A Kepler Study of Starspot Lifetimes with Respect to Light Curve Amplitude and Spectral Type
Helen A.C. Giles, Andrew Collier Cameron, Raphaëlle D. Haywood
MMNRAS , 1–11 (2016) Preprint 28 July 2017 Compiled using MNRAS L A TEX style file v3.0 A Kepler
Study of Starspot Lifetimes with Respect toLight Curve Amplitude and Spectral Type
Helen A. C. Giles, , (cid:63) Andrew Collier Cameron and Rapha¨elle D. Haywood Observatoire de Gen`eve, Universit´e de Gen`eve, Chemin des Maillettes 51, Versoix, 1290, Switzerland Centre for Exoplanet Science, SUPA, School of Physics and Astronomy, University of St Andrews, North Haugh, St Andrews, KY16 9SS, UK Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138, USA
Accepted XXX. Received YYY; in original form ZZZ
ABSTRACT
Wide-field high precision photometric surveys such as
Kepler have produced reamsof data suitable for investigating stellar magnetic activity of cooler stars. Starspotactivity produces quasi-sinusoidal light curves whose phase and amplitude vary asactive regions grow and decay over time. Here we investigate, firstly, whether there isa correlation between the size of starspots - assumed to be related to the amplitude ofthe sinusoid - and their decay timescale and, secondly, whether any such correlationdepends on the stellar effective temperature. To determine this, we computed theautocorrelation functions of the light curves of samples of stars from
Kepler and fittedthem with apodised periodic functions. The light curve amplitudes, representing spotsize were measured from the root-mean-squared scatter of the normalised light curves.We used a Monte Carlo Markov Chain to measure the periods and decay timescalesof the light curves. The results show a correlation between the decay time of starspotsand their inferred size. The decay time also depends strongly on the temperatureof the star. Cooler stars have spots that last much longer, in particular for starswith longer rotational periods. This is consistent with current theories of diffusivemechanisms causing starspot decay. We also find that the Sun is not unusually quietfor its spectral type - stars with solar-type rotation periods and temperatures tend tohave (comparatively) smaller starspots than stars with mid-G or later spectral types.
Key words: techniques: photometric – stars: activity – stars: starspots – stars:rotation
The
Kepler mission was designed to search for extrasolarplanet transits in stars (within a single field of view) inparticular small, Earth-like planets around Sun-like stars(Borucki et al. 2010; Koch et al. 2010; Jenkins et al. 2010).It has provided insight into planet formation as well as newexoplanet discovery, which also allowed to determine occur-rence rates (Howard et al. 2012; Petigura et al. 2013; Kaneet al. 2014; Burke et al. 2015; Dressing & Charbonneau 2015;Santerne et al. 2016) and further probe the statistics of ex-oplanet population and system architectures.
Kepler has also revolutionised stellar physics. Tens ofthousands of stars have 4 years worth of almost contin-uous, high precision photometry, allowing for a thoroughstudy of stellar brightness modulations across different stel-lar ages and types. From
Kepler , fields such as asteroseismol- (cid:63)
E-mail: [email protected] ogy (Bastien et al. 2013) and differential rotation studies(Reinhold et al. 2013; Aigrain et al. 2015; Balona & Abe-digamba 2016) of main sequence stars have evolved throughthe study of such a large sample of stars. McQuillan et al.(2013, 2014) (hereafter known as McQ14) made the firstlarge-scale surveys of stellar rotation by analysing the auto-correlation functions of stellar light curves.This unprecedented wealth of high-precision, continu-ous photometric data for thousands of main-sequence starshas enabled us to take a new look at our own Sun, result-ing in comparisons between it and stars which are Sun-like.Gilliland et al. (2011) (and pre-
Kepler , Radick et al. 1998)found that the Sun appears to be unusually inactive whencompared to other solar-type stars, but it has since beensuggested that this may in fact not be the case (Basri et al.2013). This is discussed in § © a r X i v : . [ a s t r o - ph . S R ] J u l H. A. C. Giles, A. Collier Cameron & R. D. Haywood how the lifetime of an active region depends on its size andon the stellar photospheric temperature.We define stellar activity, and active regions, in this con-text as meaning phenomena that introduce surface bright-ness inhomogeneities, giving rise to apparent flux modula-tion as the star rotates. Measurements of solar irradiance asa function of wavelength show that bright faculae and darkstarspots are the main contributors to solar flux modulationon timescales