Asteroseismology of solartype stars
Abstract
Until the last few decades, investigations of stellar interiors had been restricted to theoretical studies only constrained by observations of their global properties and external characteristics. However, in the last 30 years the field has been revolutionized by the ability to perform seismic investigations of stellar interiors. This revolution begun with the Sun, where helioseismology has been yielding information competing with what can be inferred about the Earth’s interior from geoseismology. The last two decades have witnessed the advent of asteroseismology of solarlike stars, thanks to a dramatic development of new observing facilities providing the first reliable results on the interiors of distant stars. The coming years will see a huge development in this field. In this review we focus on solartype stars, i.e., cool mainsequence stars where oscillations are stochastically excited by surface convection. After a short introduction and a historical overview of the discipline, we review the observational techniques generally used, and we describe the theory behind stellar oscillations in cool mainsequence stars. We continue with a complete description of the normal mode analyses through which it is possible to extract the physical information about the structure and dynamics of the stars. We then summarize the lessons that we have learned and discuss unsolved issues and questions that are still unanswered.
Keywords
Asteroseismology Stellar oscillations Solar analogs1 Introduction
Helio and asteroseismology allow us to study the internal structure and dynamics of the Sun and other stars by means of their resonant oscillations (e.g. Gough 1985; TurckChièze et al. 1993; ChristensenDalsgaard 2002; Aerts et al. 2010; Basu 2016, and references therein). These vibrations manifest themselves by motions of the stellar photosphere and by temperature and density changes implying modulations of the positions of the Fraunhofer lines and of the stellar luminosity respectively.
Repeated sequences of stochastic excitation and damping by turbulent motions in the external convective layers lead to a suite of resonant modes in the Sun (Goldreich and Keeley 1977; Goldreich and Kumar 1988; Balmforth 1992; Goldreich et al. 1994; Samadi and Goupil 2001; Belkacem et al. 2008). The stars where their modes are excited in this way are usually called “solarlike pulsators” or simply “solarlike” stars even though their structure and dynamics could be different compared to the actual Sun, covering mainsequence (MS), subgiant and redgiant stars (e.g. De Ridder et al. 2009; Bedding et al. 2010a; García and Stello 2015; Hekker and ChristensenDalsgaard 2017). The oscillation periods of solarlike stars range from minutes to years (e.g. Mosser et al. 2010, 2013; Stello et al. 2013, 2014; Chaplin et al. 2014a).
There are other mechanisms exciting stellar oscillations in more massive and luminous mainsequence stars: (a) the heatengine mechanism (also known as \(\kappa \) or opacitydriven mechanism), related to the changes in the opacity profile due to temperature variations, and responsible for the pulsation in the instability strip and white dwarfs (e.g. Eddington 1926; b) the \(\varepsilon \) mechanism, where the nuclear reaction rate changes as a consequence of the contraction and expansion of the star (e.g. Lenain et al. 2006).; (c) tidal effects, where nonradial oscillations can be forced in stars belonging to multiple systems (e.g. Welsh et al. 2011). All of these pulsating stars are usually referred to by the generic term “classical pulsators” (e.g. \(\delta \) Scuti, \(\gamma \) Doradus, RR Lyrae, Cepheids, etc) and their study is out of the scope of this review (for more information on these variable stars see, for example, Aerts et al. 2010).
2 Asteroseismology of solartype stars in a helioseismic context
Seismic analysis tools were first applied to our closest star, the Sun, in order to infer its radial and latitudinal internal structure and dynamics. Therefore, the soundspeed profile (e.g. Basu et al. 1997; TurckChièze et al. 1997), the density profile (e.g. Basu et al. 2009), the internal rotation in the convective zone (e.g. Thompson et al. 1996) and in the radiative region (e.g. Elsworth et al. 1995; Basu et al. 1997; Chaplin et al. 1999a; Couvidat et al. 2003; García et al. 2004a, 2008c; EffDarwich et al. 2008) or the conditions and properties of the solar core (e.g. TurckChièze et al. 2001, 2004; García et al. 2007, 2008a, b; Basu et al. 2009; Appourchaux et al. 2010) have been studied and well determined. Moreover, the characterization of the pmode properties has led to the determination of other quantities such as the position of the base of the convection zone (e.g. ChristensenDalsgaard et al. 1985; Ballot et al. 2004) or the helium abundance (e.g. Gough 1983; Vorontsov et al. 1991) with high precision. With all of these observational constraints, the standard solar and stellar evolution models have been significnatly improved, reducing the uncertainties in the calculation of the stellar ages when individual pmode frequencies are considered (see for more detail the reviews by Lebreton et al. 2014a, b). However, new asteroseismic observations of many other stars (e.g. Chaplin et al. 2011b; Huber et al. 2011; Lund et al. 2017) covering a larger fraction of the HR diagram, allow us to test stellar evolution under many different conditions (e.g. ChristensenDalsgaard and Houdek 2010) while putting the Sun in its evolutionary context.
In asteroseismology, due to the absence of spatial resolution in the observations, only lowdegree modes (those with a small number of nodal lines at the surface of the star) are measured. Therefore compared to the Sun, less detailed information is available on stellar interiors. On the other hand, some pulsating solarlike stars offer the possibility to observe mixed modes, i.e., modes with mixed character resulting from the coupling between p and g modes (Arentoft et al. 2008; Bedding et al. 2010b; Chaplin et al. 2010; Deheuvels et al. 2010a; Beck et al. 2011). The study of these modes allows us to better constrain the structure and dynamics of the deep radiative interiors (e.g. Deheuvels et al. 2010b; Metcalfe et al. 2010; Bedding et al. 2011). Unfortunately, neither mixedmodes nor pure g modes have been identified individually in mainsequence solarlike stars so far because they become evanescent in the convective region and their surface amplitudes are small compared to the granulation signal. Thus, all of the information that we are obtaining for these stars comes from the characterization of p modes. However, it is important to note that for the special case of the Sun, the global signatures of the dipolar g modes have been measured (García et al. 2007) with GOLF/SoHO, as well as some individual lowfrequency g modes through the study of the perturbations induced by the g modes on the acoustic modes (Fossat et al. 2017). Both results are still controversial as shown for example by Schunker et al. (2018), who demonstrated that the latter detection of individual modes is highly dependent on the selection of the parameters used in the analysis.
3 Asteroseismic observations of solarlike stars
The requirements needed to perform asteroseismic studies of distant stars are shared with helioseismology and any other seismic studies. Stable and uninterrupted observations are ideal because most of the analyses are performed in the frequency domain, requiring long observations to increase the frequency resolution.
When preparing the observations it is also mandatory to choose a sampling rate rapid enough that the Nyquist frequency is well above the acoustic cutoff frequency of the oscillation modes. Conversely, when the stellar oscillations are just above the Nyquist frequency, aliased peaks are reflected from the Nyquist frequency leaking into lower frequencies. In this case, it is still possible to do asteroseismology for “superNyquist” oscillations as first shown by Murphy et al. (2013) and then applied to solarlike stars by Chaplin et al. (2014b). Lowmass cool mainsequence and subgiant stars have a frequency of maximum power above \(\sim \) 500–8000 \(\upmu \)Hz, i.e., in the range of \(\sim \) 2 to \(\sim \) 30 min. Therefore, a sampling rate faster than \(\sim \) 1 min is recommended to avoid dealing with superNyquist asteroseismology.
Continuity is needed to reduce the effect of gaps in the data. In particular, regular gaps—seen as a Dirac Comb function—should be avoided. Regular gaps are typical in single groundbased observations due to the day/night cycle or from a satellite due to regular operations such as angular momentum dumps of the reaction wheels used to stabilize the spacecraft. When regular gaps are present in the time series, the power of every stellar oscillation peak leaks into surrounding sidelobes due to the convolution by the Fourier transform of the signal with the window function. Examples of the impact of the NASA Kepler window function on stellar oscillations can be found in García et al. (2014b). If the gaps are sparse, the level of noise in the spectrum increases as a function of the duty cycle (see examples in Pires et al. 2015) and the signaltonoise ratio decreases.
Finally, stable instruments are necessary to minimize any possible instrumental modulations that could generate peaks in the same frequency domain as the expected stellar oscillations. In the case of multisite observations, it is recommended to have instruments as similar as possible. However, global asteroseismic observing campaigns have shown that it is possible to use very different instruments, normalize the data, and recover the stellar pulsations (e.g. Bedding et al. 2010b).
