z* / *X* in the solar envelope. The solar model constructed in this manner does not have any free parameters, since these are determined to match solar constraints. Such a model is known as a “standard” solar model (SSM).

The concept of standard solar models is very important in solar physics. Standard solar models are those where only standard input physics such as equations of state, opacity, nuclear reaction rates, diffusion coefficients etc., are used. The parameters \(\alpha \) and \(Y_0\) (and sometime \(Z_0\)) are adjusted to match the current solar radius and luminosity (and surface *Z* / *X*). No other input is adjusted to get a better agreement with the Sun. By comparing standard solar models constructed with different input physics with the Sun we can put constraints on the input physics. One can use helioseismology to test whether or not the structure of the model agrees with that of the Sun.

There are many physical processes that are not included in SSMs. These are processes that do not have a generally recognised way of modelling and rely on free parameters. These additional free parameters would make any solar model that includes these processes “non standard”. Among the missing processes are effects of rotation on structure and of mixing induced by rotation. There are other proposed mechanisms for mixing in the radiative layers of the Sun, such as mixing caused by waves generated at the convection-zone base (e.g., Kumar et al. 1999) that are not included either. These missing processes can affect the structure of a model. For example, Turck-Chièze et al. (2004) found that mixing below the convection-zone base can change both the position of the convection-zone base (gets shallower) and the helium abundance (abundance increases). However, their inclusion in stellar models require us to choose values for the parameters in the formulaæ that describe the processes. Accretion and mass-loss at some stage of solar evolution can also affect the solar models (Castro et al. 2007), but again, there are no standard formulations for modelling these effects.

Standard solar models constructed by different groups are not identical. They depend on the microphysics used, as well as assumption about the heavy element abundance. Numerical schemes also play a role. For a discussion of the sources of uncertainties in solar models, readers are referred to Section 2.4 of Basu and Antia (2008).

## 3 The equations governing stellar oscillations

To a good approximation, solar oscillations and solar-like oscillations in other stars, can be described as linear and adiabatic. In the case of the Sun, modes of oscillation have amplitudes of the order of \(10\,\hbox {cm s}^{-1}\) while the sound speed at the surface is more like \(10\,\hbox {km s}^{-1}\), putting the amplitudes squarely in the linear regime. The condition of adiabaticity can be justified over most of the Sun—the oscillation frequencies have periods of order 5 min, but the thermal time-scale is much larger and hence, adiabaticity applies. This condition however breaks down in the near-surface layers (where thermal time-scales are short) resulting in an error in the frequencies. This error adds to what is later described as the “surface term” (see Sect. 5) but which can be filtered out in many cases. We shall retain the approximations of linearity and adiabaticity in this section since most helioseismic results have been obtained under these assumptions after applying a few corrections.

Details of the derivation of the oscillation equations and a description of their properties has been described well by authors such as Cox (1980), Unno et al. (1989), Christensen-Dalsgaard and Berthomieu (1991), Gough (1993), etc. Here we give a short overview that allows us to derive some of the properties of solar and stellar oscillations that allow us to undertake seismic studies of the Sun and other stars.

*G*the gravitational constant. The heat equation is written in the form

In the subsequent discussion, we drop the subscript ‘0’ for the equilibrium quantities, and only keep the subscript for the perturbations.

### 3.1 The spherically symmetric case

*r*being the radial distance, \(\theta \) the co-latitude, and \(\phi \) the longitude. The different quantities can be decomposed into their radial and tangential components. Thus, the displacement \(\mathbf {\xi }\) can be decomposed as

*r*direction. Finally, the Poisson’s equation becomes

*t*does not appear explicitly in coefficients of any of the derivatives. This implies that the time dependent part can be separated out from the spatial part, and the time-dependence can be expressed in terms of \(\exp (-i\omega t)\) where \(\omega \) can be real (giving an oscillatory solution in time) or imaginary (a solution that grows or decays). Thus, we may write:

*Lamb frequency*defined by

*N*is

*Brunt–Väisälä*or

*buoyancy frequency*defined as

*n*or

*m*dependence in the Eqs. (35)–(41). The different eigenvalues for a given value of \(\ell \) are given the label

*n*. Conventionally

*n*can be any signed integer and can be positive, zero or negative depending on the type of the mode. In general |

*n*| represents the number of nodes the radial eigenfunction has in the radial direction. As mentioned in the Introduction, values of \(n > 0\) are used to specify acoustic modes, \(n < 0\) label gravity models and \(n=0\) labels f modes. Of course, this labelling gets more complicated when mixed modes (see Sect. 11) are present, but this works well for the Sun. It is usual to denote the eigenfunctions with

*n*and \(\ell \), thus, for example, the total displacement is denoted as \(\mathbf {\xi }_{n,\ell }\) and the radial and transverse components as \(\xi _{r,n,\ell }\) and \(\xi _{t,n,\ell }\) respectively. The lack of the

*m*dependence in the equations has to do with the fact that we are considering a spherically symmetric system, and to define

*m*one needs to break the symmetry. Thus, in a spherically symmetric system, all modes are of a given value of

*n*and \(\ell \) are degenerate in

*m*. Rotation, magnetic fields and other asphericities lift this degeneracy.

*M*is the total mass and

*R*is the total radius.

### 3.2 Boundary conditions

The discussion of the equations of stellar evolution cannot be completed without discussing the boundary conditions that are need to actually solve the equations. For details of the boundary conditions and how they can be obtained, the reader is referred to Cox (1980) and Unno et al. (1989). We give a brief overview.

A complete solution of the equations requires four boundary conditions. An inspection of the equations shows that they have a singular point at \(r=0\), however, it is a singularity that allows both regular and singular solutions. Given that we are talking of physical systems, we need to look for solutions that are regular at \(r=0\). As is usual, the central boundary conditions are obtained by expanding the solution around \(r=0\). This reveals that as \(r\rightarrow 0\), \(\xi _r \propto r\) for \(\ell =0\) modes, and \(\xi _r \propto r^{\ell -1}\) for others. The quantities \(P_1\) and \(\varPhi _1\) vary as \(r^\ell \).

*A*is a constant. It follow that

*R*is the radius of the star and

*M*the mass. Similarly the dimensionless radius \({\hat{r}}=r/R\), dimensionless pressure is given by \({\widehat{P}}=(R^4/GM^2)P\) and the dimensionless density by \({\hat{\rho }}=(R^3/M)\rho \). These relations lead us to the dimensionless expressions for the perturbed quantities. It can be shown very easily that \({\hat{\xi }}_r\) the dimensionless displacement eigenfunction, \({\widehat{P}}_1\) the dimensionless pressure perturbation, and \({\widehat{\varPhi }}_1\) the dimensionless perturbation to the gravitational potential can be written as

### 3.3 Properties of stellar oscillations

Equations (35), (37) and (39) can appear rather opaque when it comes to understanding the properties of stellar oscillations. The way to go about trying to distil the properties is to apply a few simplifying assumptions.

The first such assumption is to assume that the perturbation to the gravitational potential, \(\varPhi _1\), can be ignored, i.e., the Cowling approximation can be applied. It can be shown that this approximation applies to modes of oscillation where |*n*| or \(\ell \) is large.

*r*in the right-hand side of Eq. (50) can be neglected. The third assumption is that for high |

*n*| oscillations, the eigenfunctions vary much more rapidly than the equilibrium quantities. This assumption applies away from the stellar surface. The implication of this assumption is that terms containing \(H^{-1}_p\) in Eqs. (50) and (51) can be neglected when compared with the quantities on the left hand sides of the two equations. Thus, Eqs. (50) and (51) reduce to

*g modes*and their restoring force is gravity through buoyancy. Modes that satisfy the second condition, i.e., \(\omega ^2 > S^2_\ell \), and \(\omega ^2 > N^2\), are oscillatory in the outer regions, though low-degree modes can penetrate right to the centre. These are the

*p modes*and their restoring force is predominantly pressure. One can see that p modes of different degrees penetrate to different depths within the Sun. High degree modes penetrate to shallower depths than low degree modes. For a given degree, modes of higher frequency penetrate deeper inside the star than modes of lower frequencies. Thus, modes of different degrees sample different layers of a star. Note that high-\(\ell \) modes are concentrated in the outer layers, justifying to some extent the neglect of the 2 /

*r*term in Eq. (50).

#### 3.3.1 P modes

*K*(

*r*) in Eq. (57) can be approximated as

*K*(

*r*). In particular, the requirement that the modes have a lower turning point \(r_t\) and an upper turning point at \(r_u\) requires

*K*(Eq. 59) but the analysis that lead to it had no notion of an upper turning point. We just assume that the upper turning point is at \(r=R\). Thus, the “reflection” at the upper turning point does not necessarily produce a phase-shift of \(\pi /2\), but some unknown shift which we call \(\alpha _p\pi \). In other words

*r*, Eq. (63) can be rewritten as

*w*. As we can see from Eq. (58),

*w*is related to the lower turning point of a mode since

*n*. Equation (68) shows us that p modes are equidistant in frequency. It can be shown that the large spacing scales as the square root of density (Ulrich 1986; Christensen-Dalsgaard 1988). The phase shift \(\alpha _p\) however, is generally frequency dependent and thus the spacing between modes is not strictly a constant.

#### 3.3.2 G modes

G modes are low frequency modes with \(\omega ^2 < N^2\) and \(\omega ^2 < S^2_\ell \). The turning points of these modes are defined by \(N=\omega \). Thus, in the case of the Sun we would expect g modes to be trapped between the base of the convection zone and the core.

*N*. The radial wavenumber can be shown to be

#### 3.3.3 Remaining issues

The analysis presented thus far does not state anything explicitly about the upper turning point of the modes. It is assumed that all modes get reflected at the surface. This is not really the case and is the reason for the inclusion of the unknown phase factors \(\alpha _p\) and \(\alpha _g\) in Eqs. (63) and (68). The other limitation is that we do not see any f modes in the analysis.

*K*is given by

*k*being the wavenumber. Thus, f-mode frequencies are almost independent of the stratification of the Sun. As a result, f-mode frequencies have not usually been used to determine the structure of the Sun. However, these have been used to draw inferences about the solar radius (e.g., Schou et al. 1997; Antia 1998; Lefebvre et al. 2007)

## 4 A brief account of the history of solar models

The history of solar models dates back to the 1940s. Among the first published solar models is that of Schwarzschild (1946). This model was constructed at a time when it was believed that the CNO cycle was the source of solar energy. By construction, this model did not have a convective envelope, but had a convective core instead. This model does not, of course, fall under the rubric of the standard solar model—that concept was not defined till much later. As the importance of the *p*-*p* chain came to be recognised, Schwarzschild et al. (1957) constructed models with the *p*-*p* chain as the source of energy; this model included a convective envelope. These models showed how the model properties depended on the heavy-element abundance and how the initial helium abundance could be adjusted to construct a model that had the correct luminosity. The central temperature and density of the models fall in the modern range, however, the adopted heavy element abundance is very different from what is observed now. In the intervening years, several others had constructed solar models assuming radiative envelopes and homogeneous compositions (see e.g., Epstein and Motz 1953; Naur 1954; Ogden Abell 1955); of course we now know that none of these models represent the Sun very well.

The 1960s saw a new burst of activity in terms of construction of solar models. The development of new numerical techniques such as the Henyey method for solving the stellar structure equations (Henyey et al. 1959) made calculations easier. An added impetus was provided by the development of methods to detect solar neutrinos (e.g., Davis 1955). This resulted in the construction of models to predict neutrino fluxes from the Sun, e.g., Pochoda and Reeves (1964), Sears (1964) and Bahcall et al. (1963). This was a time when investigations were carried out to examine how changes to input parameters change solar-model predictions (e.g., Demarque and Percy 1964; Ezer and Cameron 1965; Bahcall et al. 1968; Bahcall and Shaviv 1968; Iben 1968; Salpeter 1969; Torres-Peimbert et al. 1969). This was also the period when nuclear reaction rates and radiative opacities were modified steadily.

The 1970s and early 1980s saw the construction of solar models primarily with the aim of determining neutrino fluxes. This is when the term “standard solar model” was first used (see e.g., Bahcall and Sears 1972). It appears that the origin of the term was influenced by particle physicists working on solar neutrinos who, even at that time, had a standard model of particle physics (Pierre Demarque, *private communication*). The term “non-standard” models also came into play at this time. An example of an early non-standard model, and classified as such by Bahcall and Sears (1972), is that of Ezer and Cameron (1965) constructed with a time-varying gravitational constant *G*. Improvements in inputs to solar models led to many new solar models being constructed. Bahcall et al. (1982), Bahcall and Ulrich (1988) and Turck-Chièze et al. (1988) for instance looked at what happens to standard models when different microphysics inputs are changed. For a history of solar models from the perspective of neutrino physics, readers are referred to Bahcall (2003).

The 1980s was when helioseismic data began to be used to examine what can be said of solar models, and by extension, the Sun. Christensen-Dalsgaard and Gough (1980) compared frequencies of models to observations to show that none of the models examined was an exact match for the Sun. Bahcall and Ulrich (1988) compared the global seismic parameters of many models. During this time investigators also started examining how the p-mode frequencies of models change with model inputs. For instance, Christensen-Dalsgaard (1982) and Guenther et al. (1989) examined how the frequencies of solar models changed with change in opacity. This was also when the first solar models with diffusion of heavy elements were constructed (see e.g., Cox et al. 1989). Ever since it was demonstrated that the inclusion of diffusion increases the match of solar models with the Sun (Christensen-Dalsgaard et al. 1993), diffusion has become a standard ingredient of standard solar models.

Standard solar models have been constructed and updated continuously as different microphysics inputs have become available. Descriptions of many standard models have been published. Helioseismic tests of these models have helped examine the inputs to these models. Among published models are those of Bahcall and Pinsonneault (1992), Christensen-Dalsgaard et al. (1996), Guzik and Swenson (1997), Bahcall et al. (1995, 1998, 2005a, b), Guenther et al. (1996), Guenther and Demarque (1997), Brun et al. (1998), Basu et al. (2000a), Neuforge-Verheecke et al. (2001a, b), Couvidat et al. (2003), Bahcall and Serenelli (2005), etc.

