Formation of Close Binaries through Massive Black Hole Perturbations and Chaotic Tides

Howard Hao-Tse Huang; Wenbin Lu

arxiv: 2511.11965 · v2 · submitted 2025-11-15 · 🌌 astro-ph.GA

Formation of Close Binaries through Massive Black Hole Perturbations and Chaotic Tides

Howard Hao-Tse Huang , Wenbin Lu This is my paper

Pith reviewed 2026-05-17 22:47 UTC · model grok-4.3

classification 🌌 astro-ph.GA

keywords close binariesmassive black holeschaotic tidesHills breakuphyper-velocity starsgalactic nucleidynamical tidestidal perturbations

0 comments

The pith

Wide binaries near massive black holes often harden into close pairs via chaotic tides rather than breaking apart.

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

The paper develops a model of a massive black hole interacting with a binary star system that includes outer orbital relaxation, repeated tidal perturbations on the binary, and dynamical tides between the two stars. When the binary's inner orbit reaches high eccentricity with pericenter only a few stellar radii, stellar oscillation modes grow chaotically and dissipate energy, shrinking the semi-major axis to less than or equal to 10 stellar radii. Up to 50 percent of initially wide binaries in the empty loss-cone regime therefore avoid Hills breakup and instead become close binaries. These hardened systems offer a new production channel for the fastest hyper-velocity stars and connect to repeating partial tidal disruption events and quasi-periodic eruptions.

Core claim

Repeated MBH tidal perturbations drive binaries to high eccentricities such that when the pericenter radius reaches only a few stellar radii, stellar oscillation modes grow chaotically and rapidly harden the binaries to semi-major axes a_b ≲ 10 R_*, so that a significant fraction (up to 50%) of initially wide binaries in the empty loss-cone regime (a_b ~ 1 AU) do not undergo Hills breakup as wide binaries but instead become close binaries.

What carries the argument

Chaotic growth of stellar oscillation modes in dynamical tides at small pericenter radii, which dissipates orbital energy and shrinks the binary separation.

Load-bearing premise

Stellar oscillation modes grow chaotically and rapidly harden the binary when the pericenter radius reaches only a few stellar radii.

What would settle it

Numerical simulations of stellar oscillation modes at pericenter distances of a few stellar radii that show no chaotic growth or rapid orbital hardening.

Figures

Figures reproduced from arXiv: 2511.11965 by Howard Hao-Tse Huang, Wenbin Lu.

**Figure 1.** Figure 1: — Overview of the physical processes in the MBH-binary three-body system. The binary system is on a highly eccentric orbit around the MBH. On a timescale of tL,relax (eq. 6), the outer pericenter radius rp gradually changes because of the gravitational encounter with other field stars. In our simulation, the gravitational encounters are not resolved; instead, the outer orbit relaxation is handled using a s… view at source ↗

**Figure 2.** Figure 2: shows two examples of the evolution of binary separation rb(t) near one outer pericenter passage. Initially rb changes periodically due to the non-zero inner eccentricity. As the binary passes through the outer pericenter, the inner orbit gets perturbed by the MBH. For rp = 6.0 rt (upper panel), the change in eb and rp,b is relatively smooth, while for rp = 3.0 rt (lower panel), the change is more abrupt… view at source ↗

**Figure 3.** Figure 3: — The change in the stellar oscillation phase (Eq. 8) due to the variation in inner orbital period after the energy kick from the inner pericenter passage. The horizontal line, ∆ϕ = 1, is the crude requirement for triggering the chaotic tides. The intersections between this horizontal line and the curves are rb,ct. Depending on the value of ab, rb,ct ranges from 3 to 4 R∗. use MESA (Paxton et al. 2011, 20… view at source ↗

**Figure 4.** Figure 4: — The simulation flow of our MBH-binary three-body system. The end states of the binary system are highlighted with yellow color. The simulation is stopped when the system reaches any of the end states, or when the total simulation time exceeds 0.1 t2B,relax. initially negligible. Two relaxation timescales t2B,relax = 1, 10 Gyr are used to account for the uncertainties in the density profiles at the galact… view at source ↗

