Theoretical Radii of Extrasolar Giant Planets: the Cases of TrES-4, XO-3b, and HAT-P-1b
aa r X i v : . [ a s t r o - ph ] J u l A CCEPTED TO A P J, J
ULY
26, 2008
Preprint typeset using L A TEX style emulateapj v. 08/22/09
THEORETICAL RADII OF EXTRASOLAR GIANT PLANETS:THE CASES OF TRES-4, XO-3B, AND HAT-P-1B X IN L IU , A DAM B URROWS , AND L AURENT I BGUI Accepted to ApJ, July 26, 2008
ABSTRACTTo explain their observed radii, we present theoretical radius-age trajectories for the extrasolar giant planets(EGPs) TrES-4, XO-3b, and HAT-P-1b. We factor in variations in atmospheric opacity, the presence of an innerheavy-element core, and possible heating due to orbital tidal dissipation. A small, yet non-zero, degree of coreheating is needed to explain the observed radius of TrES-4, unless its atmospheric opacity is significantly largerthan a value equivalent to that at 10 × solar metallicity with equilibrium molecular abundances. This heatingrate is reasonable, and corresponds for an energy dissipation parameter ( Q p ) of ∼ . to an eccentricity of ∼ × solar atmospheric opacity and a heavy-element core of M c = 30 M ⊕ . For XO-3b, whichhas an observed orbital eccentricity of 0 .
26, we show that tidal heating needs to be taken into account toexplain its observed radius. Furthermore, we reexamine the core mass needed for HAT-P-1b in light of newmeasurements and find that it now generally follows the correlation between stellar metallicity and core masssuggested recently. Given various core heating rates, theoretical grids and fitting formulae for a giant planet’sequilibrium radius and equilibration timescale are provided for planet masses M p = 0.5, 1.0, and 1.5 M J with a = 0.02-0.06 AU, orbiting a G2V star. When the equilibration timescale is much shorter than that of tidalheating variation, the “effective age” of the planet is shortened, resulting in evolutionary trajectories more likethose of younger EGPs. Motivated by the work of Jackson et al. (2008a,b), we suggest that this effect couldindeed be important in better explaining some observed transit radii. Subject headings: planetary systems — planets and satellites: general — stars: individual (HAT-P-1, GSC02620-00648, XO-3) INTRODUCTION
As of the writing of this paper, an astounding 47 transit-ing extrasolar giant planets (EGPs) have been discovered .For transiting planets, the inclination/planet-mass degeneracyis resolved and the photometric dip in the stellar flux dur-ing the transit yields the planet’s radius ( R p ). Theory thenattempts to explain the measured radii (Guillot et al. 1996;Burrows et al. 2000, 2003, 2004, 2007; Bodenheimer et al.2001, 2003; Baraffe et al. 2003, 2004, 2008; Chabrier et al.2004; Fortney et al. 2007; Laughlin et al. 2005). Importantly,the comparison between theory and measurement must bedone for a given stellar type, orbital distance, planet mass,and age. The latter is poorly measured, but crucially impor-tant (see, e.g. §5 re HD 209458b).Burrows et al. (2007) modeled the theoretical evolutionof the radii of the 14 transiting EGPs known at the time.These authors suggest that there are two radius anoma-lies in the transiting EGP family, of which the smaller-radius anomaly is a result of the presence of dense cores(Mizuno 1980; Pollack et al. 1996; Hubickyj et al. 2004),whereas the larger-radius anomaly might be explained bythe enhanced atmospheric opacities which slow down theheat loss of the core. They also discussed the effects onplanet structure of the possible extra heat source in the in-terior, yet found no obvious correlation between the req- Department of Astrophysical Sciences, Princeton University, PeytonHall – Ivy Lane, Princeton, NJ 08544; [email protected], [email protected] Department of Astronomy and Steward Observatory, The University ofArizona, Tucson, AZ 85721; [email protected] See J. Schneider’s Extrasolar Planet Encyclopaedia athttp://exoplanet.eu, the Geneva Search Programme at http://exoplanets.eu,and the Carnegie/California compilation at http://exoplanets.org uisite power and the intercepted stellar power. Note thatnone of the transiting EGPs modeled by Burrows et al. (2007)is known to have a highly eccentric orbit. However, re-cently several transiting EGPs with significantly non-zeroeccentricities ( e & .
15) have been discovered, includingthe EGPs XO-3b (Johns-Krull et al. 2008), HAT-P-2b (a.k.a.HD 147506b; Bakos et al. 2007a; Loeillet et al. 2008), HD17156b (Barbieri et al. 2007; Gillon et al. 2008; Narita et al.2008; Irwin et al. 2008), and the “hot Neptune” GJ 436b(Gillon et al. 2007; Deming et al. 2007; Demory et al. 2007).At least for those systems with highly eccentric orbits, heat-ing due to orbital tidal dissipation (Bodenheimer et al. 2001,2003) should be incorporated into theoretical models for theradius-age trajectories. A preliminary exploration of this in arestricted context motivates the present paper. When the theo-retical radius evolution calculations are tailored to a system’sspecific planet mass, age, primary stellar properties, and or-bital distance (as they must), the current radii of most of theknown transiting EGPs can be explained by the theoretical ra-dius models of Burrows et al. (2007). In many instances, ahigher-density core mass provides an even better fit, and ex-tra internal heat sources are not required. However, at leastthree transiting EGPs seem to be exceptional in some way.The planets TrES-4 and XO-3b (for its large-radius solutionbased on stellar parameters from spectral synthesis model-ing) are cited by their discoverers (Mandushev et al. 2007;Johns-Krull et al. 2008) as anomalously large and inconsis-tent with extant theoretical models. In addition, Burrows et al.