We briefly describe the general physical picture of tidally excited waves in early-type stars. Early seminal studies used asymptotic approximations of gravity waves (Zahn, 1975, 1977; Goldreich and Nicholson, 1989), and it was extended to include the effect of rotation (Mathis, 2009). Later numerical calculations include the effect of non-adiabaticity and rotation (Savonije et al., 1995; Papaloizou and Savonije, 1997; Savonije and Papaloizou, 1997). Dedicated calculations (Witte and Savonije, 1999a,b) on massive stars studied the binary evolution and the intricate effects such as resonance locking. Other studies implemented the mode decomposition approach (Alexander, 1987; Lai et al., 1993; Lai, 1997; Schenk et al., 2002; Fuller, 2017).

Intermediate and massive stars possess a convective core and radiative envelope. In binaries containing these stars, internal gravity waves (IGW) are generated by the tidal potential (also by the convective motion in the core) at the radiative-convective boundary and propagate outward (Goldreich and Nicholson, 1989; Lecoanet and Quataert, 2013; Rogers et al., 2013; Edelmann et al., 2019; Lecoanet et al., 2019; Horst et al., 2020). They suffer from linear damping due to radiative diffusion (Press, 1981; Garcia Lopez and Spruit, 1991; Zahn et al., 1997). The low-frequency, short-wavelength waves are damped strongly and behave like traveling waves (Ratnasingam et al., 2019). Higher-frequency waves can be reflected at the outer turning points and interfere constructively to form global normal modes (Prat et al., 2016).

Most of the observed prominent TEOs in HBs are standing waves, suffering from less damping. When the TEO amplitudes surpass the parametric instability threshold, the TEOs begin to suffer from non-linear mode coupling and transfer energy to daughter modes or multiple pairs of daughter modes (Weinberg et al., 2012; Yu et al., 2020). The daughter modes may again become unstable and couple with grand-daughter modes. In general, a mode coupling network can be formed. Observationally, this can be seen as mode triplets or multiplets satisfying the resonance conditions. This weakly-nonlinear regime will be discussed in the next section.

The amplitudes of the tidally excited gravity waves, when propagating to the near-surface layers with smaller densities, increase significantly. If the waves become significantly non-linear (the multiplication of the radial wavenumber and radial displacement k_rξ_r ≳ 1), they overturn the stratification and break (Su et al., 2020). Thus, they deposit their energy (tidal heating) and angular momentum (tidal synchronization), and turn into small-scale turbulence. Thus, the surface layers are synchronized first and a differential rotation profile may be produced (Goldreich and Nicholson, 1989), although the hydromagnetic effects tend to smooth out differential rotation (Rüdiger et al., 2015; Townsend et al., 2018). Critical layers where the Doppler-shifted wave frequency approaches zero, may form and move inward (Alvan et al., 2013). The subsequent gravity waves cannot pass the critical layer and waves dissipate strongly. Observationally, single upper main-sequence stars tend to have a nearly uniform rotation profile in the radiative envelope, inferred from asteroseismology (Bowman, 2020; Aerts, 2021)³. The g-mode pulsating γ Dor stars (spectral type A-F-) in close binaries with P_orb ≤ 10 d show a convective-core-boundary rotation period that is similar to the orbital period, suggesting that the tidal synchronization has already reached the deep interior (Guo et al., 2019; Li et al., 2020a; Saio, 2020). Nevertheless, Kallinger et al. (2017) found a Slowly Pulsating B-star in a triple system that appears to show a faster-rotating surface layer, which may fall into the Goldreich and Nicholson's outside-in synchronization scenario.

Since the TEOs are direct manifestation of dynamical tides, they are crucial for our understanding of the above physical processes. First, we show some general properties of the TEOs in heartbeat stars.

The overall strength of the tidal response of star 1 due to star 2 is determined by the tidal parameter ϵ_l:

where D_peri = a(1 − e). Since R1Dperi≪1, it is usually sufficient to consider the dominant l = 2 component.