of order days to weeks (Foukal & Lean 1986).These modulations have a greater amplitude when the Sunis near the maximum of its 11-year activity cycle. The solarirradiance variations are complex; solar active regions oftencomprise a bipolar spot group surrounded by an extendedfacular region of enhanced surface brightness. As an activeregion crosses the solar disc, the limb brightening of the fac-ulae and foreshortening of the dark spots tends to cause anet initial flux increase. This is followed by a decrease asthe spot visibility increases and the facular limb brighteningdeclines (Fligge et al. 2000). A similar pattern is seen in Ke-pler light curves. At times of high activity, the amplitude ofvariability is often seen to increase with no obvious changein the mean flux level in the Kepler bandpass. Solar irradi-ance measurements, however, show clearly that the facularflux increase outweighs the dark spot deficit at times of highactivity (Lockwood et al. 2007).For the Sun, a range of activity levels have been ob-served since telescopic records began (from the MaunderMinimum to large-amplitude cycles in the mid-20th century)and there are many differing opinions on what constitutes‘typical’ solar activity levels (Hanslmeier et al. 2013; Usoskinet al. 2016; Inceoglu et al. 2015; Krivova et al. 2007; Wehrliet al. 2013; Livingston et al. 2007). The consensus appearsto be that the average level of solar activity lies in betweenthe extremes observed in the past 400 years. For our pur-pose, we will use the activity levels seen in the last 3 to 4sunspot cycles as typical levels.Furthering our understanding of stellar activity is notonly important to the stellar community; it is crucial tomany other areas of investigations, particularly in the ex-oplanet society. The presence of starspots and other mag-netic active regions can induce quasi-periodic variations overtimescales of weeks to months. These activity signatures areseen as major sources of noise in the search for small exo-planets (Earths and super-Earths); spots can lead to wrongplanet radius measurements (Barros et al. 2014). The pres-ence of starspots and other magnetically active regions area real nuisance in RV exoplanet observations. As well asstarspots, faculae and granulation produce signals modu-lated by the star’s rotation. They evolve over time, givingrise to quasi-periodic signals with varying amplitudes andphases. This induces RV variations of 1-2 ms − even in thequietest stars (Isaacson & Fischer 2010). Stellar noise canconceal and even mimic planetary orbits in RV surveys, andhas resulted in many false detections (eg. CoRoT-7d, Hay-wood et al. 2014; Alpha Centauri Bb, Rajpaul et al. 2015;HD166435, Queloz et al. 2001; HD99492, Kane et al. 2016;HD200466, Carolo et al. 2014; TW Hydra, Hu´elamo et al.2008; HD70573, Soto et al. 2015; HIP13044, Jones & Jenk-ins 2014; Kapteyn’s Star, Robertson et al. 2015; Gliese 667d,Robertson & Mahadevan 2014; and GJ 581d Robertson et al.2014). It also significantly affects our mass estimates, whichare routinely determined from RVs. A number of methods have been developed to account for activity-induced RV sig-nals and have been quantitatively tested to review theirperfomance (Dumusque 2016; Dumusque et al. 2017; Hay-wood et al. 2016; Rajpaul et al. 2015). Therefore, know-ing the active region lifetimes can provide significant con-straints for models used to determine exoplanet properties,such as mass (see L´opez-Morales et al. 2016). Additionally,planet radii and masses are central to theoretical modelsof planet composition and structure (e.g. Zeng & Sasselov2013) and are essential to interpreting observations of at-mospheres (see Winn 2010). When it comes to studying at-mospheric transmission spectroscopy of planet atmospheres,un-occulted spots serve to increase the ratio of the area of theplanet’s silhouette to that of the bright photosphere, makingthe transit look deeper than it really is. On the other hand,un-occulted faculae have the opposite effect. Since the con-trast of both faculae and spots against the quiet photospheredepends on wavelength, particular care has to be taken inthe interpretation of atmospheric transmission spectroscopy(Pont et al. 2007; Oshagh et al. 2016; Chen et al. 2017).As the effects of starspots and suppression of the granularblueshift in faculae are expected to diminish towards longerwavelengths (Marchwinski et al. 2015), forthcoming infraredRV spectrometers such as CARMENES (Quirrenbach et al.2014) and SPIRou (Delfosse et al. 2013) may help to separateplanetary reflex motions from stellar activity signals. How-ever, until recently only optical spectrometers were reachingthe precision needed to determine the masses