(2007) found that HAT-P-1b deviated from the core-massstellar-metallicity relationship followed by many of the tran-siting EGPs that they modeled (for the core-mass stellar-metallicity correlation, also see Guillot et al. 2006).Therefore, with this paper, we focus on this small subset LIU ET AL.of three interesting objects to determine how their radii canindeed be explained with minimal assumptions, that never-theless can include tidal heating. We find that tidal heat-ing in the core, given measured (XO-3b) and possibly non-zero (TrES-4 and HAT-P-1b) eccentricities, which never-theless are still consistent with the upper limits, can nat-urally explain the measured radii. For HAT-P-1b with itsnew age estimate, we find that a core of reasonable sizecan now be accommodated. Importantly, when tidal heat-ing needs to be invoked, a canonical tidal dissipation pa-rameter, “ Q p ”(Goldreich & Soter 1966) , with a value near10 - , along with the measured eccentricity (or reasonablevalues consistent with its current bounds), suffices to explainthe measurements. Hence, only simple extensions of the de-fault evolution models that incorporate a known process withcanonical parameters are required. We postulate that all mea-sured transiting EGP radii can be explained by available the-ory when proper account is taken of the measured planet-star system parameters, reasonable core masses that followthe relationship between core mass and stellar metallicity(Burrows et al. 2007; Guillot et al. 2006), and tidal heating atthe expected rate for non-zero, but measured, eccentricities.Exceptions to this might arise if it is determined that tidal dis-sipation occurs predominantly in the atmosphere, not the con-vective core, and/or if the eccentricity and semi-major axishistory must be factored into the tidal heating history of theplanet. The latter effect is intriguing and has been suggestedby Jackson et al. (2008a,b).We describe our computational techniques and model as-sumptions in §2. In §3, we review the measurements of thesethree EGP systems, identify the discrepancies between the ob-served planetary radii and those predicted by previous the-oretical models and present new theoretical radius-age tra-jectories for them using tailored atmospheric boundary con-ditions. These new trajectories and theoretical radii includethe effects of tidal heating in the convective core for mea-sured or reasonable values of the orbital eccentricities and fora range of values for the tidal parameter, Q p . In §4, we pro-vide theoretical grids and fitting models for the equilibriumplanetary radius and the equilibration timescale, given a cer-tain set of planet mass, orbital distance, and tidal heating ratefor a G2V primary star. Finally, in §5 we summarize our re-sults for each system, discuss the relevant constraints obtainedon their structural properties, and list caveats concerning ourmodel assumptions. COMPUTATIONAL TECHNIQUES AND MODEL ASSUMPTIONS
A detailed discussion of our computational techniques canbe found in Burrows et al. (2003) and Burrows et al. (2007).Here, we present only a brief summary, along with our modelassumptions. We generate realistic atmospheres customizedfor the three EGPs, their time-averaged orbital separations,and primary stars. The adopted atmospheric boundary con-ditions incorporate irradiation using the observed stellar lu-minosity and spectrum, and the measured planet orbital dis-tance. For planets with eccentric orbits, the time-averagedinsolation flux is employed in constructing the atmosphericboundary conditions. The theoretical stellar spectra of Kurucz(1994) are adopted. For the given stellar spectrum and flux(inferred from the luminosity and the planet orbital distance),an S - T e f f - g grid is calculated for the core entropy, S , effec- Note that the Q p parameter used throughout this paper corresponds to the Q ′ p in Mardling (2007). tive temperature, T e f f , and gravity, g , using the discontinuousfinite element (DFE) variant of the spectral code TLUSTY(Hubeny & Lanz 1995). It is assumed that both the stellarspectrum and flux are constant during the evolution.We employ the Henyey evolutionary code of Burrows et al.(1997) for the radius-age evolutionary calculations, using thefunction T e f f ( S , g ) for the interior flux, inverted from the ta-ble of S , T e f f , and g , referred to above. The helium fraction Y He is assumed to be 0.25. We calculate models with differ-ent atmospheric opacities, the effect of which can be conve-niently mimicked by using 1 × solar, 3 × solar, and 10 × solarabundance atmospheres. Note that the increase in atmo-spheric opacity does not need to, and should not be duesolely to, increased metallicity. The effects of increased atmo-spheric opacity and increased envelope heavy-element abun-dances are decoupled, so that the implied increases in theheavy-element burden of the envelope, if any, will not can-cel the expansion effect of enhanced atmospheric opacity(Burrows et al. 2007). To model the presence of a heavy-element core, a compressible ball of olivine is placed inthe center of the model planet, and pressure continuity be-tween the heavy-element core and the gaseous envelope is en-sured throughout the evolution. We adopt the Saumon et al.(1995) equation of state (EOS) for the H /He envelope andthe ANEOS by Thompson & Lauson (1972) for olivine.As noted, to model the atmospheric opacity, we use su-persolar metallicities (e.g., 3 × solar, 10 × solar) to mimicthe expansion effects of enhanced atmospheric opacity(Burrows et al. 2007). Possible causes for such enhancedopacities might be supersolar metallicities in the atmosphere,nonequilibrium chemistry, errors in the default opacities, andthick hazes or absorbing clouds. Note that the expansion ef-fects of enhanced atmospheric opacity and the shrinkage ef-fects of increased envelope metallicities conceptually decou-pled in our models and that an increase of the envelope heavy-element burden, will not necessarily cancel the expansion ef-fect due to enhanced atmospheric opacity. The effect of acentral heavy-element core on the planet radius is to shrink itmonotonically with core mass.Given a non-zero orbital eccentricity, the tidal dissipationrate is calculated using the formulation summarized in the Ap-pendix (Bodenheimer et al. 2001, 2003; Gu et al. 2004). Weassume that the planet is in synchronous rotation and that allthe tidal heating is in the convective core. Note that the ef-fects of other core energy dissipation mechanisms on planetstructural evolution are also implicitly addressed. As indi-cated in the Appendix, the tidal heating rate is proportionalto f ( e ) / Q p , where f ( e ) = e when e ≪
1. Also, the valuesof Q p for EGPs with masses M p ∼ M J , although very un-certain, are estimated to be ∼ - (Adams & Laughlin2006; Gu et al. 2003; Jackson et al. 2008a,b). Therefore, wecalculate typical heating rates using the combination e / Q p for TrES-4 and HAT-P-1b, of which the orbital eccentricitieshave been estimated to be ≪
1, if nonzero at all. Models with-out external heat sources ( e / Q p = 0) are also presented. OBSERVED PROPERTIES AND THEORETICAL PLANETARYRADII
