On the non-dissipative orbital evolution of a binary system comprising non-compact components with misaligned spin and orbital angular momenta

J. C. B. Papaloizou; P. B. Ivanov; Y. A. Lazovik

REVIEW 2 major objections 4 minor 16 references

On the non-dissipative orbital evolution of a binary system comprising non-compact components with misaligned spin and orbital angular momenta

T0 review · 2 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read Tidal torque can make the apsidal angle librate instead of precess in a binary of two ordinary stars with misaligned spins.

desk verdict A careful two-spin extension of the IP framework; the main unproven step is real but likely not fatal, and the paper deserves a serious referee. read the letter →

arxiv 2411.09112 v2 pith:36C26P4D submitted 2024-11-14 astro-ph.SR astro-ph.EP

Y. A. Lazovik , P. B. Ivanov , J. C. B. Papaloizou This is my paper · ORCID

classification astro-ph.SRastro-ph.EP

keywords tidalinteractionapsidalprecessionspin-orbitmisalignmentnon-dissipativetorquecriticalcurveslibrationbinarystarsDIHer

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 extends the theory of non-dissipative tidal evolution in misaligned binaries from the case of one compact, spinless companion to two ordinary non-compact stars, each carrying its own spin angular momentum. It argues that the parallel component of the tidal torque, arising from inertial and Coriolis forces on the tidal response, comes to dominate the orbital dynamics near critical curves in parameter space where the time-averaged apsidal precession rate vanishes. Near these curves the apsidal angle librates rather than circulating, and the spin-orbit inclination angles undergo bounded oscillations whose amplitude is set by the distance from the critical curve. In the small-spin limit, critical curves exist only for binaries with at least one retrograde component. The paper provides the full governing equations, a forced-pendulum approximation for the near-critical motion, and numerical simulations for a DI-Her-like system that reproduce the analytic libration-circulation transition.

What carries the argument

The central object is the parallel tidal torque $T_{\parallel,k}$ acting on each component, defined through eq. (17) with the explicit form in eq. (B1). This torque depends on the apsidal angle $\hat\varpi_k=\varpi+\gamma_k$ and arises from inertial and Coriolis forces acting on the tidal bulge. The critical curves are the loci in parameter space where the time-averaged apsidal precession rate $\dot\varpi_T+\dot\varpi_E+\dot\varpi_R+\dot\varpi_{NI}$ vanishes; near these curves the system reduces to the forced simple pendulum of eq. (82), whose pendulum frequency $\Omega_\parallel$ is given by eq. (83), predicting both librating and circulating solutions separated by a separatrix.

What would settle it

A direct numerical computation of the tidal response of two extended, deformable stars in a misaligned eccentric binary, without assuming each star feels only the point-mass field of the other, would settle whether the torque decomposition is valid; a mismatch in the magnitude or phase of the parallel torque would shift the critical curves and change the libration predictions.

Watch

Extended reading notes

Core claim

The central claim is that for a binary with two non-compact components, the long-term evolution of the apsidal angle and of the spin-orbit inclinations is controlled by the parallel tidal torque, and that the most interesting behaviour occurs on critical curves where the time-averaged total apsidal precession rate is zero. On those curves the apsidal angle satisfies a forced-pendulum equation, so it can librate between prograde and retrograde orientations instead of steadily precessing, while the obliquities oscillate with amplitude bounded by the distance from the critical curve. In the small-spin limit such critical states require at least one retrograde component, because the non-inertial spin-precession contribution to the apsidal rate changes sign for retrograde spins. Numerical integrations for parameters similar to DI Her confirm the expected transition between librating and circulating solutions, and show that near-polar, rapidly rotating systems can still exhibit libration even when the timescale separation assumed by the pendulum model breaks down.

Load-bearing premise

The load-bearing premise is that the torque expressions derived for a binary with one compact, spinless companion apply independently to each non-compact star, so that the total non-dissipative torque is simply the sum of the two identical functional forms applied separately.

