A comparison of gyrochronological and isochronal age estimates for transiting exoplanet host stars
aa r X i v : . [ a s t r o - ph . E P ] M a r Astronomy & Astrophysicsmanuscript no. paper-arxiv c (cid:13)
ESO 2015April 1, 2015
A comparison of gyrochronological and isochronal age estimatesfor transiting exoplanet host stars
P. F. L. Maxted , A. M. Serenelli , and J. Southworth Astrophysics Group, Keele University, Keele, Sta ff ordshire ST5 5BG, UKe-mail: [email protected] Instituto de Ciencias del Espacio (CSIC-IEEC), Facultad de Ciencias, Campus UAB, 08193, Bellaterra, SpainReceived ; accepted
ABSTRACT
Context.
Tidal interactions between planets and their host stars are not well understood, but may be an important factor in theirformation, structure and evolution. Previous studies suggest that these tidal interactions may be responsible for discrepancies betweenthe ages of exoplanet host stars estimated using stellar models (isochronal ages) and age estimates based on the stars’ rotation periods(gyrochronological ages). Recent improvements in our understanding of the rotational evolution of single stars and a substantialincrease in the number of exoplanet host stars with accurate rotation period measurements make it worthwhile to revisit this question.
Aims.
Our aim is to determine whether the gyrochronological ages for transiting exoplanet host stars with accurate rotation periodmeasurements are consistent with their isochronal ages, and whether this is indicative of tidal interaction between the planets and theirhost stars.
Methods.
We have compiled a sample of 28 transiting exoplanet host stars with measured rotation periods, including two stars (HAT-P-21 and WASP-5) for which the rotation period based on the light curve modulation is reported here for the first time. We use ourrecently-developed Bayesian Markov chain Monte Carlo method to determine the joint posterior distribution for the mass and age ofeach star in the sample. We extend our Bayesian method to include a calculation of the posterior distribution of the gyrochronologicalage that accounts for the uncertainties in the mass and age, the strong correlation between these values, and the uncertainties in themass-rotation-age calibration.
Results.
The gyrochronological age ( τ gyro ) is significantly less than the isochronal age for about half of the stars in our sample. Tidalinteractions between the star and planet are a reasonable explanation for this discrepancy in some cases, but not all. The distributionof τ gyro values is evenly spread from very young ages up to a maximum value of a few Gyr, i.e., there is no obvious “pile-up” of starsat low values or very high of τ gyro as might be expected if some evolutionary or selection e ff ect were biasing the age distribution ofthe stars in this sample.. There is no clear correlation between τ gyro and the strength of the tidal force on the star due to the innermostplanet. There is clear evidence that the isochronal ages for some K-type stars are too large, and this may also be the case for someG-type stars. This may be the result of magnetic inhibition of convection. The densities of HAT-P-11 and WASP-84 are too largeto be reproduced by any stellar models within the observed constraints on e ff ective temperature and metallicity. These stars mayhave strongly enhanced helium abundances. There is currently no satisfactory explanation for the discrepancy between the young agefor CoRoT-2 estimated from either gyrochronology or its high lithium abundance, and the extremely old age for its K-type stellarcompanion inferred from its very low X-ray flux. Conclusions.
There is now strong evidence that the gyrochronological ages of some transiting exoplanet host stars are significantlyless than their isochronal ages, but it is not always clear that this is good evidence for tidal interactions between the star and the planet.
Key words. stars: solar-type – planet-star interactions
1. Introduction
Stars are born rotating rapidly and can then lose angular mo-mentum if they have a magnetised stellar wind. This observa-tion leads to methods to estimate the age of single late-type starsfrom their rotation period, a technique known as gyrochronol-ogy (Barnes 2007). It is unclear whether exoplanet host stars canbe considered as single stars in this context. If significant orbitalangular momentum is transferred by tides from the orbit of aplanet to the rotation of the host star (“tidal spin-up”) then thestar may rotate faster than a genuine single star of the same age,i.e., its gyrochronological age ( τ gyro ) will be an underestimate ofits true age. There is currently no quantitative global theory tocalculate the e ffi ciency of tidal spin-up for stars with convectiveenvelopes because the dissipation of tidal energy by the exci-tation and damping of waves in convective atmospheres is not fully understood (Zahn 1975, 1977; Goodman & Lackner 2009;Damiani & Lanza 2014; Ogilvie 2014). Alternatively, the planetmay disrupt the magentic field geometry of the star and therebyreduce the e ffi ciency of angular momentum loss in the magne-tised stellar wind, although this is only likely to be a significante ff ect for F-type stars ( T e ff > P <
10 d, Lanza 2010).If a planet transits its host star then an analysis of the eclipselight curve can yield an accurate estimate for the radius of thestar relative to the semi-major axis of the planet’s orbit, R ⋆ / a ,provided that the eccentricity of the orbit is known. This esti-mate can be combined with Kepler’s laws to estimate the densityof the host star (Seager & Mallén-Ornelas 2003). The densitycan be combined with estimates for the e ff ective temperatureand metallicity of the star to infer a mass and age for the starby comparison with stellar models. The age derived by compar- Article number, page 1 of 11page.11 & Aproofs: manuscript no. paper-arxiv ing the properties of a star to a grid of stellar models is knownas the isochronal age ( τ iso ). It is not straightforward to calculatethe statistical error on τ iso because it is strongly correlated withthe estimate of the stellar mass ( M ⋆ ) and the distribution can bestrongly non-Gaussian, e.g., for stars near the end of their main-sequence lifetime the probability distribution for τ iso given theobserved constraints (the posterior probability distribution) canbe bi-modal (Maxted et al. 2015). There are also systematic er-rors in stellar models due to uncertainties in the input physics.For example, the e ffi ciency of energy transport by convectionis usually parameterised by the mixing length parameter α MLT (the path length of a convective cell in units of the pressure scaleheight) that is not known a-priori but that can be estimated byfinding the value of α MLT for which the models match the ob-served properties of the Sun. The initial helium abundance forthe Sun is estimated in a similar way. The radii of some M- andK-type dwarf stars are not accurately predicted by these “solar-calibrated” stellar models (Hoxie 1973; Popper 1997). This “ra-dius anomaly” for low mass stars is an active research topic thatis motivated by a need to better understand exoplanet host starsand is driven by advances in simulating convection in low-massstars (Ludwig et al. 2008) and incorporating magnetic fields intostellar models (Feiden & Chaboyer 2013).Pont (2009) found suggestive empirical evidence for excessrotation in stars with close-in giant planets (“hot Jupiters”) basedon a sample of 28 stars. For 3 of these stars the rotation periodwas measured from the rotational modulation of the light curveby star spots. For the other 25 stars the rotation period was esti-mated from the projected equatorial rotation velocity measuredfrom rotation line broadening ( V rot sin i ⋆ ). For 12 of the stars, V rot sin i ⋆ was below the instrumental resolution. It is di ffi cult tomake accurate estimates of V rot sin i ⋆ in these cases because itrequires the deconvolution of the instrumental profile from theobserved spectral line profiles and the results are sensitive to de-tails of