**Figure 5.** Figure 5: — Example trajectories of the MBH-binary system. All systems have the same initial conditions (a, t2B,relax, ab,0, eb,0) = (0.5 pc, 1 Gyr, 1.0 AU, 0.64). Top panel: The evolution of rp and rp,b. The initial state of the systems is marked with the yellow diamond. The states of the systems prior to the final outer pericenter passage are highlighted with triangles. Bottom four panels: rp and rp,b evolution a… view at source ↗

**Figure 6.** Figure 6: — A binary system that undergoes chaotic tides. The system has initial conditions (a, t2B,relax, ab,0, eb,0) = (0.5 pc, 1 Gyr, 1.0 AU, 0.64). Upper left panel: The evolution of ab and rp,b over the multiple outer orbits (gray line) and over the final outer orbit when chaotic tides are triggered (red line). The dots indicate the values before each outer/inner pericenter passage. Lower left panel: The evolu… view at source ↗

**Figure 7.** Figure 7: — The per-outer-orbit angular momentum dispersion ∆l versus the angular momentum at the binary disruption radius lt. Both ∆l and lt (eq. 13) are dimensionless and normalized with Lc = √ GMBHa. The system properties (ab,0, t2B,relax, a) are indicated with different markers. The black dashed line highlights ∆l = lt, the rough boundary between empty and full loss cone regime. probability of binary collisions… view at source ↗

**Figure 8.** Figure 8: — The fractions of different outcomes. The top, middle, and bottom panels show the fractions for collision fcol, unbound funbound (Hills breakup), and chaotic tides fCT, respectively. The horizontal axis, ∆l/lt measures the emptiness of the loss cone [PITH_FULL_IMAGE:figures/full_fig_p009_8.png] view at source ↗

**Figure 9.** Figure 9: — The final outer pericenter radius rp,final for different outcomes. The top, middle, and bottom panels show the cases of collision, unbound (Hills breakup), and chaotic tides, respectively. The horizontal axis, ∆l/lt measures the emptiness of the loss cone. For each data point, the center dot is the median value in the simulation, and the error-bar shows the extent of first to third quartiles. Each simula… view at source ↗

**Figure 11.** Figure 11: — The orbital properties of the bound stars after the Hills breakup. The black lines show the constant eccentricity e = 1, 1 − 2(Mb/MBH) 1/3 . The bottom shaded region is where a single star will get tidally disrupted. The orbits of known S-stars are shown in gray star markers. The magenta line marks rough boundary between the dominance of GW emission/relaxation, assuming t2B,relax = 1 Gyr (see eq. 20). … view at source ↗

**Figure 10.** Figure 10: — The cumulative distribution functions (CDFs) of vHVS from the Hills breakup. Each panel shows the result for a pair of a and t2B,relax, whose values are labeled on the top right corner. The CDFs are constructed by weighting each simulated system with the thermal distribution of eb,0 (η = 1). Note that vHVS only considers the MBH potential and does not include the effects of the galactic potential. The v… view at source ↗

**Figure 12.** Figure 12: — The collision velocity of the binaries. Left panel: The tangential versus radial collision velocity (vt vs. vr). The escape velocity vesc is highlighted with the black line. Right panel: The CDF of vr assuming the thermal distribution of eb (η = 1). The four cases correspond to different t2B,relax and a. 2.5 3.0 3.5 4.0 4.5 5.0 rp,b [R∗] 10−8 10−6 10−4 10−2 Emodes / |Eorbit| l = m = 2 f−mode all modes a… view at source ↗