Many non-standard models have been constructed with a variety of motives. For instance, Ezer and Cameron (1965), as well as Roeder and Demarque (1966), constructed solar models with a time-varying value of the gravitational constant *G* following the Brans–Dicke theory. More modern solar models with time-varying *G* were those of Demarque et al. (1994) and Guenther et al. (1995) who were investigating whether solar oscillation frequencies could be used to constrain the time-variation of *G*. Christensen-Dalsgaard et al. (2005) on the other hand, tried to examine whether helioseismic data can constrain the value of *G* given that \(GM_{\odot }\) is known extremely precisely. Another set of non-standard models are ones that include early mass loss in the Sun. The main motivation for these models is to solve the so-called “faint Sun paradox”. Models in this category include those of Guzik et al. (1987) and Sackmann and Boothroyd (2003).

A large number of non-standard models were constructed with the sole purpose of reducing the predicted neutrino flux from the models and thereby solving the solar neutrino problem (see Sect. 7.1 for a more detailed discussion of this issue). These include models with extra mixing (Bahcall et al. 1968; Schatzman 1985; Roxburgh 1985; Richard and Vauclair 1997, etc.). And some models were constructed to have low metallicity in the core with accretion of high-*Z* materials to account for the higher metallicity at the surface (e.g., Christensen-Dalsgaard et al. 1979; Winnick et al. 2002). Models that included effects of rotation were also constructed (Pinsonneault et al. 1989).

More recently, non-standard solar models have been constructed to try and solve the problems solar models face if they are constructed with solar abundances as advocated by Asplund et al. (2005a, 2009). This issue has been reviewed in detail by Basu and Antia (2008) and more recent updates can be found in Basu and Antia (2013), and Basu et al. (2015). We discuss this issue in Sect. 7.5.

There is another class of solar models, the so-called ‘seismic models’ that have also been constructed. These models are constructed with helioseismically derived constraints in mind. We discuss those in Sect. 7.4.

## 5 Frequency comparisons and the issue of the ‘surface term’

*n*and frequency \(\nu \) defined in Eq. (43), and \(E_{\ell =0}(\nu )\) is the mode inertia of a radial mode at the same frequency. The quantity \(E_{\ell =0}(\nu )\) is obtained by interpolation between the mode inertias of radial modes of different orders (and hence different frequencies). The scaled frequency-differences between the Sun and Model S can be seen in Fig. 4b. Note that most of the \(\ell \)-dependence disappears and one is left with frequency differences that are predominantly a function of frequency. If we were using these difference to determine whether Model S is a good model of the Sun, we could not draw much of a conclusion.

Similar issues are faced if we are to compare other physics, such as the equation of state. In Fig. 6, we show the frequency difference between the Sun four models, each constructed with identical inputs, except for the equation of state. The EFF model has an RMS frequency difference of 5.5, the CEFF model of 4.8, the MHD model has an RMS difference of 5.3 and the OPAL model (this is Model S) has an RMS difference of 6.0. This we could be tempted to say that the CEFF model is the best and OPAL the worst, and the CEFF equation of state is the one we should use. However, as we shall see later, that is not the case: while CEFF is not too bad, OPAL is actually the best. In fact the spread in the differences of modes of different degrees is the clue to which model is better—generally speaking, the larger the spread, the worse the model. The models shown in Fig. 5 were constructed using a different code and different microphysics and atmospheric inputs than the models shown in Fig. 6, this gives rise to the large difference in the value of the RMS frequency differences for the different sets of models, again highlighting problems that make interpreting frequency differences difficult.

The reason behind the inability to use frequency comparisons to distinguish which of the two models is twofold. The first is that the frequencies of all the models were calculated assuming that the modes are fully adiabatic, when in reality adiabaticity breaks down closer to the surface. This of course, can be rectified by doing non-adiabatic calculations, as have been done by Christensen-Dalsgaard and Gough (1975), Guzik and Cox (1991, 1992), Guenther (1994), Rosenthal et al. (1995), etc. However, this is not completely straightforward since there is no consensus on how non-adiabatic calculations should be performed. The main uncertainty involves accounting for the influence of convection, and often damping by convection is ignored. The second reason is that there are large uncertainties in modelling the near-surface layers of stars. These uncertainties again include the treatment of convection. Models generally use the mixing length approximation or its variants and as a result, the region of inefficient convection close to the surface is modelled in a rather crude manner. The models do not include the dynamical effect of convection and pressure support due to turbulence either; this is important again in the near surface layers. The treatment of stellar atmospheres is crude too, one generally uses simple atmospheric models such as the Eddington *T*–\(\tau \) relation or others of a similar nature, and these are often, fully radiative, grey atmospheres and do not include convective overshoot from the interior. Some of the microphysics inputs can be uncertain too, in particular low-temperature opacities that have to include molecular lines as well as lines of elements that cause line-blanketing. Since all these factors are relevant in the near-surface regions, their combined effect is usually called the “surface effect” and the differences in frequencies they introduced is referred to as the “surface term.” Even if we could calculate non-adiabatic frequencies properly and alleviate some of the problems, the issues with modelling will still give rise to a surface term.

For low and intermediate degree modes, the surface term depends only on the frequency of the mode once mode-inertia is taken into account. A simple way to visualise this would be to look at modes close to their upper turning points. In that regions, these modes travel almost radially, regardless of their degree. The only difference is brought about by the fact that modes of different frequencies are reflected at different layers in the near surface layers as was shown in Fig. 3. The effect of near-surface uncertainties is the least in low-frequency modes that have upper turning points deeper insider a star, and increases with mode-frequency since increasing mode frequencies push the upper turning point to progressively shallower layers were the uncertainties dominate. This is why the scaled frequency differences between Sun and the models increase with frequency. The fact that in Fig. 6 the EFF model shows large difference with respect to the Sun at low frequencies points to serious issues with the model in the deeper layers. However, this is not full picture. Christensen-Dalsgaard and Thompson (1997) investigated the reasons for the small frequency shifts in low-frequency modes that result from near-surface changes and challenged the view that the deeper upper-turning points of the low-frequency modes caused the small difference at low frequencies. They showed that the small shifts could be a result of near-cancellation of different contributions, which are individually much larger than the results shifts. The small shifts can be more easily explained if Lagrangian differences (i.e., differences at fixed mass), rather than Eulerian differences (differences at fixed radius) are considered; they found that the Lagrangian differences are indeed confined closer to the surface and explain the frequency shifts more naturally.

*smooth*function of frequency that can be modelled as a low-degree polynomial. Gough (1990) showed that any localised feature in the sound speed inside a star introduces an oscillatory term in frequencies as a function of radial order

*n*which is proportional to

*T*–\(\tau \) relation and the other the Krishna Swamy

*T*–\(\tau \) relation. Also shown is the relative difference between the sound-speed profiles between the two models. Note that the sound-speed differences are significant only close to the surface, and that the frequency differences are smooth. Similar results are obtained when we have models with different formulations of convection but otherwise identical inputs (Fig. 9).

While the surface term is a nuisance, it is by no means devoid of information. Asymptotically, the influence of the upper layers of a star on p mode frequencies can be described as a phase function which essentially depends on the frequency of the modes. Christensen-Dalsgaard and Pérez Hernández (1992), and Pérez Hernández and Christensen-Dalsgaard (1994a, b) did extensive studies of the surface term and derived kernels for the term relating it to near-surface differences in structure. Pérez Hernández and Christensen-Dalsgaard (1994b) even used it to determine the convection zone helium-abundance of the Sun. Antia and Basu (1994a), and Basu and Antia (1995) also used the surface term, though in a different manner, to determine the solar helium abundance; they however, found that the effect of the equation of state on the surface term resulted in systematic uncertainties in the estimate of the solar convection-zone helium abundance.

Since one of the major sources of the surface term is the treatment of convection, there have been attempts to change and improve the treatment of convection in solar models. Rosenthal et al. (1995) showed that the non-local mixing length formulation of (Gough 1977a) as applied by Balmforth (1992) reduces the surface term with respect to the Sun. These non-local formulations however, often use more than one free parameter, which is unsatisfactory. The most promising avenue however, seems to be to inclusion of dynamical effects from numerical simulations in models. Demarque et al. (1997) showed that by parameterizing the structure of the super-adiabatic layer seen in simulations and applying that to solar models improves frequencies. Instead of parameterizing the effects of convective dynamics, Rosenthal et al. (1998, 1999) attached their simulation results to models and showed that the surface term improves. This approach was also used by Piau et al. (2014) who, in addition to normal hydrodynamic simulations, also looked at the result of using magnetohydrodynamic simulations. This way of “pasting” the simulation results on a solar model however, does not allow one to construct evolutionary models of the Sun. Li et al. (2002) tried a different approach. They explicitly include the effects of turbulence in stellar structure equations and determined the quantities from simulations. However, this too has the problem that models can only be constructed for the present day Sun. Simulations of convection show that properties of stellar convection change with \(\log g\) and \(T_\mathrm{eff}\) (Ludwig et al. 1999; Robinson et al. 2004; Tanner et al. 2013; Trampedach et al. 2013; Magic et al. 2013, etc.), thus a consistent treatment of realistic convection properties in solar models constructed from the zero-age main sequence is difficult.

The difficulty in drawing conclusions about models and their constituent microphysics by comparing frequencies have led to the development of other techniques, such as inversions, to determine solar structure and physics of the solar interior. Most helioseismic results are a result of inversions or similar techniques. In Sect. 6 we describe the common inversion techniques and some of the major results are described in Sect. 7.

## 6 Inversions to determine interior structure

The dominance of the surface effects in frequency differences between the Sun and solar models led to efforts in the direction of ‘inverting’ solar oscillation frequencies. The idea is simple: the frequencies of solar oscillations depend on the internal properties of the Sun in a known manner, thus if we had enough frequencies, we could determine solar structure from the frequencies.

Initial inversions for solar sound speed were done using the asymptotic expression given by Duvall Law (Eq. 65). However, limitations of the asymptotic expression and improvements in computational resources led to the development of inversion techniques using the full set of oscillation equations. We discuss both below. Earlier discussions of these techniques can be found in Gough and Thompson (1991), Thompson (1998a) and Christensen-Dalsgaard (2002).

### 6.1 Asymptotic inversions

*F*(

*w*) required in this equation can be obtained from observations using the Duvall Law by fitting a function (like a spline) though the points. The surface term can be accounted for by making \(\alpha _p\) in Eq. (65) a function of frequency, though early inversions did not do so (see Christensen-Dalsgaard et al. 1985). One of the issues in this inversion is that it requires the value of

*F*(

*w*) at the surface. Most observed modes have their lower-turning points well below the surface and hence

*F*(

*w*) needs to be extrapolated. Inversions of frequencies of solar models also show that results near the core are unreliable. Brodsky and Vorontsov (1987, 1988) also used the Duvall Law as the starting point, but their way of determining

*F*(

*w*) was different. They assumed that mode frequencies are a continuous function of

*n*and

*L*that can be determined from observations. They also used a frequency-dependent \(\alpha _p\). Asymptotic inversions were also carried out by Kosovichev (1988b) who showed that the solar sound speed could be successfully determined between radii of 0.4 and \(0.7R_{\odot }\). Shibahashi (1988) and Sekii and Shibahashi (1989) had a different starting point: instead of using the Duvall Law, they used the quantisation condition obtained by Eq. (78). They did however, assume that \(\omega ^2\gg N^2\) and neglected the buoyancy frequency.

*S*(

*w*) is a function of the reference model, and is given by

*S*(

*w*) is the sound travel time along a ray between successive deflections at the surface. For properties of Eq. (87), see Christensen-Dalsgaard et al. (1988b). Note that the first term in the RHS of Eq. (87) is a function of

*w*, while the second term is a function of \(\omega \). The scaled frequency differences \(S(w)\delta \omega /\omega \) can therefore be expressed as

*w*and one of \(\omega \) to the frequency differences between the Sun and the reference model. Normally this is done by fitting splines (see, e.g., Christensen-Dalsgaard et al. 1989b; Basu and Antia 1994).

### 6.2 Full inversions

While asymptotic inversions, in particular differential asymptotic inversions, were successful in revealing some of the details of the solar interior, their inherent limitation was that they assumed that solar oscillations could be explained solely through differences in sound speed alone. The role of density perturbations was generally ignored. Although some attempts were made to include higher order terms that did not neglect the density perturbations (Vorontsov and Shibahashi 1991; Roxburgh and Vorontsov 1993), the methods did not become popular. Other limitations arise from the fact that the asymptotic relations generally assume that the eigenfunctions vary more rapidly than the pressure scale height, but this breaks down close to the surface. As a result, these days it is more common to do inversions based on the full set of equations. The increase in computing power also encouraged this change.

*P*is related to \(\rho \) through the equation of hydrostatic equilibrium, and \({\mathbf {g}}\) can be derived from \(\rho \)), we do not have any means of determining the displacement eigenfunction \(\mathbf {\xi }\) except at the surface. The way out of this stalemate is to recognise that Eq. (95) is an eigenvalue equation of the form

*x*and

*y*be two eigenfunctions of the operator, then

*a*be the eigenvalue corresponding to the eigenfunction

*x*, then

*reference*model. Thus, if \(\mathcal {L}\) be the operator for a model, we assume that the operator for the Sun or other star can be expressed as \(\mathcal {L}+\delta \mathcal {L}\). The corresponding displacement eigenfunctions are \(\mathbf {\xi }\) and \(\mathbf {\xi }+\delta \mathbf {\xi }\) respectively and the corresponding frequencies are respectively \(\omega \) and \(\omega +\delta \omega \). Thus

#### 6.2.1 Inversions kernels

*E*is the mode inertia and

We show a few sound speed and density kernels in Figs. 10 and 11. Note that modes of higher \(\ell \) are restricted closer to the surface, while modes of lower \(\ell \) penetrate deeper. This is consistent with our earlier discussion of the lower turning points of different modes. Also note that sound-speed kernels are positive at all radii, while density kernels oscillate between positive and negative values. This oscillation reduces the contribution of the second term of the RHS of Eq. (113) and explains why density inversions are more difficult than sound speed inversions. This also explains why we can invert frequencies reasonably when using the asymptotic form of the oscillation equations where it is assumed that the frequency differences between a model and the Sun can be explained by differences in sound speed alone. The small discrepancies in the results obtained using asymptotic relation is a result of ignoring the small contribution from density.

*B*(

*r*) is the kernel \(K_{\rho ,\varGamma _1}\).

