Orbital Decay of Short-period Exoplanets via Tidal Resonance Locking

Resonances between tidal forcing frequencies and stellar oscillation mode frequencies can greatly enhance tidal dissipation rates. Specifically, the orbital energy loss rate due to a tidally forced mode with angular frequency ω_α excited by the tidal potential of a circularly orbiting planet with forcing frequency ω_f (each measured in a frame corotating with the star) is given by (Fuller 2017)

$$\begin{eqnarray}&&{\dot{E}}_{\mathrm{orb},\mathrm{tide}}=\displaystyle \frac{m{\omega }_{\alpha }{{\rm{\Omega }}}_{\mathrm{orb}}| {\gamma }_{\alpha }{\left|{M}_{* }{R}_{* }^{2}| {{ \mathcal Q }}_{\alpha }\right|}^{2}{\omega }_{{\rm{f}}}^{2}}{{\left({\omega }_{\alpha }-{\omega }_{{\rm{f}}}\right)}^{2}+{\gamma }_{\alpha }^{2}}{\left(\displaystyle \frac{{M}_{{\rm{p}}}}{{M}_{* }}\right)}^{2}{\left(\displaystyle \frac{{R}_{* }}{a}\right)}^{6},\end{eqnarray} \tag{ 3 }$$

where γ_α is the mode growth rate and m is the mode's azimuthal index (m = 2 corresponds to the strongest tidal forcing for aligned orbits). ${{ \mathcal Q }}_{\alpha }$ is a dimensionless number describing the spatial coupling between oscillations and the tidal potential defined in Fuller (2017). The denominator is smallest near resonance, when ω_α ≃ ω_f, leading to the greatest tidal dissipation and the smallest tidal migration timescale, as shown in the left panel of Figure 1.

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The thick radiative zones in main-sequence stars cause internal gravity modes (g modes) to have a dense spectrum in frequency space (see Figure 2, left panel). Because the star's internal Brunt–Väisälä frequency typically increases on a stellar evolution timescale, the g mode frequencies increase on a similar timescale, which we define as the mode evolution timescale ${t}_{\alpha }\equiv {\omega }_{\alpha }/{\dot{\omega }}_{\alpha }$. A planet at angular orbital frequency Ω_orb produces tidal forcing at the frequency ${\omega }_{{\rm{f}}}\,=m\left({{\rm{\Omega }}}_{\mathrm{orb}}-{{\rm{\Omega }}}_{{\rm{s}}}\right)$, where Ω_s is the stellar spin frequency. As the stellar oscillation mode frequencies increase, one of them will quickly encounter a resonance with the tidal forcing frequency, i.e., ω_α → ω_f (see Figure 2, right panel).

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As the planet falls into resonance, it can become "trapped" in resonance (resonantly locked) in the following manner, as shown in Figure 1. If the orbit is perturbed outward such that ω_f decreases, it falls deeper into resonance, which increases the tidal dissipation, such that the planet migrates inward and away from exact resonance. If the orbit is perturbed inward such that ω_f increases, it moves further from resonance, which decreases the tidal dissipation, allowing the increasing mode frequency to catch up with the planet and sustain the resonant lock. The planet is thus forced to "ride the mode" and evolve inward at the same pace as the mode's resonant location (i.e., a resonance lock), and the planet's orbital frequency increases as the star's oscillation mode frequency increases (see Figure 2, right panel). The mode evolution timescale ${t}_{\alpha }\equiv {\omega }_{\alpha }/{\dot{\omega }}_{\alpha }$ hence determines the tidal migration timescale t_tide, which is directly related to Q' by Equation (1).

2.1.1. Stellar Models

To make quantitative predictions, we construct solar-metallicity stellar models with the MESA stellar evolution code (Paxton et al. 2011, 2013, 2015, 2018, 2019), and we compute their non-adiabatic oscillation modes with the GYRE pulsation code (Townsend & Teitler 2013; Townsend et al. 2018; Goldstein & Townsend 2020). Example inlists are given in the supplementary materials. The models start at zero-age main sequence (ZAMS) with a spin period of 3 days, though we do not include rotational effects within the MESA model. The nontidal angular momentum loss rate is assumed to be similar to Skumanich's law and is calculated via (Skumanich 1972; Krishnamurthi et al. 1997)

$$\begin{eqnarray}&&{\dot{J}}_{* ,\mathrm{ex}}={K}_{w}{I}_{* }{{\rm{\Omega }}}_{{\rm{s}}}^{3}{\left(\displaystyle \frac{M}{{M}_{\odot }}\right)}^{-1/2}{\left(\displaystyle \frac{R}{{R}_{\odot }}\right)}^{1/2}\end{eqnarray} \tag{ 4 }$$

where K_w ≈ −6 × 10⁻¹² day is a constant fitted by the Sun's spin period and age.

In Figure 3 we plot mode evolution timescales t_α for g modes in stellar models with different masses. The example g modes we plot have periods of 1.5 days in the inertial frame (i.e., they would be resonant with a planet at P_orb = 3 days) at a stellar age of 700 Myr. We see that t_α is usually comparable to the star's main-sequence lifetime t_MS. The evolution timescale is slightly shorter near the beginning and end of the main sequence when the star's structure changes more rapidly. For the 1.2 M_⊙ model, the g mode frequencies first increase and then decrease with time due to a growing convective core, causing the value of t_α to diverge and then become negative. During that time, inward migration via resonance locking cannot occur because the resonance locations move away from the star rather than toward it.

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2.1.2. Tidal Migration Timescale and Quality Factor

As discussed above, during resonance locking a star's mode frequency remains nearly equal to the tidal forcing frequency (Fuller et al. 2016): ¹

$$\begin{eqnarray}&&{\omega }_{\alpha }\simeq {\omega }_{{\rm{f}}}=m\left({{\rm{\Omega }}}_{\mathrm{orb}}-{{\rm{\Omega }}}_{{\rm{s}}}\right)\end{eqnarray} \tag{ 5 }$$

at all times. Differentiating this equation over time leads to the locking criterion

$$\begin{eqnarray}&&{\dot{\omega }}_{\alpha }\simeq {\dot{\omega }}_{{\rm{f}}}=m\left({\dot{{\rm{\Omega }}}}_{\mathrm{orb}}-{\dot{{\rm{\Omega }}}}_{{\rm{s}}}\right).\end{eqnarray} \tag{ 6 }$$

Combining the above equations and defining the spin evolution timescale ${t}_{{\rm{s}}}\equiv {{\rm{\Omega }}}_{{\rm{s}}}/{\dot{{\rm{\Omega }}}}_{{\rm{s}}}$, we immediately arrive at the (inverse of) the tidal migration timescale

$$\begin{eqnarray}&&{t}_{\mathrm{tide}}^{-1}\equiv -\displaystyle \frac{{\dot{a}}_{\mathrm{tide}}}{a}=\displaystyle \frac{2}{3}\displaystyle \frac{{\dot{{\rm{\Omega }}}}_{\mathrm{orb}}}{{{\rm{\Omega }}}_{\mathrm{orb}}}=\displaystyle \frac{2}{3}\left({t}_{\alpha }^{-1}-\displaystyle \frac{{{\rm{\Omega }}}_{{\rm{s}}}}{{{\rm{\Omega }}}_{\mathrm{orb}}}\left({t}_{\alpha }^{-1}-{t}_{{\rm{s}}}^{-1}\right)\right).\end{eqnarray} \tag{ 7 }$$