of super-Earthplanets but CARMENES has been achieving − whichis sufficient for measuring super-Earths (Quirrenbach et al.2016) which would therefore suggest that others will be ableto perform similarly, according to their specifications.Sunspot (and by association, starspot) decay lifetimeshave been a point of interest for decades, with many the-ories for the cause of their decay and what function it fol-lows. Numerical investigations such as those by Petrovay& Moreno-Insertis (1997); Petrovay & van Driel-Gesztelyi(1997); Litvinenko & Wheatland (2015, 2016) indicate thatsunspot decay is consistent with a parabolic decay law,where the area of the spots decreases as a quadratic functionof time. Observations of the Sun (Moreno-Insertis & Vazquez1988; Martinez Pillet et al. 1993; Petrovay & van Driel-Gesztelyi 1997; Petrovay et al. 1999; Hathaway & Choud-hary 2008) have similarly reflected the same behaviour. Thisrelationship would imply that the main factor in spot de-cay is granulation, which was first hypothesised by Simon &Leighton (1964). Extrapolating the physics observed to oc-cur on the Sun, only a few attempts have been made to mea-sure starspot decay lifetimes. These studies would allow us totest our theories for sunspot decay on other Sun-like stars.As we cannot resolve the surfaces of others stars directlyand at high-resolution like we can for the Sun, their sizesover time have to be inferred from indirect indicators. Brad-shaw & Hartigan (2014); Davenport et al. (2015); Aigrainet al. (2015) have recovered the decay lifetime of starspotsfrom both real and simulated Kepler data. However, therehas not been a large-scale survey of starspot decay lifetimesuntil now.In this paper, we determine the starspot lifetimes in alarge sample of stars selected to have rotation periods closeto 10 days and 20 days. Our technique, based on MCMCparameter estimation, allows us to determine estimates and
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Kepler Study of Starspot Lifetimes uncertainties for the stellar rotation period and starspot life-time of each star. We then investigate how the decay life-times relate to extrapolated spot sizes and whether the stel-lar spectral type has a role in this relationship. In §
2, wejustify the choice of stellar targets. In § § Our samples are drawn from the sample of stars analysedby McQ14. They analysed over 34,000 main sequence starstaken from the
Kepler mission stellar archive at the NASAExoplanet Archive (Akeson et al. 2013). All of the stars inMcQ14 were less than 6500K in temperature and excludedknown eclipsing binaries (EBs) and
Kepler
Objects of In-terest (KOIs). McQ14 utilised T eff - log g and colour-colourcuts used by Ciardi et al. (2011) to select only main sequencestars. The boundary of 6500K was selected by McQ14 toensure that only stars with convective envelopes, that spindown during their lifetime, were included.To keep computational time to manageable levels, twosamples were drawn from the +34,000 McQ14 stars basedon the measured rotation periods. Sample 1 has a range ofperiods between 9.5 and 10.5 days, and sample 2 with arange of 19.5 to 20.5 days. This resulted in 1089 and 1155stars in each respectively. Unlike in McQ14 where they usedquarters 3-14 from the Exoplanet Archive, quarters 1 to 17were used here. This was done to extend the temporal spanof the light curves as much as possible. We created ACFs in the same fashion as McQuillan et al.(2013, 2014) who cross-correlated each
Kepler light curvewith itself at a series of discrete timeshifts (time lags). Thecorrelation increases and decreases dependening on the pres-ence of a large dominant starspot. As a light curve can beapproximated as sinusoidal in shape (Jeffers & Keller 2009),a time lag at an integer multiple of the stellar rotation pe-riod correlates strongly meaning the first side lobe of anACF corresponds to the stellar rotation period with furtherside lobes as harmonics of the period. The decrease in sidelobe amplitude at higher time lags occurs as the light curvegradually varies in amplitude and phase due to starspot for-mation and decay. Therefore the decay rate of the side lobesdescribes the decay rate of the starspots. By visual inspec-tion, this appears to be comparable to an exponential decay.With this knowledge, ACFs were fitted with a simple analyt-ical function. This is an improvement on what was reportedby McQuillan et al. (2013, 2014) as it establishes furtherparameters of the stellar activity but also determines errorsfor them.Many autocorrelation algorithms require the data to beuniformly sampled in time –
Kepler data is close to unifor-mity but has variation in exact observation times and has
Figure 1.
