Discrepancies between isochrone fitting and gyrochronology for exoplanet host stars?

Abstract

1 INTRODUCTION

Determining stellar ages is notoriously difficult, but they are becoming increasingly important in the field of exoplanetary science as a stepping stone to a better understanding of the evolution of planetary systems. In order to fully characterize the time-scales involved in processes such as planet formation and destruction, orbital migration and circularization, and intrasystem dynamical interactions, it is vital that we are able to accurately assess the ages of exoplanet host stars. A wide range of methods exist for the evaluation of stellar age, making use of a disparate array of phenomena. Two that are particularly prevalent in the exoplanet literature are gyrochronology and isochrone fitting.

1.1 Isochrone fitting

Stellar model fitting, also known as isochrone fitting, is widely used owing to its relative ease of implementation. Traditionally, either absolute stellar magnitude, M_v (e.g. Edvardsson et al. 1993; Lachaume et al. 1999), or stellar surface gravity, log (g_s) (e.g. Bouchy et al. 2005; Konacki et al. 2005), is interpolated through theoretical models of stellar evolution along with the stellar effective temperature, T_eff. However, for exoplanetary studies, it has become common practice to replace M_v and log (g_s) with the cube root of the stellar density, as this can be constrained to high precision through transit photometry. This leads to a parameter space of [T_eff, (ρ_s/ρ_⊙)^{− 1/3}] (Sozzetti et al. 2007).

In principle, isochrone fitting is applicable to stars across the spectral range, but it can be difficult to determine ages for stars with spectral type later than mid-to-late G owing to the fact that they evolve very slowly, having nuclear burning time-scales that are longer than the age of the Galactic disc. The complex shape of isochrones close to the main-sequence (MS) turn-off can also pose problems, and linearly interpolating through isochrones is not always a valid approach owing to their non-uniform spacing (Soderblom 2010).

1.2 Gyrochronology

Gyrochronology is a method for determining a cool star's age through measurement of its rotation period and colour, and arose from observations showing that by the age of the Hyades the rotation of stars in stellar clusters tends to converge to a single period–colour–age relation. First suggested by Barnes (2003), it builds on the simple relationship between rotation period and age described by Skumanich (1972) to provide a model-independent alternative to age estimation methods that require distance measurements for the stars under examination. Subsequent development of the method in Barnes (2007) showed that gyrochronology provides age estimate that are more self-consistent than those derived through isochrone fitting. It has been demonstrated that, if rotation periods have been measured and the equations correctly calibrated, gyrochronology can provide ages with an accuracy of 10 per cent for F, G, K, and M spectral types (Mamajek & Hillenbrand 2008; Collier Cameron et al. 2009; Delorme et al. 2011b).

One drawback with the method is that it assumes that the natural rotational evolution of the star progresses free from any outside influence. This is not always the case; in both binary star systems and hot Jupiter exoplanetary systems, tidal torques between nearby bodies in close proximity can potentially overwhelm the natural spin-down that results from magnetic braking, at least for short periods of time. In addition, gyrochronology is not calibrated for hot, rapidly rotating, early-type stars, and is only limited to ‘solar-type (FGKM) stars’ (Barnes 2007). As transit searches prioritize stars of F or G spectral type (e.g. Bentley 2010, for WASP targets; Batalha et al. 2010, for Kepler targets), this is not particularly limiting, but Lanza (2010) suggests that gyrochronology may not always provide accurate age estimates for planetary systems. Lanza found that plotting P_rott^{− ζ} as a function of T_eff for planet-hosting stars gives a poor fit to the period–colour relation of Barnes (2007), and that the rotation periods of hot Jupiter hosts were, on average, a factor of 0.7 faster than non-planet-hosting stars; such a discrepancy would clearly lead to underestimation of the gyrochronology ages of stars with known planets with respect to their true age.

In this work, I investigate the ages of a sample of planetary systems primarily discovered by transit searches. I first discuss the methods that I have used to determine the ages of the stars in my sample, before comparing the results obtained using isochrone fitting to those obtained through gyrochronology. I also investigate the subset of my sample for which the sky-projected spin–orbit alignment angle has been measured, to check for biases in either of the age estimation methods that might be induced in misaligned systems.