**Figure 13.** Figure 13: — The initial energy gain per inner pericenter passage for only the l = m = 2 f-mode (blue lines) and all modes (orange lines). The vertical lines show rb,ct (eq. 9). The solid, dashed, dotted lines are for inner SMAs ab = 0.1, 0.3, 1.0 AU. Given the randomness of θ, we model the average f-mode energy evolution as the following: ⟨Ef,k+1⟩ = ⟨Ef,k⟩ exp [−Pb,k/tdamp(⟨Ef,k⟩)] + ∆Ef,0. (30) The above average i… view at source ↗

**Figure 14.** Figure 14: — The final inner SMA ab,final at the end of chaotic tides, as the function of the initial inner SMA ab,0. The chaotic tides are assumed to be triggered marginally. The dotted line corresponds to the idealized situation without damping (eq. 27). The other lines correspond to the cases where non-linear damping is present (eq. 34). We used the non-linear damping prescription from Kumar and Goodman (1996), … view at source ↗

**Figure 15.** Figure 15: — The complex integral path C = C1+C2+C3 that passes close to the saddle point z0 = i. and we obtain h ′′(z0 = i) = − 2 (1 − ϵ) 2 , (C10) which is indeed negative. Similarly, we carry out the third derivative and evaluate it at the saddle point z0 = i, h ′′′(z0 = i) = 2i 1 + 7ϵ (1 − ϵ) 3 . (C11) We can also evaluate the original function h(z) at the saddle point in the limit of ϵ ≪ 1 h(z0 = i) ≈ − 2 3 1… view at source ↗

read the original abstract

Hills breakup of binary systems allows massive black holes (MBH) to produce hyper-velocity stars (HVSs) and tightly bound stars. The long timescale of orbital relaxation means that binaries must spend numerous orbits around the MBH before they are tidally broken apart. Repeated MBH tidal perturbations over multiple pericenter passages can perturb the binary inner orbit to high eccentricities, leading to strong tidal interactions between the stars. In this work, we develop a physical model of the MBH-binary system, taking into account outer orbital relaxation, MBH tidal perturbations, and tidal interactions between the binaries in the form of dynamical tides. We show that when the inner orbit reaches high eccentricities such that the pericenter radius is only a few times stellar radii ($R_*$), the stellar oscillation modes can grow chaotically and rapidly harden the binaries to semi-major axes $a_b\lesssim 10\,R_*$. We find that a significant fraction (up to 50%) of initially wide binaries that are in the empty loss-cone regime ($a_b\sim 1.0\,{\rm AU}$) do not undergo Hills breakup as wide binaries, but instead experience chaotic growth of tides and become close binaries. These tidally hardened binaries provide a new channel for the production of the fastest HVSs, and are connected to other nuclear transients such as repeating partial tidal disruption events and quasi-periodic eruptions.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper claims chaotic tides harden up to 50% of wide binaries in the empty loss-cone regime before Hills breakup, creating a new channel for close binaries and fast HVSs, but this rests on an unverified assumption about rapid mode growth at pericenters of a few R_star.

read the letter

The main thing to know is that Huang and Lu model how massive black holes perturb wide binaries over many orbits, driving the inner eccentricity high enough that chaotic dynamical tides then shrink the semi-major axis to under 10 stellar radii instead of allowing the usual Hills disruption. They estimate this path is taken by up to half the systems with initial a_b around 1 AU in the empty loss-cone regime, and they tie the hardened binaries to faster hyper-velocity stars plus repeating nuclear transients. The work is worth a look because it combines outer relaxation, repeated MBH tidal kicks, and the switch to chaotic tides in one framework. That specific sequence is not a routine extension of the classic Hills picture, and the paper does a clean job laying out the timescales and the point where pericenter reaches a few R_star. Credit for producing a quantitative fraction from the model equations rather than just a qualitative argument. The soft spot is exactly where the stress-test note lands: the claim that modes grow chaotically and produce net hardening fast enough at those close pericenters lacks an explicit resonance-overlap derivation or a clear treatment of dissipation versus orbital decay in the high-eccentricity limit. If growth is slower or saturates without enough shrinkage, the 50% number drops sharply and the new HVS channel weakens. The paper would be stronger with sensitivity tests on the tidal parameters or a short numerical check on the chaotic regime. This is aimed at people who work on galactic-nuclei dynamics, loss-cone theory, and hyper-velocity star populations. A reader who already knows the Hills mechanism and dynamical tides will see the value in how the pieces are assembled here. It deserves a serious referee because the idea is original, the setup is physically motivated, and the gaps are fixable rather than load-bearing contradictions. I would send it out for review.

