TIDAL EVOLUTION OF CLOSE-IN PLANETS

It is common to describe the dissipation inefficiency of tides in terms of the tidal quality factor instead of a constant lag angle ϕ, or a constant time lag Δt. The specific dissipation function is defined as follows (Goldreich 1963):

$$\begin{equation} Q\equiv \frac{2\pi E^{*}}{\oint -({\it dE}/{\it dt}){\it dt}}=\frac{1}{\tan \phi }, \end{equation} \tag{ 13 }$$

where E* in the numerator is the peak energy stored in tides during one tidal cycle, while the denominator represents the energy dissipated over the cycle. When the phase angle ϕ (which is twice the geometrical lag angle) is small, this is simplified as Q ∼ 1/ϕ, which implies a large Q value. The estimated tidal quality factors are ≳10⁴ for Jupiter and Neptune (e.g., Lainey et al. 2009; Zhang & Hamilton 2008), and even larger values are expected for synchronized close-in exoplanets (Ogilvie & Lin 2004). On the other hand, the values usually adopted are ≳10⁶ for main-sequence stars (e.g., Trilling et al. 1998). Thus, we are generally interested in the case of Q ≫ 1, and the above approximation is reasonable. Using the weak friction approximation (ϕ ∼ σΔt), we can redefine Q as

$$\begin{equation} Q\equiv \frac{1}{\sigma \Delta t} . \end{equation} \tag{ 14 }$$

In the following sections, we study some limiting cases of tidal frequencies. Before the spin–orbit synchronization, the tidal dissipation is generally dominated by the semidiurnal tide with the forcing frequency of σ = |2ω − 2n| (Ferraz-Mello et al. 2008). In this case, the tidal quality factor can be written as

$$\begin{equation} Q\sim \frac{1}{ |2\omega -2n|\Delta t} . \end{equation} \tag{ 15 }$$

As the system approaches synchronization, the effect due to the semidiurnal tide diminishes, and the annual tide with σ = |2ω − n| prevails (Ferraz-Mello et al. 2008). The corresponding tidal quality factor can be written in a similar manner to the above as

$$\begin{equation} Q\sim \frac{1}{ |2\omega -n|\Delta t} . \end{equation} \tag{ 16 }$$

Note that when ω = n, the most efficient energy dissipation occurs on the same timescale as the orbital period.

In Sections 4 and 5, we rewrite Equations (7)–(11) by assuming that both stellar and planetary tidal quality factors change as Q ∝ 1/n, while in Section 6, we adopt Q_* ∝ 1/|2ω − 2n| and Q_p ∝ 1/|2ω − n|. The former is chosen to compare our results with the study of LWC09. For the latter, we are implicitly assuming that close-in exoplanets are pseudosynchronized, while their host stars are not. We will see that this is a reasonable approximation in Section 4.

Throughout this paper, we discuss our results in terms of the modified tidal quality factor Q' ≡ 1.5Q/k₂, where k₂ is the Love number for the second-order harmonic potential (Love 1944).

In this section, we revisit the tidal stability problem for transiting extrasolar planets and show that there are two distinct evolutionary paths for Darwin-unstable systems. Throughout this section, we assume that the total angular momentum is conserved (i.e., magnetic braking is neglected). We take into account the magnetic braking effects later in Section 4.3.

The existence and stability of tidal equilibrium states were investigated by many authors (e.g., Hut 1980; Peale 1986; Chandrasekhar 1987; Lai et al. 1994) for binary systems and the solar system. Minimizing the total energy under the constraint of conservation of the total angular momentum, Hut (1980) found that all equilibrium states are characterized by orbital circularity, spin–orbit alignment, and synchronization of the stellar rotation with the orbit. He also showed that equilibrium states exist only when the total angular momentum of the system L_tot is larger than some critical value L_crit, and that such equilibrium states are unstable when the orbital angular momentum is less than three times the total spin angular momentum (L_spin/L_orb>1/3). In this paper, we are interested in the ultimate fate of close-in exoplanets, and thus we call a system "Darwin stable" only when it is expected to evolve ultimately toward a stable equilibrium state. All other systems are called "Darwin unstable."

First, we check whether the stable tidal equilibrium states exist for the currently known transiting planets listed in Table 1. Tidal equilibrium can only exist when both primary and secondary, with zero obliquities, are synchronously rotating with their orbital motion. Under these constraints, the total angular momentum

$$\begin{equation*} L_{\rm tot}=L_{\rm orb}+C_*\omega _*+C_{{{\rm p}}}\omega _{{{\rm p}}} =L_{\rm orb}+(C_*+C_{{{\rm p}}})n \end{equation*}$$

has a minimum as a function of the semimajor axis a when L_orb = 3L_spin and

$$\begin{equation*} L_{\rm tot}=L_{\rm crit}=4\left[\frac{G^2}{27}\frac{M_*^3M_{{{\rm p}}}^3}{M_*+M_{{{\rm p}}}}(C_* +C_{{{\rm p}}})\right]^{\frac{1}{4}}. \end{equation*}$$

Here, C = αMR² is the moment of inertia and $L_{\rm orb}=\frac{M_*M_{{{\rm p}}}}{\sqrt{M_*+M_{{{\rm p}}}}}\sqrt{Ga(1-e^2)}$ is the orbital angular momentum. For L_tot < L_crit, there can be no tidal equilibrium; for L_tot>L_crit, two equilibrium states exist. The inner state (L_orb < 3L_spin) is unstable, so the only stable tidal equilibrium is the outer state, which requires L_orb>3L_spin (e.g., Hut 1980; Peale 1986; also see the top panel of Figure 3). A local example of dual synchronous rotation is the Pluto–Charon system (e.g., Peale 1986).

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The results are summarized in Table 2. As already pointed out by LWC09, most systems have no tidal equilibrium states (i.e., L_tot/L_crit < 1), and thus are Darwin unstable (i.e., the planet will eventually fall all the way to the Roche limit of the central star, even when the total angular momentum is strictly conserved). Note that, using the refined parameters in Pál et al. (2010), even HAT-P-2, which was the only system with tidal equilibria in LWC09, in fact now appears to have no such states (L_tot/L_crit = 0.995).