Observational asteroseismology of mainsequence cool dwarfs developed during the 1990s and the first years of the twentyfirst century. The first solarlike star for which pulsations were observed was \(\alpha \) Cen A. It was first observed in photometry by Schou and Buzasi (2000, 2001) using the WideField Infrared Explorer satellite (WIRE, Buzasi et al. 2000; Fletcher et al. 2006). Later, it was reobserved using Doppler velocity techniques from the ground (Bouchy and Carrier 2001; Martić et al. 2001; Bouchy and Carrier 2002).
Two other more evolved stars (subgiants) were studied at that time too: \(\alpha \) CMi (Procyon) and \(\eta \) Boo. Procyon was observed by several teams from the ground using Dopplervelocity measurements (Brown et al. 1991; Mosser et al. 1998; Martić et al. 1999; Bouchy et al. 2004). Procyon was also studied in photometry from space (Matthews et al. 2004) by the Canadian satellite Microvariability and Oscillations of STars (MOST, Matthews 1998; Guenther et al. 2007), providing some controversial results (see for example the discussions in Bedding et al. 2005; Régulo and Roca Cortés 2005; Bedding and Kjeldsen 2007). Today, however, mode detection for Procyon is well established thanks to a multisite groundbased campaign (Arentoft et al. 2008; Bedding et al. 2010b). Several individual modes were identified and the internal structure of the star was extracted using these asteroseismic constraints. \(\eta \) Boo was first asteroseismically observed in the equivalent width of the Balmer lines by Kjeldsen et al. (1995) (reobserved in radial velocity by Kjeldsen et al. 2003) and its oscillations were independently confirmed by the MOST space photometric observations (Guenther et al. 2005).
Although the highest signaltonoise asteroseismic results are obtained by observing stars in Doppler velocity, most of the pulsating mainsequence, solarlike stars have been observed using photometric techniques. Indeed, photometric instruments have better performance outside the Earth’s atmosphere. They require fewer photons per star allowing a high sampling rate while keeping the telescope size small. Moreover, many objects can be studied at a time. To give an example, the NASA Kepler mission (Borucki et al. 2010) allowed observing 512 stars at any time with a short cadence of 60 s.
Modern spacebased asteroseismology of solartype stars and subgiants started with the Convection Rotation and Planetary Transits space mission (CoRoT Baglin et al. 2006), which observed around a dozen such targets. Originally, the objective was to study hot F stars because they were expected to have higher amplitude modes when compared to G and K dwarfs. Unfortunately, the widths of the modes in these stars were also very large. Although this was expected, it was found that the modes overlapped each other and it was extremely difficult to properly identify the modes and to extract precise pmode parameters (see the discussions in Appourchaux et al. 2008; Benomar et al. 2009b). Hence, the observing strategy evolved and cooler stars were prioritized. The same strategy was then followed later with Kepler.
So far, cool mainsequence dwarfs and early subgiants have been observed during five space missions: WIRE, MOST, CoRoT, Kepler, including its second’s life as K2 (Howell et al. 2014), and the Transiting Exoplanet Survey Satellite (TESS, Ricker et al. 2014). TESS is primarily a mission for subgiant stars as demonstrated by the theoretical studies already done (Campante 2017; Schofield et al. 2019) and corroborated by the first marginal detection of pulsations in the solartype star \(\pi \) Mensae (Gandolfi et al. 2018) and the clear detection of pulsations in the late subgiant TOI197 (Huber et al. 2019). The small fraction of solartype stars observed by TESS will be extremely useful as these targets will be very bright and groundbased complementary studies will contribute to better characterizing them. An additional spacebased observatory, the ESA M3 Planetary Transits and Oscillations of stars (PLATO) mission (Rauer et al. 2014) is expected to be launched around 2026. From the ground, the SONG network (Grundahl et al. 2011) is already running with two sites, Spain and China, with a site in Australia expected to be operative in 2020. SONG will also be able to study solartype stars although it will be best suited for subgiants and red giants (e.g. Grundahl et al. 2017; Arentoft et al. 2019).
3.1 Structure of the power spectrum density of a solartype star
Starting from the lowfrequencies (< 10 \(\upmu \mathrm{Hz}\)), the spectrum is dominated by a series of high peaks and their harmonics. These peaks correspond to the surface differential rotation of the star because of the modulation induced in the mean stellar luminosity by dark spots crossing the visible face of the stellar disk (e.g. Berdyugina 2005). The average surface rotation of this star is \(12.3 \pm 0.15\) days (Ballot et al. 2011b). At higher frequencies, between 50 and 1000 \(\upmu \)Hz, the spectrum is dominated by a continuum (Harvey 1985a), which is the result of the turbulent movements at the surface of the star due to convection, such as granulation or supergranulation. At even higher frequencies, the pmode envelope is visible. For this star, the acoustic modes are centered around 2000 \(\upmu \)Hz, i.e., around 8 min. For reference, the oscillations of the Sun are centered around 3000 \(\upmu \)Hz (5 min). Finally, close to the Nyquist frequency, the spectrum is flat and it is dominated by the photon noise of the instrument. This noise level depends on the properties of the instrument and it would eventually be above the pmode hump in stars for which the modes have low amplitudes or when the star is distant and faint.
4 Theory of oscillations
After describing the different observations, we present in this section the theory developed to interpret the stellar oscillation spectra. We focus here on theoretical concepts that are useful for this review. More detailed descriptions may be found in, e.g., Cox (1980), Unno et al. (1989), ChristensenDalsgaard (2002), Aerts et al. (2010).

the radial order, n, indicates the number of nodes along the radius. By convention, we denote the p modes by positive numbers and g modes by negative ones (see Sect. 4.1).

The angular degree, \(\ell \), is a nonnegative integer denoting the number of nodal lines at the surface of the sphere. Thus, modes with \(\ell =0\) are radial modes while those with \(\ell \ge 1\) are the nonradial modes. More specifically \(\ell =1\) are called dipole modes, those with \(\ell =2\) are the quadrupole modes, the \(\ell =3\) are the octupole modes, etc.

The azimuthal order, m, gives the number of nodal lines passing through the poles. It can take values from \(\ell \) to \(+\ell \) including zero. Positive and negative values corresponding to retrograde and prograde waves respectively. When \(m=0\), modes are axisymmetric. These modes are usually called zonal modes; modes with \(m=\ell \) are called sectoral modes.
We usually denote the frequency \(\nu _{n,\ell ,m}=\omega _{n,\ell ,m} / 2 \pi \), expressed in Hz, where \(\omega _{n,\ell ,m}\) is the angular frequency in \(\mathrm {rad\,s}^{1}\). Due to the spherical symmetry, modes are degenerate relative to m, making the frequencies independent on m. Thus, we can write \(\nu _{n,\ell ,m}=\nu _{n,\ell }\). By considering such a symmetry, we implicitly neglect the impact of stellar rotation. We will introduce rotation in Sect. 4.2.
4.1 Acoustic, gravity and mixed modes
4.1.1 p modes
As indicated by their name, p modes are pressure, or acoustic, waves for which the restoring force arises from the pressure gradient. They are the most important modes in solarlike star seismology since they are by far the most observed ones, with periods of several minutes. Normally stable regarding nonadiabatic processes such as \(\kappa \) mechanism, these modes are stochastically excited by the turbulent convective envelope. Observed modes correspond to highorder lowdegree modes. The asymptotic (in the sense that \(n\gg \ell \)) theory of p modes has initially been developed by Vandakurov (1967) and continued in the 1980s by Tassoul (1980) and Deubner and Gough (1984) for example. Asymptotic theory described also the structure of pmode spectra. The theory predicts regular patterns: modes are organized in a comb structure as observed. These different regularities are detailed in Sect. 4.4.
