Tidal Dissipation Due to Inertial Waves Can Explain the Circularization Periods of Solar-type Binaries

Tidal interactions drive orbital and spin evolution in planetary systems and binary stars (e.g., Mazeh 2008; Zahn 2008; Ogilvie 2014). Observations of various populations of stars with different ages provide strong evidence for efficient tidal dissipation in solar-type binary stars, both during the pre-main sequence (PMS) and also later on the main sequence (MS) (e.g., Meibom & Mathieu 2005; Van Eylen et al. 2016; Triaud et al. 2017; Nine et al. 2020; Justesen & Albrecht 2021). The maximum orbital period out to which binary orbits are preferentially circular ¹ is referred to as the circularization period, P_circ. From analyzing samples of stars of different ages, P_circ has been found to increase with age on the MS. There is also evidence for spin synchronization in solar-type binaries (e.g., Meibom et al. 2006; Lurie et al. 2017) and circularization of evolved stars (e.g., Verbunt & Phinney 1995).

Tidal flows in stars are often decomposed into two components (e.g., Zahn 1977; Ogilvie 2014): a non-wavelike quasi-hydrostatic bulge and its associated flow that is referred to as the equilibrium tide and a wavelike component that is referred to as the dynamical tide. The dynamical tide is thought to consist mainly of inertial waves in convection zones of rotating stars and internal gravity waves in radiation zones.

Previous theoretical work has been unable to explain the observed P_circ and its variation with age. Early work by Zahn & Bouchet (1989; see also Zahn 2008) suggested that dissipation of the equilibrium tide by the turbulent viscosity due to convection on the PMS could explain prior observations. However, their models assumed an optimistic turbulent viscosity acting on equilibrium tides, ² which is incompatible with the latest hydrodynamical simulations (e.g., Ogilvie & Lesur 2012; Duguid et al. 2020a, 2020b; Vidal & Barker 2020a, 2020b). In addition, it appears that another mechanism must operate later on the MS. Other works that accounted for a more realistic frequency reduction of the turbulent viscosity have claimed that equilibrium tide dissipation is far too weak to explain the observed P_circ on the MS (Goodman & Oh 1997; Terquem et al. 1998). This suggests that dissipation due to the dynamical tide must be responsible instead, in the absence of a more efficient mechanism to damp small-amplitude equilibrium tides. ³

An obvious candidate is internal gravity-wave dissipation in radiation zones. This appears to be more efficient than equilibrium tide dissipation for short orbital periods but is also unable to explain the observed P_circ (Goodman & Dickson1998; Terquem et al. 1998). Because eccentricity tides in spin-synchronized (or possibly pseudo-synchronized; e.g., Hut 1981) binaries have forcing frequencies that are equal to (or similar to) the stellar spin frequency, this suggests that inertial waves (restored by Coriolis forces) could be important for tidal dissipation. Prior theoretical work has indicated that inertial waves can significantly enhance the dissipation of eccentricity tides, but this mechanism still appears to be insufficient to explain the observed P_circ in the MS models of Ogilvie & Lin (2007).

In this paper, we build upon prior theoretical work by studying tidal dissipation in solar-type and low-mass binaries following their evolution from the PMS until they evolve off the MS. We demonstrate that inertial wave dissipation is very efficient in PMS phases, and it subsequently evolves with MS age in accordance with observations. Inertial wave dissipation can therefore explain the observed P_circ and its variation with age (with equilibrium tides and gravity waves possibly contributing at the end of the MS). We also demonstrate that both equilibrium tide dissipation and internal gravity-wave dissipation are unlikely to account for the observed P_circ, in agreement with the conclusions of prior work.

We consider tides in both the convection and radiation zones of an approximately spherically symmetric star with mass M and radius R in hydrostatic equilibrium that rotates with angular velocity Ω = 2π/P_rot (where P_rot is the rotation period). We define the dynamical frequency ${\omega }_{\mathrm{dyn}}=\sqrt{{GM}/{R}^{3}}$ and timescale P_dyn = 2π/ω_dyn. We also define ${\epsilon }_{{\rm{\Omega }}}^{2}={P}_{\mathrm{dyn}}^{2}/{P}_{\mathrm{rot}}^{2}$. The star is described by the standard equations of stellar structure, and we consider models with masses in the range 0.2–1.6 M_⊙ computed using MESA (e.g., Paxton et al. 2011, 2015, 2019), evolving them throughout the PMS phase until the end of the MS. Details of our calculations are relegated to Appendices A and B, and further details are reported in Barker (2020).

We compute the dissipation of equilibrium tides in convection zones by assuming a turbulent viscosity due to convection. This mechanism is believed to be strongly frequency dependent, being inhibited in the regime of fast tides, when the tidal frequency ω exceeds the dominant convective turnover frequency ω_c (Zahn 1966; Goldreich & Nicholson 1977; Zahn 1989). We employ a frequency-dependent effective viscosity (ν_E ) that is compatible with the latest hydrodynamical simulations (Duguid et al. 2020a, 2020b; Vidal & Barker 2020a, 2020b), which at high frequency matches ${\nu }_{E}\propto {(\omega /{\omega }_{c})}^{-2}$, the scaling law of Goldreich & Nicholson (1977).