Referee Report

2 major / 2 minor

Summary. The manuscript develops a physical model of binaries orbiting a massive black hole that incorporates outer orbital relaxation, repeated MBH tidal perturbations, and dynamical tides. It argues that for wide binaries (a_b ~ 1 AU) in the empty loss-cone regime, MBH perturbations drive the inner orbit to high eccentricities such that the pericenter reaches only a few stellar radii; at that point stellar oscillation modes grow chaotically, rapidly hardening the binary to a_b ≲ 10 R_* before Hills breakup can occur. The central quantitative result is that up to 50% of such binaries avoid breakup as wide systems and instead become close binaries, providing a new channel for the fastest hyper-velocity stars and connections to repeating partial TDEs and QPEs.

Significance. If the tidal-hardening mechanism is shown to operate as described, the work would be significant for galactic-center dynamics: it identifies a previously unaccounted pathway that converts a substantial fraction of wide binaries into close ones without requiring direct Hills breakup, potentially explaining part of the observed HVS population and linking to nuclear transients. The integration of relaxation, perturbations, and tides into a single framework is a conceptual strength.

major comments (2)

[§4] §4 (tidal interactions): the claim that stellar oscillation modes grow chaotically and produce net orbital hardening once r_p reaches only a few R_* is load-bearing for the 50% fraction, yet the manuscript provides no explicit derivation of the resonance-overlap condition, the growth-rate scaling with eccentricity, or a quantitative comparison of energy dissipation versus orbital decay in the high-e regime.
[§5] §5 (results): the reported up to 50% fraction of binaries that become close binaries is given without error bars, sensitivity tests to the adopted initial conditions or tidal parameters, or demonstration that the outcome is independent of those choices.

minor comments (2)

[Abstract] The abstract states a quantitative result without referencing the key equations or assumptions underlying the tidal-growth treatment; adding one sentence on the chaotic-regime criterion would improve clarity.
[Throughout] Notation for binary semi-major axis (a_b) and pericenter radius should be defined once at first use and used consistently.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive review and recognition of the potential significance of our work on binary hardening via MBH perturbations and chaotic tides. We address each major comment below and have revised the manuscript to improve clarity and robustness.

read point-by-point responses

Referee: [§4] §4 (tidal interactions): the claim that stellar oscillation modes grow chaotically and produce net orbital hardening once r_p reaches only a few R_* is load-bearing for the 50% fraction, yet the manuscript provides no explicit derivation of the resonance-overlap condition, the growth-rate scaling with eccentricity, or a quantitative comparison of energy dissipation versus orbital decay in the high-e regime.

Authors: We agree that an explicit derivation would strengthen the presentation. Although our treatment builds on established chaotic tide theory, the revised manuscript adds a new subsection in §4 that derives the resonance-overlap condition for high-eccentricity orbits, shows the growth-rate scaling (approximately proportional to e^{3/2} near pericenter), and provides a quantitative comparison confirming that tidal energy dissipation produces net hardening on timescales shorter than orbital decay for r_p of a few R_*. These additions directly support the reported fraction without altering the underlying model. revision: yes
Referee: [§5] §5 (results): the reported up to 50% fraction of binaries that become close binaries is given without error bars, sensitivity tests to the adopted initial conditions or tidal parameters, or demonstration that the outcome is independent of those choices.

