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Tidally distorted stars are triaxial pulsators

T0 review · 3 major / 6 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read Tidal pull makes pulsating stars ring along three axes.

desk verdict The triaxial-pulsator picture is likely right and worth building on, but the paper illustrates it at P_orb = 1 d where its own isolated-multiplet assumption breaks down. read the letter →

arxiv 2411.09743 v1 pith:J7EPHO6C submitted 2024-11-14 astro-ph.SR

Jim Fuller , Saul Rappaport , Rahul Jayaraman , Don Kurtz , Gerald Handler This is my paper · ORCID
classification astro-ph.SR
keywords tidallytiltedpulsationstriaxialpulsatorsasteroseismologyclosebinariesdipolemodestidalcouplingamplitudemodulationdeltaScutistars
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper argues that stars in close, tidally synchronized binaries do not pulsate in the usual spherical-harmonic patterns tied to the spin axis. Instead, the combined tidal and centrifugal distortion makes the star triaxial, and for dipole pressure modes the tidal force couples the $m=+1$ and $m=-1$ members of each multiplet so strongly that the modes become three standing oscillations aligned with the three principal axes of the ellipsoid. In the observer's frame, the two modes lying in the orbital plane are seen at a changing angle and their amplitudes peak twice per orbit, so each appears in the power spectrum as an equal-amplitude doublet spaced by exactly twice the orbital frequency, while the third mode, along the spin axis, is a singlet. If correct, this gives a clean observational signature and a way to identify mode geometries in close binaries, turning a complication into a tool for asteroseismology, and it predicts that gravity modes remain aligned with the spin axis rather than being tidally tilted.

What carries the argument

The central object is the $3\times 3$ (and analogous $5\times 5$) matrix eigenvalue problem for the modes of a multiplet, equation 23, whose entries combine the Coriolis self-terms, centrifugal self-coupling, and tidal self- and cross-coupling terms built from overlap integrals $T_{\rm int}$ and $V_{\rm int}$ over the stellar model. The off-diagonal entry $\delta\omega^2_{1,-1}=(3/10)\epsilon(V_{\rm int}-\omega_\alpha^2 T_{\rm int})$ is what mixes $m=+1$ and $m=-1$; when it dominates, the eigenvectors become $(1,\pm1)$, i.e., equal superpositions that form standing waves along the tidal ($x$) and intermediate ($y$) axes, while the $m=0$ component remains a standing wave along the spin ($z$) axis. The dimensionless tidal distortion $\epsilon=(M_c/M)(R/a)^3$ controls everything: it sets the size of the frequency shifts and the threshold where tidal tilting beats the Coriolis force.

What would settle it

Compute the same multiplet coupling including angular degree 0 and 2 modes at an orbital period of 1 day: if the tidally tilted eigenvectors and frequency splittings change substantially, the isolated-multiplet prediction fails in a regime the paper uses for its figures. Observationally, measure the power spectrum of a tidally synchronized delta Scuti binary near a 3-day orbital period and check whether each dipole radial order shows the predicted equal-amplitude doublets at exactly twice the orbital frequency plus a singlet; absence of that pattern would contradict the central claim.

Watch

Extended reading notes

Core claim

Within linear perturbation theory for an isolated multiplet of fixed angular degree and radial order, the $\ell_t=2$, $m_t=\pm 2$ part of the tidal potential couples the $m=\pm1$ modes through off-diagonal matrix elements $\delta\omega^2_{1,-1}=(3/10)\epsilon(V_{\rm int}-\omega_\alpha^2 T_{\rm int})$, where $\epsilon=(M_c/M)(R/a)^3$ is the dimensionless tidal distortion. When this coupling exceeds the Coriolis-induced difference between the $m=1$ and $m=-1$ frequencies, the eigenmodes become the equal superpositions $(Y_{1,1}\pm Y_{1,-1})/\sqrt{2}$, which are standing waves proportional to $y$ and $x$, while the $m=0$ mode remains proportional to $z$. The paper calls these the $Y_{10x}$, $Y_{10y}$, and $Y_{10z}$ modes and shows that the $x$- and $y$-modes are amplitude-modulated twice per orbit and produce doublets split by exactly $2\nu_{\rm orb}$, while the $z$-mode is a singlet; a full dipole triplet therefore produces five peaks. Quadrupole modes split into $Y_{21\pm}$ doublets at $2\nu_{\rm orb}$, $Y_{22\pm}$ doublets at $4\nu_{\rm orb}$, and a $Y_{20z}$ singlet, with some mass-ratio-dependent mixing. Applied to a $\delta$ Scuti model, the calculation yields simple formulae for the tidal frequency shifts and predicts that p modes are tidally tilted in synchronized binaries with periods below roughly 3--6 days, while g modes, whose Coriolis terms dominate, stay aligned with the spin axis.