Table 1 displays the relevant observed quantities and thecorresponding references for the EGPs TrES-4, XO-3b, andHAT-P-1b and their parent stars. These properties includesemi-major axis ( a ), orbital period and eccentricity ( e ), stel-lar mass ( M ∗ ), radius ( R ∗ ), effective temperature T e f f , metal-licity ([Fe/H] ∗ ), age, and planetary mass ( M p ) and radiusHEORETICAL RADII OF EXTRASOLAR GIANT PLANETS 3( R p ). Also shown are the stellar flux at the planet’s sub-stellar point, F p , in units of 10 erg cm - s - , and the ra-tio between the possible tidal energy dissipation rate withinthe planet and the insolation rate ˙ E tide / ˙ E insolation in the unitof ( Q p / ) - . Q - p ≡ π E H ( - dEdt ) dt is the specific dissipa-tion function of the planet, where E is the maximum energystored in the tidal distortion and - dEdt is the rate of dissipa-tion (Goldreich & Soter 1966). The parameters for TrES-4and HAT-P-1b are drawn from Torres et al. (2008). These au-thors provide a uniform analysis of transit light curves andstellar parameters based on stellar evolution models, and acritical examination of the corresponding errors. Since XO-3b has not been studied by Torres et al. (2008), we generatemodels using parameters reported by Johns-Krull et al. (2008)and Winn et al. (2008).Theoretical evolutionary trajectories are presented forTrES-4, XO-3b, and HAT-P-1b under various assumptionsabout the atmospheric opacity, the presence of a heavy-element core and possible tidal heating. The transit radiuseffect (Burrows et al. 2007) is also included in the models. TrES-4
TrES-4 is the current record-holder for the lowest EGPdensity (Mandushev et al. 2007; Torres et al. 2008). In thediscovery work, Mandushev et al. (2007) carried out spec-troscopic observations with the CfA Digital Speedometer(Latham 1992) , radial velocity (RV) measurements withKeck, and transit photometry in the z band with Kepler-Cam at the F. L. Whipple Observatory (FLWO) and in the B band using NASACam on the 0.8-m telescope at the Low-ell Observatory. Assuming [Fe/H] ∗ = 0 . + . - . dex, Mandu-shev et al. derived T e f f = 6100 + - K, M ∗ = 1 . + . - . M ⊙ , R ∗ = 1 . + . - . R ⊙ , and an age of 4 . + . - . Gyr for the star,and M p = 0 . + . - . M J and R p = 1 . + . - . R J for the planet.The orbital eccentricity was assumed to be exactly zero in thefit. Mandushev et al. (2007) suggested that its observed radiusis too large to be explained by the theoretical EGP models ofBurrows et al. (2007), given its estimated mass, age and inso-lation, even when the effects of higher atmospheric opacitiesand the transit radius correction are considered.The parameters of TrES-4 listed in Table 1 are fromTorres et al. (2008). These authors derived these parame-ters using the RV measurements and transit photometry fromMandushev et al. (2007), and the stellar atmospheric proper-ties from Sozzetti et al. (2008; in preparation). Note that theirestimated planetary radius R p = 1 . + . - . R J is ∼ σ largerthan the value of R p = 1 . + . - . R J of Mandushev et al.(2007). Results for TrES-4
We calculate the radius-age trajectories for TrES-4 usingthe parameters from Torres et al. (2008), taking into accountthe possible effects of enhanced atmospheric opacities, andthe presence of tidal dissipation given a small, yet non-zero,orbital eccentricity. Models with the presence of a heavy-element core are also calculated. The value of ˙ E tide / ˙ E insolation quoted for TrES-4 in Table 1 is calculated assuming e = 0 . F IG . 1.— Theoretical planet radius R p ( R J ) versus age (Gyr) for TrES-4. Also shown on both panels with error bars are the observed radiusand age from Torres et al. (2008). Various values are assumed concern-ing the atmospheric opacity, the presence of a heavy-element core, and thecore heating due to tidal dissipation. Different colors correspond to dif-ferent tidal heating rates, which are proportional to e / Q p when e ≪ Top : This panel demonstrates the ef-fect of enhanced atmospheric opacity under various heating powers. Mod-els assuming 3 × solar (10 × solar) atmospheric opacities are plotted as solid(dashed) curves, whereas the black, blue, and red curves correspond to( e / . (10 / Q p ) = 0 .
0, 0.1, and 6.0, all without a heavy-element core ( M c = 0). Bottom : The effect of the presence of a heavy-element core is illustrated,where the dashed (solid) curves denote models with M c = 30 (0) M ⊕ . If TrES-4 follows the core-mass stellar-metallicity relation found by Burrows et al.(2007), then it should contain a heavy-element core with M c ∼ - M ⊕ ,given its stellar metallicity [Fe/H] ∗ ( + . + . - . dex). Assuming 3 × solar at-mospheric opacity, the model with Q p = 10 . , e ∼ M c = 30 M ⊕ ( black-dashed curve) explains the observed radius well. The first panel of Fig. 1 shows the theoretical radii R p (inunits of R J ) as a function of age (in units of Gyr) for TrES-4,under various assumptions concerning the atmospheric opac-ity and the level of tidal dissipation, without a heavy-elementcore ( M c = 0). Models assuming 3 × (10 × ) solar atmosphericopacities are shown as solid (dotted) curves. The black curvesshow the models without any heat sources, whereas the blue(red) curves depict those with a tidal heating rate assuming( e / . (10 / Q p ) = 0 . . ∼ σ too small. As-suming the Mandushev et al. (2007) parameters, the theoret- LIU ET AL.ical radii are still ∼ σ too small. The discrepancy will be-come smaller for models with even higher atmospheric opac-ities. So it is concluded that either the atmospheric opacityof TrES-4 is unusually large (much higher than the equivalentof a 10 × solar metallicity, equilibrium mixture), or there areextra heat sources in the core. The required heating poweris very modest; the model with ( e / . (10 / Q p ) = 0 . × solar atmospheric opacity produces theoretical radii con-sistent with the 1- σ lower bound of R p from Torres et al.(2008).The models shown in the first panel of Fig. 1 do not includeany heavy-element cores. If TrES-4 follows the core-massstellar-metallicity relation studied by Burrows et al. (2007),then there should be a heavy element core with M c ∼ - M ⊕ , given its stellar metallicity [Fe/H] ∗ ( + . + . - . dex). Thepresence of such a heavy-element core will shrink the plane-tary radius, and, therefore, would require a higher tidal heat-ing rate than do models without a core to explain the ob-served radius. This effect of including a heavy-element corein the models for TrES-4 is shown in the second panel of Fig.1. Assuming 3 × solar atmospheric opacity, the model with Q p = 10 . , e ∼ M c = 30 M ⊕ ( black-dashed ) explains the observed radius well. Within 1 σ uncertainties, the model with Q p = 10 . , e ∼ M c = 30 M ⊕ ( red-dashed curve) canalso fit the observed radius.In summary, unless the atmospheric opacity of TrES-4 isunusually large, core heating is required to explain its ob-served radius. However, the required heating power is mod-est. A non-core model with ( e / . (10 / Q p ) = 0 . σ lower boundary of R p fromTorres et al. (2008), assuming 10 × solar atmospheric opac-ity. The required energy dissipation rates become larger formodels with a heavy-element core, but are still reasonable.For instance, the model with Q p = 10 . , e ∼ M c = 30 M ⊕ produces the observedradius well, assuming 3 × solar atmospheric opacity. To bet-ter constrain models of TrES-4, definitive measurements of,or stronger limits to, its orbital eccentricity are needed. XO-3b