However, when tidal migration occurs, the planet adds angular momentum to the stellar spin, which means t_s and t_tide are related. This is especially important for systems with massive planets, as shown in Lainey et al. (2020). Additionally, the system may lose angular momentum due to magnetic braking of the host star. To account for these factors, recall that the total angular momentum J_tot of the star–planet system is

$$\begin{eqnarray}&&{J}_{\mathrm{tot}}={J}_{* }+{J}_{{\rm{p}}}={I}_{* }{{\rm{\Omega }}}_{{\rm{s}}}+{M}_{{\rm{p}}}\sqrt{{{GM}}_{* }a},\end{eqnarray} \tag{ 8 }$$

where I_* is the moment of inertia of the star. Defining the system's change in total angular momentum as ${\dot{J}}_{* ,\mathrm{ex}}$, we have

$$\begin{eqnarray}&&{\dot{J}}_{* ,\mathrm{ex}}={\dot{I}}_{* }{{\rm{\Omega }}}_{{\rm{s}}}+{I}_{* }{\dot{{\rm{\Omega }}}}_{{\rm{s}}}-\displaystyle \frac{1}{2}{J}_{{\rm{p}}}{t}_{\mathrm{tide}}^{-1},\end{eqnarray} \tag{ 9 }$$

assuming constant stellar/planetary masses as appropriate in most exoplanet systems. If we define the evolution timescale for the moment of inertia ${t}_{I}={I}_{* }/{\dot{I}}_{* }$ and the external stellar spin evolution timescale ${t}_{{\rm{s}},\mathrm{ex}}={J}_{* }/{\dot{J}}_{* ,\mathrm{ex}}$, this leads us to the relation

$$\begin{eqnarray}&&{t}_{{\rm{s}}}^{-1}={t}_{{\rm{s}},\mathrm{ex}}^{-1}-{t}_{I}^{-1}+\displaystyle \frac{{J}_{{\rm{p}}}}{2{J}_{* }}{t}_{\mathrm{tide}}^{-1}.\end{eqnarray} \tag{ 10 }$$

Substituting Equation (10) into Equation (7), we get the final expression for t_tide:

$$\begin{eqnarray}\begin{array}{rcl}{t}_{\mathrm{tide}} & = & \displaystyle \frac{3}{2}\displaystyle \frac{{{\rm{\Omega }}}_{\mathrm{orb}}}{{\dot{{\rm{\Omega }}}}_{\mathrm{orb}}}\\ & = & \displaystyle \frac{3}{2}\left(1-\displaystyle \frac{{I}_{{\rm{p}}}}{3{I}_{* }}\right){\left[\displaystyle \frac{1}{{t}_{\alpha }}-\displaystyle \frac{{{\rm{\Omega }}}_{{\rm{s}}}}{{{\rm{\Omega }}}_{\mathrm{orb}}}\left(\displaystyle \frac{1}{{t}_{\alpha }}-\displaystyle \frac{1}{{t}_{{\rm{s}},\mathrm{ex}}}+\displaystyle \frac{1}{{t}_{I}}\right)\right]}^{-1},\end{array}\end{eqnarray} \tag{ 11 }$$

where I_p = M_p a² is the moment of inertia of the planet's orbit. The prefactor 1 – I_p/3I_* accounts for the angular momentum transport from the planet's orbit to the stellar spin, indicating that the tidal migration timescale becomes very short for massive planets as I_p approaches 3I_*, and resonance locking cannot occur if I_p > 3I_*. In practice, one can combine Equations (9) and (11) to get a set of coupled differential equations for Ω_s(t) and Ω_orb(t), and hence solve the full evolution of the spin and orbit numerically. For most short-period exoplanet systems, Ω_s is usually negligible compared to Ω_orb. I_p is usually negligible compared to I_*, except for high-mass or long-period planets.

In Figure 4, we plot the relevant evolution timescales for a 1 M_⊙ star with a 10 M_⊕ planet. The external spin evolution timescale t_s,ex is usually comparable to the stellar age, and t_I is always long during the main sequence. At early times when the star is rapidly rotating, the second term in brackets in Equation (11) contributes, increasing the value of t_tide. The star quickly spins down such that Ω_s ≪ Ω_orb, at which point ${t}_{\mathrm{tide}}\sim \tfrac{3}{2}{t}_{\alpha }$ until the end of the main sequence. Hence, the tidal migration timescale is primarily determined by the evolution timescale of the stellar oscillation mode in resonance with the orbit.

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The corresponding effective tidal quality factor during resonance locking is

$$\begin{eqnarray}&&\begin{array}{l}{Q}_{\mathrm{RL}}^{{\prime} }=\displaystyle \frac{9}{2}\displaystyle \frac{{M}_{{\rm{p}}}}{{M}_{* }}{\left(\displaystyle \frac{{R}_{* }}{a}\right)}^{5}\left(1-\displaystyle \frac{{I}_{{\rm{p}}}}{3{I}_{* }}\right)\\ \ \times \ {\left[\displaystyle \frac{1}{{t}_{\alpha }}-\displaystyle \frac{{{\rm{\Omega }}}_{{\rm{s}}}}{{{\rm{\Omega }}}_{\mathrm{orb}}}\left(\displaystyle \frac{1}{{t}_{\alpha }}-\displaystyle \frac{1}{{t}_{{\rm{s}},\mathrm{ex}}}+\displaystyle \frac{1}{{t}_{I}}\right)\right]}^{-1}\left({{\rm{\Omega }}}_{\mathrm{orb}}-{{\rm{\Omega }}}_{{\rm{s}}}\right).\end{array}\end{eqnarray} \tag{ 12 }$$

When Ω_s ≪ Ω_orb as appropriate at most stellar ages, this reduces to

$$\begin{eqnarray}\begin{array}{rcl}{Q}_{\mathrm{RL}}^{{\prime} } & \simeq & \displaystyle \frac{9}{2}\displaystyle \frac{{M}_{{\rm{p}}}}{{M}_{* }}{\left(\displaystyle \frac{{R}_{* }}{a}\right)}^{5}{t}_{\alpha }{{\rm{\Omega }}}_{\mathrm{orb}}\\ & \approx & \displaystyle \frac{9}{2}\displaystyle \frac{{\left(2\pi \right)}^{13/3}{M}_{{\rm{p}}}{R}_{* }^{5}}{{G}^{5/3}{M}_{* }^{8/3}}{t}_{\alpha }{P}_{\mathrm{orb}}^{-13/3}.\end{array}\end{eqnarray} \tag{ 13 }$$

By solving for the evolution of internal oscillation mode frequencies in stellar models, we can quickly compute the corresponding tidal quality factor resulting from resonance locking. Equation (13) evaluates to

$$\begin{eqnarray}&&\begin{array}{l}{Q}_{\mathrm{RL}}^{{\prime} }\simeq 2\times {10}^{6}\\ \ \times \ \left(\displaystyle \frac{{M}_{{\rm{p}}}}{{M}_{{\rm{J}}}}\right){\left(\displaystyle \frac{{M}_{* }}{{M}_{\odot }}\right)}^{-8/3}{\left(\displaystyle \frac{{R}_{* }}{{R}_{\odot }}\right)}^{5}\left(\displaystyle \frac{{t}_{\alpha }}{5\,\mathrm{Gyr}}\right){\left(\displaystyle \frac{{P}_{\mathrm{orb}}}{2\,\mathrm{days}}\right)}^{-13/3}.\end{array}\end{eqnarray} \tag{ 14 }$$

That is, ${Q}_{\mathrm{RL}}^{{\prime} }$ is proportional to the planet mass, and it has a −13/3 power-law dependence on the orbital period. This is very different from the prescription of constant Q' that is often assumed in the literature.

Figure 5 shows the value of ${Q}_{\mathrm{RL}}^{{\prime} }$ for a 1 M_⊙ star with a 10 M_⊕ planet as a function of orbital period and stellar age. ${Q}_{\mathrm{RL}}^{{\prime} }$ decreases sharply as the orbital period increases, and it increases somewhat as a function of age primarily because the stellar radius increases slightly, as we would expect from Equation (12). In contrast, the value of ${t}_{\mathrm{tide}}\simeq \tfrac{3}{2}{t}_{\alpha }$ remains nearly constant within this parameter space.