Load-bearing premise

The central calculation assumes each pulsation mode can be studied alone, ignoring how tides mix modes of different shapes and frequencies, an effect the paper itself says becomes important in the shortest-period systems it models.

Editorial extensions

If this is right

  • In any tidally synchronized, circular binary with p-mode pulsations, each radial order of dipole modes should show two equal-amplitude doublets spaced by exactly $2\nu_{\rm orb}$ plus one singlet, instead of the triplet expected for spin-aligned modes.
  • The amplitude and phase modulation of the Y10x and Y10y modes provides mode identification: their peak phases differ by a quarter orbit, and phase jumps of half a cycle signal standing rather than traveling modes.
  • If true, asteroseismology of close binaries becomes feasible: matching measured tidal frequency shifts to equations 64-67 constrains the stellar structure and the tidal distortion $\epsilon$.
  • The prediction that g modes stay spin-aligned means that orbitally modulated g-mode amplitudes, such as those reported by Van Reeth et al., should be interpreted as tidal amplification with little phase change, not tidal tilting.
  • At $P_{\rm orb} \lesssim 2$ d the same tidal perturbation approaches the large frequency spacing, so coupling across different angular degrees should produce single-sided, tidally trapped pulsations and more complex spectra; the present five-peak pattern is the weak-to-moderate distortion regime.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Going beyond the paper: the exact $2\nu_{\rm orb}$ spacing is independent of tidal strength and mode frequency, so a blind search for equal-amplitude doublets spaced by twice the orbital period in light curves of short-period binaries could identify triaxial pulsators even when individual modes are not otherwise recognizable.
  • Going beyond the paper: because the three dipole frequencies shift in opposite directions with the three axis lengths (x lower, z higher, y intermediate), measuring all three members of a radial-order multiplet gives a direct probe of the star's tidal and centrifugal flattening, potentially constraining the internal density profile.
  • Going beyond the paper: the same triaxial basis should apply to any tidally locked oscillating body dominated by pressure-like restoring forces, so the doublet signature is a candidate diagnostic for oscillations of strongly distorted exoplanets or white dwarfs in close binaries, although the paper only mentions the white-dwarf case in passing.
  • A testable extension: compute the full coupled-angular-degree system at $P_{\rm orb}=1$ day; if cross-multiplet coupling destroys the equal-amplitude doublet pattern in that regime, the paper's figures that use 1-day orbits would not represent the true observable spectra there.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 6 minor

Summary. The paper develops a linear perturbation theory for the pulsation modes of a tidally distorted, synchronously rotating star in a circular binary, coupling the azimuthal orders m within an isolated (l, n_pg) multiplet while including the Coriolis force, the tidal potential, and the centrifugal distortion (Sec. 2, Eqs. 1-22). For dipole modes, when the m_t = +/-2 tidal coupling dominates the Coriolis splitting, the m = +/-1 pair hybridizes into two standing modes aligned with the tidal axis (Y10x) and with the intermediate axis (Y10y), while the m = 0 mode remains aligned with the spin axis (Y10z); the multiplet is thus 'triaxial'. Because the standing-wave patterns are fixed in the corotating frame, the observed amplitude of the x- and y-modes is modulated twice per orbit, producing, in the observer's frame, doublets spaced by exactly twice the orbital frequency, whereas the z-mode is a singlet (Sec. 2.1.1, Figs. 2-4). Analogous results are derived for l = 2 modes, with Y21+/- modes producing 2 nu_orb doublets and Y22+/- modes 4 nu_orb doublets, along with the Y20z singlet (Sec. 2.2, Figs. 5-8). Simple asymptotic frequency shifts for p modes (Eqs. 64-67) and criteria for when tilting occurs (Sec. 5.1) are derived and applied to a 1.74 M_sun MESA/GYRE delta Scuti model with a 1.3 M_sun companion at P_orb = 1-3 d, yielding synthetic light curves, amplitude/phase-modulation curves, power spectra, and echelle diagrams (Figs. 9-13).