Thus, to have a larger tidal amplitude, one could (1) make the mass ratio M₂/M₁⁴ larger (i.e., close to 1.0); (2) make the stellar radius R₁ bigger; (3) have a smaller periastron distance D_peri. And indeed, observationally: (1) many heartbeat stars have a mass ratio close to unity⁵; (2) lots of heartbeat stars with tidally excited oscillations are slightly evolved main-sequence stars (e.g., KIC 4142768 has a primary star with M = 2.05M_⊙, R = 2.96R_⊙, Guo et al., 2019); (3) heartbeat binaries have a high eccentricity (≈ 0.2−0.9) and short periastron distance. The D_peri of 19 HBs in Shporer et al. (2016) ranges from 0.05 to 0.1AU, and the corresponding tidal parameter ϵ₂ values are ≈ 10⁻³.

The observed TEOs correspond to the frequencies of stellar g-modes with radial orders from ≈ 10 to a few tens. For example, the primary star in KIC 3230227 (M = 1.84M_⊙, R = 2.01R_⊙, Guo et al., 2017a) shows orbital-harmonic oscillations corresponding to l = 2, m = 2 g modes, with radial order n_g ~ 10−30; KOI 54 (M = 2.05M_⊙, R = 2.33R_⊙, O'Leary and Burkart, 2014) shows TEOs that mostly have l = 2, m = 0, corresponding to radial order n_g ~ 10−50. The slightly evolved primary in KIC4142768 (Guo et al., 2017a) shows TEOs that are in agreement with n_g ≈ 30 − 70 g modes.

In Figure 2, we stack the observed TEOs in 22 heartbeat binaries together, with decreasing orbital eccentricities from the top to the bottom. These Fourier spectra show that the TEOs generally have oscillation frequencies <~5 d⁻¹. They mostly correspond to orbital harmonics N from 4 to 40, although in some special cases the N can reach much larger values (N ~ 300 in KIC 8164262). The TEO amplitudes can be as large as > 10 milli-mag although the majority are lower than 0.5 milli-mag (right panel in Figure 2).

In general, most of the observed TEOs are likely (linearly) excited by the dynamical tide (exact orbital-harmonic frequencies (Nf_orb, see next section for nonlinear non-harmonic TEOs), with the stellar response dominated by the closest frequency g-mode. But which orbital harmonics N are favorably excited? Following Burkart et al. (2012), the favorable range of orbital harmonics depends essentially on the multiplication of Q_nl and X_lm (see immediately later in this paragraph for the definitions). First, not all g-modes couple with the tidal potential equally. The weights are described by the tidal overlap integral Q_nl, which peaks around the dynamical frequency of the star. Q_nl decreases toward lower frequencies since higher order g-modes have shorter wavelength and cannot couple well spatially with the tidal potential. Secondly, stars in eccentric orbits experience a series of forcing frequencies (Nf_orb, with |N| < ∞) which are weighted by the eccentricity-dependent Hansen coefficient X_lm. X_{l = 2, m = 2} peaks at the periastron-passage frequency and decreases toward larger N; X_{l = 2, m = 0} monotonically decreases as N increases (Willems, 2003, Figures 1, 2; Fuller, 2017, Figure 3). Thus, the favored range of orbital harmonics N is between the peaks of Q_nl and X_lm (Burkart et al., 2012, Figures 2, 3).

In Figure 3, we show the observed TEO amplitudes (in magnitude variation Δmag or luminosity variation ΔL/L, related by ΔL/L ≈ 1.086Δmag) as a function of the orbital harmonic number (N) in four HBs KIC4142768, KIC3230227, ι Ori and KOI-54 (gray squares, circles, red squares, and red crosses, respectively). These observed TEOs should be compared with the theoretical amplitudes for m = 2, m = 2, m = 2, m = 0 modes (blue diamonds, open diamonds, blue circles, and open triangles, respectively, same ordering as above). It can be seen that the observed TEO range (between the two vertical lines) matches well with the theoretical expectations (the “bump” formed by background symbols. For ι Ori, the theoretical TEO amplitudes of m = 0 modes (cyan open circles) are below the detection limit and much lower than observed TEOs (m = 2). For KOI-54, the Δmag from the temperature effect (Δmag_T; upper) and geometrical effect (Δmag_G; lower) are distinguished. Note that the observed magnitude variations are primarily due to the temperature perturbations (slightly overestimated in Fuller and Lai, 2012).