Editorial extensions

If this is right

Binary systems whose parameters lie near a critical curve will show libration of the apsidal line rather than steady precession, so the orbit can oscillate between prograde and retrograde apsidal states.
Spin-orbit inclination angles undergo bounded oscillations whose amplitude is controlled by the distance to the critical curve, and near the separatrix high-frequency forcing can generate chaotic motion.
In the small-spin regime, a critical state is possible only with at least one retrograde component, so the predicted apsidal libration is a signature of retrograde rotation in the binary.
For a close, eccentric binary like DI Her, the measured apsidal precession rate and spin inclinations can be used to assess proximity to a critical curve; if the secondary is slightly retrograde, libration about a critical curve is a realistic outcome.
The closed system of equations (25)-(30) provides a directly integrable non-dissipative model for arbitrary eccentricity and two spinning non-compact stars, replacing the earlier compact-companion restriction.

Reading between the lines

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

Beyond the paper: the same pendulum mechanism should apply to star-planet systems where the star is non-compact and the planet can be treated as a point mass, so the critical-curve condition may explain observed spin-orbit misalignments in close-in exoplanet systems that linger near apsidal resonances.
Beyond the paper: a direct hydrodynamical simulation of two extended, deformable stars with misaligned spins would test whether the torque decomposition into two independent point-mass-companion expressions remains valid when the two tidal responses can interact.
Beyond the paper: if apsidal libration near critical curves is common, the distribution of apsidal angles in a large sample of misaligned eclipsing binaries should cluster near the librating phase rather than being uniformly distributed.
Beyond the paper: the requirement of at least one retrograde component in the small-spin limit creates a selection effect: binaries with anomalously slow measured apsidal precession may be those currently undergoing near-critical libration, so their observed $\dot\varpi$ is systematically smaller than the steady-precession prediction.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Short version: this paper is IP plus a second spinning star, done with the same level of care as the earlier papers. The genuinely new content is the two-spin generalization: both spin-orbit inclinations evolve under parallel and perpendicular torques, the relative azimuthal angle nu_1 - nu_2 enters as a dynamical variable, and the critical-curve/pendulum analysis is carried out for two non-compact components. There is a real result here: near critical curves the tidal torque drives apsidal libration and inclination oscillations; in the small-spin limit the critical curves exist only when at least one component rotates retrograde.

What is done well: the derivation of the governing equations is transparent. The compact limit reduces to IP1, checked explicitly. The analytic secular theory is compared with direct integration in Case A and the libration/circulation boundary tracks E/E_crit as predicted. The MESA/GYRE inputs are described in enough detail that the stellar parameters are reproducible, and the convergence of the mode sums with N_r is shown.

The soft spots are real but smaller than the stress-test note implies. The torque on each star is taken from IP, where the companion was a point mass. Applying those expressions independently to each component assumes that the two tidal responses add linearly and that each spin magnitude is conserved. The conserved spin magnitude is standard for conservative non-dissipative torques; the additivity is the natural leading-order weak-tide approximation, since quadrupole-quadrupole terms are higher order. But the paper never states this, and a referee should ask for a short justification or an estimate of the neglected cross-tidal terms. The more concrete limitation is the timescale separation |Delta_omega| >> Omega_parallel behind the pendulum description, eq. (84). Case B violates it, and the authors are upfront about that; the analytic model is tested in Case A, where it works.

Minor: no code or data is shipped, only 'available on reasonable request.' That is acceptable for a theory paper but it would help a referee check the torque additivity assumption.

Bottom line: this deserves a serious referee and, after a modest revision, publication. If I were working on misaligned binaries, I would cite it.

Referee Report

2 major / 4 minor

Summary. This paper extends the non-dissipative tidal torque formalism of Ivanov & Papaloizou (IP, IP1, IP2) from binaries with one compact component to binaries in which both components are non-compact and carry spin angular momentum. The authors write down a system of ordinary differential equations for the orbital angular momentum, the spin vectors, and the apsidal angle, using torque components taken from IP for each component. In the limit of small spin-to-orbit angular momentum ratio, they reduce the system to a smaller set of equations, identify critical curves in parameter space where the time-averaged apsidal precession rate vanishes, and show that near these curves the apsidal angle can librate, described by a forced pendulum equation. They compute critical curves for parameters appropriate to DI Her using MESA/GYRE stellar models and present numerical integrations for two representative cases, finding agreement with the analytic theory in one case (A) and qualitative libration/circulation behavior in the other (B).