XO-3b has been observed to be supermassive and onan eccentric orbit ( M p = 13 . + . - . M J , e = 0 . + . - . ;Johns-Krull et al. 2008). The discoverers obtained transitlight curves with relatively small 0.3-m telescopes, spectro-scopic observations using the 2.7-m Harlan J. Smith (HJS)telescope and the 11-m Hobby-Eberly Telescope (HET) , andRV measurements with the HJS telescope. Based on theoreti-cal spectral models of the HJS data, Johns-Krull et al. (2008)derive T e f f = 6429 + - K, [Fe/H] ∗ = - . + . - . dex, andlog g ∗ = 3 . + . - . for the star. Combined with the RV mea-surements, they arrive at M ∗ = 1 . + . - . M ⊙ and R ∗ = 2 . + . - . R ⊙ for the star, and M p = 13 . + . - . M J and R p = 1 . + . - . R J for the planet. These authors have commented that XO-3b is observed to be so large that in all cases analyzed byFortney et al. (2007), their models predict a much smaller ra-dius. However, due to the absence of a precise trigonometricparallax of XO-3, its distance is very uncertain. Assuming asmaller distance, and, hence, a reduced stellar mass and radiusthan obtained using the isochrone method, Johns-Krull et al.(2008) found a best fit to their transit light curves with log g ∗ =4 . M ∗ = 1 . M ⊙ , and R ∗ = 1 . R ⊙ , with the correspond- ing estimates for M p of 12 . + . - . M J and R p of 1 . + . - . R J .These light-curve-based results have recently been strength-ened by Winn et al. (2008) using larger aperture telescopes.These authors observed 13 transits photometrically using the1.2-m telescope at the FLWO, along with 0.4-0.6-m tele-scopes. Based on these more precise transit light curves, theyconcluded that log g ∗ = 4 . M ∗ = 1 . M ⊙ , and R ∗ = 1 . R ⊙ , with the corresponding estimates for M p of 11 . + . - . M J and for R p of 1 . + . - . R J . Since a trigonometric parallaxmeasurement of XO-3 is still lacking that could distinguishthese two different methods, we make models for both thespectroscopically-determined and light-curve-based parame-ter sets.Both the spectroscopical results of Johns-Krull et al. (2008)and the light-curve-based results of Winn et al. (2008) arelisted in Table 1. We note that the M p and R p values derivedusing the two different methods differ significantly from oneanother ( ∼
10% in M p and ∼
50% in R p ). Therefore, separatemodels and discussions for these different sets of planetaryproperties are in order and we calculate theoretical radii forXO-3b for both estimates of the planetary radius and mass. Results for XO-3b
We include the possible heating due to orbital tidal dissipa-tion, assuming reasonable values of Q p . Our results for XO-3b are shown in Fig. 2, where in the first panel a planet massof M p = 13 . M J , derived from the spectral synthesis methodof Johns-Krull et al. (2008), and an atmospheric opacity asso-ciated with 3 × solar metallicity are assumed, whereas in thesecond panel the corresponding values are M p = 11 . M J from the light-curve fitting results of Winn et al. (2008) and1 × solar atmospheric opacity. In both cases, we assume that amassive heavy-element core is absent. For such massive plan-ets, the effect of reasonable core masses on planetary radii issmall.As shown in the first panel of Fig. 2, the very large radiusderived by Johns-Krull et al. (2008), based on spectral synthe-sis modeling, can be explained by energy dissipation due totidal heating when the Q p parameter of XO-3b is ∼ . . Forthe case of the light-curve-based planetary mass (second panelof Fig. 2), the inferred radius is broadly consistent to within1- σ errors with the absence of an internal heat source, or withtidal heating with Q p & . These specific constraints on Q p assume 3 × solar in the former case and 1 × solar atmosphericopacity in the latter. For our first model of XO-3b assuming M p = 13 . M J and 3 × solar atmospheric opacity, it would besurprising to find a planet with R p / R J . . & .
2, sincethis will require Q p to be either too large or too small. Forthe second model assuming M p = 11 . M J and 1 × solar at-mospheric opacity, such a region would be R p / R J . . & .
6, for the same reason. Given significant eccentricity, itis important to appropriately account for tidal heating in orderto model the planet’s structural evolution. We want to empha-size our adopted model assumptions that 1) the planet is insynchronous rotation and that 2) all the tidal heating is in theconvective core. Even though the e / Q p degeneracy is brokendue to the known value of e , detailed radius evolution modelscould be used to constrain Q p for EGPs, but only if it is deter-mined that tidal dissipation occurs predominantly either in theconvective core or in the atmosphere, and if the uncertaintiesin the core mass and atmospheric opacity are both resolved. HAT-P-1b
HEORETICAL RADII OF EXTRASOLAR GIANT PLANETS 5 F IG . 2.— Theoretical planet radius R p ( R J ) versus age (Gyr) for XO-3b.In the first panel, a planet mass of M p = 13 . M J based on the spectralsynthesis model of Johns-Krull et al. (2008) and 3 × solar atmospheric opac-ity are assumed, whereas in the second panel the corresponding values are M p = 11 . M J according to the light-curve fitting results of Winn et al.(2008) and 1 × solar atmospheric opacity. Also shown with error bars onboth panels are the observed radii and age estimates from Johns-Krull et al.(2008) and Winn et al. (2008), based on the two different analyses. In bothpanels, models assuming various tidal heating rates corresponding to Q p = ∞ (no heating), 10 . , 10 . , 10 . , and 10 . are color coded. In all the mod-els, e = 0 .