4.1.2 g modes
At the lowfrequency part of the spectrum of solarlike stars, we find g modes, for which the restoring force is the buoyancy due to density fluctuations and gravity. These lowfrequency modes are confined in the radiative interior of solarlike stars since gravity waves can only propagate in a region where \(N^2>0\) (see Eq. (1)), i.e., in nonconvective zones by definition. They are thus evanescent in solarlike envelopes and reach the surface with very low amplitudes relative to the p modes. For the Sun, expected periods for g modes are of hours and longer than 35 min for the shortest period. The gmode spectrum also has a regular pattern: g modes of the same degree \(\ell \) are evenly spaced in period, the period increasing when the absolute value of the radial order increases. Moreover, radial gravity modes do not exist. Up to now, the only claims of gmode detection in solarlike stars concerns the Sun (e.g. TurckChièze et al. 2004; García et al. 2007; Fossat et al. 2017).
4.1.3 Mixed p/g modes
When solarlike stars reach the end of the main sequence, due to the buildup of strong density gradients in the core, the Brunt–Väisälä frequency increases there. As a consequence there exists a frequency range where g modes in the core and p modes in the envelope may coexist. If the evanescent region between the p and gmode cavities is small enough, a coupling between them occurs. In such a case we get mixed modes with both p and g characteristics. Their properties were initially discussed theoretically by Scuflaire (1974) and the first observations of mixed modes in solarlike stars were reported from groundbased observations of \(\eta \) Bootis, by Kjeldsen et al. (1995) and confirmed later by Kjeldsen et al. (2003), and Carrier et al. (2005). They were in very good agreement with theoretical predictions by, e.g., ChristensenDalsgaard et al. (1995). More recently, many observations of mixed modes have been made from groundbased observations but also from space thanks to CoRoT (e.g. Deheuvels et al. 2010a) and Kepler observations (Chaplin et al. 2010; Campante et al. 2011; Mathur et al. 2011a). Beck et al. (2011) first reported the existence of mixed modes in redgiant stars. Later, Bedding et al. (2011) and Mosser et al. (2011a) showed the power of these modes to measure the evolutionary status of red giants, with a clear difference between stars ascending the redgiant branch (RGB) and those in the socalled “clump”.
Mixed modes are very useful to put constraints on the internal structure and dynamics of stars since they are very sensitive to the core due to their gmode behaviour, whereas their pmode properties make their surface amplitudes high enough to be detected.
4.2 Effects of rotation
Equation 7 induces symmetrical multiplets. However asymmetrical splittings may be observed if differential rotation in latitude is strong enough (e.g. Gizon and Solanki 2004), or if the star is oblate due to fast rotation, or in mixed modes due to neardegeneracy effects (Deheuvels et al. 2017).
It is important to note that magnetic fields can also produce asymmetries in the multiplets in both amplitudes and frequencies (e.g. Gough and Thompson 1990; Goode and Thompson 1992; Shibahashi and Takata 1993; Kiefer and Roth 2018; Augustson and Mathis 2018).
4.3 Mode visibility
To be able to separate \(r_{\ell ,m}\) and \(V_\ell \) factors in Eq. (8), W must depend on \(\mu \) only. It is, for example, well known that the amplitude ratio in \(\ell =2\) and 3 multiplets in solar data observed, for instance, by GOLF does not follow Eq. (10) since the spatial response of the instrument does not depend on \(\mu \) only. Recent detailed measurements have been presented and discussed in Salabert et al. (2011a).
4.4 Frequency separations
5 Spectral analysis
In this section, we introduce various practical tools that are used to analyze seismic observations of solarlike stars. We do not deal with the computation of a power spectrum density from a velocity curve or a light curve.
5.1 The échelle diagram
5.2 Modelled spectrum
5.2.1 Background model
This model \(B(\nu )\) is the background limit spectrum that would be obtained after an infinite observing time, which would average all statistical fluctuations. The observed background is then this limiting spectrum multiplied by a random noise following a twodegreeoffreedom (2dof) \(\chi ^2\) statistic. Random processes in time series tend to produce normal (Gaussian) noises in the Fourier domain, both for the real and imaginary parts of the Fourier transform, due to the central limit theorem. Thus, a power spectrum being the sum of squared real and imaginary parts follows a 2dof \(\chi ^2\) statistic. It is true for the background but also for stochastically excited modes (see next section). This statistical distribution has been well verified in observations.
5.2.2 Mode model
Such a Lorentzian model is sufficient to reproduce the observed modes, even if asymmetries were reported for the Sun since Duvall et al. (1993). These asymmetries are small enough for lowdegree modes—typically of the order of a few percent (e.g. Toutain et al. 1998; Chaplin et al. 1999b; Thiery et al. 2000)—to consider such Lorentzian description as sufficiently accurate for analysing observations shorter than a year. For longer time series, including asymmetries may be relevant (Benomar et al. 2018).
This Lorentzian model is also valid as long as the modes are resolved, i.e., the spectra resolution is finer than the mode width. It means that the observing time is longer than the mode lifetime. This point is verified for observed p modes in solarlike stars, but it may be invalidated at very low frequency.
5.3 Maximum likelihood estimators
5.3.1 Fitting spectra
5.3.2 Errors and correlations
To estimate the covariance matrix of parameters \(\mathbf {p}\), a typical method is to approximate it by the inverse of the Hessian matrix. The uncertainties on fitted parameters are therefore taken as the square roots of the diagonal elements of the inverted matrix. These estimates are based on the Kramer–Rao theorem. By using them, we have to keep in mind a few crucial points: (i) these error estimates are only lower limits of the statistical errors and (ii) it is only asymptotically valid: the statistical distribution of parameters are assumed to be normal, that is not necessarily the case (e.g. Ballot 2010). Thus, the variables are to be carefully chosen, for example \(h=\ln H\) and \(\gamma =\ln \varGamma \) are more suitable variables than H and \(\varGamma \) to estimate the errors through the Hessian (see discussion in Toutain and Appourchaux 1994). The prefered way to estimate the errors remains in performing Monte Carlo simulations.
It is also important to notice that some parameter estimates are strongly correlated. It is specially the case between the mode height H and width \(\varGamma \), which are strongly anticorrelated, making the quantity \(H\varGamma \), hence the mode energy P, better determined than H and \(\varGamma \) individually (e.g. Toutain and Appourchaux 1994). Similar correlations exist between the inclination angle i and the splitting \(\nu _{\mathrm{s}}\) when \(\nu _{\mathrm{s}}\) is not significantly larger than the mode width. The correlation is well visible in a likelihood mapped in the \((\nu _{\mathrm{s}},i)\) plane. The likelihood maximum show a banana shape structure (Fig. 14). In this case, the projected splitting \(\nu _{\mathrm{s}}\sin i\) is better determined than the splitting itself, even when the inclination is poorly measured (see Ballot et al. 2006, 2008).
5.4 Bayesian methods
5.4.1 Bayesian inference
5.4.2 Priors: knowledge and ignorance
Bayesian methods sample the function \(p(\mathbf {p}\mathbf {Y},I)\) to give a global picture of the problem. Moreover, Bayesian approaches allow us to include relevant priors in fitting procedures. Even more, priors are needed: posterior probabilities only make sense when prior probabilities are set. Sometimes, when our knowledge is limited we look for priors that take that ignorance into account. As an example, let us consider the prior for the inclination angle i. We may certainly consider our prior independent of the other stellar quantities. When we do not have any complementary observations (or we do not want to use them), we would naturally assumed that all orientations are evenly probable. This does not mean that prior probability is uniform between \(0^{\circ }\) and \(90^{\circ }\). If we assume isotropy, the prior probability distribution for i is \(p(i)\,{\mathrm{d}}i = \sin i\,{\mathrm{d}}i\). A uniform distribution for i would favour a rotation axis oriented toward us. Ignorance priors for frequency (or splitting) are uniform probability distributions whereas it is uniform in logarithm for heights and widths (see, e.g., Benomar et al. 2009a; Handberg and Campante 2011, for more detailed discussions).
5.4.3 Markov chain Monte Carlo
From a mathematical point of view, the Markov chain asymptotically represent the target function (i.e., \(p(\mathbf {p}\mathbf {Y},I)\)) independently from the choice of the proposal probability distribution \(p_p\). However, since we will get a chain with a limited size, the \(p_p\) law must be chosen to ensure an efficient sampling of \(p(\mathbf {p}\mathbf {Y},I)\) in reasonable computing time. As we consider \(p_p\) as a multivariate normal law, we must find an suited covariance matrix for this law. Different automated algorithms are used to constructed such covariance matrices, for example in Benomar et al. (2009a) and Handberg and Campante (2011).