We calculate dynamical tide dissipation in radiation zones by assuming that internal gravity waves are launched from the radiative/convective interface and are then subsequently fully damped, following, e.g., Goodman & Dickson (1998; who applied the ideas of, e.g., Zahn 1977 to solar-type stars). This provides a simple estimate of the dissipation due to these waves, which likely corresponds with the maximum dissipation in between resonances. Eccentricity tides in solar-type binaries are thought to be strongly nonlinear near the stellar center (Goodman & Dickson 1998; Ogilvie & Lin 2007), causing wave breaking (Barker & Ogilvie 2010; Barker 2011), following which these waves are expected to be efficiently damped.

Inertial wave dissipation is computed by employing a frequency-averaged formalism that fully accounts for the realistic internal structure of the star (following Ogilvie 2013 and Barker 2020, which builds upon but is more realistic than the two-layer model considered by, e.g., Mathis 2015; Gallet et al. 2017). This represents a crude measure of the tidal dissipation due to these waves that is primarily useful for population-wide studies such as this. This mechanism operates if the tidal frequency ω satisfies ∣ω∣ < 2Ω, which is expected in spin-synchronized (or pseudo-synchronized) binaries because the relevant tidal frequencies for eccentricity tides are ω = ±Ω (or ∣ω∣ ∼ Ω). We note that the dissipation due to inertial waves has been found to be strongly frequency dependent in prior linear calculations (e.g., Savonije & Papaloizou 1997; Ogilvie & Lin 2007; Papaloizou & Ivanov 2010; Rieutord & Valdettaro 2010). The frequency-averaged dissipation ignores this complicated behavior but nevertheless provides a useful way to quantify the importance of inertial waves. It is also much simpler to calculate in stellar models.

We first introduce our results for the frequency-averaged dissipation due to inertial waves by plotting a modified tidal quality factor ⁴ for this component $Q^{\prime} \equiv \langle {Q}_{\mathrm{IW}}^{{\prime} }\rangle$, which is an inverse measure of the dissipation, as a function of stellar age in Figure 1. We have shown results for stars with masses in the range 0.2–1.6 M_⊙. The top panel indicates ${\epsilon }_{{\rm{\Omega }}}^{2}\langle {Q}_{\mathrm{IW}}^{{\prime} }\rangle$. If the rotation period P_rot of a star is known (along with its mass and radius), then this figure allows the typical level of tidal dissipation due to inertial waves to be computed. We complement this in the bottom panel, where we have shown $\langle {Q}_{\mathrm{IW}}^{{\prime} }\rangle$ by setting P_rot = 10 days for all stars.

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Figure 1 shows that inertial wave dissipation is very efficient during PMS phases and evolves to become less efficient on the MS (until the latest stages). This conclusion will be further strengthened by considering that young stars rotate more rapidly. Cooler stars are generally more dissipative than hotter stars, in that they have a smaller $\langle {Q}_{\mathrm{IW}}^{{\prime} }\rangle$ on the MS, with F stars (with masses M > 1.3M_⊙) being the least dissipative in this mass range. For stars with masses in the range 0.2–1.2 M_⊙, we typically find $\langle {Q}_{\mathrm{IW}}^{{\prime} }\rangle \approx {10}^{7}{({P}_{\mathrm{rot}}/10\,\mathrm{days})}^{2}$.

Tidal timescales strongly depend on the stellar radius, it is therefore essential to explore how this quantity varies with age. In Figure 2, we plot the evolution of the stellar radius with age, which highlights that the PMS and later evolutionary phases at the end of the MS are those in which these stars have the largest radii. Together with the results of Figure 1, we might therefore expect the PMS and the later evolutionary phases at the end of the MS to be the most important for tidal dissipation.

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We now apply the results in Figures 1 and 2 to calculate theoretically the circularization periods of binary stars due to inertial waves. We then calculate P_circ due to equilibrium tides and internal gravity waves for comparison. For a conservative estimate, we consider dissipation of the l = m = 2 (spherical harmonic degree l and azimuthal wavenumber m) tide in only the primary star of mass M that has a companion of mass M₂ = M_⊙ (for all primary stars), and we assume spin–orbit synchronization (P_rot = P_orb, the orbital period, which we will later justify for most evolutionary stages). For small e, the eccentricity evolves according to ⁵

$$\begin{eqnarray}&&\displaystyle \frac{d\mathrm{ln}e}{{dt}}=-\displaystyle \frac{225\pi }{8}\displaystyle \frac{1}{Q^{\prime} }\left(\displaystyle \frac{{M}_{2}}{M}\right){\left(\displaystyle \frac{M}{M+{M}_{2}}\right)}^{\tfrac{5}{3}}\displaystyle \frac{{P}_{\mathrm{dyn}}^{\tfrac{10}{3}}}{{P}_{\mathrm{orb}}^{\tfrac{13}{3}}},\end{eqnarray} \tag{ 1 }$$

where $Q^{\prime}$ is an appropriate tidal quality factor. This equation can be integrated from a (uncertain) starting age t₀ to a final age t, following stellar evolution, to give

