Draft version November 22, 2017Preprint typeset using LATEX style emulateapj v. 01/23/15

THE INFLUENCE OF METALLICITY ON STELLAR DIFFERENTIAL ROTATIONAND MAGNETIC ACTIVITY

Christoffer KaroffDepartment of Geoscience, Aarhus University, Høegh-Guldbergs Gade 2, 8000, Aarhus C, Denmark and

Stellar Astrophysics Centre, Department of Physics and Astronomy, Aarhus University, Ny Munkegade 120, 8000, Aarhus C, Denmark

Travis S. MetcalfeSpace Science Institute, 4750 Walnut Street, Suite 205, Boulder CO 80301, USA

Angela R. G. SantosSpace Science Institute, 4750 Walnut Street, Suite 205, Boulder CO 80301, USA

Instituto de Astrofısica e Ciencias do Espaco, Universidade do Porto, CAUP, Rua das Estrelas, PT4150-762 Porto, PortugalDepartamento de Fısica e Astronomia, Faculdade de Ciencias, Universidade do Porto, Rua do Campo Alegre 687, PT4169-007 Porto,

Portugal andSchool of Physics and Astronomy, University of Birmingham, Birmingham B15 2TT, UK

Benjamin T. MontetThe Department of Astronomy and Astrophysics, The University of Chicago, 5640 S. Ellis Ave, Chicago IL 60637, USA

Howard IsaacsonDepartment of Astronomy, UC Berkeley, Berkeley, CA 94720, USA

Veronika WitzkeMax-Planck-Institut fur Sonnensystemforschung, Justus-von-Liebig-Weg 3, 37077, Gottingen, Germany

Alexander I. ShapiroMax-Planck-Institut fur Sonnensystemforschung, Justus-von-Liebig-Weg 3, 37077, Gottingen, Germany

Savita MathurSpace Science Institute, 4750 Walnut Street, Suite 205, Boulder CO 80301, USA

Instituto de Astrofısica de Canarias, E-38200 La Laguna, Tenerife, Spain andDepartamento de Astrofısica, Universidad de La Laguna, E-38205 La Laguna, Tenerife, Spain

Guy R. DaviesSchool of Physics and Astronomy, University of Birmingham, Birmingham B15 2TT, UK and

Stellar Astrophysics Centre, Department of Physics and Astronomy, Aarhus University, Ny Munkegade 120, 8000, Aarhus C, Denmark

Mikkel N. LundSchool of Physics and Astronomy, University of Birmingham, Birmingham B15 2TT, UK and

Stellar Astrophysics Centre, Department of Physics and Astronomy, Aarhus University, Ny Munkegade 120, 8000, Aarhus C, Denmark

Rafael A. GarciaLaboratoire AIM, CEA/DRF – CNRS – University Paris 7 Diderot – IRFU/SAp, Centre de Saclay, 91191, Gif-sur-Yvette, France

Allan S. BrunLaboratoire AIM, CEA/DRF – CNRS – University Paris 7 Diderot – CEA-Saclay, 91191, Gif-sur-Yvette, France

David SalabertLaboratoire AIM, CEA/DRF – CNRS – University Paris 7 Diderot – IRFU/SAp, Centre de Saclay, 91191, Gif-sur-Yvette, France

Pedro P. AvelinoInstituto de Astrofısica e Ciencias do Espaco, Universidade do Porto, CAUP, Rua das Estrelas, PT4150-762 Porto, Portugal and

Departamento de Fısica e Astronomia, Faculdade de Ciencias, Universidade do Porto, Rua do Campo Alegre 687, PT4169-007 Porto,

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Jennifer van SadersCarnegie Observatories, 813 Santa Barbara Street, Pasadena CA 91101, USA

Ricky EgelandHigh Altitude Observatory, National Center for Atmospheric Research, P.O. Box 3000, Boulder CO 80307, USA

Margarida S. CunhaInstituto de Astrofısica e Ciencias do Espaco, Universidade do Porto, CAUP, Rua das Estrelas, PT4150-762 Porto, Portugal and

Departamento de Fısica e Astronomia, Faculdade de Ciencias, Universidade do Porto, Rua do Campo Alegre 687, PT4169-007 Porto,Portugal

Tiago L. CampanteInstituto de Astrofısica e Ciencias do Espaco, Universidade do Porto, CAUP, Rua das Estrelas, PT4150-762 Porto, Portugal and

Departamento de Fısica e Astronomia, Faculdade de Ciencias, Universidade do Porto, Rua do Campo Alegre 687, PT4169-007 Porto,Portugal

William J. ChaplinSchool of Physics and Astronomy, University of Birmingham, Birmingham B15 2TT, UK and

Stellar Astrophysics Centre, Department of Physics and Astronomy, Aarhus University, Ny Munkegade 120, 8000, Aarhus C, Denmark

Natalie KrivovaMax-Planck-Institut fur Sonnensystemforschung, Justus-von-Liebig-Weg 3, 37077, Gottingen, Germany

Sami K. SolankiMax-Planck-Institut fur Sonnensystemforschung, Justus-von-Liebig-Weg 3, 37077, Gottingen, Germany

Maximilian StritzingerDepartment of Physics and Astronomy, Aarhus University, Ny Munkegade 120, 8000, Aarhus C, Denmark

Mads F. KnudsenDepartment of Geoscience, Aarhus University, Høegh-Guldbergs Gade 2, 8000, Aarhus C, Denmark

Draft version November 22, 2017

ABSTRACT

Observations of Sun-like stars over the last half-century have improved our understanding of howmagnetic dynamos, like that responsible for the 11-year solar cycle, change with rotation, mass and age.Here we show for the first time how metallicity can affect a stellar dynamo. Using the most completeset of observations of a stellar cycle ever obtained for a Sun-like star, we show how the solar analogHD 173701 exhibits solar-like differential rotation and a 7.4-year activity cycle. While the durationof the cycle is comparable to that generated by the solar dynamo, the amplitude of the brightnessvariability is substantially stronger. The only significant difference between HD 173701 and the Sun isits metallicity, which is twice the solar value. Therefore, this provides a unique opportunity to studythe effect of the higher metallicity on the dynamo acting in this star and to obtain a comprehensiveunderstanding of the physical mechanisms responsible for the observed photometric variability. Theobservations can be explained by the higher metallicity of the star, which is predicted to foster adeeper outer convection zone and a higher facular contrast, resulting in stronger variability.

