Inflating and Deflating Hot Jupiters: Coupled Tidal and Thermal Evolution of Known Transiting Planets

Inflating and Deflating Hot Jupiters:
Coupled Tidal and Thermal Evolution of Known Transiting Planets

N. Miller11affiliation: Department of Astronomy and Astrophysics, University of California, Santa Cruz (, , J. J.  Fortney11affiliation: Department of Astronomy and Astrophysics, University of California, Santa Cruz (, , and B. Jackson22affiliation: Lunar and Planetary Laboratory, University of Arizona (

We examine the radius evolution of close-in giant planets with a planet evolution model that couples the orbital-tidal and thermal evolution. For 45 transiting systems, we compute a large grid of cooling/contraction paths forward in time, starting from a large phase space of initial semi-major axes and eccentricities. Given observational constraints at the current time for a given planet (semi-major axis, eccentricity, and system age) we find possible evolutionary paths that match these constraints, and compare the calculated radii to observations. We find that tidal evolution has two effects. First, planets start their evolution at larger semi-major axis, allowing them to contract more efficiently at earlier times. Second, tidal heating can significantly inflate the radius when the orbit is being circularized, but this effect on the radius is short-lived thereafter. Often circularization of the orbit is proceeded by a long period while the semi-major axis slowly decreases. Some systems with previously unexplained large radii that we can reproduce with our coupled model are HAT-P-7, HAT-P-9, WASP-10, and XO-4. This increases the number of planets for which we can match the radius from 24 (of 45) to as many as 35 for our standard case, but for some of these systems we are required to be viewing them at a special time around the era of current radius inflation. This is a concern for the viability of tidal inflation as a general mechanism to explain most inflated radii. Also, large initial eccentricities would have to be common. We also investigate the evolution of models that have a floor on the eccentricity, as may be due to a perturber. In this scenario we match the extremely large radius of WASP-12b. This work may cast some doubt on our ability to accurately determine the interior heavy element enrichment of normal, non-inflated close-in planets, because of our dearth of knowledge about these planet’s previous orbital-tidal histories. Finally, we find that the end state of most close-in planetary systems is disruption of the planet as it moves ever closer to its parent star.

Subject headings:
planetary systems – planets and satellites: general

1. Introduction

The precise mass and radius measurements for transiting exoplanets provide information about the planets’ interior structure and composition, which are often apparently unlike that of Jupiter and Saturn. Indeed, it is the incredible diversity of measured radii of transiting planets that has been most surprising. In the solar system, Jupiter and Saturn differ in mass by a factor of three while their radii differ by only 18%. However, amongst exoplanets, planets with the same mass can differ in radius by a factor of two. A hope amongst planetary astrophysicists was that the measurement of the mass and radius, when compared to models, would cleanly yield information on planetary interior composition. Although there are clearly examples where this has been done successfully, including heavy element rich planets such as HD 149026b (Sato et al., 2005; Fortney et al., 2006) and GJ 436b (Gillon et al., 2007), in general modelers have been foiled by planets with very large radii, larger than can be accommodated by “standard” cooling/contraction models.

Considerable work has been done in the past several years to understand the large radii of some planets, as well as the radius distribution of the planets as a whole. Explanations for the “anomalously” large planets have fallen into three categories: those that are a current or recent additional internal energy source, which has stalled the interior cooling and contraction (Bodenheimer et al., 2001; Guillot & Showman, 2002; Bodenheimer et al., 2003; Gu et al., 2003; Winn & Holman, 2005; Liu et al., 2008; Jackson et al., 2008b; Ibgui & Burrows, 2009), those that instead merely delay the contraction by slowing the transport of interior energy (Burrows et al., 2007; Chabrier & Baraffe, 2007), and those that invoke various evaporation mechanisms (Baraffe et al., 2004; Hansen & Barman, 2007). These are briefly reviewed in Fortney (2008).

Tidal heating as an explanation for these large-radius planets was suggested by Bodenheimer et al. (2001) for HD 209458b and has been revisited frequently by other authors (e.g. Bodenheimer et al., 2003; Winn & Holman, 2005; Liu et al., 2008; Gu et al., 2003, 2004; Jackson et al., 2008a, b; Ibgui & Burrows, 2009). We note that the mechanism of heating by obliquity tides (Winn & Holman, 2005) has been cast in considerable double by several authors (Levrard et al., 2007; Fabrycky & Tremaine, 2007; Peale, 2008).

At this time, tidal heating by orbit circularization is generally believed to be the most important type. The largest uncertainties in the standard tidal theory is the “tidal ” value, a standard parameterization of the rate of tidal effects. In this work, we use the standard notation for the planet tidal value as  and the stellar tidal value as  . Jupiter’s  value has been constrained to be between and (Goldreich & Soter, 1966). For tidal heating by circularization to take place, the planet must either initially have an eccentric orbit or the system must be driving the eccentricity of the planet at recent times.

The former scenario would have the following qualitative stages. The planet is left with an eccentric orbit through planet-planet interactions (Rasio & Ford, 1996; Chatterjee et al., 2008; Ford & Rasio, 2008). Tides on the star gradually reduce the semi-major axis. These tidal effects accelerate as the semi-major axis decreases. Tides on the planet become more important and the planet’s orbit circularizes; at the same time depositing orbital energy into the planet’s interior. Scattering/tidal evolution models of this sort were recently computed by Nagasawa et al. (2008). At this point, the system might be observed to have a fairly circular orbit and a larger-than-expected radius. Ibgui & Burrows (2009) use a coupled tidal-thermal evolution model, quite similar to the one we present here, to show that this scenario might be possible for the HD 209458 system, and by extension, many hot Jupiter planets. Such a model is necessary to self-consistently explain a planet’s radius in this picture. One potential issue with this scenario is that it can require large-radius planets to be observed at a “special time” since after the orbit is circularized, the planet may rapidly contract.

Alternatively, some planets might be found in an equilibrium state where their eccentricity is being forced by a third body while at the same time tides on the planet are damping the eccentricity (Mardling, 2007). This is an attractive explanation because the planet might be found in an inflated state for a long period of time. Previously, Bodenheimer et al. (2001) calculated the tidal power required to maintain the radius for HD 209458 b, Ups And b, and Tau Boo b, as a function of the assumed core size, in a stationary orbit. Recently, thermal evolution calculations with constant heating have been performed for TrES-4, XO-3b and HAT-P-1b by Liu et al. (2008), who placed constraints on - where is the recent time-averaged eccentricity of the orbit. These calculations are useful for estimating the required recent tidal heating. In some cases, where the eccentricity is non-zero and a perturber is necessary to invoke, then this constant heating picture might accurately describe the recent thermal history of the planet. In many cases the eccentricity is observed to be close to zero, which either implies that a) the planet’s eccentricity is at a non-zero equilibrium, but the planet’s  value is much smaller than inferred from Jupiter or b) the planet’s orbit is circularized and this calculation does not apply.

Clearly it is important to accurately measure the eccentricity of inflated systems to determine if either scenario is plausible. For many transiting systems, the eccentricity has been only weakly constrained with several radial velocity points and it is very difficult to distinguish a small eccentricity from one that is truly zero (Laughlin et al., 2005). For systems with an observed secondary eclipse, stronger upper limits on eccentricity can be found based on the timing of the eclipse (Deming et al., 2005; Charbonneau et al., 2005; Knutson et al., 2009). Note that secondary eclipse timing only constrains so it is possible that some of these systems have much larger eccentricity, but it is unlikely.