the stellar atmosphere models used to calculate the intrin-sic stellar line profiles. Brown (2014) used a maximum likehoodmethod to estimate the ages of 68 transiting exoplanet host starsusing 5 di ff erent grids of stellar models and also applied four dif-ferent methods to calculate the age using gyrochronology. Therotation periods for 8 of the stars in that study were based on ro-tational modulation of the light curve, the remainder were esti-mated from V rot sin i ⋆ . Estimates of the rotation period based on V rot sin i ⋆ are generally much less precise than those measureddirectly from the lightcurve and can be a ff ected by systematicerrors when the line broadening due to rotation is small com-pared to other poorly-understood line-broadening e ff ects such asmacro-turbulence. Additional uncertainty is introduced by theunknown inclination of the star’s rotation axis, i ⋆ . Brown founda “slight tendency for isochrones to produce older age estimates”but that the “evidence for any bias on a sample-wide level is in-conclusive.”Planet host stars are an interesting test case for our under-standing of the tidal interactions because tidal spin-up will in-crease the star’s magnetic activity, which may increase the lossof rotational angular momentum through a magnetised stellarwind. The balance of these two e ff ects may lead to a quasi-stationary state in which the planet’s orbit is stable in the longterm (Damiani & Lanza 2015). However, if tidal interactionsbetween stars and planets are strong enough to e ffi ciently de-stroy massive, short-period planets then we might expect to onlyfind hot Jupiters around young stars. To test this hypothesis itwould be useful in these cases to be able to estimate an accurateisochronal age, because it is likely that the star will be spun-upduring the destruction of the planet and so its gyrochronological age will not be an accurate estimate of its true age. Comparingthe isochronal and gyrochronological ages is not a straightfor-ward problem because the uncertainties on these two age esti-mates are strongly non-Gaussian and τ gyro depends on the as-sumed stellar mass, so τ iso and τ gyro are also correlated.In Maxted et al. (2015) we developed a Markov chain MonteCarlo (MCMC) method that enables us to estimate the jointposterior probability distribution for τ iso and M ⋆ based on theobserved density, e ff ective temperature (T e ff ) and metallicity([Fe / H]) of a star. In this paper we select a sample of 28 tran-siting exoplanet host stars with accurately measured rotation pe-riods (Section 2). For two of the stars in this sample the rotationperiods are new estimates based on the rotational modulation ofthe light curves. We then describe how we have extended themethod of Maxted et al. (2015) to include a calculation of theposterior probability distribution for τ gyro and present the resultsof applying this technique to our sample of 28 planet host stars(Section 3). The implication of these results for our understand-ing of tidal spin-up of exoplanet host stars and the reliability ofsolar-calibrated stellar models is discussed in Section 4 and ourconclusions are given in Section 5.
2. Sample definition and data selection
We have compiled a list of 28 transiting exoplanet host stars forwhich the rotation period has been measured directly from therotational modulation of the light curve due to star spots or, inthe case of 55 Cnc, spectroscopic variability in the Ca ii H and Kemission line fluxes. For all these stars the existence of a transit-ing planet has been confirmed by multiple radial velocity mea-surements based on high-resolution spectroscopy or dynamicalanalysis of transit timing variations in multi-planet systems (e.g.,Kepler-30). The observed properties of these stars are given inTable 1. For each star we list the rotation period, P rot , the orbitalperiod, P orb , the e ff ective temperature, T e ff , the logarithmic sur-face iron abundance relative to the Sun, [Fe / H], the mean stellardensity, ρ ⋆ , the flux from the star at the top of the Earth’s atmo-sphere, f ⊕ , and the logarithm of the stellar luminosity, log( L ⋆ ).We have used only one or two sources of input data for thisanalysis rather than attempting to reconcile or take the averageof multiple studies for each system. We have excluded stars withmasses that are too low to be covered by our grid of stellar mod-els ( < . M ⊙ ) or too high for gyrochronology to be applicable( > . M ⊙ ). We have also excluded stars where the density es-timate is based on a Kepler long-cadance (LC) light curve only,because these density estimates appear to be biased comparedto the density estimated using asteroseismology (Huber et al.2013).
We can include the observed luminosity of the star ( L ⋆ ) as anadditional constraint in the analysis to derive the mass and ageof the star. For stars that have a trigonometrical parallax in vanLeeuwen (2007) with precision σ π /π < ≈ . L ⋆ us-ing the flux at the top of the Earth’s atmosphere ( f ⊕ ) estimatedby integrating a synthetic stellar spectrum fit by least-squares tothe observed fluxes of the star. Optical photometry is obtainedfrom the Naval Observatory Merged Astrometric Dataset (NO-MAD) catalogue (Zacharias et al. 2004), the Tycho-2 catalogue(Høg et al. 2000), The Amateur Sky Survey (TASS, Droege et al. Article number, page 2 of 11page.11. F. L. Maxted et al.: Ages of transiting exoplanet host stars
Table 2.
Periodogram analysis of the WASP light curves for WASP-5.Observing dates are JD-2450000, N is the number of observations usedin the analysis, A is the semi-amplitude of the sine wave fit by least-squares at the period P found in the periodogram with false-alarm prob-ability FAP. Camera Dates
N P [d] A [mag] FAP227 3870 – 4054 4429 8.18 0.003 0.027228 3870 – 4020 3918 16.35 0.003 0.094228 4284 – 4433 4673 8.52 0.002 0.052225 5352 – 5527 7720 15.04 0.006 < . and Deep Near In-frared Survey of the Southern Sky (DENIS) catalogues (TheDENIS Consortium 2005; Skrutskie et al. 2006). The syntheticstellar spectra used for the numerical integration of the fluxesare from Kurucz (1993). Reddening can be neglected for thesenearby stars given the accuracy of the measured fluxes and paral-laxes. Standard errors are estimated using a simple Monte Carlomethod in which we generate 65,536 pairs of π and f ⊕ valuesfrom Gaussian distributions and then find the 68.3% confidenceinterval of the resulting log( L / L ⊙ ) values. The 2MASS photom-etry for 55 Cnc is not reliable so for this star we have used thevalue of f ⊕ from Boyajian et al. (2013). The rotation periods for the stars in Table 1 are taken from thepublished sources noted except for HAT-P-21 and WASP-5, forwhich we have measured the rotation periods using observa-tions obtained by the WASP project (Pollacco et al. 2006) andthe method described by Maxted et al. (2011). For HAT-P-21we used 4185 observations obtained between 2007 Jan 2 and2007 May 15. These data show a very clear sinusoidal signalwith a period of 15.9 d with an amplitude of 0.012 magnitudes.This period agrees very well with the estimate P rot = . ± . V rot sin i ⋆ = . ± . − , Bakos et al. 2011) and theradius of the star, assuming that the inclination of the star’s rota-tion axis is i ≈ ◦ . The WASP data for HAT-P-21 are shown asa function of rotation phase in Fig. 1. The standard error on thisvalue has been estimated from the full-width at half-maximumof the peak in the periodogram.For WASP-5 we analysed 6 sets of data obtained with 3 dif-ferent cameras in 4 di ff erent observing seasons, as detailed inTable 2. In all 6 data sets we detect a periodic signal consistentwith a rotation period P rot ≈
16 d if we allow for the possibilitythat the distribution of star spots can produce a signal at the firstharmonic of the rotation period (i.e., at P rot / P rot in Table 1 is the mean of the observed values calculated withthis assumption and the error quoted in the standard error of themean.Mohler-Fischer et al. (2013) report a factor of 2 ambiguityin the rotational period of HATS-2. We have used P rot = .