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An exception is at early ages, where the stellar spin is larger than a critical frequency such that the second term in the square bracket of Equation (11) is larger than the first, which occurs at approximately

$$\begin{eqnarray}&&{{\rm{\Omega }}}_{{\rm{s}}}={{\rm{\Omega }}}_{{\rm{s}},\mathrm{crit}}\simeq \displaystyle \frac{| {t}_{{\rm{s}},\mathrm{ex}}| }{| {t}_{{\rm{s}},\mathrm{ex}}| +{t}_{\alpha }}{{\rm{\Omega }}}_{\mathrm{orb}},\end{eqnarray} \tag{ 15 }$$

where we assumed t_s < 0 for main-sequence magnetic braking (spin-down). This would lead to a divergence of t_tide and ${Q}_{\mathrm{RL}}^{{\prime} }$. Physically, the divergence signals the boundary where tidal migration due to resonance locking no longer occurs: for a higher spin frequency (or lower orbital frequency) the resonant locations move outward rather than inward. Since Ω_s is still less than Ω_orb according to Equation (15), tidal dissipation would still push the planet inward, and the planet would evolve through the resonances rather than becoming locked in resonance. The colored hatched regions of Figure 5 indicate these regions where resonance locking cannot occur.

At even higher spin frequencies where Ω_s > Ω_orb (gray regions of Figure 5), the planet would migrate outward, in the same direction as the resonant locations, such that outward migration via resonance locking could occur. This effect could potentially drive rapid outward migration of short-period planets at very young ages, but we do not study that process in this paper.

2.2.1. Validity of Linear Theory

The whole theory of resonance locking is based on a linear analysis of dynamical tides (Fuller 2017). After the waves get excited at the radiative/convective interface inside a star, they propagate toward the center and are geometrically focused such that their amplitudes increase. We thus expect that resonance locking may not occur if the waves become sufficiently nonlinear near the star's center. Specifically, the dominant nonlinear term in the fluid momentum equation is ξ · ∇ξ ∼ ξ∣d ξ_r /dr∣ for g modes. Hence, the quantity ∣d ξ_r /dr∣ naturally serves as a measure of nonlinearity: if ∣d ξ_r /dr∣ ≳ 1, then nonlinear effects become very strong, typically causing wave breaking near the center of the star (Barker & Ogilvie 2011) such that standing g modes no longer exist, and resonance locking cannot occur. In fact, nonlinear g mode damping occurs at smaller g mode amplitudes (see Section 2.2.3), further limiting the situations in which resonance locking can operate.

When resonance locking does occur, the amplitude of oscillating modes can be calculated as follows: the energy and angular momentum dissipation rates are determined by the tidal migration rate of the planet (Equation (11)). Since energy and angular momentum are conserved, this allows us to compute the corresponding wave amplitude, given a wave damping rate (see Fuller 2017). ² The result is

$$\begin{eqnarray}&&{\left|{a}_{\alpha }\right|}_{\mathrm{RL}}=\displaystyle \frac{1}{2}{\left|\displaystyle \frac{m{{\rm{\Omega }}}_{\mathrm{orb}}}{{\chi }_{\alpha }{\omega }_{\alpha }{\gamma }_{\alpha }{t}_{\alpha ,\mathrm{in}}}\right|}^{1/2}\end{eqnarray} \tag{ 16 }$$

where χ_α ≡ 12(M_* + M_p)R²/(M_p a²) − 10/(3κ) for l = m = 2 modes and $\kappa \equiv {I}_{\ast }/({M}_{\ast }{R}_{\ast }^{2})$ is the dimensionless moment of inertia of the star. Above, ${t}_{\alpha ,\mathrm{in}}\equiv {\sigma }_{\alpha }/{\dot{\sigma }}_{\alpha }$ is the mode evolution timescale in the inertial frame, with σ_α = ω_α + mΩ_s ≃ mΩ_orb.

To test the linear approximation for resonantly locked modes in our models, we evaluate the magnitude of ∣d ξ_r /dr∣. Figure 6 shows ∣d ξ_r /dr∣ as a function of radius for a 1.0 M_⊙ model at 3.2 Gyr and a 1.2 M_⊙ model ³ at 1.1 Gyr. For each model we include the mode excited by an off-resonance 1 M_J hot Jupiter and an on-resonance 10 M_⊕ mini-Neptune (with amplitude calculated via Equation (16)), both of which are put in a 2 day orbit. We find that for all models, the g mode nonlinearity indeed increases near the center of the star due to geometrical focusing. However, in the model with a convective core (1.2 M_⊙ model), gravity waves do not propagate into the stellar core, and the g mode always remains linear (∣d ξ/dr∣ ≲ 10⁻³). For the model with a radiative core (1.0 M_⊙ model), the resonantly locked mode excited by a mini-Neptune comes close to the wave-breaking threshold (∣d ξ/dr∣ ∼ 1) but does not exceed it. Interestingly, this mode has a larger amplitude than the nonresonant mode excited by a hot Jupiter, demonstrating the great enhancement in amplitude produced by the resonant forcing.

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We note that the resonant locking amplitude computed above depends on the damping rate γ_α in Equation (16). Weak nonlinear damping may increase the effective value of γ_α , decreasing the necessary mode amplitude for resonance locking. We revisit this issue in Section 2.2.3.

2.2.2. Wave Breaking

When nonlinear wave breaking occurs near the stellar center, the waves overturn the stratification and are efficiently absorbed (Barker & Ogilvie 2010, 2011). The tidally excited gravity waves can then be treated as traveling waves rather than standing g modes, and the corresponding energy dissipation rates have been closely examined in several works (Zahn 1975; Goldreich & Nicholson 1989; Goodman & Dickson 1998; Barker 2020). Specifically, Barker & Ogilvie (2010) compute the corresponding tidal quality factor for wave breaking,

$$\begin{eqnarray}&&{Q}_{\mathrm{WB}}^{{\prime} }={10}^{5}\left(\displaystyle \frac{{{ \mathcal G }}_{\odot }}{{ \mathcal G }}\right){\left(\displaystyle \frac{M}{{M}_{\odot }}\right)}^{2}\left(\displaystyle \frac{{R}_{\odot }}{R}\right){\left(\displaystyle \frac{{P}_{\mathrm{tide}}}{0.5\,\mathrm{days}}\right)}^{8/3}\end{eqnarray} \tag{ 17 }$$

where P_tide = 2π/ω_f is the tidal forcing period and ${ \mathcal G }$ is a parameter that depends on the stellar structure and is defined in Barker (2020).

Nonlinear wave breaking in Sun-like stars only occurs for planets with M ≳ 3 M_J(P/1 day)^−1/6, according to Barker & Ogilvie (2010). This appears roughly consistent with the calculation of linear mode amplitude in Figure 6. Hence, in our calculations of planetary evolution in Section 4, we only use Equation (17) for the most massive exoplanets.

2.2.3. Weakly Nonlinear Damping

Essick & Weinberg (2016) have examined the nonlinear damping of g modes tidally excited by hot Jupiters with periods P ≲ 4 days in Sun-like stars. They examined the weakly nonlinear case where the g waves do not break, but they are sufficiently nonlinear to excite daughter and granddaughter modes that dissipate their energy. They found that even for off-resonance hot Jupiters (like the model shown in Figure 6), nonlinear damping is sufficient to wipe out resonances, i.e., the energy dissipation rate is the same for resonant and nonresonant modes. Our on-resonance mini-Neptune in Figure 6 excites even larger oscillations than an off-resonance hot Jupiter, meaning that nonlinear damping will dominate over linear damping before the resonant amplitude of Equation (16) is reached.