MCMC is not the only sampling method developed for seismology of solarlike stars. For example, Corsaro and De Ridder (2014) proposed a spectrum analysis method based on a nested sampling Monte Carlo algorithm.
5.5 Fitting strategy: local versus global fits
A global fit of the background is generally previously performed by ignoring the frequency range of p modes, or modelling the pmode hump as a Gaussian profile or two Lorentzian profiles (as has been shown for the solar case by Lefebvre et al. 2008). An example of such a fit for the two solar analogs 16 Cyg A and B observed by Kepler is shown in Fig. 17 (Metcalfe et al. 2012).
5.6 Hypothesis tests and model comparisons
The fitting techniques described in the previous section provide the best parameters of a model, assuming that the hypotheses are correct. Hypotheses include especially the choice of the model, and some a priori information such as the mode identification. It is however frequently necessary to test several competiting hypotheses to make a final decision. The most outstanding one is to test the mode identification. The first solarlike star observed by CoRoT was the Ftype star HD49933. Earlytype solarlike stars have broad modes due to strong mode damping, thus mode widths are not smaller than the small separation, making the even modes \(\ell =0,2\) overlap. As a consequence, identifying the \(\ell =1\) ridge from the \(\ell =0,2\) ridge is not obvious in an échelle diagram. Both hypotheses have to be tested with a frequentist (Appourchaux et al. 2008) or Bayesian approach (Benomar et al. 2009a). Nevertheless to complement statistical approaches, White et al. (2012) propose a method based on physical properties of the stars to disentangle both scenarios using the variation of \(\varepsilon \) (see Eq. (12)) with effective temperature.
It is also very useful to use model comparisons to validate the significance of a splitting. Thus, we can verify if a mode is significantly better fitted with a multiplet than with a single Lorentzian profile (e.g. Deheuvels et al. 2015).
When the two hypotheses \(H_0\) and \(H_1\) have the same number of free parameters (typically when we want to compare the two different mode identifications), the ratio of their likelihoods \({\mathscr {L}}_0\) and \({\mathscr {L}}_1\) gives a direct comparison of the two hypotheses. The pvalue of favouring \(H_0\) over \(H_1\) is then \(p=(1+{\mathscr {L}}_1/{\mathscr {L}}_0)^{1}\). However, as already mentioned, since MLE may be biased for low SNR, such a comparison may be skewed.
5.7 Global seismic parameters
We mainly detailed in the previous sections how individual mode properties can be extracted. Of course, global parameters, especially the mean large separation \(\varDelta \nu \), the frequency at maximum amplitude \(\nu _{\max }\) and the maximum amplitude of radial mode \(A_{\max }\) can be derived from a detailed fit. However, there exist quick methods to recover these main features. Such methods are useful when we must deal with a large number of stars. This is the reason why they have been massively used to analyse several thousand red giants observed by Kepler, but they have also been applied to mainsequence stars. Various pipelines have been developed during the last decade by several teams around the world (e.g. Roxburgh 2009a; Hekker et al. 2010; Huber et al. 2010; Kallinger et al. 2010; Mathur et al. 2010b; Mosser et al. 2011b).
The most common technique to measure \(\nu _{\max }\) is to fit a bellshape curve to the pmode hump and define its maximum as \(\nu _{\max }\). \(A_{\max }\) can be derived from the total mode power around the maximum. By measuring the total power of the modes over a large separation interval around \(\nu _{\max }\), we measure the total power of one mode of each degree. Using mode visibilities and assuming that all modes in the interval have the same intrinsic amplitude, we recover the power of a radial mode. The square root of this quantity provides the rms amplitude (see Kjeldsen et al. 2008a). Generally, we convert the observed amplitude into a bolometric amplitude using bolometric corrections derived for each instrument (see, e.g., for CoRoT and Kepler Michel et al. 2009; Ballot et al. 2011a).
To determine \(\varDelta \nu \), two techniques may be used to recover the regular pattern of p modes. This first one is to find a maximum of the autocorrelation of the spectrum in the pmode region. The autocorrelation lag providing the largest peak is the large separation, another peak (generally slightly smaller) occurs at \(\varDelta \nu /2\) when the \(\ell =1\) modes coincide with the \(\ell =0,2\) modes. The second technique, which has appeared to be more robust in practice, is to consider the Fourier transform of the power spectrum in the pmode region. Doing so, the largest peak of the Fourier transform is \(\tau =2/\varDelta \nu \), which corresponds to the main regularity visible in spectra produced by the alternation of even and odd modes. Since products in Fourier domain are convolutions in the time domain, we can easily show that this technique is equivalent to looking at the autocorrelation of the time series. We can refer to Roxburgh and Vorontsov (2006) for detailed discussions of its use and Mosser and Appourchaux (2009) for discussions on measurement errors with this technique.
6 Inferences on stellar structure
6.1 Scaling relation for masses and radii
If detailed modelling of a star to reproduce the observed frequencies is the most accurate way to determine its mass and radius, a simpler approach using scaling relations from solar values have been massively used these last few years. These scaling relations link the mass M and radius R of a star to the large separation \(\varDelta \nu \), the frequency at maximum amplitude \(\nu _{\max }\), and the effective temperature \(T_{\mathrm{eff}}\).
6.2 Modelindependent determination of masses and radii
It has also been possible to test the global seismic scaling relations using independent measurements of radii for mainsequence and giant stars using interferometry. Huber et al. (2012) performed such an analysis comparing asteroseismic radii obtained with Kepler with interferometric radii obtained with the CHARA array and found an agreement of around 4% (see also White et al. 2013). Unfortunately, because mainsequence stars have small angular diameters, it is extremely difficult to properly extract their radius, these stars being more prone to systematic errors in the adopted calibrator diameters than subgiants and red giants. This is the reason why it has been recommended to restrict any comparison of asteroseismic diameters with interferometry to stars with angular diameters larger than \(> 0.3\) mas excluding many mainsequence solarlike stars (Huber et al. 2017).
Another independent validation of the asteroseismic radius can be done using astrometric results (e.g. Silva Aguirre et al. 2012) from Hipparcos (van Leeuwen 2007) or Gaia (Perryman et al. 2001). Indeed, Huber et al. (2017) compared asteroseismic radii of 2200 oscillating stars observed by Kepler (including 440 mainsequence stars and subgiants from Chaplin et al. 2011b) with Gaia DR1 results included in the TychoGaia Astrometric Solution (TGAS, Michalik et al. 2015). The overall agreement found was excellent, which helped to empirically demonstrate that asteroseismic radii computed using global seismic scaling relations were accurate to \(\approx \,10 \%\) for stars ranging from \(\approx \,0.8\) to \(10\,R_\odot \) without any visible offset between the two radii determinations for mainsequence stars (1 to \(1.5\,R_\odot \)). Moreover, no significant trends were found with metallicity \(\mathrm{[Fe/H]} = 0.8\) to \(+0.4\ \mathrm{dex}\).
6.3 Model dependent determination of masses and radii
6.4 Ensemble observational asteroseismology
Mode amplitudes, heights, and linewidths are more difficult to measure. Appourchaux et al. (2014) showed that systematic effects between 8 different groups of “fitters” were mainly due to the way that the convective background was treated, as well as on the fitted values of the rotational splittings and inclination angles. Following a correction scheme based on the onefit approach of Toutain et al. (2005), they demonstrated that the systematic effects could be reduced to less than \(\pm 15\%\) for the linewidths and heights, and to less than \(\pm \,5\%\) for the amplitudes. It is worth mentioning that different convective background models introduce frequencydependent systematic errors that could bias any comparison with theoretical predictions and between different groups of fitters. Once frequencies, amplitudes, and linewidths are measured with precision and accuracy, it is possible to look for global trends with, for example, age, effective temperature, or mass.
6.5 The CD diagram
The ChristensenDalsgaard diagram (or simply CD) is the representation of the small separation \(\delta \nu _{0,2}\) as a function of the large separation \(\varDelta \nu \) (ChristensenDalsgaard 1988). By placing stars on this diagram (see Fig. 26, it is possible to directly determine their masses and ages. Indeed the tracks for different masses and ages are well separated. However, it is important to note that this diagram depends on metallicity. Nevertheless, if the metallicity of a star is known, such a diagram is an interesting tool to derive its age. A CD diagram has been constructed for 76 Kepler stars and several CoRoT targets by White et al. (2011a). Results are shown in Fig. 26. We can see how evolution tracks are well split during the main sequence but converge at the terminal age main sequence. It also gives an estimate of the mass within 4 – 7% when the metallicity is determined within 0.1 dex. The agedetermination accuracy depends strongly on the smallseparation determination.