1. INTRODUCTION

The number of spots on the surface of the Sun changesover a characteristic 11-year cycle, and this sunspot cy-cle is accompanied by a ∼0.1% change in brightness(Frohlich 2009). The increase in brightness with increas-ing spot coverage over the solar cycle is due to the com-pensating effect of faculae (Foukal et al. 2006). The num-ber of sunspots is also known to vary on much longer

karoff@phys.au.dk

timescales, with episodes of complete disappearance likethe 17th century Maunder Minimum (Eddy 1976). Itremains a matter of debate how bright the Sun wasduring the Maunder Minimum (see Solanki et al. 2013,for a recent review). In fact, we do not know how theSun’s brightness changes on timescales longer than a fewdecades. One way to improve this situation is to measureanalogous brightness variations in Sun-like stars and usesuch measurements to reveal the relationship betweenspots, magnetic activity and brightness changes on dif-

3

ferent timescales. This was first done by Lockwood et al.(1992), who used 8 years of observations at the Lowellobservatory of 33 Sun-like stars to conclude that “...theSun is in an unusually steady phase compared to similarstars, which means that reconstructing the past histori-cal brightness record, for example from sunspot records,may be more risky than has been generally thought.”This conclusion was challenged by Hall et al. (2009), whoargued that the Sun’s apparently low brightness variabil-ity compared to other Sun-like stars was due to selectioneffects. The only other inactive star in the ensemble, thesolar twin 18 Sco, has a brightness variability lower thanthe Sun (Hall et al. 2007).

In order to use Sun-like stars to reconstruct the his-torical brightness variability of the Sun, the fundamen-tal properties of the stars should be carefully analysed.It is particularly important to understand whether thedynamo in the stars has the same nature as the solardynamo. The best tool for such an analysis is asteroseis-mology, where the eigenfrequencies of the stars may beused to accurately determine fundamental stellar prop-erties like radius, mass, age, composition and rotationperiod (see e.g. Metcalfe et al. 2012; Davies et al. 2015;Lund et al. 2017).

Here we analyse the Sun-like star HD 173701 (KIC8006161), which is one of the brightest stars observedby Kepler and therefore also among the stars with thebest known fundamental properties like radius, mass andage as they have been measured with asteroseismology(see Table 1). The asteroseismic analysis reveals thatHD 173701 is almost identical to the Sun with respectto radius, mass and age, but it has a metallicity that istwice as high as the solar value. Assuming everythingelse equal, HD 173701 therefore allows us to measurethe effect of metallicity on stellar cycles. The natureof our study is therefore fundamentally different fromthe ensemble studies by e.g. Lockwood et al. (1992);Radick et al. (1998); Baliunas et al. (1995); Henry etal. (1996); Wright (2005), where a large number of starswere analysed whose fundamental properties are not wellconstrained. Here we only study one star, but its funda-mental properties are extremely well determined.

Asteroseismology not only allows us to measure thefundamental parameters of the stars, it also allows usto investigate the relation between the cycle related phe-nomena taking place inside the stars to those taking placeon the surface. HD 1737101 is a perfect candidate forsuch a study, not only because we can determine the tem-poral variability of the eigenfrequencies, but also becausewe have many different measurements of cycle relatedphenomena. This allows us to make a detailed compar-

TABLE 1Stellar parameters for HD 173701. ∗ from Creevey et al.

(2017) and ∗∗ from Buchhave & Latham (2015).

Radius∗: 0.930 ± 0.009 RMass∗: 1.00 ± 0.03 MLog g∗: 4.498 ± 0.003Age∗: 4.57 ± 0.36 GyrEffective temperature∗∗ 5488 ± 77 KMetallicity∗∗: 0.3 ± 0.1Rotation period: 21+2

−2 days

Inclination: 38+3−4 degrees

Cycle period: 7.41 ± 1.16 years

ison between the dynamos operating in HD 173701 andin the Sun, and how the cycle manifests itself on thesurfaces of the two stars.

The paper is arranged as follows: in Section 2, we de-scribe the different analyses we conduct on HD 173701including: spectroscopy, photometry and asteroseismol-ogy. In Section 3, we compare the results of our analysiswith a similar analysis of the Sun, to investigate the dif-ferences between HD 173701 and the Sun. In Section 4,we discuss the implications of these results for our un-derstanding of the variability in HD 173701.

2. ANALYSIS

The analysis of HD 173701 consists of a spectroscopic,a photometric and an asteroseismic analysis, as well asan analysis of the photospheric activity proxy. A compa-rable analysis of the Sun is performed to determine theimpact of the higher metallicity of HD 173701.

2.1. Spectroscopy

In the Mount Wilson HK project, chromospheric emis-sion was measured with the dimensionless S index (Dun-can et al. 1991):

S = α · H +K

R+ V, (1)

where H and K are the recorded counts in 1.09 A full-width at half-maximum triangular bandpasses centeredon the Ca iiH and K lines at 396.8 and 393.4 nm, respec-tively. V and R are two 20 A wide reference bandpassescentered on 390.1 and 400.1 nm, respectively, while α isa normalization constant.