$$\begin{eqnarray}&&{\rm{\Delta }}\mathrm{ln}e=\displaystyle \frac{225\pi }{8}\left(\displaystyle \frac{{M}_{2}}{M}\right){\left(\displaystyle \frac{M}{M+{M}_{2}}\right)}^{\tfrac{5}{3}}\displaystyle \frac{1}{{P}_{\mathrm{orb}}^{\tfrac{13}{3}}}{\int }_{{t}_{0}}^{t}\displaystyle \frac{{P}_{\mathrm{dyn}}^{\tfrac{10}{3}}}{Q^{\prime} }\,{dt},\end{eqnarray} \tag{ 2 }$$

where ${\rm{\Delta }}\mathrm{ln}e=\mathrm{ln}e({t}_{0})-\mathrm{ln}e(t)$. For inertial wave dissipation, assuming spin–orbit synchronization, we set $Q^{\prime} =\left[{\epsilon }_{{\rm{\Omega }}}^{2}\langle {Q}_{\mathrm{IW}}^{{\prime} }\rangle \right]{P}_{\mathrm{orb}}^{2}/{P}_{\mathrm{dyn}}^{2}$, such that

$$\begin{eqnarray*}&&{\rm{\Delta }}\mathrm{ln}e=\displaystyle \frac{225\pi }{8}\left(\displaystyle \frac{{M}_{2}}{M}\right){\left(\displaystyle \frac{M}{M+{M}_{2}}\right)}^{\tfrac{5}{3}}\displaystyle \frac{1}{{P}_{\mathrm{orb}}^{\tfrac{19}{3}}}{\int }_{{t}_{0}}^{t}\displaystyle \frac{{P}_{\mathrm{dyn}}^{\tfrac{16}{3}}}{{\epsilon }_{{\rm{\Omega }}}^{2}\langle {Q}_{\mathrm{IW}}^{{\prime} }\rangle }\,{dt}.\end{eqnarray*}$$

Note that for low-mass stars in which ${\epsilon }_{{\rm{\Omega }}}^{2}\langle {Q}_{\mathrm{IW}}^{{\prime} }\rangle$ is approximately constant as they evolve from the PMS to the end of the MS (see Figure 1), $\langle {Q}_{\mathrm{IW}}^{{\prime} }\rangle \propto 1/{R}^{3}$, indicating that ${\rm{\Delta }}\mathrm{ln}e\propto {R}^{11}$, an extremely strong function of R. This indicates that even though the PMS phase is very short, because the star then has a much larger R, it can dominate the circularization (as also hypothesized for equilibrium tides by Zahn & Bouchet 1989).

For a conservative estimate, here we assume an initial eccentricity of 1 (albeit strictly invalidating the assumption of small e) at t₀ = 0.15 Myr and a final eccentricity of 0.01 at each age t > t₀, so that ${\rm{\Delta }}\mathrm{ln}e\approx 4.6$. This equation is rearranged to obtain the critical orbital period P_circ ≡ P_orb out to which circularization is expected, assuming the required value of ${\rm{\Delta }}\mathrm{ln}e$ and computing the integral following the evolution of the star. The integrand is computed using hundreds of snapshots following stellar evolution, which are interpolated to a finer grid in time to calculate the integral numerically.

Our results for P_circ as a function of age are shown in the top panel of Figure 3 for stars with masses 0.2–1.6 M_⊙. This demonstrates that efficient dissipation of inertial waves on the PMS can circularize orbits out to 10 days prior to a few Myr. There is then negligible evolution of P_circ on the MS until a few Gyr (depending on M), after which P_circ increases again as the star evolves toward the end of its life on the MS. These theoretical predictions are in accordance with the observations compiled by Nine et al. (2020), which are indicated as the black symbols with labels on this figure. Figure 3 therefore demonstrates that inertial wave dissipation is able to explain the observed circularization periods of close binary stars. We do not need to invoke any artificial enhancement of tidal dissipation to explain these observations. Note that t₀ was chosen to explain M35 because this requires the most efficient dissipation; choosing, e.g., t₀ ≈ 0.3 Myr instead allows the PMS and Pleiades to be explained neatly with the correct stellar masses without affecting later MS stages.

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In the middle panel of Figure 3, we show the corresponding prediction for P_circ due to internal gravity-wave dissipation. This is obtained by solving Equation (2) after setting $Q^{\prime} ={Q}_{\mathrm{IGW}}^{{\prime} }$, as defined by Equation (A26), which leads to ${\rm{\Delta }}\mathrm{ln}e\propto {P}_{\mathrm{orb}}^{-7}$. This mechanism is insufficient to explain the observed P_circ until later ages on the MS, when it also predicts an increase in P_circ with age, though only for the most massive stars with radiative cores.