However, to model the TEO amplitudes individually, one needs to consider the Lorentzian term Δ_nlmN (a term describing the resonances, see Equation 13, Burkart et al., 2012), which depends sensitively on the frequency detuning, i.e., the closeness of a certain forcing frequency (Nf_orb) to the nearest eigenmode frequency. Unfortunately, even a change of 0.001M_⊙ in stellar models can significantly change the detuning parameter, thus the Lorentzian term. A better way is to treat the detuning parameter as a random variable, which is uniformly distributed between its minimum value (= 0, perfect resonance) and maximum values (half of the adjacent g-mode spacing). In this way, a credible interval can be calculated for the Lorentzian term and thus the observed TEO amplitude (Fuller, 2017). For example, the 95% credible interval (±2σ) of theoretical TEO amplitude for KIC 4142768 is shown as the shaded region in the upper left panel of Figure 3.

TEO phases, measured with respect to the periastron, deserve a particular discussion. We expect most observed TEOs are standing waves and nearly adiabatic, and their phases are close to the adiabatic expectations which are essentially only a function of ω (argument of periastron) and m (Burkart et al., 2012; Guo et al., 2020). In Figure 4, we show the observed TEO phases (symbols) and the theoretical adiabatic phases (vertical lines) for five HBs. As expected, low inclination HBs tend to show m = 0 modes (top two systems), and intermediate/high inclination HBs usually present both m = 0 and m = 2 modes. The low-frequency TEOs experience more radiative damping, and they can be distinguished by their relatively large phase offset from adiabatic phases (O'Leary and Burkart, 2014, Figure 4; Guo et al., 2019, Figure 7). Weakly non-linear TEOs that experience non-linear mode coupling also show deviations from the adiabatic phases. It is possible that TEOs locked in resonance with the orbit still have relatively large frequency detuning compared with the mode damping rate, and thus they do not show arbitrary phases as in the perfect-resonance case.

To summarize, the tidal response to a forcing frequency (Nf_orb) is a summation of the mode eigenfunctions weighted by the mode amplitude A_nlmN ∝ ϵ_lQ_nlX_lmΔ_nlmN (Burkart et al., 2012; Fuller, 2017). Roughly speaking, ϵ_l determines the overall strength, Q_nlX_lm controls the range of excited orbital harmonics N, and Δ_nlmN sets the detailed amplitude of each TEO. TEO phases are primarily determined by the orbital orientation. Most observed TEOs in HBs are chance resonances (i.e., random frequency detuning) with g-modes and can be modeled by the above theoretical framework. In fact, the aforementioned statistical approach is good for finding TEOs larger than expectation. These TEOs may be locked in resonance with the orbit and require a different modeling approach (see section 2.3).

Modes near resonances can non-linearly interact and observationally, this can generate combination frequencies in the form of mf_a ± nf_b, with m and n being integers. Resonance mode couplings have been observed and studied in free oscillations for B-type pulsators (Degroote et al., 2009), δ Scuti pulsators (Breger and Montgomery, 2014; Bowman et al., 2016) as well as compact pulsators (Zong et al., 2016a,b) and other types of variables. Theoretical studies include, e.g., Dziembowski and Krolikowska (1985), Van Hoolst (1994), and Buchler et al. (1997).

In the context of tidal oscillations, a striking feature in the observed TEOs is that some of them are not orbital harmonics. And the anharmonic frequencies can pair up and sum to an orbital harmonic (f_a + f_b ≈ Nf_orb). This can be explained by the non-linear resonance mode coupling.

