inferences on stellar activity and stellar cycles from asteroseismology

Space Sci RevDOI 10.1007/s11214-014-0090-2

Inferences on Stellar Activity and Stellar Cyclesfrom Asteroseismology

William J. Chaplin · Sarbani Basu

Received: 25 March 2014 / Accepted: 19 August 2014© Springer Science+Business Media Dordrecht 2014

Abstract The solar activity cycle can be studied using many different types of observations,such as counting sunspots, measuring emission in the Ca II H&K lines, magnetograms, radioemissions, etc. One of the more recent ways of studying solar activity is to use the changingproperties of solar oscillations. Stellar activity cycles are generally studied using the Ca IIlines, or sometimes using photometry. Asteroseismology is potentially an exciting means ofstudying these cycles. In this article we examine whether or not asteroseismic data can beused for this purpose, and what the asteroseismic signatures of stellar activity are. We alsoexamine how asteroseismology may help in more indirect ways.

Keywords Stars: activity · Stars: oscillations

1 Introduction: Observing Stellar Cycles

Magnetic fields are ubiquitous in the universe. Effects of magnetic fields are detected acrossvirtually the whole of the Hertzsprung–Russell diagram. Observations indicate that the pres-ence of magnetic activity is common in solar-type stars (e.g., Radick 1994), i.e., cool main-sequence and sub-giant or “late-type” stars. Magnetic effects are also common in early-typestars, such as those of spectral types O, B and A (e.g., Linsky 1999).

Solar magnetic activity and magnetic-activity cycles can be observed in many differentways, with the most obvious being the rise and fall of the number of sunspots on the solar

W.J. Chaplin (B)School of Physics and Astronomy, University of Birmingham, Edgbaston, B15 2TT Birmingham, UKe-mail: [email protected]

W.J. ChaplinStellar Astrophysics Centre (SAC), Department of Physics and Astronomy, Aarhus University,Ny Munkegade 120, 8000 Aarhus C, Denmark

S. BasuDepartment of Astronomy, Yale University, P.O. Box 208101, New Haven, CT 06520, USAe-mail: [email protected]

mailto:[email protected]

mailto:[email protected]

W.J. Chaplin, S. Basu

surface. Detailed magnetograms, such as those from the Kitt Peak National Observatory,also reveal the solar cycle quite clearly, as do other activity-related proxies such as the radioflux at 10.7 cm. Observing activity in stars is admittedly more difficult. The most popularway is to look at the cores of Ca II H&K absorption lines. What makes these lines interestingis that their cores show emission, and the emission is correlated with activity. Observationsof these lines for the Sun clearly show the 11-year solar cycle (see Baliunas et al. 1995,Fig. 1d). Thus these chromospheric lines can be used to monitor stellar activity.

One of the first papers looking at chromospheric variations in main-sequence stars wasthat of Wilson (1978) based on observations that started in 1966. Since then, there have beenother surveys of stellar cycles, such as the current Mt. Wilson HK project (Duncan et al.1991), the Vienna-Kitt Peak National Observatory Ca II H&K survey (Strassmeier et al.2000), the surveys done at the Lowell Observatory (Hall et al. 2007) etc., targeting northernstars, and the surveys by Henry et al. (1996), Gray et al. (2006), Jenkins et al. (2006) etc.,for southern stars. What the data show is that some stars show stellar activity cycles (seee.g., Baliunas et al. 1995), like the Sun; whilst some others may be in a Maunder-minimumlike state, showing no detectable variability.

Observations showing that changes in solar irradiance are correlated with solar rotationand solar activity led to campaigns of high-precision photometry of stars (see e.g., Lock-wood et al. 1997; Radick et al. 1998). These campaigns were indeed able to observe stellarvariability in photometry. However, Lockwood et al. (1997) concluded that the chromo-spheric activity produces a stronger and more easily detected signal than photometric vari-ability. This was not really surprising, for two reasons: the first is that in the case of the Sunthe Ca II K emission varies by more than 20 % over a solar cycle, while the smoothed irra-diance changes by less than 0.1 % and hence we should expect something similar in otherstars. The second reason is that the Lockwood et al. (1997) results were from ground-basedobservations and atmospheric turbulence increases noise. The best means of measuring pho-tometric variability is through space missions. NASA’s Kepler mission and the CNRS/ESACoRoT satellite have helped immensely in this regard.

Kepler (Koch et al. 2010) observed a single field in the Cygnus–Lyra region for about4 years. In total it monitored just under 200,000 stars and for most determined their bright-ness every 30 minutes, though a subset of stars were observed with a cadence of 1 minute.Although the primary goal of Kepler is to detect extra-solar planets through transit measure-ments, the extreme precision of Kepler photometry has led to other scientific investigationsincluding stellar variability. Gilliland et al. (2011) examined Kepler photometry and con-cluded more than half of the solar-type stars in the Kepler field were more active than theSun. Although the result that the Sun is less active than other solar-type stars was disputedby Basri et al. (2013), it is clear that photometry can detect stellar activity. We cite moreexamples later in Sect. 4.