260 (Johns-Krull et al. 2008) is assumed. Note that the observedradii, based on the two different methods, differ quite a bit from one another( ∼ Q p ∼ . , whereas the much smaller one inferred fromthe light-curve fit can be explained by models with Q p values down to 10 ,within 1- σ errors. See §3.2.1 for more discussion. Using photometry conducted by the Hungarian-made Au-tomated Telescope Network (HATNet) project, Bakos et al.(2007b) discovered HAT-P-1b transiting one member of thestellar binary ADS 16402. These authors suggested that HAT-P-1b was too large to be explained by theoretical EGP models.Spectral synthesis modeling of the parent star ADS 16402B,based on its Keck spectra, yielded T e f f = 5975 + - K and[Fe/H] ∗ = + . + . - . dex. Bakos et al. (2007b) also fit bothstellar members in the binary to evolutionary tracks and basedon the Subaru and the Keck spectra derived M ∗ = 1 . + . - . M ⊙ and R ∗ = 1 . + . - . R ⊙ for ADS 16402B and a best-fit ageof 3 . z -band transit curvesfrom KeplerCam (Holman et al. 2006), combined with RVmeasurements from Subaru and Keck, these authors derive M p = 0 . + . - . M J and R p = 1 . + . - . R J , where the errors in theplanetary radius include both statistical and systematic errorsin both the stellar radius and mass. Note that in the above fits, a circular orbit ( e = 0) was assumed. However, the χ fittingof the RV data in Bakos et al. (2007b) favors a small, yet non-zero, eccentricity: e = 0 . + . - . . These authors did estimatethe heating rate assuming e = 0 .
09 and suggested that if thisnon-zero orbital eccentricity is confirmed, the observed large R p could be explained by tidal heating. However, note thata non-zero eccentricity is only suggestive, mainly due to thesmall number of RV observations (13 velocities, for whichthe typical S/N is about 150 per pixel). Since RV-based ec-centricity estimates are positively biased due to noise (e.g.,Shen & Turner 2008), it is very likely that the true e is smallerthan 0 .
09. In fact, Johnson et al. (2008) find a upper limit on e of 0.067, with 99% confidence, by combining their new andprevious RV measurements. Therefore, we assume a smallervalue, e = 0 .
01, in our baseline model and see where such anassumption leads. Such a small eccentricity could result fromKozai cycles with tidal friction (e.g., Fabrycky & Tremaine2007), though there is evidence that the spin-orbit misalign-ment is small (Johnson et al. 2008).Based on more high-precision transit observations, how-ever, Winn et al. (2007) report that HAT-P-1b is less “bloated”than originally thought. Their observations include three tran-sits observed in z band with the 1.2-m telescope at the FLWO,three observed through the “Gunn Z” filter (Pinfield et al.1997) using the Nickel 1-m telescope at Lick Observatory,and three observed through the Johnson I filter using the 1-m telescope at the Wise Observatory. Winn et al. (2007) de-rived R ∗ / M / ∗ by fitting the transit light curves, and con-cluded that R ∗ = 1 . + . - . R ⊙ and R p = 1 . + . - . R J . Notethat in their fits the orbital eccentricity was assumed to bezero. These authors suggest that the updated radius can beexplained by the structural models of Burrows et al. (2007),unless the planet has a very massive core of heavy elements.Indeed, Burrows et al. (2007) calculated radius-age trajec-tories for HAT-P-1b. They included different core massesand atmospheric opacities in their models, and found thatin order to fit the observed radius, HAT-P-1b deviates fromthe stellar-metallicity versus core-mass sequence otherwiseroughly followed by the transiting EGPs included in their pa-per (see Fig. 9 of Burrows et al. 2007). However, the stel-lar and planetary parameters of the HAT-P-1 system adoptedby Burrows et al. (2007) are from the discovery work ofBakos et al. (2007b). The parameters of the HAT-P-1 sys-tem derived by Torres et al. (2008) are listed in Table 1.These authors compile the z -band light curves of Winn et al.(2007) and the RV data and the atmospheric parameters ofBakos et al. (2007b), but with increased uncertainties for T e f f ,[Fe/H] ∗ , and log g ∗ . As we show in §3.3.1, using thesenew parameters and the new planet radius, we now find in-ferred core masses that are roughly consistent with the stellar-metallicity versus core-mass relationship followed by theEGPs studied by Burrows et al. (2007). Results for HAT-P-1b
Given the new measured radii and stellar age of Torres et al.(2008), we have reexamined the best-fit core masses for HAT-P-1b. The effects of the possible heating due to tidal dissi-pation on the planet’s structural evolution are considered, as-suming a small yet non-zero eccentricity e = 0 .
01. Theoreticalevolutionary trajectories of planet radius with age for HAT-P-1b are shown in Fig. 3, where the first (second) panel displaysthe results assuming a 3 × solar (10 × solar) atmospheric opac-ity. Models with different tidal heating rates proportional to LIU ET AL. F IG . 3.— Theoretical planet radius R p ( R J ) versus age (Gyr) for HAT-P-1b.Also shown with error bars are the observational stellar age and planet ra-dius from Torres et al. (2008), as listed in Table 1. We reexamine the best-fitcore masses for HAT-P-1b, under various assumptions concerning the atmo-spheric opacity and the possible core heating rate due to tidal dissipation.The top (bottom) panel shows models assuming 3 × solar (10 × solar) atmo-spheric opacities. In both panels, the black, blue, and red curves denotemodels assuming ( e / . (10 / Q p ) = 0, 0.8, and 8, respectively. Differ-ent line styles correspond to models with various heavy-element core massesin units of the Earth mass, M ⊕ , as labeled on the plot. If there is tidal heat-ing assuming reasonable values of ( e / . (10 / Q p ) ∼ .