Example of a fitted auto-correlation function for
KIC8869186 using eq. 2. Selecting the positive time lag half of anACF, it follows a similar pattern as an underdamped simple har-monic oscillator (uSHO) which has a functional form that can befitted using Monte Carlo Markov Chains (MCMC).. significant data gaps. Therefore to generate ACFs, the lightcurves were binned and weighted as described by Edelson &Krolik (1988), which has the added advantage of providingerror estimates. Once the ACFs were generated, they wereorthogonalised by subtracting the inverse variance-weightedmean, to ensure there were no unwanted correlations be-tween the ACF power and the time lag.The behaviour of an ACF at zero time lag ≥ daysresembles the displacement of an underdamped simple har-monic oscillator ( uSHO ), described by y ( t ) = e − t / τ AR (cid:18) A cos (cid:18) π tP (cid:19)(cid:19) + y . (1)Many ACFs have an additional ‘interpulse’ close to halfof the stellar rotation period (Fig. 1). This corresponds tothere being another large but less dominant starspot on theopposite side of the star. Therefore the uSHO equation wasadapted to include an inter-pulse term, y ( t ) = e − t / τ AR (cid:18) A cos (cid:18) π tP (cid:19) + B cos (cid:18) π tP (cid:19) + y (cid:19) . (2) τ AR is the decay timescale [days] of the ACF which repre-sents the decay timescale of the dominant starspot. P is thestellar rotation period [days − ]. (Parameters A , B and y donot represent physical properties of the star, but are neededto fit the uSHO equation.) A and B are the amplitudes of thecosine terms and y is the offset of the uSHO from y = .The stellar rotation period is taken to be the time lag atwhich the largest side lobe occurs at and is found by search-ing for all peaks in the ACF and establishing which is thehighest (besides the peak at time lag = 0 days).Brewer & Stello (2009) used a damped, stochastically-driven harmonic oscillator model to emulate the quasi-periodic behaviour of solar p-modes. They also computedthe autocorrelation function of the resulting time series, ob-taining an expression equivalent to eq. 1 above. They usedthis as the kernel for a gaussian-process regression analysis MNRAS000
KIC8869186 using eq. 2. Selecting the positive time lag half of anACF, it follows a similar pattern as an underdamped simple har-monic oscillator (uSHO) which has a functional form that can befitted using Monte Carlo Markov Chains (MCMC).. significant data gaps. Therefore to generate ACFs, the lightcurves were binned and weighted as described by Edelson &Krolik (1988), which has the added advantage of providingerror estimates. Once the ACFs were generated, they wereorthogonalised by subtracting the inverse variance-weightedmean, to ensure there were no unwanted correlations be-tween the ACF power and the time lag.The behaviour of an ACF at zero time lag ≥ daysresembles the displacement of an underdamped simple har-monic oscillator ( uSHO ), described by y ( t ) = e − t / τ AR (cid:18) A cos (cid:18) π tP (cid:19)(cid:19) + y . (1)Many ACFs have an additional ‘interpulse’ close to halfof the stellar rotation period (Fig. 1). This corresponds tothere being another large but less dominant starspot on theopposite side of the star. Therefore the uSHO equation wasadapted to include an inter-pulse term, y ( t ) = e − t / τ AR (cid:18) A cos (cid:18) π tP (cid:19) + B cos (cid:18) π tP (cid:19) + y (cid:19) . (2) τ AR is the decay timescale [days] of the ACF which repre-sents the decay timescale of the dominant starspot. P is thestellar rotation period [days − ]. (Parameters A , B and y donot represent physical properties of the star, but are neededto fit the uSHO equation.) A and B are the amplitudes of thecosine terms and y is the offset of the uSHO from y = .The stellar rotation period is taken to be the time lag atwhich the largest side lobe occurs at and is found by search-ing for all peaks in the ACF and establishing which is thehighest (besides the peak at time lag = 0 days).Brewer & Stello (2009) used a damped, stochastically-driven harmonic oscillator model to emulate the quasi-periodic behaviour of solar p-modes. They also computedthe autocorrelation function of the resulting time series, ob-taining an expression equivalent to eq. 1 above. They usedthis as the kernel for a gaussian-process regression analysis MNRAS000 , 1–11 (2016)