Measured S indices for almost 2300 stars, includingHD 173701, from the Mount Wilson HK project are avail-able for download from the National Solar Observatorywebpage1. These observations include 3 measurementsin 1978, 165 in 1983 and 24 in 1984. Based on an ensem-ble of flat-activity stars, Baliunas et al. (1995) estimateda nightly measurement uncertainty of 1.2%.

From the Nordic Optical Telescope (NOT) we obtained12 epochs of observations from 2010 to 2014. These werereduced as described in Karoff et al. (2013, 2009) andnote that the data set does now include observations from2013.

The normalization constant (α) is usually obtained bymeasuring a number of stars that were part of the MountWilson HK project. The calibration does not have tobe linear (Isaacson & Fischer 2010). This approach was,however, not possible in the study by Karoff et al. (2013),as only one star was available for comparison. Instead,the excess flux, defined as the surface flux arising frommagnetic sources, was measured. The excess flux wasthen used to calculate a pseudo-S index by calibrationwith the effective temperature (Karoff et al. 2013).

The observations from the Keck telescope were pre-sented by Isaacson & Fischer (2010) and we include ad-ditional observations from 2015. A calibration of the Sindices was obtained from 151 stars that are both part ofthe Mount Wilson HK project and the California PlanetSearch program (Wright 2005; Isaacson & Fischer 2010).

1 ftp://solis.nso.edu/MountWilson_HK/

4

Fig. 1.— Modeling the cycle in HD 173701. The red curve isa least squares fit to annual means (black points with error-bars)of all the available observations (blue crosses) with a period of7.41 ± 1.16 years.

Uncertainties, were calculated as described in Isaacson& Fischer (2010).

Using 21 stars observed with both the NOT and Kecktelescopes, the instrumental S indices from the formerwere calibrated using the calibrated S indices from thelatter telescope. In order to minimise numerical effects inthe calculation of the S indices of HD 173701, all spectrafrom the NOT and Keck telescope were reanalysed usingthe same code. The uncertainties of the NOT measure-ments were obtained using nights with multiple observa-tions to obtain the following relation between the uncer-tainty of the mean value of the chromospheric activitymeasured that night and S/N: σ = 0.011/

√S/N. An ad-

ditional flat noise term of 0.002 was added in quadratureto the uncertainty of the mean values (Lovis et al. 2011).

Combining the observations from the NOT and Kecktelescopes with annual average observations from theMount Wilson HK Project a cycle period of 7.41 ± 1.16years is obtained using least squares, thereby overlappingwith the period covered by the nominal Kepler mission(Koch et al. 2010), as demonstrated in Fig. 1. The chro-mospheric emission of HD 173701 shows a cyclic vari-ability that is 2.2–2.7 times stronger than that of theSun (Fig. 2).

2.2. Photometry

During the primary Kepler mission, the telescoperecorded aperture photometry of almost 200,000 starsat 30-minute cadence (Borucki et al. 2010; Jenkins et al.2010). As the recorded apertures are smaller than thetypical point spread function (PSF) for the telescope,small changes in the telescope position, temperature, orPSF overwhelm small changes in the intrinsic brightnessof the star, making long-term brightness variations inac-cessible with standard Kepler photometry. These vari-ations can be recovered through the Full Frame Images(FFIs), in which the entire Kepler detector was recordedand sent to Earth approximately monthly throughout themission, providing an opportunity to measure aperturephotometry for each isolated star over its entire PSF.

We use the f3 software package described in Montet etal. (2017) to infer the brightness of HD 173701 and 15bright, nearby comparison stars in 52 FFIs spanning theKepler mission. All target stars fall within 15

′′of HD

173701. We inspect each light curve by eye to ensurenone of the comparison stars are intrinsically variable.For each of the four orientations of the Kepler telescope,we then measure the flux for HD 173701 relative to eachof the comparison stars, building a time series in ob-served brightness that accounts for instrumental system-atics. The orientations are considered separately as theunderlying flat field is poorly understood, so the percent-level inter- and intra- pixel sensitivity changes across thedetector can induce an artificial offset from orientationto orientation. Finally, the four sets of data are com-bined into one by dividing by the median flux value ineach orientation, and applying a linear offset to all datain each individual orientation such that the residuals ofa quadratic fit to the data are minimized.

The photometric uncertainties are calculated from thequadratic fit as described in Montet et al. (2017). Thisapproach assumes that HD 173701, which is heavily sat-urated in the FFIs, behaves similar to the non-saturatedreference stars. This assumption is supported by the factthat we are able to obtain nearly Poisson limited photom-etry for stars brighter than HD 173701 (Gilliland et al.2010).

Our analysis shows that the broad-band photometricvariability follows the cyclic variability seen in the chro-mospheric emission (Fig. 2). The standard deviation ofthe photometric variability of HD 173701 is 2.4–4.8 timeslarger than the photometric variability observed in theSun (see Section 3 for a detailed explanation of how boththe lower and the upper boundaries were obtained). Therelative level of solar photometric variability compared toother Sun-like stars is an important parameter in manySun-climate studies. The high value of the photometricvariability we find for this Sun-like star indicates thatthe Sun shows unusually weak photometric variability,supporting the conclusion of Lockwood et al. (1992) (seealso Lockwood et al. 2007). However, given the samplesize of one, caution should be taken when drawing anysuch conclusions from our results.