In the case of the Sun, the activity cycle is not only present in chromospheric and photo-metric measurements, but we also see the cycle in the frequencies of the solar oscillations.It is now well established that the frequencies of the Sun’s p modes increase with activ-ity (Woodard and Noyes 1985; Elsworth et al. 1990; Libbrecht and Woodard 1990), and asshown in Fig. 1, they clearly show the 11-year cycle as well. The frequency shifts are alsocorrelated with other solar activity indices such as the International Sunspot Number andthe 10.7-cm radio flux (see e.g., Jain et al. 2000; Chaplin et al. 2007b).

The trend observed in the frequency shifts of the most prominent modes—the higher thefrequencies, the larger are the shifts—is a tell-tale sign that it is the impact of the near-surfacemagnetic activity that is responsible for the shifts. However, uncertainty remains over how

Inferences on Stellar Activity and Stellar Cycles

Fig. 1 Changes in solar oscillation frequencies as a function of time. We show the average shift for modes inthe frequency range of 2500 to 3500 µHz, obtained from observations made by the Birmingham Solar-Oscil-lations Network (BiSON). The frequency-shifts were calculated by subtracting the average frequency of themodes over the entire observation period from the frequency at any given epoch. This results in both positiveand negative shifts. (Figure courtesy of A.-M. Broomhall)

much of the observed shifts can be explained as being due to sunspots, active regions, orplage, and the relative influence of the direct and indirect action of the magnetic fields (seee.g., Dziembowski and Goode 2005; Thompson 2006).

The magnetic fields may act directly on the modes, via the Lorentz force: this providesan additional restoring force, the result being an increase of frequency, and the appearanceof new modes. Magnetic fields can also act indirectly by affecting the physical properties inthe mode cavities and, as a result, the propagation of the acoustic waves within them. Thisindirect effect can act both ways, to either increase or decrease the frequencies.

Now that there are seismic observations for other stars, the question we would like toanswer is whether we could learn about stellar activity and activity cycles from asteroseis-mology? We try to answer that question in the following sections.

2 Could We Get Asteroseismic Measures of Stellar Activity?

There are two issues that could hinder asteroseismic studies of stellar activity. The firstis that asteroseismic observations do not cover a long time-period. Even with Kepler, thelongest span of observation for any given star is about four years; observations of manysolar-type stars cover a shorter period. The second issue is whether activity-related variationsin asteroseismic properties can be detected at all.

2.1 The Issue of the Period Length

It has been increasingly clear for some time that many stars have fairly short cycles(see e.g., Baliunas et al. 1995; Saar and Brandenburg 1999). Stellar activity data sug-gest that there is a correlation between the rotation period of a star and the lengthof the cycle, with faster rotators having shorter activity cycles (Baliunas et al. 1996;Oláh and Strassmeier 2002, etc.). Thus, faster rotating stars would seem to be good can-didates for asteroseismic observations. The relationship between the rotation period andcycle-length is, however, complicated. Noyes et al. (1984), using dynamo models, had sug-gested that the cycle-period should be a function of the Rossby number, Ro, where Ro


Fig. 2 A modified version of Fig. 1 of Böhm-Vitense (2007) showing the relationship between a stellarrotation period and activity cycle period. Points in blue are stars with one known cycle period. Points in redare stars with known, measured secondary cycle periods, and each star is shown with a unique symbol. Addedto the data of Böhm-Vitense (2007) are those for ι Hor (Metcalfe et al. 2010, in green) and ε Eri (Metcalfeet al. 2013, in pink). The Sun is marked in orange—both 11-year and 2-year periods are marked for the Sun.The upper branch is the so-called “active” branch, the lower is the “inactive” branch. Note that for many starswith secondary periods, the two different periods lie on the two different branches

∝ Prot/τc , Prot being the rotation period and τc the convective turnover time-scale. Whileexamining the correlation between stellar activity cycles and the Rossby number, Branden-burg et al. (1998) showed that the stars could be separated into two branches with a markedseparation at log(Prot/Pcyc) ∼ −2.3, one branch defining a young/active cohort and the otheran old/inactive cohort. The designations “active” and “quiet” were made using the amountof emission in the Ca II H&K bands. The “inactive” stars have a higher log(Prot/Pcyc) ratioat a given Rossby number than do the “active stars.”

The issue of the “active” and “inactive” branches gained more attention when Böhm-Vitense (2007) showed that stars could be separated into the branches even in the Pcyc–Prot

plane. More interesting is where stars with secondary cycles lie. Many stars are known tohave two distinct cycles, a longer primary cycle and a shorter secondary cycle (Baliunaset al. 1995; Saar and Brandenburg 1999; Oláh et al. 2009; Metcalfe et al. 2013). We showa modified version of Fig. 1 of Böhm-Vitense (2007) in our Fig. 2. What is interestingis that for stars with two cycles, the longer cycle falls on one branch, and the shorter cy-cle on the other. The shorter secondary cycles should be very amenable to asteroseismicstudies; the important question of course is whether the cycle will be detectable in seismicdata.