H. A. C. Giles, A. Collier Cameron & R. D. Haywood of the waveform. Because of the N computational overheadinvolved in Gaussian-process regression, the large numberof data points in each light curve and the large number oflight curves analysed here, we elected instead to perform theparametric fit to the autocorrelation functions, as describedby eq. 2. The uSHO equation was fitted to ACFs using a Monte CarloMarkov Chain - MCMC. An MCMC is a means to ‘randomwalk’ towards the and to sample the joint posterior probabil-ity distribution of the fitted parameters. By estimating ini-tial values for the parameters, X θ , an initial fit of the uSHO equation is done and the likelihood, L , measured through ln L = − χ − N (cid:213) i = (cid:0) ln σ y i (cid:1) − N ( π ) (3)where χ = N (cid:213) i = (cid:18) y i − µσ y i (cid:19) . (4)where N is the number of ACF points, y i the value of theACF points with the error σ y i , µ is the model ACF pointvalue that corresponds to y i . As the ACFs are often moredistorted from the uSHO trend at higher time lags, due to in-terference from new starspots coming into effect, the MCMConly fits up to a time lag equivalent to . × P .The parameter values are then perturbed by a smallamount to a new position in parameter space and the fitand likelihood calculations are repeated. If the likelihood ishigher than the previous likelihood then the step is acceptedand the next step takes place from the current location inparameter space. If the likelihood is worse than previous, itmay be accepted under the Metropolis-Hastings algorithm(Metropolis et al. 1953; Hastings 1970), otherwise it will berejected and the step is not completed and it goes back tothe previous step and randomly steps again.The Metropolis-Hastings algorithm enables occasionalsteps in the wrong direction to ensure that an MCMC doesnot become trapped at a local likelihood maximum, and toenable exploration of the entire likelihood landscape. An op-timum acceptance rate for an N-dimensional MCMC is ap-proximately 0.25 (Roberts et al. 1997). Rates much lower orhigher than this may struggle to converge. To achieve this,an optimal step size is calculated from the curvature of the χ -parameter space for each parameter α , σ X i = (cid:115) ∂ χ / ∂ α , (5)where the exact step size per MCMC step is a Gaussian dis-tribution using σ X i and centred on the previous parametervalue.The initial inputs of the parameters for the MCMC areestimated from the ACF or given standard values: periodin days, determined as the time lag of the largest side lobeof the ACF, representative of the rotation period; the decaytime τ AR is based on the ratio of the first and second peaks of the ACF, τ AR = − P log (cid:16) y i ( P ) y i ( ) (cid:17) ; (6) A is the ACF value at time lag = 0; and B and y are takento be zero.As a means to encourage the MCMC not to search forsolutions in the unlikely areas of parameter space, Gaussianpriors were applied to three of the parameters: amplitude A , P and log τ AR . For τ AR , having a Gaussian prior in logspace reduces the risk of the MCMC wandering to unlikelyhigh values. Also a hard lower limit of 1 day was includedfor log τ AR to prevent a highly improbable τ AR value.To determine whether convergence has been achieved,we adopt a likelihood rule as used by Charbonneau et al.(2008) and Knutson et al. (2008). Each calculated likelihood L was stored and the current likelihood compared to themedian of all those previous. When L falls below the me-dian, the MCMC is considered to have achieved convergence.The MCMC then conducts another 5000 steps from whichthe mean and the standard deviation of each parameter aremeasured. This then launches a second MCMC routine usingthe mean and standard deviations as new initial parameters, X θ , and step sizes ( ± σ X θ ). This second MCMC explores thelikelihood maximum to find the optimal parameter values.Two final tests for convergence are applied to the final 5000steps of the second MCMC chain: we calculate the corre-lation length of this chain (and check that it is less than ∼ of the total chain) and compute the Gelman-Rubintest (Gelman & Rubin 1992). Only stars that passed bothof these tests are considered completed. These stars werethen quickly visually inspected to remove any where the fitswere obviously wrong. Additionally, a check for correlationsof all the fits of the ACFs for the targets was conducted bycomparing all the parameter values to one another.In Figs. 2 and 3, it can be seen that there are no strongunexpected correlations. The small correlation between thetwo amplitude sizes is not concerning as when there is aninterpulse present in an ACF this reduces the initial ampli-tude at zero-time lag. Therefore, the larger the interpulseamplitude, the smaller the initial amplitude. Kepler
Light Curve Morphologies
There are three distinct types of light curve morphologies(Fig. 4) that can be seen in the bulk of
Kepler data - ‘Sun-like’, ‘Beater’ and ‘Coherent’. These are purely qualitativedescriptions. On the other hand, inspecting the autocorre-lation functions, a distinction can be seen. ‘Sun-like’ starsappear to have starspot decay lifetimes that last approxi-mately a rotational period, ‘Beaters’ have lifetimes that lasta few rotations and the ‘Coherent’ stars have spots thatpersist for many rotations. Thereby taking the ratio of theactivity starspot lifetime versus rotational period, τ AR / P rot (AR=Active Region, rot=rotation), we can define the ratiofor each light curve morphology as ∼ for Sun-like stars, > for ‘Beaters’ and (cid:29) for the ‘Coherent’ stars. It isknown from Doppler imaging studies that many very active,fast-rotating stars have large, dark polar spots (Vogt & Pen-rod 1983; Strassmeier 2009, and references therein). Unlessthey are perfectly axisymmetric, such large polar features MNRAS , 1–11 (2016)
Kepler Study of Starspot Lifetimes Figure 2.