*u*,

*Y*),

*Y*being the helium abundance. Kernels for

*Y*are non-zero only in the ionisation zone, which makes inversions easier. The adiabatic index \(\varGamma _1\) can be expressed as

#### 6.2.2 Taking care of the surface term

### 6.3 Inversion techniques

For *N* observed modes, Eq. (122) represents *N* equations that need to be “inverted” or solved to determine the sound-speed and density differences. The data \({\delta \omega _i/\omega _i}\) or equivalently \(\delta \nu _i/\nu _i\) are known. The kernels \({K}_{c^2,\rho }^i\) and \({K}_{\rho , c^2}^i\) are known functions of the reference model. The task of the inversion process is to estimate \({\delta c^2/c^2}\) and \(\delta \rho /\rho \) after somehow accounting for the surface term \(F_\mathrm{surf}\).

There are two popular methods of inverting Eq. (122): (1) The Regularised Least Squares (RLS) method, and (2) the method of Optimally Localised Averages (OLA). These are complementary techniques (see Sekii 1997, for a discussion) that have different aims. RLS aims to find the \(\delta c^2/c^2\) and \(\delta \rho /\rho \) profiles that give the best fit to the data (i.e., give the smallest residuals) while keeping the errors small; the aim of OLA is not to fit the data at all, but to find linear combinations of the frequency differences in such a way that the corresponding combination of kernels provides a localised average of the unknown function, again while keeping the errors small. We discuss the two inversion techniques in some detail below. Other descriptions may be found in Gough and Thompson (1991) and Basu (2014). Properties of inversion techniques were investigated by Christensen-Dalsgaard et al. (1990); although their investigation was specifically for inversions of solar dynamics, the general principles apply to structure inversions too.

#### 6.3.1 The regularised least squares technique

*r*. Thus, for

*N*observed modes, Eq. (122) represent

*N*equations of the form:

#### 6.3.2 Implementing RLS

There are many ways in which the coefficients \(a_i\), \(b_i\) and \(c_i\) may be determine. The implementation of Antia and Basu (1994b) and Basu and Thompson (1996) is discussed below.

*N*equations in \(k=2n+m\) unknowns. These equations can thus be represented as:

*A*is an \(N\times (2n+m)\) matrix, \({\mathbf {x}}\) is a vector of length \((2n+m)\) consisting of the unknown coefficients \(a_i\), \(b_i\) and \(c_i\), and \({\mathbf {d}}\) is the vector with the data, i.e., \(\delta \omega /\omega \). To take data errors into account, each row

*i*of each matrix is divided by \(\sigma _i\), where \(\sigma _i\) is the uncertainty of the

*i*th data point. The elements of vector \({\mathbf {x}}\) can be determined using singular value decomposition (SVD).

*A*results in the decomposition of

*A*into three matrices, \(A=U\varSigma V^T\).

*U*and

*V*have the property that \(U^TU=I\), \(V^TV=VV^T=I\), and \(\varSigma = \mathrm{diag}(s_1,s_2,\ldots ,s_{2n+m})\), \(s_1\ge s_2\ge \ldots s_{2n+m}\) being the singular values of

*A*. Thus, the given set of equations can be reduced to

*V*.

*M*uniformly spaced points and adding the corresponding equations to the system of equations to be solved. Thus, the following terms are added:

*q*(

*r*) in the above equation allows us to smooth \(\delta c^2/c^2\) and \(\delta \rho /\rho \) preferentially in any given part of the Sun or the star. Thus, there are now \(N'=N+2M\) equations in \(k=2n+m\) unknowns which can be solved using SVD.

The ‘art’ of doing an inversion lies in choosing the correct parameters. There are at least three parameters that need to be specified to do an RLS inversion: (a) the number of basis functions in frequency used to define the surface term, (b) the number of basis functions in radius to describe the two functions, and (c) the regularisation or smoothing parameter \(\alpha \). We have the option of increasing the number of parameters by having different numbers of basis functions the two unknown functions (e.g., \(c^2\) and \(\rho \)) and/or different values of \(\alpha \) for the two functions. How these parameters can be determined have been described by Basu and Thompson (1996) and Basu (2014), a brief outline is given here.

The number of basis functions needed depends on the type of basis functions used. It is most advantageous to have well localised basis functions; this ensure that the solution at one radius is not correlated much with the solution at other radii. Antia and Basu (1994b) and Basu and Thompson (1996) used cubic B-splines (de Boor 2001). Unlike normal cubic splines, these functions are localised. They also have continuous first and second derivatives making it easy to apply smoothing. The positions where B-splines are defined are known as knots and each spline is defined over five knots. \(m+2\) knots need to be defined for *m* basis functions. For inversions of solar frequencies, knots in frequency are usually kept equidistant in frequency. Knots in *r* are defined so that they are equidistant in the acoustic depth \(\tau \). This takes into account the fact that modes spend more time in regions of lower sound speed making those regions easier to resolve.

It is usual to determine the number of frequency knots first. This is done by fixing the number of *r* knots to a reasonably large value and then examining the residuals left when the number of frequency-knots is changed; the aim is to eliminate structure in the residuals when plotted against frequency. To determine the number of knots in *r*, one can use a combination of the reduced \(\chi ^2\) obtained for reasonable smoothing, as well as the condition number (the ratio of the largest to smallest singular value) of the matrix *A* in Eq. (131). The two quantities have different behaviours when plotted as a function of the number of knots. There is a sharp decrease in \(\chi ^2_\nu \) as the number of *r* knots is increased. The number of knots selected should lie after the decrease, where the curve flattens out. The behaviour of the condition number is the opposite, and there is a steep rise beyond a certain number of knots. Experience shows that it is best to select the number of knots from the part of the curve just after the steep jump in the condition number. This position does depend on the smoothing, and hence the selection of the number of knots and the smoothing parameter have to be done in an iterative manner.

The smoothing parameter \(\alpha \) is usually determined by inspecting the so-called L-curve (Hansen 1992) that shows the balance between how smooth the solution is and how large the residuals are by plotting the smoothing constraint \(||L||^2\) (Eq. 130) as a function of the \(\chi ^2\) per degree of freedom. Increasing the smoothing parameter increases the mismatch between the data and the solution, i.e., increases \(\chi ^2\), while decreasing the oscillations in the solution (i.e., lowering \(||L||^2\)). Decreasing the smoothing parameter reduces \(\chi ^2\), but increases \(||L||^2\). The optimum value of smoothing should lie somewhere in between. The smoothing parameter not only influences how smooth a solution is, it also affects the uncertainties in the solution caused by uncertainties in the data. The higher the smoothing, the lower the uncertainties.

The inversion parameters not only depend on the type of basis function, but also on the number of modes available, their uncertainties, and the distribution of the lower turning points of those modes. Thus one cannot use the same inversion parameters if the mode-set changes.

#### 6.3.3 Optimally localised averages

The method of optically localised averages was originally developed for application in geophysics (Backus and Gilbert 1968, 1970). Gough (1985) described how it could be used for solar data. In the following discussion, we will assume that we want to determine the sound-speed differences by inverting Eq. (122).

*i*.

\(\mathcal {K}(r_0,r)\) is the “averaging kernel” or “resolution kernel” at \(r=r_0\). The width of the resolution kernel is a measure of the resolution of the inversions since the kernels represent the region over which the underlying function is averaged. \(\mathcal {C}(r_0, r)\) is the “cross-term kernel” and measures the contribution of the second variable (in this case density) on the inversion results. The aim of OLA inversions is to obtain the narrowest possible averaging kernels that the data allow, keeping the uncertainty in the solution to acceptable levels, at the same time minimising the effect of the cross-term kernel on the solution.

There are two widely used implementations of OLA. The first is the original one, which helioseismology community often calls the *Multiplicative* Optimally Localised Averages, or MOLA. MOLA is computation-intensive and requires a large matrix to be set up and inverted at reach target radius \(r_0\). A less computationally intensive implementation, the *Subtractive* Optimally Localised Averages (SOLA) was introduced by Pijpers and Thompson (1992). SOLA requires just one matrix inversion.

#### 6.3.4 Implementing OLA

*J*is defined as \(12(r-r_0)^2\)) and \(E_{ij}\) are elements of the error-covariance matrix. If the uncertainties in the mode frequencies are uncorrelated, then \(E_{ij}=\sigma _i\delta _{ij}\). This implementation is called the ‘Multiplicative’ OLA because the function

*J*multiplies the averaging kernel in the first term of Eq. (141). The parameter \(\beta \) ensures that the cross-term kernel is small, and \(\mu \) ensures that the error in the solution is small. The constraint of unimodularity of the averaging kernel \(\mathcal {K}\) is applied through a Lagrange multiplier. The surface term is expressed as

*A*is a normalisation factor that ensures that \(\int \mathcal {T} {\,\mathrm {d}}r =1\). However, this form has the disadvantage that for small \(r_0\), \(\mathcal {T}\) may not be equal to 0 at \(r=0\) where the kernels \(K_{c^2,\rho }\) and \(K_{\rho , c^2}\) are zero. This leads to a forced mismatch between the target and the resultant averaging kernel. One way out is to force the target kernels to go to zero at \(r=0\) using a modified Gaussian such as

*A*is the normalisation factor that ensures that the target is unimodular.

The width of the averaging kernel at a given target radius \(r_0\) depends on the amount of information present for that radius, the widths are usually smaller towards the surface than towards the core. This is a reflection of the fact that (for p modes at least) the amplitudes of the kernels are larger towards the surface. Thus the target kernels need to be defined such that they have a variable width. The usual practice is to define the width \(\varDelta _f\) at a fiducial target radius \(r_0=r_f\) and specify the widths at other locations as \(\varDelta (r_0)=\varDelta _fc(r_0)/c(r_f)\), where *c* is the speed of sound. This inverse variation of the width with sound speed reflects the ability of modes to resolve stellar structure (see Thompson 1993).

*M*observed modes the matrix elements \(A_{ij}\) for SOLA inversions are given by

*A*for \(i,j \le M\) change, and are given by

*A*is a function of the target radius \(r_0\). As a result, the matrix has to be set up and inverted at every point where we need a result. This makes MOLA very computationally expensive.

Choosing inversion parameters for OLA is a bit more difficult than that for RLS. For MOLA inversions we need to determine at least three parameters—the number of surface terms \(\varLambda \), the error suppression parameter \(\mu \) and the cross-term suppression parameter \(\beta \). A fourth parameter, the width of the target averaging kernels \(\varDelta (r_f)\) at a fiducial radius of \(r_f\), is needed for SOLA. Rabello-Soares et al. (1999) and Basu (2014) give extensive discussions on how parameters can be selected. In Fig. 13 we show a few sample averaging kernels from the inversions reported in Basu et al. (2009).

As in the case of RLS inversions, the surface term is the first parameter to be determined. It is done by fitting different numbers of basis functions \(\varPsi \) to the frequency differences to be inverted. The fit is done to the scaled differences. The number of functions is increases till there is no large-scale structure discernible when the residuals are plotted against frequency.

It should be noted that it is almost impossible to reduce error correlations for density inversions. The conservation of mass condition forces the solution in one part of the star to be sensitive to the solution at other parts of the star. There is an anti-correlation of errors between the core (where density is the highest) and the outer layers (where density is the lowest), and the cross-over occurs around the radius at which \(r^3\rho \) has the largest value.

There are two other parameters that need to be determined, the error-suppression parameter \(\mu \) and the cross term suppression parameter \(\beta \). Increasing \(\mu \) decreases the error correlation \(E(r_1,r_0)\) and the uncertainty in the solution \(e(r_0)\), but increases \(\chi (r_0)\) the mismatch between the target kernels and the averaging kernel. The influence of the cross term as measured by \(C(r_0)\) also increases. Increasing \(\beta \) increases \(E(r_1,r_0)\), \(e(r_0)\), and \(\chi (r_0)\), but decreases \(C(r_0)\). Given that the cross term parameter reduces systematic errors while the error-suppression parameter reduces random errors, it is usual to try and reduce \(C(r_0)\) even if that results in a somewhat larger uncertainty in the solution.

Given the complexity of the parameter finding process, it is advisable to start the inversion process by first to examining the behaviour of the solutions obtained by inverting frequency differences between two models using the observed modeset.

#### 6.3.5 Ensuring reliable inversions

OLA inversions are notoriously prone to systematic errors caused by outliers, while RLS results are relatively insensitive. Unless both types of inversions are done, one often cannot detect the error due to bad data points. An example of this is shown in Fig. 14. The figure shows the inverted density difference between the Sun and model BP04 of Bahcall et al. (2005a). The modeset was a combination of \(\ell = 0, 1, 2\), and 3 modes with frequencies less than \(1800\,\upmu \hbox {Hz}\) obtained by GOLF as listed in Bertello et al. (2000) supplemented with modes from the first year of MDI (Schou et al. 1998b). This was the so-called ‘GOLF1low’ modeset of Basu et al. (2009). RLS inversions show extremely large residuals for the \(\ell =0,n=3\) and \(\ell =0,n=5\) modes. Almost identical results were obtained with RLS inversions whether or not these two modes are included in the modeset that is inverted. SOLA inversion results on the other hand changed drastically, especially at very low radii. This discrepancy would not have been noticed had both RLS and SOLA inversions had not been performed, and erroneous results about the solar core would have been drawn.

*u*,

*Y*) kernels had been used extensively in early inversion because

*Y*kernels are non-zero only in the helium ionisation zone. This made it very easy to suppress the cross-term while keeping the error low even with the early, limited data sets. The price paid was the systematic error caused by differences between the equation of state of solar material and that used to construct solar models. Basu and Christensen-Dalsgaard (1997) showed that it is still possible to use \((\rho ,Y)\) and (

*u*,

*Y*) kernels without introducing errors by adding another term in Eq. (119) that accounts for the intrinsic differences between the equations of state. Thus

*Y*kernels may have given.

Although structure inversions rely on a linearisation of the oscillation equations around a reference model, given the quality of current solar models, this does not appear to add any errors. Basu et al. (2000a) examined the influence of the reference model on the estimated sound-speed, density and \(\varGamma _1\) profile of the Sun. They found that the effect is negligible, much smaller than the uncertainties caused by data errors, when one uses modern solar models.