6.6 Modelling a star and surface effects
Nevertheless, the use of frequencies without specific care provides biased results due to surface effects. It is known for the Sun (e.g. ChristensenDalsgaard and Berthomieu 1991) that significant discrepancies between observed frequencies and those computed from 1D model occur above \(\approx \,2000\,\upmu \mathrm{Hz}\), as shown in Fig. 27. This is due to the poor modelling of upper layers. Low frequency modes are almost unaffected because their outer turning point are deeper in the star, as discussed in Sect. 4.1.1 (Fig. 8). Upper layers are affected by dynamical processes missing in 1D modelling: 3D model atmospheres show that turbulent pressure cannot be neglected (at the photosphere it reaches about 15% of the total pressure) and the structure is also affected by convective backwarming (e.g. Trampedach et al. 2013). Moreover the \(\beta \) plasma becomes small, letting the magnetic field play a role (it is the reason why its variations during stellar cycles are visible on frequencies, see Sect. 8). There are not only structural effects, but also modal effects: non adiabatic effects must be included as well as fluctuations of the turbulent pressure.
To improve the modelling of outer layers, 1D models patched with 3D simulations have been proposed for decades (e.g. Stein and Nordlund 1991; Rosenthal et al. 1999; Yang and Li 2007; Piau et al. 2014; Bhattacharya et al. 2015; Sonoi et al. 2015, 2017; Ball et al. 2016; Magic and Weiss 2016; Houdek et al. 2017; Trampedach et al. 2017). To compute the oscillations primarily two different assumptions were proposed to model the Lagrangian fluctuations \(\delta p_t\) of turbulent pressure. The first one, called the Gas \(\varGamma \) model (GGM), consists in assuming that they vary as the total pressure, i.e., \(\delta p_{\mathrm{t}}/p_{\mathrm{t}} \approx \delta p_\mathrm{tot}/p_\mathrm{tot} \approx \delta p_\mathrm{gas}/p_\mathrm{gas} = \varGamma _1 \delta \rho /\rho \). In the second model, called the Reduced \(\varGamma \) model (RGM), the turbulent pressure vanishes \(\delta p_{\mathrm{t}}/p_{\mathrm{t}} \approx 0\), hence \(\delta p_\mathrm{tot}/p_\mathrm{tot} \approx (\varGamma _1 p_\mathrm{gas}/p_\mathrm{tot}) \delta \rho /\rho \). Rosenthal et al. (1999) shows that GGM gives better results than RGM. Using patched models with GGM improves the computed high frequencies, but a residual of several \(\upmu \)Hz remains. Recently Sonoi et al. (2017) and Houdek et al. (2017) used timedependent convection models to compute the oscillations of patched models. The first team used prescriptions from Grigahcène et al. (2005) and Dupret et al. (2005), whereas the second one used a model developed by Gough (1977). Doing so, the discrepancies do not almost depend on frequency and remain limited to a few \(\upmu \)Hz (Fig. 27).
6.7 Constraints on the internal structure
Sharp variations of the sound speed (or derivative) occur in stars where (i) the adiabatic exponent \(\varGamma _1\) quickly decreases, which occurs in the ionisation of abundant elements (H, He i, He ii) (ii) when the thermal gradient changes, especially when energy transport processes change from radiative to convective (or the opposite). Measuring glitches allow us to probe (i) the helium ionisation zone, possibly giving constraints on the helium abundance in the envelope, (ii) the position and the structure of the base of the convective envelope (BCE). Glitches generated by convective cores are too deep—in terms of acoustic depth—to be seen as oscillations in frequency. We will discuss convective cores later on.
Glitches are also visible in small separations and ratios, but oscillation periods of signatures are not linked to the acoustic depth but to the acoustic radius (time for a sound wave to propagate to the glitch from the stellar centre). As a consequence, the helium ionisation signature almost disappears in these variables (for more details, see Roxburgh and Vorontsov 2003; Roxburgh 2005, 2009b).
Measurement of glitches requires high precision on individual mode frequencies. In CoRoT observations of HD 52265, a BCE glitch was visible (Ballot et al. 2011b) and has been analysed by Lebreton and Goupil (2012). Their analysis suggests that the penetrative distance of the convective envelope in the radiative zone reaches \(0.95\,H_{\mathrm{p}}\) (6% of the total radius), which is significantly larger than what is found for the Sun. Measuring glitches in Ftype stars is challenging because uncertainties of frequencies are high due to mode broadening (Eq. (31)), reinforced by the overlap of \(\ell =0\) and 2 modes. However, Brito and Lopes (2017) succeeded in measuring the position of helium ionisation zone in HD 49933 using CoRoT data. Glitches have also been intensively studied in Kepler solartype stars. A first work on 19 stars with oneyear observations has been carried out by Mazumdar et al. (2014), then extended to the LEGACY sample (66 stars observed for up to 4 years) by Verma et al. (2017). Figure 28 shows an example of second differences fitted with oscillatory components due to BCE and He ii ionisation zone. Helium glitches can be robustly measured in many stars, whereas BCE is more difficult to measure for supersolar mass stars. These measurements can now be used to constrain properties of the helium ionisation zone and helium abundance in the envelope.
In the case of helioseismology, fine constraints on the internal structure were achieved by inversion techniques. This was possible thanks to the observations of many intermediate and highdegree modes. Structure inversions for other stars, based only on lowdegree modes without strong external constraints on mass and radius are challenging. In this context, Reese et al. (2012) proposed an inverse method to accurately (0.5% accuracy) estimate stellar mean densities. Inversion efforts have been pursued for several Kepler targets, especially 16 Cyg A and B by Buldgen et al. (2016a, b). Doing so, they reduced the uncertainties to 2% on mass, 1% on radius, and 3% on the age for 16 Cyg A. New inversion methods to constrain convective regions have been proposed by Buldgen et al. (2018) and should be tested soon on Kepler observations. Finally, a new method called “Inversion for agreement” has been proposed by Bellinger et al. (2017). This method takes into account imprecise estimates of stellar mass and radius as well as the relatively small amount of modes available. Because the result is independent of models, it can be used to test their inferences. Thus, the results obtained on 16 Cyg A and B showed that the core sound speed in both stars exceeds that of the models.
7 Stellar rotation
One of the main results obtained with helioseismology is the accurate determination of the solar differential rotation in the convective zone (e.g. Thompson et al. 1996, 2003; Howe 2009), as well as the nearly constant rotation rate in the radiative interior down to \(\approx \,0.2\,R_\odot \), where the measured lowdegree p modes do not provide enough sensitivity below this radius (e.g. Chaplin et al. 1999a; García et al. 2004a, 2008c). Gravity modes are needed to properly constrain the rotation within the core (e.g. Mathur et al. 2008). However, although the detection of individual g modes in the Sun is still controversial (e.g. Appourchaux et al. 2010; Schunker et al. 2018) the latest results obtained considering these modes or their period spacing suggest a faster rotation rate in the core (García et al. 2007, 2008a, 2011; Fossat et al. 2017).
When extending the analysis of internal rotation to other stars, the precision and the extension of the region to be explored depends on our ability to probe their inner regions. In MS stars, only pure acoustic modes have been unambiguously characterized so far and, therefore, only the outer convective zone and the outer part of the inner radiative zone can be probed by asteroseismology. Only when stars enter the subgiant region, can mixed modes be measured (e.g. Deheuvels et al. 2010a; Benomar et al. 2012) and rotation of the inner zones and the core can be obtained (e.g. Deheuvels et al. 2012, 2014).
This example illustrates all of the information provided by the study of continuous highprecision photometry. On the one hand, a direct determination of the rotation period can be obtained by the analysis of the light curve, either in the time domain, or by studying the lowfrequency part of the temporal power spectrum. In addition, starspot modeling can also provide additional information such as the rotation inclination angle (e.g. Mosser et al. 2009a; Lanza et al. 2014). On the other hand, seismology yields a weighted average of the internal rotation heavily biased towards the surface during the MS, as well as the rotation inclination angle. In the next two sections, we will describe in more detail these two types of analyses.