However, Essick & Weinberg (2016) also found that nonlinear energy dissipation is much smaller for off-resonance planets with M_p ≲ 0.3 M_J. For a 10 M_⊕ planet, this implies that the nonlinear energy dissipation rate is small away from resonance and will greatly increase as the planet moves toward a resonance, such that the mode amplitude increases and nonlinear dissipation ramps up. In essence, the total damping rate γ in the expression for the mode amplitude is itself a function of the mode amplitude in this situation. If γ becomes too large near resonance, the resonance will be "saturated" (i.e., the blue curve in Figure 1 will be moved upward near resonance) such that the resonance locking fixed point does not exist.

To address this possibility, in Appendix C we attempt to extrapolate the results of Essick & Weinberg (2016) to low-mass planets below ≃0.3 M_J. Using the resulting nonlinear damping rate in place of the linear damping rate in Equation (3) results in the orange curve in Figure 1, in which the nonlinear damping makes the resonance wells much shallower and wider. The planet may become trapped in resonance if its orbital period is short enough (P_orb ≲ 1.7 days in Figure 1), though we emphasize that more detailed nonlinear coupling calculations are needed for reliable results. This suggests that resonance locking may occur for sufficiently low-mass planets at sufficiently short periods, though the exact mass threshold requires a more accurate calculation of nonlinear damping.

Essick & Weinberg (2016) find that the following quality factor provides a good fit to their calculations for sufficiently massive planets around Sun-like stars:

$$\begin{eqnarray}&&{Q}_{\mathrm{EW}}^{{\prime} }=2\times {10}^{5}{\left(\displaystyle \frac{{M}_{{\rm{p}}}}{{M}_{{\rm{J}}}}\right)}^{1/2}{\left(\displaystyle \frac{{P}_{\mathrm{tide}}}{0.5\,\mathrm{days}}\right)}^{2.4}.\end{eqnarray} \tag{ 18 }$$

Based on their results, Equation (18) breaks down for planet masses with M ≲ 0.3 M_J, so we only use this formula for planets in the range 0.3 M_J ≤ M_p ≤ 3 M_J in our calculations of orbital evolution in Section 4.

We see that in both Equations (17) and (18), the effective tidal quality factor increases with the orbital period, in stark contrast to the prediction of resonance locking where Q' decreases with orbital period. This entails that resonance locking may be more important at longer orbital periods (so long as it can operate), while nonlinear dissipation is likely to dominate at short orbital periods, with important differences in long-term behavior of real systems (see Section 4).

We conclude that resonance locking will not be prevented by nonlinear effects in stars with convective cores, but nonlinear damping will prevent resonance locking from occurring for hot Jupiters around Sun-like stars. It is unclear whether nonlinear damping will prevent resonance locking of low-mass (M ≲ 0.3 M_J) planets, and this should be studied in future work.

Penev et al. (2018) analyzed 188 known hot Jupiter systems to constrain their effect tidal factors based on an improved method from Penev et al. (2014). They managed to constrain two-sided limits on Q' for 35 systems, and to derive lower bounds on Q' for another 40 systems, while the remaining systems in their sample did not lead to meaningful constraints. Of the 75 systems they studied, they found a clear trend toward lower Q' for larger P_tide, where ${P}_{\mathrm{tide}}\equiv {\left({P}_{\mathrm{orb}}^{-1}-{P}_{\mathrm{spin}}^{-1}\right)}^{-1}/m$. The trend is then fitted by the following power-law formula (see Figure 7 or Figure 2 in Penev et al. 2018):

$$\begin{eqnarray}&&{Q}^{{\prime} }=\max \left[{10}^{6.0}{\left(\displaystyle \frac{{P}_{\mathrm{tide}}}{1\,\mathrm{day}}\right)}^{-3.1},{10}^{5}\right].\end{eqnarray} \tag{ 19 }$$

When resonance locking occurs and stellar spin is negligible compared to the orbit, Equation (14) predicts ${Q}^{{\prime} }\approx 2\,\times {10}^{6}{\left({P}_{\mathrm{tide}}/1\,\mathrm{day}\right)}^{-13/3}$ for fiducial hot Jupiter parameters. This simple analysis immediately leads to a similar power-law trend to the fitted formula (19), but with no free parameters. In reality, we expect significant scatter due to the variation of other factors in Equation (14) away from fiducial parameters (e.g., variations in R_* and M_p translate to variations in Q').

To compute the exact value of Q' as a function of P_tide for resonance locking, we construct a 1 M_⊙ stellar model with MESA and compute non-adiabatic oscillation modes. Assuming a stellar rotational evolution based on the modified Skumanich law (Equation (4)), we track the run of Q' upon P_tide for a typical 1 M_J planet at the stellar age of 5 Gyr, using Equation (12). The results are shown in Figure 7. We find that the predicted trend is remarkably similar to the power-law fit by Penev et al. (2018), though slightly offset to higher values of Q'. Overall, the resonance locking prediction fits the data very well. We also plot the predicted relations from wave breaking (Equation (17)) and weakly nonlinear dissipation (Equation (18)). Those models predict that Q' increases as P_orb increases, opposite to the trend inferred by Penev et al. (2018).

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While these results at first glance appear to provide compelling evidence for the operation of resonance locking in hot Jupiter systems, we caution that other explanations for the trend in Q' from Penev et al. (2018) should be examined. From a theoretical perspective, resonance locking likely cannot operate for Jupiter-mass planets around Sun-like stars due to the nonlinear damping discussed in Section 2.2. Hence, we are hesitant to ascribe the trend in Q' from Penev et al. (2018) to resonance locking, though resonance locking could provide a nice explanation if nonlinear mode dissipation is much less efficient than found by Essick & Weinberg (2016).

Another possibility worth considering is that the inferred trend in Q' from Penev et al. (2018) does not arise from tidal spin-up of the host stars. Penev et al. (2018) infer the value of the host stars' Q' by combining age estimates with measurements of spin period. Rapidly rotating host stars (compared to typical field stars) are often inferred to have been tidally spun up by their hot Jupiter companions, entailing that the tidal spin-up timescale is comparable to the main-sequence lifetime of the host star. This can be seen because much shorter tidal spin-up times would result in the synchronization of the star or the destruction of the hot Jupiter, while much longer ones would not increase the star's rotation significantly. Assuming ${t}_{s}={{\rm{\Omega }}}_{{\rm{s}}}/{\dot{{\rm{\Omega }}}}_{{\rm{s}}}=10\,\mathrm{Gyr}$, some algebra shows that this requires a scaling Q' ∼ 2 × 10⁶(P_tide/1 day)⁻⁴(P_spin/10 days)(M_p/M_J)², almost identical to the trend shown in Figure 7. Hence, if one does not have accurate age estimates and assumes that hot Jupiter hosts have similar ages to typical field stars, it could yield a spurious scaling of Q' that is very similar to the trend found by Penev et al. (2018).

Instead, we speculate that moderately rotating host stars of hot Jupiters are (in some cases) simply young stars that are still spinning down, and that they have not been substantially spun up by tides. This may be consistent with the young average ages of hot Jupiter host stars found by Hamer & Schlaufman (2019). In this case, it is very difficult to observationally constrain the host star's Q', except perhaps to place a lower limit. For massive planets where nonlinear effects prevent resonance locking, we expect a trend in Q' similar to that predicted by Essick & Weinberg (2016) and Barker (2020). Future work could aim to more accurately constrain the ages of the hot Jupiter host stars in Penev et al. (2018) to determine whether their rapid rotation arises from youth or tidal spin-up.

Here we summarize our prediction for the effective tidal Q' of 15 real systems based on resonance locking. Most of the systems are chosen from Patra et al. (2020), who argue that the orbital decay of these systems should be the easiest to observe. We also study TRES-3b and WASP-4b, which have new observational constraints (Bouma et al. 2019; Mannaday et al. 2020), and we include WASP-128b, a system with a very massive hot Jupiter.