## 7 Results from structure inversions

### 7.1 Solar structure and the solar neutrino problem

The inversion results show that modern standard solar models agree quite well with the Sun, certainly by usual astronomical standards. In Fig. 16, we show the relative sound-speed and density differences between the Sun and three standard solar models. Note that the sound-speed difference is fractions of a percent, while the density difference is about 2 % at the maximum. There are however, statistically significant differences in some regions. In the case of sound speed, we find a large difference just below the base of the convection zone. The peak in the sound-speed difference has been attributed to the steep gradient in the helium and heavy-element abundances just below the convection zone of the models. All models shown in the figure include diffusion. Diffusion increases the abundance of helium and metals just below the base of the convection zone, which in turn decreases the sound speed in the models (since \(c^2\propto 1/\mu \), \(\mu \) being the mean molecular weight). This localised difference can be reduced if some mixing is included below the base of the convection zone (e.g., Richard et al. 1996; Basu et al. 2000a; Brun et al. 2002). There are indeed some helioseismic investigations that suggest that the increase in *Y* and *Z* below the base of the convection zone is not as steep as that predicted by standard solar models (Kosovichev 1996; Antia and Chitre 1998). Inversions for solar rotation (see e.g., Thompson et al. 1996; Schou et al. 1998a, and references therein. More in Sect. 8.) show that there is a strong shear layer at the base of the convection zone that is usually referred to as the “tachocline”. This shear layer could easily cause mixing in the radiative-region just below the convection zone, thereby smoothing out abundance gradients (e.g., Zahn 1992). Models that include rotationally-induced mixing do indeed have a smoothed out profile (see e.g., Richard et al. 1996). The dip in the sound-speed difference around 0.2\(R_{\odot }\) has not been completely explained yet, though it is often interpreted as being caused by extra mixing in the early life of the Sun that is not present in the models (Gough et al. 1996). Density differences are more difficult to interpret since the conservation of mass condition implies that a large difference in one part of the Sun has to be compensated by an opposite difference elsewhere.

The extremely small difference between the structure of standard solar models and the Sun gave the first hints to the solution of the so-called solar neutrino problem. Solar neutrino measurements started with a Chlorine based detector (Davis 1964, 1994). Other early experiments included the water Cerenkov experiments Kamiokande (Totsuka 1992; Suzuki 1995) and Super Kamiokande (Suzuki 1991) and gallium experiments GALLEX (Anselmann et al. 1995) and SAGE (Abdurashitov et al. 1994). The solar neutrino problem arose in the 1970s when it was found that the observed flux of neutrinos from the Sun using the Chlorine-based detector was only about a third of the neutrino flux predicted by models. The problem was confirmed by later observations using other detectors based on water and gallium. The small difference in structure between solar models and the Sun led many helioseismologists to conclude that the solution to the solar neutrino problem must lie with the standard model of particle physics which postulates mass-less neutrinos. Early claims were based purely on frequency comparisons (Elsworth et al. 1990a) and comparisons of small separations (Elsworth et al. 1995a); later claims were based in inversion results (e.g., Bahcall et al. 1997; Takata and Shibahashi 1998; Watanabe and Shibahashi 2001, etc.). In fact if a non-standard solar model is constructed so that the neutrino constraints from the Chlorine experiment are satisfied, the sound-speed difference between the model and the Sun at the core would be about 10 % (Bahcall et al. 1998), which is much larger than what is seen for standard solar models. The solar neutrino problem had a second component, it was found that the number of neutrinos detected by the Chlorine detector was not consistent with those detected by the water-based detectors, which in turn were inconsistent with the gallium-based measurements. Antia and Chitre (1997) showed that it was not possible to construct a solar model that satisfied all three solar neutrino constrains (i.e., those given by the Chlorine, water and Gallium experiments) simultaneously even if some of the solar constraints were relaxed. Hata et al. (1994), Haxton (1995), Castellani et al. (1997) and Heeger and Robertson (1996) made similar inferences using the observed data alone, without involving solar models. Thus it was evident that the solution to the solar neutrino problem did not lie in deficiencies of modelling the Sun, but in the assumption regarding properties of neutrinos. The particle-physics solution to the solar neutrino problem has since been confirmed by results from the Sudbury Neutrino Observatory (Ahmad et al. 2002), and the solar neutrino problem is one of the few examples of fully solved problems in astrophysics.

### 7.2 Some properties of the solar interior

While inversions have allowed us to determine solar structure in general, the finite resolution of the inversions has often meant that specialised techniques have had to be applied to determine some of the finer details about the solar interior, these include determining the exact position of the base of the solar convection zone (\(r_b\)), the amount of overshoot below the solar convection zone as well as the amount of helium in the convection zone (\(Y_s\)).

#### 7.2.1 The base of the convection zone

Ulrich and Rhodes (1977) and Rhodes et al. (1977) compared early helioseismic data with models to show that the Sun has a deeper convection zone than the models of that era, and that the solar \(r_b\) was between 0.62\(R_{\odot }\) and 0.75\(R_{\odot }\). Berthomieu et al. (1980) estimated the depth of the solar convection zone to be 200Mm. Precise estimates of solar \(r_b\) had to wait for better data sets.

*W*(

*r*) given by

*W*(

*r*) is nearly constant, and equal to \(-0.67\), in the deeper part of the convection zone. It increases abruptly below the convection-zone base. Christensen-Dalsgaard et al. (1991) used the position of this increase to find that the base of the solar convection zone was located at \(r_b=0.713\pm 0.001R_{\odot }\). Kosovichev and Fedorova (1991) obtained similar results. Basu and Antia (1997) did a detailed study of the systematic errors involved in the process. They used the function \(H_1(w)\) (Eq. 89) obtained between the Sun and models with different values of \(r_b\) to determine position of the solar convection-zone base. Their results, despite using much better data, was identical to that of Christensen-Dalsgaard et al. (1991). Basu (1998) studied whether or not errors in measurement of the solar radius would have any effect on \(r_b\) determinations, she found none, and using data from the GONG and MDI projects she obtained \(r_b=0.7133\pm 0.0005R_{\odot }\). Basu and Antia (2004) examined whether the adopted value of the solar heavy-element abundance has any effect on the estimated value of solar \(r_b\) and found none. Thus the position of the solar convection zone base is known sufficiently precisely to provide a strong constraint that solar models need to satisfy. The question of whether the base of the solar convection zone has any latitudinal dependence has also been studied by different groups. Gough and Kosovichev (1995) inverted for \(\delta u/u\) between the Sun and a spherically symmetric model using BBSO data to find that the equator may be somewhat deeper than that at the poles, but stated that the difference did not exceed 0.02\(R_{\odot }\). However, neither Monteiro and Thompson (1998) nor Basu and Antia (2001a) found any such latitudinal variation.

The abrupt change of the temperature gradient at the convection-zone base is an acoustic glitch that leaves its signature on the frequencies [see Sect. 5, Eq. (83)], and this signature can be isolated to study the region. The amplitude of the signal is a function of the size of the glitch. Simple models of convective overshoot model the phenomenon by extending the adiabatic temperature gradient artificially and then changing over to the radiative gradient below the overshoot region. This increases the size of the acoustic glitch and that can be used to determine the extent of overshoot below the solar convection zone, an idea put forward by Gough and Sekii (1993). In Fig. 18, we show this for two solar models. Assuming that this model of overshoot is correct, then overshoot below the solar convection-zone base is very small (e.g., Monteiro et al. 1994; Basu et al. 1994; Basu and Antia 1994; Roxburgh and Vorontsov 1994; Basu 1997), and Basu (1997) put an upper limit of \(0.05H_p\) on the amount of overshoot. Of course, such a simple model of overshoot is unlikely to be correct and there have been attempts to modify the change of the temperature gradient to make it more realistic (Christensen-Dalsgaard et al. 1995, however, what is realistic is still a matter of debate (see, e.g., Rempel 2004). Two-dimensional numerical simulations suggest that overshoot could lead to a slightly extended mildly adiabatic temperature gradient beneath the convection zone up to \(0.05H_p\) followed by a rapid transition to a strongly sub-adiabatic region (Rogers et al. 2006). There are few 3D simulations that are focused on studying the overshooting region, and there are issues with those (see Canuto 2011, for an account).

#### 7.2.2 The convection-zone helium abundance

Another important property of the Sun is its convection-zone helium abundance \(Y_s\). Since the Sun is a cool star, \(Y_s\) cannot be measured spectroscopically. Yet, this is an important parameter that controls solar evolution. The convection-zone helium abundance is expected to be lower than the initial helium abundance \(Y_0\) of the Sun because of diffusion and gravitational settling, but nevertheless it can constrain models of the current Sun. Helium leaves its signature on solar oscillation frequencies because it changes the mean molecular weight (and hence sound speed), it also contributes an acoustic glitch in the form of the He ii ionisation zone (the He i ionisation zone for the Sun overlaps with the H i ionisation zone and as a result causes large systemic error when determining the helium abundance, and hence it is not used for such studies). It is the second effect that is usually exploited in different ways to determine the solar helium abundance.

*u*,

*Y*) to obtain \(Y_s=0.268\pm 0.01\); Dziembowski et al. (1991) on the other hand found \(Y_s=0.234\pm 0.005\). Kosovichev et al. (1992) found \(Y_s=0.232\pm 0.005\); they also investigated the reason for the large variation in the estimated values of \(Y_s\) and found that a major source of systematic error is the equation of state needed to convert the \((c^2,\rho )\) kernels to the (

*u*,

*Y*) kernels. Kosovichev (1993) used low degree data from the IPHIR instrument (Toutain and Froehlich 1992), and used the \(A^*\)–

*Y*kernel combination to examine whether low-degree modes are enough to determine

*Y*; \(A^*\) is negligible in the CZ and hence its contribution to the inversion should be small. He found that \(Y_s=0.251\) and 0.256 for the two data sets he used. The spread in the values of \(Y_s\) remained even when better data from GONG and MDI became available. Using reference models constructed with the MHD equation of state Richard et al. (1998) estimated \(Y_s\) to be \(0.248\pm 0.002\) which using models constructed with the OPAL equation of state Di Mauro et al. (2002) determined it to be \(0.2539\pm 0.0005\).

Performing direct inversions is only one way to determine the solar helium abundance. Vorontsov et al. (1992) by using an asymptotic description of the oscillation equations concluded that \(Y_s=0.25\pm 0.01\). Guzik and Cox (1992) did a straightforward frequency comparison to find \(Y_s=0.24\pm 0.005\). Antia and Basu (1994a) used a different method altogether—following the idea of Gough (1984b) they used the dimensionless sound speed gradient *W*(*r*) (Eq. 156) to determine \(Y_s\). The height of *W*(*r*) in the helium ionisation zone is a function of the helium abundance, unfortunately it also depends on the equation of state (see Fig. 19). They found that *W*(*r*) for models constructed with the EFF equation of state did not match observations at all. Using models constructed with MHD equations of state they estimated \(Y_s\) to be \(0.252\pm 0.003\). Once the OPAL equation of state was made available, Basu and Antia (1995) showed that \(Y_s\) estimates using calibration method, when either *W*(*r*) or \(H_1(w)\) were calibrated against models of know helium abundance, were less sensitive to equation-of-state effects than full inversions. Using the Libbrecht et al. (1990) data they obtained \(Y_s=0.246\) using MHD models, and \(Y_s=0.249\) using OPAL models. Taking into account systematic errors due to the techniques and equation of state effects, they estimated \(Y_s\) to be \(0.249\pm 0.003\). Basu (1998) used data from the GONG and MDI projects to revise the number to \(Y_s=0.248\). Basu and Antia (2004) revisited the issue. Instead of looking at \(Y_s\), they looked at the hydrogen abundance \(X_s\) and found \(X_s=0.7389\pm 0.0034\) using models with \(Z/X=0.0171\) in the convection zone.

Most of the effort into determining the solar helium abundance has been directed at determining the present-day helium abundance of the Sun. In terms of stellar evolution though, it is the initial helium abundance that is important. As a result of gravitational settling, we cannot directly estimate what the solar initial helium abundance was. However, using current constraints, it is possible to work backwards to estimate \(Y_0\) for the Sun. Serenelli and Basu (2010) determined how the initial and present-day helium abundances depend on the parameters used to construct solar models, and then they determined what the limits on the initial helium abundance of the Sun should be given our knowledge of the current structure of the Sun. They found that when only standard solar models are considered, the estimate of the initial helium abundance for the Sun is \(Y_0=0.278\pm 0.006\), independent of the solar model or its metallicity. If non-standard models with turbulent mixing below the convection-zone base are used, then \(Y_0=0.273\pm 0.006\).

### 7.3 Testing input physics

As can be seen from Fig. 20, the model without diffusion is much more discrepant than the one with diffusion. The primary reason for the large discrepancy of the non-diffusion model is that it has a much shallower convection zone than the Sun. Additionally, models without diffusion also have a much higher value of \(Y_s\) than the Sun. As a result, modern standard solar models include the diffusion and gravitation settling of helium and heavy elements.

In Fig. 6, we showed the frequency differences between models constructed with different equations of state and the Sun and were overwhelmed by the surface term. In Fig. 21, we show the sound-speed difference between the Sun and the different solar models. It is pretty clear that EFF is discrepant, and it appears that OPAL works the best.

Elliott and Kosovichev (1998) examined the equation of state for conditions present in the solar core. They determined the difference in \(\varGamma _1\) between models constructed with the MHD and OPAL equations of state and the Sun and found a discrepancy in the core. The cause of the discrepancy was identified as the use of the non-relativistic approximation to describe partially degenerate electrons instead of the relativistic Fermi–Dirac integrals. The deficiency has since been corrected for both OPAL (Rogers and Nayfonov 2002) and MHD (Gong et al. 2001) equations of state.

Somewhat more indirect means have been used to test opacities and nuclear energy generation rates. The stellar oscillation equations do not depend directly on these inputs. What we look for are the effects of changes in these inputs on solar models and how these models look in comparison with the Sun. The earliest tests of opacities were simple. Guzik and Cox (1991) compared frequencies of models constructed with the then available opacity tables (those of Cox and Tabor 1976) to suggest that the opacities were too low. Saio (1992) put opacity results on a more quantitative basis by assuming that the sound-speed differences between the models and the Sun were caused by opacity alone. He found that the discrepancy between his models and the Sun can be reduced if the Los Alamos opacities (Weiss et al. 1990) were increased by 20–50 %. The publication of the OPAL opacity tables (Rogers and Iglesias 1992) and the OP tables (Badnell et al. 2005) confirmed the results of Saio (1992). Similar methods have been used to look at differences between OPAL opacities and the opacity of the solar material. Tripathy and Christensen-Dalsgaard (1998) using the assumptions of Saio (1992) calculated kernels linking sound-speed changes to opacity changes. Christensen-Dalsgaard et al. (2009) used this method to calculate the opacity changes needed to make low-*Z* solar models, more specifically models constructed with the Asplund et al. (2005a) metallicities, agree with the Sun and found that changes of up to 30 % may be required. Basu and Antia (1997) looked at density differences rather than sound-speed differences, and found that the density difference between solar envelope models that have the correct convection-zone depth and helium abundance could be used to look at changes in opacity. They concluded that the OPAL opacities were consistent with seismic constraints. It should be noted though that their results were for models with the then accepted solar metallicity of \(Z/X=0.0245\). The authors used the same method to later quantity opacity changes needed for models constructed with \(Z/X=0.0171\) and found that an increase of about 20 % would be needed to match the density profiles (Basu and Antia 2004).