Significance. If the central claim holds, this is a significant advance for the asteroseismology of close binaries: it replaces the tidal-axis-aligned traveling-wave picture with triaxial standing modes and converts an empirical pattern (the doublets at about twice the orbital frequency seen in TIC 184743498 and similar stars) into a parameter-free geometric prediction, since the doublet spacing of exactly twice the orbital frequency follows from the viewing geometry rather than from any fit to observed power spectra. The paper is admirably explicit and reproducible in method: the coupled eigenvalue problem is written out in full (Eqs. 23 and 43), the asymptotic formulas of Eqs. (64)-(67) are derived rather than fitted, and the predictions are falsifiable (p modes tilted while g modes remain spin-aligned; nearly equal-amplitude doublets spaced by exactly 2 nu_orb; tilting confined to synchronized systems with P_orb <~ 3 d, and preferentially in higher-order p modes). The frank discussion in Sec.

major comments (3)
  1. [Sec. 5.2 vs. Secs. 2.1.1/4, Figs. 2-8, 10-13] The central observable claim of the paper - that tidally tilted dipole p modes produce nearly equal-amplitude doublets spaced by exactly twice the orbital frequency - is demonstrated with synthetic light curves and amplitude spectra computed at P_orb = 1 d (Sec. 2.1.1, Figs. 2-8, using the delta Scuti model of Sec. 4 with a 1.3 M_sun companion). However, Sec. 5.2 states that the isolated-multiplet approximation on which the entire Sec. 2 calculation rests fails at P_orb <~ 2 d: once the tidal frequency perturbation becomes comparable to the large spacing Delta nu, coupling to modes of different angular degree (in particular l = 0 with l = 2) becomes strong and 'will need to be accounted for'. The manuscript itself lists the consequences of that omitted coupling: doublets with unequal peak amplitudes, central frequency components, or more than three peaks - precisely the departures that would corrupt the clean signature claimed in the abstract. Moreover, at P_orb = 1 d the model has R/a ~ 0.37, close to Roche-lobe filling, where the paper's own truncation to the l_t = 2 tidal component (Sec. 2) is stated to be inappropriate. Because the observed tidally tilted delta Scuti systems that motivate the paper have orbital periods near 1 d, the quantitative predictions are presented in the regime the paper itself declares invalid. The qualitative triaxial standing-wave geometry may survive the inclusion of cross-l coupling, but the equal-amplitude doublet signature, the five-peak multiplet counting, and the frequency and echelle predictions of Figs. 10-13 are not self-consistently derived at P_orb = 1 d. I request that the demonstration figures be recomputed at a period where the isolated-multiplet treatment is valid (the paper itself treats P_orb = 3 d as such in Figs. 9 and 13), or that the calculation be extended to include cross-l coupling at short periods, with the spectral predictions re-derived accordingly.
  2. [Sec. 3, Eqs. (64)-(67); Sec. 4] Equations (64)-(67) are presented as simple and accurate expressions for the tidal frequency perturbations of dipole p modes, and Sec. 4 reports agreement with the numerical solutions at P_orb = 1 d and 3 d. The numerical verification, however, is performed within the truncated model of Eq. (23), whose range of validity is limited by Sec. 5.2 to P_orb >~ 2 d. At P_orb = 1 d, the frequency shifts from the omitted cross-l coupling are comparable to the quoted perturbations themselves (this is the paper's own criterion for strong coupling in Sec. 5.2, and Fig. 13 shows the resulting scatter of the modes), so the claimed accuracy of Eqs. (64)-(67) has not been established at the period most relevant to the motivating observations. I ask that the accuracy claim be restricted to the regime in which the truncated model is self-consistent, or that the formulas be tested against a calculation that includes the l-coupling.
