Pith. sign in

REVIEW 3 major objections 5 minor 54 references

Nonlinear coupling between magnetar QPOs

T0 review · 3 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Nonlinear three-mode coupling can explain why magnetar QPOs appear and disappear and drift in frequency, the paper argues.

desk verdict New mechanism for QPO intermittence/drift via triplet coupling, but the quantitative timescale match relies on an unjustified 1000-fold detuning reduction; worth reviewing as a proof-of-concept. read the letter →

arxiv 2501.04556 v2 pith:QC4KKDVY submitted 2025-01-08 astro-ph.HE astro-ph.SRgr-qc

Pantelis Pnigouras , Samuel K. Lander This is my paper · ORCID
classification astro-ph.HEastro-ph.SRgr-qc
keywords magnetarsquasi-periodicoscillationsnonlinearmodecouplingthree-modeaxial-axial-polargiantgamma-rayflaresneutronstarinternalmagneticfieldgeometry
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 the intermittent appearance and frequency drift of magnetar quasi-periodic oscillations are not the behaviour of independently ringing modes but signatures of nonlinear coupling among triples of stellar oscillation modes. It selects a representative triplet of magneto-elastic modes that matches the frequencies observed after the giant flares of SGR 1806-20 and SGR 1900+14, solves the three-mode equations analytically, and shows that periodic energy exchange produces both amplitude modulation on 10–100 s timescales and frequency shifts up to a few hertz. The authors conclude that the dominant coupling is axial-axial-polar in character, and that the small polar fraction of one mode, parametrised by $\zeta$, can be tied to the strength of the toroidal component of the magnetar's internal magnetic field. If correct, the messy temporal behaviour of QPOs becomes a diagnostic of internal magnetic field geometry and a test for nonlinear physics in neutron-star oscillations.

What carries the argument

The load-bearing object is the three-mode coupling system with amplitude equations $\dot{Q}_\alpha = i\omega_\alpha \kappa Q_\beta Q_\gamma e^{-i\Delta\omega t}$ and cyclic permutations, where $\kappa$ is an overlap integral over the eigenfunctions of the three modes. For an axial-axial-polar triplet, $\kappa$ separates into an angular selection integral requiring $m_\alpha=m_\beta+m_\gamma$ and $l_\alpha=l_\beta+l_\gamma-2\lambda$, and a radial integral involving the polar divergence $f_\alpha\propto\zeta$. Each magneto-elastic mode is represented by the axisymmetric axial ansatz of Eq. (11), with sinusoidal radial dependence and nodes fixed by the mode indices; $\zeta$, the relative amplitude of the polar piece, is the single free parameter and is associated with the toroidal field strength. The analytic solution uses three conserved quantities ($E_1$, $E_2$, $L$) to reduce the dynamics to a cubic, solved with Jacobi elliptic functions, from which the modulation period and frequency shifts follow.

What would settle it

A calculation of the coupling coefficient $\kappa$ from the actual magneto-elastic eigenfunctions of Gabler et al. (2016), instead of the ansatz of Eq. (11), would settle the mechanism: if the overlap integrals are negligibly small or yield modulation periods far outside 10–100 s for realistic $\zeta$ and $\Delta\omega$, the claim fails. Observationally, continuous amplitude and frequency tracking in a future giant flare should show frequency drift peaking at amplitude minima; a QPO that reappears with no such correlated drift, or with aperiodic timing, would contradict the single-triplet model.

Watch

Extended reading notes

Core claim

The central claim is that a leading-order nonlinear resonance among one high-frequency parent mode and two lower-frequency daughter modes reproduces the observed disappearance, reappearance, and frequency drift of giant-flare QPOs. For the two best-studied magnetars, the paper identifies the triplet $2U_2$, $6U_2$, $6U_4$ from the magneto-elastic mode classification, with inferred field strengths $8.6\times10^{14}$ G and $7.2\times10^{14}$ G that agree with spin-down estimates. The coupled amplitude equations are solved exactly with Jacobi elliptic functions, giving a modulation period $T=2K(k)/\sqrt{E_1(q_3-q_1)}$ and frequency shifts that peak when the mode amplitude is smallest. Because the divergence of an axial mode vanishes, three purely axial modes cannot couple; the paper therefore argues the coupling is axial-axial-polar, with one mode carrying a small polar piece $\zeta$ proportional to the toroidal magnetic field. Small values of $\zeta$ and very small detuning $\Delta\omega/2\pi\sim 0.003$ Hz produce the 10–100 s modulation periods inferred from the flare tails.