Correlation of all five MCMC parameters for the 10day period sample. Most show Gaussian distributions apart fromthose associated with the offset – they indicate that the offsetvalue is dependent on other parameters. There is also a correla-tion between the two ACF amplitudes, which is not surprising astypically if there was an interpulse present in a target’s ACF thenthe larger the interpulse, the smaller the primary amplitude. are likely to give rise to quasi-sinusoidal modulation. Sincepolar spots are generally large, we might expect them tohave long lifetimes, producing modulations that would re-main coherent for many rotation cycles. At the modest ac-tivity levels of most
Kepler stars, however, such large polarspots are not expected to be widespread.
Whilst it is possible to determine approximate spot sizes forFGK-stars from Doppler imaging (Collier Cameron 1995;Barnes et al. 2002), there is currently no direct method tomeasure them from light curves. However, light curves dohave continuous variations – these occur due to asymme-try between two sides of the star. It is worth making thepoint that the amplitude of solar photometric variability in-creases with overall activity levels through the magnetic cy-cle (Krivova et al. 2003). This implies that the power-lawdistribution of active-region sizes is such that the largestindividual active regions dominate the modulation. If all ac-tive regions were of similar size, an increase in the num-ber of active regions at different longitudes would cause thelight curve modulation amplitude to decrease rather thanincrease.(Bogdan et al. 1988) Therefore, as a proxy, the root-mean-square (RMS) scatter of the light curve can be extrap-
Figure 3.
Correlation of MCMC parameters of the 20 day pe-riod sample. All have Gaussian distributions apart from the twoACF amplitudes, which typically have smaller primary ampli-tudes when the interpulse amplitude increases. olated to be representative of starspot size.
RMS = (cid:118)(cid:117)(cid:116) N N (cid:213) i = y i (7)N is the total number of points in the light curve and y i thevalue of the flux at each data point. For a target, the 2- σ range of the RMS (which encompasses ∼
95% of points) iscalculated, as this encompasses the majority of the sinuousstructure of the light curve but ought not include the erro-neous outliers which may not have all been removed duringpost-observation processing.
Generally the quality of the fits produced by the MCMC rou-tine were good, though some were poorer and a couple wereentirely spurious fits. Therefore all of the results were alsoinspected by eye and those with significantly different fits,therefore not representative, were rejected from the sample.With 1089 stars for the 9.5-10.5 day (i.e. 10 day) periodsample and 1154 stars for the 19.5-20.5 day (i.e. 20 day)period sample, the ACF fitting program returned 913 (83.8%success rate) and 861 (74.6% success rate) acceptable ACFfits for the 10 day and 20 day sample respectively.In Fig. 5 the targets have been partitioned by spectraltype (from M- to F-stars) as determined from Pecaut &Mamajek (2013), and are represented by different coloursand symbols which are detailed in the attached key. The
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H. A. C. Giles, A. Collier Cameron & R. D. Haywood
Figure 4.
Three example light curves showing the three distinct light curve morphologies often seen in
Kepler data, and their auto-correlation functions. The Sun-like star,
KIC 2985814 , shows starspots which have a decay lifetime similar to the rotational period. Thebeating star (“beater”),
KIC 11802642 , has starspots which manage to survive a couple of stellar rotations and presents with a beatingeffect in the light curve. The long τ AR star, KIC 8869186 , is very coherent and has starspots which last many rotations. All of thesedecay lifetimes can be quite easily seen in how the peaks decay away in the autocorrelation functions. Taking a ratio of the decay lifetimeover the rotational period, each morphology can be defined as ∼ , > and (cid:29) for Sun-like, ‘Beaters’ and coherent stars respectively. first row shows the how the RMS amplitude of the rotationalmodulation (proxy for the starspot size for a star) varies withthe stellar effective temperature for each of the two samples.The second row displays how the decay lifetime depends onthe effective stellar temperature. In McQ14 the periods were determined using an autocor-relation function routine, and these were used during sam-ple selection. Comparing the periods from McQ14 and thosegenerated by the MCMC (Fig. 6), there is some variationwith the 10 day sample varying less than the 20 day sample.This range will reflect upon the difference in autocorrelationfunction generation as the routines used in McQ14 and thispaper are different, meaning variation in stellar rotations pe-riods is to be expected. Further, as a point of interest, theresiduals for the 10 day sample are asymmetric, with our al-gorithm generally finding longer periods than McQ14. Dueto not fitting the decay envelope, McQ14 will have underes-timated the period, biasing the first sidelobe to a lower time lag. Therefore, the shorter the decay lifetime, the larger adiscrepancy seen in Fig. 7. Interestingly, this becomes sym-metric for the 20 day sample, but with the same trend thatshorted decay lifetimes have larger range.