2 IMPLEMENTATION

I consider a sample of 68 planet-hosting stars with 6226 ≤ T_eff ≤ 5273 K. These limits were chosen to restrict my sample to spectral types F7–G9 (inclusive), and are based on the values given in table B1 of Gray (2008). This restriction on the available parameter space avoids the problems encountered when isochrone fitting for stars with long MS lifetimes, and has an upper limit that coincides with the magnetic braking boundary at mid-to-late F spectral type observed by Kraft (1967). Stars with earlier spectral types than this show little-to-no relation between P_rot and age (Wolff, Boesgaard & Simon 1986), and are therefore poor targets for gyrochronology.

The majority of the sample, which is described in Table 1, consists of the host stars of sub-stellar companions discovered by the WASP project (Pollacco et al. 2006), with the remaining systems selected from the Holt–Rossiter–McLaughlin data base of René Heller ¹ as of 2013 October 24. Fig. 1 displays a colour–magnitude diagram for the sample as compared to the Yonsei–Yale (YY) isochrones for the zero-age MS (ZAMS) and other representative ages. I note that there are three systems which seem to lie to the left of the ZAMS (as well as a further two for which the uncertainties are such that agreement with the ZAMS is possible) in a position which seems to be somewhat unphysical. Either the (B − V) colours for these systems are substantially wrong or they are very young systems, although this seems unlikely given the selection constraints placed on my sample.

Figure 1.

Colour–magnitude diagram for the sample of stars detailed in Table 1. Solid circles represent stars with measured rotation periods and open circles represent those for which the rotation period was derived from stellar data. The solid star represents the location of the Sun. Various isochrones from the YY set are also represented: the ZAMS (black, solid line); 1 Gyr (red, dashed line); 5 Gyr (green, dot–dashed line), and 10 Gyr (blue, dotted line). Note that the position of the isochrones shifts slightly depending on the choice of models, and that the absolute magnitudes are calculated using estimated distances in the majority of cases.

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2.1 Isochrone ages

The choice of isochrones being used can have a large impact on the derived properties of planetary systems. Southworth (2009, 2010) suggests that multiple sets of isochrones should be used if at all possible, in preference to relying on a single formulation, as each model introduces its own systematic errors into the derived stellar parameters. I selected five sets of stellar models for my analysis: Padova isochrones (Marigo et al. 2008; Girardi et al. 2010); YY isochrones (Demarque et al. 2004); Teramo isochrones (Pietrinferni et al. 2004); Victoria-Regina isochrones (VRSS; VandenBerg, Bergbusch & Dowler 2006), and Dartmouth Stellar Evolution Database isochrones (DSED; Dotter et al. 2008).

The main difficulty of isochrone fitting is that it is an attempt to fit a single point to a three-dimensional [[M/H], T_eff, (ρ_s/ρ_⊙)^{− 1/3}] parameter space in order to derive associated parameters (age and stellar mass). The problem can trivially be reduced to a two-dimensional one by considering only a single metallicity value at a time, which I achieve by neglecting the uncertainty in [M/H]. To convert between [M/H] and Z, I use a value of Z_⊙ = 0.0189.

There are many possible fitting procedures. The simplest is to merely take the closest isochrone as the age of the system, but this often provides only crude estimates and has an accuracy that is constrained by the ages for which isochrones have been provided. A more involved approach would be to find the two closest isochrones and interpolate between them. Another alternative would be the Bayesian approach of Pont & Eyer (2004).

I have chosen to describe the [T_eff, (ρ_s/ρ_⊙)^{− 1/3}] surface to which the stellar data is being fitted, and then to use this description to define a small plane over which I can interpolate the stellar data. For this purpose, I use a Delaunay triangulation, computed for a sub-region of the full isochrone parameter space that is centred on the measured stellar parameters. For details, please see Appendix A. Uncertainties in my interpolated ages are calculated by interpolating combinations of the 1σ limits on both T_eff and (ρ_s/ρ_⊙)^{− 1/3} using the same procedure.