The observational properties of these systems are summarized in Table 2. For each system, we construct a number of stellar models to fit their host stars with different initial masses and metallicities within the observational errors and locate the model that matches the other observed properties of the star. We are able to fit the masses, radii, metallicities, ages, and effective temperatures within the observational error (see Table 2 for a summary) for all the host stars except KELT-16b. A typical inlist file is given in the supplementary material.

For each stellar model, we compute ℓ = 2 non-adiabatic oscillation modes with GYRE, with typical inlist files given in the supplementary material. For each system, we assume a negligible spin of the host star and identify the oscillation mode resonant with the tidal forcing, and we calculate the value of t_α for that mode. The effective tidal Q' are then calculated via Equation (12) using our model properties. We summarize the predictions for resonance locking in Table 1, along with predictions for tidal migration rates due to wave breaking (Barker 2020) and nonlinear dissipation by Equation (18) (Essick & Weinberg 2016). Below are detailed discussions for each system.

• 1.  
  HAT-P-23b: A planet of 2.09 M_J in a 1.21 day orbit around a G-type dwarf (Bakos et al. 2011). Our best-fit model is a 1.10 M_⊙ star with a radiative core. Lack of detected orbital decay requires Q' > (3.6 ± 1.1) × 10⁵ (Patra et al. 2020). Our resonance locking calculation predicts ${Q}_{\mathrm{RL}}^{{\prime} }=6.0\times {10}^{7}$, with t_tide = 5.6 Gyr, but resonance locking is not expected due to nonlinear effects in the radiative core. Barker (2020) predicts ${Q}_{\mathrm{WB}}^{{\prime} }=3\times {10}^{5}$ from calculations of wave breaking, but the best-fit model in that work has a convective core. Weak nonlinear mode damping gives ${Q}_{\mathrm{EW}}^{{\prime} }=4.6\times {10}^{5}$. This is a promising system in which to observe tidal decay if the core is indeed radiative.
• 2.  
  HATS-18b: A planet of 1.98 M_J in a 0.84 day orbit around a G-type star (Penev et al. 2016). Our best-fit model is a 1.03 M_⊙ star with a radiative core. No reliable constraint on Q' could be found due to the lack of data (Patra et al. 2020). Our resonance locking calculation predicts ${Q}_{\mathrm{RL}}^{{\prime} }=2.1\times {10}^{8}$, with t_tide = 9.7 Gyr, but resonance locking is not expected due to nonlinear effects since the star has a radiative core and a massive planet. Barker (2020) predicts ${Q}_{\mathrm{WB}}^{{\prime} }\approx 7\times {10}^{4}$ from calculations of wave breaking; weak nonlinear mode damping predicts ${Q}_{\mathrm{EW}}^{{\prime} }=1.9\times {10}^{5}$. The results make HATS-18b a very promising candidate in which to observe orbital decay.
• 3.  
  KELT-16b: A planet of 2.75 M_J in a 0.97 day orbit around an F-type star (Oberst et al. 2017). Our best-fit model is a 1.18 M_⊙ star with a convective core. Lack of detected orbital decay requires Q' > (0.5 ± 0.1) × 10⁵ (Patra et al. 2020). Our resonance locking calculation predicts ${Q}_{\mathrm{RL}}^{{\prime} }=5.1\times {10}^{8}$, with t_tide = 9.8 Gyr, which is consistent with the observed lower limit, indicating no tidal decay should have been observed.
• 4.  
  OGLE-TR-56b: A planet of 1.39 M_J in a 1.21 day orbit around an F-type star (Sasselov 2003; Torres et al. 2008). Our best-fit model is a 1.23 M_⊙ star with a convective core. Lack of detected orbital decay requires Q' > (4.4 ±1.3) × 10⁵ (Patra et al. 2020). Our resonance locking calculation predicts ${Q}_{\mathrm{RL}}^{{\prime} }=1.1\times {10}^{8}$, with t_tide = 12.9 Gyr, which is consistent with the observed lower limit, indicating no tidal decay should be observed.
• 5.  
  TRES-3b: A planet of 1.92 M_J in a 1.306 day orbit around a G-type dwarf (O'Donovan et al. 2007). Our best-fit model is a 0.89 M_⊙ star with a radiative core. Observations indicate Q' ≈ (5.5 ± 4.2) × 10⁴ (Mannaday et al. 2020), potentially detecting rapid orbital decay. Our resonance locking calculation predicts ${Q}_{\mathrm{RL}}^{{\prime} }=1.2\times {10}^{7}$, with t_tide = 8.0 Gyr, but resonance locking is not expected due to nonlinear damping in the radiative core. Barker (2020) predicts ${Q}_{\mathrm{WB}}^{{\prime} }=4.3\times {10}^{5}$ from calculations of wave breaking, slightly larger than the measured value. Weak nonlinear mode damping predicts ${Q}_{\mathrm{EW}}^{{\prime} }\,=5.3\times {10}^{5}$. Further observations should attempt to verify the result of Mannaday et al. (2020) and will help calibrate models of orbital decay via nonlinear g mode damping in the core.
• 6.  
  WASP-4b: A planet of 1.186 M_J in a 1.338 day orbit around a main-sequence star (Bouma et al. 2019; Southworth et al. 2019). Our best-fit model is a 0.83 M_⊙ star with a radiative core. Observations suggest Q' = (1.8 ± 0.2) × 10⁴ (Bouma et al. 2019), but Bouma et al. (2020) recently discovered a third massive companion that might cause the shift in transit times. Our resonance locking calculation predicts ${Q}_{\mathrm{RL}}^{{\prime} }=1.1\times {10}^{7}$, with t_tide = 9.9 Gyr, but resonance locking is not expected due nonlinear damping in the radiative core. Barker (2020) predicts ${Q}_{{\rm{WB}}}^{{\rm{{\prime} }}}=(2\text{--}3)\times {10}^{5}$ from calculations of wave breaking. Weak nonlinear mode damping predicts ${Q}_{\mathrm{EW}}^{{\prime} }=4.4\times {10}^{5}$. Further observations will shed more light on the system and have a good chance of confirming the detection of orbital decay.
• 7.  
  WASP-12b: A planet of 1.47 M_J in a 1.09 day orbit around a late F-type main-sequence star or a subgiant (Hebb et al. 2009; Collins et al. 2017; Weinberg et al. 2017). Our best-fit model is a 1.44 M_⊙ star with a convective core, but subgiant models without convective cores are also compatible with the data (Bailey & Goodman 2019). Observations indicate orbital decay with a quality factor Q' = (1.1 ± 0.1) × 10⁵ (Patra et al. 2017, 2020). Our resonance locking calculation predicts ${Q}_{\mathrm{RL}}^{{\prime} }=1.9\times {10}^{8}$, three orders of magnitude too high, with t_tide = 7.2 Gyr. Barker (2020) finds ${Q}_{\mathrm{WB}}^{{\prime} }$ ranging from 1.8 × 10⁵ to 3 × 10⁶ assuming that gravity waves break near a radiative core, based on different stellar models they choose. Weak nonlinear mode damping predicts ${Q}_{\mathrm{EW}}^{{\prime} }=3.0\times {10}^{5}$. The measured decay rate thus indicates that the star is indeed a subgiant undergoing orbital decay via nonlinear gravity wave damping. We speculate that WASP-12b was previously migrating inward slowly via resonance locking when the host star was on the main sequence and had a convective core. When the core became radiative at the end of the main sequence, nonlinear damping became effective, driving the much faster orbital decay we see today. This may help alleviate fine-tuning problems in formation models for WASP-12b, allowing the planet to survive until the end of the main sequence, while also explaining the rapid inward migration at the start of the subgiant phase.
• 8.  
  WASP-18b: A massive planet of 11.4 M_J in a 0.94 day orbit around a relatively hot (T_eff = 6431 K) F-type star (Hellier et al. 2009; Stassun et al. 2017). The mass of the host star is a bit uncertain, with a measurement of 1.46 ± 0.29 M_⊙ reported. Our best-fit model falls at the low end of the mass measurement, with a mass of 1.17 M_⊙ and a convective core. Lack of detected orbital decay requires Q' > (1.0 ±0.2) × 10⁶ (Patra et al. 2020). Our resonance locking calculation predicts ${Q}_{\mathrm{RL}}^{{\prime} }=3.3\times {10}^{8}$, with t_tide = 4.7 Gyr, which is consistent with the observed lower limit, indicating no tidal decay should have been observed.
• 9.  
  WASP-19b: A planet of 1.139 M_J in a 0.79 day orbit around a Sun-like star, making it the hot Jupiter system with the shortest period yet observed (Hebb et al. 2010; Mancini et al. 2013). Our best-fit model is a 0.91 M_⊙ star with a radiative core. Observations indicate Q' = (3.1 ±0.9) × 10⁵, but the authors encourage caution due to the scanty data (Patra et al. 2020). Our resonance locking calculation predicts ${Q}_{\mathrm{RL}}^{{\prime} }=2.2\times {10}^{8}$, with t_tide =9.7 Gyr, but resonance locking is not expected due to nonlinear effects in the radiative core. Barker (2020) predicts ${Q}_{{\rm{WB}}}^{{\rm{{\prime} }}}\approx (4\text{--}5)\times {10}^{4}$ from calculations of wave breaking, smaller than the observational constraint. Weak nonlinear mode damping predicts ${Q}_{\mathrm{EW}}^{{\prime} }=1.2\times {10}^{5}$. Further observations of this system will hence be very useful to constrain tidal theories.
• 10.  
  WASP-43b: A planet of 2.034 M_J in a 0.81 day orbit around a K-type dwarf (Hellier et al. 2011; Gillon et al. 2012). Our best-fit model is a 0.70 M_⊙ star with a radiative core. Lack of detected orbital decay requires Q' > (2.5 ± 0.2) × 10⁵ (Patra et al. 2020). Our resonance locking calculation predicts ${Q}_{\mathrm{RL}}^{{\prime} }=2.3\times {10}^{8}$, with t_tide = 26.6 Gyr, but resonance locking is not expected due to nonlinear effects in the radiative core. Barker (2020) predicts ${Q}_{\mathrm{WB}}^{{\prime} }\approx {10}^{5}$ from calculations of wave breaking, comparable to the lower limit from observations. Weak nonlinear mode damping predicts ${Q}_{\mathrm{EW}}^{{\prime} }=1.7\times {10}^{5}$. This is another good candidate for orbital decay to be detected in the near future.
• 11.  
  WASP-72b: A planet of 1.546 M_J in a 2.22 day orbit around an F-type star (Gillon et al. 2013). Our best-fit model is a 1.33 M_⊙ subgiant with a radiative core. Lack of detected orbital decay requires Q' > (1.2 ± 0.8) × 10³ (Patra et al. 2020). Our resonance locking calculation predicts ${Q}_{\mathrm{RL}}^{{\prime} }=4.5\times {10}^{6}$, with t_tide = 0.8 Gyr, but resonance locking is not expected due to nonlinear damping in the radiative core. Barker (2020) predicts a very large ${Q}_{\mathrm{WB}}^{{\prime} }\gt {10}^{12}$ from calculations of wave breaking. However, that work appears to use a stellar model with surface temperature much higher than the observed temperature of T = 6250 ± 100 K from Gillon et al. (2013), likely translating to a predicted value of Q' that is far too high. Weak nonlinear mode damping predicts ${Q}_{\mathrm{EW}}^{{\prime} }=1.7\times {10}^{6}$. Future models should re-examine the theoretical predictions.
• 12.  
  WASP-103b: A planet of 1.51 M_J in a 0.93 day orbit around a late F-type star (Gillon et al. 2014; Delrez et al. 2018). Our best-fit model is a 1.18 M_⊙ star with a convective core. Lack of detected orbital decay requires Q' > (6.4 ± 0.6) × 10⁴ (Patra et al. 2020). Our resonance locking calculation predicts ${Q}_{\mathrm{RL}}^{{\prime} }=4.0\times {10}^{8}$, with t_tide = 9.7 Gyr, which is consistent with the observed lower limit, indicating no tidal decay should have been observed.
• 13.  
  WASP-114b: A planet of 1.769 M_J in a 1.55 day orbit around an early G-type star (Barros et al. 2016). Our best-fit model is a 1.24 M_⊙ star with a convective core. Being a newly discovered system, no reliable constraint on Q' could be found due to its lack of data (Patra et al. 2020). Our resonance locking calculation predicts ${Q}_{\mathrm{RL}}^{{\prime} }=4.5\times {10}^{7}$, with t_tide = 7.9 Gyr, and the linearity of resonant modes shows resonance locking could occur. Orbital decay is unlikely to be detected for this system unless the star is less massive and contains a radiative core.
• 14.  
  WASP-122b: A planet of 1.284 M_J in a 1.71 day orbit around a G-type star (Turner et al. 2016). Our best-fit model is a 1.25 M_⊙ star with a convective core. Being a newly discovered system, no reliable constraint on Q' could be found due to its lack of data (Patra et al. 2020). Our resonance locking calculation predicts ${Q}_{\mathrm{RL}}^{{\prime} }=6.3\times {10}^{6}$, with t_tide = 2.4 Gyr, and the linearity of resonant modes shows resonance locking could occur. Orbital decay is unlikely to be detected for this system unless the star is less massive and contains a radiative core.
• 15.  
  WASP-128b: A brown dwarf of 37.19 M_J in a 2.209 day orbit around an G-type dwarf (Hodžić et al. 2018). Our best-fit model is a 1.13 M_⊙ star with a radiative core. No observational constraint on Q' is available at current time. The large companion mass makes the tidal stellar spin-up important, and we do not expect resonance locking in this system because I_p > 3I_* in Equation (11) such that resonance cannot be maintained. Barker (2020) predicts ${Q}_{\mathrm{WB}}^{{\prime} }$ from 3 × 10⁶ to 1.3 × 10⁸ from calculations of wave breaking, based on the different rotation periods in the stellar models in that work. Weak nonlinear mode damping predicts ${Q}_{\mathrm{EW}}^{{\prime} }=8.2\times {10}^{6}$. Further observations should attempt to measure the stellar spin rate and could potentially detect orbital decay.