There are several systems which have tidal equilibria (L_tot/L_crit ≳ 1), and thus could be Darwin stable, namely, CoRoT-3, CoRoT-6, HAT-P-9, HD 80606, Kepler-8, WASP-7, and WASP-17. Among these, CoRoT-3, CoRoT-6, and HD 80606 are clearly evolving toward the stable equilibrium state, since they all have L_*,spin/L_orb < 1/3. Therefore, these systems are Darwin stable. Out of the systems with L_*,spin/L_orb>1/3, WASP-17 is Darwin unstable, independent of the 1/3 criterion, since it is in a retrograde orbit (L_orb < 0) and thus has L_spin>L_orb. For the other systems (HAT-P-9, Kepler-8, and WASP-7), we checked their tidal equilibrium states as in Figure 3. Here, we compare the total angular momentum and total energy for the dual synchronous state with the current values. As clearly seen in the figure, HAT-P-9 is Darwin unstable, since the system is inside the inner unstable equilibrium state. Most of the angular momentum in the system is in the spin, and the planet falls toward the central star as a result of tidal energy dissipation. Similarly, Kepler-8 is Darwin unstable since the system is far inside the inner unstable equilibrium state. On the other hand, WASP-7 exists just outside the inner unstable equilibrium state, and thus is migrating outward, toward the stable tidal equilibrium state. Similar plots for CoRoT-3, CoRoT-6, and HD 80606 reveal that CoRoT-3 and CoRoT-6 are migrating outward toward the stable equilibria, while HD 80606 is migrating inward toward the stable state. For systems with non-zero eccentricities (CoRoT-3, HD 80606, and WASP-17), we show the evolution explicitly in Figure 6. In short, we find that all planetary systems in Table 1 except CoRoT-3, CoRoT-6, HD 80606, and WASP-7 are Darwin unstable.

The rest of the transiting systems have no tidal equilibrium (L_tot/L_crit < 1) with the fiducial orbital and spin parameters listed in Table 1, and thus should be Darwin unstable. However, some of them are borderline systems with L_tot/L_crit ≃ 1, and they could be either Darwin stable or unstable within the observational uncertainties or depending on the value of the unknown parameters (e.g., stellar obliquity). As an example, we show the tidal evolution of a hypothetical planet with M_p = 3M_J and R_p = 1.2R_J orbiting a Sun-like star. We assume the initial semimajor axis, eccentricity, and stellar velocity of a = 0.06 AU, e = 0.3, and v_* = 10 km s⁻¹. Such a system would have a stable tidal equilibrium if the stellar obliquity is small, because L_tot/L_crit ∼ 1.040 and L_*,spin/L_orb ∼ 0.193 < 1/3. In Figure 4, we show the results of tidal evolution for two different stellar obliquities of _* = 20° and 60°. Here the tidal quality factors are scaled as Q' = Q'₀n₀/n, where 0 indicates the initial/current values. This scaling ensures that our model is consistent with the constant time lag model as shown in Section 3.1. For each obliquity, we performed three different runs with the same Q'_*,0 = 10⁶ and different Q'_p,0 of 10⁵, 10⁶, and 10⁷. For the smaller stellar obliquity (_* = 20°), we find that the system arrives at the stable tidal equilibrium as circularization, synchronization, and alignment are achieved. On the other hand, for the larger stellar obliquity (_* = 60°), the system turns out to be Darwin unstable, and the planet spirals into the star on a ∼10 Gyr timescale. The examples of these borderline systems include HAT-P-2, WASP-10, and XO-3. XO-3 was a Darwin-stable system with previously obtained observed parameters, while HAT-P-2 and WASP-10 can be Darwin stable within the uncertainties as we see in Figure 6.

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The tidal evolution is not completely simple even for clearly Darwin-unstable systems. The bottom panels of Figure 4 demonstrate that such systems can take either of two different evolutionary paths, depending on the relative efficiency of tidal dissipation inside the star and the planet. With the smaller planetary tidal quality factors and thus with a more efficient tidal dissipation in the planet (Q'_p,0 = 10⁵ and 10⁶ in the figure, which correspond to the dotted and solid curves, respectively), the planetary orbit circularizes before the planet spirals into the Roche limit (τ_e < τ_a), while with the larger planetary tidal quality factors (Q'_p,0 = 10⁷ in the figure, corresponding to the dashed curves), the circularization time becomes comparable to the orbital decay time (τ_e ∼ τ_a). The difference occurs because the orbital decay time is largely determined by the dissipation inside the star, while the circularization time can be determined by either the dissipation inside the star, or that inside the planet. This is apparent from the figure—the orbital decay times are similar for different Q'_p,0 values, which means that the semimajor axis evolution is largely independent of the tidal dissipation in the planet and instead is determined entirely by the dissipation in the star (τ_a ∼ τ_a,*). On the other hand, the eccentricity damping timescales change significantly for different Q'_p,0 values, which suggests that the dissipation in the planet plays a significant role in circularization. However, the circularization time is never longer than the orbital decay time. Therefore, when the expected circularization time from Q'_p is longer than the orbital decay time, the eccentricity damping time is also determined by the stellar tidal dissipation (τ_e ∼ τ_e,*, which corresponds to Q'_p,0 = 10⁷ case in the figure). We show that these results are generally true in Section 4.3.

Unfortunately, it is nontrivial to constrain tidal quality factors and determine which type of evolution each system would follow. This is because tidal quality factors depend sensitively on the detailed interior structure of the body (either star or planet), as well as the tidal forcing frequency and amplitude, and are unlikely to be expressed as a simple constant value (e.g., Ogilvie & Lin 2004, 2007; Wu 2005). However, since we observe a sharp eccentricity decline within a ≲ 0.1 AU, and many Darwin-unstable extrasolar planets are observed to be on nearly circular orbits, it is likely that the eccentricity damping time tends to be short compared to the orbital decay time (τ_e < τ_a). We discuss this issue further in Section 5.2.