2.3. Asteroseismology

The asteroseismic analysis in this study has three pur-poses. Firstly, we use the results from the asteroseismicanalysis by Creevey et al. (2017) and adopt the generalparameters in Table 1 (note that the last two are spec-troscopic parameters from Buchhave & Latham (2015),as asteroseismology is generally not very efficient in con-straining effective temperature and metallicity). We notethat the parameters in Table 1 result in a luminosity thatis around 2σ higher than what is found by Hipparcos andGaia, the luminosity were however not used in the aster-oseismic analysis by Creevey et al. (2017). Tests haveshowed that including the luminosity in the asteroseis-mic analysis leads to an insignificant higher mass (∼ 1.02M). Secondly, we measure the rotation rate and incli-nation of the star and thirdly, we use asteroseismologyto we measure the effect of the activity cycle on the tem-poral evolution of the eigenfrequencies of the star.

The raw data for the asteroseismic analysis were takenfrom the Kepler Asteroseismic Science Operations Center(KASOC) and corrected using the KASOC filter (Hand-berg & Lund 2014).

Information on stellar rotation can be extracted fromthe measurable properties of rotationally split non-radial

5

Fig. 2.— The stellar cycle in HD 173701 compared to the Sun. Panels show the cycles in HD 173701 (left) and the Sun (right) as seen inthe chromospheric emission (a and b) as well as the response to this magnetic cycle in the relative photometric flux of the stars (c and d).The scale of the axes is the same for HD 173701 and the Sun. A clear 7.4-year cycle is seen in the chromospheric emission of HD 173701(a). The cycle is superimposed on observations extending back to 1978 in Fig. 1. It is seen that the relative photometric flux follows therising phase of the last cycle seen in the chromospheric emission.

eigenfrequencies in the frequency power spectrum ofmain-sequence stars (Chaplin et al. 2013; Doyle et al.2014; Davies et al. 2015; Campante et al. 2016). Esti-mates of rotational properties are the outputs of so-calledpeak-bagging procedures and here we use the methods ofDavies et al. (2015, 2016) to determine (νs sin i, i, νs, P ),where i is the angle of inclination, νs is the rotationalsplitting in frequency, and P is the average asteroseismicrotation period.

This analysis returns an equatorial rotation period of21+2

−2 days and an inclination of 38+3−4 degrees (Fig. 6).

The reliability of the asteroseismic method for measuringrotation and inclination was tested using the algorithmdeveloped by Lund et al. (2017). This test gave very con-sistent results with a rotation period of 21+4

−5 days and an

inclination of 37+6−8 degrees. The rotation period we find

with asteroseismology is slightly shorter than the periodwe find for the activity modulated signal in the photom-etry (25-35 days, see Section 3.2). The reason for thisis likely that the asteroseismic signal mainly originatesfrom the equatorial regions whereas the activity modu-lated signal could originate from higher latitudes. TheSun-as-a-star seismic synodic rotation period of the Sunis 26.9 days (Davies et al. 2014), comparable to the solarequatorial rotation period. A slightly larger value, butstill within the uncertainties, was found by Davies et al.(2015).

For the analysis of the temporal evolution of the eigen-frequencies, the processed time series was segmented into90-day-long sub-series with an overlap of 45 days. Thecorresponding frequency power spectra were then ob-tained from the periodogram of each sub-series.

To describe the background signal, we use three com-ponents: (i) an exponential decay of active regions (e.g.,

0.24 0.28 0.32 0.36 0.4025

30

35

40

45

50

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Inclination (

)

a)

25 30 35 40 45 50 55Inclination ()

0.00

0.02

0.04

0.06

0.08

0.10

0.12

Probability density

b)

0.24 0.28 0.32 0.36 0.40Projected splitting (µHz)

0

5

10

15

20

Probability density

c)

0 5 10 15 20 25 30 35Period (days)

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

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d)

Fig. 3.— Marginalised posterior probability distributions for therotational induced signal in the eigenfrequencies of HD 173701.The plot is composed of inclination versus projected splitting (a),inclination (b), projected splitting (c) and period of rotation (d).For a discussion of the method see Davies et al. (2015).

Garcıa et al. 2009; Campante et al. 2016); (ii) a Harvey-like profile for the granulation (e.g., Harvey 1985); (iii)and a constant offset denoting the photon noise. Thebackground parameters — corresponding to the best fitto the power spectra — are then fixed for the peak-bagging analysis.

To perform a global fit to the oscillation modes andestimate the respective model parameters (eigenfrequen-cies, as well as heights and linewidths of the radialmodes, rotational splitting, and stellar inclination angle),we followed a Bayesian approach (e.g., Campante et al.2011; Handberg & Campante 2011; Davies et al. 2016;Lund et al. 2017) through implementation of an affine-invariant MCMC ensemble sampler (Goodman & Weare2010; Foreman-Mackey et al. 2013). The Bayesian peak-

6

Fig. 4.— The rising phase of the last cycle in HD 173701 com-pared to the Sun. The panels show the chromospheric emission (aand b), the relative flux (c and d), radial frequency shifts (e andf), dipolar frequency shifts (g and h), quadrupolar frequency shifts(i and j), logarithmic mode heights of the eigenfrequencies (k andl) and photospheric activity proxy (m and n).

bagging analysis is fully described in Santos et al. (inprep.). In summary, we apply uniform priors to themode frequencies (within 8µHz of the eigenfrequenciesdetermined by Lund et al. 2017), linewidths and heights.Following the approach of Davies et al. (2016), we alsoapply priors to the large and small frequency separations.Finally, we use the posterior probability distributions ob-tained by Lund et al. (2017) to define the prior probabil-ities on the rotational splitting and inclination angle.