98 dfor our analysis since this gives a value of τ gyro that is more con-sistent with the value of τ iso . http://cdsweb.u-strasbg.fr/denis.html Fig. 1.
WASP photometry for HAT-P-21 plotted as a function of rotationphase assuming P rot = .
3. Analysis and Results
We used version 1.1 of the program bagemass (Maxted et al.2015) to calculate the joint posterior distribution for the massand age of each star based on the observed values of T e ff , [Fe / H],the mean stellar density ρ ⋆ and, if available, L ⋆ . The stellar mod-els used for our analysis were produced with the garstec stellarevolution code (Weiss & Schlattl 2008). The initial compositionof the models is computed assuming a cosmic helium-to-metalenrichment ∆ Y / ∆ Z = ( Y ⊙ − Y BBN ) / Z ⊙ , where Y BBN = . Z ⊙ = . Y ⊙ = . ∆ Y / ∆ Z = . <
2% of breakupvelocity in all but two cases) and so it is not expected that rota-tion would play a direct role in the structure of these stars. We donot rule out other, indirect e ff ects, e.g. low convective e ffi ciencyassociated to large magnetic fields, but these have to be modelledin an ad-hoc manner. This is discussed later in this work, in rela-tion to specific stars in the sample. The methods used to calculateand interpolate the stellar model grid are described in Serenelliet al. (2013) and Maxted et al. (2015). We set lower limits of80 K on the standard error for T e ff and 0.07 dex for the standarderror on [Fe / H] (Bruntt et al. 2010) and assume flat prior dis-tributions for the stellar mass and age. The results are given inTable 3, where the maximum-likelihood (best-fit) values of thestellar mass and age are denoted M b and τ iso , b , respectively. Themean and standard deviation of the posterior distributions for themass and age are listed under h M ⋆ i and h τ iso i . Also listed in thistable are our estimates of the systematic errors in these valuesdue to an assumed error of 0.2 for α MLT and an assumed errorof 0.02 for the initial helium abundance, Y . The change in theestimated mass and age due to increasing Y by its assumed errorare given by the quantities σ M , Y and σ τ, Y , respectively. Similarly, σ M ,α and σ τ,α quantify the change in the estimated mass and agedue to the error in α MLT . We show the best-fit values of M ⋆ and τ iso and the e ff ects of changing Y and α MLT by their assumeduncertainties for all the stars in the sample in Fig. 2. http://sourceforge.net/projects/bagemass Article number, page 3 of 11page.11 & Aproofs: manuscript no. paper-arxiv
Table 1.
Observed properties of stars in our sample.
Star P rot P orb T e ff [Fe / H] ρ ⋆ f ⊕ log( L ⋆ ) Ref.[d] [d] [K] [ ρ ⊙ ] [pW m − ] [ L ⊙ ]55 Cnc 39 . ± .
00 0.74 5234 ± + . ± .
04 1 . + . − . . ± . − . ± .
009 1, 2, 3, 4CoRoT-2 4 . ± .
02 1.74 5598 ± + . ± .
05 1 . ± .
064 5, 6CoRoT-4 8 . ± .
12 9.20 6190 ± + . ± .
07 0 . + . − .
7, 8CoRoT-6 6 . ± .
50 8.89 6090 ± − . ± .
10 0 . ± .
064 9, 8CoRoT-7 23 . ± .
62 0.85 5313 ± + . ± .
07 1 . ± .
073 10, 11, 12CoRoT-13 13 . + . − . ± + . ± .
07 0 . ± .
072 13, 8CoRoT-18 5 . ± .
40 1.90 5440 ± − . ± .
10 1 . ± .
160 14, 6HAT-P-11 30 . + . − . ± + . ± .
05 2 . ± .
097 5 . ± . − . ± .
035 15, 8HAT-P-21 15 . ± .
80 4.12 5634 ± + . ± .
08 0 . ± .
150 16, 12HATS-2 24 . ± .
04 1.35 5227 ± + . ± .
05 1 . ± .
060 17, 17HD 189733 11 . ± .
01 2.22 5050 ± − . ± .
05 1 . ± .
170 27 . ± . − . ± .
024 18, 19HD 209458 10 . ± .
75 3.52 6117 ± + . ± .
05 0 . ± .
008 23 . ± . . ± .
041 20, 19Kepler-17 12 . ± .
56 1.49 5781 ± + . ± .
10 1 . + . − .
21, 6Kepler-30 16 . ± .
40 29.33 5498 ± + . ± .
27 1 . ± .
070 22, 23, 22Kepler-63 5 . ± .
01 9.43 5576 ± + . ± .
08 1 . + . − .
24, 24Qatar-2 11 . ± .
50 1.34 4645 ± − . ± .
08 1 . ± .
016 25, 25WASP-4 22 . ± .
30 1.34 5540 ± − . ± .
09 1 . ± .
022 26, 6WASP-5 16 . ± .
40 1.63 5770 ± + . ± .
09 0 . ± .
080 6WASP-10 11 . ± .
05 3.09 4675 ± + . ± .
20 2 . + . − .
27, 28, 29WASP-19 11 . ± .
09 0.79 5460 ± + . ± .
11 0 . ± .
006 30, 31WASP-41 18 . ± .
05 3.05 5450 ± − . ± .
09 1 . ± .
140 32, 32WASP-46 16 . ± .
00 1.43 5600 ± − . ± .
13 1 . ± .
100 33, 33WASP-50 16 . ± .
50 1.96 5400 ± − . ± .