7.1 Photospheric rotation from the study of the photometric light curves
The crossing of cool spots over the visible disk of a star produces a modulation in the photometric signal which is proportional to the rotation period of the star, \(P_{\mathrm{rot}}\), at the latitudes where spots and active regions exist. Several works have been dedicated to the study of the extraction of these rotation periods for CoRoT, Kepler and the K2 missions. Thus, in the last decade, it has been possible to retrieve the rotation periods of thousands of stars (e.g. McQuillan et al. 2013; Nielsen et al. 2013; Reinhold et al. 2013; Walkowicz and Basri 2013; McQuillan et al. 2014; Leão et al. 2015).
There are several different—but complementary—ways to extract the information on the rotation period from light curves. In this section, we do not pretend to provide an exhaustive review of all of the techniques and results on this topic; we focus mostly on what has been done related to asteroseismic studies of MS solarlike stars.
The careful study of the lowfrequency part of the power spectrum can be used to extract the rotation period by selecting the highest peak in this frequency region (e.g. Barban et al. 2009; Campante et al. 2011; Nielsen et al. 2013). However, sometimes the highest peak can be the second or even the third harmonic of the rotation period instead of the first corresponding to the true rotation period. An example is given in Fig. 30, where the light curve and the longer periods of the period–power spectrum of KIC 4918333 is shown. On the lefthand panel, the first 310 days of the light curve are analyzed and the tallest peak corresponding to a rotation period of \(9.8 \pm 0.8\) days is shown. A second peak is also visible at a period of \(19.5 \pm 1.2\) days but with a smaller amplitude. When this analysis is extended to 1459.5 days, the period of \(19.5 \pm 1.2\) days is the most prominent one without ambiguity. Hence, when analyzing the rotation period from the study of the power spectrum, it is important to check any possible signal at twice the period corresponding to the highest peak.
Since the different methods used to calibrate the data can filter the stellar signal in different ways, while leaving other instrumental signals in the final light curves, it is recommended to use several types of data calibrations when studying the rotation of a star (e.g. García et al. 2013a, 2014a; Ceillier et al. 2016; Buzasi et al. 2016). It has also been shown that each method to extract the rotation (e.g. ACF, time–period diagrams, the direct analysis of the lowfrequency part of the spectrum) works better in some circumstances or for some type of stars. Hence, the most reliable procedures to retrieve the rotation periods are those that combine different calibration procedures and analysis techniques (Aigrain et al. 2015).
The average rotation is not the only quantity that can be inferred from the analysis of the light curves. Differential rotation in latitude \({{\Delta \Omega }}\) can also be obtained in some cases (e.g. Fröhlich et al. 2012; Reinhold and Reiners 2013; Lanza et al. 2014; Reinhold and Gizon 2015), by directly studying the number and structure of the peaks in the lowfrequency part of the spectrum or by performing starspot modeling (e.g. Croll 2006; Fröhlich 2007; Nielsen et al. 2013; Walkowicz et al. 2014; Lanza et al. 2016). Thanks to these analyses, general trends have been found. For example, the dependence of \({{\Delta \Omega }}\) with the rotation period is weak and it slightly increases with \(T_{\mathrm{eff}}\) in the range 3500–6000 K (Reinhold et al. 2013), while the relative differential rotation, \({\Delta \Omega /\Omega }\), increases with the rotation period (Reinhold and Gizon 2015). In addition, Reinhold and Arlt (2015) were able to discriminate solar and antisolar differential rotation (i.e., to identify the sign of the differential rotation at the stellar surface) using peakheight ratios of the first harmonics of the differential rotation peaks. However, a theoretical and numerical study by Santos et al. (2017) showed that the peakheight ratios are essentially a function of the fraction of time the spots are visible. This time is related to how strongly the spot modulation follows a sinusoidal form. Hence, depending on the rotation inclination angle with respect to the line of sight and on the location of the spots, the inferred sign of the differential rotation can be wrong.
7.2 Internal rotation through asteroseismic measurements
As we have already said in this review, only acoustic modes have been characterized in mainsequence solarlike dwarfs so far. Therefore, the information that one can obtain from seismology comes from the careful analysis of acoustic modes. Unfortunately, although lowdegree p modes penetrate deep in the stellar interior, they spend only a small fraction of their time in the deep interior and the amount of information that they can provide from these regions is small. This is illustrated in Fig. 33 where we show the rotational kernels, \(K_{n,\ell } \), of the dipolar and quadripolar modes of radial orders \(n=10\) and \(n=25\) for three planethosting stars: Kepler25 (KIC 4349452, \(M=1.26 \pm 0.03\,M_\odot \), Benomar et al. 2014), HATP7 (KIC 10666592, \(M \sim 1.59 \pm 0.03\,M_\odot \), Benomar et al. 2014) and the Sun. The two radial orders \(n=10\) and 25 cover the typical observational range for theses stars. In spite of the different masses of the three stars and the different position of the base of the convective zone, the kernels of these modes are nearly identical (see also Lund et al. 2014). However, the position of the base of the convective zone depends on the mass of the star. Therefore, more massive stars have shallower convective zones and thus, they have a larger contribution from the radiative zone to the average internal rotation rate. It is, however, possible to extract some general properties. These kernels have larger amplitudes near the surface. They are also denser in these outer regions implying that the waves spend more time there than in the inner layers. As a consequence, the sensitivity to the rotation is larger towards the surface of the star. Moreover, the kernels in the radiative zone are almost linear functions of the radius above \(\sim \,0.15\,R_\odot \) up to the base of the convective zone. Therefore, each observed mode probes the radiative zone uniformly.
The first consequence is that the rotation rate extracted from seismology and the rotation period extracted from the surface (either from the study of the photometric variability or from spectroscopy once the inclination angle is known) should be similar as already explained when described the first asteroseismic detection of the rotation in HD 52265 (see Fig. 29). Interestingly, any significant difference between the surface and the asteroseismic rotation rate should indicate the existence of differential rotation, either from the surface or from the external convective zone inside the stars. Gizon et al. (2013) found that both determinations of the rotation agreed within one sigma in all cases, suggesting that the differential rotation should be weak in these stars.
Based on similar studies of the rotational kernels but using inversion methods, Schunker et al. (2016a) suggested that it could be interesting to use many stars in order to reduce the observational errors and be able to, for example, constrain the sign of the radial differential rotation. Moreover, it has been demonstrated using an “ensemblefit” of 15 stars across the main sequence, that it would be possible to distinguish between solid rotation and radial differential rotation of around 200 nHz using observable splittings of angular degrees 1 and 2 (Schunker et al. 2016b).
It is important to note that only one star, KIC 9139163, significantly departed from the onetoone relation shown in Fig. 35, offering a scenario where the interior could be spinning much faster than the surface. Interestingly, this star is among the youngest and more massive of the sample suggesting a scenario where the angular momentum transport processes that are responsible for the quasiuniform internal rotation might not have had enough time to complete their work.
8 Stellar magnetic activity and magnetic cycles
In distant stars, as was the case for rotation, asteroseismic observations provide two different, but complementary, ways to study magnetic activity in general and magnetic activity cycles in particular. On the one hand, magnetic variability can be measured on different time scales by directly analyzing the average luminosity flux modulation in the light curve or its fluctuation as a function of time (e.g. Basri et al. 2010, 2011; García et al. 2014a). It is out of the scope of this review to describe in details the methodologies and results associated to these studies. However, in the next paragraph we will provide a brief overview of them. On the other hand, magnetic variability can be studied through the characterization of the temporal evolution of the oscillation modes, i.e., measuring the variations of the frequencies, amplitudes, and line widths of the acoustic modes.
As discussed in the section about internal rotation, to reach the core, mixed or gravity modes are required. Unfortunately, those modes have not been unambiguously detected in mainsequence solartype dwarfs. Therefore, nothing can be said about the magnetic field in the core of these stars. However, studying red giant stars, Stello et al. (2016) evoked the possibility of having magnetic fields of dynamo origin in the convective core of stars with masses greater than \(\sim \,1.2\,M_\odot \). The origin of the depressed dipolar modes seen in Kepler observations (e.g. Mosser et al. 2012; García et al. 2014c) has been suggested to be magnetic, although the exact nature of phenomena is still debated (e.g. Fuller et al. 2015; Mosser et al. 2017; Loi and Papaloizou 2017, 2018).