The above possibilities are also consistent with the popular planet formation and migration theories. These planets form while the protoplanetary disk is still present at much larger orbital distances and migrate early in their life to small orbital distances (e.g. Lin et al., 1996). After this initial phase, tidal evolution between the star and the planet occurs on Gyr time scales. The migration mechanism is important because it determines the initial orbital parameters for tidal evolution. There are multiple postulated migration mechanisms.

  1. Planet-disk interaction: Gravitational interactions between the planet and protoplanetary disk can exert torque on the planet (Ward, 1997a, b). These mechanisms tend to circularize the planet’s orbit very early on and decrease the semi-major axis. The disk migration time scales is significantly shorter than the lifetime of the disk, as described in Papaloizou et al. (2007) and references therein.

  2. Planet-planet interaction: Gravitational interactions with other nearby planets can transfer orbital energy and angular momentum between the two bodies. This can result in quickly decreasing or increasing the orbital distance of one of the planets as well as producing non-zero initial eccentricity orbits. Using N-body simulations, (Rasio & Ford, 1996; Weidenschilling & Marzari, 1996; Chatterjee et al., 2008; Ford & Rasio, 2008) have shown that this effect can be important and can result in the inner bodies having initial eccentricity as large as 0.8, before tidal damping ensues. Other authors have investigated migration with coupled secular driving and tidal friction, which can operate on similar timescales (Wu & Murray, 2003; Faber et al., 2005; Ford & Rasio, 2006; Fabrycky & Tremaine, 2007). There are also a handful of transiting planets that have non-zero eccentricity today, which can be explained by planet-planet interactions. It is also suggestive that many of these eccentric planets are more massive and have longer circularization time scales. Since the circularization time is longer for massive planets, this observation is consistent with the idea that planets of all masses can have large initial eccentricity, but that the lower mass planets have circularized while the massive planets may still be circularizing.

We expect that both of these mechanisms do happen to some extent. Therefore, we assume that a wide range of initial orbital parameters are possible, and we following the orbital and structural evolution of planets from a wide range of possible initial eccentricities, as described below. In the absence of a theory to predict likely initial eccentricities for a given planetary system, we seek to understand the physics of the evolution from a variety of initial states.

Most of the detected transiting planets currently have small eccentricities consistent with zero. These can be explained by either migration mechanism. If the planet migrated through planet-disk interactions, then it would have zero eccentricity when tidal evolution began. If the planet migrated through planet-planet interactions, then the orbit may have circularized due to tides on the planet.

2. Model: Introduction

In this work we would like to test the possibility that tidal heating by orbit circularization can explain the transit radius observations for each particular system. A necessary condition for this model is that a self-consistent evolution history can be found that agrees with all of the observed system parameters. To check this condition, we forward-evolve a coupled tidal-thermal evolution model over a large grid of initial semi-major axis and eccentricity for each system. We perform this test for , and , . Also, for each system with non-zero current eccentricity, we emulate an eccentricity driving source by performing runs with an eccentricity floor equal to the observed value. Later we also explore some higher  cases.

To properly understand the planet’s thermal evolution, it is necessary to couple the planet thermal evolution to the orbital-tidal evolution. Generally planets with initial semi-major axis of 0.1 AU or less will spiral into the star in Gyr timescales (Jackson et al., 2008a). This has a large impact on the incident flux on the planet, and therefore the loss of intrinsic luminosity of the planet. For some systems, this more efficient cooling at early times makes it possible to achieve smaller radii at the present. As the planet moves closer to the star, the tidal effects accelerate. If the orbit is eccentric, then at some point the planet’s orbit undergoes a period of circularization. At this time a significant amount of orbital energy is deposited into the planet, which increases its radius. The question of this work is whether, at this stage, the system’s observables (, , age, ) can simultaneously be achieved in the model. After this stage, the planet may lose mass by Roche lobe overflow (Gu et al., 2003), which can temporarily prevent the planet from falling into the star. However, the planet’s destiny is to fall into the star (Levrard et al., 2009; Jackson et al., 2009). These final stages of the planet’s life, including the mass loss stage, are not modeled in this work.

We typically find that tides on the star are the dominant source of semi-major axis evolution (Jackson et al., 2008a, b). When the eccentricity is large and damping, the tides on the planet can be the dominant semi-major axis damping source (Jackson et al., 2008a; Ibgui & Burrows, 2009). After surveying our suite of systems, we find that tidal heating can usually provide sufficient energy to inflate planetary radii as large as observed, but we do not always find an evolutionary history where the radius, semi-major axis, eccentricity and age all simultaneously fall within the observed error bars. Regardless, we find that tidal processes are an important aspect of planet evolution, particularly for hot Jupiter systems.

3. Model: Implementation

The Fortney et al. (2007) giant planet thermal evolution model has been coupled to the Jackson et al. (2008b) tidal evolution model. Therefore, the semi-major axis, eccentricity, and radius of the planet all evolve simultaneously. The tidal power is assumed to be deposited uniformly into the envelope of the planet. The planet structure model is assumed to be composed of four parts:

  1. a 50% rock/ 50% ice core (by mass) with the ANEOS equations of state (Thompson, 1990). The core does not participate in the thermal evolution of the planet, as in Fortney et al. (2007).

  2. a H/He envelope with , which uses the equation of state of Saumon et al. (1995). The envelope is assumed to be fully convective and thus has constant specific entropy throughout. At each time step the envelope is assumed to be in hydrostatic equilibrium.

  3. a series of radiative-convective, equilibrium chemistry, non-grey atmosphere models described in more detail in Fortney et al. (2007) and Fortney et al. (2008). These grids are computed for the incident fluxes at 0.02, 0.045, 0.1, and 1 AU from the Sun. This correctly determines the atmospheric structure and luminosity of the planet as a function of the planet’s surface gravity, incident flux from the host star, and interior specific entropy. In cases where the planet migrates to a semi-major axis with more incident flux than the innermost grid, then the boundary condition at the innermost grid is used.

  4. an extension of the atmosphere to a radius where the slant optical depth in a wide optical band (the Kepler bandpass) reaches unity. Therefore, all plotted radii are at the “transit radius,” as discussed by several authors (Hubbard et al., 2001; Baraffe et al., 2003; Burrows et al., 2003). The slant optical depth as a function of pressure is computed with the code described in Hubbard et al. (2001) and Fortney et al. (2003). We have found that the atmosphere height approximately follows the following relation


    where is the height in cm of the atmosphere from 1 kbar (approximately the depth where the radiative/convective zone boundary lies) to 1 mbar (where the planet becomes optically thin), is the planet’s surface gravity (cgs), and is the effective temperature in Kelvin. Taking into account this atmosphere height is significant when the planet has low gravity or high effective temperature. In Fortney et al. (2007), the radii at 1 bar were presented.