Finally, we show the same result for equilibrium tide dissipation in the bottom panel of Figure 3, which is obtained by solving Equation (2) with $Q^{\prime} ={Q}_{\mathrm{eq}}^{{\prime} }$, as defined by Equation (A25), which gives ${\rm{\Delta }}\mathrm{ln}e\propto {P}_{\mathrm{orb}}^{-10/3}$ in the high-frequency regime for ν_E = ν_FIT in Equation (A21) (the same conclusion is reached using ν_E based on Goldreich & Nicholson 1977). This is shown using solid lines, and a similar prediction that instead uses the (unphysical) conventionally adopted mixing-length expectation (with no frequency reduction) ν_E = u_c l_c /3, for which ${\rm{\Delta }}\mathrm{ln}e\propto {P}_{\mathrm{orb}}^{-16/3}$, is shown using dashed lines. This mechanism is clearly inefficient and is unable to circularize orbits outside 2 days prior to Gyr ages due to the reduction in the turbulent viscosity for high frequencies. This disagrees with the conclusions of Zahn & Bouchet (1989) because they adopted a less drastic frequency reduction for ν_E , which is incompatible with the latest (albeit idealized) simulations (Ogilvie & Lesur 2012; Duguid et al. 2020a, 2020b; Vidal & Barker 2020a, 2020b). On the other hand, we also show that even if there is no reduction in ν_E for high frequencies, while this mechanism is then much more efficient, it is still unable to explain the observations even during the PMS. This mechanism becomes more efficient however as the star evolves toward the end of the MS, when it can provide a comparable contribution to inertial waves (for either choice of ν_E ).

Our estimates for P_circ here are probably overly conservative estimates for several reasons: (1) We have considered tides in only the primary star (considering tides in both stars would raise P_circ due to inertial waves by approximately 2^3/19 ≈ 12%), (2) we assume a large ${\rm{\Delta }}\mathrm{ln}e$, (3) we start at t₀ = 0.15 Myr, meaning that we ignore earlier (short) PMS phases when the star had an even larger radius, (4) we ignore tidal components other than l = m = 2, and (5) we assume spin–orbit synchronization, which may overestimate the rotation periods for young stars. We have found that using an earlier t₀ < 0.1 Myr can lead to significantly larger P_circ for inertial wave dissipation, i.e., this mechanism is then too efficient to fit the observations. Our choice of t₀ = 0.15 Myr here was found to give a good fit with the observations for the stellar models considered, as we have shown in Figure 3, though values t₀ ≲ 0.8 Myr also work well (except for M35). On the other hand, the frequency-averaged measure may over- or underestimate the dissipation of inertial waves at a given tidal frequency by an uncertain factor that is hard to quantify. Further work is required to explore this matter by directly solving the linear tidal response for eccentricity tides in a wide range of stellar models.

We can crudely demonstrate that our assumption of spin–orbit synchronization above is not unreasonable by solving ⁶

$$\begin{eqnarray*}&&{\rm{\Delta }}\mathrm{ln}{P}_{\mathrm{rot}}=\displaystyle \frac{9\pi {r}_{g}^{2}}{2}{\left(\displaystyle \frac{{M}_{2}}{M+{M}_{2}}\right)}^{2}\displaystyle \frac{1}{{P}_{\mathrm{rot}}{P}_{\mathrm{orb}}^{4}}{\int }_{{t}_{0}}^{t}\displaystyle \frac{{P}_{\mathrm{dyn}}^{4}}{{\epsilon }_{{\rm{\Omega }}}^{2}\langle {Q}_{\mathrm{IW}}^{{\prime} }\rangle }\,{dt}.\end{eqnarray*}$$

We assume P_rot = 5 days, and we define P_sync ≡ P_orb by assuming a logarithmic change in the stellar rotation period ${\rm{\Delta }}\mathrm{ln}{P}_{\mathrm{rot}}=1$. The results are shown in Figure 4, which indicates the typical orbital period out to which the tidal evolution of stellar spin is efficient. Because P_circ is generally shorter than P_sync, it is reasonable to assume spin–orbit synchronization for most ages in Figure 3.

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The efficiency of inertial wave PMS circularization is further evidenced by Figure 5. Here we model the rotational evolution of our solar-mass star (ignoring tides) through the PMS to the end of the MS by following the approaches of, e.g., Amard et al. (2019) and Lazovik (2021), including star-disk locking and magnetic braking (following e.g., Matt et al. 2015). The top panel shows the evolution of P_rot as a function of age for three initial rotation periods, and the bottom panel shows the corresponding evolution of P_circ due to inertial waves by solving Equation (2) (using t₀ = 0.32 Myr) for these P_rot(t) (instead of assuming P_rot = P_orb). This can be compared with Figure 3 and indicates that efficient inertial wave PMS circularization is predicted without assuming spin–orbit synchronization. A more realistic calculation should really account for the tidal evolution of Ω along with e, accounting for multiple tidal components. For example, eccentricity can be briefly excited by tides when the star rotates sufficiently rapidly relative to the orbit (e.g., Zahn & Bouchet 1989), and stellar magnetic braking rates are probably modified by tidal evolution of the stellar rotation, which may both modify slightly our conclusions for P_circ. However, such a more detailed study is outside the scope of this Letter.