Southworth (2010, 2012) homogeneously studied large samples of exoplanet host stars, as part of which he carried out age determinations using a range of isochrones that included the YY isochrones. Southworth uses a more traditional isochrone interpolation method, and as such these papers provide a reasonable comparison to my results. Cross-matching the results of those two papers to my own (see Fig. 2) shows that there are eight systems in common in both cases. My results are generally compatible with those of Southworth (2012), although it is immediately apparent that the uncertainties in my ages are smaller. I suspect that this partly results from Southworth's use of multiple isochrones to determine the systematic contribution to their age uncertainties, inflating their error bars somewhat. My uncertainties are also likely to be underestimated owing to my disregard for the uncertainty in metallicity. Comparing to Southworth (2010), I find similar ages for the younger stars, whilst for the two oldest systems in common (WASP-4 and WASP-5), I find younger ages, although the uncertainties on the ages are substantial.

Figure 2.

Left: ages from Southworth (2010) as a function of ages calculated using the YY isochrones in conjunction with my Delaunay triangulation interpolation technique. Right: ages from Southworth (2012) as a function of ages calculated using the YY isochrones in conjunction with my Delaunay triangulation interpolation technique. The dotted line denotes y = x. The maximum age on both axes is set to the age of the Universe. Direct measurements of the stellar rotation period were available for systems marked by solid symbols. Open symbols mark stars for which the rotation period has been derived using stellar parameters. In both cases, the ages are broadly similar, although the uncertainties that I find are significantly smaller. This arises due to Southworth's use of multiple sets of isochrones to determine systematic contributions to the uncertainties on his ages.

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Takeda et al. (2007) studied a large sample of stars from the Spectroscopic Properties of Cool Stars (SPOCS) catalogue, calculating ages using the YY isochrones. Unfortunately, of the 1074 stars in their sample there are only 2 in common with this work – HD 209458 and HD 80606. Our age estimates for HD 209458 agree well, but for HD 80606, the age that I calculate is much younger. As with the Southworth studies, my uncertainties are smaller than those of Takeda et al.

2.2 Gyrochronology calculations

Broad-band colour indices for the WASP systems were derived using magnitude data from the AAVSO Photometric All-Sky Survey (APASS; Henden et al. 2012), accessed through the UCAC4 catalogue (Zacharias et al. 2013) and 2MASS (Skrutskie et al. 2006) for the (B − V) and (J − K) colours, respectively. For systems with available stellar rotation period measurements, I created a Gaussian distribution with mean and variance set to the known period and 1σ error, respectively. The distribution was sampled 10⁴ times, and for each sampling I calculated age estimates using equations (1)–(4). Final ages for each method were taken to be the median of the appropriate set of results, with 1σ uncertainties set to the values which encompassed the central 68.3 per cent of the data set.

Nine of the systems in my sample have directly measured rotation periods available. For these systems, I also calculated the rotation period, allowing me to compare the derived periods to the measured values. In five out of the nine cases, the two periods agree, although in the case of WASP-46 this is due to the substantial uncertainty in the derived period. For the remaining three systems, the periods disagree by more than 5σ. This is likely due to misalignment of the stellar rotation axis along the line of sight such that vsin I_s is not a good representation of the true rotation speed of the stars involved. Three of the systems for which the two periods disagree have a derived period that is longer than the measured period, supporting the case for a misaligned star. The exception is WASP-19, for which the derived period is significantly shorter than the measured period for reasons unknown.

It is also possible that differential rotation in the star has led to the rotation period being measured at a latitude other than the stellar equator. Evidence for differential rotation has in fact been observed for CoRoT-2 (Fröhlich et al. 2009; Huber et al. 2010), with possible indications also present for WASP-19 (Hellier et al. 2011; Tregloan-Reed, Southworth, & Tapper 2013), but the scale of the effect is insufficient to explain the discrepancy between the measured and derived rotation periods. Misalignment along the line of sight thus seems a more probable explanation for discrepant derived periods among my stellar sample, but without measurements of the stellar inclination the reliability of the derived periods is difficult to ascertain. Assuming i_orb = I_s initially seems reasonable, but it has been shown that some systems will have stellar inclinations such that this assumption is invalid (Schlaufman 2010).