The orbital-tidal evolution model is described in detail by Jackson et al. (2008b, 2009) and references therein. The equations used in this work are


where is the semi-major axis, is the eccentricity, and is the tidal power deposited into the planet. This model attempts to describe tidal heating only by orbit circularization and ignores other forms of tides such as spin synchronization or obliquity tides, which are not believed to be as important. This model assumes that the star is rotating slowly relative to the orbit of the planet and is second order in eccentricity. Therefore the evolution histories that include periods when the orbit has high eccentricity should be regarded with caution. Because there is a lot of other uncertainty with regard to tidal theory, we choose to use this simple model instead of more complex models such as Wisdom (2008). For at least 1 of the 45 systems, HAT-P-2, the planet-star system may be able to achieve a double tidally locked equilibrium state (star is tidally locked to the planet and the planet is tidally locked to the star) as shown by Levrard et al. (2009); in this system it is not a good assumption that the star is rotating slower than the period of the orbit. However, Levrard et al. (2009) find that this assumption is valid for most stars. We find that tidal heating is largest where is not particularly large ( falling towards zero) so this theory suffices for our purposes.  is the tidal parameter of the planet and  is the tidal parameter of the star. In this work we have predominantly investigated cases when as well as the case of , . Since the value is in principle a function of the driving frequency (Ogilvie & Lin, 2004), amplitude of the distortion, and internal structure of the body, the value for close-in extra solar giant planets is potentially not equal to the value for Jupiter. If the value is a very “spiky” function of the driving frequency, then the system might spend a lot of time in a state where the tidal effects are occurring at a slow rate and quickly pass through states where tidal effects are rapid. The stellar value is typically estimated through the observed circularization of binary stars orbits, but has also been estimated by modeling the dissipation inside of a star (Ogilvie & Lin, 2007).

We assume that the tidal power is uniformly deposited into the envelope of the planet. The net energy loss is given by the following equation:


where is the luminosity at the planet’s surface, is some small nonzero time step, and is the specific entropy. If , then the planet’s envelope will be increasing in entropy and the planet’s radius will increase. More typically, and the planet’s entropy is decreasing and thus the planet is contracting. The power ratio is a useful measure of how important tidal effects are. It clearly indicates whether there is a net energy input (ratio larger than unity) or net energy loss (ratio smaller than unity).

For a given radius, assumed core size and average incident flux of the planet, . Therefore, if we calculate , the radius contraction rate when there is no internal heat source, we can use the following relationship to calculate when there is an assumed tidal heating (or an input power of another source).


Due to tidal migration the incident flux upon the planet increases with time. Based on the planet’s incident flux at a given time, we interpolate in the 4 grids which include the incident flux level from the Sun at 0.02, 0.045, 0.1, and 1 AU. Here we neglect the more minor effect that parent star spectra can differ somewhat from that of the Sun.

In order to examine all the plausible evolutionary tracks for each of the 45 transiting planets studied, we modeled their thermal evolution over a range of

  1. initial semi-major axis: the observed semi-major axis to five times the observed value.

  2. initial eccentricity: from 0 to 0.8.

  3. core mass: 0, 10 , 30 , 100 . For very massive planets we also consider core masses of 300 and 1000 . Except for GJ 436b, HAT-P-11b, and HD 149026b, the core was required to be at most 70% of the mass of the planet. For GJ 436b, we sample up to 21 and for HAT-P-12, we sample up to 23 .

Each of these possible evolution histories were run until either a) the time reached 14 Gyr, b) the entropy of the envelope became larger or smaller than the range of entropy values in the grid of hydrostatic equilibrium structures, or c) the planet reaches a small orbital distance (realistically, the planet would be disrupted before this stage, but in this work we do not model the mass loss process).

For each run, we searched the evolution history during the estimated system age range for times when the orbital parameters were also within their observed range. If this occurred we then recorded the transit radius during these times and compared the range of achieved values to observed values. In situations where a good estimate on the age is not available, we searched within 1 to 5 Gyr. When a secondary eclipse constraint on the eccentricity is not available we assume that the eccentricity value is (i.e. the likely range is between 0 and 0.05). In cases where the eccentricity is observed to be consistent with zero from a secondary eclipse, we assume that the eccentricity value is (ie. the likely range is between 0 and 0.01). We use the observed semi-major axis and error. We then search for instances of evolution histories during the possible age range that have an error-normalized distance less than 3 to the observed value. This distance is defined as


where and are the orbital parameters for the instance of a particular run and , , , and are the measured/assumed semi-major axis, semi-major axis sigma, eccentricity, and eccentricity sigma. Planet orbital parameters, transit radii, and stellar parameters are from F. Pont’s website at and The Extrasolar Planets Encyclopedia at

4. General Examples

Here we add different components of the model step-by-step, such that each effect can be appreciated independently. The two opposing effects of tidal evolution are late-time heating that is associated with eccentricity damping and more efficient early-time cooling due to initial semi-major axes that are larger then the present value. The four cases present are for a 1 planet orbiting a 1 star at 0.05 . In each of these cases we assume that the planet has a 10  core.

Case 1: no tidal effects, Figure 1. In the left panel, the solid line is the planet transit radius and the dot-dashed line is the radius at 1 kbar (near the convective-radiative boundary). In the right panel, the intrinsic planet luminosity is plotted as a function of time. As the planet contracts the luminosity of the planet significantly decreases. Without an internal heat source or semi-major axis evolution the planet’s radius monotonically decreases with time.

Figure 1.— Radius and intrinsic planet luminosity evolution for a 1 planet at 0.05 AU around a 1 star without any tidal effects. In the left panel, the dashed line is the radius at 1 kbar, near the convective/radiative boundary at gigayear ages. The solid line is the radius where the atmosphere reaches 1 mbar - approximately the radius that would be observed in transit.

Case 2: no orbital evolution, constant interior heating, in Figure 2. In this case the net output power is the difference between the intrinsic luminosity and a constant interior heating source of unspecified origin. In these evolution runs, the planet stops contracting when the intrinsic luminosity is equal to the constant heating source. This is equivalent to when the ratio between the input power and the luminosity of the planet is equal to unity. The upper 3 evolution tracks (purple, cyan, and blue) all reach an equilibrium between the interior heating and luminosity of the planet within 2 Gyr, but the evolution runs with lower input power do not reach an equilibrium state in the 6 Gyr plotted. As expected, when there is more input power, the equilibrium radius is larger. In practice, the input power through tides or other processes will not be constant over gigayears, but a planet may be inflated to a radius such that it is in a temporary equilibrium state.

Figure 2.— Similar to Figure 1, but with various constant heating applied in the interior of the planet. Moving from bottom to top, the constant heating rates are , , , and erg s.

Case 3: tidal orbital evolution, but without tidal heating, Figure 3. This case demonstrates how the orbital evolution due to tides effects the thermal evolution of the planet. Here we plot both the (tidal effects on the planet occur faster) and (tidal effects on the planet occur slower) cases with in black and red respectively. These curves exactly track each other because the tides on the planet do not significantly contribute to the migration when the eccentricity is small (here ). When comparing Figure 1 to Figure 3, notice that in the second case, the power drops off more rapidly as the semi-major axis decreases. This is due to the increase in insolation by the parent star, which deepens the atmospheric radiative zone, lessening transport of energy from the interior (e.g. Guillot et al., 1996). Another result of moving the planet closer to the star is that there is an up-tick in the transit radius. This is due only to an increase in the effective temperature, which increases the atmosphere height. The semi-major axis evolution accelerates as the planet moves inward due to the tidal migration rate’s strong dependence on semi-major axis.