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Our results are in conflict with Zahn & Bouchet (1989), who claim that "there is little doubt that the circularization in these stars is due to the action of the equilibrium tide early on the PMS." We do agree that the PMS phase is probably the most important for tidal circularization, mainly because the stars then have much larger radii than on the MS (see also Gallet et al. 2017)—at least based on standard models for PMS evolution (e.g., Stahler & Palla 2005); the picture may be different in "cold accretion" models where the stars do not attain as large radii (e.g., Kunitomo et al. 2017), such that by understanding tidal evolution better we may even be able to constrain PMS evolution. However, while significant uncertainties remain, based on our current understanding of tidal dissipation, inertial waves are probably much more important in this problem, and they can readily explain the observed P_circ and its evolution with age.

It is interesting to briefly discuss our results in light of the recent analysis of observational data for solar-mass binaries by Zanazzi (2021), who argues against the validity of the longer P_circ from observations that we have reproduced in Figure 3. They found evidence for P_circ ≈ 3 days for eccentric binaries, approximately independent of MS age, but with a population of circular binaries out to 10 days that they suggest could bias previous studies with smaller sample sizes (e.g., Meibom & Mathieu 2005) and perhaps produce their larger inferred values of P_circ. We have shown here that inertial waves are efficient enough to be able to explain the P_circ previously reported, as we show in Figure 3. The two populations identified by Zanazzi (2021) could in principle be produced by a distribution in initial stellar rotation periods, which we have shown can potentially lead to very different P_circ in Figure 5. Because stars that have undergone efficient circularization would generally have synchronized even more rapidly (Figure 4), analyzing stellar rotation periods provides a possible way to observationally infer whether inertial waves have been responsible or whether the nearly circular orbits are produced without strong stellar tidal dissipation, e.g., by eccentricity damping during gas-disk migration, ⁷ perhaps with eccentric binaries produced by dynamical interactions. Our results for solar-mass stars are consistent with Zanazzi's (2021) observation that most tidal evolution occurs during the PMS and that there is negligible evolution on the MS until the latest evolutionary stages ≳5 Gyr.

We have demonstrated that the circularization periods of solar-type binary stars (e.g Meibom & Mathieu 2005; Triaud et al. 2017; Nine et al. 2020) can be explained by tidal dissipation due to inertial waves in convection zones. This mechanism is very efficient during the PMS; it also predicts an increase in P_circ with age for spin-synchronized stars on the MS in accordance with observations.

We have shown that this component of the dynamical tide is much more dissipative than convective damping of equilibrium tides, both during the PMS and until the latest evolutionary stages on the MS. This is primarily because the latter mechanism is strongly inhibited by the frequency reduction of the turbulent viscosity (Goldreich & Nicholson 1977). The latter result agrees with Goodman & Oh (1997) and Terquem et al. (1998) but disagrees with Zahn & Bouchet (1989), who adopted a much weaker frequency reduction that is inconsistent with the latest simulations (Ogilvie & Lesur 2012; Duguid et al.2020a, 2020b; Vidal & Barker 2020a, 2020b). We also show that internal gravity-wave dissipation is unlikely to explain the observed circularization periods, in agreement with Goodman & Dickson (1998).

Previous theoretical work studying tidal dissipation in rotating solar-type stars suggested that inertial wave dissipation could be the dominant mechanism for binary circularization (Ogilvie & Lin 2007). However, this mechanism was previously thought to be too weak to account for the observed circularization periods of solar-type binary stars on the MS (see also Barker 2020). Our new conclusion here has arisen following inertial wave dissipation along the evolution of solar-type stars. We have demonstrated that inertial wave dissipation is very efficient during PMS phases and that it evolves with MS age, such that it can readily produce the observed circularization periods of solar-type binaries. Inertial waves have the additional benefit that they do not pose a problem for the survival of the shortest-period hot Jupiters; indeed, they are not linearly excited for aligned orbits unless P_rot ≤ 2P_orb and so cannot drive orbital decay around slowly rotating stars (e.g., Ogilvie & Lin 2007).

Our idealized proof-of-concept calculations have focused on the tidal evolution of binary eccentricities by modeling the dissipation due to inertial waves using a highly simplified frequency-averaged formalism that properly accounts for the realistic internal structure of the star (following Ogilvie 2013). Future work should study the impact of this mechanism of tidal dissipation in more sophisticated calculations that also study the dynamical evolution of stellar populations (e.g., Geller et al.2013) and simultaneously model tidal dissipation with models for the evolution of stellar rotation from the PMS until the end of the MS (e.g., Gallet et al. 2017). It would also be worthwhile to explore the excitation of inertial waves in calculations that directly compute the frequency-dependent linear tidal response in a range of stellar models and to explore further the possibility of resonance locking ⁸ following stellar evolution (e.g., Witte & Savonije 2002; Zanazzi & Wu 2021).

This work was funded by STFC grants ST/R00059X/1 and ST/S000275/1. We would like to thank the reviewer, Robert Mathieu, and Gordon Ogilvie for their very helpful comments and suggestions.