Load-bearing premise

The load-bearing premise is that real magnetar oscillations resemble the paper's simplified model modes closely enough, and can sit close enough to a three-way frequency resonance (about 0.003 Hz), for the computed 10–100 s cycles and frequency shifts to match the data.

Editorial extensions

If this is right

  • The 26 and 30 Hz QPOs of SGR 1806 are the same mode seen at different phases of the coupling cycle, not independent modes.
  • The observed 10–100 s modulation timescales require a small polar fraction $\zeta$ and near-resonant detuning, so QPO timing data constrain the toroidal component of the internal magnetic field.
  • Frequency drift and amplitude should be anticorrelated: the largest frequency shift occurs just as the mode becomes hardest to detect.
  • The same triplet mechanism can be extended to the $\sim 57$, $\sim 90$, and $\sim 150$ Hz QPOs (since $57+90\approx 150$), implying coupling between the two triplets and richer, less strictly periodic variability.
  • High-frequency ($\gtrsim 600$ Hz) QPOs, if their eigenfunctions permit coupling, and QPOs seen in gamma-ray burst precursors and extragalactic magnetar flares could share the same explanation.

Reading between the lines

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

  • If the mechanism is correct, reanalysis of existing giant-flare tails for correlated amplitude and frequency behaviour could measure $\kappa$ without waiting for a new flare.
  • The requirement of a detuning near 0.003 Hz suggests either a fortuitous near-degeneracy in the magneto-elastic spectrum or a selection effect favouring the most resonant triplets; counting how many candidate triplets satisfy both the selection rules and this tolerance would give a testable probability.
  • Coupling between overlapping triplets is a natural route from the model's strictly periodic behaviour to the observed aperiodicity, and could be checked by adding the $\sim 150$ Hz mode to the coupled system.
  • The same analytic solution could be transferred to other neutron-star oscillation contexts, such as post-merger remnants, where mode energy exchange and frequency drift might be observable.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper proposes that the intermittent detectability and frequency drifts of magnetar QPOs arise from nonlinear three-mode coupling. Selecting a specific triplet of magneto-elastic modes (2U2, 6U2, 6U4) from Gabler et al. (2016), the authors derive a coupling coefficient for axial-axial-polar interactions, solve the three-mode amplitude equations analytically in terms of Jacobi elliptic functions, and demonstrate periodic energy exchange and frequency shifts. They argue that the coupling is likely axial-axial-polar and that the observed phenomenology could constrain the internal magnetic field geometry. The analytical solution and the selection rules are internally consistent, and the paper is clearly written; however, the quantitative demonstration of 10–100 s modulation periods relies on an ad hoc reduction of the detuning by a factor of 1000 and on a simplified sinusoidal eigenfunction ansatz.

Significance. If the central claim holds, this work opens a new observational window: nonlinear mode coupling would explain not just the presence of QPO frequencies but also their time-dependent behaviour, potentially linking QPO intermittence and drift to the star's internal magnetic field geometry. The analytic solution of the three-mode system is elegant and may be useful beyond magnetar QPOs, e.g., in other astroseismic contexts. The authors also correctly identify the limitation that their model predicts strictly periodic variability, whereas the observed QPO behaviour is irregular. The main significance is conditional on the quantitative match to observed timescales, which is currently not supported by the stellar-model frequencies without a 1000-fold arbitrary reduction of the detuning.