Significance. If the underlying torque model is accepted, this is a valuable extension because real close binaries such as DI Her have two non-compact components. The paper recovers the compact limit of IP1, provides transparent derivations of the angle evolution equations, and uses independent stellar-structure calculations for the parameters. The analytic pendulum description gives a falsifiable prediction: near critical curves, libration of the apsidal angle and oscillations of spin-orbit inclinations should occur, and in the small-spin limit such curves require a retrograde component. The numerical comparison in Case A is a good consistency check of the secular reduction. The main caveat is that the torque expressions are assumed to carry over independently from the compact-companion calculation, and the simulations do not test this assumption.

major comments (2)

[Section 2.4, Eq. (17), Appendix B] The paper assumes that the torque on each component has the form dS_k/dt = T_k with T_k perpendicular to S_k, and that the components T_parallel,k and T_perpendicular,k are exactly the expressions derived in IP for a binary with one compact companion. This is not derived for the two-non-compact case. If the tidal response of one star is affected by the deformation of the other beyond the point-mass monopole, or if the torque has a component along S_k so that |S_k| changes, then Eqs. (25)-(30), the critical-curve condition, and the pendulum Eq. (82) are not the correct dynamical system. I request an explicit ordering argument or an order-of-magnitude estimate showing that the neglected cross-tidal and along-spin torque contributions are higher order in (R/a)^5 or (Omega_r/Omega_*)^2, or a statement of the approximation regime in which the IP expressions are valid for each component.
[Sections 3.3 and 4.4] The numerical simulations integrate the same assumed torque model, so they validate the reduction from the full ODE system to the secular and pendulum descriptions, but they cannot validate the torque model itself. The agreement in Case A is evidence that the secular theory correctly represents the assumed equations, not that the assumed equations are the true two-deformable-body dynamics. The abstract and conclusions should be phrased as conditional on the torque model, and the paper should explicitly state that the torque additivity and perpendicularity are inputs rather than outputs of the numerical comparison.

minor comments (4)

[Section 5.1] The statement that the semi-major axis is 'conserved' is not consistent with Eq. (25), which allows dL/dt to be nonzero. In the small-spin limit the change is negligible, but the statement should be qualified as an approximation.
[Section 4.3.1] The discussion of the possibility that only the secondary of DI Her is retrograde should be connected to the condition derived in Section 3.3, which requires the component with the larger perpendicular spin projection to be retrograde for a critical state to exist.
[Abstract] The phrase 'at least one component with retrograde rotation' is imprecise; the derived condition is that the component that dominates the averaged non-inertial apsidal precession must be retrograde, not merely that some component is retrograde.
[Equations (26) and (27)] In the typeset version, Eq. (26) and Eq. (27) appear to have unbalanced parentheses in the right-hand sides; please re-check the formulas to ensure they are typeset correctly.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the libration and critical-curve behavior are derived consequences of the stated torque equations, not renamed inputs; the main risks are the imported two-component torque assumption and an acknowledged timescale limitation.