Figure 3.— Planet thermal evolution with orbit evolution, but without tidal heating. Transit radius, semi-major axis and the planet’s intrinsic luminosity are plotted from left to right. and cases are plotted in black and red respectively.

Case 4: tidal orbital evolution and tidal heating, Figure 4. We now put both the orbital evolution and corresponding tidal heating together. Black is the case and red is case. Notice that in the low case, the planet circularizes quickly and tidal heating becomes less important. In the high case, the planet is still undergoing circularization and significant tidal heating at late times. As a result, the radius in the high  case (slower rate of tidal effects in planet) can be larger than the low  case (faster rate of tidal effects in planet) at late times. Both trials start out with fairly modest eccentricity ().

Figure 4.— Coupled planet thermal evolution and orbital evolution. and cases both with are plotted in black and red respectively. We plot, the radius evolution in the upper left, semi-major axis evolution in the upper right, ratio between tidal heating and intrinsic planet luminosity in the lower left, and eccentricity in the lower right.

In Figure 5, we compare the radius evolution in all four of these cases: Case 1 (no tidal effects, black), Case 2 (no orbital evolution, constant heating, blue), Case 3 (tidal orbital evolution, but not tidal heating, red), and Case 4 (full tidal evolution model, cyan). The cases with tidal evolution are plotted for the high  case. Clearly, when tidal heating is included (cyan or blue), it can result in a radius larger than achieved without including tidal heating (red or black). Since tidal heating is a time-varying quantity, the planet’s radius when tidal heating will not be as simple as in Case 2. Generally, the planet will experience significant tidal heating when the orbit is being circularized. At this time, the radius will increase, but after this time the radius of the planet will contract again. Also, because the planets in Case 3 (red) start at larger orbital distance than that of Case 1 (black), the radius contracts marginally faster when the planet is at larger semi-major axis. This is why the red line is lower than the black line before 2 Gyr. After this point, the transit radius increases in the red line case because the planet has moved close to the star, the effective temperature of the planet increases, and the atmosphere height also increases.

Figure 5.— Radius evolution in different cases. Cases 1, 2, 3, 4 (see text) are plotted in black, blue, red, and cyan.

To examine how different levels of internal heating affect the radius of the planet, we plot the planet radius after 5 Gyr as a function of mass in Figure 6. Again, these models assume a 10  core, at a orbital distance of 0.05 AU around a 1 Solar Mass star. In this figure, the black dotted line is the prediction of the thermal evolution model without tidal heating. The red dashed line is the base of the atmosphere at 1 kbar. Clearly, the height of the atmosphere is much larger for smaller planets due to their smaller gravities. The solid blue line is the radius relation from (Fortney et al., 2007). The solid black lines are the radius of the planet given a constant heating rate after 5 Gyr of evolution. The pink dotted curves are constructed in the same manner as the solid black curves, but required extrapolation (here, quadratic) off of the calculated atmosphere grid. At this point in time, most of these planets have reached an equilibrium state where an equal amount of internal heating is balanced by the planet’s intrinsic luminosity. Clearly, the effect on the radius for a given heating is larger for smaller mass planets.

Figure 6.— The dotted black line is the transit radius without any internal heating as a function of mass assuming a 10  core. In these models, we hold the planet at 0.05 AU around a 1 Solar Mass star. The dashed red line is the 1 kbar radius—near the convective/radiative zone boundary. The blue line is the relation from Fortney et al. (2007). The solid black lines are the radius one would find if there were a constant heating source (values between and erg s). The pink dotted lines were calculated in the same way, but required extrapolation (quadratic) off of the grid of atmosphere models.

5. Results

5.1. Specific Systems

While we have computed the evolution history of 45 systems, here we show representative calculations for particular samples of planets. These are TrES-1b, XO-4b, HD 209458b, and WASP-12b, and are shown in Figures 7, 8, 10, and 11 respectively. These four cases demonstrate qualitatively different cases. TrES-1b is a circularized planet with a “normal” radius value. XO-4b, HD 209458b, and WASP-12b are large-radii planets with a small relatively unconstrained eccentricity, zero eccentricity, and a nonzero value, respectively. In Figures 7 - 11, the transit radius evolution is plotted in the upper left panel, the semi-major axis evolution is plotted in the upper right panel, the ratio between the tidal power and luminosity is plotted in the lower left panel, and the eccentricity evolution is plotted in the lower right panel. The observed semi-major axis, eccentricity, and transit radius are plotted on each of the respective panels. The power ratio, tidal power to luminosity, describes how important tidal effects are to the energy flow of the planet. When this ratio is somewhat smaller than unity, tidal heating is relatively un-important for the thermal evolution of the planet and when this ratio reaches or surpasses unity, tidal heating plays a more significant role in the thermal evolution. In each of these figures, a set of runs were selected such that the orbital parameters and transit radius are closest to the observed values.

TrES-1b is a transiting hot-jupiter planet with zero or small eccentricity and a typical radius observation. The system is composed of a 0.76  planet orbiting a 0.89 star with a 0.04 AU semi-major axis. Tidal heating is not necessary to invoke to explain this system; we demonstrate that this tidal model can still explain these kinds of modest radius systems. Possible evolution histories with tidal effects are shown in Figure 7. These possible histories are selected such that their orbital parameters at the current age agree with the observed values and the transit radius that is close to the observed value. We show various core sizes in different colors: black for zero core, red for a 10  core, and blue for a 30  core. The cyan dotted line is the evolution history of a non-tidal thermal evolution model with a 10  core. Notice the radius evolution of the non-tidal model doesn’t differ significantly from the radius evolution of the corresponding 10  (red) tidal model. In these possible evolution histories with tidal effects, the initial eccentricity is relatively small and tidal heating doesn’t dominate the energy flux budget (in the lower left panel, the power ratio is always less than 1). However, the orbit decays significantly due to tides raised on the star by the planet, which continues even at . These tides cause these planet to migrate from an initial semi-major axis of 0.05 AU to 0.04 AU with the assumed . Figure 7 demonstrates that this model easily explains the radius of TrES-1b with a core between 10  and 30 .

Figure 7.— Possible tidal/thermal evolution tracks for the planet around the star TrES-1. Black: no core. Red: 10  core. Blue: 30  core. Cyan dotted: 10  core evolution history without tidal effects. This is a 0.76  planet orbiting a 0.89 star. Upper left panel: transit radius evolution. Upper right panel: semi-major axis evolution. Lower left panel: ratio between tidal power injected into the planet and intrinsic planet luminosity. Lower right panel: eccentricity evolution. Observed semi-major axis, eccentricity and observed radius are plotted in their respective panels. These evolution tracks were selected to have orbital parameters that agree with the observed values. , .

There is a slight upturn in radius just before an age of 4 Gyr. This is due to the heating of the planet’s atmosphere at very small semi-major axis, and is not due to tidal power. As the planet reaches smaller orbital distances the incident flux it intercepts increases dramatically, leading to an enlarged atmospheric extension, and greater transit radius. This feature is also present in the recent paper by Ibgui & Burrows (2009). The tracks end when we stop following the evolution, with the assumption that the planet is disrupted or collides with the parent star. This is merely the first of many evolution tracks that we present with the end state being the disruption of the planet. This finding is essentially quite similar to that of Levrard et al. (2009) who find that all of the known transiting planets, save HAT-P-2b, will eventually collide with their parent stars. Robust observational evidence for this mechanism was recently detailed by Jackson et al. (2009).