8, then the coremass required to fit the observed radius is ∼ M ⊕ ( ∼ M ⊕ ), assuming3 × solar (10 × solar) atmospheric opacity, in which case HAT-P-1b does fol-low the core-mass stellar-metallicity relation found by Burrows et al. (2007).See §3.3.1 for a discussion. ( e / . (10 / Q p ) are color-coded as labeled. For both ofthe panels, different line styles represent “no heavy-elementcores," or the presence of a heavy-element core with a rangeof masses in units of Earth masses, M ⊕ . For clarity, only se-lected models are presented in the figure. Table 2 lists the bestestimates for the core mass under various assumptions.As demonstrated in Fig. 3 and Table 2, there are multiplesolutions to explain the observed radius of HAT-P-1b. In allthe cases considered, a non-zero heavy-element core mass isneeded, which, without any external heat sources, is ∼ M ⊕ ( ∼ M ⊕ ) assuming 3 × solar (10 × solar) atmospheric opac-ity,. This best-fit core mass becomes larger when there isexternal heating. If HAT-P-1b does follow the approximatecore-mass/stellar-metallicity relation found by Burrows et al.(2007), then given its [Fe/H] ∗ ( + . + . - . dex), the core masswould be ∼ M ⊕ ( ∼ M ⊕ ) assuming 3 × solar (10 × solar)atmospheric opacity. These core masses correspond to thecases with ( e / . (10 / Q p ) = 0 .
8. In summary, if there is tidal heating with reasonable val-ues of Q p , the core-mass estimates suggest that HAT-P-1b fol-lows the correlation between stellar metallicity and core massfound by Burrows et al. (2007), or if there is no extra heating,deviates mildly from the correlation sequence. However, alarger core mass, more in keeping with the correlation foundby Burrows et al. (2007), is more consistent with reasonablevalues of ( e / . (10 / Q p ), as long as e is non-zero and Q p is not anomalously small. EQUILIBRIUM PLANETARY RADII AND EQUILIBRATIONTIMESCALES FOR VARIOUS HEATING RATES: A PARAMETERSTUDY
In this section, we investigate the effects of generic coreheating on the planet’s equilibrium radius, and the time toreach this equilibrium. We refer to the latter as the equilibra-tion timescale. The heat source discussed in this section couldbe due to orbital tidal heating, but does not have to be. Figure4 shows equilibrium planetary radii ( R eq ) and equilibrationtimescales ( τ eq ) with various core-heating powers, quantifiedby ˙ E heating / ˙ E insolation . Models are calculated for planets withmasses M p = 0.5, 1.0, and 1.5 M J , with semi-major axes a =0.02, 0.03, 0.04, 0.05, and 0.06 AU, orbiting a G2V star. Equi-librium is defined as the state after which the planet radius isconstant to within a part in 10 . We define the equilibrationtimescale as the time it takes the planet to evolve from 1 . R eq to 1.05 R eq . Note that for those rare models for which theplanetary radii still change by more than 10 - R eq at the end ofcalculation (10 Gyr), the equilibrium state is assumed to havebeen reached at the final age of the evolutionary trajectory.In Fig. 4, filled circles represent the values of R eq and τ eq calculated from our theoretical trajectories, whereasthe curves are least-square fits to them. We adopt afourth-order polynomial in fitting R eq / R J as a function oflog( ˙ E heating / ˙ E insolation ), and a linear model for log( τ eq / Gyr)versus log( ˙ E heating / ˙ E insolation ), given by: R eq R J = C + C x + C x + C x + C x , log (cid:18) τ eq Gyr (cid:19) = b + kx , (1)where x ≡ log( ˙ E heating / ˙ E insolation ). The model fits of the pa-rameters are given in Table 3.For an extreme close-in EGP with M p = 0 . M J at a = 0 . ∼ R J for little heating ( ˙ E heating / ˙ E insolation . - ) to ∼ R J for strong heating ( ˙ E heating / ˙ E insolation ∼ - ). Thecorresponding timescales for the planet to reach these radiiare ∼ ∼ a = 0 .
06 AU, the equilibrium radii are smaller and the relevanttimescales are longer - from R eq ∼ . R J and τ eq ∼ ˙ E heating / ˙ E insolation . - to R eq ∼ . R J and τ eq ∼ ˙ E heating / ˙ E insolation ∼ - . For more massive planets, the equi-librium radii are smaller and the timescales are longer. For anEGP with M p = 1 . M J at a = 0 .
02 AU, the values are R eq ∼ . R J and τ eq ∼ ˙ E heating / ˙ E insolation ∼ - . Our theoret-ical model grids along with the fitting curves provided in eq.(1) and the parameters listed in Table 3 can be used to calcu-late the equilibrium planetary radius and the typical timescaleto reach it, given different combinations of planet mass, or-bital distance, and the ratio of core-heating power to insola-HEORETICAL RADII OF EXTRASOLAR GIANT PLANETS 7 F IG . 4.— Equilibrium planetary radius R eq ( R J ) and equilibration timescale (Gyr) assuming various ratios between the core heating power and the insolationpower. The equilibration timescale is defined as the time it takes the planet to evolve from 1.25 R eq to 1.05 R eq . Models are calculated for planets with masses M p = 0.5, 1.0, and 1.5 M J , and semi-major axes a = 0.02, 0.03, 0.04, 0.05, and 0.06 AU, orbiting a G2V star. Filled circles represent results calculated fromradius-age trajectories, whereas the curves are fits to them given by eq. (1) and the corresponding parameters in Table 3. See §4 for more information. tion power. SUMMARY AND DISCUSSION
We have calculated theoretical radius-age trajectories forthree EGPs: TrES-4, XO-3b, and HAT-P-1b, under variousassumptions concerning atmospheric opacity, the presence ofan inner heavy-element core, and possible heating due to or-bital tidal dissipation. The main model results are the follow-ing:1. Unless the atmospheric opacity of TrES-4 is unusu-ally large (much higher than 10 × solar equivalent), coreheating is required to explain its observed radius ( R p =1 . + . - . R J ; Torres et al. 2008). However, the re-quired heating power is modest. A non-core model with( e / . (10 / Q p ) = 0 . σ lower boundary of R p from Torres et al. (2008),assuming 10 × solar atmospheric opacity. The requiredenergy dissipation rates become larger for models witha heavy-element core. The model with Q p = 10 . , e ∼ M c = 30 M ⊕ , re-produces the observed radius well, assuming 3 × solaratmospheric opacity. If TrES-4 follows the core-massstellar-metallicity correlation found by Burrows et al.(2007), then the models with a non-zero heavy-elementcore mass are favored, considering its stellar metallic-ity [Fe/H] ∗ = + . + . - . dex (Torres et al. 2008). Ongo-ing Spitzer photometry of its secondary eclipse will putmore stringent constraints on e and can either confirmor rule out these possibilities. 2. For XO-3b, we have shown that orbital tidal heat-ing is a key factor in explaining the planet radius.The very large radius ( R p = 1 . + . - . R J ) derived byJohns-Krull et al. (2008) based on spectral synthesismodeling can be explained by energy dissipation due totidal heating. In this case, the Q p parameter of XO-3bis near ∼ . , a not unreasnoable value. On the otherhand, the much smaller radius (1 . + . - . R J ) based onlight-curve fit by Winn et al. (2008) is consistent withno core heating sources, or with tidal heating assum-ing Q p & , within 1- σ errors. These constraints on Q p assume 3 × solar atmospheric opacity for the formercase and 1 × solar for the latter, but are only weakly de-pendent on this.3. We have reexamined the core mass required for HAT-P-1b using the updated data (importantly, its radius) fromTorres et al. (2008), and now find it generally followsthe correlation between core mass and stellar metal-licity found by Burrows et al. (2007). In all the casesconsidered, a non-zero heavy-element core mass isneeded to explain the observed radius ( R p = 1 . + . - . R J ; Torres et al. 2008). The core mass is ∼ M ⊕ ( ∼ M ⊕ ) assuming 3 × solar (10 × solar) atmosphericopacity when there is no external heating. If thereis tidal heating corresponding to reasonable values of( e / . (10 / Q p ) ∼ .