full rationale

No circular step can be exhibited. The paper's input-to-output mapping is: (i) torque components T_parallel,k and T_perpendicular,k taken from IP (Appendix B, Eqs. B1-B2), (ii) stellar structure parameters computed with MESA and GYRE (Appendix A), and (iii) standard apsidal-motion terms (Sterne 1939; Barker & O'Connell 1975). From these inputs the paper derives the governing system (Eqs. 25-30), the small-inclination reduction (Eqs. 44-51), and the pendulum description near critical curves (Eqs. 82-83). Critical curves are defined, not fitted, by setting the time-averaged apsidal precession rate to zero, and the libration/circulation criterion E/Ecrit is a mathematical consequence that is then checked against numerical integrations of the same system. No constant is adjusted to reproduce the claimed libration; the numerical simulations serve as a consistency check of the secular reduction, not as a source of fitted parameters. The paper relies heavily on prior work by the same authors (IP, IP1, IP2) for the torque and precession formulas, but those are published derivations with stated assumptions, and the present extension to two non-compact components is transparent. The main caveats are correctness risks rather than circularities: the two-component torque model assumes that IP's compact-companion torque expressions apply additively and independently to each non-compact star (Section 2.4 and Appendix B), and the analytic pendulum model requires the timescale separation |Delta_omega| >> Omega_parallel (Eq. 84), which the paper explicitly acknowledges is violated in Case B (Section 4.4.3). These are identifiable limitations in the derivation, but the central prediction is not equivalent to its inputs by construction.

Assumptions & free parameters 1 free parameters · 8 assumptions · 0 invented entities

The central derivation adds no fitted constants to the target dynamics. Torque expressions, apsidal precession formulas, and stellar structure parameters are taken from prior work or computed independently from MESA and GYRE models. The main assumptions are the perpendicular torque condition, additivity of the two components' torques, and the small-spin and timescale-separation approximations used for the analytic theory. No new physical entities are introduced.

free parameters (1)

Secondary dimensionless moment of inertia in Case A = 4.3e-2
In Section 4.4.2 the secondary's moment of inertia is reduced below its MESA value to make S1 B2/(S2 B1) close to 2. This is a demonstration parameter chosen by hand, not fitted to observations, and it does not enter the general theory.

assumptions (8)

standard math Total angular momentum J = L + S1 + S2 is conserved (Eq. 1).
Isolated binary with no external torques; standard Newtonian conservation used throughout the derivation.
domain assumption Tidal torque on each spin is perpendicular to that spin, dS_k/dt = T_k with T_k = T_parallel,k s_parallel,k + T_perpendicular,k s_perpendicular,k (Eq. 17).
Adopted from IP; this is the foundation for the spin-evolution equations (25)-(29).
domain assumption Torque expressions T_parallel,k and T_perpendicular,k from IP (Appendix B) remain valid for both non-compact components with the companion treated as a point mass.
The paper generalizes apsidal precession to two components but imports the torque formulas without rederiving them for two deformable stars; see Section 2.8 and Appendix B.
domain assumption Spin magnitudes S_k are constant in the non-dissipative regime.
Stated in Section 2.5 before deriving Eqs. (28)-(29).
domain assumption For the analytic theory, spin angular momenta satisfy S_k << L so the orbital inclination i is small.
Enables the approximations in Section 3, including Eqs. (42)-(50) and the critical-curve analysis; not required for the general equations.
ad hoc to paper Near critical curves, the timescale separation |Delta_omega| >> Omega_parallel (Eq. 84) holds, permitting averaging and a pendulum description.
Required for Eq. (82). The paper notes Case B violates this assumption yet still finds libration, so the analytic description is not universally applicable.
domain assumption Apsidal precession contributions are additive: d_omega/dt = d_omega_T/dt + d_omega_E/dt + d_omega_R/dt + d_omega_NI/dt (Eq. 30).
Standard in celestial mechanics, but here assumed additive for both components; see Section 2.7.
domain assumption Stellar normal-mode expansions truncated at N_r = 10 give converged Q_eq, omega_eq, and beta_star.
Numerical convergence is shown in Appendix A3, Fig. A3; the truncation is assumed to represent the tidal response adequately.