One of the striking features of the Böhm-Vitense (2007) plot is that the Sun does not fallon either of the two branches—the 11-year cycle lies mid-way between the two. The Sun tooappears to exhibit a secondary cycle. One can detect a shorter period of about 2 years in solaroscillations data (Fletcher et al. 2010; Broomhall et al. 2012). The two-year modulation inthe solar frequencies can be seen in Fig. 1.


2.2 Asteroseismic Predictions

Whether or not asteroseismic data can reveal anything about stellar activity cycles dependsto a large extent on: first, whether oscillations are observable; and second, if oscillations areobservable, whether their characteristics show solar-cycle related changes.

The first point is an important one. Solar observations have consistently shown that am-plitudes of oscillations decrease as solar activity increases (e.g., Komm et al. 2000). This hasalso been demonstrated on local scales by looking at oscillation-amplitudes inside and out-side sunspots (e.g., Braun et al. 1988). Thus it is quite possible that stellar activity will hinderthe detection of stellar oscillations. Chaplin et al. (2011a) had investigated the detectabilityof stellar oscillations as a function of the global properties of the stars, using data fromKepler. They showed that there were fewer asteroseismic detections than were predicted,and presented evidence to suggest that the most plausible reason for the lack of detectionsis elevated levels of stellar activity. These results have since been confirmed, using data ondirect measurements of oscillation amplitudes in a cohort of Kepler stars (Campante et al.2014). Of course, Kepler and CoRoT observations have led to the detection of oscillationsin many stars (see Sect. 3 below), thus while high levels of activity may lead us to miss afew stars, the cohort of stars with detections offers many promising targets for stellar-cyclestudies.

Metcalfe et al. (2007) derived a way to predict asteroseismic signatures of stellar activityusing the Mg II activity index. They assumed that the relation between the solar Mg II ac-tivity index and changes in solar oscillation frequencies could be applied to all stars. Solarfrequency changes as a result of the solar cycle are a function predominantly of frequency,once the effect of mode-inertia is removed. This has led to the general idea that the sourceof solar-cycle induced frequency changes lies close to the solar surface (e.g., Nishizawa andShibahashi 1995). Metcalfe et al. (2007) thus assumed that the same would apply to otherstars, and that a scaling similar to the solar one between the Mg II index and the changes instellar frequencies could be applied. They tested this on ground-based asteroseismic obser-vations of β Hyi at two different epochs.

Chaplin et al. (2007a, 2008a) used a different approach and tried to determine the pre-cision with which it will be possible to extract activity-related shifts in asteroseismic dataof solar-type stars along the lower main sequence. They used models of damping rates ofstochastically excited p-modes based on non-local mixing length formulations and usedthe averaged shift across the highest-amplitude peaks to reduce uncertainties. They madepredictions of the expected frequency shifts based on existing stellar Ca II H&K data andconcluded that basic properties of the asteroseismic signatures of stellar cycles could bemeasured with a few multi-month segments of data.

Chaplin et al. (2007a) also found that it should be possible to make inferences on thedistribution of activity on stellar surfaces, as we now go on to explain. When the near-surfacemagnetic activity is distributed in a non spherically symmetric manner, i.e., into active bandsof latitude, it will give rise to acoustic asphericity, i.e., the magnitudes of the frequencyshifts will depend on the angular degree l and the azimuthal order m of the mode. Hence,measurements of the relative sizes of the shifts shown by low-degree modes having different(l, |m|) combinations may be used to place constraints on the active latitudes shown by asolar-type star. It is worth noting that the stellar rotation may also contribute to asphericity,even for moderate rates of rotation. The two effects can also be difficult to disentangle (e.g.,see discussion in Gizon 2002). Here, we consider just the impact of near-surface magneticactivity.


Fig. 3 Predicted frequency shifts δνlm of different azimuthal components, relative to the radial-modeshift δν00, as a function of maximum latitude of magnetic activity, λmax (see text for explanation ofmodel of activity). Linestyles: dotted for δν10/δν00; short dashes for δν11/δν00; dot-dashed for δν20/δν00;dot-dot-dot-dashed for δν21/δν00; and long dashes for δν22/δν00

If the active bands lie at low latitudes, as for the Sun, the largest shifts of non-radial,low-degree modes (i.e., l = 1, 2 and 3) will be those shown by the sectoral (l = |m|) com-ponents. Analysis of Sun-as-a-star data has uncovered the expected variations in the sizesof the frequency shifts (e.g., see Chaplin et al. 2004a; Jiménez-Reyes et al. 2004). Here, weuse a simple toy model to predict the relative sizes of the average frequency shifts of var-ious low-l components, relative to the radial-mode shifts, for different assumed latitudinalbands of near-surface stellar activity. These predictions are based first and foremost on theassumption that changes in the sound speed contributing to the frequency shifts are locatedvery close to the surface. In what follows we assume that we average frequency shifts overseveral radial orders, hence the frequency dependence of the shifts does not appear explicitlybelow. However, that dependence may also in principle be calculated, given a set of modeeigenfunctions and, as noted above, under the assumption that the magnetic perturbationsare located close to the surface.

The expected (radial-order averaged) shifts, δνlm, may be approximated (e.g., Moreno-Insertis and Solanki 2000) by

δνlm ∝(

l + 1

2

)(l − m)!(l + m)!