We consider a single l = m = 2 tidal potential component

$$\begin{eqnarray}&&{\rm{\Psi }}({\boldsymbol{r}},t)=\mathrm{Re}\left[{{\rm{\Psi }}}_{l}(r){Y}_{l}^{m}(\theta ,\phi ){{\rm{e}}}^{-i\omega t}\right]\end{eqnarray} \tag{ A1 }$$

inside the primary star of mass M, due to another star of mass M₂ in a close binary system, where we define Ψ_l = Ar^l , where A is the tidal amplitude and (r, θ, ϕ) are the usual spherical coordinates centered on the primary (e.g., Ogilvie 2014). The linearized Eulerian gravitational potential response is

$$\begin{eqnarray}&&{\rm{\Phi }}^{\prime} ({\boldsymbol{r}},t)=\mathrm{Re}\left[{{\rm{\Phi }}}_{l}^{{\prime} }(r){Y}_{l}^{m}(\theta ,\phi ){{\rm{e}}}^{-i\omega t}\right],\end{eqnarray} \tag{ A2 }$$

with a similar expansion for other variables.

In a convective region with efficient convection such that it is adiabatically stratified, the low-frequency equilibrium tide is irrotational (Goodman & Dickson 1998; Terquem et al. 1998),

$$\begin{eqnarray}&&{{\boldsymbol{\xi }}}_{\mathrm{nw}}={\rm{\nabla }}X,\end{eqnarray} \tag{ A3 }$$

and its displacement satisfies, after expanding X in terms of spherical harmonics (Ogilvie 2013),

$$\begin{eqnarray}&&\displaystyle \frac{1}{{r}^{2}}\displaystyle \frac{d}{{dr}}\left({r}^{2}\rho \displaystyle \frac{{{dX}}_{l}}{{dr}}\right)-\displaystyle \frac{l(l+1)}{{r}^{2}}\rho {X}_{l}=-\rho \displaystyle \frac{d\rho }{{dp}}({{\rm{\Phi }}}_{l}^{{\prime} }+{{\rm{\Psi }}}_{l}),\end{eqnarray} \tag{ A4 }$$

where ρ(r) is the density and p(r) is the pressure. The boundary conditions on a convective core are

$$\begin{eqnarray}&&{\xi }_{\mathrm{nw},r}=\displaystyle \frac{{{dX}}_{l}}{{dr}}=0\quad \mathrm{at}\quad r=0,\end{eqnarray} \tag{ A5 }$$

and for all other boundaries of any convection zone,

$$\begin{eqnarray}&&{\xi }_{\mathrm{nw},r}=\displaystyle \frac{{{dX}}_{l}}{{dr}}=-\displaystyle \frac{{\rm{\Phi }}^{\prime} +{\rm{\Psi }}}{g},\end{eqnarray} \tag{ A6 }$$

which also applies at r = R for a convective envelope, where g(r) is the gravitational acceleration. The perturbation to the gravitational potential satisfies

$$\begin{eqnarray}&&\begin{array}{l}\displaystyle \frac{1}{{r}^{2}}\displaystyle \frac{d}{{dr}}\left({r}^{2}\displaystyle \frac{d{{\rm{\Phi }}}_{l}^{{\prime} }}{{dr}}\right)-\displaystyle \frac{l(l+1)}{{r}^{2}}{{\rm{\Phi }}}_{l}^{{\prime} }\\ \quad +\,4\pi G\displaystyle \frac{d\rho }{{dp}}\rho ({{\rm{\Phi }}}_{l}^{{\prime} }+{{\rm{\Psi }}}_{l})=0,\end{array}\end{eqnarray} \tag{ A7 }$$

where ${{\rm{\Phi }}}_{l}^{{\prime} }$ must satisfy the boundary conditions

$$\begin{eqnarray}&&\displaystyle \frac{d\mathrm{ln}{{\rm{\Phi }}}_{l}^{{\prime} }}{d\mathrm{ln}r}=l\quad \mathrm{at}\quad r=0,\end{eqnarray} \tag{ A8 }$$

$$\begin{eqnarray}&&\displaystyle \frac{d\mathrm{ln}{{\rm{\Phi }}}_{l}^{{\prime} }}{d\mathrm{ln}r}=-(l+1)\quad \mathrm{at}\quad r=R.\end{eqnarray} \tag{ A9 }$$

The equilibrium tide defined here differs from the conventional equilibrium tide of, e.g., Zahn (1989). The latter is strictly invalid in large parts of convection zones, and we find that it also typically overpredicts the dissipation by a factor of 2–3 (Barker 2020). Our main conclusions would however be unchanged if we were to adopt the conventional equilibrium tide here instead.