In Fig. 3, I plot effective temperature, (B − V) colour, and (J − K) colour as functions of rotation period, and overplot both the position of the break in the Kraft rotation period curve (Kraft 1967) and relevant relationships between colour and period. As already noted, my sample is selected on T_eff with an upper limit that approximates to the temperature at which the Kraft break occurs. This is clear from the upper panel of Fig. 3, with only nine systems having uncertainties such that they might lie slightly ‘above’ the break. The location of the break in (J − K) space is also close to the edge of the sample; five systems appear to have (J − K) colour such that they lie ‘above’ the break, though the uncertainties are such that that number could be anywhere between 0 and 11. But when translated into (B − V) colour space, the Kraft break seems to shift within the sample, with three systems displaying lower (B − V) colour index than the position of the break even when uncertainties are accounted for. However, these three systems are the same as those that were out of place in Fig. 1; checking other sources suggests that the APASS calibration for these three stars is likely to be inaccurate, and I therefore exclude them when analysing ages calculated using equation (1).

Figure 3.

Upper: rotation period as a function of T_eff. Lower left: rotation period as a function of (B − V) colour. The solid, black curve represents the period–colour–age relation from equation (1), computed at the 0.625 Gyr age of the Hyades cluster. Lower right: rotation period as a function of (J − K) colour. The solid, black curve represents the period–colour relation from Collier Cameron et al. (2009), as in equation (2). In all three figures, the position of the Sun is denoted by the solid star, and the location of the break in the Kraft curve is represented by the vertical dotted line. Solid symbols denote stars with measured rotation periods and open symbols mark stars for which the rotation period has been derived using stellar parameters.

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3 COMPARING THE AGE CALCULATION METHODS

Although I have carried out similar analyses for all 20 combinations of the isochrones and gyrochronology relations mentioned above, in the discussion that follows I will concentrate on the results obtained using the YY isochrones and the Barnes (2010) gyrochronology formulation described by equation (4). The comparison utilizes only those systems with valid results for both methods. The maximum permitted age for any star was set to the current best estimate of the age of the Universe (Planck Collaboration et al. 2013, and other papers in the series), and systems with calculated ages greater than this were disregarded. This is perhaps a somewhat unrealistic upper bound; the age of the Galactic disc might be more suitable (and is thought to be somewhat younger than the Universe), but introduces its own set of problems. Do the thick and thin discs have the same age, and if not, which should be used? Or should the sample be split up by population, and if so, how would that be done (disc component membership is a difficult attribute to characterize)? For simplicity, I have stuck to the age of the Universe.

The null hypothesis of this work is that the two age calculation methods are equally accurate, and therefore that the ages calculated using the two different methods will agree. This may not always be true on a case-by-case basis, but when viewed as an overall sample, then agreement is the expected outcome.

3.1 Isochrones versus gyrochronology

As a starting point, I plot gyrochronology age as a function of isochrone age. If the two methods provided similar answers, I would expect a tightly correlated sequence centred on the line age_gyro = age_iso (within errors). However, there appears to be a preponderance of points lying towards the isochronal side of the line, suggesting that isochrone fitting tends to return ages that are older than those preferred by gyrochronological methods. Neglecting uncertainties, there are twice as many systems for which the isochrone age is older than for which the gyrochronology age is older; this ratio increases once uncertainties are taken into account, in a large part owing to the large uncertainties on my calculated gyrochronology ages. But the number of systems with error ellipses consistent with equal ages is, at 30 systems, more than half of the 57 systems for which valid ages were returned by both methods. This is a significant fraction of the sample, and indeed one would hope that this would be the case given the stated null hypothesis. However, the number of systems for which the isochrone age is still greater than the gyrochronology age once the uncertainties are taken into account is 22, whilst the converse case includes only 5 systems.

Another interesting facet of Fig. 4 is the distribution of the points along both axes. Just over half of the systems lie within a region defined by age_gyro < 4 Gyr and age_iso < 6 Gyr. This is not entirely surprising given the region of parameter space to which I have restricted the study. Rough estimates of τ_MS for stars at the limit of my parameter space are τ_MS = 3.5 Gyr for an F7 star and τ_MS = 11.4 Gyr for a G9 star (using masses from table B1 of Gray 2008). A drop-off after roughly 4 Gyr is consistent with this, as systems at the hotter end of the parameter range start to evolve off the MS, and are therefore no longer targeted by transit search programs. Including uncertainties in this analysis lowers the number of systems that are definitively within this high-density region to 21, with a further 19 which have error ellipses at least partially within this region of parameter space.

Figure 4.