In summary, we find that there are two evolutionary paths for Darwin-unstable planets—the "stellar-dissipation-dominated" case in which all the parameters except planetary spin evolve on a similar timescale ($\tau _{e}\sim \tau _{a}\sim \tau _{\epsilon _{*}}\sim \tau _{\omega _{*}}$), and the "planetary-dissipation-dominated" case in which circularization occurs before any significant orbital decay ($\tau _{e}<\tau _{a}\sim \tau _{\epsilon _{*}}\sim \tau _{\omega _{*}}$). In the former case, not only the evolution of stellar spin and obliquity, but also that of the semimajor axis and eccentricity, are controlled by the efficiency of tidal dissipation in the star. In the latter case, eccentricity damping is driven by the dissipation in the planet, while orbital decay is largely determined by the dissipation in the star.

Note that, although the stellar-dissipation-driven case (τ_e ∼ τ_a) looks similar to the one illustrated by LWC09, there are fundamental differences. In Figure 2 of LWC09, the exponential eccentricity damping approximation, in which they only integrate the planetary dissipation term in the eccentricity evolution, shows a faster damping time than the tidal evolution involving both stellar and planetary energy dissipation. Generally, however, this approximation should provide a timescale comparable to, or longer than, what is obtained with the full integration. This is because most planet-hosting stars are spinning slower than the orbital frequency (see Table 2), and thus energy dissipation in the star accelerates the eccentricity damping, rather than slowing it down (see also Dobbs-Dixon et al. 2004). Therefore, for most systems, the exponential damping approximation involving only the dissipation in the planet should provide an upper limit for eccentricity damping.

We demonstrate this in Figure 5 by taking HD 209458 (the same system considered in LWC09) as an example. Note however that, with current data, HD 209458 has an orbital eccentricity consistent with zero, and thus is not included in the analysis presented in the rest of this paper. We adopt the same initial conditions and assumptions and use the same set of tidal equations as LWC09. Here, the tidal quality factors are initially Q'_p,0 = Q'_*,0 = 10⁶ and scale as Q' = Q'₀n₀/n. Our result is shown in the top panel of Figure 5. We find that the tidal evolution in Figure 2 of LWC09 is recovered except for the eccentricity. In our results, the eccentricity evolves on a timescale consistent with the exponential eccentricity damping approximation (dashed line, which is completely overlapped with solid line in the top left panel). In other words, the eccentricity damps on the timescale determined by the tidal dissipation in the planet, as previously shown by many authors (e.g., Rasio et al. 1996; Dobbs-Dixon et al. 2004; Mardling & Lin 2004). For the middle panels, we assume a less efficient tidal dissipation for the planet (Q'_p,0 = 10⁹). In this case, we obtain an eccentricity evolution similar to LWC09, but the planetary spin–orbit synchronization, as expected, occurs more slowly. Note that the dashed curve is the eccentricity damping approximation with Q'_p,0 = 10⁶. Finally, for the bottom panels, we reproduce Figure 2 of LWC09 by artificially reducing the planetary contribution term in the eccentricity evolution equation (i.e., the second term in Equation (8)), by a factor of 10³, i.e., completely suppressing the eccentricity damping effect due to the planet.⁵ As already mentioned, this discrepancy occurs because the numerical results in LWC09 indeed underestimated the eccentricity damping in the planet, since their code incorrectly had an additional factor of n multiplied in the second term of Equation (8) (B. Levrard 2009, private communication).

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In Section 4.1, we identify two characteristic evolutionary paths for Darwin-unstable planets. We now study the tidal evolution of eccentric transiting planets by integrating the tidal equations forward in time with various tidal quality factors, and further investigate the implications of these two paths.

We use the currently observed parameters as initial conditions (see Table 1). The orbital and stellar spin parameters are taken from The Extrasolar Planets Encyclopedia (http://exoplanet.eu/) and references therein. For the planetary spin period, we assume a small initial value (0.2 days), but the overall results are not affected by the exact choice of this value. This is because the planetary spin carries a very small angular momentum, and thus pseudosynchronization with the orbit is achieved very quickly (see also LWC09). For the stellar obliquity, we use the observed projected value λ when RM measurements are available. Here, we implicitly assume that both the angle between the stellar spin axis and the line of sight (i_*) and the angle between the orbital angular momentum and the line of sight (i₀) are ≃90°. Thus, from cos _* = sin i_*cos λsin i₀ + cos i_*cos i₀ (Fabrycky & Winn 2009), the stellar obliquity becomes comparable to the projected one (_* ≃ λ). For systems without RM measurements, we assume an initial stellar obliquity _*,0 = 2°. This choice is rather arbitrary, but is motivated by the typical projected obliquity observed (see Table 2).

First, we show the tidal evolution for typical tidal quality factors (Q'_*,0 = Q'_p,0 = 10⁶), with the scaling of Q' = Q'₀n₀/n, and without magnetic braking effect. Our goal here is to show how different the evolution can be within observational uncertainties. We exclude WASP-12 and HAT-P-1 from our analysis because of the uncertainty in their age. The evolution of the semimajor axis and eccentricity of each system is shown in Figure 6. In each panel, we show the results of five different integrations. The solid curves represent the results with fiducial values of a and e, while the dotted curves correspond to four different combinations of maximum and minimum a and e values within their error bars. As expected from Section 4.1, the Darwin-stable systems CoRoT-3 and HD 80606 migrate outward and inward, respectively, and arrive at their tidal equilibrium, with orbital decay eventually stopping. The borderline systems HAT-P-2 and WASP-10 are likely Darwin unstable, but may arrive at a stable tidal equilibrium within the observed uncertainties. Thus, it is important to know the orbital and spin parameters as well as possible to determine the final fate of these borderline systems. The other systems are definitely Darwin unstable within the current observational accuracy, and their planets spiral toward the central star on different timescales. The vertical lines indicate the estimated ages with uncertainties. With Q'_*,0 = Q'_p,0 = 10⁶ some systems undergo orbital decay too quickly to be compatible with their likely age (e.g., WASP-18, XO-3). Therefore, these results clearly imply that a single set of values for the tidal quality factors cannot reasonably apply to all systems (also see, for example, Matsumura et al. 2008).