When the tidally excited oscillations surpass the linear regime and become weakly non-linear, the oscillation mode can suffer from parametric instability (Schenk et al., 2002; Arras et al., 2003), which is the leading-order non-linear effect (Weinberg et al., 2012). In the dominant three-mode resonance coupling scenario (f_A = f_a + f_b), the parent mode A is resonantly excited by a linear dynamical tide (an orbital harmonic Nf_orb); and when its amplitude exceeds the three-mode instability threshold, it can transfer energy to two daughter modes (a, b). In Figure 5, we show different mode coupling patterns. The upper left is the aforementioned basic three-mode coupling. The mode-coupling triplet (A, a, b) can also couple to other mode triplets, either by sharing the parent (lower left) or sharing a daughter mode (upper middle panel). Multiple mode-triplets can also couple together (lower middle panel). In KIC 3230227, we find that two mode-coupling triplets share one daughter mode. And the resonance conditions for the two mode triplets are labeled in Figure 5: f_A(22) = f_a(9.88) + f_b(12.12) and f_B(26) = f_d(13.88) + f_b(12.12), where all frequencies are in units of the orbital frequency. In KOI-54, it is found that the 91st orbital harmonic resonantly excites an eigenmode very close to it, and this parent mode has at least four pairs of daughter modes. Some of the pairs even share daughters (O'Leary and Burkart, 2014). Furthermore, third-order mode coupling is also evidenced (the rightmost panel in Figure 5).

Observationally, these non-linear coupled TEOs can offer us lots of information beyond the linear theory. Firstly, the selection rules (relating the modes' l and m, Dziembowski, 1982) in mode coupling can help to identify the daughter modes (O'Leary and Burkart, 2014; Guo, 2020). If we have a large number of these mode triplets, since the linearly driven parent modes can be identified from phases, we may be able to discern the g-mode period spacings among the daughter modes and possibly among the close-to-resonance parent modes. The asymptotic period spacing pattern has been routinely found in self-excited g-modes in γ Dor stars and SPB stars, but not in tidally excited modes. Secondly, parent modes that suffer from mode-coupling instability should be close to the eigenmode frequency (although they do not necessarily have a larger amplitude than the daughter modes). Together with the anharmonic daughter modes, these mode frequencies provide a list of eigenmodes that can be compared with stellar models. This kind of tidal asteroseismology can thus be performed. After the preliminary effort by Burkart et al. (2012), we are still waiting for its first concrete application in a real star. Thirdly, the observed parent mode amplitude can be compared to the mode-coupling threshold. This helps to determine the nature of coupling. For KOI-54, O'Leary and Burkart (2014) showed that five-mode coupling can decrease the threshold amplitude of parametric instability (i.e., smaller than three-mode coupling threshold). This explains the observed parent-mode amplitude being much smaller than the three-mode instability threshold. In fact, Weinberg et al. (2012) showed that, if the parent mode couples to N daughter-mode pairs, the threshold amplitude can decrease by a factor of N. Fourthly, the mode coupling systems can have different behavior (Wersinger et al., 1980; Wu and Goldreich, 2001), depending on the frequency detuning (the difference between the parent-mode frequency f_A and the resonant linear tide at Nf_orb ≈ f_A, and also between the parent-mode frequency and the daughter-mode frequency sum f_a + f_b) and the daughter-mode damping rates. Guo (2020) showed that the observed stable amplitudes/phases of the parent and daughter modes in KIC3230227 indicate the five-mode-coupling system has settled into an equilibrium state, and this agrees with the theoretical mode damping rates. The mode-coupling systems can show limit cycles (Moskalik, 1985) and observables such as cycle period can help to constrain mode parameters (e.g., damping rates). In ZZ Ceti (DAV) type pulsating white dwarfs, it is in fact limit cycles that explain the observed outbursts in the light curves (Luan and Goldreich, 2018). Non-linear mode coupling is believed to be the dominant amplitude limitation mechanism in many types of pulsators such as δ Scuti (Dziembowski and Krolikowska, 1985; Dziembowski et al., 1988) and SPB stars (Lee, 2012). The same limitation mechanism applies to the tidally excited g modes. Unfortunately, relevant studies in this direction are rare.