Gyrochronology age, calculated using equation (4), as a function of isochrone-fitting age, found using the YY isochrones. The dashed line denotes y = x; systems clustered around this line show similar age values for different methods of calculation. The maximum age on both axes is set to the age of the Universe. Direct measurements of the stellar rotation period were available for systems marked by solid circles. For systems marked by open circles, P_rot was derived from vsin I and R_s according to equation (5). It appears that the gyrochronology ages have a slight tendency to be younger than isochrone-fitting ages, particularly for systems with measured rotation periods.

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In terms of the different methods, 60 per cent of the gyrochronology estimates are less than 4 Gyr, with possible stellar ages ranging from 0.3 Gyr up to the age of the Universe. For the isochrone-fitting estimates, 54 per cent are younger than 6 Gyr, with the estimates covering a similar range. It therefore seems, at first glance, that gyrochronology tends to return stellar age estimates which are slightly biased towards younger ages than the results from isochrone fitting. Whilst this conclusion is tempered somewhat by the magnitude of the uncertainties on the ages that I have calculated, particularly for the gyrochronology ages, it may be true even accounting for these.

To further examine the different distributions, I computed kernel density estimates (KDEs; Parzen 1962; Rosenblatt 1956) for the two data sets, additionally disregarding systems for which one or both of the two methods returned only an upper or lower limit on the age. One of the advantages of this visualization method compared to cumulative probability distributions or histograms is that it intrinsically accounts for the uncertainties in the measured parameters, giving a more accurate idea of the shapes of the distributions and allowing more concrete comparison between them. For each system, I took 10⁴ random samples from a normal distribution with a mean of 0 and a standard deviation of 1, scaling these random numbers according to the system's age and 1σ uncertainties. Combining these sets of sampled ages across all of the systems being examined, I used Scott's Rule (Scott 1992) to compute the KDEs. These can be seen in Fig. 5, sampled at 100 points evenly spaced between 0.0 Gyr and the age of the Universe.

Figure 5.

KDEs for the results that I obtained from isochrone fitting to the YY isochrones (solid, black line) and from gyrochronology using equation (4) (dashed, blue line). The highest peaks of the two distributions are at 2.8 Gyr for the isochrone-fitting results and 1.5 Gyr for the gyrochronology results. This suggests a small overarching offset between the sets of age estimates provided by the two methods.

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The highest peaks of the two KDEs lie at 2.8 Gyr and 2.0 Gyr for isochrone fitting and gyrochronology, respectively, again suggesting that there may be a slight difference in the age estimates being returned by the two methods. This visualization technique also provides another look at the different regions of parameter space occupied by the two sets of results, as the relative heights and widths of the two peaks again indicate that the gyrochronology results are concentrated in a slightly smaller region of parameter space than the isochrone-fitting results.

3.2 Δage analysis

To further investigate this bias, I calculated Δage = age_Iso − age_Gyro for each of the systems in my sample and computed a KDE for the set of results. Fig. 6 shows a small apparent offset towards positive Δage, in line with the suggestion from the previous section that isochrone fitting is returning ages which are slightly older than those from gyrochronology. The peak of the KDE lies at 1.8 Gyr, and the average upper and lower error bars on Δage are 4.0 and 2.1 Gyr, respectively, so this is an inconclusive 0.9σ effect. However, the KDE is asymmetrical, with a narrower peak but broader shoulder in positive Δage than in negative Δage; comparison to Fig. 5 shows that this derives from the isochrone-fitting KDE. This matches the distribution of the data in Fig. 4: there are more data for which the isochronal age is older than the gyrochronological age, but the uncertainties are large enough that they dilute the effect.

Figure 6.

Kernel density estimation for the difference between the age results obtained by isochrone fitting using the YY isochrones and by gyrochronology using equation (4). The vertical dotted line denotes Δage = 0, and the peak of the KDE is offset towards positive Δage. This again suggests that isochrone fitting is returning ages which are slightly older than those from gyrochronology.