∫ π

0

∣∣P ml (cos θ)

∣∣2B(θ) sin θ dθ, (1)

where P ml (cos θ) are Legendre polynomials, and B(θ) describes the distribution of the ac-

tivity as a function of co-latitude, θ . Following Chaplin et al. (2007a) and Chaplin (2011),we assume in our toy model that B(θ) is described by:

B(θ) ={

const., for λmin ≤ |λ| ≤ λmax,

0, otherwise,(2)

i.e., in effect a uniform band of activity. We fixed the minimum latitude at λmin = 5 degreesin all models, whilst λmax was varied.1

Figure 3 the shows predicted frequency shift ratios δνlm/δν00, as a function of max-imum latitude of magnetic activity, λmax. The linestyles show results as follows: dottedfor δν10/δν00; short dashes for δν11/δν00; dot-dashed for δν20/δν00; dot-dot-dot-dashed forδν21/δν00; and long dashes for δν22/δν00.

1The total surface magnetic flux is proportional to λmax −λmin. One could instead normalise B(θ) so that thetotal flux is conserved for all values of λmax; however, this will not affect the ratios of the frequency shifts.


Chaplin (2011) used low-l solar p-mode frequency shifts measured by the BirminghamSolar-Oscillations Network (BiSON) to make an estimate of λmax,� based on use of the toymodel above. He compared the observed ratios given by average frequency shifts of thel = 2, 1 and 0 modes to the predicted ratios in Fig. 3. A weighted combination of his resultsgives λmax,� = 37 ± 8 deg, a reasonable estimate for the Sun.

This type of analysis is complicated by the fact that the disc-averaged visibility of thenon-radial modes depends on the angle of inclination of the star, is, a point we shall returnto in Sect. 4 below. In the solar case—where the rotation axis lies close to the plane of thesky when viewed from within the ecliptic plane—the zonal (m = 0) component of the l = 1modes is all but absent, and so one may obtain a clean estimate of the (1,1) shift. However,this is not so at l = 2, where the shifts will include a contribution from the (2,0) components.It actually turns out that this contribution is fairly weak, so that the observed Sun-as-a-starl = 2 shifts approximate quite closely those of the (2,2) components. But things would notbe so simple for Sun-like stars observed at other viewing angles. The relative contributionscan nevertheless be modelled (e.g., see Chaplin et al. 2004b), given good constraints on is.Observational uncertainties for other stars are likely to be too large to detect the impact ofanother effect: that one might expect there to phase offsets in time between the frequencyshifts of different (l, |m|), i.e., reflecting the spatio-temporal dependence of the evolution ofnear-surface activity during an activity cycle (e.g., see Moreno-Insertis and Solanki 2000).

Even in the absence of cyclic variations, the presence of persistent, strong near-surfacemagnetic field will give rise to frequency asymmetries of the components in the non-radialmodes. In the absence of very rapid rotation—which will also give rise to asymmetry ofmultiplet frequencies—measurement of any asymmetry could be used to place constraintson levels of near-surface activity. One does not require exotic levels of activity: frequencyasymmetries of l = 2 modes have been measured in Sun-as-a-star data (Chaplin et al. 2003).

Broomhall et al. (2011) have also looked at how the large frequency separation �ν,which is the separation in frequency between modes with the same degree but consecutiveradial order, changes with activity. They looked at solar data from BiSON, and showed thatsolar-cycle related changes in �ν could be detected using �ν derived by different means,although the signature was somewhat different for each of the fitting methods. Since thelarge separation can be determined relatively precisely even when individual frequenciescannot, being able to detect the signature of stellar activity in �ν potentially opens up moretargets for study.

Extraction of estimates of stellar-cycle induced shifts of the p-mode parameters—be theythe frequencies, large separations or amplitudes and mode damping rates—usually proceedsby dividing a long dataset into shorter segments for analysis. This makes it possible to trackseismic responses to varying levels of activity. However, one may also look for signaturesof changing activity in the frequency-power spectrum of the complete dataset. If the modesundergo significant frequency shifts over the duration of the dataset, the mode peaks willbe distorted from their expected, Lorentzian-like shape. Chaplin et al. (2008b) discussedthis effect in some detail, deriving analytical expressions for distorted profiles given bydifferent functional forms for the time-dependent frequency variations. The significance ofthe distortion effect is determined by the ratio

ε = δν/Γ, (3)

where δν is the shift in frequency and Γ is the FWHM of the mode peak. The higher thisratio, the more distorted the mode peak will be.

Figure 4 shows the distorted profiles given by frequency shifts of different sizes, for amode having Γ = 1.5 µHz. Plotted in different colours are the profiles given by cosinusoidal


Fig. 4 Distorted profiles givenby frequency shifts of differentsizes (see text), for a modehaving Γ = 1.5 µHz. The variouscoloured lines show profilesgiven by cosinusoidal(minimum-to-maximum)frequency shifts of zero (grey),0.6 µHz (black), 1.5 µHz (red)and 2.25 µHz (blue)

(minimum-to-maximum) frequency shifts of zero (grey), 0.6 µHz (black), 1.5 µHz (red) and2.25 µHz (blue). The ratios ε vary from zero up to 1.5, with the black profile having a ratioapproximately equal to that of the low-degree solar p modes. The peaks are clearly flattenedin appearance as a result of the frequency shifts. Similar distorted shapes are given by, forexample, a simple linear frequency shift in time (again, see Chaplin et al. 2008b).