There have also been attempts to constrain the rate of the *p*-*p* reaction. This is the reaction which controls energy generation inside the Sun. Given that the rate of this reaction has not been measured, many helioseismologists turned to seismic data to put constraints on this reaction (see e.g., Antia and Chitre 1998, 1999, 2002; Degl’Innocenti et al. 1998; Schlattl et al. 1999; Turck-chièze et al. 2001). For instance, the seismic constraint on the cross-section of the *p*-*p* reaction obtained by Antia and Chitre (2002) (see also Brun et al. 2002) is \(S_{11}=(4.07\pm 0.07)\times 10^{-25}\,\hbox {MeV barns}\). This should be compared to \((4.00\pm 0.03)\times 10^{-25}\,\hbox {MeV barns}\) (Adelberger et al. 1998) and \((4.01\pm 0.04)\times 10^{-25}\,\hbox {MeV barns}\) (Adelberger et al. 2011) that has been recommended by nuclear physicists.

Other “secondary” inversion results are those for the abundance profiles inside the Sun. These inversions are done under the assumption that we know the opacity of stellar material perfectly. Kosovichev (1995) determined \(\delta X/X\) between the Sun and the model of Christensen-Dalsgaard et al. (1993) and found that the results indicated that the Sun has a smoother gradient of *Z* below the convection zone than the model. This was confirmed by others. For example, Shibahashi and Takata (1996) assumed \(Z/X=0.0277\) and \(Y=0.277\) in the solar envelope and varied the *X* abundance below to determine the *X* abundance. Antia and Chitre (1998) did a similar analysis. The results showed that the hydrogen abundance below the convection-zone base is smoother than that found in standard solar models. Models constructed with these “seismic” abundance profiles do not have the sharp difference in sound speed at the base of the convection zone that we see in Fig. 16 for standard solar models.

### 7.4 Seismic models

The fact that the structure of SSMs do not agree completely with that of the Sun have resulted in the construction of so-called “seismic models.” These models are constructed so that their structure agrees with that of the Sun. The primary motivation behind early seismic models was better predictions of the solar neutrino fluxes and estimating other properties of the Sun.

There are two types of seismic models. The more common type are seismic models of the present day Sun. These are obtained by solving the stellar structure equations using the helioseismically determined solar sound-speed and density profiles as constraints. The other type consist of models obtained by the usual evolution from the zero-age main sequence, but in these models one or more physics inputs are tuned so that the structure of the final model agrees with that of the Sun. The second type of seismic model is not particularly common; the models of Turck-Chièze et al. (2001) and Couvidat et al. (2003) fall into this category.

One of the earliest examples of the first category of seismic models is that of Kosovichev and Fedorova (1991). Dziembowski et al. (1994, 1995) soon followed with their own models. Antia (1996) constructed a seismic model using an elaborate iterative technique. Seismic models were also constructed by Takata and Shibahashi (1998), Watanabe and Shibahashi (2001) and Shibahashi et al. (1998). Such models are still being constructed as helioseismic data get better; these include the seismic model of Choubey et al. (2001) which was used to estimate properties related to mixing of different neutrino flavours.

### 7.5 The solar abundance issue

In Fig. 16, we showed the sound-speed and density differences between some solar models and the Sun. All the differences were small and we could assume that we are quite good at constructing solar models. All models shown is Fig. 16 however, were constructed assuming that the Sun has a relatively high heavily element abundance. Model S was constructed using \(Z/X=0.0245\) as per the estimates of Grevesse and Noels (1993). The other two models were constructed with \(Z/X=0.023\) as per Grevesse and Sauval (1998).

The solar heavy-element abundance is one of the most important inputs to solar models. Since heavy elements increase opacity, the heavy-element abundance affects the structure of solar models. The solar heavy-element abundance is also used as a standard for calibrating the abundances of other stars, as a result, there have been many attempts to determine what the solar metallicity is, and some of the more recent estimates have generated a lot of debate and discussion in the community. In a series of papers Allende Prieto et al. (2001, 2002), and Asplund et al. (2004, 2005b, c) revised the spectroscopic estimates of the solar photospheric composition downwards. The main feature of their analysis was the use of three-dimensional model atmospheres that included the dynamics of the gas and hence obviated the use of micro- and macro-turbulence parameters; they also incorporated some non-LTE effects in their analysis. That resulted in lowering the carbon, nitrogen and oxygen abundances in the Sun by 35–45 % of those listed in Grevesse and Sauval (1998). The revision of the oxygen abundance leads to a comparable change in the abundances of neon and argon since these abundances are generally measured through the abundances ratio for Ne/O and Ar/O. Additionally, Asplund (2000) also determined a somewhat lower value (by about 10 %) for the photospheric abundance of silicon compared with the value of Grevesse and Sauval (1998). As a result, all the elements for which abundances are obtained from meteoritic measurements have seen their abundances reduced by a similar amount. These measurements have been summarised by Asplund et al. (2005a). The net result of these changes is that *Z* / *X* for the Sun reduces to 0.0165 (or \(Z\sim 0.0122\)), about 28 % lower than the previous value of Grevesse and Sauval (1998) and almost 40 % lower than the old value of Anders and Grevesse (1989). These changes in the solar abundance led to very large changes in the structure of solar models. Models constructed with these abundances do not agree well with the Sun. The models have very shallow convection zones and low convection-zone helium abundances.

*Z*solar models and the Sun led to numerous attempts at modifying the models, and their inputs. There have also been attempts to determine solar abundances from helioseismology. These attempts and their results have been reviewed thoroughly by Basu and Antia (2008). What we give below is an update of the situation. With further improvement in analysis, abundances were updated by Asplund et al. (2009) to \(Z/X = 0.018\). While this improved the models somewhat, they were by no means as good as those with higher values of

*Z*/

*X*(see e.g., Serenelli et al. 2009). There were other, independent, attempts to determine solar heavy element abundances using three-dimensional model atmospheres, and this led to \(Z/X=0.0209\) (Caffau et al. 2010, 2011). In Fig. 23, we show the relative sound-speed and density differences between the Sun and solar models constructed with different heavy-element abundances. It is clear that the low-

*Z*models do not match. Some of the properties of these models are listed in Table 3. Basu and Antia (2013) used solar envelope models and density inversion results to determine the amount how much opacity increase would be needed to make the models in Fig. 23 agree with the Sun; they found changes of 6 % for the model with Caffau et al. (2010, 2011) abundances, 16.5 % for the Asplund et al. (2009) model and 26.5 % for the Asplund et al. (2005a) models. A recent review of the issue can be found in Bergemann and Serenelli (2014).

*Z*models can be brought in agreement with the Sun without too many changes to the inputs. One such attempt was by Zhang (2014) who showed that including the dynamical effects of convection, in particular the kinetic energy flux, could make low-

*Z*models agree with the Sun. While the scope of that investigation was very limited and not applied to SSMs but only to envelope models, and it is not clear if the kinetic energy profile used is realistic, it is certainly an interesting, physically motivated idea. Another, somewhat unconventional, idea was proposed, and ultimately rejected, by Vincent et al. (2013). They examined the impact of particles with axion-like interactions with photons and whether they affect the spectroscopically determined values of the solar heavy-element abundance. They found that these interactions could in principle resolve the problem by inducing a slight increase in the continuum opacity at line-forming heights in the solar atmosphere. This in turn would reduce the computed equivalent widths of solar absorption lines for any given elemental abundance, which would result in an underestimation of the abundances. However, the authors found that the coupling necessary to obtain the required changes are ruled out by current experiments.

Properties of standard solar models shown in Fig. 23

Mixture |
| \(R_\mathrm{CZ}\) | \(Y_\mathrm{CZ}\) | \(Y_0\) |
---|---|---|---|---|

Helioseismic | – | \(0.713\pm 0.001^\mathrm{a}\) | \(0.2485\pm 0.0034^\mathrm{b}\) | \(0.273\pm 0.006^\mathrm{c}\) |

GS98 | 0.023 | 0.7139 | 0.2456 | 0.2755 |

AGS05 | 0.0165 | 0.7259 | 0.2286 | 0.2586 |

AGSS09 | 0.018 | 0.7205 | 0.2352 | 0.2650 |

C+11 | 0.0209 | 0.7150 | 0.2415 | 0.2711 |

There have been more efforts to determine the solar abundance independently of spectroscopy. Vorontsov et al. (2013), using solar envelope models and the adiabatic index \(\varGamma _1\) claim that helioseismic data are consistent with low (and even very low) metallicities in the range \(Z=0.008\)–0.013. This contradicts the results of Antia and Basu (2006). These low-*Z* results also contradict what was found by Villante et al. (2014) who did a joint statistical analysis of helioseismic and solar neutrino data to find that the solar abundances of oxygen and iron are consistent at the \({\sim }1\sigma \) level with (high) values of Grevesse and Sauval (1998). Solar *Z* values as low as 0.008 are also inconsistent with current models of Galactic chemical evolution.

Thus the solar abundance issue is still alive and well! Reliable high-degree mode frequencies will help in resolving the near-surface layers of the stars where the ionisation zones of different elements leave their imprint. Until then we have to look at multiple diagnostics and determine whether we can reach a consensus on this issue.

### 7.6 How well do we really know the structure of the Sun?

Better results for the solar core can be obtained with g modes (see e.g., Kosovichev 1988a). These modes have high amplitudes in the core and hence, sample the core well. However, as mentioned in Sect. 3, g-mode amplitudes decay in convection zones and the Sun has a fairly deep one, making the task of detecting g modes very difficult. There is a long history of trying to detect solar g modes and a short report of the rather frustrating quest can be found in Appourchaux and Pallé (2013). For a more technical review see Appourchaux et al. (2010).

García et al. (2007) claim that they detect periodic structures that was consistent with the expected splitting for dipole g modes in periodograms constructed with GOLF data. That work did not list individual frequencies or frequency-splittings. García et al. (2008) showed that there was statistically significant power in the same frequency range in the VIRGO data. There is no consensus yet as to whether or not individual g mode peaks can be detected with the data we have today, and for that matter, whether or not their frequencies can be measured. García (2010) claims tentatively that the individual modes can be seen. This is the only group that has claimed to have measured frequencies of individual g modes and not merely their signature in modern, high-precision data. If their results are confirmed independently, and the frequencies are measured to a good precision, we should be able to get a better handle on the structure of the solar core and the density of all layers of the Sun.

The other part of the Sun which we have not probed very well yet is close to the solar surface, i.e., the near-surface layers. The reason is the dearth of reliable high-degree modes. Most helioseismic data sets have mode frequencies of up to \(\ell =200\) (GONG) or \(\ell =250\) (MDI and HMI). These modes do not probe the outer layers very well. Having high-degree modes assists the inversions by better defining the surface term by resolving differences close to the surface (Rabello-Soares et al. 2000). The reason for the lack of modes is technical: above about \(\ell =200\), individual modes have large line-widths due to their short lifetimes, and they merge into ridges when combined with the observational spatial leakage. The spatial leakage is the spherical-harmonic transform equivalent of the side lobes seen in Fourier transforms. The leakage occurs because a spherical harmonic decomposition is not orthogonal when using data over the visible solar disc, which is less than half a hemisphere. This results in modes of similar degrees and azimuthal orders leaking into the estimate of a specific degree and azimuthal order. Other reasons of leakage include line-of-sight projection effects, distortion of modes by differential rotation, etc. For low-\(\ell \) models the leakages are separated in frequency domain, but the large widths of high-degree modes means that the leakages and the central peak form a ridge. Asymmetries in the leakage could mean that the highest point of the ridge may not be the eigenfrequency of the Sun. Thus determining frequencies of high degree modes is challenging (see, e.g., Rhodes et al. 2003; Reiter et al. 2003; Korzennik et al. 2013b). At lot of effort is being put into this, however, progress has been slow.

## 8 Departures from spherical symmetry

The Sun is spherical in shape—the measured difference between the polar and equatorial radii is only about 1 part in \(10^5\) as per the analysis by Kuhn et al. (1998) of MDI data. Updated results obtained with data from the PICARD satellite puts the number slightly lower, a difference of 6.1 km, i.e., about 9 parts in \(10^6\) (Irbah et al. 2014) or perhaps even lower (\(5.7\pm 0.2\,\hbox {km}\), Meftah et al. 2015). However, although spherical, the Sun is not spherically symmetric. The main deviation from spherical symmetry in the solar interior is caused by rotation; on the solar surface, deviations from spherical symmetry are also seen in magnetic fields.

Rotation, magnetic fields and anything else that breaks spherical symmetry causes oscillation frequencies to split and lifts the degeneracy between frequencies of modes that have the same degree \(\ell \) and radial order *n* but that have different values of the azimuthal order *m* (Ledoux 1949; Ledoux and Walraven 1958; Cowling and Newing 1949). These splittings can be used to determine solar rotation. The frequency splittings \(\nu _{n,\ell ,m}-\nu _{n,\ell ,0}\) can be separated into two components, one that is odd in *m* and one that is even in *m*. The odd component arises from rotation, and in particular from advection and Coriolis force. The component even in *m* is caused by centrifugal forces, large-scale magnetic fields or any non-spherically symmetric distortion of the star.

*m*, i.e.,

*a*-coefficients are sensitive to rotation. This is a result of the structure of the kernels that relate the frequency splittings to rotation (for a description of the kernels see Christensen-Dalsgaard 2003). The \(a_1\) coefficient encodes the rotation rate averaged over all latitudes; \(a_3\) and higher order coefficients describe differential rotation. The even-order coefficients are sensitive to the second-order effects of rotation (i.e., the centrifugal forces) as well as magnetic fields and other deviations from spherical symmetry. The actual values of the

*a*coefficients at a given epoch depends on the polynomials \(\mathcal {P}\) used in the expression given in Eq. (157). The most common practice today is to use either the Clebsch–Gordon coefficients or the Ritzwoller and Lavely (1991) formulation, or polynomials proportional to those. Connections between some of the more common definitions of the coefficients may be found in Pijpers (1997).

### 8.1 Solar rotation

*Living Reviews in Solar Physics*article by Howe (2009); we describe the basic features here.