major comments (3)
  1. [Section 5, Figure 2 (and text near Eq. 23)] The reduction of the detuning from Δω/2π = 3 Hz to 0.003 Hz is a load-bearing ad hoc step with no physical justification. As the authors note, at the physical detuning of 3 Hz the system is already near the coupling threshold ζlim ≈ 4×10^-4 (Figure 1), and the values of ζ needed to obtain 10–100 s modulation periods (ζ ≈ 5×10^-6 to 3×10^-5) are below that threshold, so the amplitudes would remain effectively constant (Eq. 23). Only by multiplying the detuning by 10^-3 does the nonlinear resonance become strong enough at small ζ. This is not a minor parameter adjustment; it changes the resonance condition qualitatively. The authors must either provide a physical mechanism that reduces the effective detuning (e.g., uncertainties in the mode frequencies, magnetic-field evolution, or coupling to other modes) or present the model as qualitative only, explicitly abandoning the quantitative match to observed 10–100 s timescales.
  2. [Section 4, Eq. (11)] The replacement of the Gabler et al. (2016) magneto-elastic eigenfunctions with the axisymmetric axial ansatz of Eq. (11), using the same sinusoidal radial dependence for all components, is a severe simplification. The coupling coefficient κ in Eq. (14) depends directly on this ansatz through the functions fα, gβ, gγ and the radial integrals; consequently, the magnitude of κ, the threshold ζlim, and the modulation periods T (Eq. 28) are all sensitive to the assumed radial and angular structure. The paper does not test whether a different, more realistic eigenfunction shape (e.g., with nodes/maxima following magnetic field lines) changes the coupling timescales by orders of magnitude. Without such a sensitivity test, the claimed quantitative agreement with observed intermittence and drift timescales is not established.
  3. [Section 6 (Discussion)] The abstract and Section 6 assert that observed intermittence and frequency drifts 'provide a way to infer details of the magnetar's internal magnetic field geometry'. This is premature because ζ is treated as a completely free parameter; the paper only argues qualitatively that ζ could be related to the toroidal-to-poloidal field ratio, without computing ζ from any magnetic-field model or demonstrating that the required values (≈10^-5) are plausible for realistic field geometries. To support the claim, the authors should either connect ζ to a concrete field model and estimate its expected range, or soften the conclusion to state that the mechanism is consistent with magnetic-field geometry inference only if such a connection is established.
minor comments (5)
  1. [Section 3, Eq. (9)] The selection rule (9) is incomplete: it enforces l_a = l_b + l_c - 2λ, which gives the upper bound and parity, but not the lower bound l_a ≥ |l_b - l_c|. The Gaunt integral also vanishes unless the triangle inequality holds. The error does not affect the specific triplet used, but the displayed rule should be corrected for completeness.
  2. [Section 4, Eq. (11)] The notation 'lUn modes' is not defined on first use; the authors later explain that n and l are not the usual spherical-harmonic degrees, but a sentence of clarification would help readers not familiar with Gabler et al. (2016).
  3. [Section 5, Figure 2 caption] The phrase 'detuning dropped by a factor of 1000' should be 'reduced by a factor of 1000' for clarity; 'dropped' is informal.
  4. [Section 5, Eq. (18d)] In the expression for φ̇, the term '+ Δω' could be mistaken for a frequency variable; adding a tilde or a comment that Δω is the detuning would improve readability.
  5. [Section 6] The discussion of the second triplet (57, 90, 150 Hz) and of dissipation is appropriate, but the paper does not quantify how these effects would alter the strict periodicity of the solution; a brief comment that the real data's aperiodicity is an expected consequence of such extensions would strengthen the presentation.

Circularity Check

2 steps flagged · score 4.0 of 10

The core three-mode coupling mechanism is derived independently, but the quantitative match to observed 10-100 s intermittence is obtained by arbitrarily reducing the detuning from 3 Hz to 0.003 Hz, and the QPO grouping presumes the very frequency drift the model claims to explain.

  1. fitted input called prediction [Section 5, paragraph introducing Figure 2 (after Eq. 28)]
    "In order to explain the appearance and disappearance of modes over timescales of order 10−100 s, which is what QPO observations suggest, we need lower values of ζ. Let us, therefore, reduce the original value of the detuning (∆ω/2π = 3 Hz) by a factor of 1000. For this case, we plot |Q| and ∆ν in Figure 2 ... for ζ = 5 × 10−6 we obtain T ≈ 50 s"

    The target quantity (10-100 s intermittence timescales) is not an output of the model but the criterion used to pick the detuning. Equation (28) makes T a function of the roots q_i of the cubic in Eq. (23), which contains (Δω/2 q − L/E1)^2; reducing Δω/2π from 3 Hz to 0.003 Hz is what lowers ζlim and allows T ≈ 50 s at ζ = 5 × 10^-6, whereas at the original detuning of 3 Hz Figure 1 shows that such a small ζ would fall below ζlim and freeze the amplitudes. The subsequent Figures 2-3 therefore demonstrate a parameter choice, not a derivation of the observed timescale. Since the reduction is admitted with no physical mechanism, the 'explanation' of the observed intermittence is fitted by construction.