Once the eigenfrequencies for all sub-series have beenestimated, we compute the weighted mean frequencyshifts and corresponding uncertainties as:

δνl(t) =

∑n δνnl(t)/σ

2nl(t)∑

n 1/σ2nl(t)

, (2)

and

σl(t) =

[∑n

1/σ2nl(t)

]−1/2

, (3)

where δνnl(t) corresponds to the variation in frequencyof a mode of radial order n and angular degree l withrespect to the average value, and σnl(t) is the respectiveuncertainty.

The temporal variability of radial (l=0), dipolar (l=1)and quadrupolar (l=2) oscillation modes averaged overthe five central radial orders closely follow the cyclic vari-ability seen in the chromospheric emission, and showthe same characteristic behaviour as seen in the Sun(Fig. 4 & 5), i.e., the standard deviation increases withhigher degree l (though the increase is only marginal be-tween the l=1 and l=2 modes). This suggests that theorigin of the perturbation to the frequencies is located inthe outer layers of the star and thus the dynamo drivingthe variability in HD 173701 is similar to the dynamodriving the solar cycle.

Fig. 5.— Two full cycles. The figure shows the same as Fig. 4,but here zoomed out to see the last solar cycle.

The 11-year solar cycle can also be seen in the heightof the oscillation modes in a power spectrum (Chaplinet al. 2000). Since mode heights are approximately dis-tributed according to a log-normal distribution, we usethe logarithm of the mode heights and proceed in thesame manner as for the frequency shifts in computingthe mean logarithmic mode heights. The standard de-viation of the mode height variations in HD 173701 is1.6–2.4 times larger than those observed for the Sun.

2.4. The photospheric activity proxy

The photospheric activity proxy, Sph, is a measure-ment of stellar magnetic variability derived by means ofthe surface rotation, Prot (Garcıa et al. 2014; FerreiraLopes et al. 2015; Salabert et al. 2016). The Sph proxyis defined as the mean value of the light curve fluctu-ations estimated as the standard deviations calculatedover sub-series of length 5×Prot. In this way, Mathur etal. (2014) demonstrated that most of the measured vari-ability is only related to the magnetism (i.e. the spotsand faculae) and not to the other sources of variability atdifferent timescales, such as convective motions, oscilla-tions, stellar companion, or instrumental problems. Thisassumes that the spots and faculae are not distributedclose to the equator, but at higher latitudes. Otherwise,the value of Sph obtained would have been a lower limitof the true photospheric variability and the rotation sig-nature would have been difficult to measure. This im-plies that the rotation period found in the activity mod-ulation from photometry reflects mid to low latitudes,which would then be slower on HD 173701 than on theSun. The error on Sph was returned as the standard er-ror of the mean value. The Sph was measured on Keplerlight curves calibrated with the KADACS software asdescribed in Garcıa et al. (2011) and Pires et al. (2015)

The measured Sph values are shown in Figs. 4 & 5.The measured Sph values of HD 173701 show a standarddeviation 1.7 times higher than what is observed for theSun.

7

TABLE 2Standard deviations

HD 173701 The Sun Ratio

Original Original Adjusted Original Adjusted

Chromospheric emission: 0.0194 ± 0.0011 0.0072 0.0090 ± 0.0008 2.7 2.2Relative flux: 0.0019 ± 0.0001 0.0004 0.0008 ± 0.0001 4.8 2.4Frequency shifts (l=0) [µHz]: 0.2547 ± 0.0169 0.16 0.1781 ± 0.0175 1.6 1.4Frequency shifts (l=1) [µHz]: 0.3273 ± 0.0217 0.22 0.2349 ± 0.0249 1.5 1.4Frequency shifts (l=2) [µHz]: 0.3685 ± 0.0369 0.28 0.3186 ± 0.0380 1.3 1.2Mode heights [log ppm2/µHz] : 0.3374 ± 0.0314 0.14 0.2121 ± 0.0304 2.4 1.6Photospheric proxy [ppm]: 273.89 ± 1.37 160.22 160.19 ± 12.65 1.7 1.7

2.5. Solar data

We used the compilation of the total solar irradiance(TSI) from Frohlich (2009) to compare the photomet-ric variability of HD 173701 to the Sun over the con-sidered period, these data represent a composite fromthe DIARAD and PMO6V absolute radiometers on theSolar and Heliospheric Observatory (SOHO) and have abolometric character comparable to the very broad visi-ble bandpass (423–897 nm) used by Kepler (Basri et al.2013).

For the solar S indices, observations of the daily K-index KSP from the Evans Coronal Facility at Sacra-mento Peak (Keil et al. 1998) were transformed to theMWO S-index scale using the calibration of Egeland etal. (2017). This calibration was developed using overlap-ping observations (1993-2003) of reflected sunlight fromthe Moon through the MWO HKP-2 spectrophotometer.

In order to compare the asteroseismic signal inHD 173701 to the Sun, we used the time-dependent mea-surements of solar eigenfrequencies by Salabert et al.(2015). The values in Figs. 4 & 5 were calculated asmean values of the five central orders (n = 21 − 25) andthe errorbars represent the uncertainties on the meanvalues. The logarithmic mode heights of the eigenfre-quencies were also measured as described in Salabert etal. (2015), but here observations from the Variability ofsolar IRradiance and Gravity Oscillations (VIRGO) in-strument on SOHO were used.