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Although the effect in Δage is small, the comparison of the individual methods suggests that there might be a disagreement between the ages that are produced by gyrochronology and isochrone fitting. Does this possible discrepancy correlate with a physical parameter in the systems that I am studying? Is it that isochrone fitting is overestimating ages, or that gyrochronology is underestimating ages (or a combination of the two)? Fig. 4 certainly seems to imply the former, as the systems with measured rotation periods, and thus the most reliable gyrochronology ages, all show a tendency towards an older isochronal age, in some cases with strong significance. This could be an indication that my new method for determining isochrone ages is overestimating the ages of my systems; with the comparison sample that is available (see Fig. 2), it seems that this is not the case on average, although the comparison is very limited in scope. As noted in Section 1.2 though, gyrochronology is only applicable if no external factors act to modify the natural stellar spin-down. If hot Jupiter host stars are rotating more rapidly than expected, then their age would be underestimated.

3.3 The influence of tidal interactions

One possibility might be that the spin rate of the star is being modified somehow, and angular momentum exchange between the star and the planet's orbit provides one route by which such a scenario might occur. The chief method of angular momentum exchange within planetary systems is through tidal interaction, which has well-documented consequences for stellar spin. For this work, I am interested in the possibility of a link between the strength of the tidal interactions and the magnitude of the difference between my age estimates.

In Fig. 7, I plot Δage as a function of tidal time-scale. If angular momentum exchange is the cause of the discrepancy between the two age estimation methods, then I would expect the difference to be greatest for systems with the shortest tidal time-scales (i.e. the strongest tides). Unfortunately, the evidence is inconclusive owing to the size of the uncertainties on Δage, which means that any conclusion would be tentative at best. Any evidence for a trend is also countered by the number of systems for which Δage is apparently negative, although it is worth noting that several of these systems have substantial uncertainties such that they are consistent with Δage = 0.

Figure 7.

Δage as a function of τ_tidal, the tidal realignment time-scale. The shorter the time-scale, the stronger the tidal interactions within the system, and the greater the angular momentum exchange. No trend is apparent between Δage and τ_tidal, although two of the systems with measured rotation period (solid symbols) show significantly greater age difference and significantly shorter tidal time-scale than the other systems for which P_rot has been measured. The horizontal dashed line marks Δage = 0.

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Considering only the stars in my sample with directly measured rotation period (the solid data in Fig. 7) reveals a possible trend, with two systems exhibiting an age difference of a few Gyr, and tidal time-scales close to the minimum value for the sample. A sample size of only nine systems means that this is far from clear however, and any possible trend hinges on two of the systems. It is interesting, however, to further consider the WASP-19 system, which is the system with the shortest tidal time-scale, 3.89 Gyr, of those for which the rotation period has been measured.

WASP-19 b has the shortest orbital period of the WASP planets, and has an orbital semi-major axis of only 0.016 53 ± 0.000 13 au. |$\Delta {\rm age}=8.17^{+2.21}_{-0.92}$| Gyr for this system, implying that the gyrochronology age is being underestimated. Brown et al. (2011) investigated the possibility of tidal interactions, finding that it is possible that the star is undergoing tidal spin-up which would explain the underestimation of the gyrochronology age (assuming that the isochrone age is correct).

3.3.1 Tidal effects in the WASP-19 system

To check the plausibility of tidal spin-up, I consider a range of evolutionary scenarios for the WASP-19 system. A full investigation of the tidal effects is beyond the scope of this paper, but it is simple to compare the rotation periods produced by different combinations of parameters at the expected age of the system.

I used the tidal equations and integration procedure described in Brown et al. (2011), starting from a set of defined initial conditions. I fixed the stellar and secondary body tidal quality factors at the values determined for the WASP-19 system by Brown et al., set the orbital eccentricity to 0, and set P₀ = 1.1 d to determine the initial stellar rotation frequency. The secondary body's spin was assumed to be synchronized to its orbit. a₀ and M₂ were varied to produce different rotational histories for the star, with R₂ = R_Jup for planetary mass companions, and R₂ = R_⊙ for stellar mass companions.

I first calibrated the simulations by turning off tidal interactions such that the only influence on the rotation of the primary body was magnetic braking. At the age of |$8.91^{+2.21}_{-0.92}$| Gyr provided by the YY isochrones, this gives a rotation period of |$33.1^{+5.0}_{-3.0}$| d and a gyrochronology age of |$3.4^{+0.4}_{-0.5}$| Gyr using equation (4). There is still a substantial offset between this and the isochronal age estimate, with isochrone fitting again overestimating the age; indeed the quoted isochronal age lies towards the upper bound of that given in Brown et al. (2011). The rotation period is also significantly longer than the measured period of 10.2 ± 0.5 d., which is at odds with the period derived using equation (5) in Section 2.2. This might suggest that the initial rotation rate of the star is poorly estimated, but I found that changing the initial rotation period had little effect on the rotation period derived using this calibration scenario. This is unsurprising given previous gyrochronology work which shows that stars tend to converge to a single period–colour–age relation within a few hundred Myr.