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We repeated the above calculations for various initial stellar and planetary tidal quality factors ranging over the interval 10⁴ ⩽ Q'₀ ⩽ 10⁹, with the scaling of Q' = Q'₀n₀/n. In Figure 7, we show the range of values for the tidal quality factors that allow planets to survive (a ≳ R_*) and stay eccentric (e ≳ 0.001) for 0.1, 1, and 10 Gyr. Here, magnetic braking is not included. For Darwin-stable systems (CoRoT-3 and HD 80606 in our list), the minimum stellar and planetary tidal quality factors correspond to the orbital circularization times. For all Darwin-unstable transiting systems with non-circular orbits, we find the same trend as in Section 4.1: the circularization time is largely determined by the dissipation in the planet, while the survival time is largely determined by the dissipation inside the star. In other words, for Darwin-unstable planets, the minimum planetary tidal quality factors can be inferred from the circularization time, while the minimum stellar tidal quality factors can be inferred from the orbital decay time. We demonstrate this below.

The approximate minimum planetary tidal quality factor that allows a planet to keep a non-circular orbit (e ≳ 0.001) for a certain time (in our examples, 0.1–10 Gyr) can be determined by assuming that the eccentricity evolution depends only on the tidal dissipation inside the planet (de/dt) ≃ (de/dt)_p. We rewrite Equation (8) as follows by assuming pseudosynchronization of the planetary spin (dω_p/dt = 0), as well as conservation of angular momentum (a(1 − e²) = const):

$$\begin{eqnarray} \frac{{\it de}}{{\it dt}} &\sim& \frac{81}{2}\frac{n_0}{Q^{\prime }_{{\rm p},0}}\frac{M_{*}}{M_{{\rm p}}}\frac{R_{{\rm p}}^{5}}{a_0^{5}}\big(1-e_0^2\big)^{-8} e(1-e^{2})^{3/2}\nonumber\\ &&\times\left[\frac{11}{18}\frac{f_{2}(e^{2})f_{4}(e^{2})}{f_{5}(e^{2})}-f_{3}(e^{2})\right]. \end{eqnarray} \tag{ 17 }$$

By integrating the above equation from the currently observed eccentricity to e = 0.001, and solving for Q'_p,0, we obtain the minimum planetary tidal quality factors to keep a non-circular orbit for 0.1, 1, and 10 Gyr. These values are plotted as the vertical dashed lines in Figure 7.

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Similarly, we can determine the approximate minimum stellar tidal quality factor that allows a planet to survive for a certain time before plunging into the central star by assuming that the semimajor axis evolution depends only on the tidal dissipation inside the star (da/dt) ≃ (da/dt)_*. Note that this is a reasonable approximation when the pseudosynchronization of the planetary spin is achieved, and the orbit is nearly circular. By setting e = 0, we rewrite Equation (7) as follows:

$$\begin{equation} \frac{{\it da}}{{\it dt}} \sim \frac{9}{Q^{\prime }_{*,0}}\frac{M_{{\rm p}}}{M_{*}}\frac{R_{*}^{5}}{a^{4}}\left(\frac{a_0}{a}\right)^{3/2} [\omega _{*,0}\cos \epsilon _{*,0}-n]. \end{equation} \tag{ 18 }$$

We integrate the above equation from a = a₀(1 − e²₀) to R_* and solve for Q'_*,0 to obtain the horizontal dashed lines. Here, we make use of the fact that the difference in eccentricity damping times does not strongly affect the orbital decay time and assume that the orbital decay time of any eccentric Darwin-unstable system can be well described by that of a system with equal angular momentum and a circular orbit.

As seen in Figure 7, the agreement of both the horizontal and vertical dashed lines with the calculations for Darwin-unstable systems is very good. Since Darwin-stable systems (CoRoT-3 and HD 80606) arrive at the stable tidal equilibria and stop migrating, the horizontal dashed lines for these systems significantly differ from the calculated results. Now we present some example runs along the horizontal and vertical lines for WASP-17 to better understand their implications. The left panel of Figure 8 shows three runs along the lowermost horizontal line that corresponds to the survival time of 0.1 Gyr. The dotted, solid, and dashed curves show the evolutions with the same initial stellar tidal quality factor Q'_*,0 = 9.13 × 10⁴, and different initial planetary tidal quality factors Q'_p,0 = 7.43 × 10⁴, 7.43 × 10⁵, and 7.43 × 10⁶, respectively. The figure confirms that the orbital decay time is largely determined by the tidal dissipation in the star, since there is no obvious difference depending on Q'_p,0 values. At the vertex of the vertical and horizontal dashed lines in Figure 7 (i.e., (Q'_*,0,  Q'_p,0) = (9.13 × 10⁴,  7.43 × 10⁵)), we find that the semimajor axis and eccentricity damp roughly on the similar timescale (τ_e ∼ τ_a ∼ 0.1 Gyr). For the smaller Q'_p,0, the eccentricity damps much faster than the orbital decay (τ_e < τ_a ∼ 0.1 Gyr), while for the larger Q'_p,0, the eccentricity damps slower than expected from the other two curves and on a similar timescale to the orbital decay (τ_e ∼ τ_a ∼ 0.1 Gyr). Similarly, the right panel of Figure 8 shows three runs with the same initial planetary tidal quality factor Q'_p,0 = 7.43 × 10⁵ and different initial stellar tidal quality factors Q'_*,0 = 9.13 × 10³, 9.13 × 10⁴, and 9.13 × 10⁵. When Q'_*,0 is smaller than the vertex value, the orbit decays much faster than 0.1 Gyr, and the circularization occurs on the similar timescale (τ_e ∼ τ_a < 0.1 Gyr). On the other hand, when Q'_*,0 is larger than the vertex value, the orbit decays much slower than 0.1 Gyr, and the circularization time is shorter than the orbital decay time (τ_e ∼ 0.1 Gyr < τ_a). The figure also implies that the orbital circularization time is largely determined by the dissipation in the planet unless τ_e ∼ τ_a. Thus, we find that τ_e ∼ τ_a is a good approximation along the horizontal dashed lines, while τ_e < τ_a is a good approximation along the vertical lines. In other words, the region below the diagonal line drawn by connecting the vertices of the vertical and horizontal lines is the stellar-dissipation-dominated region, while the region above the line is the planetary-dissipation-dominated region.