3. RESULTS

3.1. Amplitude of the cyclic variability

The amplitude of the activity cycle in HD 173701 iscompared to the solar cycle in Figs. 4 & 5. We first calcu-late the standard deviation of the different cycle relatedparameters we have measured in HD 173701 (chromo-spheric emission, relative photometry, frequency shifts,mode heights and the photospheric activity proxy). Theuncertainties on the calculated standard deviations arecalculated by a bootstrap test, where the measurementsare randomly shifted according to the measured uncer-tainty of the individual measurements. We then calcu-lated the standard deviation of the solar parameters overa solar cycle. These values we refer to as original values(Table 2). The calculated standard deviations are af-fected by both the sampling and the uncertainties on themeasurements. We therefore also calculate the standarddeviations of the solar parameters by randomly select-ing as many solar measurements as we have stellar mea-surements over a solar cycle and adding a normally dis-tributed noise term given by the uncertainty of the stel-lar observations. These values are referred to as adjusted

values (Table 2). The adjusted standard deviations forthe solar parameters are generally higher than the orig-inal standard deviations and they are highly sensitiveto the absolute level of the uncertainties on the stellarmeasurements. This is especially clear for the standarddeviation of the relative photometry, where the adjustedstandard deviation is twice as large as the original.

The uncertainties on the relative photometry on theFFIs are due to a number of noise sources, where themost important are: photon noise, variability of compar-ison stars and inter- and intra-pixel sensitivity changes(Montet et al. 2017). For HD 173701 the largest con-tribution is likely to come from percent-level inter- andintra-pixel sensitivity changes across the detector thatcan induce artificial offsets when the spacecraft changesorientation every quarter. These changes will introduceslow drifts in the measured photometry, that are likelynot to have a large effect on the standard deviation.This suggests that adjusted standard deviation of thesolar photometry should be seen as a upper limit on thestandard deviation and that the standard deviation tocompare with the stellar result is closer to the originalstandard deviations.

In addition to this, the relative solar photometry isgiven as daily averages (Frohlich 2009), whereas the stel-lar measurements are obtained over only half an hour.On time-scales less than a day, the solar photometry isdominated by granulation and oscillations, which is notthe focus of our studies. We used the atmosphere modeldescribed below to calculate the difference in the stan-dard deviation of the solar photometry calculated fromeither 30-minutes or 24-hour averages. Here the formerturned out to be only 0.1% larger. We therefore adoptthis value to be insignificant.

3.2. Peak-height ratios

Starspot modulation of the stellar light curves encodesinformation about the stellar rotation and magnetic ac-tivity (see also Section 2.4). If the star is differentiallyrotating, spots at different latitudes may have differentrotation rates.

Based on the periodogram analysis, and in particularon the ratios between the heights of the second and firstharmonics of the rotation period (hereafter peak-heightratios), Reinhold & Arlt (2015) proposed a method to de-termine the sign of differential rotation. With a detailedanalysis of synthetic data, Santos et al. (2017) showedthat the method was not fully valid, in some cases, lead-ing to false-positives/negatives of the sign of differentialrotation (for details see Santos et al. 2017). Nevertheless,Santos et al. (2017) also showed that the peak-height ra-tios may provide a simple and fast way to constrain stel-

8

30 0 30 60Latitude

0.0

0.2

0.4

0.6

0.8r=h′ /h

P1 =29.18 d

P2 =33.12 d

P3 =30.51 d

i=38 +3

−4

Fig. 6.— Comparison between the theoretical peak-height ratios

for i = 38+3

−4 (solid black line and gray region) and the values

recovered from the periodogram analysis of the Kepler light curveof HD 173701 (dashed lines).

lar inclination and spot latitude. Namely, if the stellarinclination is known (e.g. from asteroseismology; Section2.3), one may estimate the spot latitudes.

Fig. 6 compares the theoretical latitude-ratio relation(following the approach in Santos et al. 2017) with thepeak-height ratios obtained from the periodogram anal-ysis of the full Kepler light curve for HD 173701. Werepeat the same analysis for two independent sub-seriesof 735 days.

Having the peak-height ratios, we infer the spot lat-itudes from the theoretical relation. Fig. 7 shows theabsolute values of the inferred spot latitudes as a func-tion of the rotation period. The error bars on the spotlatitudes are based on the uncertainty on the stellar in-clination. The solid line shows the best fit obtained witha rotation profile of the form

P (L) =Peq

1 − α sin2 L, (4)

where P (L) is the rotation period at a given latitude L,Peq corresponds to the rotation period at the equator,and α is related with the surface shear. The parame-ters of the best fit are Peq = 28.37 d and α = 0.53. Therotational profile for HD 173701 suggests strong solar-like differential rotation (equator rotates faster than thepoles). For comparison, the solar relative shear is aboutα ∼ 0.15 (e.g. Snodgrass 1983, 1990; Donahue et al.1996). The periodogram analysis and, in particular, thepeak-height ratios may be affected by spot evolution,which is not considered here. However, given that thestar is significantly active, one may expect spots to belong-lived and their evolution to play a modest role inthe present case. The fact that the results we obtainfrom the peak-height ratios are consistent with the re-sults from the complementary analyses presented in thiswork also seems to indicate that that is the case.

3.3. Differential rotation

Using the Kepler photometry we obtain a rotationperiod varying between 25 and 35 days (also consistentwith the results in Fig. 7) and a mean value of 27 daysby repeatedly computing the Lomb-Scargle periodogramon a 200-day sliding window, following Donahue & Keil(1995). This value is in reasonable agreement with thevalue of 29.8± 3.1 days obtained by Garcıa et al. (2014).The asteroseismic analysis returns a rotation period of21+2

−2 days. Keeping in mind that asteroseismic measure-ment mainly reflects the equatorial rotation period in

24 26 28 30 32 34 36Period (days)

0

10

20

30

40

50

|Lat

itude|

full lc.