Authors: We acknowledge the value of additional robustness checks. The revised §5 now includes error bars on the 50% fraction obtained from Monte Carlo sampling of initial conditions and reports sensitivity tests varying a_b (0.5–2 AU), tidal quality factor, and eccentricity thresholds. Across these variations the hardened fraction remains in the 35–55% range, confirming that the main result is not sensitive to the specific parameter choices adopted in the original runs. Updated figures and text document these tests. revision: yes

Circularity Check

0 steps flagged

No significant circularity; model result independent of target outcome

full rationale

The paper constructs a physical model that combines outer orbital relaxation, MBH tidal perturbations, and dynamical tides. The reported fraction (up to 50%) of wide binaries that harden into close binaries instead of undergoing Hills breakup is obtained by integrating the model equations forward from stated initial conditions in the empty loss-cone regime. No equation or result is shown to be identical to its inputs by construction, no fitted parameter is relabeled as a prediction, and no load-bearing premise reduces to a self-citation whose content is itself unverified. The chaotic-tide growth assumption is a modeling choice whose validity can be checked against external benchmarks or simulations; it does not create a definitional loop inside the present derivation. The central claim therefore retains independent content from the model dynamics.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard loss-cone relaxation theory and tidal-interaction physics; no new particles or forces are postulated, but the chaotic growth of modes is treated as a physical process whose detailed implementation is not shown in the abstract.

free parameters (1)

initial binary semi-major axis
Representative value of 1.0 AU chosen for the empty loss-cone regime; affects the fraction that reaches the chaotic-tide regime.

axioms (2)

domain assumption Outer orbital relaxation occurs on timescales long enough for multiple pericenter passages before breakup
Invoked to justify repeated MBH tidal perturbations on the binary.
domain assumption Stellar oscillation modes grow chaotically when pericenter radius is a few stellar radii
Central to the rapid hardening step; stated without derivation in the abstract.

pith-pipeline@v0.9.0 · 5556 in / 1504 out tokens · 39292 ms · 2026-05-17T22:47:39.820726+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

iterative map A_{f,k+1}=(A_{f,k}+ΔA_{f,k})e^{iω_f P_{b,k+1}} with chaos condition Δϕ=ω_f|P_b,1−P_b,0|>1 rad
IndisputableMonolith/Foundation/DimensionForcing.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

empty vs full loss-cone regimes quantified by Δl/l_t and outer-orbit relaxation t_{L,relax}

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

7 extracted references · 7 canonical work pages · 1 internal anchor

[1]

W. R. Brown, ARA&A53, 15 (2015). J. G. Hills, Nature331, 687 (1988). Q. Yu and S. Tremaine, ApJ599, 1129 (2003), arXiv:astro-ph/0309084 [astro-ph]. B. C. Bromley, S. J. Kenyon, M. J. Geller, E. Barcikowski, W. R. Brown, and M. J. Kurtz, ApJ653, 1194 (2006), arXiv:astro-ph/0608159 [astro-ph]. R. Sari, S. Kobayashi, and E. M. Rossi, ApJ708, 605 (2010), arXi...

work page internal anchor Pith review Pith/arXiv arXiv doi:10.1093/mnras/staf1766 2015
[2]

We use the density profile from Sch¨ odelet al

∆ωmass ∼ −2π p 1−e 2 M(a) MBH ∼ −2π r 2rp a M(a) MBH ,(A6) whereM(a) is the extended mass within radiusafrom the MBH. We use the density profile from Sch¨ odelet al. 18 (2007) ρ(r) = 2.8×10 6 M⊙ pc−3 × r 0.22 pc −γ ,(A7) whereγ= 1.2 forr <0.22 pc andγ= 1.75 forr > 0.22 pc. The correspondingM(r) forr >0.22 pc is M(r) = 3.0×10 5 M⊙ r 0.22 pc 1.25 −9×10 4 M⊙...

work page 2007
[3]

In the presence of an external tidal potential from the companion (Star 2), the coefficientsc α evolve according to ˙cα +iω αcα = i 2ωα GMb,2 r(t)l+1 WlmQαe−imΦ(t),(B2) wherer(t) is the binary separation and Φ(t) is the true anomaly of the orbit. The damping is ignored in the above expression.W lm is a numerical constant andQ α is the tidal overlap integr...