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We repeat these integrations by including magnetic braking effects. As can be seen in Figure 9, there is little difference between the cases with and without magnetic braking. Note that the dashed lines are the same as the ones in Figure 7, and thus do not take account of the magnetic braking effects. Barker & Ogilvie (2009) suggested that the effect of magnetic braking can be important for systems with rapidly spinning stars (ω_*cos /n ≫ 1). From Table 2, there are five such systems (CoRoT-3, CoRoT-6, HAT-P-2, HD 80606, and WASP-7). Among them, CoRoT-3, CoRoT-6, HD 80606, and WASP-7 are Darwin-stable systems, while HAT-P-2 is a borderline case that can be either Darwin stable or unstable within observational uncertainties. Out of these systems with fast spinning stars, CoRoT-3, HAT-P-2, and HD 80606 have eccentric planets and are shown in Figure 9. Excluding Darwin-stable cases (CoRoT-3 and HD 80606), HAT-P-2 indeed shows a significant difference between Figures 7 and 9 for the 1 and 10 Gyr cases.

In Section 1, we mentioned the two main scenarios to form close-in planets—planet migration in a disk and tidal circularization of an eccentric orbit, which may be obtained as a result of planet–planet scattering or Kozai-type perturbations. It is nontrivial to determine which formation mechanism dominates, but there are at least a few observational indications that support the second scenario.

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One of them relates to the orbital distribution of planetary systems. Wright et al. (2009) compared the properties of multiple-planet systems with single-planet systems (i.e., with no obvious additional giant planet) and showed that their semimajor axis distributions are different. While single-planet systems have a double-peaked distribution, which is characterized by a pileup of giant planets between 0.03 and 0.07 AU (the so-called three-day pileup) as well as a jump in the number of planets beyond 1 AU, multiple-planet systems have a much more uniform distribution. They also pointed out that the occurrence of close-in (a < 0.07 AU) planets is lower for multiple-planet systems and that planets beyond 0.1 AU in multiple-planet systems exhibit somewhat smaller eccentricities compared to the corresponding single ones. If confirmed by future observations, this trend would favor planet–planet interaction scenarios over a migration one, because there is no obvious reason why the orbital distributions of single- and multiple-planet systems should be different for planet migration. On the other hand, gravitational interactions combined with tidal circularization may be able to explain such a difference, because strong gravitational interactions tend to disrupt the system, and thus currently observed multiple-planet systems are unlikely to have been strongly perturbed by stellar/planetary companions, and/or to have undergone catastrophic scattering events.

Another indication comes from the similarity between the stellar obliquity distribution derived from the observed systems and the distribution predicted by the Kozai migration scenario (Fabrycky & Tremaine 2007; Wu et al. 2007). Triaud et al. (2010) observed the RM effect for six transiting hot Jupiters and found that four of their targets appear to be in retrograde orbits with a projected stellar obliquity >90°. Combining the previous 20 systems with such measurements, they pointed out that 8 out of 26 systems are clearly misaligned and that 5 out of 8 misaligned systems exhibit retrograde orbits. They also derived the stellar obliquity distribution by assuming an isotropic distribution of the stellar spin with respect to the line of sight and found that the distribution closely matches that expected from the Kozai migration scenario (Fabrycky & Tremaine 2007; Wu et al. 2007). Fabrycky & Tremaine (2007) and Wu et al. (2007) independently studied the possibility of forming a close-in planet by considering the combined effects of secular perturbations due to a highly inclined distant companion star (i.e., Kozai-type perturbations) and tidal interactions with the central star. In this scenario, a Jupiter-mass planet which was initially on a nearly circular orbit at ∼5 AU can become a hot Jupiter. The mechanism involves a companion star on a highly inclined orbit (≃90°), which perturbs the planetary orbit and increases its eccentricity. Once the pericenter distance of the planet becomes small enough for tidal interactions with the central star to be important, energy dissipation leads to the circularization of the planetary orbit and eventually to the formation of a hot Jupiter with a small, or nearly zero, eccentricity. They found that hot Jupiters formed this way tend to be in misaligned orbits and frequently in retrograde orbits. Planet–planet interactions around a single star (without a binary companion) could also form hot Jupiters via Kozai migration (Nagasawa et al. 2008).

Yet another clue regarding the origin of close-in planets is related to the above scenario and comes from the inner edge of the orbital distribution for hot Jupiters. Ford & Rasio (2006) proposed that the observed inner cutoff for hot Jupiters is defined not by an orbital period, but by a tidal limit, and studied such a cutoff of the distribution of close-in giant planets in the mass–period diagram by performing a Bayesian analysis. Assuming that the cutoff slope in such a diagram follows the Roche limit, they found that the observations suggest an inner cutoff close to twice the Roche limit. This can be explained naturally if the planetary orbits were initially highly eccentric and later circularized via tidal dissipation while conserving orbital angular momentum. They suggested that this result is inconsistent with the migration scenario, because migration would lead to an inner edge right at the Roche limit (a factor of 2 closer to the star than what is observed).