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Cite this review

Pith. "Pith review of On the non-dissipative orbital evolution of a binary system comprising non-compact components with misaligned spin and orbital angular momenta." pith.science (2026). https://pith.science/paper/36C26P4D

@misc{pith2026241109112,
  author       = {Pith},
  title        = {Pith review of: On the non-dissipative orbital evolution of a binary system comprising non-compact components with misaligned spin and orbital angular momenta},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/36C26P4D}},
  note         = {Machine review of arXiv:2411.09112}
}

read the original abstract

In this Paper we determine the non-dissipative tidal evolution of a close binary system with an arbitrary eccentricity in which the spin angular momenta of both components are misaligned with the orbital angular momentum. We focus on the situation where the orbital angular momentum dominates the spin angular momenta and so remains at small inclination to the conserved total angular momentum. Torques arising from rotational distortion and tidal distortion taking account of Coriolis forces are included. This extends the previous work of Ivanov & Papaloizou relaxing the limitation resulting from the assumption that one of the components is compact and has zero spin angular momentum. Unlike the above study, the evolution of spin-orbit inclination angles is driven by both types of torque. We develop a simple analytic theory describing the evolution of orbital angles and compare it with direct numerical simulations. We find that the tidal torque prevails near 'critical curves' in parameter space where the time-averaged apsidal precession rate is close to zero. In the limit of small spin, these curves exist only for systems that have at least one component with retrograde rotation. As in our previous work, we find solutions close to these curves for which the apsidal angle librates. As noted there, this could result in oscillation between prograde and retrograde states. We consider the application of our approach to systems with parameters similar to those of the misaligned binary DI Her.

Figures

Figures reproduced from arXiv: 2411.09112 by the authors.

**Figure 1.** Top left panel. We show the inclination angles 𝛿𝑘 between the unit vector e𝑧 pointing in the direction of the total angular momentum and the unit vector in the direction of the spin vector of a particular component, s𝑘, the inclination angle 𝑖 between e𝑧 and the unit vector in the direction of the orbital angular momentum, l, as well as the angle 𝛽𝑘 between l and s𝑘. The ’rotational’ angles 𝜈𝑘 and 𝛼 define the posit… view at source ↗

**Figure 2.** Critical curves in the (𝑎, 𝜎 ˜ ) plane. Dark solid, red dashed, and green dash-dotted lines correspond to 𝑒 = 0.2, 0.5, and 0.9, respectively. Each panel represents a different 𝛿 value specified in the top right corner. Note that, in order to be on a critical curve for the values of 𝑎˜ and 𝜎 considered, it is necessary that 𝛿 > 𝜋/2. The actual location of DI Her is depicted by the dark cross in the top left panel. 1… view at source ↗

**Figure 3.** The rate of precession of the longitude of periapsis for a system on the critical curves. The parameters and designations are the same as in [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗

Figures from the paper (2 more)

**Figure 4.** Figure 4: Critical curves in the (𝑎, 𝛿 ˜ ) plane. Each panel represents a different 𝜎 value specified in the top right corner. conserved quantities and the initial values of the orbital parameters. These are denoted as case A and case B. Case A is used to illustrate the theory d…

**Figure 5.** Figure 5: The orbital evolution of four systems in the vicinity of a critical curve. The left panels correspond to case A and the right panels to case B. The values in the legends are the average values of 𝐸/𝐸crit over the time interval illustrated and the initial values of 𝛿. T…

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Pith tools

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2 𝑒2(1−𝑒2)3/2 1+ 𝑒2 6 Ω𝑟,𝑘 𝜔𝑒𝑞,𝑘 2 sin𝛽𝑘 sin 2 ˆ𝜛𝑘, with 𝑇∗,𝑘 = 6𝜋 5 𝐺𝑀𝑗𝑄𝑒𝑞,𝑘 𝑎3(1−𝑒2)3√︁𝑁0,𝑘𝜔𝑒𝑞,𝑘 !2 = 3𝑘2,𝑘𝑀2 𝑗 𝑀T 𝑅∗,𝑘 𝑎 5 𝑛2 𝑜𝑎2(1−𝑒2)−6. (B1) Here we remind that𝑒 and𝑎 are orbital eccentricity and semi-major axis, 𝑛0 is the mean motion,𝑀𝑘, 𝑅∗,𝑘, and Ω𝑟,𝑘 are the mass, radius, and rotational frequency of the𝑘𝑡ℎ component, respectively, and 𝑗 ≠ 𝑘. The ...

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