08 1 . ± .
032 34, 35, 34WASP-69 23 . ± .
16 3.87 4700 ± + . ± .
08 1 . ± .
130 36, 36WASP-77 15 . ± .
40 1.36 5500 ± + . ± .
11 1 . + . − .
37, 37WASP-84 14 . ± .
35 8.52 5300 ± + . ± .
10 2 . ± .
070 36, 36WASP-85 14 . ± .
47 2.66 5685 ± + . ± .
10 1 . ± .
010 38, 38WASP-89 20 . ± .
40 3.36 4960 ± + . ± .
14 1 . + . − .
39, 39
References. (1) Henry et al. (2000); (2) Dragomir et al. (2014); (3) Valenti & Fischer (2005); (4) Boyajian et al. (2013); (5) Lanza et al. (2009);(6) Southworth (2012); (7) Aigrain et al. (2008); (8) Southworth (2011); (9) Fridlund et al. (2010); (10) Lanza et al. (2010); (11) Barros et al.(2014); (12) Torres et al. (2012); (13) Cabrera et al. (2010); (14) Hébrard et al. (2011); (15) Sanchis-Ojeda & Winn (2011); (16) Bakos et al.(2011); (17) Mohler-Fischer et al. (2013); (18) Henry & Winn (2008); (19) Southworth (2010); (20) Silva-Valio (2008); (21) Béky et al. (2014);(22) Sanchis-Ojeda et al. (2012); (23) Fabrycky et al. (2012); (24) Sanchis-Ojeda et al. (2013); (25) Mancini et al. (2014); (26) Sanchis-Ojedaet al. (2011); (27) Smith et al. (2009); (28) Christian et al. (2009); (29) Barros et al. (2013); (30) Tregloan-Reed et al. (2013); (31) Mancini et al.(2013); (32) Maxted et al. (2011); (33) Anderson et al. (2012); (34) Gillon et al. (2011); (35) Tregloan-Reed & Southworth (2013); (36) Andersonet al. (2014); (37) Maxted et al. (2013); (38) Brown et al. (2014); (39) Hellier et al. (2014).
Our Bayesian MCMC method produces a large set of points inthe mass-age parameter space (“Markov chain”) that has thesame distribution as the posterior probability distribution forthese parameters. We used equation (32) from Barnes (2010) tocalculate τ gyro from the rotation period and the mass of the starfor every point in the Markov chain for each star. The convec-tive turn-over time scale that encapsulates the mass dependenceof this mass-age-rotation relation was interpolated from Table1 of Barnes & Kim (2010). This table suggests that stars withmasses M ⋆ > ≈ . M ⊙ do not have convective envelopes so wehave restricted our analysis to stars with P ( M ⋆ < . M ⊙ ) > . τ gyro requiresan estimate for the value of the parameter P . We account for theuncertainty in this value by randomly generating a value of P uniformly distributed in the range 0.12 d to 3.4 d for each pointin the Markov chain. In principle, there is some additional uncer-tainty in τ gyro due to the stars’ surface di ff erential rotation com-bined with the variation in the latitude of the active regions thatproduce the star spot modulation (Epstein & Pinsonneault 2014).We assume that this uncertainty a ff ects the calibration sampleused by Barnes (2010) to the same extent that it a ff ects the stars Article number, page 4 of 11page.11. F. L. Maxted et al.: Ages of transiting exoplanet host stars
Fig. 2.
Change in the best-fitting masses and ages of transiting exoplanet host stars due to a change in the assumed helium abundance or mixinglength parameter. Dots show the best-fitting mass and age for the default values of Y and α MLT and lines show the change in mass and age due toan increase in helium abundance ∆ Y = + .
02 (left panel) or a change in mixing length parameter ∆ α MLT = − . in our sample so that this uncertainty is already accounted forby randomly perturbing the parameter P . The uncertainties inthe observed values of P rot are accounted for in a similar way,but using Gaussian random distributions for the standard errorsshown in Table 1. Fig. 3 shows the joint posterior distributionsfor ( M ⋆ , τ iso ) and ( τ iso , τ gyro ) calculated from the Markov chainusing this method for CoRoT-13.For each star we compare the values of τ gyro calculated usingthe mass value at every point in the Markov chain to the cor-responding τ iso values for the same point in the Markov chain.To quantify the di ff erence between τ iso and τ gyro we calculatethe fraction of points p τ in the chain for which τ gyro > τ iso , i.e., p τ = P ( τ gyro > τ iso ) is the probability that the gyrochronogicalage is greater than the isochronal age. The results of this com-parison are given in Table 4 and are shown in Fig. 4. Also givenin Table 4 is τ tidal , which is a very approximate estimate of thetime scale for tidal interactions between the star and the planet(Albrecht et al. 2012). It must be emphasized that the actual timescale for tidal spin-up in these systems is uncertain by a few or-der of magnitude (Ogilvie 2014). For multi-planet systems, τ tidal applies to the innermost planet. The values of τ gyro as a functionof τ tidal are shown in Fig. 5.
4. Discussion
It is clear that the gyrochronological age is significantly less thanthe isochronal age for about half of the stars in this sample. Thisdiscrepancy is apparent from Fig. 4, but the errors on these val-ues are correlated and non-Gaussian so to accurately quantifythis discrepancy we need to use the values of p τ in Table 4. Thereis no obvious relation between the gyrochronological age of thestars and the estimated time scale for tidal interactions betweenthe star and the planet. This may not be surprising given that τ tidal is uncertain by a few orders of magnitude (Ogilvie 2014). Itis also apparent that there is an even spread of τ gyro values fromvery young ages up to a maximum value of a few Gyr, i.e., thereis no obvious “pile-up” of stars with very low or very high τ gyro values as might be expected if some evolutionary or selection ef-fect were biasing the age distribution of the stars in this sample.The discrepancy between isochronal and gyrochronologicalages for some planet host stars has been noted for more lim-ited samples of planet host stars before, and has been cited as Fig. 4.
Comparison of gyrochronological ages ( τ gyro ) to isochronal ages( τ iso ) for planet host stars with measured rotation periods. Points witherror bars indicate the mean and standard deviation of the posterior agedistribution. The straight line is the relation τ gyro = τ iso . evidence for tidal spin-up of the stars by their planetary com-panions, or for the disruption of the normal spin-down processfor these stars (Pont 2009; Lanza 2010; Poppenhaeger & Wolk2014). However, there are other possibile explanations in somecases. It may be that the isochronal ages for some stars are notaccurate because of missing physics in the stellar models. It isalso possible for the distribution of active regions on a star toproduce a modulation of the light curve with a period of halfthe rotation period. This is not a viable explanation for the lowvalues of p τ in Table 4 in general, but may be an issue for afew stars with limited photometric data. We have been careful toselect stars for our sample for which the observational data arerobust, so the general appearance of distribution in Fig. 4 can-not be ascribed to observational errors, although there may beissues with the observed data for a few stars. The limitations ofthe observed data, the stellar models and the estimated value of τ tidal a ff ect di ff erent stars in this sample in di ff erent ways, so tounderstand the implication of these results we need to look at theresults on a case-by-case basis. Article number, page 5 of 11page.11 & Aproofs: manuscript no. paper-arxiv
Table 3.