The motions of sunspots on the solar disk had revealed that the surface of the Sun rotated differentially with the solar equator rotating once every 25 days and the higher latitudes taking a longer time to rotate. This had been confirmed from Doppler shifts of photospheric lines (Snodgrass 1983, 1984; Ulrich et al. 1988, etc.), as well as measurements with other tracers (e.g., Duvall 1980; Snodgrass and Ulrich 1990; Komm et al. 1993). It had been predicted that the rotation would be constant on cylindrical surfaces throughout the convection zone, matching onto the observed surface rotation at the top of the convection zone, i.e., the Sun iso-rotation contours of the Sun will be cylinders. Helioseismology revealed a very different, and more complicated, picture (see e.g., Thompson et al. 1996; Schou et al. 1998a, and references therein). The mean rotation profile of the solar interior is shown in Fig. 26. A few features stand out immediately. The Sun rotates like a solid body below the base of the convection zone, though the rotation in the innermost parts of the Sun is not really known because it is so poorly constrained by the observations of p modes. Within the convection zone rotation changes with both latitude and radius. For most of the convection zone, rotation is nearly constant on cones, i.e., at fixed latitude the rotation is almost independent of depth, however, there is some hint of rotation on cylinders in the near-equatorial regions of the convective envelope (Gilman and Howe 2003). There are two distinct shear layers, one in the immediate sub-surface layers, and a more pronounced one at the base of the convection zone. The shear layer at the base of the convection zone is now called the “tachocline.” For a review of different methods of determining solar rotation, see Beck (2000).

*H*, of the Sun, its total rotational kinetic energy,

*T*, as well as the gravitational quadrupole moment, \(J_2\). Pijpers (1998) showed that the error-weighted mean of these quantities obtained with GONG and MDI data are:

### 8.2 Other deviations from spherical symmetry

While solar rotation has been studied in great detail, other asphericities in the Sun have not been the subject of many detailed studies. The reason for this is twofold, the first being that the signature of such asphericities in the Sun is small, as is clearly seen in Fig. 25. The second reason is perhaps the more fundamental one: the even-*a* coefficients have the disadvantage that the signal in the coefficients could be a result of sound-speed asphericities or magnetic fields or both. Zweibel and Gough (1995) showed that even-order frequency splittings cannot be unambiguously attributed to the effect of a magnetic field, and the effect of a magnetic field on frequency splittings can be reproduced by a perturbation of the sound speed. Zweibel and Gough (1995) argued that the ambiguity in the signature of global models arises because the range of latitudes sampled by the modes depends on their azimuthal order *m*. This allows both magnetic and acoustic perturbations to produce the same frequency shifts. However, the spatial configuration of a magnetic field that produces a given splitting is not the same as the spatial configuration of an acoustic perturbation that produces the same splitting. As a result, studies of asphericity using frequency splitting proceed by assuming that the even-order splittings or *a* coefficients are caused by magnetic fields alone or by aspherical perturbations to structure alone. Most modern investigations however, do correct for the second-order effects of rotation. It should be noted that asphericities in the shape of the Sun can also produce even-order splittings (Gough and Taylor 1984), however, the measured asphericity of the Sun’s limb is smaller than what is needed to explain the observed *a* coefficients. For a review on how magnetic fields can affect mode frequencies and splittings see Thompson (2006).

#### 8.2.1 Magnetic fields

One of the first estimates of solar interior magnetic fields obtained with frequency splittings was that of Dziembowski and Goode (1989). They developed a formalism relating the frequencies to the magnetic field and used the data from Libbrecht (1989) to find that there should be an axisymmetric quadrupole toroidal magnetic field of \(2\pm 1\,\hbox {MG}\) centred near the base of the convection zone. They confirmed the results using data from Libbrecht and Woodard (1990) (Dziembowski and Goode 1991). Goode and Thompson (1992) tested the hypothesis that the radiative interior of the Sun could be hiding a large-scale magnetic field, which might not be axisymmetric about the observed rotation axis and found that the strength of an axisymmetric relic field must be less than 30 MG. Such a field will cause an oblateness of 5–\(10\times 10^{-6}\). Goode and Dziembowski (1993) revisited their older work and using updated data for the splittings found that the significance of the mega-Gauss field at the convection-zone base was lower than what they had suggested in their earlier papers. Their result was revised further by Basu (1997) who used different data (those obtained by the GONG project), and a somewhat different method, to find an upper limit 0.3 MG on the toroidal magnetic field concentrated below the convection-zone base.

Gough and Thompson (1990) presented a perturbation method to calculate the effects of magnetic fields and rotation on the frequency splittings. Modified versions of that formulation have been used extensively to estimate the magnetic fields in the interior of the Sun. Using the precise *a* coefficients obtained by the GONG and MDI projects, Antia et al. (2000b) used the Gough and Thompson (1990) formulation to do a forward analysis and showed that the even-order *a* coefficients, when corrected for the second-order effects of rotation, can be explained by a 20 KG field located at a depth of 30,000 km below the solar surface. They also found an upper limit of 0.3 MG for a toroidal field at the base of the convection zone. Antia (2002) used the data to put an upper limit of \(10^{-5}\) on the ratio of magnetic pressure to gas pressure in the solar core. Baldner et al. (2009) did a more detailed analysis; they used multiple data sets and found that the data are best fit by a combination of a poloidal field and a double-peaked near-surface toroidal field. The toroidal fields are centred at \(r_0=0.999R_{\odot }\) and \(r_0=0.996R_{\odot }\) and have peaks strengths of \(380 \pm 30\,\hbox {G}\) and \(1.4 \pm 0.2\,\hbox {KG}\) respectively. The poloidal field is best described by a dipole field with a peak strength of \(124 \pm 17\,\hbox {G}\). It should be noted that the idea of a shallow field goes back a long way—in an IAU Symposium in 1986, Gough and Thompson showed how the then-available data have hints of a shallow perturbation (see Gough and Thompson 1998).

#### 8.2.2 Acoustic asphericity

As mentioned earlier, even-order splitting coefficients can also be caused by aspherical acoustic perturbations, and hence, there have been a number of investigations aimed at estimating the acoustic asphericity, if any, of the Sun. Early works (Kuhn 1988a, 1989) related the frequency splittings to temperature perturbations, but the data were not good enough to make firm conclusions. Data from GONG and MDI changed the situation somewhat, however, there are still very few studies of asphericity of solar internal structure.

Basu and Antia (2001b) used a combination of the central frequencies and the even-order *a* coefficients to make frequency combinations pertaining to different latitudes and used those to determine the position of the base of the convection zone as a function of latitude. They did not find any statistically significant difference in the position with latitude. Antia et al. (2001a) did a detailed analysis of all the sets of splittings available to determine the latitudinally dependent sound-speed profile of the Sun by doing a two-dimensional inversion of the even-order *a* coefficients. They found that it was possible to determine structural asphericities within the convection zone, though the errors became progressively larger the deeper they went. They found sound-speed asphericities of the order of \(10^{-4}\). The most notable feature is a peak in the asphericity around \(0.92R_{\odot }\). The other feature is that sound-speed perturbation at the equator is negative, but it is positive at mid-latitudes, a results which can be interpreted as the equator being cooler than the mid-latitude region. The amount of the temperature asphericity is similar to that found by Dziembowski et al. (2000) who inverted the different orders of the coefficients, i.e., \(a_2\), \(a_4\) etc., separately.

Asphericity in the near-surface layers is expected to be larger because the low densities there make the layers susceptible to perturbations. However, the absence of reliable high-degree mode-frequencies makes it difficult to study these regions.

## 9 Solar-cycle related effects

Woodard et al. (1991) showed that changes in solar frequencies were strongly correlated with the changes in the magnetic field strength. This was subsequently confirmed by Bachmann and Brown (1993), Elsworth et al. (1994) and Régulo et al. (1994). The variation of the frequencies with the changing magnetic activity is now well established and its nature being looked into in detail. Although solar oscillation frequencies increase with increase in solar activity, the rise and fall does not always follow the same path. Jiménez-Reyes et al. (1998) analysed low-degree cycle 22 data to find a “hysteresis” effect. This was also seen by Tripathy et al. (2001) in intermediate-degree data. Like Jiménez-Reyes et al. (1998), they found that some of the solar activity indices also show this effect. However, the interesting feature shown by Tripathy et al. (2001) is that while magnetic indices (like KPMI, the Kitt Peak Magnetic Index and MPSI, the Magnetic Plage Strength Index) show the same type of hysteresis as solar frequencies, some of the radiative indices of solar activity do not. For the 10.7-cm flux as well as the equivalent width of the He 10830 Å line, the ascending and descending branches cross each other.

The Sun also exhibits variability of time scales much shorter than the usual 11-year solar cycle. Over the last few decades, it has been suggested that the Sun shows significant variability on a shorter time scale of about 2 years. Benevolenskaya (1995) and Benevolenskaya (1998b) showed this using low-resolution magnetic field observations taken at the Wilcox Solar Observatory. Mursula et al. (2003) reported periods between 1 and 2 years in various heliospheric parameters such as solar wind speed, interplanetary magnetic fields, geomagnetic activity etc. Valdés-Galicia and Velasco (2008) showed that such 1–2 year variabilities are also seen in coronal holes and radio emissions. Readers are referred to the *Living Reviews in Solar Physics* article of Hathaway (2015) for details of the quasi-biennial period in different observations. Such short-term variations are also seen in solar frequencies. Fletcher et al. (2010) showed that solar frequencies are modulated with a 2 year period. This was seen in data from both BiSON and GOLF (see Fig. 29). Note that the amplitude of the modulation is almost independent of the frequency of the mode. This is quite unlike the main solar-cycle related modulation where high-frequency modes show a larger change of frequencies than low-frequency modes. However, the amplitude of the modulation shows a time-variation: the amplitude is larger at solar maximum than at solar minimum. Fletcher et al. (2010) and Broomhall et al. (2011) speculated that the 2-year signal is that from a second dynamo whose amplitude is modulated by the stronger 11 year cycle. The almost frequency-independent nature of the 2-year signal points to a second dynamo whose seismic effects originate at layers deeper than those of the primary dynamo. The presence of two different types of dynamos operating at different depths has already been proposed to explain the quasi-biennial behaviour observed in other proxies of solar activity (Benevolenskaya 1998a, b). However, not everybody agrees at the second signal is that of a second dynamo; Simoniello et al. (2013) claim that the signal could result from the beating between a dipole and quadrupole magnetic configuration of the solar dynamo.

### 9.1 Were there solar cycle-dependent structural changes in the Sun?

Direct inversions of changes in the structure of the solar interior probed by the spherically symmetric global modes have yielded inconsistent results about changes inside the Sun. Dziembowski et al. (2000), looking at differences in mode frequencies between 1999.2 and 1996.4 claimed to detect a significant change at a depth of 25–100 Mm which they interpreted as either being due to a magnetic perturbations of \((60\,\hbox {kG})^2\) or a relative temperature perturbation of about \(1.2\times 10^{-4}\). However, neither Basu (2002) nor Eff-Darwich et al. (2002) could find any measurable differences in the solar interior. Eff-Darwich et al. (2002) were able to put an upper limit of \(3\times 10^{-5}\) for any change in the sound speed at the base of the convection zone. Chou and Serebryanskiy (2005) and Serebryanskiy and Chou (2005) tried used a different technique—they plotted the smoothed scaled frequency differences as a function of the *w* which is related to the lower turning point (Eq. 67), and showed that there was a discernible time-variation at \(\log (w)\approx 2.7\). This corresponds approximately to the position of the convection-zone base. Chou and Serebryanskiy (2005) estimated the change to be caused by a magnetic field variation of \((1.7{-}1.9)\times 10^5\,\hbox {G}\) at the convection-zone base assuming equipartition, or a temperature perturbation of about 44–132 K.

Baldner and Basu (2008) tried an altogether different approach. Instead of inverting data sets of different epochs and then comparing the results, they took 54 MDI data sets covering the period from 1996 May 1 to 2007 May 16, and 40 GONG data sets spanning 1995 May 7 to 2007 April 14 and did a principal component analysis (PCA) to separate the frequency differences over solar cycle 23 into a linear combination of different time-dependent components. They found that the dominant component was indeed predominantly a function of frequency, however, it also showed a small dependence on the lower turning point of the modes, implying a time dependent change in the interior. Baldner and Basu (2008) inverted the appropriately scaled first principal component to determine the sound-speed difference between solar minimum and maximum. They found a well-localised change of \(\delta c^2/c^2=(7.23\pm 2.08)\times 10^{-5}\) at \(r=(0.712^{+0.0097}_{-0.0029})R_{\odot }\). The change near the convection-zone base can be interpreted as a change in the magnetic field of about 290 kG. They also found changes in sound speed correlated with surface activity for \(r > 0.9R_{\odot }\). Rabello-Soares (2012) examined the surface variations further by determining the frequencies of modes with \(\ell \le 900\) with MDI data centred around 2008 April (solar minimum) and 2002 May (solar maximum) and using those to resolve the surface structure better. She found a two-layer configuration—the sound speed at solar maximum was smaller than at solar minimum at round 5.5 Mm, the minimum-to-maximum difference decreased until \({\sim }7\,\hbox {Mm}\). At larger depths, the sound speed at solar maximum was larger than at solar minimum and the difference increases with depth until \({\sim }10\,\hbox {Mm}\). She could not detect deeper changes.

One of the interesting features of solar-cycle related frequency changes is the nature of the change in the frequencies of f modes. As mentioned in Sect. 3.3, the frequencies of f modes are essentially independent of stratification and depend on the local value of gravitational acceleration *g*. Given that we do not expect the mass of the Sun to change as a function of the solar cycle, the f-mode changes led to investigations of changes in the solar “seismic radius”. Dziembowski et al. (1998) looked into f-mode frequency changes in the period between 1996 May 1 and 1997 April 25 and arrived at a value of 4 km. However, Goode et al. (1998) looked at the time variation of the f-mode frequencies between 1996 May 1 and 1997 April 25, to finding a relative radius difference of \(6\times 10^{-6}\) or 4 km. They did not find any monotonic trend, instead they found a 1-year cycle in the frequencies, which of course is probably an artefact. Antia et al. (2000a) examined the f-mode frequency changes from 1995 to 1998 to conclude that if the changes were due to a change in radius, the change correspond to a change of 4.7 km.