Following Ogilvie (2013), the pressure perturbation of inertial waves (proportional to ${W}_{l}{Y}_{l}^{m}$, but obtained using an impulsive formalism) satisfies

$$\begin{eqnarray}&&\displaystyle \frac{1}{{r}^{2}}\displaystyle \frac{d}{{dr}}\left({r}^{2}\rho \displaystyle \frac{{{dW}}_{l}}{{dr}}\right)-\displaystyle \frac{l(l+1)}{{r}^{2}}\rho {W}_{l}=\displaystyle \frac{2{im}{\rm{\Omega }}}{r}\displaystyle \frac{d\rho }{{dr}}{X}_{l},\end{eqnarray} \tag{ A10 }$$

subject to the boundary conditions of vanishing radial velocity at the boundaries of each convection zone, i.e.,

$$\begin{eqnarray}&&\displaystyle \frac{{{dW}}_{l}}{{dr}}=\displaystyle \frac{2{im}{\rm{\Omega }}{X}_{l}}{r}.\end{eqnarray} \tag{ A11 }$$

The components of the flow are

$$\begin{eqnarray}&&{a}_{l}=\displaystyle \frac{2{im}{\rm{\Omega }}}{r}{X}_{l}-\displaystyle \frac{{{dW}}_{l}}{{dr}},\end{eqnarray} \tag{ A12 }$$

$$\begin{eqnarray}&&{b}_{l}=\displaystyle \frac{2{im}{\rm{\Omega }}}{l(l+1){r}^{2}}\left(r\displaystyle \frac{{{dX}}_{l}}{{dr}}+{X}_{l}\right)-\displaystyle \frac{{W}_{l}}{{r}^{2}},\end{eqnarray} \tag{ A13 }$$

$$\begin{eqnarray}&&{c}_{l-1}=\displaystyle \frac{2{\rm{\Omega }}{q}_{l}}{{r}^{2}}\left(r\displaystyle \frac{{{dX}}_{l}}{{dr}}+(l+1){X}_{l}\right),\end{eqnarray} \tag{ A14 }$$

$$\begin{eqnarray}&&{c}_{l+1}=-\displaystyle \frac{2{\rm{\Omega }}{q}_{l+1}}{{r}^{2}}\left(r\displaystyle \frac{{{dX}}_{l}}{{dr}}-{{lX}}_{l}\right),\end{eqnarray} \tag{ A15 }$$

where

$$\begin{eqnarray}&&{q}_{l}=\displaystyle \frac{1}{l}{\left(\displaystyle \frac{{l}^{2}-{m}^{2}}{4{l}^{2}-1}\right)}^{\tfrac{1}{2}}.\end{eqnarray} \tag{ A16 }$$

The associated tidal quality factor representing the frequency-averaged dissipation of inertial waves is then

$$\begin{eqnarray}&&\displaystyle \frac{1}{\langle {Q}_{\mathrm{IW}}^{{\prime} }\rangle }=\displaystyle \frac{32{\pi }^{2}G}{3(2l+1){R}^{2l+1}| A{| }^{2}}({E}_{l}+{E}_{l-1}+{E}_{l+1}),\end{eqnarray} \tag{ A17 }$$

where

$$\begin{eqnarray}&&{E}_{l}=\displaystyle \frac{1}{4}\int \rho {r}^{2}\left(| {a}_{l}{| }^{2}+l(l+1){r}^{2}| {b}_{l}{| }^{2}\right){dr},\end{eqnarray} \tag{ A18 }$$

$$\begin{eqnarray}&&{E}_{l-1}=\displaystyle \frac{1}{4}\int \rho {r}^{2}l(l-1){r}^{2}| {c}_{l-1}{| }^{2}{dr},\end{eqnarray} \tag{ A19 }$$

$$\begin{eqnarray}&&{E}_{l+1}=\displaystyle \frac{1}{4}\int \rho {r}^{2}l(l+2){r}^{2}| {c}_{l+1}{| }^{2}{dr}.\end{eqnarray} \tag{ A20 }$$

Note that $\langle {Q}_{\mathrm{IW}}^{{\prime} }\rangle \propto {{\rm{\Omega }}}^{-2}$, so that relatively rapidly rotating stars are more dissipative. In a given stellar model, we solve Equations (A4), (A7), and (A10) numerically using a Chebyshev collocation method with a sufficiently large number of points to ensure accurate solutions. The integrals required to compute Equation (A17) are then straightforward to compute numerically. Note that we fully account for the continuous interior profiles of ρ(r) and p(r); we do not assume a piecewise homogeneous two-layer model for the star, unlike, e.g., Mathis (2015) and many others.

To compute equilibrium tide dissipation in convection zones, we assume that at each radius in the star, turbulent convection acts like an isotropic effective viscosity ν_E (r). The latest simulations of Duguid et al. (2020b) suggest that we consider the piecewise power-law fit:

$$\begin{eqnarray}{\nu }_{\mathrm{FIT}}={u}_{c}{l}_{c}\left\{\begin{array}{ll}5\quad & \Space{0ex}{0.25ex}{0ex}(\displaystyle \frac{| \omega | }{{\omega }_{c}}\lt {10}^{-2}\Space{0ex}{0.25ex}{0ex}),\\ \displaystyle \frac{1}{2}{\left(\displaystyle \frac{{\omega }_{c}}{| \omega | }\right)}^{\tfrac{1}{2}}\quad & \Space{0ex}{0.25ex}{0ex}(\displaystyle \frac{| \omega | }{{\omega }_{c}}\in [{10}^{-2},5]\Space{0ex}{0.25ex}{0ex}),\\ \displaystyle \frac{25}{\sqrt{20}}{\left(\displaystyle \frac{{\omega }_{c}}{| \omega | }\right)}^{2}\quad & \Space{0ex}{0.25ex}{0ex}(\displaystyle \frac{| \omega | }{{\omega }_{c}}\gt 5\Space{0ex}{0.25ex}{0ex}),\end{array}\right.\end{eqnarray} \tag{ A21 }$$