Bayesian mass and age estimates for the host stars of transiting extrasolar planets using garstec stellar models assuming α MLT = . p MS ) is the probability that the star is still on the main sequence. The systematic errors on the mass and age due touncertainties in the mixing length and helium abundance are given in columns 9 to 12. Star τ iso , b [Gyr] M b [ M ⊙ ] [Fe / H] i , b χ h τ iso i [Gyr] h M ⋆ i [ M ⊙ ] σ τ, Y σ τ,α σ M , Y σ M ,α
55 Cnc 10.9 0.91 + .
378 0.63 10 . ± .
62 0 . ± .
020 0 .
48 3 . − . − . + .
057 0.02 2 . ± .
62 0 . ± .
034 1 .
24 1 . − . − . + .
076 0.01 2 . ± .
07 1 . ± .
044 0 .
42 0 . − . − . − .
160 0.01 3 . ± .
49 1 . ± .
048 0 .
39 1 . − . − . + .
046 0.02 2 . ± .
87 0 . ± .
029 1 .
87 0 . − . − . + .
083 0.02 5 . ± .
40 1 . ± .
053 0 .
30 1 . − . − . − .
011 0.01 10 . ± .
27 0 . ± .
043 1 .
37 3 . − . − . ⋆ + .
238 9.26 0 . ± .
83 0 . ± .
016 0 . − . − .
023 0 . + .
121 0.01 9 . ± .
26 0 . ± .
045 0 .
66 2 . − . − . + .
214 0.01 9 . ± .
77 0 . ± .
037 0 .
99 3 . − . − . − .
014 0.02 4 . ± .
15 0 . ± .
023 2 .
45 1 . − . − . + .
065 0.20 2 . ± .
80 1 . ± .
038 0 .
30 1 . − . − . ⋆ + .
252 0.04 1 . ± .
07 1 . ± .
034 0 .
53 0 . − .
040 0 . + .
192 0.01 4 . ± .
24 0 . ± .
065 1 .
23 0 . − .
049 0 . + .
074 0.00 3 . ± .
88 0 . ± .
037 1 .
08 1 . − . − . ⋆ + .
149 3.56 15 . ± .
36 0 . ± .
013 0 .
00 0 . − .
018 0 . + .
019 0.04 6 . ± .
34 0 . ± .
045 0 .
95 2 . − . − . + .
150 0.00 5 . ± .
86 1 . ± .
048 0 .
62 2 . − . − . + .
033 0.03 6 . ± .
12 0 . ± .
031 2 .
56 0 . − .
039 0 . + .
222 0.00 9 . ± .
49 0 . ± .
048 0 .
88 3 . − . − . − .
022 0.01 8 . ± .
59 0 . ± .
049 1 .
21 3 . − . − . − .
294 0.01 10 . ± .
51 0 . ± .
051 0 .
50 3 . − . − . − .
054 0.02 8 . ± .
86 0 . ± .
041 1 .
15 3 . − . − . ⋆ + .
251 0.13 15 . ± .
55 0 . ± . − .
04 2 . − . − . + .
055 0.04 7 . ± .
53 0 . ± .
049 0 .
46 3 . − . − . ⋆ − .
072 0.73 1 . ± .
61 0 . ± . − .
09 0 . − .
010 0 . + .
104 0.02 2 . ± .
38 0 . ± .
036 1 .
12 0 . − .
049 0 . + .
236 0.01 12 . ± .
11 0 . ± .
037 1 .
00 4 . − . − . ⋆ Best-fit is for age near the edge of the stellar model grid – σ τ, Y and σ τ,α may not be reliable. Fig. 3.
Left panel: Joint posterior distribution for the mass and age of CoRoT-13 estimated by our isochrone fitting technique. Right panel: Jointposterior distribution for the age of CoRoT-13 estimated by our isochrone fitting technique and using gyrochronology. For clarity, only 10% of thepoints from the Markov chain are plotted.
There is no satisfactory fit to the observed properties of HAT-P-11 at any age or mass for our grid of standard stellar models,i.e. models using a mixing length calibrated on the solar model( α MLT = .
78) and ∆ Y / ∆ Z = . Kepler short-cadence (SC) data for this star. The
Kepler data clearly show distortions to the light curve due to theplanet crossing dark spots on the face of the host star. Di ff er- Article number, page 6 of 11page.11. F. L. Maxted et al.: Ages of transiting exoplanet host stars
Table 4.
Stellar ages measured using our stellar models, τ iso , and using gyrochronology, τ gyro , and the probability p τ = P ( τ gyro > τ iso ). A veryapproximate estimate for the time scale for tidal interaction between the star and the inner-most planet is given under log( τ tidal ). Notes includeconstraints on the age of the star and any companion stars from the X-ray luminosity ( τ X ), and age constraints from the star’s surface lithium (Li)abundance τ Li Star τ iso τ gyro p τ log( τ tidal ) Notes[Gyr] [Gyr] [yr]55 Cnc 10 . ± .
62 8 . ± .
54 0.23 12.8 Companion star τ X > . ± .
62 0 . ± .
06 0.02 10.3 τ Li ≈ . τ X ≈ .
25 Gyr, but companion τ X > . ± .
06 1 . ± .
17 0.39 14.3 τ iso consistent with τ gyro but large relative error on both.CoRoT-6 3 . ± .
49 0 . ± .
10 0.01 13.0 Fast rotator.CoRoT-7 2 . ± .
87 2 . ± .
82 0.52 13.1 τ iso consistent with τ gyro but large relative error on both.CoRoT-13 5 . ± .
40 2 . ± .
92 0.08 11.9 Tidal spin-up?CoRoT-18 10 . ± .
28 0 . ± .
08 0.00 10.2 Tidal spin-up?HAT-P-11 0 . ± .
83 3 . ± .
89 0.98 15.7 Poor isochrone fit – helium-rich?HAT-P-21 9 . ± .
26 1 . ± .
29 0.00 11.1 τ X = τ X > ⇒ tidal spin-up.HATS-2 9 . ± .
77 3 . ± .
30 0.01 10.5 Tidal spin-up?HD 189733 4 . ± .
15 0 . ± .