XO-4b is an inflated planet where the eccentricity has not been well constrained, due to sparse radial velocity sampling (McCullough et al., 2008). In these cases we search for instances over the evolution histories where the eccentricity is between 0 to 0.05, because we assume that a larger value would have been clearly noticed in radial velocity data. With this eccentricity constraint we show in Figure 8 that there is a narrow period of time when we can explain the inflated state with a recent circularization of the orbit that has deposited energy into the interior of the planet. The evolution curves shown here are for tidal parameters and ; in the higher case, the radius evolution curves do not agree with the observed value. In Figure 8, we show black, red, and blue curves for evolution runs with no core, 10  core, and 30  core respectively. The pink curve is an evolution history for low initial eccentricity with a 30  core. Again, the cyan curve is a no-tidal evolution history with 10  core. Since tidal power is deposited mainly when the planet is being circularized, high initial eccentricity orbits are required for these planets to experience significant later tidal inflation. Another interesting feature of this plot, is that when comparing the radius of the runs for different cores at any given time, we find that the radius is not always monotonically decreasing with core size. This shows that uncertain past orbital-tidal history can lead to uncertainly in derived structural parameters such as the core mass.

Figure 8.— Possible tidal/thermal evolution tracks for the planet around the star XO-4. This is a 1.72  planet orbiting a 1.32 star. Black: no core. Red: 10  core. Blue: 30  core. Magenta: 30  core with a low initial eccentricity. Cyan dotted: 10  evolution history without tidal effects. Panels are analogous to Figure 7. The eccentricity that is marked in the lower right panel is our assumed possible range (0 to 0.05). These evolution tracks were selected to have orbital parameters that agree with the observed values. . Notice that the tidal models initially have smaller radii than the non-tidal model because the tidal models are able to more efficiently cool at early times due to their larger semi-major axis.

As an example of the kind of calculation that was performed for every planet, in Figure 9 we show snapshots of the orbital parameters ( and ) of the ensemble of systems that are at some point consistent with the observed orbital parameters and age of XO-4b. Note that we do not require that the radius simultaneously also agree with the observed radius, but rather compare the range of possible radius values achieved by the model to the actual observed value. The black points are the original orbital parameters. The red points are the orbital parameters for one of these runs at a later point in time (0.5 Gyr, 1.5 Gyr, and 2.1 Gyr). The filled green circle marks the 1 observed orbital parameters, while the dashed region is the 3 zone.

Figure 9.— Grid of evolution histories for XO-4b that were found to be consistent with the orbital parameters at a later time. These histories are not required to also have a radius value that is consistent with the observed value. These evolution runs assume a core size of 10  ,  , and  . This serves as a sample for the type of calculation that was performed for every planet. Black: original orbital parameters of each run. Red: orbital parameters at a later marked time (0.5 Gyr, 1.5 Gyr, and 2.1 Gyr). The filled green circle is the 1 zone, while the dashed region is the 3 zone.

HD 2094598b is a large-radius planet with eccentricity that has been observed to be very close to zero (Deming et al., 2005). The planet is observed to have a radius of 1.32  and mass of 0.657 . Therefore we require evolution histories where the current eccentricity is . Evolution histories for this system are shown in Figure 10 with   and  . With these chosen values, we find that the planet could have experienced tidal heating at a previous time, however by the time it has an eccentricity of 0.01 or less the planet’s radius has since deflated below the observed value. It is possible to find an evolution histories that agrees with the observations by allowing different values, as shown by Ibgui & Burrows (2009). Although the tidal value is not strongly constrained and may even vary depending on the configuration of the system (Ogilvie & Lin, 2004), it is our view that it makes the most sense to fix the value close to prior inferred values. Again, the black, red, and blue curves correspond to no core, 10  core, and 30  core sizes respectively. The cyan curve is a non-tidal thermal evolution history for a 10  core. In these cases, tidal power is sufficient to inflate the planet’s radius to its observed value, however we do not find evolution histories that also agrees with the other observed parameters—especially the eccentricity. In the semi-major axis evolution, there is a clear transition knee where the rate of orbital evolution decreases. The first phase is due to tidal effects of both the star and planet while the eccentricity is nonzero. The second phase is mainly due to tides on the star when the eccentricity is zero.

Figure 10.— Possible tidal/thermal evolution tracks for the planet around the star HD 209458. This is a 0.657  planet orbiting a 1.101 star. The planet has a radius of 1.32  and an observed eccentricity of zero. Black: no core. Red: 10  core. Blue: 30  core. Purple: 30  core with low initial eccentricity. Cyan dotted: 10  core evolution model without tidal effects. Panels are analogous to Figure 7. .

WASP-12b is a planet with an especially large radius of 1.79  with a non-zero eccentricity of 0.05 (Hebb et al., 2009). An interesting property of this system is that the planet is filling at least 80 % of its Roche lobe by radius (Li et al., 2009). Figure 11 shows evolution curves in black, red, and blue for no core, 10  core, and 30  core cases respectively when an eccentricity floor is imposed. Also, in cyan is the non-tidal model. In these tidal cases the tidal power increases in strength as the semi-major axis decays until the planet undergoes a rapid expansion. When the semi-major axis gets small enough, the tidal power exceeds the luminosity and the planet’s radius rapidly increases. This happens both because the incident flux decreases the intrinsic luminosity of the planet and tidal heating has a strong semi-major axis dependence (). We do not model the mass loss process, which is likely to occur at late times for systems such as these (Gu et al., 2003) This should only be taken as evidence that if there was an eccentricity driving companion similar to mechanisms suggested by Mardling (2007), then it may be possible to heat this planet to quite large radii.

Figure 11.— Possible tidal/thermal evolution for WASP-12b. This is a 1.41  planet orbiting a 1.35 star. The planet has a very large observed transit radius of 1.79  and an eccentricity of 0.05. In these evolution histories, we impose an eccentricity floor mimicking the effects of an eccentricity driving force. Black: no core. Red: 10  core. Blue: 30  core. Cyan dotted: 10  core evolution history without tidal effects. Panels are analogous to Figure 7.   and  .
Figure 12.— Observed planet radius (black) compared to a range of achieved model radii (colors) using ; . Planets are ordered by increasing incident flux according to their current observed parameters. Planets are marked with a * if they have nonzero observed eccentricity. The range of possible radius values under the full tidal evolution model is plotted in purple with initial eccentricity between 0 and 0.8. The radius range for a model with tidal-orbital evolution, but without the tidal heating into the interior of the planet is plotted in green. The radius range for a standard stationary model without any tidal effects is plotted in blue. The radius range for the full tidal evolution model with a maximum initial eccentricity of 0.4 is plotted in orange. In cases where a nonzero eccentricity has been observed, the radius range with an eccentricity floor equal to the observed value is shown in red.
Figure 13.— Observed planet radius (black) compared to a range of viable model radii (colors) using ; . Qualitatively, we observe the same trends that were observed in Figure (a)a and ((b)b). A larger value decreases the rate of tidal effects via tides on the planet. Typically the tides on the planet from the star are responsible for circularizing the orbit, while tides on the star from the planet are responsible for decreasing the semi-major axis. In the larger case, the tidal circularization can be delayed for longer, which can make the possible radius of the planet larger. On the other hand, a larger also decreases the power deposited into the planet.