  2. self definitional [Section 2, mode-selection paragraph before 'Our ansatz...']
    "Since mode coupling is known to cause frequency drift (see Section 5), we assume that QPOs of similar frequency represent the same underlying mode; i.e., for SGR 1806, we regard the strong 92.5 Hz and weak 95 Hz QPOs of Israel et al. (2005) and the 90 Hz QPO of Strohmayer & Watts (2006) as the same mode; likewise for the 30 Hz (Israel et al. 2005) and 26 Hz (Watts & Strohmayer 2006) QPOs."

    The paper defines which observed QPOs belong to the same mode by invoking the frequency-drift effect that Section 5 derives from the nonlinear-coupling model. The grouped QPO frequencies then constitute the input triplet, and in Section 6 the same effect is presented as 'providing a concrete physical reason to believe that, e.g., the 26 and 30 Hz QPOs of SGR 1806 are the same oscillation mode.' The observational input is thus constructed from the conclusion it is used to support. This circularity affects mode identification rather than the ODE solution itself, so it is a secondary rather than primary issue.

full rationale

The central mechanism is not circular: the three-mode equations (4), the selection rules (8)-(9), and the exact solution (27)-(28) are derived from standard perturbation theory and reproduce the known periodic amplitude exchange independently of any parameter choice; the axial-axial-polar selection argument also follows from the vanishing divergence of axial modes. The paper does not pass off a self-citation chain as proof: the magneto-elastic mode frequencies come from external Gabler et al. (2016) fitting formulae, and the coupling formalism from external Schenk et al. (2001). However, two steps are circular in a partial, quantitative sense. First, the observed 10-100 s intermittence timescale is used as the target for choosing Δω/2π = 0.003 Hz, a factor-1000 reduction from the 3 Hz detuning implied by the fitted triplet; Eq. (23) shows the detuning controls the coupling threshold, so Figures 2-3 exhibit a fitted parameter rather than a prediction. Second, the grouping of 92.5/95/90 Hz and 30/26 Hz QPOs as single modes is justified by the frequency-drift effect that the paper then claims to explain. These are real but limited circularities; the existence and qualitative nature of the nonlinear coupling remain independent content, so the overall score is 4 rather than higher.

Assumptions & free parameters 5 free parameters · 6 assumptions · 0 invented entities

The central mechanism rests on standard three-mode perturbation theory plus several domain assumptions: the lUn mode identification, the sinusoidal eigenfunction ansatz, the axial-polar mixing parameter zeta, and an artificially reduced detuning. The fitted numerical parameters are B, zeta, initial amplitudes, and the detuning reduction; these control the quantitative match to observations.