09 0.07 11.9 τ X = τ X > ⇒ tidal spin-up.HD 209458 2 . ± .
79 1 . ± .
85 0.29 12.5 τ iso consistent with τ gyro but large relative error on both.Kepler-17 1 . ± .
07 1 . ± .
43 0.53 10.3 τ iso consistent with τ gyro but large relative error on both.Kepler-30 4 . ± .
24 1 . ± .
24 0.22 19.0 τ iso consistent with τ gyro but large relative error on both.Kepler-63 3 . ± .
88 0 . ± .
06 0.02 15.8 Fast rotator.Qatar-2 15 . ± .
36 0 . ± .
10 0.00 10.2 Inflated K-dwarf.WASP-4 6 . ± .
34 2 . ± .
83 0.09 10.6 Tidal spin-up?WASP-5 5 . ± .
86 2 . ± .
52 0.05 10.5 Tidal spin-up?WASP-10 6 . ± .
12 0 . ± .
10 0.06 11.6 Tidal spin-up?WASP-19 9 . ± .
49 0 . ± .
12 0.00 9.5 Tidal spin-up?WASP-41 8 . ± .
59 1 . ± .
21 0.03 12.3 Magnetically active G-type star – τ iso unreliable?WASP-46 10 . ± .
51 1 . ± .
20 0.01 10.3 Tidal spin-up?WASP-50 8 . ± .
86 1 . ± .
15 0.00 11.2 Tidal spin-up?WASP-69 15 . ± .
55 2 . ± .
12 0.00 13.9 Inflated K-dwarf.WASP-77 7 . ± .
53 1 . ± .
18 0.00 10.3 Companion V rot sin i ⋆ ⇒ age ≈ τ iso unreliable?WASP-84 1 . ± .
61 0 . ± .
10 0.36 14.7 Poor isochrone fit – helium-rich?WASP-85 2 . ± .
37 1 . ± .
33 0.41 12.0 τ iso consistent with τ gyro but large relative error on τ iso .WASP-89 12 . ± .
11 1 . ± .
18 0.00 10.9 Tidal spin-up?
Fig. 5.
Gyrochronological ages as a function of log( τ tidal ), a very ap-proximate estimate of the time scale for tidal interaction between thestar and the innermost planetary companion. Note that the time scalefor tidal interactions is uncertain by a few orders of magnitude. Sys-tems with significant di ff erences between τ iso and τ gyro ( p < .
05) areplotted with filled symbols. 55 Cnc ( τ iso ≈ τ tidal / y) ≈
13) isnot shown here. ent investigators have accounted for these spot-crossing eventsin di ff erent ways, but the stellar densities derived are all consis-tent with the value we have used in our analysis. There are alsothree independent analyses of the spectrum of this star that havebeen used to derive the T e ff and [Fe / H]. Again, all three valuesare consistent with each other, including the value we have usedfor our analysis.In principle, the
Kepler short-cadence photometry for HAT-P-11 can be used to estimate the density of this star using aster-oseismology. Christensen-Dalsgaard et al. (2010) analysed the
Kepler data for HAT-P-11 from the commissioning period andthe first month of regular observations and claimed definite evi-dence for solar-like oscillations, yielding a preliminary estimateof its mean density. However, an analysis of the complete
Kepler data set for this star shows no convincing evidence for solar-likeoscillations because HAT-P-11 has a much higher level of pho-tometric noise than a typical star of the same magnitude (Davies,priv. comm.).In general, the analysis of the light curve for a transiting ex-trasolar planet makes the assumption that, apart from the e ff ectsof limb-darkening, the mean surface brightness of a star is thesame as the surface brightness in the regions obscured by theplanet. This would not be the case if, for example, the planettransits a chord near and approximately parallel to the stellarequator of a star with dark spots near its poles. If the star spots Article number, page 7 of 11page.11 & Aproofs: manuscript no. paper-arxiv
Table 5.
Independent measurements of the mean density ( ρ ⋆ ), e ff ective temperature (T e ff ) and metallicity ([Fe / H]) for HAT-P-11 and their weightedmean values. ρ ⋆ /ρ ⊙ T e ff [K] [Fe / H] Source Notes1.75 + . − . ±
50 0 . ± .
05 Bakos et al. (2010) Discovery paper2 . ± .
10 Southworth (2011)
Kepler
Q0 – Q22 . + . − . Sanchis-Ojeda & Winn (2011)
Kepler
Q0 – Q22 . ± .
11 Deming et al. (2011)
Kepler
Q0 – Q2 plus B-band and J-band4792 ±
69 0 . ± .
07 Torres et al. (2012)4624 ±
225 0 . ± .
08 Mortier et al. (2013)2 . ± .
08 Müller et al. (2013)
Kepler
Q0 – Q6that are not occulted are equivalent to completely black spotsthat cover a fraction f of the stellar disc, then from equation (19)of Seager & Mallén-Ornelas (2003), we can estimate that thedensity will be systematically too high by a factor (1 − f ) − . Thedensity of HAT-P-11 predicted by our stellar models based on theobserved values of T e ff , L ⋆ and [Fe / H] assuming τ iso <
10 Gyris ρ ⋆ = . ± . ρ ⊙ . If this were to be explained by unocculteddark spots this would require f ≈ .
3. Such a large value of f can be ruled out in the case of HAT-P-11 because of the unusualorbit of the planet relative to the rotation axis of the star. Therotation and orbital axes are almost perpendicular and the ratioof the orbital and rotation periods is close to 1:6. This meansthe planet e ff ectively scans the stellar disc along 6 di ff erent linesof longitude. Distortions to the light curve are seen due to theplanet crossing star spots with a typical size of about 5 ◦ locatedin two bands of latitude, similar to the pattern of spots seen onthe Sun (Sanchis-Ojeda & Winn 2011; Béky et al. 2014). Thesespots are too small to explain the large value of ρ ⋆ inferred fromthe transit light curve of HAT-P-11.In principle, we could find a set of stellar models that matchthe properties of HAT-P-11 by increasing the assumed mixinglength parameter, but energy transport by convection is expectedto be less e ffi cient in magnetically active stars like HAT-P-11,not more e ffi cient as a larger value of α MLT would imply (Feiden& Chaboyer 2013). A more plausible explanation for the highdensity of HAT-P-11 is that this star has an abnormally high he-lium abundance. We used garstec to calculate a grid of stellarmodels identical to the grid of standard stellar models describedabove, but with the helium abundance increased by an amount ∆ Y = + .