Table 1. Observed/Calculated Tidal Quality Factor Q', Core Status, and t_tide,RL Induced by Resonance Locking for the Systems We Study

Name	${Q}_{\mathrm{obs}}^{{\prime} }$	${Q}_{\mathrm{RL}}^{{\prime} }$	${Q}_{\mathrm{WB}}^{{\prime} }$	${Q}_{\mathrm{EW}}^{{\prime} }$	Core Status	ttide,RL (Gyr)
HAT-P-23b	>(3.6 ± 1.1) × 105	6.0 × 107	3 × 105	4.6 × 105	radiative?	5.6
HATS-18b		2.1 × 108	7 × 104	1.9 × 105	radiative	9.7
KELT-16b	>(0.5 ± 0.1) × 105	5.1 × 108	5 × 105	3.1 × 105	convective	9.8
OGLE-TR-56b	>(4.4 ± 1.3) × 105	1.1 × 108	106	3.7 × 105	convective	12.9
TRES-3b	(5.5 ± 4.2) × 104	1.2 × 107	4.3 × 105	5.3 × 105	radiative	8.0
WASP-4b	(1.8 ± 0.2) × 104	1.1 × 107	(2–3) × 105	4.4 × 105	radiative	9.9
WASP-12b	(1.1 ± 0.1) × 105	1.9 × 108	(0.18–3) × 106	3.0 × 105	convective?	7.2
WASP-18b	>(1.0 ± 0.2) × 106	3.3 × 108	2 × 106	5.8 × 105	convective	4.7
WASP-19b	(3.1 ± 0.9) × 105	2.2 × 108	(0.4–0.5) × 105	1.2 × 105	radiative	9.7
WASP-43b	>(2.5 ± 0.2) × 105	2.3 × 108	1 × 105	1.7 × 105	radiative	26.6
WASP-72b	>(1.2 ± 0.8) × 103	4.5 × 106	>2 × 1012	1.7 × 106	radiative	0.8
WASP-103b	>(6.4 ± 0.6) × 104	4.0 × 108	2 × 105	2.1 × 105	convective	9.7
WASP-114b		4.5 × 107	2 × 106	7.6 × 105	convective	7.9
WASP-122b		6.4 × 106	2.3 × 105	8.2 × 105	convective	2.4
WASP-128b		(no RL)	(0.03–1.3) × 108	8.2 × 106	radiative	20.9