free parameters (5)
  • zeta (polar-to-axial amplitude ratio of parent mode) = 4e-4, 5e-6, 1e-5, 3e-5 (values chosen per figure)
    Controls coupling strength kappa and hence modulation period T and frequency-shift magnitude; set by hand to illustrate timescales, not measured or predicted independently.
  • Reduced detuning Delta-omega/2pi = 0.003 Hz (original 3 Hz divided by 1000)
    In Figure 2 the detuning is artificially reduced by a factor of 1000 solely to push modulation periods into the observed 10-100 s range; no physical mechanism for this fine-tuning is given.
  • Initial mode amplitudes |Q_alpha|, |Q_beta|, |Q_gamma| = 10^-3, 10^-5, 10^-5
    Chosen initial conditions; the authors note that increasing amplitudes is equivalent to increasing zeta, so this choice helps set the nonlinearity depth.
  • Magnetic field strength B = 8.6e14 G (SGR 1806), 7.2e14 G (SGR 1900)
    Adjusted so the Gabler et al. (2016) mode frequencies best match observed QPO frequencies; consistent with spindown estimates but still a fit, not an independent prediction.
  • Amplitude detection cutoff = |Q| = 4e-4
    Arbitrary visibility threshold used in Figure 3 to mimic mode disappearance; any cutoff changes the apparent on/off pattern.
assumptions (6)
  • standard math Second-order perturbation theory with the three-mode amplitude equations (4) is adequate for magnetar QPO amplitudes.
    The coupled-mode formalism of Dziembowski (1982) and Schenk et al. (2001) is taken as given; no higher-order terms or dissipation are included.
  • ad hoc to paper The Gabler et al. (2016) lUn modes used for the triplet are the correct physical modes behind the observed QPOs, and their eigenfunctions can be approximated by Eq. (11).
    Section 4 replaces the true magneto-elastic eigenfunctions with an axisymmetric axial mode with sinusoidal radial dependence, an approximation the authors acknowledge makes the calculation qualitative.
  • domain assumption A linked poloidal-toroidal background magnetic field induces a polar piece in axial modes with relative amplitude zeta.
    Used to justify the axial-axial-polar coupling and to identify zeta with toroidal field strength; referenced to Colaiuda and Kokkotas (2012) rather than demonstrated for this model.
  • domain assumption The chosen QPOs at similar frequencies from different observations are the same underlying mode despite frequency shifts.
    Section 2 merges 26/30 Hz and 90/92.5/95 Hz QPOs into single modes; this identification is necessary for the triplet but not independently confirmed.
  • domain assumption Background star is a Gamma=2 polytrope with Gamma1=2.1, M=1.4 solar masses, and R=10 km.
    Section 4 sets the background model to allow analytic evaluation; results may depend on this choice.
  • domain assumption Core continuum damping and dissipation are negligible over the modelled timescales.
    The equations of motion (4) omit damping; the discussion mentions dissipation only as future complexity, so the periodic energy exchange may be modified in reality.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Nonlinear coupling between magnetar QPOs." pith.science (2026). https://pith.science/paper/QC4KKDVY

@misc{pith2026250104556,
  author       = {Pith},
  title        = {Pith review of: Nonlinear coupling between magnetar QPOs},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QC4KKDVY}},
  note         = {Machine review of arXiv:2501.04556}
}
abstract

The quasi-periodic oscillations (QPOs) observed in the tails of magnetar giant $\gamma$-ray flares have long been interpreted as normal oscillation modes of these stars. However, most studies modelling QPOs have neglected some key features in the analyses of the signals, namely that QPOs appear to be detectable only intermittently and exhibit drifts in their frequencies. These are typical characteristics of nonlinear mode coupling, where, at leading order, the modes couple and evolve collectively as triplets. Using a representative triplet of modes, we solve the system's nonlinear equations of motion analytically and argue that the coupling is likely axial-axial-polar in nature, with the observed intermittence and frequency drifts providing a way to infer details of the magnetar's internal magnetic field geometry.

Figures

Figures reproduced from arXiv: 2501.04556 by the authors.

Figure 1
Figure 1. Evolution of our representative coupled mode triplet, with α labelling the high-frequency (parent) mode and β, γ the mid- and low-frequency (daughter) modes, re￾spectively. We plot the mode amplitude |Q| (top) and mode frequency shift ∆ν ≡ ϑ/˙ 2π in Hz (bottom) as functions of time, for ζ = 4 × 10−4 and initial conditions |Qα| = 10−3 , |Qβ| = |Qγ| = 10−5 . |Qβ| = |Qγ| = 10−5 . The energy exchange between the modes, … view at source ↗
Figure 2
Figure 2. The effect of varying ζ on |Q| (top panels) and ∆ν (bottom panels), for the same initial conditions as for [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Evolution of |Q| and ∆ν from the first column of [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

54 extracted references · 17 canonical work pages

  1. [1]

    , " * write output.state after.block = add.period write newline

    ENTRY address archivePrefix author booktitle chapter doi edition editor eprint howpublished institution journal key month number organization pages publisher school series title misctitle type volume year version url label extra.label sort.label short.list INTEGERS output.state before.all mid.sentence after.sentence after.block FUNCTION init.state.consts ...

  2. [2]

    write newline

    " write newline "" before.all 'output.state := FUNCTION format.url url empty "" new.block "" url * "" * if FUNCTION format.eprint eprint empty "" archivePrefix empty "" archivePrefix "arXiv" = new.block " " eprint * " " * new.block " " eprint * " " * if if if FUNCTION format.doi doi empty "" " " doi * " " * if FUNCTION format.pid doi empty eprint empty ur...