05. With this “very helium enhanced” stellar modelgrid we find a good fit to the observed properties of HAT-P-11( χ = .
34) for a mass h M ⋆ i = . ± . M ⊙ and an age h τ ⋆ i = . ± . Y and Z that we have used. Another star in oursample with τ iso , b ≈ χ ≈ . ν = It has long been known that some K-dwarfs appear to be largerby 5 per cent or more than the radius predicted by standard stel-lar models (Hoxie 1973; Popper 1997). This “radius anomaly”is correlated with the rotation rate of the star, but also showssome dependence on the mass and metallicity of the star (López-Morales 2007; Spada et al. 2013). The dependence on rotationis thought to be the result of the increase in magnetic activityfor rapidly rotating stars. Magnetic activity can a ff ect the struc-ture of a star by producing a high coverage of starspots, whichchanges the boundary conditions at the surface of the star, or byreducing the e ffi ciency of energy transport by convection. What-ever the cause of the radius anomaly in K-dwarfs, the existenceof inflated K-dwarfs is one likely explanation for the appear-ance of stars in Table 3 that seem to be older than the Galacticdisc (10 Gyr, Cojocaru et al. 2014) or older than the Universe(13.75 Gyr, Hinshaw et al. 2013).One method that has been proposed to deal with the radiusanomaly is to simulate the magnetic inhibition of convectionby reducing the mixing length parameter (Chabrier et al. 2007).This phenomenological approach has some support from stellarmodels that incorporate magnetic fields in a self-consistent way(Feiden & Chaboyer 2013). Qatar-2 is one K-dwarf in our sam-ple that has an isochronal age that is clearly older than the age ofthe Galactic disc ( P ( τ iso <
10 Gyr) = . α MLT from 1.22 to 2.32.We find that models with α MLT < ≈ . α MLT is a source of systemic error thatpotentially a ff ects the mass and age estimates for all the stars thatwe have studied. This includes the stars for which we have founda satisfatory fit to the observations for some age within the rangeexpected (0 to 10 Gyr), i.e., it may be that we have assumed aninappropriate value of α MLT for some stars but that this is hiddenby deriving incorrect (but plausible) values for the age and mass.
From Table 4 we see that there are nine stars for which there isgood agreement between τ iso and τ gyro ( p τ > . Article number, page 8 of 11page.11. F. L. Maxted et al.: Ages of transiting exoplanet host stars is partly due to the large uncertainty in one or both of theseage estimates. The time scale for tidal spin-up of the star de-pends sensitively on the structure of the star and whether thereis any resonance between the orbit of the planet and internalgravity wave modes in the star (Ogilvie 2014), so we shouldnot expect that there is a precise value of log( τ tidal / yr) that di-vides systems with and without tidal spin-up. Nevertheless, thegood agreeement between the isochronal and gyrochronologi-cal age for HD 209458 and CoRoT-4 suggests that tidal spin-up for planet host stars may be ine ffi cient for systems withlog( τ tidal / yr) > ∼ . Maxted et al. (2015) compared the observed masses of stars indetached eclipsing binaries to the masses predicted using bage - mass based on their density, e ff ective temperature and metallic-ity. They found that masses of some stars with orbital periodsless than about 6 days were under-predicted by about 0.15 M ⊙ .The rotation periods of these stars are expected to be equal totheir orbital periods due to strong tidal interactions between thestars. There were no stars in the sample of eclipsing binaries usedby Maxted et al. (2015) with orbital periods in the range 7 – 14days, so it is not known whether stars with rotation periods inthis range are a ff ected by the same problem. The cause of thisdiscrepancy is not known so the isochronal ages for some starswith rotation periods less than about 6 days will be unreliable,and there is also the possibility that this problem a ff ects starswith rotation periods up to about 14 days.There are four stars in our sample of planet host stars withrotation periods less than about 6 days (CoRoT-2, CoRoT-6,CoRoT-18 and Kepler-63). All four stars are clear examplesof the gyrochronological age being significantly lower than theisochronal age ( p τ < . ff ect on our conclusions as thereare several other examples of stars with gyrochronological agessignificantly lower than their isochronal age.If we take the more cautious approach and exclude all starsin our sample with rotation periods less than 14 days and HAT-P-11 (discussed above), we are left with a sample of 14 stars.Of these, about half have a gyrochronological ages significantlylower than their isochronal age ( p τ < . τ tidal / yr) > ∼ . ff ected by magnetic inhibition of convection in a sim-ilar way to K-dwarfs. WASP-41 is another G-dwarf for whichthe gyrochronological age is significantly less than its isochronalage ( p τ = .
03) despite having a very long tidal time scale(log( τ tidal / yr) = . P rot = . Brown (2014) found a “slight tendency for isochrones to pro-duce older age estimates” but that the “evidence for any bias ona sample-wide level is inconclusive.” All 8 stars in that samplewith directly measured rotation periods have been re-analysedhere. For 7 of these stars, our results are consistent with thoseof Brown (2014). The exception is WASP-50, for which Brownuses a value of T e ff based on photometric colour (Brown, priv.comm.), rather than the lower and more accurate value that wehave used here based on an analysis of the spectrum from Gillonet al. (2011). From a comparison of the gyrochronological andisochronal ages, Brown notes that of these 8 stars, 2 show “anage di ff erence of a few Gyr”. We see very clear discrepanciesbetween τ iso and τ gyro for 7 of these 8 stars, i.e., p τ < .
1, oftenmuch lower. We have restricted our analysis to stars with directlymeasured P rot values and increased the number of such starsstudied to 28 so we very clearly see that τ iso and τ gyro disagreefor the majority of stars common to both studies. A key di ff er-ence between our study and the study by Brown is the way thatthe uncertainties on the ages have been calculated. Brown cal-culated the range of isochronal ages corresponding to the “1- σ ”error ellipse on T e ff and ρ ⋆ (the error on [Fe / H] was neglected)and then used this range as the standard deviation of a normaldistribution to represent the posterior probability distribution for τ iso . The correlation between τ iso and τ gyro via their mutual cor-relation with the assumed stellar mass was not considered. Withour Markov chain method we can accurately account for the er-rors in τ iso and τ gyro from the errors in T e ff , ρ ⋆ and [Fe / H] andcalculate the joint posterior distribution for these values (e.g.,Fig. 3). This enables us to accurately calculate a statistic like p τ that has greater statistical power than the method used by Brown.Pont (2009) used the full sample of 41 transiting exoplanethost stars known at that time to investigate whether there wasempirical evidence for tidal evolution in these planetary systems.He identified two stars (HD 189733 and CoRoT-2) that showlarge excess rotation among the late-type stars in this sample(T e ff < p τ < . p τ < . p τ = . Poppenhaeger & Wolk (2014) took a very di ff erent approach totesting whether hot Jupiters a ff ect the rotation rate of their hoststars. They compared the stellar coronal X-ray emission of 5planet host stars with their companion stars in wide binary sys-tems and used this activity indicator to estimate the age of bothstars in each binary system. They found much higher magneticactivity levels in HD 189733 and CoRoT-2 than in their compan-ion stars. The estimated age of HD 189733 based on the X-rayflux is τ X = τ X = . τ X > ff erence in age between the com-panion and the planet host star for τ Boo ( τ X = ν And
Article number, page 9 of 11page.11 & Aproofs: manuscript no. paper-arxiv ( τ X > τ X > τ gyro and τ X for CoRoT-2 and τ gyro is consistent with τ X in the case of 55 Cnc. The probability that τ iso for CoRoT-2 calculated using our grid of standard stellarmodels is consistent with the value of τ X for its companion is P ( τ iso > = .