One may therefore look for evidence of distorted profiles in the real observations. A sim-ple approach would be to fit the usual, Lorentzian-like models to the observed peaks, andthen to interogate the form of the residuals given by dividing the observed spectrum by thebest-fitting Lorentzian-based model. One may of course also attempt to fit distorted profilesdirectly to the data. This approach depends in part on the accuracy or appropriateness of theassumed form for the distortion.

Next, we go on to discuss the asteroseismic data that are available for stellar cycle studies.

3 Asteroseismic Data for Studies of Stellar Activity

The availability of asteroseismic data for studies of solar-type stars has been revolutionisedby new satellite and telescope observations, in particular those of the CNRS/ESA CoRoTsatellite and the NASA Kepler Mission.

Some attempts had already been made to look for stellar-cycle-related frequency shiftsof bright solar-type stars in data collected in relatively short, episodic campaigns widelyseparated in time. Fletcher et al. (2006) used data from the startracker on the WIRE satel-lite to search for frequency shifts of α Cen A; while, as noted previously, Metcalfe et al.(2007) looked at data from ground-based Doppler velocity data on β Hyi. Data on the brightsolar twin 18 Sco (Bazot et al. 2011, 2012), collected at two epochs separated in time bya few years, are also in the process of being analysed. While the planned SONG network(Grundahl et al. 2011, 2014) will also present an excellent opportunity to perform regular,episodic campaigns on the brightest solar-type stars.

The first multi-month datasets for solar-type stars were provided by CoRoT (Michel et al.2008). CoRoT observed 12 solar-type targets for asteroseismology having apparent magni-tudes in the range 5.4 ≤ mv ≤ 9.4, with dataset lengths from 20 to 170 days. Some of thesetargets were observed more than once, with significant gaps between observations, therebymaking it possible to search for possible changes of the p-mode parameters. The brightF-type subgiant HD49933 was the first of these stars to be subjected to detailed searchesfor seismic signatures of stellar cycle variations. García et al. (2010) uncovered systematic


changes in the frequencies of the star’s most prominent p modes. The rapid variations sug-gested a short cycle period, between about 1 and 2 years. They also found evidence forchanges in the amplitudes of the modes, anti-correlated with the changes detected in thefrequencies. This matches observations of the solar p modes: enhanced levels of magneticactivity lead to an increase of the frequencies of the Sun’s most prominent low-degree pmodes, while at the same time the mode amplitudes are diminished. Salabert et al. (2011)followed with a more detailed analysis of the frequency shifts of HD49933, presenting evi-dence that the shifts were strongest for the highest mode frequencies. This trend is again inbroad agreement with the solar results, suggesting that the perturbations responsible for theshifts are localised in the near-surface layers of the star.

Mathur et al. (2013a) repeated the analysis on three other stars observed by CoRoT—HD49385, HD52265 [a known planet host; see Gizon et al. 2013] and HD181420—but theydid not find any evidence of seismic signatures due to stellar cycles. They also consideredother types of data, including rotationally modulated signatures in the photometric stellarlightcurves, and spectropolarimetric data. We come back to discuss the combined use ofdifferent types of data in Sect. 4 below.

Kepler has provided significantly longer datasets on a much larger sample of solar-typestars. During the first few months of science operations, a substantial fraction of the short-cadence target slots required to conduct asteroseismology of solar-type stars were selectedby the Kepler Asteroseismic Science Consortium (KASC; see Gilliland et al. 2010). Targetswere observed for one month at a time, and this “asteroseismic survey” yielded detectionsin more than 500 solar-type stars (Chaplin et al. 2011b, 2014). Around 150 of the stars werethen selected to be observed for durations of at least 3 months (i.e., for at least one Keplerobserving Quarter). Of these stars, 91 had at least 24 months and 58 had at least 36 monthsof data by the time Kepler’s nominal mission ended.

Kepler Objects of Interest (KOIs) having short-cadence data provide another source ofmulti-month and multi-year datasets for asteroseismology (e.g., see Huber et al. 2013a).These targets are candidate, validated or confirmed exoplanet host stars and asteroseismol-ogy has already played a key rôle in helping to better characterise many of these systems.A total of 43 solar-type KOIs have at least 24 months of asteroseismic data, while 19 haveat least 36 months of data.

Tables 1 and 2 list solar-type stars observed by Kepler that have at least 36 months ofasteroseismic data. The stars in Table 1 are the field stars that were part of the KASC aster-oseismic survey; while those in Table 2 are KOIs from the planet-hosting sample. Figure 5plots both cohorts on a Hertzsprung–Russell diagram. Grey symbols show KASC field stars,while red symbols show the KOIs. The two black symbols mark the locations of the twocomponents of the bright binary 16 Cyg (Metcalfe et al. 2012), which was also targeted byKepler (KIC numbers 12069242 and 12069449). Also plotted are the CoRoT targets thathave been analysed for asteroseismic signatures of stellar cycles, namely HD49933 (filledblue symbol; see García et al. 2010) and HD49385, HD181420 and HD52265 (square sym-bols; Mathur et al. 2013a).