  3. [3]

    ange subappendix Anhang Anh\

    thebibliography [1] 20pt to REFERENCES 6pt =0pt -12pt 10pt plus 3pt =0pt =0pt =1pt plus 1pt =0pt =0pt -12pt =13pt plus 1pt =20pt =13pt plus 1pt \@M =10000 =-1.0em =0pt =0pt 0pt =0pt =1.0em @enumiv\@empty 10000 10000 `\.\@m \@noitemerr \@latex@warning Empty `thebibliography' environment \@ifnextchar \@reference \@latexerr Missing key on reference command E...

    2021

  4. [4]

    Abramowitz , M., & Stegun , I. A. 1972, H andbook of M athematical F unctions (New York: Dover). http://adsabs.harvard.edu/abs/1972hmfw.book.....A

    work page 1972

  5. [5]

    A., & Pons , J

    Akg \"u n , T., Cerd \'a -Dur \'a n , P., Miralles , J. A., & Pons , J. A. 2018, , 481, 5331, 10.1093/mnras/sty2669

    work page doi:10.1093/mnras/sty2669 2018

  6. [6]

    2020, , 101, 083001, 10.1103/PhysRevD.101.083001

    Andersson , N., & Pnigouras , P. 2020, , 101, 083001, 10.1103/PhysRevD.101.083001

    doi:10.1103/physrevd.101.083001 2020

  7. [7]

    \'E ., Morsink , S

    Arras , P., Flanagan , \'E . \'E ., Morsink , S. M., et al. 2003, , 591, 1129, 10.1086/374657

    doi:10.1086/374657 2003

  8. [8]

    J., stgaard , N., G \"o g \"u s , E., et al

    Castro-Tirado , A. J., stgaard , N., G \"o g \"u s , E., et al. 2021, , 600, 621, 10.1038/s41586-021-04101-1

    doi:10.1038/s41586-021-04101-1 2021

Show all 54 references
  1. [10]

    Chirenti , C., Dichiara , S., Lien , A., & Miller , M. C. 2024, , 967, 26, 10.3847/1538-4357/ad3bb7

    2024 doi

  2. [11]

    Colaiuda , A., & Kokkotas , K. D. 2012, , 423, 811, 10.1111/j.1365-2966.2012.20919.x

    2012

  3. [12]

    Duncan , R. C. 1998, , 498, L45, 10.1086/311303

    1998 doi

  4. [13]

    1982, Acta Astron., 32, 147

    Dziembowski , W. 1982, Acta Astron., 32, 147. http://adsabs.harvard.edu/abs/1982AcA....32..147D

    1982

  5. [14]

    A., M \"u ller , E., & Stergioulas , N

    Gabler , M., Cerd \'a -Dur \'a n , P., Font , J. A., M \"u ller , E., & Stergioulas , N. 2013 a , , 430, 1811, 10.1093/mnras/sts721

    2013 doi

  6. [15]

    A., & M \"u ller , E

    Gabler , M., Cerd \'a -Dur \'a n , P., Stergioulas , N., Font , J. A., & M \"u ller , E. 2013 b , , 111, 211102, 10.1103/PhysRevLett.111.211102

    2013 doi

  7. [16]

    2014, , 443, 1416, 10.1093/mnras/stu1263

    ---. 2014, , 443, 1416, 10.1093/mnras/stu1263

    2014 doi

  8. [17]

    2016, , 460, 4242, 10.1093/mnras/stw1272

    ---. 2016, , 460, 4242, 10.1093/mnras/stw1272

    2016 doi

  9. [18]

    Hambaryan , V., Neuh \"a user , R., & Kokkotas , K. D. 2011, , 528, A45, 10.1051/0004-6361/201015273

    2011 doi

  10. [19]

    M., Watts , A

    Huppenkothen , D., Heil , L. M., Watts , A. L., & G \"o g \"u s , E. 2014, , 795, 114, 10.1088/0004-637X/795/2/114

    2014 doi

  11. [20]

    L., Belloni , T., Stella , L., et al

    Israel , G. L., Belloni , T., Stella , L., et al. 2005, , 628, L53, 10.1086/432615

    2005 doi

  12. [21]

    1998, , 393, 235, 10.1038/30410

    Kouveliotou , C., Dieters , S., Strohmayer , T., et al. 1998, , 393, 235, 10.1038/30410