The f mode changes are \(\ell \) dependent. Antia et al. (2001b) looked into the \(\ell \) dependence of the f-mode frequency changes over the period 1995–2000 and concluded that the most likely reason for the change is not a change in radius, but effects of magnetic fields. Sofia et al. (2005) however, argued that the \(\ell \)-dependent perturbations to f-mode frequencies points to a non-homologous change in the radius. Lefebvre et al. (2007) inverted the f-mode differences to determine the change in solar radius as a function of depth; their results showed a smaller change than was predicted by the Sofia et al. (2005) model but nonetheless showed a clear variation. Surprisingly, the issue has not been investigated using HMI data and for that matter reprocessed MDI data, and perhaps it is time that we do so!

### 9.2 What do the even-order splittings tell us?

The even-order frequency splittings and splitting coefficients change with time; even early helioseismic data showed so. Kuhn (1988a) had first suggested that the even-order splittings showed a time variation; Libbrecht (1989), Libbrecht and Woodard (1990) and Woodard and Libbrecht (1993) agreed. Kuhn (1988b, 1989) and Libbrecht (1989) studied the temporal variation of the \(a_2\) and \(a_4\) coefficients and its relation to changes in the latitudinal dependence of limb temperature measurements, and Kuhn (1989) predicted that the same relationship should extend to higher order coefficients. The inference from the inversions of the BBSO even-order *a* coefficient data carried out by Libbrecht and Woodard (1990) and Woodard and Libbrecht (1993) was that most of the variation in the even coefficients was localised close to the surface at the active latitudes, with a near-polar variation anti-correlated to the global activity level. Gough et al. (1993) reported latitude-dependent variations in *u* between 1986 and 1989. Although they did not give a quantitative estimate, their figures suggest a change of a few parts in \(10^{-4}\).

*a*coefficients have been put on a much firmer basis with data collected by the GONG, MDI and HMI projects. In particular, one can show that the even

*a*-coefficients are extremely well correlated with the corresponding Legendre component of the average magnetic field on the Sun (see e.g., Antia et al. 2001a). Howe et al. (2002) used the frequency changes of individual rotationally split components within an (\(\ell ,n\)) multiplet to carry out one-dimensional inversions for frequency shifts as a function of latitude; they showed that the frequency shifts were closely associated with regions of higher magnetic flux both as a function of time and latitude. There have been other attempts to determine whether the north–south symmetric latitudinal distribution of sound speed changes as a function of time. Antia et al. (2001a) did not find any time-variation at least below a radius of \(0.98R_{\odot }\) which is where the most reliable results are obtained. However, Baldner and Basu (2008) using a principal component analysis found significant differences in the speed between solar maximum and minimum at latitudes below \(30^\circ \), their inversions were unstable at higher latitudes. They found an increase in the sound speed at the lower latitudes during solar maximum compared with solar minimum.

As mentioned earlier in Sect. 8.2, the even-order splitting coefficients can be interpreted as being either due to structural asphericities of a result of magnetic fields. Baldner et al. (2009) analysed the changes in these coefficients in terms of magnetic fields. They found at that each epoch, they could explain the data as being due to a combination of a poloidal and a near-surface toroidal field, with a changing strength (see Fig. 31). They also found that the strengths of both components tracked the 10.7-cm radio flux, however, there were hints of saturation at high activity (see Fig. 32). They also found that the relation between the magnetic field strengths during the rising phase was different from that of the falling phase.

### 9.3 Changes in solar dynamics

*Living Reviews in Solar Physics*article by Howe (2009) for details, we describe a few highlights.

*r*, the radius, \(\theta \) the latitude and

*t*time. \(\langle \varOmega (r, \theta , t)\rangle \) is the time average of the rotation rate. From the zonal flow rate one can define the zonal flow velocity as

While the most visible changes in solar dynamics are in the zonal flows, and which are highly latitudinally dependent, there are also changes in the component of the rotation that is independent of latitude. We can see this by inverting just the \(a_1\) coefficient. In Fig. 35, we show the changes in the latitude-independent part of the solar rotation in the outer layers of the Sun as determined using MDI data. The results were obtained by subtracting the time-averaged \(a_1\) inversion result from the \(a_1\) result at any given epoch. Note that there appears to be a solar-cycle related change, with the fastest rotation at the cycle 23 solar maximum and the slowest at the two minima. Rotation during cycle 24 also appears to be somewhat slower than that during cycle 23, at least in the outer layers. The results for deeper layers are too noisy to draw any conclusion. There is significant uncertainty as to whether the dynamics of the solar radiative zone shows any time dependence (Eff-Darwich and Korzennik 2013).

While the dynamics within the convection zone clearly changes with time, the tachocline does not show much change. Antia and Basu (2011) did a detailed analysis of the then available GONG and MDI data and found that of the various properties of the tachocline, the jump in the rotation rate across the tachocline is the only parameter that shows a significant change with solar activity, but even that change is small.

### 9.4 The “peculiar” solar cycle 24

Solar cycle 24 is showing all signs of being a very weak cycle. The minimum that preceded it was extremely long and very quiet and unusual in its depth—it had more sunspot free days and any recorded in the space age, and the 10.7-cm flux was the lowest ever recorded. The polar fields observed by the Wilcox solar Observatory and other instruments were also very weak (e.g., Lo et al. 2010; Gopalswamy et al. 2012). The structure of the solar corona was very different during the minimum—instead of having a simple, equatorial configuration as is normally seen during solar minima, the corona was bright at higher latitudes too. Another noteworthy feature is that the fast solar winds were confined to higher latitudes (Manoharan 2012). In fact the minimum was unusual enough to have multiple workshops dedicated to it and many of the measured peculiarities are discussed in the workshop proceedings edited by Cranmer et al. (2010).

Of course one could ask if the minimum was really peculiar, and it is possible that the lack of extensive data for earlier solar cycles is fooling us. After all, the minimum between cycles 23 and 24 was the not the only one with many sunspot free days, there were sunspot free minima in the early part of the twentieth century (the minima of 1900 and 1910). Going even further back in time, we can find a lot of other minima that were as deep at the cycle 24 minimum, at least as far as sunspot numbers are concerned (see, e.g., Sheeley 2010; Basu 2013). As mentioned earlier, a peculiarity of the minimum was the configuration of the solar corona. However, as Judge et al. (2010) have pointed out, there are eclipse records that show that the solar corona during the solar minimum circa 1901 was very similar to that of 2009 as seen in eclipse photographs. Thus the seeming peculiarity of the solar minimum is simply a result of not having enough data; after all the Sun is 4.57 Gyr old, the sunspot data go back to only about 400 years and solar magnetic field data to date back to only 40 years.

There were early signs that the Sun was heading towards an unusual phase, unfortunately, we missed a chance to predict the peculiarity and only saw the signs after the onset of cycle 24 (Basu et al. 2012a). BiSON has been collecting data for over 30 years and its observations cover cycles 22 and 23 in their entirely and all of cycle 24 to date. They also cover a few epochs of cycle 21. Basu et al. (2012a) found that the frequency-dependence of the solar-cycle related shifts were different in cycle 23 compared with cycle 21. This can be seen in Fig. 36 where we plot the frequency shifts of low-degree solar modes for three different frequency bands. The shifts are calculated with respect to the frequencies at the maximum of cycle 22. Note that the high- and intermediate-frequency bands show a different pattern of the shifts compared with that of the low-frequency band. The difference is particularly notable for cycle 23. Also note that the frequency shifts in the high- and intermediate-frequency bands stop following the sunspot number (in red) and the 10.7-cm flux (in blue dashes) sometime just before the maximum of cycle 23. This figure thus shows that cycle 23 itself was quite different from cycle 22, and perhaps the “peculiarity” of cycle 24 had its seeds in the descending phase of the previous cycle.

Another way to show the differences between the cycles is to plot the shifts in the different frequency ranges against one another, as has been done in Fig. 37. Note that the shifts of the intermediate frequency range appears to follow that of the high frequency range for all cycles shown. The situation is different for the low-frequency modes. Cycle 23 is very different, and although cycle 24 the low-frequency shifts in cycle 24 have the same slope with respect to the cycle 22 shifts, they are following a different path altogether. It is therefore quite possible that as far as the signatures from the solar interior are concerned, each cycle is distinct in it own way and that the assumption that we can have a “typical” solar cycle is simply a reflection of the lack of data.

## 10 The question of mode excitation

All the results discussed in the earlier sections were obtained by analysing and interpreting data on the frequencies of solar oscillation. However, the frequencies are not the only observable—the height of the individual modes as well as the width of the peaks in the power spectra can be measured as well. Using the information contained within these observations requires us to understand how modes are excited and damped. The two possibilities for exciting the modes are (1) self excitation and (2) stochastic excitation by turbulent convection of intrinsically damped oscillations.

In the solar case when self-excitation due to over-stability^{1} was considered, it was argued that the limiting amplitude could be determined by non-linear processes such as mode coupling (e.g., Vandakurov 1979; Dziembowski 1982). Ando and Osaki (1975) performed one of the first calculations to determine whether or not solar modes are unstable, and found them to be so. They found that the driving of oscillations is mainly due to the \(\kappa \) mechanism (Cox 1967) of the hydrogen ionisation zone with damping being provided by radiative loses in the optically thin layers. However, their results ignored the coupling of convection to pulsations. Similar results were obtained by Goldreich and Keeley (1977b) who found that if damping by turbulent viscosity was neglected, all modes with periods greater than about 6 min were unstable, and that the \(\kappa \) mechanism was responsible for driving the modes. Modes with shorter periods were stabilised by radiative damping. However, when they included a model (albeit a crude one) of the interaction between convection and oscillations, they found that the turbulent dissipation of pulsational energy made all the modes stable. There have been many investigations to determine whether or not solar modes are stable (e.g., Gough 1980; Christensen-Dalsgaard and Frandsen 1983; Antia et al. 1982, 1988), the results have been contradictory and have depended on details of how energy dissipation was treated. Goldreich and Keeley (1977b) had concluded that the lack of a reliable time-dependent theory of convection the process of determining the stability of solar modes an uncertain one.

*A*satisfies the equation

*f*(

*t*) is a random forcing function that is independent of

*A*. The complexity of the model lies in determining the damping term \(\eta \) and the forcing function

*f*. These depend on entropy fluctuations and fluctuations of the Reynolds stress produced by convection in stellar envelopes.

A tremendous amount of work has been done since the Goldreich–Keeley papers with regards to mode excitation in the Sun. Gough (1980) showed that turbulent-excitation models predict the right order of magnitude for the p-mode amplitudes and also explains the excitation of many modes simultaneously. Kumar et al. (1988) calculated how the mode energy should be distributed, Goldreich and Kumar (1988) derived expressions for the emissivity and absorptivity of acoustic radiation by low Mach number turbulent fluids, and Goldreich et al. (1994) investigated the rates at which energy is supplied to modes as a function of their frequencies and degree. The stochastic excitation model has also been applied to solar-like oscillations in other stars Sun-like stars (Christensen-Dalsgaard and Frandsen 1983; Balmforth 1992; Houdek et al. 1999; Dupret et al. 2009; Samadi and Goupil 2001; Belkacem and Samadi 2013, etc.)

Observations do guide us somewhat. For instance, even within the rubric of stochastic mode excitation there was a question as to whether the convective flux perturbations that are induced by oscillations play a role. Christensen-Dalsgaard et al. (1989a) used the then available data on line-widths and showed that calculations that took some account of the perturbation of convective fluxes fared much better when compared with observations than calculations that did not. The data Christensen-Dalsgaard et al. (1989a) used for their tests were crude by today’s standards, but they were able to demonstrate the diagnostic potential of p-mode line widths. Elsworth et al. (1995b) and Chaplin et al. (1995) looked into the distribution energy in the modes, and found the energy distribution followed a negative exponential form. These results agree with the prediction of Kumar et al. (1988). However, both Elsworth et al. (1995b) and Chaplin et al. (1995) found some high-energy events that did not follow the trend. Chaplin et al. (1997) did an analysis of mode linewidths using BiSON data and found that the linewidths predicted by theory (in particular Goldreich and Murray 1994; Balmforth 1992) were in line with the BiSON results.

Another question that could be asked about the excitation mechanism is exactly where the excitation sources lie. Goode et al. (1992) analysed high-\(\ell \) data to show that the likely source of excitation are ‘exploding granules’ about 200 m below the base of the photosphere. Kumar and Lu (1991) looked above the acoustic cut-off frequency and place the sources somewhat lower, at \(\log (P)=5.7\), deeper than what would be suggested by the mixing length approximation. The Goode et al. results were based on the mixing length approximation that postulates that upflows and downflows are symmetric, simulations of convection have shown that this is not the case at all, and in fact that the downflows are fast and cold and are what are observed as intergranular lanes, and that upflows are slow and hot. Observations now suggest that the sources of excitation are not exploding granules, instead they are located in the dark intergranular lanes (Rimmele et al. 1995) and are related to the collapse of these lanes (Goode et al. 1999).

One of the biggest obstacles in calculated mode-excitation related properties of the Sun and Sun-like stars is the lack of a good description of convection. This makes using the information in mode amplitudes and mode widths difficult to use. The conventional mixing length approximation is clearly wrong when compared with simulations, even variations such as that of Canuto and Mazzitelli (1991) do not describe the dynamics of convection. Non-local formalisms, even those that include the anisotropy of flows, have been proposed (e.g., Gough 1977a, b), but all such formalisms have at least one, and sometimes more, free parameters and thus lack true predictive power. As a result, it is becoming increasingly common to use simulations of stellar convection to perform such calculations. Stein and Nordlund (2001) first derived expressions for the excitation of radial modes. Other works, such as those of Samadi et al. (2003a, b, 2007), Stein et al. (2004), etc., have followed. Stein et al. (2004) for instance have shown that they can reproduce the excitation rates of modes of \(\ell =0\)–3. They have also applied their method to a few other stars. Given the difficulty in modelling convection analytically, we expect that the study of mode excitation in simulations of convection to become the usual way will study mode-excitation in solar-like oscillators.

## 11 Seismology of other stars

Pulsating stars, such as Cepheids and RR Lyræ stars have been known for a long time. These stars lie on the instability strip and their pulsations are excited by a “heat engine” such as the \(\kappa \)-mechanism (see e.g., Cox 1967). We shall not discuss these stars here. This section is devoted to solar-like oscillations in stars, i.e., oscillations in stars that, like in the case of the Sun, are excited stochastically by a star’s convective envelope. For recent reviews of this subject, readers are referred to Chaplin and Miglio (2013), Christensen-Dalsgaard (2016), etc.