Work is now underway to search the long Kepler lightcurves for evidence of asteroseis-mic signatures of stellar cycles.

4 Combining Asteroseismic and Other Data

A much more complete picture of the activity of the star may be obtained once astero-seismology data are combined with high-precision spectroscopic or photometric data. For


Table 1 Solar-type starsobserved by Kepler havingquantities of short-cadence datafor asteroseismology in excess of36 months. Stars in this table arefield stars selected specificallyfor asteroseismic observations byKASC (see text)

KICnumber

Coverage, Q1 to Q17(months per quarter)

Total time(months)

1435467 01003333333333331 38

2837475 01003333333333331 38

3424541 01003333333333331 38

3427720 10003333333333331 38

3733735 01003333333333331 38

3735871 01003333333333331 38

5607242 01003333333333331 38

5955122 10003333333333331 38

6116048 01003333333333331 38

6508366 10003333333333331 38

6603624 10003333333333331 38

6679371 01003333333333331 38

6933899 01003333333333331 38

7103006 01003333333333331 38

7174707 10003333333333331 38

7206837 01003333333333331 38

7341231 10003333333333331 38

7584900 10003333333333331 38

7747078 10003333333333331 38

7799349 01003333333333331 38

7871531 10003333333333331 38

7976303 10003333333333331 38

8006161 01003333333333331 38

8228742 10003333333333331 38

8394589 01003333333333331 38

8524425 01003333333333331 38

8561221 01003333333333331 38

8694723 01003333333333331 38

8702606 10003333333333331 38

8760414 10003333333333331 38

9025370 01003333333333331 38

9098294 01003333333333331 38

9139151 10003333333333331 38

9139163 01003333333333331 38

9574283 01003333333333331 38

9812850 10003333333333331 38

10018963 01003333333333331 38

10124866 01003333333333331 38

10454113 01003333333333331 38

10644253 10003333333333331 38

11026764 10003333333333331 38

11081729 10003333333333331 38

11193681 01003333333333331 38


Table 1 (Continued)KICnumber


Total time(months)

11244118 01003333333333331 38

11253226 01003333333333331 38

11414712 01003333333333331 38

11554100 01003333333333331 38

11717120 10003333333333331 38

11771760 10003333333333331 38

11772920 01003333333333331 38

11968749 00103033333333331 36

12009504 10003333333333331 38

12069424 00000033333333331 31

12069449 00000033333333331 31

12258514 10003333333333331 38

12307366 01003333333333331 38

12317678 01003333333333331 38

12508433 01003333333333331 38

Table 2 The same as Table 1 butfor Kepler Objects of Interest,i.e., candidate, validated orconfirmed planet host stars withasteroseismic detections (seetext)

KICnumber

KOInumber


Total time(months)

3544595 69 01333333333333331 44

3632418 975 01003333333333331 38

4141376 280 01100333333333331 36

4349452 244 01003333333333331 38

5094751 123 00333333333333300 39

5866724 85 00333333333333331 43

6521045 41 00333333333333331 43

7051180 64 00333333333330331 40

7199397 75 10333333333333331 44

8292840 260 00003333333333331 37

8349582 122 00333333333333331 43

8478994 245 00003333333333331 37

8554498 5 03333333333333331 46

8866102 42 10333333333333331 44

9955598 1925 01003333333333331 38

10593626 87 01333333303333331 41

10666592 2 13333333333333331 48

11295426 246 01003333333333331 38

11512246 168 00333333333333331 43

example, Karoff et al. (2013) have been monitoring 20 bright solar-type Kepler targets since2009 using the FIES spectrograph on the Nordic Optical Telescope. It will be possible tocombine the resulting measures of activity in the Ca II H&K lines (see Sect. 1) with astero-seismic results on the stars.


Fig. 5 Hertzsprung–Russell diagram of a selected subset of the solar-type stars observed by Kepler andCoRoT for asteroseismology. The plotted Kepler targets all have quantities of short-cadence data for aster-oseismology in excess of 36 months (listed in Table 1). Grey symbols show field stars selected specificallyby KASC for asteroseismic observations. Red symbols show Kepler Objects of Interest, i.e., candidate, vali-dated or confirmed planet host stars with asteroseismic detections (see Huber et al. 2013a, 2013b). The twoblack symbols mark the locations of the two components of the bright binary 16 Cyg (also observed by Ke-pler). Plotted CoRoT stars are targets that have been analysed for asteroseismic signatures of stellar cycles:HD49933 (filled blue symbol) and HD49385, HD181420 and HD52265 (square symbols). (See text for furtherdetails)

From the CoRoT or Kepler data alone it is possible to combine asteroseismology withsignatures of activity in the raw, time-domain lightcurves. Photometric variability of solar-type stars has shorter-timescale contributions due to granulation or oscillations. At longertimescales, variability is given by signatures of magnetic activity, e.g., the evolution and de-cay of starspots, or modulation associated with rotation. Rotational modulation of signaturesof starspots and active regions makes it possible to measure not only the average surface ro-tation, but also the latitudinal gradient of any surface differential rotation. Some analysesalso use the measured photometric variations to attempt a reconstruction of the surface dis-tribution of spots and active regions, but one must take great care over the degeneraciesinherent in the method (e.g., see Walkowicz et al. 2013).