Note. ${Q}_{\mathrm{obs}}^{{\prime} }$ shows the observed constraints from Patra et al. (2020), Bouma et al. (2019), and Mannaday et al. (2020). ${Q}_{\mathrm{RL}}^{{\prime} }$ is the predicted tidal factor from the best-fit resonance locking model. ${Q}_{\mathrm{WB}}^{{\prime} }$ is the predicted tidal factor from gravity wave breaking (Barker 2020), and ${Q}_{\mathrm{EW}}^{{\prime} }$ is the predicted tidal factor from nonlinear g mode dissipation (Essick & Weinberg 2016). We also show the convective/radiative core status of our models. We emphasize that for massive planets like these, resonance locking is only expected to occur in stars with convective cores, and wave breaking/nonlinear dissipation is expected to dominate stars with radiative cores. We use bold to show the Q' values of our inference of the appropriate tidal theory for each system, while the other values are left for reference. The t_tide,RL values show that most system have long tidal migration timescales if they experience resonance locking. Some authors use a different definition of Q' from ours. We have corrected their results to make them consistent with our definition, so the numbers here may appear different from the original literature.

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To summarize, for the hot Jupiter systems above, we predict orbital decay timescales of a few gigayears for host stars with convective cores in which resonance locking can operate. Orbital decay via nonlinear mode damping (Essick & Weinberg 2016) or wave breaking (Barker 2020) is likely to operate in stars with radiative cores, causing shorter tidal decay timescales that can be more easily observed. Future observations can help confirm our prediction of more rapid orbital decay of hot Jupiters in stars with radiative cores.

We have argued that resonance locking, nonlinear g mode dissipation, and gravity wave breaking can all operate in short-period exoplanet systems. In general, one must solve for the angular momentum evolution (Equation (9)) with appropriate tidal dissipation physics (i.e., the appropriate value of Q' from Equation (1)) to track the full orbital evolution of the system. We expect that resonance locking is the dominant tidal dissipation mechanism for stars massive enough to have convective cores (M ≳ 1.1 M_⊙), where nonlinear damping is weak and g modes can be resonantly excited. In these stars, Equation (11) can be used to estimate the tidal dissipation rate. Heartbeat stars with large-amplitude, tidally excited g modes (e.g., Fuller 2017) are proof that nonlinear damping does not prevent resonant mode excitation in these stars, even for stellar-mass companions.

For stars with radiative cores, a planet more massive than a few Jupiter masses causes gravity wave breaking near the core (Barker & Ogilvie 2010) such that tidal dissipation is determined by Equation (17). For Jupiter-mass exoplanets approximately in the range 0.3 M_J ≲ M_p ≲ 3 M_J, wave breaking does not occur, but nonlinear mode damping prevents resonant excitation and produces tidal dissipation according to Equation (18) (Essick & Weinberg 2016). Resonance locking is likely to be the dominant dissipation mechanism for less massive planets with M ≲ 0.3 M_J, though future work is needed to quantify this number more accurately (see Section 2.2.3).

We hence study the orbital evolution of three fiducial exoplanet systems with masses of 10 M_⊕, 1 M_J, and 5 M_J, in which resonance locking, nonlinear damping, and wave breaking apply, respectively. We initialize calculations of orbital evolution for planets in 3 day orbits around a 1 M_⊙ star at an age of 700 Myr. We integrate the combined equations of orbital decay (Equation (2)), spin evolution (Equation (10)), and tidal theories (Equation (12), (17), or (18)) to track the full evolution of the systems. We also integrate a system without planets (i.e., a star spinning down purely by magnetic braking) for comparison. The results are shown in Figure 8.

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For the 10 M_⊕ mini-Neptune, resonance locking causes significant orbital decay during the main sequence, with t_tide ∼ t_MS as typically expected from resonance locking. We note that the effective tidal quality factor driving this planet's inward migration is initially quite small, with Q' < 10⁵. Consequently, the planet migrates much farther than typical parameterized tidal models with Q' ∼ 10⁵–10⁶. Compared to prior work, resonance locking typically predicts substantially more tidal migration for low-mass planets at orbital periods P ≳ 2 days. In this example, the planet has migrated to ultrashort periods by the end of the main sequence, but it does not plunge into its host star because the value of Q' increases at short periods in order to maintain t_tide ∼ t_α . Due to the relatively low mass of the planet, the stellar spin is hardly influenced by the angular momentum input from the planet's orbit.

For the 1 M_J hot Jupiter where nonlinear mode damping dominates the dissipation, the orbit initially decays more slowly than what resonance locking would predict during the first ∼7 Gyr, because resonance locking would predict a path very similar to that of the 10 M_⊕ planet above. The slow initial migration is due to the strong period dependence of Q' from Equation (18), producing an initially large value of Q'. However, the orbit decays very rapidly after the planet migrates to a critical period P ≲ 2 days, after which the planet is quickly tidally disrupted. Hence, nonlinear mode coupling predicts rapid orbital decay for hot Jupiters with the shortest periods of P_orb ≲ 2 days, so that such systems are expected to be rare around Sun-like stars with radiative cores. The final plunge also spins up the host star by a factor of ≈3.

A similar evolution occurs for the 5 M_J planet, which is massive enough to trigger wave breaking. The mass dependence of this mechanism (Equation (17)) entails shorter migration times for more massive planets, so it takes less time (∼3.5 Gyr) for tidal disruption to occur. The host star is highly spun up during the final plunge.

Hence, we conclude that hot Jupiters on short-period orbits around Sun-like stars are likely to be destroyed during the main sequence. Short-period super-Earths and mini-Neptunes are more likely to survive, though their orbits are expected to decay significantly due to resonance locking.

Our main predictions appear to be consistent with the recent finding that hot Jupiter host stars are on average slightly younger than field stars (Hamer & Schlaufman 2019), implying that a substantial fraction of hot Jupiters are destroyed before their host star evolves off the main sequence. More detailed population modeling will be required to predict an exact number, but we also predict that host stars of short-period (e.g., P_orb ≲ 3 days) hot Jupiters will be younger than host stars of long-period (e.g., P_orb ≳ 4 days) hot Jupiters. Noting also the well-known trend that higher-mass hot Jupiters have shorter orbital periods on average (e.g., Owen & Lai 2018), we predict that host stars of high-mass hot Jupiters will be younger than host stars of low-mass hot Jupiters, for host stars of nearly the same mass.

In stars with radiative cores, we predict that orbital decay for massive planets (M ≳ 0.3 M_J) at short periods (P ≲ 2 days) proceeds much more rapidly due to nonlinear damping processes. Tidal destruction takes much longer if these processes do not operate (Figure 8). Our resonance locking models indicate that wave breaking rarely occurs in stars with convective cores, and the damping produced by nonlinear mode coupling is also likely to be strongly reduced relative to Sun-like stars. Hence, we predict slower orbital decay at short orbital periods and a higher main-sequence survival fraction of hot Jupiters in stars with M ≳ 1.2 M_⊙ than of their lower-mass counterparts.