09, i.e., if we assume that the value of τ X forthe companion is an accurate estimate of actual age of CoRoT-2then the isochronal age is likely to be an underestimate of the trueage. Increasing the helium abundance by about + α MLT by about 0.4, or changing both these factors by abouthalf as much, would be su ffi cient to bring the τ iso and τ X intoagreement. For 55 Cnc, both τ iso and τ gyro are consistent witheach other and with the lower limit τ X > ff ected by their planetary companions or whatproperties determine the strength of this interaction, but it is clearthat τ gyro is not a reliable estimate of the age for some planet hoststars.The lithium abundance at the surface of a star can provideuseful constraints on the age of a star, particularly for starsyounger than about 600 Myr where there are good data in var-ious open clusters that can be used to calibrate the dependanceof the lithium depletion rate with mass. We have compared theobserved lithium abundances for the stars listed in the Table 4 tothe calibration data from Sestito & Randich (2005) to see if theresulting constraints on the age are consistent with the values of τ iso and τ gyro . In general we can only set a lower limit to the age > ≈ . ≈ . . + . − . Gyr, con-sistent with their estimate of 1.0 + . − . Gyr for the gyrochronologi-cal age of WASP-77. Maxted et al. (2013) used a di ff erent cali-bration for the gyrochronological age to the one used here, but itis clear that the gyrochronological age of companion to WASP-77 is inconsistent with the age for WASP-77 estimated using agrid of standard stellar models. If we assume a lower heliumabundance for WASP-77 (Fig. 2) we will obtain a slightly lowerestimate for the isochronal age, but it is not possible to reconcilethe gyrochronological age and isochronal age without using ahelium abundance less than the lower limit set by the primordialcontent of the Universe. It is possible to reconcile the isochronalage with the gyrochronological age by using a reduced value ofthe mixing-length parameter α MLT much lower than the solar-calibrated value. WASP-77 shows chromospheric emission linescharacteristic of magnetically active stars despite its moderaterotation period, P rot = . α MLT isconsistent with the idea of magnetic inhibition of convection thathas been succesfully applied to explain the radius anomaly in K-dwarfs, but extends the idea in this case to a G-type star. We take the dividing line between K-type and G-type dwarfs to be0.8 M ⊙ in mass or T e ff = Meibom et al. (2015) have recently used
Kepler photometry of30 stars in the open cluster NGC 6811 to measure their rota-tion periods. The calibration used here (Barnes 2010) predictsrotation periods for stars in good agreement with the observedvalues at the age of the cluster (2.5 Gyr) across the entire massrange studied (0.85 – 1.25 M ⊙ ). Epstein & Pinsonneault (2014)criticised the method used by Barnes (2010) to derive their cali-bration. We used Fig. 7 and Fig. 8 of their paper to estimate “by-eye” the gyrochronological ages of all the stars in our sampleand found that these generally agree well with the values derivedhere, particularly if the calibration using the Kawaler style windmass loss rate is used. For rapidly rotating stars the calibrationof Epstein & Pinsonneault (2014) provides only an upper limitof 0.5 Gyr. This is consistent with the young ages that we derivefor these stars, but it may be that these very young gyrochrono-logical ages are not reliable to the precision quoted here. This isunlikely to a ff ect our conclusions substantially because an up-per limit of 0.5 Gyr is generally su ffi cient to show that theserapidly rotating stars have gyrochronological ages inconsistentwith their isochronal ages. The calibration of Epstein & Pinson-neault (2014) was also applied to the data for stars in NGC 6811by Meibom et al. (2015) and found to give rotation periods thatare too low by about 10% for stars with masses ≈ . M ⊙ butthat agree well with the observed rotation periods for stars withmasses near 1 M ⊙ .
5. Conclusions
By using new data and improved analysis methods we haveshown that there is now good evidence that some exoplanethost stars rotate more rapidly than expected. For our sam-ple of 28 transiting exoplanet host stars, about half the sam-ple have gyrochronological ages that are significantly less thantheir isochronal ages. In a few such cases there are indepen-dent constraints on the age of the star that are consistent withthe isochronal age, which suggests that tidal spin-up of the hoststar has occured in these systems. However, in several cases it isnot clear that tidal interactions between the star and the planetare responsible for this discrepancy. For some K-type stars thisis a result of the well-known radius anomaly that may be due tomagnetic inhibition of convection. We find some evidence thatthis anomaly may also a ff ect some of the rapidly rotating and / ormagnetically active G-type stars in our sample, either from in-dependent age constraints on the age of the star (WASP-77) orbecause it may be implausible that the strength of the tidal in-teraction with the planet is strong enough to spin-up the star(CoRoT-6, Kepler-63 and, perhaps, WASP-41).Some planet-host stars (HAT-P-11 in particular) appear to bemuch denser than predicted by stellar models. These stars maybe significantly enhanced in helium. This makes it di ffi cult toassess the reliability of isochronal mass and age estimates forthese stars. There is currently no simple explanation for the in-consistency between the young age of CoRoT-2 implied by stel-lar models, gyrochronology, its X-ray flux and its high lithiumabundance with the very old age inferred for its K-type compan-ion based on its lack of X-ray flux.Our improved analysis methods have enabled us to show thatthere is now clear evidence that the gyrochronological ages ofsome transiting exoplanet host stars are significantly less thantheir isochronal ages. However, a careful consideration of all theavailable data on a case-by-case basis shows that it is not always Article number, page 10 of 11page.11. F. L. Maxted et al.: Ages of transiting exoplanet host stars clear that this is good evidence for tidal spin-up of the host starby the planet.
Acknowledgements.
JS acknowledges financial support from the Science andTechnology Facilities Council (STFC) in the form of an Advanced Fellowship.AMS is supported by the MICINN ESP2013-41268-R grant and the Generalitatof Catalunya program SGR-1458. PM is grateful to Prof. Rob Je ff ries for discus-sions about the lack of X-ray flux from the companion to CoRoT-2. We thankthe anonymous referee for their careful reading of the manuscript and commentsthat have helped to improve this paper. References
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