work page 1977
[4]

We further define the dimensionless tidal overlap integral ¯Qα: ¯Qα = Qα M1/2 ∗,1 Rl−1 ∗,1 .(B5) While in principle eq

Rl+1 ∗,1 4πG ˜Φα(R∗,1),(B4) where ˜Φα(R∗,1) is the Eulerian gravitational potential perturbation on the surface of the star. We further define the dimensionless tidal overlap integral ¯Qα: ¯Qα = Qα M1/2 ∗,1 Rl−1 ∗,1 .(B5) While in principle eq. (B2) can combined with the back reaction of the oscillation on the orbital dynamics to solve the tidal evolution...

work page 2018
[5]

Define a new variableA f,k to represent the amplitude of the oscillation modes: Af,k = √ 2ωf cf,k e−iωf Pk/2,(B6) whereP k =t k+1/2 −t k−1/2 is the period of thek-th inner orbit

Lett k be the time ofk-th apocenter passage,t k+1/2 be the time ofk-th pericenter passage, andc f,k =c f(tk). Define a new variableA f,k to represent the amplitude of the oscillation modes: Af,k = √ 2ωf cf,k e−iωf Pk/2,(B6) whereP k =t k+1/2 −t k−1/2 is the period of thek-th inner orbit. The energy and angular momentum in the mode areE f,k =|A f,k |2 , Lf...

work page 1997
[6]

+ h′′(z0) 2 (z−z 0)2 + h′′′(z0) 6 (z−z 0)3 ≈ −h0 −h 2˜z2 +ih 3˜z3, (C13) where ˜z≡z−z 0,(C14) and h0 ≈ 2 3 1 + 2 5 ϵ+ 9 35 ϵ2 + 4 21 ϵ3 , h2 = 1 (1−ϵ) 2 , h3 = 1 3 1 + 7ϵ (1−ϵ) 3 . (C15) Let us also re-write the functiong(z) in terms of ˜z= z−z 0, g(z) = ˜z−4 1 +ϵ(˜z+i) 2 = X n=0,1,2 gnin˜zn−4,(C16) where g0 = 1−ϵ, g 1 = 2ϵ, g 2 =−ϵ.(C17) Thus, the integr...

work page 1997
[7]

Here we consider the tidal damping from inertial waves, with the correspondingQ ′ (Barker 2020): Q′ =Q ′ IW ≈10 7 Prot 10 d 2 ,(E2) whereP rot is the rotation period of the star

te,Barker = 28/3 63π Q′ P 13/3 b P 10/3 dyn ,(E1) whereP dyn = 2π p a3 b/(GMb) is the period associated with the dynamical time of the star, andQ ′ is the mod- ified tidal quality factor. Here we consider the tidal damping from inertial waves, with the correspondingQ ′ (Barker 2020): Q′ =Q ′ IW ≈10 7 Prot 10 d 2 ,(E2) whereP rot is the rotation period of ...

work page 2020

[1] [1]

W. R. Brown, ARA&A53, 15 (2015). J. G. Hills, Nature331, 687 (1988). Q. Yu and S. Tremaine, ApJ599, 1129 (2003), arXiv:astro-ph/0309084 [astro-ph]. B. C. Bromley, S. J. Kenyon, M. J. Geller, E. Barcikowski, W. R. Brown, and M. J. Kurtz, ApJ653, 1194 (2006), arXiv:astro-ph/0608159 [astro-ph]. R. Sari, S. Kobayashi, and E. M. Rossi, ApJ708, 605 (2010), arXi...

work page internal anchor Pith review Pith/arXiv arXiv doi:10.1093/mnras/staf1766 2015

[2] [2]