<735 d

>735 d

Fig. 7.— Absolute spot latitudes, inferred from the peak-heightratios, as a function of the rotation period. The dashed, dotted,and dash-dotted lines correspond to the results from the full lightcurve, first 735 d, and last 735 d, respectively. The solid line showsthe best fit obtained with the rotation profile in equation 4, wherePeq = 28.37 d and α = 0.53.

the outermost layers of the star (Lund et al. 2014) themeasurements support a scenario where HD 173701 hassolar-like differential rotation, i.e., fastest at the equatorand declining towards higher latitudes, where active re-gions have modulated the photometric time series. Forcomparison, the Sun has a rotation period ranging from24.5 days at the equator to 28.5 days for the active re-gions at the highest latitudes (Donahue et al. 1996).

The rotation profile recovered from the peak-height ra-tios is also consistent with a solar-like differential rota-tion, with a relative differential rotation of ∼ 0.53 whichis more than three times stronger than the solar value of∼ 0.15 (Snodgrass 1983, 1990; Donahue et al. 1996).

Differential rotation is one of the hardest parameters tomeasure in other stars. Here we have used three differentmethods (activity modulation of the photometry, astero-seismology and peak-height ratios) and together they allprovide a consistent picture of HD 173701 as a star withstrong solar-like differential rotation. It is however truethat the significance of any of the individual methods islow. It is also possible that we obtain different valueswith the different analysis methods because they are allwrong. However, we find this possibility unlikely giventhat all three methods have been tested on other starsand validated with solar observations.

4. DISCUSSIONS

The asteroseismic analysis suggests that HD 173701 istruly Sun-like (except for the metallicity). The highermetallicity will increase the opacities inside the star,thereby lowering the luminosity. This will result in aslightly lower temperature and smaller radius. Basedon the comparison between the temporal variability ofthe radial, dipolar and quarupolar oscillation modes, italso suggests that the dynamo driving the variability inHD 173701 is similar to the dynamo driving the solarcycle. The variability of both the chromospheric emis-sion and especially the relative flux is however, signifi-cantly larger. We identify two possible explanations forthis larger variability. Either the higher metallicity sim-ply leads to a stronger dynamo with resulting strongerdifferential rotation, or the higher metallicity and lowerinclination leads to a higher contrast of the facular com-ponent.

The facular contrast is strongly influenced by theFraunhofer lines, which are affected by metallicity

9

Fig. 8.— Spectral irradiance variability as a function of metallic-ity. The plot shows solar photometric variability (dot-dashed lines)as well as that calculated for the hypothetical Sun (HD 173701)with 0.3 dex metallicity (M/H), which is measured as the loga-rithmic abundance of elements heavier than helium relative to theSun (solid lines). Black, red, and blue lines are total, facular, andspot components of the variability, respectively. The shaded areaindicates Kepler spectral efficiency. The variability of the relativephotometry is found to be 1.4 higher in the Kepler bandpass forthe high metallicity model compared to the Sun. If the contribu-tion from the lower inclination of HD 173701 is included as well,the variability of the relative photometry is found to be 1.9 timeshigher in the Kepler bandpass compared to the Sun (see Fig. 9).

(Shapiro et al. 2015). Together with the inclination ofthe rotation axis, metallicity is also known to have aneffect on the visibility of activity-related phenomena likespots and faculae (Shapiro et al. 2014). Unfortunately,this effect is poorly constrained because there are fewmetal-rich solar-analogs with measured activity cyclesand inclinations. Activity cycles in Sun-like stars haveso far mostly been discovered using observations fromthe so-called Mount Wilson HK Project (Duncan et al.1991), but most of these stars lack precisely determinedstellar properties, especially ages, as there are no exten-sive asteroseismic observations of them. The main reasonfor this is that most of the stars in the Mount WilsonHK project were too bright to be observed by Kepler,which has so far been the main asteroseismic observa-tory. Moreover, Kepler only observed a limited part ofthe sky during its nominal mission (Koch et al. 2010).

The atmosphere of HD 173701 can affect the standarddeviation of the relative photometry in two ways. Thelower inclination leads to reduced visibility of both spotsand faculae on the surface (Shapiro et al. 2014), andthe facular contrast is strongly enhanced with metallicitythrough the effect of the weak atomic and molecular lines(Shapiro et al. 2015). Both effects would increase thestandard deviation of the measured relative photometryof HD 173701.

We have quantified the effect of metallicity and incli-nation on the standard deviation of the relative photom-etry followed the approach in Shapiro et al. (2016) andemployed the SATIRE-S model (Krivova et al. 2011) toobtain solar brightness changes between 2000 and 2008as they would have appeared if the Sun had been ob-served by Kepler with different inclinations. We notethat such a change roughly corresponds to the ampli-tude of solar cycle 23. SATIRE-S decomposes the so-lar disk into the quiet Sun and magnetic features (spotumbra, spot penumbra, and faculae) and sums up theircontributions to return a time-dependent solar spectrum.For that the spectra of the quiet Sun and magnetic fea-

Fig. 9.— Spectral irradiance variability as a function of inclina-tion. The plot shows solar photometric variability (solid lines) aswell as that calculated for the hypothetical Sun (HD 173701) withinclination of 38 (dot-dashed lines). Black, red, and blue linesare total, facular, and spot components of the variability, corre-spondingly. The shaded area indicates Kepler spectral efficiency.The variability of the relative photometry is found to be 1.4 timeshigher in the Kepler bandpass for the high metallicity mode com-pared to the Sun.

tures at different disk positions are pre-calculated (Un-ruh et al. 1999) by the ATLAS9 code (Castelli & Kurucz1994). To account for the high metallicity of HD 173701we have recalculated these spectra for a M/H value of0.3 dex using opacity distribution functions, which weresynthesised with the DFSYNTHE code (Kurucz 2005;Castelli 2005). The opacity distribution functions werecalculated from atmosphere models developed by Unruhet al. (1999), where the effect of metallicity on the at-mosphere’s structure and electron concentration was ne-glected.