5.2. Summary for Suite

We have summarized our results for all 45 planetary systems in Figures 12 and 13 for  equal to and . In these figures, we have plotted the observed radius range (lower limit to upper limit) in black. The achieved radius range under various assumptions is plotted in color. Possible radii are recorded in instances of the evolution histories when the orbital parameters and age all agree with the observed , , and age values (as defined previously, within 3 error-normalized distance units of the observed value). The age of each system is often quite uncertain; since the possible radius values are sensitive to the age of the system, this is a large source of uncertainty for our results. For each planet, a range of radius values is plotted for up to five different successful types of models. These are models computed as discussed in §4.

  1. The full tidal evolution model is shown in purple. In this model the initial eccentricity was sampled from 0 to 0.8 and the initial semi-major axis was sampled from the observed semi-major axis to 5 the observed value. This is case 4 in §4.

  2. The model with tidal migration but without heating is shown in green. We perform the same search procedure as in the full tidal model. This model is not meant to be physical, but to give us an understanding of how tidal orbital migration alone effects the planet’s radius. This is case 3 in §4.

  3. The “stationary” model is shown in blue with all tidal effects turned off. These are “standard” cooling/contraction models, quite similar to those in Fortney et al. (2007). These models differ slightly than the models listed in Fortney et al. (2007) in two ways. First, these models more accurately take into account the height of the atmosphere. Second, some of these models explore a wider range of core sizes. This is case 1 in §4.

  4. For planets whose current observed eccentricity is less than 0.4, the full tidal evolution with an maximum initial eccentricity of 0.4 is plotted in orange. Because tidal heating in the planet is directly connected to eccentricity damping, these runs serve as a demonstration of relatively less tidal heating due to circularization. This is a subset of case 4 from Section 4.

  5. For systems where there is a measured non-zero eccentricity, we simulate the effects of an eccentricity source by performing the full tidal evolution with an eccentricity floor equal to the observed value. These cases are shown in red. This is essentially a combination of Case 4 and Case 2.

For some planets, some of these “cases” were either not possible to compute or in no instances were the observed parameters consistent with the model parameters. For instance in cases when the observed eccentricity is larger than 0.4, the tidal evolution histories with 0.4 maximum initial eccentricity never are consistent with the observation. In these cases, no radius range is drawn. In some of the cases where tidal heating is included, an evolution history is found where a large amount of energy is deposited into the planet while the orbital parameters are consistent with observations. These result in a maximum achieved radius that sometimes exceeds 2 . In some of these cases, the planet will later cool off before the evolution stops. In other cases, the tidal power is sufficient to increase the planet’s entropy beyond the maximum entropy of our grid, which ends the evolutionary calculation. In the future we plan to include mass loss and the subsequent evolution history.

By comparing these models we find a few interesting patterns. When comparing the full tidal evolution model (purple) to the stationary model (blue), notice that there are some cases where the full tidal model has a larger maximum radius and other cases where the reverse is true. This can be understood to be caused by the two competing effects of tidal evolution. Tidal heating puts power into the planet and inflates the radius, and tidal orbital evolution allows the planet to cool more efficiently at earlier times when the planet is less irradiated by the parent star. It is also useful to compare these two cases to the no heating model. The no heating model generally has a smaller maximum radius than the stationary model because of the second effect. The tidal model has a larger maximum radius than the no heating model because of energy deposition into the planet.

Often the model achieves large radius values through a recent circularization of an originally high eccentricity orbit. During the circularization event (when the eccentricity drops significantly), tidal dissipation in the interior of the planet may deposit sufficient energy to significantly inflate the planet. The orange case (maximum initial eccentricity equal to 0.4) has been plotted to compare against the purple (initial eccentricity up to 0.8) to show how large initial eccentricity evolution histories contribute to the maximum achieved radius. Note that in the low  case in Figure 12, extremely large radii can be achieved for GJ 436b and HAT-P-11. This happens in our model through a recent rapid circularization of the orbit.

It may also be possible to have tidal heating without large initial eccentricities if there is a eccentricity driving source in the system. In some cases, such as in WASP-6b or WASP-12b, the resulting tidal heating may be enough to explain the large transit radius. By comparing the red (tidal evolution with an eccentricity floor) to the purple (regular tidal evolution), larger radius values can be achieved when the orbit is not allowed to circularize.

Tidal evolution and heating clearly have important effects on a planet’s evolution, but not all of the large-radius planets could be explained through this mechanism, given our chosen values. The planets HD 209458b, COROT-EXO-2b, HAT-P-9b, WASP-1b and TrES-4b have radii that are larger than achieved in our models in both the low and high  cases. Typically, while it is possible to inflate the radius to the observed values, it difficult to find the system with an inflated radius and low current eccentricity. WASP-12b was explained if we assume that its eccentricity is maintained.

When comparing Figure 12 to Figure 13, it is interesting that some of the planets that are not explainable in the lower  case can be explained with larger  . Although results in tidal heating being stronger than the case, it also results in circularization on a shorter time scale. In the cases, it is often common for there to be a possible recent circularization of a high initial eccentricity orbit where no such history was found in the evolution runs.

In Table 1, we have selected a set of the largest planets and listed various properties. In the left column, we list the observed parameters. For various core sizes, we list the achieved radius of the tidal model in the low  and high  cases, the estimated luminosity of the planet at its current radius, and the current contraction rate of the planet without internal heating (previously defined as ). Also, on the top row for each planet, we list the coefficient of tidal heating. This is defined as


This quantity allows one to get an order-of-magnitude idea of recent tidal heating given the more constrained properties of the system (radius of the planet, masses of the bodies, and semi-major axis). The actual tidal power will greatly depend on the eccentricity and values, which are more uncertain. The ratio between the luminosity of the planet and this coefficient of tidal heating is a dimensionless number that describes how important tidal effects can be for a given core size. Certainly, since and  is quite uncertain, this ratio is not a strong test of tidal effects, but it is a simple way of testing how important tidal effects presently can be. Notice also that for an assumed tidal power, we can compute the present contraction rate using this table and Equation 6.

When calculating the contraction rate, the planet is assumed to be located at the current observed semi-major axis, which determines the incident flux from the star, structure of the planet’s atmosphere, and thus the intrinsic luminosity of the planet at each time. For these large-radius systems, the contraction rate is often very fast. If we assume that tidal heating is the cause of large radii, but that an eccentricity driving companion is not present, then either the system is in a transient period or that this thermal evolution model is not correct. On the other hand, if we rule out transient explanations, then either a constant heating is present or it is necessary to invoke another mechanism.

5.3. High  cases

Although  is generally thought to be closer to based on the observed circularization time in binaries, it is possible that that tidal dissipation in the stars is less efficient in the planet-star case. Since tidal evolution is not fully understood, the high  case may or may not be physical. However, an advantage of this case is that it allows for orbital history solutions with a recent circularization. In this regime, the planet migrates inward at a slower rate and thus the circularization would occur at a later time. Also, after the tidal power is deposited, the planet is not rapidly migrating into the star as in the low  cases. Ibgui & Burrows (2009) have suggested that high  case can better explain the radius of HD 209458b.