For this sample, in Fig. 5 (left-hand side), there is a dis-tinct distribution of starspot sizes and decay lifetimes. Hot-ter stars with T eff greater than 6200K, have a smaller rangeof spot sizes than cooler stars. These stars also have spotswhich do not survive for very long. At effective temperaturesabove the ∼ K boundary, the limit on decay lifetime isless than 100 days. This is up to a third of starspot lifetimeson much cooler stars.For ease of viewing, the comparison between spot sizeand decay lifetime has been split into the four observed spec-tral types in Fig. 8. The coolest stars (M-stars) have a largerange of spot size vs. decay timescale but given the verysmall stellar population this is not representative. However,there are a great many more K-stars and G-stars which show
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Kepler Study of Starspot Lifetimes Figure 5.
Plot series showing both data sets in two configurations. Upper level: effective temperature of targets (as stated by McQ14)vs. the RMS of the targets’ light curve. Lower level: effective temperature of targets vs. the measured decay lifetime. All targets havebeen split in colour and symbol based on their spectral type (from M- to F-stars) determined from Pecaut & Mamajek (2013). For bothdata sets, the average spot size (RMS) and decay lifetime decrease the hotter the star. a strong trend of longer decay lifetimes for larger spots. Thegradient of the trend is greater for the K-stars, indicatingthat the hotter the star, the shorter the lifetime. Addition-ally, the range of the spot sizes associated with the G-starsis less than the K-stars. This limits spots to have no largereffect on the light curve than an RMS of 0.025 mag. The F-stars, like the M-stars are not very numerous in this sample.However, they do all cluster together at low decay lifetimesand small spot sizes suggesting that for this the hottest ofall the targets, spots rarely reach a large size or survive verylong. This would also suggest spots survive longer the biggerthey are.
The 20 day sample is similar to the previous sample with afew small differences (Fig. 5): the temperature above whichthe range of spot sizes dramatically decreases is at a lowertemperature ∼ MNRAS000
The 20 day sample is similar to the previous sample with afew small differences (Fig. 5): the temperature above whichthe range of spot sizes dramatically decreases is at a lowertemperature ∼ MNRAS000 , 1–11 (2016)
H. A. C. Giles, A. Collier Cameron & R. D. Haywood
Figure 6.
Comparison of the MCMC-measured stellar rotation period and the period determined by McQ14. The red line representsthe line where the MCMC-measured period is the same as those from McQ14. For both the 10 day period sample, and in particular the20 day period sample, there is a large range of differences in periods. However, something to note is the difference in auto-correlationfunction generation from McQ14 and that an MCMC was then applied to the different ACF.
Figure 7.
Comparison of the MCMC-measured stellar rotation period and the period determined by McQ14 with respect to τ AR . Thered line indicates where the MCMC-measured period is the same as McQ14. The 10 day period sample shows an asymmetry in theresiduals indicating that for the smaller decay lifetimes τ AR there is a larger disagreement between the two measured periods. This ismost likely due to McQ14 underestimating the true period as they did not consider the decay envelope. In the 20-day sample, shortactive-region lifetimes degrade the precision with which the rotation periods can be determined, leading to a more symmetric distributionin the differences between periods determined with the two methods. for the K-stars. However, the range of decay lifetimes andspot sizes is much more limited for these G-stars than inthe other sample. The F-stars similarly cluster in the lowerdecay lifetime, smaller spot size area, but have a little morerange than the 10 day period sample of F-stars. From investigations on stars observed by
Kepler and previ-ous surveys, there was discussion about the activity of theSun and whether it was unusually quiet (Radick et al. 1998; Gilliland et al. 2011). Comparing it to the 20 day sample(solar rotation period ∼ days), stars with Sun-like tem-peratures ( ∼ K) all have small light curve amplitudesindicating small spots. The amplitudes of solar variabilitymeasured by Krivova et al. (2003) through the solar cycleare very similar to those measured in this work for stars withsolar-like rotation periods and effective temperatures. Thiswould (as discussed in Basri et al. 2013) indicate that theSun is not suspiciously inactive.
MNRAS , 1–11 (2016)
Kepler Study of Starspot Lifetimes Figure 8.