8, then the core mass required tofit the observed radius is ∼ M ⊕ ( ∼ M ⊕ ) assuming3 × solar (10 × solar) atmospheric opacity, in which caseHAT-P-1b follows the core-mass stellar-metallicity re- LIU ET AL.lation found by Burrows et al. (2007) and Guillot et al.(2006).In addition, we have carried out a parameter study of the ef-fects of core heating and provided theoretical grids and fittingformulae for the equilibrium planet radius and equilibrationtimescale, given various core heating powers for planets withmasses M p = 0.5, 1.0, and 1.5 M J with a = 0.02-0.06 AU,orbiting a G2V star. The fitting formula for the equilibriumplanet radius can be used for a theoretical zeroth-order esti-mate, without carrying out detailed evolutionary calculations.The equilibration timescale τ eq characterizes the time it takesthe planet to adjust its structure in response to a given degreeof core heating.Recently Jackson et al. (2008a) considered the effect of theco-evolution of the orbital eccentricity and the semi-majoraxis on the tidal dissipation history. In the past, the semi-major axis had been assumed to be constant when conductingtidal evolution studies (e.g. Bodenheimer et al. 2001, 2003;Gu et al. 2004). Jackson et al. (2008b) calculate the evolu-tionary histories of the tidal dissipation rate for several EGPs,and find that in most cases the tidal heating rate increases asa planet moves inward and then decreases as the orbit circu-larizes. The relevant timescale, τ heating , is the time it takes thetidal heating rate to decay by a factor of e . If τ eq ≫ τ heating ,then it is valid to take a constant effective tidal heating ratein the planet radius-age trajectory calculation. If τ eq ∼ τ heating ,then in order to account for the effect of a varying tidal heat-ing rate, different values should be used at each time step ofthe radius-age trajectory calculation. If τ eq ≪ τ heating , then theplanet will have enough time to reach an equilibrium radiusbefore the tidal heating rate decays significantly. In this case,the planet’s structure evolves in a quasi-equilibrium manner.Theoretical planet radius-age trajectory models will easily beable to account for the effect of varying tidal heating rate byadopting different core heating rates at each time interval of τ heating during the calculation. In effect, there is a reset of the“clock” right after the time of maximum heating - the planetbecomes most extended on a timescale ∼ τ eq after the tidalheating rate achieves this maximum. Because of the intenseheating and the quick response, the planet loses the memoryof its shrinkage history before maximum heating, which is ef-fectively a reset of its “age.”Burrows et al. (2007) calculated theoretical radii for HD209458b. They found that the measured radius deviated atthe ∼ σ level for the age they assumed, even when em-ploying 10 × solar atmospheric opacity, no inner solid core,and no core heating. However, the updated age measure- ment for HD 209458b by Torres et al. (2008) of 3 . + . - . Gyr ismuch smaller than the one adopted by Burrows et al. (2007)(5 . + . - . Gyr), whereas the updated radius, 1 . + . - . , is sim-ilar to that used by Burrows et al. (2007) (1 . + . - . ). As aresult, the Torres et al. (2008) radius and age measurementfor HD 209458b can be fit by the 10 × solar-opacity model ofBurrows et al. (2007) within ∼ σ uncertainties, without theneed of any core heating sources. Moreover, if the tidal heat-ing rate of HD 209458b decayed from ∼ × erg s - to ∼ × erg s - during the past 2 Gyr (Jackson et al. 2008b),then its effective age would be 2 Gyr younger, due to the“clock reset” effect. Based on our parameter study in §4,the equilibration timescale of HD 209458b during maximumheating would have been ∼ ∼ ∼ × erg s - to ∼ × erg s - ) forour models to reproduce HD 209458b’s current radius. Inthis case, the Torres et al. (2008) radius measurement for HD209458b, along with an “effective age” of ∼ ∼ σ uncertainties, evenwith the 1 × solar-opacity model of Burrows et al. (2007) anda core mass of 10-20 M ⊕ , but without the need for any currentcore heating. The latter comports with the very small limit of ∼ a , e , and R p , but also important to factor in the feedback of theassociated tidal heating power on R p and its radius-age trajec-tory (Jackson et al. 2008a,b). Such a project is in progress.We thank Gáspár Bakos for comments on the eccentricityof HAT-P-1b, and Georgi Mandushev for insights concern-ing the possible eccentricity range of TrES-4. We also thankJosh Winn for a careful reading of an earlier version of themanuscript, and an anonymous referee for a careful and use-ful report that improves the paper. This study was supportedin part by NASA grants NNG04GL22G, NNX07AG80G, andNNG05GG05G and through the NASA Astrobiology Instituteunder Cooperative Agreement No. CAN-02-OSS-02 issuedthrough the Office of Space Science. APPENDIXEXTERNAL HEATING DUE TO TIDAL DISSIPATION
The total tidal energy dissipation rate within the planet in its rest frame assuming equilibrium tides with constant lag angle andsynchronous rotation is (e.g. Goldreich & Soter 1966; Bodenheimer et al. 2001, 2003; Gu et al. 2004): ˙ E tide = GM ∗ µ f ( e ) a τ circ ≈ . × erg s - (cid:18) e . (cid:19) (cid:20) f ( e ) e (cid:21) (cid:18) M ∗ M ⊙ (cid:19)(cid:18) M p M J (cid:19)(cid:18) a .