Figure 6 shows examples of two stars with high-quality asteroseismic data and visiblesignatures of rotation and activity in their lightcurves. HD49933, one of CoRoT’s mostimportant targets (Appourchaux et al. 2008), has already been mentioned above. The topleft-hand panel shows the frequency-power spectrum of its oscillations. The top right-handpanel comprises a smoothed 60-day segment of the time-domain lightcurve, with the or-dinate scaled to show relative brightness variations in parts per million. Variations in thelightcurve are visually dominated by periodic signatures due to rotational modulation ofstarspots. Those variations imply a mean surface rotation period of around 3.5 days. Thebottom panels have similar plots for the planet-hosting star Kepler-50 (Chaplin et al. 2013).Its lightcurve variations are not as strong, and the slower variations imply a slightly longersurface rotation period, of just under 8 days.

A growing number of studies are now exploiting the new satellite data to provide es-timates of surface rotation periods of solar-type stars. There are examples for individualtargets of interest (e.g., see CoRoT examples in Mosser et al. 2009a, 2009b) and for largecohorts of field stars (e.g., see Kepler examples in McQuillan et al. 2013; Nielsen et al. 2013;Reinhold et al. 2013; Walkowicz and Basri 2013). Some of the studies have specifically tar-geted stars with asteroseismic data (e.g., see García et al. 2013). Metrics related to the ob-served amplitudes of the photometric variability have also been used as proxies of activity


Fig. 6 Top panels: CoRoT data on HD49933, showing the frequency-power spectrum of its oscillations(left-hand panel) and a segment of the time-domain lightcurve showing rotationally modulated signatures ofstarspots. Bottom panels: Similar plots, but for the planet-hosting star Kepler-50, observed by Kepler

levels on the stars (Basri et al. 2010, 2011, 2013). Mathur et al. (2013b) presented results ona sample of 22 F-type stars observed by Kepler, all of which have surface rotation periodsfaster than 12 days and good asteroseismic detections. Some of the stars show evidence forstellar-cycle like variability in their photometry.

The photometric signatures of activity are dependent on the stellar angle of inclination, is.Having good constraints on the inclination can provide important information to help inter-pret inferences made from the photometry (and also spectroscopic observations). For exam-ple, might a lack of any significant photometric variability mean a Sun-like star is inactive?Or that it is being observed at a low is, meaning rotational modulation due to low-latitudespots and active regions is not detectable? Asteroseismology provides a means to estimate is.The asteroseismic method relies on the robust extraction of parameters related to the rota-tional splitting of non-radial modes of solar-like oscillations (see Gizon and Solanki 2003;Ballot et al. 2006, 2008) and has already been used to place constraints on the spin-orbitalignment of exoplanet systems (Chaplin et al. 2013; Huber et al. 2013b; Gizon et al. 2013;Van Eylen et al. 2014).

The degeneracy of the oscillation frequencies is lifted by rotation, giving several ob-servable multiplet components whose frequencies depend on the azimuthal order, m. Per-turbations due to non-radial modes are not spherically symmetric, hence the relative disc-integrated amplitudes of the different m components change with the viewing angle, is.Figure 7 shows the expected appearance of the limit spectrum of an idealised l = 1 mul-tiplet, as a function of is (see also Gizon et al. 2013). Here, we have assumed underlyingmultiplet parameters comparable to those observed in the oscillation spectrum of Kepler-50,the planet-hosting star whose inclination was determined by Chaplin et al. (2013). Each


Fig. 7 Expected appearance of the limit spectrum of an idealised l = 1 multiplet, as a function of the stellarangle of inclination, is. The composite mode profile is shown in black, while power due to the zonal (m = 0)and sectoral (m = ±1) components is shown, respectively, in red and blue (see text for further details)

mode peak has a FWHM of 3.0 µHz. The rotational splitting in frequency between adjacentpeaks is 1.5 µHz. When the rates of differential and absolute rotation are fairly modest, asis usually the case for solar-type stars, the splittings δνnlm of the observable high-n, low-lp modes will take very similar values.

Figure 8 is a zoom in frequency of the oscillations spectrum of Kepler-50 (from Fig. 6),here showing a single l = 1 mode. The grey line shows the spectrum after applying lightsmoothing, using a boxcar filter of width 0.25 µHz. The blue line—which shows the spec-trum after applying much heavier smoothing of width 1.5 µHz—serves to provide an ap-proximate estimate of the underlying (noise-free, or “limit”) frequency spectrum. It is ev-ident that the observed mode bears a close resemblance to the idealised high-inclinationcase in Fig. 7, i.e., even without resorting to a full analysis of the data, it is evident that therotation axis of the star must lie reasonably close to the plane of the sky.