We have explored this parameter regime as shown in Table 2 for five of the systems that we were not able to explain in the low  cases. We test the cases   and   with both   and  . In the table the radius range is reported for a given core size,  and  model parameters, as well as the number of runs that were found at some point in time to be consistent with the observed age, semi-major axis and eccentricity of the system.

Also, in Figure 14, we show snapshots in semi-major axis / eccentricity space of possible evolution histories of HD 209458 b that are consistent with the observed parameters. The black points are the original orbital parameters, while the red points are the orbital parameters at a later time. The green oval is the 1 orbital parameters. The dashed green line is the 3 orbital parameters, which we require an evolution histories to fall within during the expected age range of the system. Eccentricity was sampled from 0.2 to 0.8 in this particular case.

Figure 14.— Grid of evolution histories (with initial ) that were found to be consistent with the orbital parameters at a later time for the system HD 209458. These evolution runs assume there is no core,   and  . Black: original orbital parameters of each run. Red: orbital parameters at a later marked time (0.5 Gyr, 1.5 Gyr, and 2.1 Gyr). The filled green circle is the 1 zone, while the dashed region is the 3 zone.

We also show in Figure 15 possible radius evolution histories for the planets HD 209458b, WASP-1b, and CoRoT-Exo-2b. When  is allowed to be larger, the qualitative effect is that the planet’s semi-major axis decreases slower and thus the circularization event occurs at a later time. This makes it possible to sometimes achieve higher radius values at the expected age of the system with the model. However, even for these high  runs for these large-radius planets, only for two of the five can the observed radius be matched.

Figure 15.— Potential radius evolution histories for HD 209458b, WASP-1b, and CoRoT-Exo-2b with no core,   and   (larger than our standard case). As usual, these evolution histories have been selected from an ensemble of possible initial conditions such that at some point during the estimated age of the system, the planet has orbital parameters that are consistent with the observed values.

6. Discussion & Conclusions

This paper presents a coupled tidal and thermal evolution model applicable to close-in extrasolar giant planets. The model is tested against 45 of the known transiting systems. Generally, tidal evolution yields two competing effects on the radii of close-in EGPs:

  1. Tidal evolution requires that, after planet formation and subsequent fast migration to a relatively close-in orbit, the planet start at a larger semi-major axis than is currently observed (Jackson et al., 2008a). This results in less incident flux at earlier times, which allows the planet to cool more efficiently and contract more at a young age, which moves the range of feasible model radii at the current time to smaller values. Generally this is a minor effect, but it is more important for cases when the current incident flux is larger.

  2. Tidal evolution deposits energy into the planet when the orbit is being circularized. This typically increases the radius of the planet at this time. If there is an eccentricity driving source for the inner planet, then tidal heating can be important for the duration of the planet’s life. If the planet starts with a highly eccentric orbit, it might not circularize for gigayears. The semi-major axis of the planet’s orbit will initially slowly decrease due to tides on the star. As the planet moves closer to the star, tides on the planet become more effective. This delay of circularization can sometimes allow tidal heating to significantly inflate planets multiple gigayears after formation despite these systems having shorter “circularization” time scales.

We have shown that for the close-in giant planets that orbital history can play a large role in determining the thermal evolution and current observed radius. While the effects are larger for planets with larger initially eccentricities, tidal evolution still affects the thermal evolution of planets with zero eccentricity as well. Varying amounts of time-dependent tidal heating are degenerate with the radius effects due to the core of a planet (or more generally, a heavy element enrichment).

Since at the current time we are ignorant of the exact orbital history, it is generally not possible to determine the mass of the core with complete confidence for any specific system. However, in cases when the radius of the planet is especially small, a large core or increased heavy element abundance is required. For larger radius planets, it is not possible to determine the planet’s core size because recent tidal heating is degenerate with smaller core sizes. Furthermore, some systems likely have more complex orbital dynamics than described here due to the effects a third body. The uncertainty is increased since despite our expectation that tidal effects do occur, the rate that at which they occur (controlled by ) is uncertain to an order of magnitude.

This paper serves as a forward test of the tidal theory for close-in EGPs outlined by Jackson et al. (2008b), who had previously only investigated heating rates backwards in time, from current small eccentricities from 0.001 to 0.03. Quite often however, the forward modeling of these single-planet systems, across a wide swath of initial and , is not consistent with current eccentricities as large as Jackson et al. (2008b) assumed. If initial eccentricities were indeed large, then final circularization and tidal surge may indeed by fairly recent, but this cannot be expected to be the rule in these systems. We have taken an agnostic view as to whether initial migration to within 0.1 AU was via scattering or disk migration. In the former, initial eccentricities up to 0.8 are possible (Chatterjee et al., 2008) while in the latter the initial eccentricity would be zero. The viability of tidal heating to explain even some of the inflated planets with very small current eccentricities rests on the notion that planet scattering does occur, such that circularization (and radius inflation) can occur at gigayear ages. The detection of misalignment between the planetary orbital plane axis and stellar rotation axis via the Rossiter-McLaughlin Effect (e.g. Winn et al., 2007, 2008) is beginning to shed light on migration. Fabrycky & Winn (2009) have found tentative evidence that is consistent with two modes of migration, one which may yield close alignment (perhaps from disk migration) and one with which may yield random alignment (perhaps from scattering), although to date only XO-3b in the published literature shows a large misalignment (Hébrard et al., 2008). Further measurements will help to constrain the relative importance of these two modes of migration.

Most of the systems investigated do not require tidal heating to match their radius, but these systems can also be readily explained when including tidal evolution. Some of the planets investigated can be matched with tidal heating that could not be explained with a standard contraction model. Depending on the  value chosen, HAT-P-4, HAT-P-9, XO-4, HAT-P-6, OGLE-TR-211, WASP-4, WASP-12, TrES-3, HAT-P-7, and OGLE-TR-56 can all be explained with an evolution history with non-zero initial eccentricity. WASP-6 and WASP-12 can be explained by invoking a minimum eccentricity, which may suggest the presence of a companion. Other systems were not explained by the model for our chosen  values. This suggests that either  and  may be much different then our expectation or that other mechanisms are at work in these large-radius planets.

This work should be taken as a simplified analysis of how tidal evolution can affect a planet’s thermal evolution. Strong quantitative conclusions should not be drawn because of the large uncertainties in the tidal evolution model, especially at large eccentricity. Also, the rate of tidal effects may be a very strong function of frequency. If this is the case, the planet may spend a lot of time at certain states where tidal effects are slow and rapidly pass through states where tidal effects are more rapid. If a constant Q value can even be applied, the actual value is highly uncertain. The Q values that we choose were meant only to span the range that we considered to be likely. The rate of tidal effects may depend on the interior structure of the planet and may be different for different exoplanets. Also, this analysis only takes into account orbit-circularization tidal heating.

The conclusion that should be drawn from this work is that a planet’s tidal evolution history can play an important role on the planets’ current radius, especially for systems that are born at semi-major axis less than 0.1 AU. In some cases, tidal heating could have inflated the radius of the planet in the recent past, even though tidal heating in the present might not be happening. In other cases, we were not able to explain the large-radius observations with our coupled tidal-thermal evolution model. This suggests that tidal heating will not be able to explain all of the large-radius planets, which has been a hope of some authors (Jackson et al., 2008b; Ibgui & Burrows, 2009). For some of the planets that we are able to explain, we require a recent circularization, such that this model can only explain these observations if we at at a “special time” in its evolution. This has to be reconciled with the fraction of planets that have large radii that require such an explanation. Improved constraints on the eccentricities of these systems will better constrain recent tidal heating.