*I*is the intensity and \(\delta _0\) is a phase. In other words

*A*denotes the area of the visible surface. If we have a spherically symmetric system, we are free to choose our coordinate system. For ease of calculation we assume that the polar axis is pointed towards the observer which makes the integral 0 for all

*m*except \(m=0\), and for \(m=0\)

As had been mentioned earlier in Sect. 3.3, the propagation of modes of different frequencies in star is controlled by the Lamb and Brunt–Väisälä frequencies. For a given mode, the profiles of these two frequencies determine what type of a mode it is and where it can propagate. The propagation diagram for stars in different stages of evolution can be very different from each other because of the how the Brunt–Väisälä frequency evolves with time. In Fig. 38, we show the propagation diagram for a \(1M_{\odot }\) model at three stages of evolution—the main sequence, the subgiant, and the red giant. Note how the buoyancy frequency gets larger and larger towards the core as the star evolves.

The frequency of an observed mode is not sufficient to tell us whether or not it is a mixed mode. Its frequency needs to be examined in the context of the frequencies of its neighbouring modes, and in particular, the frequency separation between neighbouring modes. For this we make use of the fact that p modes are almost equally spaced in frequency, and that \(\ell =0\) and 2 modes have similar frequencies [Eq. (68)], as do \(\ell =1\) and 3 modes. The usual practice is to divide the observed power spectrum (or frequencies) of a star into frequency slices of width \(\varDelta \nu \) and stack the slices on top of each other. The result is usually called an ‘échelle’ diagram (Grec et al. 1983). If the \(\varDelta \nu \) used is correct, the \(\ell =0\) modes line up, with the \(\ell =2\) modes close by, the same happens to \(\ell =1\) and 3 modes. If an avoided crossing is present, it shows up as a deviation in the vertical structure of the modes of that degree in the échelle diagram. We show the échelle diagram of three stars in Fig. 39, note that for the Sun (Panel a) all modes line up and for the subgiant (panel b) the \(\ell =1\) modes show a break, and the giant (panel c) shows many \(\ell =1\) mixed modes.

### 11.1 Asteroseismic analyses

There have been numerous tests of the \(\nu _{\max }\) scaling relation. Bruntt et al. (2010) and Bedding (2014) compared asteroseismic and independently determined properties (e.g., from binaries) of a selection of bright solar-type stars and red giants. All the stars showed solar-like oscillations from ground-based or CoRoT observations. The estimated properties were found to agree at the level of precision of the uncertainties, i.e., to 10 % or better. The comparisons were extended to stars observed by *Kepler*. Silva Aguirre et al. (2012) used asteroseismic data on 22 of the brightest *Kepler* targets with detected solar-like oscillations and found agreement between stellar radii inferred from the scaling relations and those inferred from using Hipparcos parallaxes at the level of a few percent. Huber et al. (2012) combined interferometric observations of some of the brightest *Kepler* and CoRoT targets with Hipparcos parallaxes and also found excellent agreement, at the 5 % level, with the stellar radii inferred using the scaling relation. There have been other, more indirect, tests of the relation. For instance, Silva Aguirre et al. (2015) determined the properties of 33 *Kepler* objects of interest (i.e., *Kepler* targets suspected to have an exoplanetary system) obtained from using combinations of individual frequencies of those stars, as well as from using the \(\varDelta \nu \) and \(\nu _{\max }\) only (but through grid-based analyses with \(\varDelta \nu \) calculated from frequencies) and the agreement is very good, except in the case of three outliers. For the rest, the median fractional difference and standard deviation between the two estimates are \(0.5\pm 1.2\,\%\) for density, \(0.7\pm 0.8\,\%\) for radius and \(1.8\pm 2.1\,\%\) for mass. More recently Coelho et al. (2015) tested the \(\nu _{\max }\) scaling relation for dwarfs and subgiants and ruled out departures from the \(gT_\mathrm{eff}^{-1/2}\) scaling at the level of \({\simeq }1.5\,\%\) over the full 1560 K range in \(T_\mathrm{eff}\) that was tested. There is some uncertainty over the absolute calibration of the scaling, but again of the same order.

#### 11.1.1 Grid based modelling

Grid based modelling is more correctly a grid based search to determine stellar parameters. When \(\varDelta \nu \), \(\nu _{\max }\) and \(T_\mathrm{eff}\) are known, Eqs. (170) and (171) represents two equations in two unknowns (*M* and *R*), and can be solved to obtain *M*, *R* (and hence surface gravity *g* and mean density \(\rho \)) in a completely model-independent manner. However, the uncertainties in the results are often large. This is because, although the theory of stellar structure and evolution tell us otherwise, the equations assume that all values of \(T_\mathrm{eff}\) are possible for a star of a given *M* and *R*. This freedom can often result in unphysical results. The way out is to use an approach that uses the relation between *M*, *R* and \(T_\mathrm{eff}\) implicitly using grids of stellar models. For given values of \(\varDelta \nu \), \(\nu _{\max }\) and \(T_\mathrm{eff}\), *M* and *R* (as well as any parameter, such as age) is obtained by searching within a grid of models. Knowing a star’s metallicity helps in this.

*q*are the observations \(\varDelta \nu \), \(\nu _{\max }\), \(T_\mathrm{eff}\) and [Fe/H]. The estimated property, such as radius or mass, is then the likelihood-weighted average of the properties. The 10,001 values of any given parameter, such as radius, estimated from the central value and the 10,000 realisations forms the probability distribution function that parameter. In the YB pipeline we adopt the median of the distribution as the estimated value of the parameter, and we use \(1\sigma \) limits from the median as a measure of the uncertainties. At the price of making the results slightly model dependent, the method gives us results that are physical. Characteristics and systematic biases involved with grid-based analyses have been investigated in detail by Gai et al. (2011), Basu et al. (2010), Bazot et al. (2012), and Gruberbauer et al. (2012). The systematics related to determining \(\log g\) have been investigated by Creevey et al. (2013).

Grid-based modelling has been used to determine the global properties (mass, radius, density) of all the short-cadence targets observed by *Kepler* (Chaplin et al. 2014). This method has also being used to determine the global properties of many of the red giants observed by *Kepler* (Pinsonneault et al. 2014).

#### 11.1.2 Detailed modelling

There are some stars for which frequencies of individual modes are available. In such cases one resorts to modelling the star, calculating the frequencies of models and comparing the frequencies of the models with that of star. The star is modelled to have the observed \(T_\mathrm{eff}\) and metallicity. This is, in a sense, reminiscent of what used to be done in the early days of helioseismology. The biggest difference is that unlike the Sun, we do not know the mass, radius, age and luminosity of these stars independently.

One of the biggest issues in such an analysis is the surface term. In Fig. 41, we show the échelle diagrams of the Sun and 16 Cyg A and their models. The familiar surface term is seen as the frequency offset between the two stars and their models at higher frequencies. As had been described earlier, the vexing issue of the surface term led to the development of inversion techniques. In the case of the Sun, if we wished we could easily use a subset of the modes to define the surface term and use the rest to do the analysis, though of course it is more usual to fit the surface term simultaneously. We do not have this luxury in the case of asteroseismic analysis. Thus ad hoc means are usually used to correct for the surface term, usually by scaling the solar surface term.

*b*and the reference frequency \(\nu _0\). The exponent is determined using the solar surface term. The quantity

*a*, though formally the frequency difference at \(\nu _\mathrm{obs}=\nu _0\) is often determined in an average manner. Variants of Eq. (174) are also used, e.g., Metcalfe et al. (2012) fix \(\nu _0\) to be \(\nu _{\max }\), reducing one degree of freedom. Other ways that have been used widely depend on explicitly scaling the solar surface term; Silva Aguirre et al. (2015) describe some of these methods. While the corrections are widely used, it is unlikely that the solar scaling can be used for all stars that show solar-like oscillations.

*E*is the more inertia, and

*c*, \(c_3\) and \(c_{-1}\) are constants obtained by fitting the frequency differences. These forms are promising and tests show that Eq. (176) works very well over a large portion of the HR diagram (Schmitt and Basu 2015) and modellers are beginning to adopt this form.

Despite the ambiguity caused by the surface term, detailed modelling gives more precise estimates of the properties of a star. Of course grid modelling is also affected by the surface term, but to a lesser degree since the term gets reduced in the process of calculating the average value of \(\varDelta \nu \). Such detailed modelling has been reported by Metcalfe et al. (2010, 2012, 2014), Deheuvels and Michel (2011), Mathur et al. (2012), Silva Aguirre et al. (2013), Silva Aguirre et al. (2015), etc. Because of the uncertainty caused by the surface term, investigators are looking into using combinations of frequencies of different modes that should be relatively insensitive to the surface term (e.g., Roxburgh and Vorontsov 2003, 2013; Otí Floranes et al. 2005; Roxburgh 2014).

#### 11.1.3 Exploiting acoustic glitches

Asteroseismic analyses are not restricted to trying to fit the oscillation frequencies. As in the case of the Sun, investigators are using signatures of acoustic glitches to determine the positions of the helium ionisation zone and the convection-zone base.

Mazumdar et al. (2012, 2014) determined the acoustic depths of the He ii ionisation zone of several stars. They were also successful in measuring the acoustic depths of the convection-zone bases of some of the stars. More work needs to be done to translate the acoustic depth to a physical radius that could help modellers.

Basu et al. (2004) had proposed that the amplitude of the He ii acoustic glitch could be used to determine the helium abundance of stars. Verma et al. (2014) applied this idea to 16 Cyg A and 16 Cyg B using frequencies obtained from *Kepler* observations. These two stars are solar analogues—only slightly higher masses than the Sun and they are slightly older. The estimate of their current helium abundance is entirely consistent with the solar helium abundance with the helium abundance of 16 Cyg A in the range 0.231–0.251 and that of 16 Cyg B in the range 0.218 and 0.266.

#### 11.1.4 Analyses of internal rotation

It is difficult to determine the rotation profile of a star using asteroseismic data. The problems lie with the data themselves—\(\ell =0\) modes are not affected by rotation at all, while \(\ell =1\) modes split into at most three components and \(\ell =2\) modes into at most five. Whether all the splittings are visible also depend on the angle of inclination between the star and us (see e.g., Gizon and Solanki 2003). Thus data from which rotation can be determined are very limited. The visibility of the components as a function of inclination has been used successfully to study the spin-orbit misalignment in exoplanetary systems (Huber et al. 2013; Chaplin et al. 2013).

One of the first detailed investigations into the internal rotation profile of a star other than the Sun was that by Deheuvels et al. (2012). They compared the splittings of the mixed modes with those of the p modes of the star KIC 7341231 and found clear evidence of a radius-dependent rotation rate. They showed that the core of this star, which is just off the subgiant into the low end of the red-giant branch, rotates at least five times faster than the surface. The ratio between the core and surface rotation rates places constraints on models of angular momentum transfer. Deheuvels et al. (2014) looked at a sample of subgiants and found that the contrast between core and surface rotation increases as the star evolves; their results suggest that while the cores spin up, the envelope spins down. Data for at least two stars suggest a sharp discontinuity in rotation between the core and the envelope located at the hydrogen burning shell. The implications of these results are still being studied.

Beck et al. (2012) and Mosser et al. (2012) have concentrated on determining and interpreting the rotational splittings of red giants. Mode splittings in red giants are dominated by the rotation rate of the core. The data suggest that the core rotates faster as it ascends the giant branch and it appears that there is a slow-down in red-clump stars compared with stars on the ascending part of the giant branch.

The most Sun-like stars that have been studied are the two components of the wide binary system 16 Cyg. Davies et al. (2015) found that the data only allow the estimation of an average rotation rate. 16 Cyg A was found to have a rotation period of \(23.8^{+1.5}_{-1.8}\) days, while the rotation rate of 16 Cyg B is \(23.2^{+11.5}_{-3.2}\) days.

The study of other Sun-like stars have not yet yielded any notable results. There have however, been several investigations into how one could study the internal rotation of these stars. Studies include that of Lund et al. (2014) who studied the impact of different differential rotation profiles on the splittings of p-mode oscillation frequencies, and Schunker et al. (2016a) who examined the sensitivity of asteroseismic rotation inversions to uncertainties in the data. Schunker et al. (2016b) also examined whether one could find evidence of radial differential rotation in an average-sense by inverting data for an ensemble of stars. This has not yet been applied to real data yet.

Gizon and Solanki (2004) had shown how one might be able to determine differential rotation of a star, i.e., variation of rotation with latitude, using seismic data, however results have been elusive. The data are not sensitive enough. This is not completely surprising according to Ballot et al. (2006a), but disappointing nevertheless.

A summary of recent developments in this field can be found in Aerts (2015).

## 12 Concluding thoughts

Helioseismology has revolutionised our knowledge of the Sun. It has allowed us to know the structure and dynamics of the solar interior very well. However, what is clearly missing is better knowledge of the near-surface layers of the Sun. Accurate and precise frequencies of high-degree will resolve this issue. Additionally, a concerted push to observe low-frequency low-degree p modes and g modes will help clarify some of the issues that remain about the structure and dynamics of the solar core.

Our database of helioseismic observations does not go very far back in time. There have been continuous observations of low-degree modes for the last 30 years, and of intermediate degree modes for only 20 years. These observations have given us tantalising glimpses of how the Sun changes over a solar cycle, and on how one cycle can be very different from another. A 20-year database is definitely not long enough to learn what the interior of the Sun behaves like in a “typical” cycle, assuming of course, that a typical cycle can be defined. We thus need to continue these observations as long as we can.

Asteroseismology is in an ascending phase. These are exciting times with newer and better data being available each day. However, in the rush to exploit these data, we seem to have forgotten that we need really good analysis techniques to make optimum use of the data—a case in point is the treatment of the surface term. Now that the data stream from *Kepler* has slowed down, and TESS and PLATO are still a few years off, this is perhaps the time to figure out newer and better techniques of analysing these data.

‘Stable’ modes are not self-excited. When it comes to excitation, terms ’unstable’ and ‘overstable’ are often used interchangeably, but they have a subtle difference. Unstable modes are those whose amplitudes grow exponentially with time. Overstable modes are a special class of unstable modes and correspond to harmonic oscillations, i.e., they are periodic, and the amplitudes of these oscillations increase with time.

## Acknowledgments

I would like to thank the two anonymous referees who have helped improve this review. I would also like to thank Charles Baldner, Anne-Marie Broomhall, William Chaplin, Rachel Howe and Kiran Jain for preparing some of the figures that appear in this review, and thanks are due to Tim Larson for early access to reprocessed MDI data.