where u_c (r) and l_c (r) are the convective velocity and mixing length based on mixing-length theory MLT (computed in MESA; see Appendix B), ω_c (r) = u_c /l_c is the convective frequency, and we set ν_E = ν_FIT. Our fit here is based on the maximum value for ν_E at high frequencies, which provides the maximum dissipation due to this mechanism. This appears to hold for a wide range of Rayleigh numbers in a local model of convection. The resulting dynamic shear viscosity is μ(r) =ρ(r)ν_E (r), so that the viscous dissipation is

$$\begin{eqnarray}&&{D}_{\nu }=\displaystyle \frac{1}{2}{\omega }^{2}\int {r}^{2}\mu (r){D}_{l}(r)\,{dr},\end{eqnarray} \tag{ A22 }$$

where the integral is carried out numerically over the entire radial extent of each convection zone, and

$$\begin{eqnarray}\begin{array}{rcl}{D}_{l}(r) & = & 3{\left|\displaystyle \frac{d{\xi }_{r}}{{dr}}-\displaystyle \frac{{{\rm{\Delta }}}_{l}}{3}\right|}^{2}+l(l+1){\left|\displaystyle \frac{{\xi }_{r}}{r}+r\displaystyle \frac{d}{{dr}}\left(\displaystyle \frac{{\xi }_{h}}{r}\right)\right|}^{2}\\ & & +(l-1)l(l+1)(l+2){\left|\displaystyle \frac{{\xi }_{h}}{r}\right|}^{2},\end{array}\end{eqnarray} \tag{ A23 }$$

$$\begin{eqnarray}&&{{\rm{\Delta }}}_{l}=\displaystyle \frac{1}{{r}^{2}}\displaystyle \frac{d}{{dr}}\left({r}^{2}{\xi }_{r}\right)-l(l+1)\displaystyle \frac{{\xi }_{h}}{r},\end{eqnarray} \tag{ A24 }$$

where ξ _nw = ξ_r e _r + ξ _h is appropriately expanded in terms of ${Y}_{l}^{m}{{\rm{e}}}^{-i\omega t}$. The associated tidal quality factor is

$$\begin{eqnarray}&&\displaystyle \frac{1}{{Q}_{\mathrm{eq}}^{{\prime} }}=\displaystyle \frac{16\pi G}{3(2l+1){R}^{2l+1}| A{| }^{2}}\displaystyle \frac{{D}_{\nu }}{| \omega | }.\end{eqnarray} \tag{ A25 }$$

We compute internal gravity-wave dissipation in the simplest manner by neglecting rotation (which would otherwise involve more complicated numerical calculations) and by assuming that these waves are launched from the convective/radiative interface and are then fully damped in the radiation zone, following, e.g., Goodman & Dickson (1998). This gives

$$\begin{eqnarray}&&\displaystyle \frac{1}{{Q}_{\mathrm{IGW}}^{{\prime} }}=\displaystyle \frac{2{\left[{\rm{\Gamma }}\left(\tfrac{1}{3}\right)\right]}^{2}}{{3}^{\tfrac{1}{3}}{(2l+1)(l(l+1))}^{\tfrac{4}{3}}}\displaystyle \frac{R}{{{GM}}^{2}}{ \mathcal G }| \omega {| }^{\tfrac{8}{3}},\end{eqnarray} \tag{ A26 }$$

where the quantities that depend on the radiative/convective interface at r = r_c in a particular stellar model are encapsulated in the quantity

$$\begin{eqnarray}&&{ \mathcal G }={\sigma }_{c}^{2}{\rho }_{c}{r}_{c}^{5}{\left|\displaystyle \frac{{{dN}}^{2}}{d\mathrm{ln}r}\right|}_{r={r}_{c}}^{-\tfrac{1}{3}},\end{eqnarray} \tag{ A27 }$$

which takes the value ${{ \mathcal G }}_{\odot }\approx 2\times {10}^{47}\mathrm{kg}\,{{\rm{m}}}^{2}\,{{\rm{s}}}^{2/3}$ for the current Sun (in which σ_c = −1.18), after fitting the local profile of N²(r). In this expression, ρ_c = ρ(r_c ), and the parameter

$$\begin{eqnarray}&&{\sigma }_{c}=\displaystyle \frac{{\omega }_{\mathrm{dyn}}^{2}}{A}{\left.\displaystyle \frac{\partial {\xi }_{d,r}}{\partial r}\right|}_{r={r}_{c}},\end{eqnarray} \tag{ A28 }$$

where the derivative of the dynamical tide radial displacement ξ_d,r is determined by integrating the linear differential equation given in Equation (3) in Goodman & Dickson (1998).

We use MESA version 12778 (e.g., Paxton et al. 2011, 2015, 2019). The inlist file that we use is given below. We alter initial_mass as required to generate a given stellar model, and the code is stopped manually at a chosen time, usually when the star has left the MS.