When the evolution of a mode frequency is in pace with that of the forcing frequency (i.e., their time derivatives are the same), the oscillation mode can be locked into resonance with the orbit. This resonance locking phenomenon can have significant consequences on the orbital evolution of not only stellar binaries, but also satellites of gaseous giant planets (Fuller et al., 2016; Lainey et al., 2020). In the context of heartbeat stars, it can have several observational implications. Firstly, the mode in resonance locking has a larger-than-expected amplitude (i.e., compared to the amplitude from the aforementioned statistical approach). This has been demonstrated for the dominant oscillation mode in the primary star of KIC 8164262 (Fuller et al., 2017; Hambleton et al., 2018). The mode in resonance locking has an orbital harmonic of N = 229 and is likely an m = 1 mode in this misaligned binary. Cheng et al. (2020) found that the oscillation at 53rd orbital harmonic is a possible candidate for resonance locking in KIC 11494130. Resonance locking can significantly enhance the tidal dissipation and orbital evolution (e.g., see Figure 3 in Witte and Savonije, 1999b for a 10M_⊙ + 1.4M_⊙ binary). For KIC 8164262, Fuller et al. (2017) estimated that the tidal quality factor Q′ is reduced from ≈ 2 × 10⁷ to ≈ 5 × 10⁴ due to the mode in resonance locking. Secondly, it remains to be seen observationally whether the TEOs in resonance locking suffer more from the (weakly) non-linear mode coupling and strong non-linear damping. In fact, the dominant TEO in KOI-54 at 91 times of orbital frequency has multiple daughter pairs, although the resonance-locking nature of this l = 2, m = 0 mode is still under debate. An examination of the resonantly-locked TEO amplitude and phase is desirable. Resonance locking depends on the evolutionary speed of the oscillation mode due to stellar evolution, and thus it is sensitive to the stellar age. A study of the resonance-locking condition spanning from ZAMS to TAMS, and for different orbital parameters would be very useful. The orbital harmonic N of a resonantly locked mode as a limited range and a predictable amplitude, allow for inference about whether resonance locking is occurring (Fuller and Lai, 2012; Burkart et al., 2014; Fuller, 2017).

In the strong-tide regime when a star fills its Roche lobe, tides induce mass transfer. Depending on the adjustment of the stellar radius and the Roche-lobe radius, the mass transfer can be in the form of stable or unstable Roche-lobe overflow (RLOF) (Vanbeveren and De Loore, 1994; Soberman et al., 1997). Mass transfer affects the evolution of the binary orbit (Dosopoulou and Kalogera, 2016a,b). But the asteroseismic consequences of accretion have not been studied. How does the mixing process modify the excitation of heat-driven pulsations? The κ-mechanism excitation occurs at the near-surface layer where the opacity due to the hydrogen/helium or iron-group elements ionization zones have a local maximum (Unno et al., 1989, chapter 5). In addition, another necessary condition for the driving is that the local thermal timescale has to be comparable to the oscillation period (Pamyatnykh, 1999). Even without material mixing, the accretion may drive the star out of thermal equilibrium, and this may also change the geometric depth of the ionization zone and thus the pulsation excitation and frequencies. If mixing (e.g., thermohaline mixing when a negative chemical composition gradient is present) does happen (Stancliffe and Glebbeek, 2008), the change of composition also needs to be taken into account.

Observationally, many mass-accreting stars do show pulsations. The Oscillating Algol (oEA) systems are a class of δ Scuti/γ Dor pulsators in Algol-type binaries with mass accretion (Mkrtichian et al., 2004, 2020). The companion is usually a low-mass star filling or nearly-filling its Roche lobe, depending on whether the mass transfer process is finished or not. To name a few, AS Eri (Mkrtichian et al., 2004), KIC 4739791 (Lee et al., 2016), KIC 8553788 (Liakos, 2018), V392 Orionis (Hong et al., 2019), KIC 10736223 (Cheng et al., 2020). Guo and Li (2019) found the period spacing pattern of dipole g modes in mass-accreting γ Dor star in KIC 9592855. A comparison with theoretical g-mode period spacings suggests that the mass of this primary star is lower than previously reported. Streamer et al. (2018) did detailed binary star evolution modeling and calculate the pulsation properties of the mass-accreting δ Scuti primary (≈ 2.2M_⊙) in TT Hor. They managed to match the observed oscillation frequencies and found a likely evolutionary history for the δ Scuti pulsator. It was initially a 1.3M_⊙ star and accreted about 0.9M_⊙ from the companion. Accretion-driven variability in oEA binaries has been studied (Mkrtichian et al., 2018) and the Fourier spectrum of accreting δ Scuti pulsators can change during the outburst. Although some of these variabilities may be attributed to the variation of mass transfer rate, the pulsation changes also contribute to the variability. A significant number of Agol-type eclipsing binaries have δ Scuti pulsating components and it seems that many of them show very high-frequency p-modes (~60 d⁻¹), which is a signature of youth and probably the result of rejuvenation from RLOF (Dray and Tout, 2007).