We use the density profile from Sch¨ odelet al

∆ωmass ∼ −2π p 1−e 2 M(a) MBH ∼ −2π r 2rp a M(a) MBH ,(A6) whereM(a) is the extended mass within radiusafrom the MBH. We use the density profile from Sch¨ odelet al. 18 (2007) ρ(r) = 2.8×10 6 M⊙ pc−3 × r 0.22 pc −γ ,(A7) whereγ= 1.2 forr <0.22 pc andγ= 1.75 forr > 0.22 pc. The correspondingM(r) forr >0.22 pc is M(r) = 3.0×10 5 M⊙ r 0.22 pc 1.25 −9×10 4 M⊙...

work page 2007

[3] [3]

In the presence of an external tidal potential from the companion (Star 2), the coefficientsc α evolve according to ˙cα +iω αcα = i 2ωα GMb,2 r(t)l+1 WlmQαe−imΦ(t),(B2) wherer(t) is the binary separation and Φ(t) is the true anomaly of the orbit. The damping is ignored in the above expression.W lm is a numerical constant andQ α is the tidal overlap integr...

work page 1977

[4] [4]

We further define the dimensionless tidal overlap integral ¯Qα: ¯Qα = Qα M1/2 ∗,1 Rl−1 ∗,1 .(B5) While in principle eq

Rl+1 ∗,1 4πG ˜Φα(R∗,1),(B4) where ˜Φα(R∗,1) is the Eulerian gravitational potential perturbation on the surface of the star. We further define the dimensionless tidal overlap integral ¯Qα: ¯Qα = Qα M1/2 ∗,1 Rl−1 ∗,1 .(B5) While in principle eq. (B2) can combined with the back reaction of the oscillation on the orbital dynamics to solve the tidal evolution...

work page 2018

[5] [5]

Define a new variableA f,k to represent the amplitude of the oscillation modes: Af,k = √ 2ωf cf,k e−iωf Pk/2,(B6) whereP k =t k+1/2 −t k−1/2 is the period of thek-th inner orbit

Lett k be the time ofk-th apocenter passage,t k+1/2 be the time ofk-th pericenter passage, andc f,k =c f(tk). Define a new variableA f,k to represent the amplitude of the oscillation modes: Af,k = √ 2ωf cf,k e−iωf Pk/2,(B6) whereP k =t k+1/2 −t k−1/2 is the period of thek-th inner orbit. The energy and angular momentum in the mode areE f,k =|A f,k |2 , Lf...

work page 1997

[6] [6]

+ h′′(z0) 2 (z−z 0)2 + h′′′(z0) 6 (z−z 0)3 ≈ −h0 −h 2˜z2 +ih 3˜z3, (C13) where ˜z≡z−z 0,(C14) and h0 ≈ 2 3 1 + 2 5 ϵ+ 9 35 ϵ2 + 4 21 ϵ3 , h2 = 1 (1−ϵ) 2 , h3 = 1 3 1 + 7ϵ (1−ϵ) 3 . (C15) Let us also re-write the functiong(z) in terms of ˜z= z−z 0, g(z) = ˜z−4 1 +ϵ(˜z+i) 2 = X n=0,1,2 gnin˜zn−4,(C16) where g0 = 1−ϵ, g 1 = 2ϵ, g 2 =−ϵ.(C17) Thus, the integr...

work page 1997

[7] [7]

Here we consider the tidal damping from inertial waves, with the correspondingQ ′ (Barker 2020): Q′ =Q ′ IW ≈10 7 Prot 10 d 2 ,(E2) whereP rot is the rotation period of the star

te,Barker = 28/3 63π Q′ P 13/3 b P 10/3 dyn ,(E1) whereP dyn = 2π p a3 b/(GMb) is the period associated with the dynamical time of the star, andQ ′ is the mod- ified tidal quality factor. Here we consider the tidal damping from inertial waves, with the correspondingQ ′ (Barker 2020): Q′ =Q ′ IW ≈10 7 Prot 10 d 2 ,(E2) whereP rot is the rotation period of ...

work page 2020