For an initial separation of only 0.05 au, none of the planetary mass secondaries survived to the isochronal age for the system listed in Table 1; all migrated inwards to the Roche limit before this, causing significant spin-up of the host as they did so such that the stellar rotation period at time of destruction was consistent with the measured value. Increasing the initial separation to a₀ = 0.0625 au revealed that secondaries of mass 0.5 M_Jup ≤ M₂ ≤ 1.5 M_Jup produced no difference in gyrochronology age compared to the calibration case, although the larger masses did produce marginally significant differences in rotation period of 1-2 d. Increasing the initial separation still further to 0.075 and 0.10 au showed that masses of M₂ ≥ 5 M_Jup and M₂ ≥ 15 M_Jup were required to produce the same, minimally measurable differences in rotation period and age. For comparison, the measured parameters for WASP-19 b are M_p = 1.14 ± 0.07 M_Jup and |$a=0.0164^{+0.005}_{-0.006}$| au.

These results imply that the initial separation must have been <0.0625 au if both the isochronal age and measured rotation period are correct, as the only way to reconcile the two is through stellar spin-up. Note also that the currently observed semimajor axis of the system is very difficult to replicate in this simplistic model, with only models causing spin-up being able to match the current orbit.

Using a stellar mass secondary showed that for a₀ < 0.25 au the rotation of the primary at the isochronal age was substantially faster than measured, with commensurately younger gyrochronology age estimates; the difference between the simulated and measured rotation period decreased as the initial separation increased. In any case, simply replacing the planet with a secondary body of stellar dimensions would have a much more severe effect on the rotation, and thus the derived age, of WASP-19 A.

3.4 A link with spectral type?

Having investigated the possibility that the gyrochronology results are too young, I turn my attention to the alternative possibility that some of the isochrone fitting results are too old. This might manifest as a bias with spectral type. To see whether any such trend is exhibited in my sample, I divide Δage by the MS lifetime of the stars, and plot the resulting age ratio as a function of effective temperature in Fig. 8. The sample as a whole shows little in the way of a trend, again due to the magnitude of the uncertainties in the age ratios. However, if I consider only the stars with measured rotation periods (the solid data), then there might be a small trend for the age ratio to increase as T_eff decreases. This conclusion is driven entirely by two of the stars in the already small set however, and should only be considered as a possibility until further rotation periods are obtained and used to recalculate gyrochronology ages for additional stars in my sample.

Figure 8.

Δage/MS lifetime as a function of T_eff, the stellar effective temperature. No clear trends are present. Consideration of only those stars with measured rotation period (solid symbols) shows a possible slight trend for Δage to increase with decreasing T_eff, towards later spectral types.

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From Fig. 9, it seems that there might be some differences in the dependence on T_eff between the two methods. The gyrochronology results are distributed evenly across the temperature range that I am considering, although the uncertainty on the age estimates increases dramatically for age ≳4 Gyr. In contrast, the isochrone results seem to show a trend with T_eff, with the oldest stars also being the coolest; this is a selection effect, as old, hot stars will have evolved off the MS, and the majority of my sample consists of hot Jupiters discovered by transit surveys which, as I have already remarked, select against older, evolved stars. There is also a noticeable trend in the uncertainties on the isochrone results, with the younger, hotter stars exhibiting more precise ages. This concurs with a study by Pont & Eyer (2004), who noted that the size of the observational uncertainties relative to the separation of the isochrones was an important parameter for isochrone fitting, and one which was most favourable for young, hot, early-type systems. The trend in Δage with temperature, if it exists, therefore seems to result from the isochronal results.

Figure 9.

Age as a function of stellar effective temperature. Solid symbols mark systems with measured rotation periods. Left: ages calculated through isochrone fitting using the YY isochrones. Right: ages calculated through gyrochronology using equation (4). Whilst the ages from isochrone fitting show similar trends with T_eff as Δage, the ages from gyrochronology show no such trends.

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