In Figure 10, we extend the work of Ford & Rasio (2006) by including all more recently discovered planets and plot planetary and stellar mass ratio against the semimajor axis in terms of the Roche-limit separation (Paczyński 1971):

$$\begin{equation} a_{\rm R}=\frac{3}{2}R_{{{\rm p}}}\left(\frac{M_{{{\rm p}}}}{3(M_*+M_{{{\rm p}}})}\right)^{-1/3} \sim \frac{R_{{\rm p}}}{0.462}\left(\frac{M_{{\rm p}}}{M_{*}}\right)^{-1/3} \. \end{equation} \tag{ 19 }$$

Here, we assume that the Roche radius of the planet, which is defined so that its spherical volume is equal to the volume within the Roche lobe, is equal to the planetary radius. Thus, the Roche-limit separation used here is the lower limit. For non-transiting planets without a measured planetary radius (orange circles), we assume a Jupiter radius for planets with mass larger than 0.01 M_J, and a Neptune radius for less massive planets. It is obvious that most planets still exist beyond twice the Roche limit (the vertical dashed line). However, there are now five planets which lie within this limit (OGLE-TR-56, CoRoT-1, WASP-4, WASP-19, and WASP-12) with a/a_R ∼ 1.70, 1.67, 1.66, 1.30, and 1.24, respectively. We need to assess whether these planets will change the claim by Ford & Rasio (2006) and support the migration scenario over the scattering/Kozai-cycle scenario. Note that the two recently discovered "extreme" close-in planets, with orbital periods less than 1 day, CoRoT-7 b and WASP-18 b, have a/a_R ∼ 2.76 and 3.52, respectively.

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The existence of at least some of these planets inside 2a_R may still be explained as a result of the tidal circularization of an eccentric orbit. One of the possibilities is that the orbits of these planets were originally circularized beyond twice the Roche limit, but the planets have migrated inward after circularization due to the tidal dissipation in the star. All of the planets within 2a_R can be potentially explained this way. Another possibility is that their orbits used to have a pericenter distance close to the Roche limit, but the initial eccentricity was smaller, e ≃ 0.7. In such a case, the expected period of the circularized orbit would be comparable to the current orbital period for OGLE-TR-56, CoRoT-1, and WASP-4, since they all have a/a_R ∼ 1.7. However, systems with smaller ratios of the semimajor axis to Roche-limit separation (WASP-12 and WASP-19) are unlikely to be explained this way. This is because their current semimajor axes (a/a_R ≃ 1.24 and 1.30) would demand small initial eccentricities (e_i ∼ 0.24 and 0.30), and thus small initial semimajor axes (a₀ ≃ 0.024 AU and 0.018 AU, respectively). This means that they have to be either born at such close-in locations, or scattered into such an orbit, which would be very difficult. Yet another possibility is that these planets used to have a smaller Roche-limit separation, due to a larger mass or a smaller radius. The orbits of these planets might have been circularized as the planet lost mass (e.g., Lammer et al. 2003; Lecavelier Des Etangs 2007), or the planetary radius expanded due to tidal heating (e.g., Bodenheimer et al. 2001; Gu et al. 2003). For OGLE-TR-56, CoRoT-1, WASP-4, WASP-19, and WASP-12 to be circularized at twice the Roche limit, either the past planetary masses must have been 2.13, 1.78, 1.53, 4.19, and 5.98 M_J, respectively, or the past planetary radii must have been 1.18, 1.24, 1.21, 0.829, and 1.11 R_J, respectively. Since the mass-loss rate can only be at most ∼10%, even for a low-density planet like WASP-12 (Lammer et al. 2009), it is unlikely that a larger mass in the past could be the correct explanation. On the other hand, tidal inflation of the planetary radius is a transient phenomenon (e.g., Ibgui & Burrows 2009). To explain the current orbital radius by invoking a smaller planetary radius in the past, we have to catch the planet just as its radius is inflating. Such a scenario may be possible, but appears unlikely. Interestingly, CoRoT-1 is observed to be strongly misaligned with λ = −77 ± 11 deg (Pont et al. 2010), which suggests a violent origin, while WASP-4 has a stellar obliquity of 4⁺³⁴₋₄₃ deg (Triaud et al. 2010), which is consistent with alignment. We urge observers to determine the projected stellar obliquity for OGLE-TR-56, WASP-19, and WASP-12. Alignment does not necessarily support the planet migration scenario over the violent origin, but misalignment would clearly imply a significant past dynamical interaction involving scattering or Kozai cycles.

As we pointed out in the previous section, planet–planet (or planet–companion star) interactions may well be responsible for the majority of close-in planets. Now we further explore this possibility by performing integrations of the tidal evolution equations backward in time. Particular focus is on the differences in evolution of Darwin-unstable planets depending on two evolution paths.

Jackson et al. (2008) first performed such a study and estimated the "initial" eccentricity distribution of close-in planets. They found that this distribution agrees well with the observed eccentricity distribution of exoplanets on wider orbits, and they proposed that most currently observed close-in planets had considerably wider and more eccentric orbits in the past. However, their study had many uncertainties. First, the possible effects of other planets and binary companions were neglected. As we pointed out in Section 2, these effects are most likely unimportant at present, but they could well have been playing a crucial role in the past, especially if another object was responsible for Kozai-type perturbations, or gravitational scattering. Second, the evolution of stars and planets is neglected. For example, both planetary and stellar radii may have been significantly different in the past. Moreover, many stars could have lost a large amount of their spin angular momentum via magnetic braking. Finally, such backward dynamical integrations of evolution equations with energy dissipation are known to be diverging, and thus small changes in initial conditions can lead to much larger changes in the calculated "initial" values.