8.1 From the direct analysis of the light curves
An example of the study of the surface dynamics (rotation and magnetism) using Kepler photometry is given in Fig. 37. In the top panel the relative measured flux is given. The envelope of the surface brightness shows an increase between the day \(\sim \) 550 and \(\sim \) 1200 of the mission that can be interpreted as an increase of the magnetic activity of the star. In the middle panel of Fig. 37 a time–period diagram is computed. It shows two main bands of power (depicted by the yellow horizontal dashed lines) at 3 and 2.54 days corresponding to the main average rotation rate of the star during these two seasons. Projecting the time–period diagram onto the time domain around the periods given before (from 2 to 6 days), a magnetic proxy can be built (see bottom panel of Fig. 37), confirming the existence of an ongoing stellar activity cycle, with a season of maximum activity where the starspots are located at longitudes of faster rotation (2.54 days), while the band at a slower rotation rate is dominating the location of the spots during the minimum of magnetic activity. This behavior is similar to that observed in the Sun which produces the socalled butterfly diagram (e.g. Hathaway 2015).
8.2 From asteroseismology
To study magnetic activity cycles with asteroseismic techniques, it is important to use the Sun as a reference because we can characterize in detail its surface magnetism and look for correlations with different observed features that are today impossible to obtain for distant stars. In this way, we can then apply this knowledge to other stars. At the very beginning of helioseismology, a correlation between the acousticmode frequencies and solar magnetic activity was found (van der Raay 1984; Woodard and Noyes 1985; Fossat et al. 1987; Pallé et al. 1989; Elsworth et al. 1990), i.e., the frequencies shifted towards higher values as the 11year magnetic cycle progressed. Later, it was discovered that the frequency shifts increased with frequency for intermediate and lowdegree modes (Libbrecht and Woodard 1990; Anguera Gubau et al. 1992), i.e., modes at higher frequency had a larger frequency shift than modes at low frequency. This frequency dependence—highfrequency modes have outer turning points compared to lowfrequency modes—is the main change of the mode parameters when the effect of the mode inertia is removed. Considering also that there was no significant dependence of the frequency shifts with the degree of the modes led to the conclusion that the perturbations related with the magnetic activity cycle were confined to a thin layer very close to the photosphere (e.g. Libbrecht and Woodard 1990; Goldreich et al. 1991; Nishizawa and Shibahashi 1995; Basu et al. 2012). Moreover, the Sun has not only a 11year periodicity. Shorter—quasibiennial—modulations have been measured in the Sun (e.g. Benevolenskaya 1995) in several magnetic activity proxies and also confirmed by seismology (Fletcher et al. 2010; Broomhall et al. 2011b; Simoniello et al. 2012). The existence of several time scales in the modulation of the magnetic proxies is not peculiar to the Sun and many other stars show several long and short periods that could be interpreted as magnetic cycles with an “active” and “inactive” phase (e.g. Baliunas and Soon 1995; Brandenburg et al. 1998, 2017; Saar and Brandenburg 1999; BöhmVitense 2007). Last but not least, important differences were found between the surface magnetic proxies and the frequency shifts during the last extended minima between Solar Cycles 23 and 24 (Broomhall et al. 2009; Salabert et al. 2009). While no activity was measured in standard magnetic proxies, the frequency shifts showed a quasi normal behavior leading to the conclusion that the magnetic perturbations in the subsurface layers were still strong (Salabert et al. 2015).
One exception to this is the characterization of lowdegree loworder p modes for which the variations induced by the magnetic activity cycle are too small, around \(\varDelta _\nu /\nu \sim 10^{5}\) nHz for the solar case (e.g. García et al. 2001).
The relation between magnetic activity cycle periods and stellar evolution is also well known (e.g. Brandenburg et al. 1998). Recently, Brandenburg et al. (2017) revisited this relation in light of the most recent and longest spectroscopic observations from Egeland (2017) who combined datasets from several instruments. Unlike previous interpretations where young stars would evolve along the active A branch, they now believe that all stars younger than 2.3 Gyr are capable of exhibiting longer and shorter cycle periods. If their calculations are correct, for the G pulsating dwarfs HD 76151 and KIC 10644253 shown in Fig. 39, longer periods in the 12–16 year range are expected, and may have already been found in HD 76151 (Egeland 2017) and in \(\iota \) Horologii, HD 17051, where Flores et al. (2017) measured a longterm activity cycle of about five years fitting the “active” branch in the BöhmVitense diagram. For the solar analogue 18 Sco, it would be interesting to discover either a second shorter modulation as in the Sun or a longer cycle period.
An interesting result was obtained by Mathur et al. (2014a) from the analysis of the temporal evolution of \(S_{\mathrm{ph}}\). They selected 22 solarlike stars with detected solarlike oscillations and with rotation periods shorter than 12 days, i.e., the limit to observe at least half of a cycle from Fig. 39. Only two of the stars show a cycliclike variation, while two others showed a decreasing or increasing trend in the \(S_{\mathrm{ph}}\) temporal variation. Five stars show modulations (or beating) due to longlived spots at two different active longitudes with different rotation rates. The rest of the stars show no cyclelike behavior although they showed surface magnetic activity. Therefore, although it is premature to infer firm conclusions from a so small sample of stars, it seems that fast rotating stars can exhibit magnetic activity but without any magnetic cycle (or at least much longer than it could be expected from Fig. 39. The only correlation found was between the \(S_{\mathrm{ph}}\) and the rotation period for stars showing a beating between long lived spots at different rotation rates, in the direction of higher \(S_{\mathrm{ph}}\) for longer \(P_{\mathrm{rot}}\).
To perform asteroseismic studies of magnetic activity, it is then necessary to first measure the oscillation properties and then study their evolution with time. As it has been said, solar magnetism reduces the amplitude of the solar modes. Therefore, a very active star would probably have oscillation modes with very small or even not detectable amplitudes. This effect was first observed by CoRoT in two mainsequence targets: HD 175726 (Mosser et al. 2009b) and in HD 49933 (García et al. 2010). Later, using a Kepler sample of mainsequence stars, Chaplin et al. (2011a) showed that there was a clear correlation between stellar magnetic activity and the amplitude of the stellar oscillations. The most active stars in the sample did not show measurable oscillation modes.
To look for the evolution with time of the seismic properties, the light curve is divided into small segments for which the characterization of the oscillation modes is carried out. The length of the subseries is found as the best tradeoff between frequency resolution and the number of subseries to be analyzed. The longer the series the better the precision on the extracted parameters (e.g. Régulo et al. 2016). However, it can be very challenging to obtain individual pmode frequency shifts that can be as small as half a \(\upmu \)Hz for short time series. This is why a global method was developed in the early days of helioseismology by Pallé et al. (1989) to obtain averaged frequency shifts by computing the cross correlation of the pmode hump computed from the PSD of each subseries in comparison to a reference one. This reference can be either the PSD of one of the subseries or the average spectrum of all of them (e.g. Pallé et al. 1989; García et al. 2010; Régulo et al. 2016; Salabert et al. 2016b; Kiefer et al. 2017; Santos et al. 2018).
The first attempts to detect differences in the seismic parameters associated with magnetic activity were reported by Fletcher et al. (2006). They analyzed observations made with the startracker on the WIRE satellite of \(\alpha \) Cen A and compared the obtained frequencies with previously obtained frequencies measured by Bouchy and Carrier (2002) and Bedding et al. (2004). Fletcher et al. (2006) conjectured that the average difference of about \(0.6 \pm 0.3\,\upmu \mathrm{Hz}\) (\(\sim \,2\sigma \)) in the oscillation frequencies could be due to an ongoing activity cycle in \(\alpha \) Cen as the WIRE observations were taken 19 months before the others.