  3. [Sec. 5.3] The reanalysis of previously published systems in Sec. 5.3 (e.g., TIC 63328020 as predominantly a Y10y mode, and modes in TIC 68495594 as primarily Y10y and Y22+ components) assigns observed modes to the clean triaxial basis of Sec. 2. These identifications are only as robust as the isolated-multiplet approximation, which Sec. 5.2 limits to P_orb >~ 2 d for the model of this paper. Since the systems being reinterpreted are close binaries of the same general type, the manuscript should either state their orbital periods and demonstrate that they lie outside the strong-coupling regime, or present the identifications as tentative and indicate which of the Sec. 5.3 conclusions would be affected by the unequal-amplitude doublets, central peaks, and extra components that the paper itself lists as signatures of the omitted coupling.
minor comments (6)
  1. [Sec. 2.2.1] The sentence 'Similarly, the the Y22-, Y22+ modes may obtain a central peak in their amplitude spectra' contains a duplicated article ('the the'); the sentence would also benefit from a cross-reference to Fig. 8, which shows the resulting triplets.
  2. [Secs. 1, 3.1] There are several missing spaces between a word and a symbol, e.g., 'examined a delta Scuti pulsator' (should be 'a delta Scuti pulsator') and 'Performing the same exercise for the delta V' (should be 'for the delta V'); these should be fixed in proof.
  3. [Figs. 1, 5] The figure captions state 'Movies showing these pulsations can be found here,' but no URL or ancillary-file information appears in the manuscript text; please provide the actual link or state that the movies are available as supplementary material.
  4. [Sec. 5.2] The claimed breakdown period of P_orb <~ 2 d for the delta Scuti model would be easier for readers to verify if the text quoted the comparison underlying the threshold (e.g., the ratio of the tidal frequency perturbation from Eq. (67) to Delta nu at P_orb = 2 d), rather than referring only to Fig. 13.
  5. [Sec. 2, Eqs. (9)-(22)] The equilibrium-tide displacement is computed with the Cowling approximation (xi_r,S = -U/g in Sec. 2), which neglects the perturbation of the star's own gravitational potential; a sentence justifying this for the p-mode overlap integrals, or an estimate of its effect on Eqs. (64)-(67), would be useful given that those equations are claimed to be accurate.
  6. [Secs. 2.2, 2.2.1, Figs. 6-8] For the quadrupole demonstration model, M_c/M = 0.75, and the ratio (delta omega^2_00 - delta omega^2_22)/delta omega^2_20 is only about 4, so the Y22-, Y22+, and Y20z modes are visibly mixed (Figs. 6-8). The text acknowledges this, but the idealized statements in Sec. 2.2.1 ('their power spectra are two amplitude peaks separated by exactly four times the orbital frequency') should be prefaced with the mixing caveat so that the analytic decomposition of Eqs. (47)-(53) is not read as the model prediction at order-unity mass ratio.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the triaxial-mode prediction is derived from a self-contained perturbation calculation, not from the observations it explains.

full rationale

The central derivation (Section 2, eq. 23) diagonalizes the tidal, centrifugal, and Coriolis perturbations within an isolated (ℓ, npg) multiplet using Wigner-3j overlap integrals and stellar-model integrals (Tint, Vint, Wint) that are computed from a MESA/GYRE model, not from observed power spectra. The triaxial eigenfunctions (eqs. 31–33) and the doublet spacing of exactly twice the orbital frequency (Section 2.1.1, Fig. 4) follow from the resulting eigenvectors combined with the rotating viewing geometry; no observed amplitude or phase values are fitted to produce those peaks. Equations 64–67 are algebraic approximations that are checked against the numerical eigensolver, so they are not fitted inputs renamed as predictions. The self-citations (Fuller et al. 2020 for overlap integrals; Zhang et al. 2024 and Jayaraman et al. 2024 as motivating observations) are used for technical machinery or as observational motivation, while the core eigenvalue problem rests on Dahlen & Tromp (1998) and the paper's own stated integrals; no load-bearing claim reduces to a self-citation chain. The paper's own caveat in Section 5.2 — that coupling between different ℓ becomes strong for P_orb ≲ 2 d while several figures use P_orb = 1 d — is an acknowledged validity limitation of the isolated-multiplet assumption, not a circular step, because the calculation remains independent of the data it aims to explain. No self-definitional, fitted-input, uniqueness-imported, or ansatz-smuggled circularity is present.