With these caveats in mind, we now re-examine the possible past histories of transiting planets. Since Equations (7)–(11) are time-invariant, we can study the past evolution by integrating the differential equations "backward in time" (i.e., by taking the negative of these differential equations and integrating them from t = 0 to τ_age). For simplicity, we assume that the planetary spin is pseudosynchronized with the orbit at all times (i.e., dω_p/dt = 0). First, we show typical results for WASP-18 in Figure 11. Here, the initial tidal quality factors are Q'_*,0 = Q'_p,0 = 10⁶ with the scaling of Q' = Q'₀n₀/n, and the magnetic braking effect is neglected. As in Figure 6, solid curves show the results of backward evolution with the fiducial orbital parameters, while four dotted curves correspond to different combinations of maximum and minimum semimajor axis and eccentricity within uncertainties. The vertical dashed and dotted lines correspond to the estimated stellar age with uncertainties. As expected, both semimajor axis and eccentricity values differ significantly within uncertainties at the zero age (the vertical dashed line), or at any specific time within the age uncertainties (between the vertical dotted lines). However, we find that the fiducial case still provides a representative evolutionary path within the stellar age uncertainties. Figure 12 presents similar backward evolutions for all the systems shown in Figure 6. It is clear that, except for the Darwin-stable CoRoT-3, WASP-18 has the largest spread in orbital parameters within uncertainties. The backward evolution of the other systems is largely independent of the exact values of the orbital parameters.

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Now we repeat backward integrations of the tidal equations by adopting various combinations of initial stellar and planetary tidal quality factors ranging over 10⁴ ⩽ Q'₀ ⩽ 10⁹. This allows us to study the "initial" orbital properties at the zero stellar age (τ_age = 0) within the estimated age uncertainties. As in Section 4.3, we also assumed that each system initially has the currently observed orbital and rotational parameters.

We select systems that have solutions such that e < 1 somewhere in the interval τ_age,min ⩽ t ⩽ τ_age,max and plot the corresponding Q'_*,0 and Q'_p,0 values in Figure 13. Again, the tidal quality factors change as Q' = Q'₀n₀/n, and magnetic braking is neglected. The blue, green, orange, and red regions represent the maximum reachable "zero-age" semimajor axis for each system being 0.1, 1, 10 AU, and ⩾10 AU, respectively. The vertical and horizontal dashed lines are the same as in Figure 7 and indicate the minimum Q'_p,0 and Q'_*,0 required to have circularization and orbital decay times (i.e., the future survival time) of 0.1, 1, and 10 Gyr. As expected, when the tidal dissipation is inefficient in both the planet and the star (top right corner area of the figure), the planets are generally expected to have stayed where they are now for a long period of time. The lack of a significant change in its orbit, especially eccentricity, may be consistent with the initial orbital properties expected from the planet migration scenario. For the future survival time much longer than 10 Gyr, only such a solution with little migration is allowed in the parameter space. On the other hand, when tidal dissipation is highly efficient (bottom left corner), a planet could have started on a wide, eccentric orbit, which was then circularized over the lifetime of the system. The property of this area is in good agreement with the expectation from Kozai cycles and/or planet–planet interactions followed by tidal dissipation. The comparison of these regions with the dashed lines implies that most planetary systems have a large region of parameter space allowed for (Q'_*,0,  Q'_p,0) which is consistent with having started on a wide, eccentric orbit, and with comfortable future survival times ∼1–10 Gyr. Thus, our results agree with the suggestion first made by Jackson et al. (2008) and show that there is a broad parameter space which supports the tidal circularization scenario as the dominant origin of close-in planets.

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When we demand that the future survival time must be comparable to or slightly longer than the estimated stellar age, Figure 13 implies an interesting trend. In the stellar-dissipation-dominated region, planets tend to have an initial orbit which is similar to the current one, while in the planetary-dissipation-dominated region, planets tend to have an initially highly eccentric and wide orbit. This is clear by comparing these two regions (below and above the diagonal line that can be drawn by connecting the vertices of the vertical and horizontal dashed lines) at around their survival times. Most systems in the figure except WASP-10, WASP-14, WASP-18, and WASP-6 have the estimated age of ∼1–10 Gyr, while WASP-10, WASP-14, and WASP-18 have ∼0.1–1 Gyr, and WASP-6 has >10 Gyr. Investigating the corresponding regions, we find that the maximum zero-age semimajor axis can be very large (up to >10 AU) in the planetary-dissipation-dominated region. On the other hand, such a solution is less likely in the stellar-dissipation-dominated region, and the area with a little change in orbital radii tends to occupy the largest region of parameter space.

Figure 14 presents similar results with magnetic braking effects included. As we can see, some systems are much less affected by magnetic braking than others. Our results show that, when either the stellar spin is slow, or the mass ratio is very low, the evolution is less affected by magnetic braking. Clearly affected systems include CoRoT-3, HAT-P-2, HD 189733, WASP-10, WASP-14, WASP-18, WASP-5, and XO-3, for which the average stellar spin period is ≃8 days and the average mass ratio is ≃0.006, while the corresponding values are about 36 days and 0.001 for the others.

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Figures 15 and 16 show similar plots to Figures 13 and 14 for the stellar obliquity. The blue, green, orange, and red areas correspond to a maximum possible zero-age obliquity of 5°, 20°, 40°, or ⩾40°. Again, also plotted are the same vertical and horizontal lines as in Figure 7.

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In Figures 15 and 16, CoRoT-3, HAT-P-2, HD 17156, HD 80606, WASP-14, WASP-17, WASP-18, WASP-5, WASP-6, and XO-3 have RM measurements, and thus known projected stellar obliquities. Among them, CoRoT-3, HD 80606, WASP-14, WASP-17, and XO-3 have a significant misalignment (>20°). Naturally, the maximum possible misalignment at zero age can be very large (⩾40°) for these systems with almost any values of (Q'_*,  Q'_p). On the other hand, for systems with small measured projected obliquities (HAT-P-2, HD 17156, WASP-18, WASP-5, and WASP-6), or systems with no RM measurements (CoRoT-5, GJ 436, HAT-P-11, HAT-P-13, HD 189733, and WASP-10), we find either a little or large change in obliquity over the stellar age.