A more robust treatment of the effects of tidal heating on transiting planet radius evolution may require a coupling of the model presented here to a scattering/disk migration model, which could derive the statistical likelihood of various initial orbital and configurations, which would then serve as the initial conditions to subsequent orbital-tidal and thermal evolution. This is important because for any particular planetary system the orbital evolutionary history of the close-in planet may be difficult to ascertain. Recently Nagasawa et al. (2008) have simulated the formation of hot Jupiters with a coupled scattering and tidal evolution code, and find a frequent occurence of hot Jupiter planets. A further coupled undertaking of this sort, to be compared with an statistically significant number of transiting planets, could be performed in the future.

JJF and NM are supported by NSF grant AST-0832769. We thank the referee, G. Chabrier, as well as E. Ford, D. Fabrycky, and S. Gaudi for their comments.


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System Core [] Radius Range ( ) Radius Range ( ) [ergs/s] [/yr]
= 0.69 0.0 1.12 - 1.19 1.13 - 1.18
= 1.32 10.0 1.08 - 1.15 1.08 - 1.15
a = 0.05 AU 30.0 1.02 - 1.08 1.02 - 1.07
100.0 0.81 - 0.90 0.81 - 0.84
= 1.03 0.0 1.14 - 1.23 1.16 - 1.79
= 1.49 10.0 1.11 - 1.21 1.13 - 1.79
a = 0.03 AU 30.0 1.07 - 1.15 1.08 - 1.52
100.0 0.95 - 1.03 0.93 - 1.07
= 3.31 0.0 1.11 - 1.23 1.11 - 1.17
= 1.47 10.0 1.11 - 1.24 1.11 - 1.16
a = 0.03 AU 30.0 1.09 - 1.23 1.09 - 1.15
100.0 1.05 - 1.20 1.05 - 1.10
= 1.72 0.0 1.15 - 1.34 1.15 - 1.17
= 1.34 10.0 1.14 - 1.30 1.13 - 1.15
a = 0.06 AU 30.0 1.11 - 1.25 1.10 - 1.13
100.0 1.02 - 1.11 1.02 - 1.03
= 1.06 0.0 1.16 - 1.29 1.16 - 1.19
= 1.33 10.0 1.14 - 1.28 1.13 - 1.16
a = 0.05 AU 30.0 1.09 - 1.28 1.09 - 1.11
100.0 0.95 - 1.09 0.95 - 0.96
= 1.78 0.0 1.14 - 1.55 1.14 - 1.21
= 1.36 10.0 1.13 - 1.56 1.12 - 1.19
a = 0.04 AU 30.0 1.11 - 1.50 1.10 - 1.16
100.0 1.01 - 1.44 1.02 - 1.06
= 0.78 0.0 1.16 - 1.49 1.16 - 1.29
= 1.40 10.0 1.13 - 1.50 1.13 - 1.25
a = 0.05 AU 30.0 1.06 - 1.36 1.06 - 1.17
100.0 0.87 - 1.00 0.87 - 0.95
= 0.93 0.0 1.15 - 1.33 1.14 - 1.17
= 1.78 10.0 1.12 - 1.32 1.11 - 1.14
a = 0.05 AU 30.0 1.07 - 1.29 1.06 - 1.09
100.0 0.91 - 0.99 0.90 - 0.92 - -
= 1.03 0.0 1.14 - 1.38 1.14 - 1.22
= 1.36 10.0 1.12 - 1.36 1.12 - 1.19
a = 0.05 AU 30.0 1.08 - 1.38 1.07 - 1.13
100.0 0.93 - 1.10 0.93 - 0.97
= 0.87 0.0 1.16 - 1.25 1.16 - 1.21
= 1.44 10.0 1.13 - 1.22 1.13 - 1.18
a = 0.04 AU 30.0 1.07 - 1.18 1.07 - 1.10
100.0 0.90 - 1.06 0.90 - 0.92
= 1.27 0.0 1.12 - 1.20 1.13 - 1.66
= 1.45 10.0 1.11 - 1.18 1.10 - 1.51
a = 0.02 AU 30.0 1.07 - 1.11 1.08 - 1.52
100.0 0.96 - 1.03 0.96 - 1.18
= 1.41 0.0 - 1.18 - 2.02
= 1.79 10.0 - 1.16 - 1.57
AU 30.0 - 1.12 - 1.37
100.0 - 1.01 - 1.11 - -

Note. – Various large-radius hot Jupiter planets have been listed. In the first column, we list the observed parameters of the system for reference. In the second column, we list an assumed core size. The achieved radius range for two different  values is liested in the third and fourth columns. In the fifth column, we list relevant power quantities. The coefficient of tidal power is listed in the first row for each system. In the following rows, we list the luminosity of the planet for the assumed core mass. In the final row, we calculate , the radius derivative when there is no internal heating source.

System Core [] Radius [] (5,6) Radius [] (5,7) Radius [] (6.5,6) Radius [] (6.5,7)
= 0.69 0.0 1.12 - 1.19 ( 683 ) 1.12 - 1.18 ( 737 ) 1.15 - 1.32 ( 816 ) 1.15 - 1.31 (1036 )
= 1.32 10.0 1.09 - 1.16 ( 931 ) 1.09 - 1.15 (1136 ) 1.12 - 1.27 ( 765 ) 1.11 - 1.25 ( 945 )
= 0.93 0.0 1.16 - 1.22 (1291 ) 1.16 - 1.21 ( 849 ) 1.24 - 1.43 ( 665 ) 1.19 - 1.37 (1205 )
= 1.78 10.0 1.13 - 1.19 (1285 ) 1.13 - 1.18 ( 959 ) 1.20 - 1.37 ( 512 ) 1.16 - 1.33 (1154 )
= 1.52 0.0 1.15 - 1.19 (1520 ) 1.15 - 1.19 (1390 ) 1.17 - 1.28 ( 538 ) 1.18 - 1.30 ( 728 )
= 1.58 10.0 1.13 - 1.18 (1515 ) 1.13 - 1.18 (1390 ) 1.16 - 1.26 ( 501 ) 1.17 - 1.28 ( 694 )
= 0.87 0.0 1.17 - 1.21 ( 835 ) 1.17 - 1.20 ( 26 ) 1.23 - 1.48 ( 656 ) 1.19 - 1.39 (1463 )
= 1.44 10.0 1.14 - 1.18 ( 829 ) 1.14 - 1.17 ( 297 ) 1.20 - 1.45 ( 636 ) 1.16 - 1.35 (1438 )
= 3.31 0.0 1.12 - 1.18 (1337 ) 1.12 - 1.19 (1069 ) 1.19 - 1.40 (1243 ) 1.13 - 1.33 (2127 )
= 1.47 10.0 1.11 - 1.17 (1334 ) 1.11 - 1.19 (1092 ) 1.18 - 1.39 (1242 ) 1.12 - 1.32 (2120 )

Note. – Achieved radius values for 5 systems with high  for core size 0.0 and 10  . The parameters used are denoted in the header with (  ). In the body of the table, the range or achieved radius values is lested along with the number of runs found in parenthesis.

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