Assumptions & free parameters 6 free parameters · 7 assumptions · 0 invented entities

The central derivation rests on the standard Dahlen and Tromp perturbative framework, the Cowling approximation, and a set of physical idealizations: circular, synchronized, aligned binaries, dominant l_t=2 tides, and isolated-multiplet coupling. The model parameters in Section 4 are illustrative inputs taken from an observed delta Scuti target rather than fitted to the predicted doublet spacings. No new physical entities are introduced; the triaxial modes are real linear combinations of spherical harmonics.

free parameters (6)
  • delta Scuti model mass = 1.74 solar masses
    MESA model mass chosen to resemble TIC 184743498 from Zhang et al. 2024; an input for the demonstration, not fitted to the theory's predictions.
  • model radius = 2.23 solar radii
    MESA model input, chosen with the mass to resemble the observed target.
  • model metallicity = Z = 0.017
    MESA model input used to construct the stellar structure.
  • model effective temperature = 7427 K
    MESA model input used as a consistency check for the stellar model.
  • companion mass = 1.3 solar masses
    Chosen in Section 4 for the orbital period range 1-3 days; the asymptotic formulas are analytic in Mc and do not depend on this value.
  • orbital period = 1 day to 3 days
    Chosen to span the synchronized-binary condition of Bashi et al. 2023; predictions are period-dependent and the term 'tidally tilted' depends on this choice.
assumptions (7)
  • standard math Linear perturbation theory via the generalized eigenvalue problem of Dahlen and Tromp (1998a) correctly describes stellar oscillation modes.
    Invoked at the start of Section 2 (equation 1); the matrix elements for kinetic, potential, and Coriolis couplings are taken from this textbook framework.
  • domain assumption The Cowling approximation (neglecting the perturbation to the gravitational potential) gives the tidal displacement via xi_r,S = -U/g.
    Used in Section 2 just before equation 15 to convert the tidal potential into the effective ellipticity; this is a standard but simplifying assumption.
  • domain assumption The pulsating star is in a circular orbit with spin synchronized and aligned with the companion's orbital angular momentum.
    Stated in the first paragraph of Section 2; the entire tidal coupling framework depends on a static distortion in the corotating frame. Non-synchronized or eccentric cases are deferred to Section 5.5 as future work.
  • domain assumption The l_t=2 component of the tidal potential dominates; l_t=3 components are neglected.
    Section 2: the l_t=3 component is smaller by a factor R/a and is dropped, with the caveat that it matters near Roche lobe filling and produces single-sided pulsations (Section 5.2).
  • ad hoc to paper Coupling is only considered within an isolated multiplet (fixed l and radial order); coupling to other l and radial orders is neglected.
    Section 2: 'we do not account for coupling with different angular orders l or radial orders npg.' The paper's Section 5.2 states this fails once the tidal perturbation approaches the large frequency spacing Delta nu, i.e., P_orb less than about 2 days.
  • domain assumption For the asymptotic p-mode formulas, the outer layer has eta approximately 3 and (d xi_r/dr)^2 approximately (omega^2/c_s^2) xi_r^2.
    Section 3.1, used to simplify delta T and delta V into equations 62-63; standard for high-order p modes whose kinetic energy concentrates near the surface.
  • domain assumption Perturbation theory for g modes remains valid only for omega_alpha greater than about 2 Omega, below which the Coriolis force transforms modes into Hough modes.
    Section 5.1.2: the paper limits its g-mode tilting analysis to omega_alpha around 2 Omega and notes the theory breaks down for lower frequencies.