05 AU (cid:19) - (cid:18) τ circ Gyr (cid:19) - (A1)where µ ≡ M ∗ M p / ( M ∗ + M p ) is the reduced mass, f ( e ) ≡ [ h ( e ) - h ( e ) + h ( e )] is a function of orbital eccentricity with h ( e ) =(1 + e + e / - e ) - / , h ( e ) = (1 + e / + e / + e / - e ) - , and h ( e ) = (1 + e / + e / + e / + HEORETICAL RADII OF EXTRASOLAR GIANT PLANETS 925 e / - e ) - / (Gu et al. 2004). Note that f ( e ) → e as e → τ circ denotes the circularization timescale, which is τ circ ≈ .
10 Gyr × (cid:18) Q p (cid:19)(cid:18) M ∗ M ⊙ (cid:19) - / (cid:18) M p M J (cid:19)(cid:18) R p R J (cid:19) - (cid:18) a .
05 AU (cid:19) / . (A2)A more informative quantity is the ratio of the tidal energy dissipation rate and the insolation rate, given by ˙ E tide ˙ E insolation = GM ∗ µ f ( e ) π F p R p a τ circ ≈ . × - „ e . « » f ( e ) e –„ Q p « - „ M ∗ M ⊙ « / „ R p R J « „ a . « - / „ F p ergcm - s - « - . (A3)REFERENCESAdams, F. C. & Laughlin, G. 2006, ApJ, 649, 1004Bakos, G. Á., Kovács, G., Torres, G., Fischer, D. A., Latham, D. W., Noyes,R. W., Sasselov, D. D., Mazeh, T., Shporer, A., Butler, R. P., Stefanik,R. P., Fernández, J. M., Sozzetti, A., Pál, A., Johnson, J., Marcy, G. W.,Winn, J. N., Sip˝ocz, B., Lázár, J., Papp, I., & Sári, P. 2007a, ApJ, 670, 826Bakos, G. Á., Noyes, R. W., Kovács, G., Latham, D. W., Sasselov, D. D.,Torres, G., Fischer, D. A., Stefanik, R. P., Sato, B., Johnson, J. A., Pál, A.,Marcy, G. W., Butler, R. P., Esquerdo, G. A., Stanek, K. 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BSERVATIONAL P ROPERTIES OF THE T RANSITING P LANET S YSTEMS . a Period M ∗ R ∗ T ef f [Fe/H] ∗ Age M p R p F p a ˙ E tide / ˙ E insolationb System (AU) (days) e ( M ⊙ ) ( R ⊙ ) (K) (dex) (Gyr) ( M J ) ( R J ) (10 erg cm - s - ) [( Q p / ) - ] ReferencesXO-3 . . . . . . 0 . + . - . . + . - . . + . - . . + . - . + - - . + . - . . + . - . . + . - . . + . - . . + . - . . × - . . . . . . . . . . . 0 . + . - . . + . - . . + . - . . + . - . + - - . + . - . . + . - . . + . - . . + . - . . + . - . . × - TrES-4 . . . . . 0 . + . - . ∼ . . + . - . . + . - . + - + . + . - . . + . - . . + . - . . + . - . . + . - . . × - HAT-P-1 . . . 0 . + . - . ∼ . . + . - . . + . - . + - + . + . - . . + . - . . + . - . . + . - . . + . - . . × - The stellar flux at the planet’s substellar point. b ˙ E tide is the total tidal energy dissipation rate within the planet in its rest frame (Gu et al. 2004). ˙ E insolation ≡ π R p F p . c Johns-Krull et al. (2008); Inferred from spectroscopically derived stellar parameters. d Winn et al. (2008); Determined from light-curve fits. e Assuming e = 0 . f Torres et al. (2008); Note that they assume e = 0 exactly in deriving the parameters, since the radial-velocity data are consistent with a circular orbit. HEORETICAL RADII OF EXTRASOLAR GIANT PLANETS 11
TABLE 2B
EST - ESTIMATE C ORE M ASSES FOR
HAT-P-1
B UNDER V ARIOUS A SSUMPTIONS .Atmospheric Opacity 3 × solar 10 × solar „ e . « „ Q p « M c / M ⊕ . . . . . . . . . . . 15 30 60 20 40 80 N OTE . — The tidal dissipation rate is proportional to e / Q p when e ≪ TABLE 3F
ITTING P ARAMETERS FOR THE E QUILIBRIUM P LANETARY R ADIUS AND E QUILIBRATION T IMESCALE FOR
EGP S U NDERGOING C ORE H EATING . M p a ( M J ) (AU) C C C C C b b k b . .
02 15 . .
69 2 .
63 0 .
328 0 . - . - . .
03 7 .
74 4 .
43 1 .
22 0 .
157 0 . - . - . .
04 6 .
42 4 .
00 1 .
25 0 .
182 0 . - . - . .
05 4 .
61 2 .
66 0 .
866 0 .
135 0 . - . - . .
06 3 .
60 1 .
79 0 .
573 0 . . - . - . . .
02 6 .
42 4 .
51 1 .
58 0 .
255 0 . - . - . .
03 4 .
44 3 .
02 1 .
16 0 .
208 0 . - . - . .
04 3 .
42 2 .
19 0 .
894 0 .
170 0 . - . - . .
05 2 .
72 1 .
54 0 .
669 0 .
137 0 . - . - . .
06 2 .
33 1 .
15 0 .
512 0 .
110 0 . - . - . . .
02 4 .
20 2 .
54 0 .
888 0 .
145 0 . - . - . .
03 3 .
15 1 .
76 0 .
663 0 .
118 0 . - . - . .
04 2 .
57 1 .
34 0 .
543 0 .
105 0 . - . - . .
05 2 .
13 0 .
924 0 .
391 0 . . - . - . .
06 1 .
91 0 .
693 0 .
292 0 . . - . - . a R eq / R J = C + C x + C x + C x + C x , where x ≡ log( ˙ E heating / ˙ E insolation ). b log( τ eq / Gyr) = b + kxkx