For mini-Neptunes or less massive planets (M ≲ 0.1 M_J), resonance locking is probably the dominant tidal dissipation mechanism for stars both with and without convective cores. Resonance locking predicts t_tide ≈ 10 Gyr for Sun-like stars regardless of planet mass and orbital period, corresponding to an effective quality factor of

$$\begin{eqnarray}&&{Q}^{{\prime} }\sim 8\times {10}^{6}\displaystyle \frac{{M}_{p}}{3{M}_{\oplus }}{\left(\displaystyle \frac{{P}_{\mathrm{orb}}}{0.5\,\mathrm{day}}\right)}^{-13/3}.\end{eqnarray} \tag{ 20 }$$

This is very close to the inferred constraint from Hamer & Schlaufman (2020) for ultrashort-period planets (USPs), though we caution against comparing Q' values because they are very sensitive to the stellar radius. Our predicted tidal migration timescale is comparable to the main-sequence lifetime, consistent with the old ages of USP host stars, but also allowing tidal orbital decay to have significantly shortened the period without destroying the planet.

We can also rule out a naive extrapolation of the nonlinear damping model of Essick & Weinberg (2016) to ultrashort-period planets, which predicts a tidal migration timescale of only ${t}_{\mathrm{tide}}\simeq 4\,\mathrm{Myr}{\left({M}_{p}/3{M}_{\oplus }\right)}^{-0.5}{\left({P}_{\mathrm{orb}}/0.5\,{\rm{d}}\right)}^{6.7}$, corresponding to a tidal quality factor of ${Q}^{{\prime} }\simeq 6\times {10}^{3}{\left({M}_{p}/3{M}_{\oplus }\right)}^{0.5}{\left({P}_{\mathrm{orb}}/0.5\,\mathrm{day}\right)}^{2.4}$. This is inconsistent with the old ages of host stars from Hamer & Schlaufman (2020), and corresponding inferred lower limits of Q' ≳ 10⁷ for most USPs. Hence, the scaling of the nonlinear migration rate (Equation (18)) must break down for lower-mass planets as predicted by Essick & Weinberg (2016), likely because the tidally excited gravity modes do not reach sufficient amplitude to transfer energy to daughter modes.

Resonance locking makes unique predictions for the statistical distributions of exoplanet orbits. Consider exoplanets born at a given orbital period _Pi at a constant rate R = dN(_Pi)/dt, and then migrating inward due to resonance locking/nonlinear dissipation. For a steady-state distribution, the rate of planets migrating through shorter periods is constant, i.e.

$$\begin{eqnarray}&&\displaystyle \frac{{dN}(P)}{{dt}}=\displaystyle \frac{{dN}({P}_{{\rm{i}}})}{{dt}}=R\end{eqnarray} \tag{ 21 }$$

or

$$\begin{eqnarray}&&\displaystyle \frac{{dN}(P)}{{da}}=\displaystyle \frac{{dN}(P)}{{dt}}\displaystyle \frac{{dt}}{{da}}=\displaystyle \frac{R}{\dot{a}}\Rightarrow \ \displaystyle \frac{{dN}(P)}{d\mathrm{ln}P}=\displaystyle \frac{2}{3}{{Rt}}_{\mathrm{tide}}.\end{eqnarray} \tag{ 22 }$$

For low-mass planets migrating inward via resonance locking, we expect t_tide is roughly constant, which entails ${dN}/d\mathrm{ln}P=\mathrm{constant}$, i.e., a uniform distribution over logP. For more massive planets migrating inward via nonlinear wave damping, t_tide becomes strongly dependent on P. For nonlinear mode coupling, this implies

$$\begin{eqnarray}&&\displaystyle \frac{{dN}(P)}{d\mathrm{ln}P}\propto {t}_{\mathrm{tide}}\propto {P}^{6.7},\end{eqnarray} \tag{ 23 }$$

such that the number of planets should fall very sharply with decreasing orbital period.

However, there are many uncertainties that complicate this simple picture. First, the observed distribution of exoplanets is not necessarily in a steady state. The number of short-period exoplanets may be growing as more numerous exoplanets born at longer periods migrate inward. Second, the birth-period distribution is also likely to be a strong function of period (Lee & Chiang 2017), which complicates interpretation. To first order, we expect resonance locking to shift the birth-period distribution to shorter periods without changing its shape. We therefore expect a flatter distribution of exoplanets at short periods when resonance locking operates, compared to nonlinear dissipation or models with a constant stellar Q', which destroy short-period planets more rapidly.

Recent studies may provide evidence for a distribution of Kepler planets sculpted by resonance locking migration. For instance, Figures 2 and 3 of Zhu & Dong (2021) show a relatively uniform occurrence rate of planets with R_p ≲ 2R_⊕ within the orbital period range 0.6 days ≲ P_orb ≲ 2 days, which agrees with the basic prediction of resonance locking. In contrast, the occurrence rate of hot Jupiters falls steeply toward short orbital periods, as expected from nonlinear g mode damping for massive planets. A prediction of resonance locking is that the occurrence rate of hot Jupiters should show a flatter trend with orbital period around slightly more massive stars with convective cores. Since resonance locking in individual hot Jupiter systems is generally hard to detect due to the long tidal migration timescale, this may serve as the best prospect to justify whether resonance locking is occurring in these systems. Future population modeling should examine the short-period exoplanetary distribution resulting from resonance locking in more detail.

As resonance locking typically predicts tidal migration timescales 2–3 orders of magnitudes longer than nonlinear g mode dissipation for hot Jupiters at orbital periods of ∼1 day (Table 1), we might expect short-period hot Jupiters orbiting stars with convective cores (where resonance locking is operating) to be more common than those orbiting Sun-like stars with radiative cores (where nonlinear dissipation dominates). However, several additional factors may complicate this picture: if the hot Jupiters are born at some minimum period (e.g., 3 days), the slow tidal migration induced by resonance locking might prevent them from reaching short orbital periods before the massive star evolves off the main sequence. The observed hot Jupiter population may also suffer from observational biases that preferentially detect systems with certain types of host stars. While disentangling these effects is beyond the scope of this paper, future population analyses may shed more light on this issue.

We have avoided modeling the early evolution (t ≲ 500 Myr) and post-main-sequence evolution of exoplanet systems in this work. At early times, it is often the case that inward migration via resonance locking cannot occur because resonant locations move outward, as discussed in Section 2. However, in this case, rapidly rotating young stars with P_s < P_orb could instead drive outward migration via resonance locking. It is not clear how far planets could be driven outward, but this possibility should be investigated in future work. For example, resonance locking via m = 0 modes during the pre-main-sequence evolution of stellar binaries likely helps to circularize their orbits (Zanazzi & Wu 2021).

After the main sequence, the timescales for stellar evolution and those for mode frequency evolution, t_α , decrease dramatically, naively resulting in much faster migration via resonance locking. However, post-main-sequence stars contain strongly stratified radiative cores, likely making nonlinear damping in the core even more efficient than in Sun-like stars. Hence, it is not clear whether resonance locking can ever occur in subgiants or stars ascending the red giant branch.

For massive planets, nonlinear damping likely dominates over linear mode damping processes. This causes the resonances to saturate at lower mode amplitudes, decreasing the maximum period P_max above which resonance locking cannot operate. The orange line in Figure 1 demonstrates this qualitatively, but the crudeness of our approximation of nonlinear damping prevents a quantitative prediction for P_max. Realistic calculations of nonlinear mode damping rates are needed to reliably predict P_max. These calculations should be performed for planets of different masses and orbital periods, as well as for stars of different masses and ages. This would allow for better predictions of the statistical distribution of exoplanets as a function of planet mass, orbital period, stellar mass, and stellar age.