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Cite this review

Pith. "Pith review of Tidally distorted stars are triaxial pulsators." pith.science (2026). https://pith.science/paper/J7EPHO6C

@misc{pith2026241109743,
  author       = {Pith},
  title        = {Pith review of: Tidally distorted stars are triaxial pulsators},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/J7EPHO6C}},
  note         = {Machine review of arXiv:2411.09743}
}
abstract

Stars in close binaries are tidally distorted, and this has a strong effect on their pulsation modes. We compute the mode frequencies and geometries of tidally distorted stars using perturbation theory, accounting for the effects of the Coriolis force and the coupling between different azimuthal orders $m$ of a multiplet induced by the tidal distortion. For tidally coupled dipole pressure modes, the tidal coupling dominates over the Coriolis force and the resulting pulsations are ``triaxial", with each of the three modes in a multiplet ``tidally tilted" to be aligned with the one of the three principal axes of the star. The observed amplitudes and phases of the dipole modes aligned orthogonal to the spin axis are modulated throughout the orbit, producing doublets in the power spectrum that are spaced by exactly twice the orbital frequency. Quadrupole modes have similar but slightly more complex behavior. This amplitude modulation allows for mode identification which can potentially enable detailed asteroseismic analyses of tidally tilted pulsators. Pressure modes should exhibit this behavior in stellar binaries close enough to be tidally synchronized, while gravity modes should remain aligned with the star's spin axis. We discuss applications to various types of pulsating stars, and the relationship between tidal tilting of pulsations and the ``single-sided" pulsations sometimes observed in very tidally distorted stars.

Figures

Figures reproduced from arXiv: 2411.09743 by the authors.

Figure 1
Figure 1. Angular pattern of ℓ = 1 triaxial pulsations in a tidally distorted star, viewed at orbital inclination i = 45◦ with the companion shown as a gray circle. The color indi￾cates the relative amplitude of the surface flux perturbation (i.e., temperature perturbation) for the three ℓ = 1 pulsa￾tions: the Y10x mode aligned with the tidal axis (top left), the Y10z mode aligned with the spin axis (bottom left), and the Y10… view at source ↗
Figure 3
Figure 3. Amplitude (top panel) and phase variation (bottom panel) of tidally tilted dipole modes over the course of one orbit, for the same modes shown in [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Power spectrum of the tidally tilted dipole modes from Figures 2 and 3. The Y10x and Y10y modes pro￾duce equal-amplitude doublets spaced by exactly 2× the or￾bital frequency, while the Y10z mode produces a singlet. model from Section 4, at an orbital period of Porb = 1 day and companion mass of 1.3 M⊙. The Y10x mode is aligned with the tidal axis and pro￾duces the largest flux modulations when the tidal axis is clos… view at source ↗
Figures from the paper (8 more)
Figure 5
Figure 5. Figure 5: Same as [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]
Figure 6
Figure 6. Figure 6: Same as [PITH_FULL_IMAGE:figures/full_fig_p009_6.png]
Figure 7
Figure 7. Figure 7: Same as [PITH_FULL_IMAGE:figures/full_fig_p010_7.png]
Figure 8
Figure 8. Figure 8: Same as [PITH_FULL_IMAGE:figures/full_fig_p011_8.png]
Figure 9
Figure 9. Figure 9: The tidal tilting fraction of dipole modes in our δ Scuti model at an orbital period of Porb = 3 d, as a function of mode radial order npg. The tidal tilting fraction is defined to be unity for tidally aligned modes, while it is zero for modes aligned with the spin axi…
Figure 10
Figure 10. Figure 10: The combined tidal and rotational frequency perturbation (measured in the star’s rotating frame) to modes in our δ Scuti model at an orbital period of Porb = 1 d (top panel) and Porb = 3 d (bottom panel). The mode ra￾dial order npg is on the y-axis. The blue lines ind…
Figure 11
Figure 11. Figure 11: Combined tidal and rotational frequency per￾turbations for modes of our δ Scuti model, but now measured in the observer’s frame. The triangle symbols indicate the contribution from different m values to each mode, whose color is arbitrary. The g modes form singlets, w…
Figure 13
Figure 13. Figure 13: Echelle diagram for our δ Scuti model, as a function of the frequency modulus relative to the large frequency spacing ∆ν of a single non-rotating star. At Porb = 3 d (bottom panel), the tidal frequency perturbations are small enough that the p modes remain confined to…

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