Consider a few specific cases. For HAT-P-2 and HAT-P-11, there are no solutions for obliquity larger than 5°. This indicates that, for HAT-P-2, the "zero-age" obliquity was likely similar to the current nominal value λ = 12. However, since HAT-P-2's measured obliquity has a very large uncertainty (λ = 12 ± 134), it is possible that the actual present value is much larger. Note that even if the stellar obliquity turns out to be small, HAT-P-2 is still likely to have the scattering/Kozai-cycle origin. This is partly because of its high eccentricity (e ∼ 0.52) and partly because of a large range of tidal quality factors that allow a much wider orbit in the past (see Figures 13 and 14). For HAT-P-11, there are no RM measurements thus far, but the system shows a similar result to HAT-P-2 with _*,0 = 2°. If the current obliquity turns out to be ≲2°, our plot indicates that HAT-P-11 would not have had a large obliquity in the past. This in turn indicates that, if HAT-P-11 initially had a large stellar obliquity, we should be able to see a clearly misaligned orbit through future observations, independent of the actual tidal quality factors for the system. The orbital eccentricity of the planet is e = 0.198 ± 0.046, which could have been produced either via planet–disk or planet–companion interactions. Interestingly, Figures 13 and 14 show that the orbital radius of HAT-P-11 b is unlikely to have been changed significantly via tidal evolution, unless the tidal dissipation inside the planet is rather efficient (Q'_p < 10⁶). If the tidal dissipation is inefficient, our result suggests that HAT-P-11 b is likely to have migrated in a disk. We have to wait for future observations to estimate whether the planet is likely formed via disk migration or tidal circularization of a highly eccentric orbit.⁶

For all the other systems, the zero-age obliquity can be as high as ⩾40° if stellar tidal quality factors are relatively small. More specifically, such solutions are allowed in the stellar-dissipation-dominated region with τ_e ≃ τ_a. The stellar obliquity generally damps on a similar timescale to the orbital decay (see, e.g., LWC09; Barker & Ogilvie 2009, as well as Section 4.1). However, this plot demonstrates that the stellar obliquity could be damped from high (≳40°) to low (∼2°) values within the current stellar age.

Note that these small stellar tidal quality factors, which allow the fast damping of stellar obliquities, also lead to relatively short survival times for planets. By comparing these zero-age high-obliquity regions with the horizontal dashed lines, we find that these tidal quality factors lead to survival times comparable at most to the current stellar age. In other words, a relatively small region of parameter space is allowed for these high zero-age obliquities. For example, when we demand that the expected survival time in forward tidal evolution must be comparable to or larger than the stellar age, we find that the stellar tidal quality factor for CoRoT-5 must be Q'_* ≳ 10⁶. This includes a small region of the red, large "zero-age" stellar obliquity area in Figure 15. However, if Q'_* ≳ 10⁷, such an area disappears. In the latter case, if the observations find a small stellar obliquity for CoRoT-5, it is unlikely that the stellar obliquity was much larger in the past. In some cases, such a region with a high zero-age obliquity and a comfortable future survival time may not even exist. For HAT-P-13, a similar comparison shows that the stellar tidal quality factor must be Q'_* ≳ 10⁷ for the survival time to be comparable to or larger than the stellar age. This corresponds to the blue, small "zero-age" stellar obliquity area in Figure 15, and thus we expect that the stellar obliquity of HAT-P-13 could not have been changed much due to tidal evolution (i.e., similar to HAT-P-2 or HAT-P-11). Interestingly, HAT-P-13 is the only system in our sample that has an additional planetary companion. Thus, the system may be an example of a close-in planet formed via migration scenario. If this is the case, we predict the future observations will find a small stellar obliquity for HAT-P-13.⁷

In short, by comparing Figure 13 with Figure 15, we find that the evolution history is largely divided into two cases, depending on the relative efficiency of energy dissipation inside the star and the planet. In the planetary-dissipation-dominated region, the planets could have had a wide, eccentric orbit in the past, and the stellar obliquity damps on a similar timescale to the orbital decay. The initial conditions implied in this region are consistent with those expected from the scattering/Kozai-cycle origin of the close-in planets. On the other hand, in the stellar-dissipation-dominated region, the system could have had a large stellar obliquity in the past, although the initial orbital radius and eccentricity are likely similar to the current values, as planet migration scenario would suggest. Further inspection of these figures shows that a unique evolution is possible in the transition region between these two, where the dissipation effects due to the planet and a star are comparable. There, the system could have had a wide, eccentric orbit, as well as a large stellar obliquity in the past. Our results also imply that if the currently observed stellar obliquity distribution is due to Kozai migration, as suggested by Triaud et al. (2010), then most exoplanetary systems have planetary-dissipation-dominated tidal interactions (i.e., τ_e < τ_a). If τ_e ≃ τ_a for most planetary systems, the current obliquity distribution should be very different from the initial one, and any memory of the dynamical history prior to the tidal dissipation should be erased.

Recently, Winn et al. (2010a) suggested that the stellar obliquity is preferentially large for hot stars with effective temperatures T_eff>6250 K and proposed that such a trend can be explained because photospheres of cool stars can realign with the orbits due to tidal dissipation in the convective zones, without affecting the orbital decay. Our results suggest that yet another possibility might be that the evolution of the systems with hot and cool stars corresponds to slow and fast obliquity damping regions, respectively. If that is the case, the stellar obliquity may stay nearly constant for the planetary systems with a hot star, because tidal dissipation is dominated by the planet (i.e., τ_e < τ_a), while the obliquity may be damped to a small value for the systems with a cool star, because tidal dissipation is either dominated by the star (i.e., τ_e ≃ τ_a), or the star and the planet have comparable dissipation. In our samples of eccentric systems, CoRoT-5, GJ 436, HAT-P-11, HAT-P-13, HD 17156, HD 189733, HD 80606, WASP-10, WASP-5, and WASP-6 have the effective temperatures less than 6250 K. For the range of tidal quality factors we use, we do not see any trend that these cool systems prefer τ_e ≃ τ_a. One possibly interesting example is HD 17156, for which a large region of parameter space is allowed for the stellar-dissipation-dominated case (τ_e ≃ τ_a).