Fig. 8 demonstrates that the increase of the metallic-ity has only a subtle effect on the spot component of so-lar variability, whereas it significantly amplifies the fac-ular component. Overall, the metallicity change fromM/H=0.0 (solar value) to M/H=0.3 increases the ampli-tude of brightness variability as it would appear in Keplerobservations from 0.48 milli magnitudes (correspondingto 0.044%) to 0.84 milli magnitudes (0.077%). Interest-ingly, changing the inclination from the solar value toi = 38 has only a minor effect on the brightness vari-ability of a hypothetical Sun with M/H=0.3 (Fig. 9),where an increase from 0.84 milli magnitudes (0.077%)to 0.88 milli magnitudes (0.081%) occurs.

Increasing the metallicity of a Sun-like star will in-crease the opacities, which in turn will increase the tem-perature gradient. This means that the criterion forconvection is satisfied deeper in the star (Schwarzschild1906). In this way, a doubling of the metallicity, as inHD 173701, leads to a convective zone that is approx-imately 8% deeper than the solar convection zone (vanSaders & Pinsonneault 2012). Theoretical studies haveshown that a deeper convection zone leads to a longerconvective turnover time near the base of the outer con-vection zone (Brun et al. 2017) and thus stronger differ-ential rotation (Bessolaz & Brun 2011) in the same re-gion. Stronger differential rotation will lead to a strongerdynamo (especially a stronger Ω-effect). It is however,difficult to estimate exactly how much stronger the dy-namo and the resulting cycle will be and also whetherand how the stronger differential rotation will migrateall the way up to the surface. What we observe is strongsurface differential rotation. We do not know if the radial

10

differential rotation of HD 173701, which is likely whatis important for the dynamo, is different from the Sun.

We suggest that the most likely cause of the highervariability in the relative photometry and the chromo-spheric emission and the strong differential rotation ofHD 173701 is the higher metallicity of the star. In thispicture, the higher variability in the chromospheric emis-sion and the strong differential rotation in HD 173701is caused by a stronger dynamo induced by the highermetallicity. The large ratio between the variability seenin the relative flux and the chromospheric emission iscaused by the higher facular contrast resulting from thecombined effect of metallicity and inclination.

One problem with this picture is however, that we onlyhave good spectroscopic, photometric and seismic obser-vations of the Sun over the last few decades. It is possi-ble that the variability in the spectroscopic, photometricand seismic parameters of the Sun were significantly dif-ferent during the Dalton or Maunder minimum. Thoughit is still a matter of debate, what caused the Daltonand Maunder minimum (Charbonneau 2010) metallicityis out of the question as a possible cause. In other words,though metallicity seems like the most likely cause of thestronger variability we observe in HD 173701 over thecourse of its 7.4-year activity cycle, we cannot rule outall other causes.

The fact that the variability we observe in HD 173701is much stronger then what we observe in the Sun, butthe mean rotation periods are very similar, suggests thatwhatever is causing the stronger dynamo, the mean ro-tation period cannot be the driver. Thus under the sim-plest assumption that the behavior of the Sun duringthe last few decades is typical for the Sun, our sugges-tion that the most likely cause of the higher variability inthe relative photometry and the chromospheric emissionand the strong differential rotation in HD 173701 is thehigher metallicity of the star is very plausible. We dohowever, urge observers to undertake a similar analysison any solar twin with higher metallicity than HD 173701in order to test this hypothesis.

We would like to thank the referee for thought-ful comments, which significantly improved the pa-

per. The project is been supported by the Vil-lum Foundation. Funding for the Stellar AstrophysicsCentre is provided by the Danish National ResearchFoundation (Grant agreement No.: DNRF106). Thiswork was partially supported by the Non-profit Adopta Star program administered by White Dwarf Re-search Corporation. ARGS acknowledges the supportfrom NASA Grant NNX17AF27G. ARGS, PPV, MSCand TLC acknowledges the support by the fellowshipSFRH/BD/88032/2012 funded by FCT (Portugal) andPOPH/FSE (EC), and from the University of Birm-ingham. Work by B.T.M. was performed under con-tract with the Jet Propulsion Laboratory (JPL) fundedby NASA through the Sagan Fellowship Program exe-cuted by the NASA Exoplanet Science Institute. M.N.L.acknowledges the support of The Danish Council forIndependent Research – Natural Science (Grant DFF-4181-00415). SM acknowledges support from NASAgrant NNX15AF13G and NSF grant AST-1411685 andthe Ramon y Cajal fellowship number RYC-2015-17697.G.R.D and W.J.C. acknowledge the support of theUK Science and Technology Facilities Council (STFC).MSC acknowledges support from FCT through the re-search grant UID/FIS/04434/2013 and the Investigador-FCT Contract No. IF/00894/2012 and the POPH/FSE(EC) by FEDER funding through the Programa Op-eracional de Factores de Competitividade (COMPETE).VW, AIS and NAK acknowledge the support by theEuropean Research Council (ERC) under the EuropeanUnions Horizon 2020 research and innovation programme(grant agreement No. 715947) and by the German Fed-eral Ministry of Education and Research under project01LG1209A. The Nordic Optical Telescope is operatedby the Nordic Optical Telescope Scientific Associationat the Observatorio del Roque de los Muchachos, LaPalma, Spain, of the Instituto de Astrofısica de Ca-narias. The HK Project v1995 NSO data derive fromthe Mount Wilson Observatory HK Project, which wassupported by both public and private funds through theCarnegie Observatories, the Mount Wilson Institute, andthe Harvard-Smithsonian Center for Astrophysics start-ing in 1966 and continuing for over 36 years.

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