The m = ±1 components have their strongest observable powers when the rotation axislies in the plane of the sky, i.e., when is � π/2. When is � 0, they are absent, and the onlyobservable mode power is that due to the m = 0 component. The dependence of the visibility(in power) follows:

Elm(is) = (l − |m|)!(l + |m|)!

[P

|m|l (cos is)

]2, (4)

where P|m|l are Legendre polynomials, and the sum over Elm(is) is normalised to unity. Mea-

suring the relative power of the azimuthal components of different |m| in a non-radial mul-tiplet therefore gives an estimate of is (actually |is| since one cannot discriminate between is

and −is, or π − is and π + is).


Fig. 8 Zoom in frequency of theoscillations spectrum ofKepler-50 from Fig. 6, showing asingle l = 1 mode. The grey lineshows the observed spectrumafter application of lightsmoothing (using a boxcar filterof width 0.25 µHz). The blue lineshows the spectrum afterapplying much heaviersmoothing (of width 1.5 µHz)

5 Other Asteroseismic Results

In addition to giving direct information about stellar cycles, asteroseismology also helpsmore indirectly. For instance, asteroseismic data can help us assess the depths of subsur-face stellar convection zones. Most dynamo theories assume that the interface between theconvection and radiative zones plays a crucial rôle in driving stellar activity cycles. Addi-tionally, it is generally believed that the convective turnover timescale, as expressed in theform of the Rossby number, Ro, plays a rôle in determining cycle properties. Hot stars thatare expected to have shallow convection zones also usually have shorter cycle periods. In-formation about the location of the boundary between the convection and radiative zones arecoded into stellar oscillation frequencies and thus one may use the frequencies to determinethe acoustic depth (i.e., in the sound travel time) of the position of the boundary.

The boundary between the outer convection zone and the radiative zone of a star ismarked by an abrupt change in the temperature gradient that causes a localised change inthe stratification of the layer. This in turn gives an abrupt change in the sound speed gra-dient commonly referred to as an “acoustic glitch.” Such a glitch introduces an oscillatorycomponent in the frequencies as a function of frequency (e.g., Gough and Thompson 1988;Vorontsov 1988; Gough 1990). The oscillatory component can be expressed as

δν ∝ sin(4πτνn,l + φ), (5)

where

τ =∫ R

r

dr

c, (6)

is the acoustic depth of the glitch, r is the corresponding location in radius, c the speed ofsound, νn,l the frequency of a mode with radial order n and degree l, and φ is a phase factor.There can be multiple acoustic glitches, for instance, the helium ionisation zone; in suchcases each glitch will contribute to the signal a specific periodicity of twice the acousticdepth of the glitch. In the case of the Sun, this oscillatory signature has been used to studyovershoot below the solar convection zone (e.g., Basu 1997, and references therein).

There are a number of different ways in which the acoustic signal can be fitted to deter-mine τ (see e.g., Mazumdar et al. 2014), but an easy way to see the signal is to take seconddifferences of the frequencies

δ2νn,l = νn−1,l − 2νn,l + νn+1,l , (7)


Fig. 9 The second differences ofthe frequencies of two models,one solar and one a higher-mass,hotter star, plotted as a functionof frequency. Both axes havebeen scaled with the ratio of thelarge frequency separation of thestar to that of the Sun in order toplot them on the same scale. Notethe longer period of theconvection-zone signal in thehotter star indicates a shallowerconvection zone

and then plot them as a function of νn,l . We show such a plot in Fig. 9. The higher-frequencycomponent is the signature of the base of the convection zone, the lower frequency compo-nent is from the He II ionisation zone.

Mazumdar et al. (2014) have shown that the oscillatory signal in real asteroseismic datacan be fitted successfully to determine the acoustic depth of the convection-zone base. Theyexamined 19 stars observed by Kepler ranging in effective temperature from about 5400 Kto 6200 K and metallicity, [Fe/H], ranging from −1.19 to +0.38. Converting acoustic depthto actual depth requires the use of models, but the same mode frequencies can be used toconstruct good models of the star, thereby providing a good estimate of the physical depthof the convection zone in those stars.

6 Concluding Thoughts

We are beginning to be able to probe stellar cycles with asteroseismic data. With twomore asteroseismology missions being planned—the Transiting Exoplanet Survey Satel-lite (TESS; launch 2017), and PLATO (launch 2024)—we should be able to extend studiesof stellar activity and stellar cycles to many more stars. Asteroseismology will also allowus to characterise the stars so that we can study the dependence between stellar interiorparameters and stellar cycles.

Acknowledgements W.J.C. acknowledges support from the UK Science and Technology Facilities Coun-cil (STFC). Funding for the Stellar Astrophysics Centre (SAC) is provided by The Danish National Re-search Foundation. S.B. acknowledges partial support from NSF grant AST-1105930 and NASA grantNNX13AE70G. The authors would like to thank R. Gilliland for help providing information on data cov-erage for Kepler short-cadence targets.

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inferences on stellar activity and stellar cycles from asteroseismology

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