Distribution of decay timescales and RMS of target light curves, split by spectral type (based on temperature boundariesfrom Pecaut & Mamajek (2013)) for the 10 ( • ) and 20 (+) day period sample. There is a slight increase in trend gradient as stellartemperature increases. There is a strong relationship between the day lifetime and RMS – the larger the RMS of the light curve, thelarger the decay lifetime. For the hottest stars, the size of spots possible appear to be very small, and they often do not survive verylong. We find that stars with large RMS-variations indicate spotswith longer lifetimes. This could lead to two interpretations:large variations could mean that there are a few big spotsdominating with smaller RMS variations meaning there areonly small spots. But it could theoretically be possible thatthere are many spots of a similar size. There is good physicalreasoning behind the hypothesis that diffusive decay takeslonger to destroy big active regions than small ones. If indeedthe lifetime is short for stars that have many spots of similarsize, short lifetimes would also be associated with small lightcurve amplitudes. Implementing Occam’s Razor, the simplerexplanation is, however, that the solar spot-size and spot-lifetime power laws can be extrapolated to other stars, andthat the same physical processes operate.
Using the two datasets together, it is possible to generate afunction using the RMS (as a spot size proxy) and effectivetemperature to generate an expected active region lifetimewhich can be used for an individual star. Orthogonalisingthe data by removing the mean value of each distributionand fitting a quadratic through regression to the data inlog-log space, the following relation is determined: log τ AR = . + . · log RMS ++ . · (cid:0) log RMS (cid:1) − . · log T eff (8) where RMS is the RMS scatter of individual
Kepler lightcurves which were normalised to a mean flux of unity, T eff is the stellar effective temperature in K, and τ AR is the re-sultant decay lifetime in days. If this is used as an estimatefor the mean of a Gaussian prior probability distribution for log τ AR then the standard deviation σ of the residuals fromthe fit should be used as the standard deviation σ of theprior: σ (cid:0) log τ AR (cid:1) = . . When considering active longitudes, evidence from the
Ke-pler light curves suggests that even if spots persistently re-cur at active longitudes, they would tend to preserve thecoherence of the light curve on timescales longer than thelifetimes of an individual active region. We cannot explicitlysay whether such an effect is present, however we note thatthe decay timescales we obtain from the light curves of thesolar-like stars are comparable with the lifetimes of the largesolar spot groups.
The subject of this paper was to determine whether there isa relationship between the sinusoidal amplitude seen in
Ke-pler light curves, as a proxy for starspot size, and the decaytimescale of starspots. Furthermore, we sought to determinewhether the lifetimes of spots of a given size depend on thestellar effective temperature.
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MNRAS000 , 1–11 (2016) H. A. C. Giles, A. Collier Cameron & R. D. Haywood
As seen within the two samples (9.5-10.5 days and 19.5-20.5 days period stars) drawn from McQ14, there are threemain conclusions:(i) Big starspots live longer on any given star,(ii) Starspots decay more slowly on cooler stars and(iii) The Sun is not unusually quiet for its spectral type.Our observation that big spots generally survive longerlonger on any given star is consistent with models of spotdecay in which turbulent diffusion is eating the edges of thespots (Simon & Leighton 1964; Litvinenko & Wheatland2015, 2016). This is also consistent with our finding thatspots generally survive longer on cooler stars. As the vigourof convection is temperature dependent, the turbulent dif-fusivity, and hence the rate of spot decay, will increase withthe convective heat flux and hence with effective tempera-ture. An analogy would be food colouring being dispersedmore slowly in cool water than in boiling water.The work presented in this paper has deepened ourknowledge of the connection between the light curve mor-phologies of Kepler stars and the physics that determineactive-region lifetimes in convective stellar photospheres.This in turn can be applied to many areas which rely onlight from stars, in particular when searching and analysingexoplanet host candidates.
ACKNOWLEDGEMENTS
We would like to thank our referee for their constructivecomments that have improved the quality of this work.This paper includes data collected by the
Kepler mis-sion. Funding for the
Kepler mission is provided by theNASA Science Mission directorate. This research has madeuse of the NASA Exoplanet Archive, which is operated bythe California Institute of Technology, under contract withthe National Aeronautics and Space Administration underthe Exoplanet Exploration Program.HACG acknowledges the financial support of the Na-tional Centre for Competence in Research ‘PlanetS’ sup-ported by the Swiss National Science Foundation (SNSF)and the financial support of the University of St Andrews;and the computer support from the University of St An-drews. ACC acknowledges support from STFC consolidatedgrant number ST/M001296/1. RDH gratefully acknowledgessupport from STFC studentship grant ST/J500744/1, agrant from the John Templeton Foundation, and NASAXRP grant NNX15AC90G. The opinions expressed in thispublication are those of the authors and do not necessarilyreflect the views of the John Templeton Foundation. Thisresearch was submitted as a Masters project for HACG su-pervised by ACC at the University of St Andrews in April2015. HACG gratefully thanks for all supervision from ACCand support from RDH and other members of the astronomygroup.
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