Previous works on how mass-transfer modifies stellar oscillations are scarce. Note that numerous oscillations of mass-accreting white dwarfs (WD) in Cataclysmic Variables (CVs) have been discovered (Mukadam et al., 2007, 2011). Arras et al. (2006) studied the g-mode pulsational instabilities of accreting WDs in CVs. He found that an envelope of solar-like composition (accreted material) on top of the pure-hydrogen layer of the WD can change the edge of the instability strip significantly. During the accretion, outbursts can heat the WD, bringing it out of the instability strip. Similar work to the pulsational instability of other types of opacity-driven pulsators would be interesting.

Post-mass transfer binaries with a δ Scuti pulsating component have been discovered, including KIC 10661783 (Southworth et al., 2011; Lehmann et al., 2013), KIC 8262223 (Guo et al., 2017c), and the aforementioned TIC 63328020 (Rappaport et al., 2021). The previous-mass-gainer δ Scuti star in KIC 8262223 pulsates at about 60 d⁻¹, suggesting that the system only just finished the mass transfer, i.e., the effect of rejuvenation is still present. In contrast, KIC 10661783 pulsates at much lower frequencies (15 d⁻¹). This is likely due to the fact that it has finished the mass transfer long ago and the δ Scuti pulsator is already evolved. Similarly, post-mass transfer γ Dor and SPB-type pulsating binaries have also been identified (Matson et al., 2015; Guo et al., 2017b). It is quite clear that δ Scuti and γ Dor type pulsations are not suppressed in close binaries. This is in contrast with red giants in binaries, in which solar-like oscillations seem to be suppressed by binarity, probably due to the enhanced magnetic activity. The majority of the aforementioned binary can be formed in the formation channel of EL CVn binaries⁸ (Maxted et al., 2013, 2014). The formation involves the evolution of two low-mass stars with stable RLOF and mass reversal (Chen et al., 2017). In Figure 6, we show typical evolutionary tracks of an EL CVn type binary. Staring with two low-mass stars (M₁ = 1.35, M₂ = 1.15M_⊙), the mass gainer can evolve to a δ Scuti/γ Dor pulsator (M = 1.72M_⊙) or even an SPB star with slightly changed initial conditions. The mass donor can become a pre-ELM WD (extremely low-mass white dwarf precursor) pulsator (M ≲ 0.2M_⊙) with p, or g-mode pulsations (Maxted et al., 2013; Gianninas et al., 2016; Istrate et al., 2016), and possibly later on the WD cooling track, become a g-mode pulsating helium WD. These five types of post-mass transfer pulsators have been marked in Figure 6 as ellipses.

Post-mass transfer RR Lyrae and Cepheid pulsators have been found by Pietrzyński et al. (2012) and Pilecki et al. (2017), respectively. Gautschy and Saio (2017) studied the binary evolution channel to form anomalous Cepheids via RLOF and merger-like evolution. Similar work by Karczmarek et al. (2017) found stars crossing the classical instability strip of RR Lyrae and Cepheids via the binary channel. The formation of recently-discovered Blue Large-Amplitude Pulsators (BLAP) also involves binary evolution with mass transfer (Pietrukowicz et al., 2017). Recently, Byrne and Jeffery (2020) studied the non-adiabatic pulsational properties by using the post-mass-transfer stellar models. We also know sub-dwarf B-stars (sdB) and blue stragglers can be formed by mass-transfer or merger (Han et al., 2002, 2003). The list of post-mass transfer pulsators can go on and on. Binary channels can generate all kinds of exotic binary systems (de Loore and Doom, 1992; Hurley et al., 2002; Eggleton, 2006). It can also generate new types of pulsating stars and also contaminate the existing pulsators (Jeffery and Saio, 2016). Pulsational analysis of post-mass transfer systems is still the frontier of asteroseismology, and it holds great promise to improve our understanding on stellar structure and evolution.