The same methodology was applied to three other CoRoT targets complemented by groundbased analysis of the chromospheric magnetic activity done with the NARVAL spectropolarimeter located at the Bernard Lyot 2 m telescope at the Pic du Midi Observatory (Aurière 2003) in order to study any possible hint of magnetic activity cycles (Mathur et al. 2013b). Interestingly one star, HD 181420 (Barban et al. 2009), seems to be in a stationary regime without any visible change of the activity during the observations. This is an unexpected result because this star rotates rapidly (2.6 days) and we were expecting to see some indications of a magnetic activity cycle. For the other two stars, HD 49385 (Deheuvels et al. 2010a) shows a small increase of activity at a 1\(\sigma \) level but not confirmed by our spectroscopic measurements, while HD 55265 (Soriano et al. 2007; Ballot et al. 2011b) presents a small variation of the seismic parameters, also at a 1\(\sigma \) level, and in the spectroscopic observations performed by NARVAL. That could indicate that this star was observed during the rising phase of a long magnetic activity cycle.
8.2.1 Influence of metallicity on magnetic activity
The longer observation period of the Kepler main mission (up to four continuous years) allows one to better track temporal changes in the seismic parameters for mainsequence solarlike stars. This is illustrated in Fig. 43 where the temporal evolution of the seismic parameters and the photospheric and chromospheric activity proxies are shown for KIC 8006161 (HD 173701) and for the Sun. All of the figures are in the same scale for both stars. KIC 8006161 is a very interesting target because it is a solar analogue (Karoff et al. 2018) with \(M=1.00 \pm 0.03\,M_\odot , R=0.93 \pm 0.009\, R_\odot \), \(T_{\mathrm{eff}} = 5488 \pm 77\ \mathrm{K}, P_{\mathrm{rot}} =21 \pm 2\ \mathrm{days}, \mathrm{age} = 4.57 \pm 0.36\ \mathrm{Gyr}\), and a metallicity that is twice the solar value (\(0.3 \pm 0.1\ \mathrm{dex}\)).
KIC 8006161 has a comparable magnetic activity cycle period (\(\sim \) 7.4 year) deduced from more than thirty years of chromospheric observations (top plot in Fig. 43), being in the rising phase of its cycle during the four years of Kepler observations. Interestingly, the amplitude of all the temporal variations recorded are much larger than the corresponding solar values. Karoff et al. (2018) conjectured that these differences could be a consequence of the higher metallicity of the star. An increase of the stellar metallicity produces larger opacities and thus a larger internal temperature gradient. Therefore, the Schwarzschild criterion for convection (Schwarzschild 1906) is satisfied deeper inside the star leading to a deeper convective zone (van Saders and Pinsonneault 2012). Theoretical studies and numerical simulations have shown that larger convective zones induce larger differential rotation (Brun et al. 2017) and thus a stronger magnetic dynamo (Bessolaz and Brun 2011). Although it is not possible with the present observations to infer how strong the dynamo is, the surface differential rotation is larger than that in the Sun, reinforcing the conclusions. Unfortunately, firm conclusions cannot be derived with only one observations and the study of a larger sample of stars with different metallicities is necessary to completely understand the influence of metallicity on the length and strength of the magnetic activity cycles.
8.2.2 Relation between the frequency shift strength with effective temperature and age
8.2.3 Relation between frequency shifts and amplitude shifts for a large sample of stars
The analysis of the temporal variations of the seismic parameters and \(S_{\mathrm{ph}}\) of a larger sample of 87 solartype Kepler stars (Santos et al. 2018), showed similar results to those of Kiefer et al. (2017). They found that about 60% of the stars in the sample show “(quasi)periodic variations” in the frequency shifts. Moreover, 20% of the stars show frequency and amplitude shifts correlated instead of anticorrelated. Although these results seem to be puzzling, they could be explained in a simple way. First, Salabert et al. (2018) showed that small variations in frequency shifts—that could be interpreted as due to magnetic origin—can be explained by different realizations of stochastic noise. Second, the presence of hysteresis effects between different magnetic proxies implies that at several stages of the cycle, two indexes could be in phase while they would be most of the time in antiphase. Longer time series covering several stellar magnetic cycles would be required to properly understand all of these observations.
8.3 On the variation of the frequency shifts with frequency
The linear dependence of the frequency shifts with frequency found in the Sun and in HD 49933 was also found in the young solar analogue KIC 10644253 (Salabert et al. 2016b). Therefore, in these three stars the perturbation inducing the variation of the mode parameters needs to be located outside of the resonant mode cavity of the modes, i.e., in a thin layer very close to the photosphere. Otherwise, an oscillatory signal would be expected as was first discussed by Goldreich et al. (1991). They explained the apparent oscillatory signal superimposed on the linear dependence of the frequency shifts with frequency in the Sun as the consequence of a perturbation located near the He i ionization zone. This depth was found by analyzing the periodicity of this oscillatory signal. Later, Gough (1994) proposed that the perturbation was much deeper and that it was due to changes in the acoustic glitch of the He ii ionization layer. Indeed several authors have found variations in the amplitude of the depression in the adiabatic index, \(\varGamma _1\), at this ionization layer that could be the consequence of the changing activity on the equation of state of the gas in that layer (Basu and Mandel 2004; Verner et al. 2006).
9 Conclusions and perspectives
In this work we have reviewed the theory behind the asteroseismic techniques applied to study mainsequence solarlike dwarfs, as well as the latest results obtained from groundbased and spaceborne missions such as CoRoT, Kepler, K2 and TESS. This is a young research topic that has reached its constant pace during the last decade.
It has been shown that asteroseismology of MS solarlike stars is providing strong constraints about the structure and dynamics from the surface to the core of stars. It is possible today to infer masses, radii and ages with a precision and accuracy never reached before. This is impacting many fields in stellar astronomy. For example, precise and accurate ages of field stars at all stages of evolution in the main sequence have shown that stars seem to stop braking when they reach a given Rossby number and from there, they seem to continue with a quite constant surface rotation (van Saders et al. 2016). This could be the consequence of a change in the properties of external magnetism, which could also impact stellar magnetic cycles (Metcalfe and van Saders 2017) and the possibility of giving a correct age to middle aged stars through gyrochronology. An extended sample of stars and complementary groundbased observations of the magnetic activity will be necessary to progress in this area.
Asteroseismology is also helping to improve exoplanet research (for a full review on the synergies between asteroseismology and exoplanetary science see Huber 2018). For example, precise stellar ages are a key parameter to date the full planetary systems and thus, better understand the theory of formation and evolution of planethosting stars and the extrasolar planet systems as a whole. The asteroseismic improvement in the determination of stellar radius is directly impacting the precision of planet radius extracted with the transit method. Thanks to this new increased precision, it is now possible to properly characterize the socalled “radius valley” or “photoevaporation desert” at around 2 \(\hbox {R}_\oplus \) (Lundkvist et al. 2016; Van Eylen et al. 2018a). Finally we mention that the combination of transit photometry with asteroseismology allows a systematic measurement of orbital eccentricities of transiting planets (e.g. Van Eylen and Albrecht 2015), which was only possible before in relatively large gasgiant planets, or for multiplanet systems where the effects of eccentricities and masses could be successfully distinguished,
New internal rotation profiles have encouraged stellar astrophysicists to study angular momentum transport and efficient mixing processes and develop new mechanisms explaining the quasiuniform rotation found in the outer part of the radiative zone, the external convective zone and the surface.
In conclusion, asteroseismology of solartype stars is in very good shape. At the time of writing these conclusions, the community is actively analyzing K2 data, where dozens of new pulsators are foreseen, as well as the first two sectors of TESS data that are already available. New missions will contribute to enhance our known sample of pulsating star. The future is already here because of the work engaged to prepare the ESA’s M3 PLATO mission, which will be able to characterize tens of thousand of these MS cool dwarfs after 2026 for which many of them will be planet hosts. Asteroseismology of MS solarlike dwarfs is just at the dawn of its potential.
Notes
Acknowledgements
The authors wish to thank the entire SoHO, CoRoT and Kepler teams, without whom many of the results presented in this review would not be possible. The authors received funding from the European Community seventh programme ([FP7/20072013]) under Grant Agreement No. 312844 (SPACEINN) and under Grant Agreement No. 269194 (IRSES/ASK). The authors also acknowledges funding from the CNES. RAG acknowledges the ANR (Agence Nationale de la Recherche, France) program IDEE (No. ANR12BS050008) “Interaction Des Étoiles et des Exoplanètes”. The authors also want to thank John Leibacher who did a thorough lecture of the manuscript helping us to improve the manuscript.
Supplementary material
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