    1998 doi

  13. [23]

    K., Jones , D

    Lander , S. K., Jones , D. I., & Passamonti , A. 2010, , 405, 318, 10.1111/j.1365-2966.2010.16435.x

    2010

  14. [24]

    2008, , 385, 2069, 10.1111/j.1365-2966.2008.12965.x

    Lee , U. 2008, , 385, 2069, 10.1111/j.1365-2966.2008.12965.x

    2008

  15. [25]

    2007, , 377, 159, 10.1111/j.1365-2966.2007.11582.x

    Levin , Y. 2007, , 377, 159, 10.1111/j.1365-2966.2007.11582.x

    2007

  16. [26]

    Markey , P., & Tayler , R. J. 1973, , 163, 77, 10.1093/mnras/163.1.77

    1973 doi

  17. [27]

    2006, , 653, 1423, 10.1086/508682

    Mereghetti , S., Esposito , P., Tiengo , A., et al. 2006, , 653, 1423, 10.1086/508682

    2006 doi

  18. [28]

    C., Chirenti , C., & Strohmayer , T

    Miller , M. C., Chirenti , C., & Strohmayer , T. E. 2019, , 871, 95, 10.3847/1538-4357/aaf5ce

    2019 doi

  19. [29]

    H., & Mook , D

    Nayfeh , A. H., & Mook , D. T. 1979, N onlinear O scillations (New York: John Wiley & Sons). http://adsabs.harvard.edu/abs/1979noos.book.....N

    1979

  20. [30]

    M., Barthelmy , S., Gehrels , N., et al

    Palmer , D. M., Barthelmy , S., Gehrels , N., et al. 2005, , 434, 1107, 10.1038/nature03525

    2005 doi

  21. [31]

    Passamonti , A., & Lander , S. K. 2013, , 429, 767, 10.1093/mnras/sts372

    2013 doi

  22. [32]

    2014, , 438, 156, 10.1093/mnras/stt2134

    ---. 2014, , 438, 156, 10.1093/mnras/stt2134

    2014 doi

  23. [33]

    Pnigouras , P., & Kokkotas , K. D. 2015, , 92, 084018, 10.1103/PhysRevD.92.084018

    2015 doi

  24. [34]

    2016, , 94, 024053, 10.1103/PhysRevD.94.024053

    ---. 2016, , 94, 024053, 10.1103/PhysRevD.94.024053

    2016 doi

  25. [35]

    Pumpe , D., Gabler , M., Steininger , T., & En lin , T. A. 2018, , 610, A61, 10.1051/0004-6361/201731800

    2018 doi

  26. [36]

    J., Baring , M

    Roberts , O. J., Baring , M. G., Huppenkothen , D., et al. 2023, , 956, L27, 10.3847/2041-8213/acfcad

    2023 doi

  27. [37]

    J., & Napolitano, J

    Sakurai, J. J., & Napolitano, J. 2011, M odern Q uantum M echanics, 2nd edn. (San Francisco: Addison-Wesley). http://adsabs.harvard.edu/abs/1985mqm..book.....S

    2011

  28. [39]

    K., Arras , P., Flanagan , \'E

    Schenk , A. K., Arras , P., Flanagan , \'E . \'E ., Teukolsky , S. A., & Wasserman , I. 2001, , 65, 024001, 10.1103/PhysRevD.65.024001

    2001 doi

  29. [40]

    2024, , 109, 023030, 10.1103/PhysRevD.109.023030

    Sotani , H. 2024, , 109, 023030, 10.1103/PhysRevD.109.023030

    2024 doi

  30. [41]

    Sotani , H., & Kokkotas , K. D. 2009, , 395, 1163, 10.1111/j.1365-2966.2009.14631.x

    2009

  31. [42]

    D., & Stergioulas , N

    Sotani , H., Kokkotas , K. D., & Stergioulas , N. 2007, , 375, 261, 10.1111/j.1365-2966.2006.11304.x

    2007

  32. [43]

    2012, , 108, 201101, 10.1103/PhysRevLett.108.201101

    Sotani , H., Nakazato , K., Iida , K., & Oyamatsu , K. 2012, , 108, 201101, 10.1103/PhysRevLett.108.201101

    2012 doi

  33. [44]

    G., & Kokkotas , K

    Sotani , H., Suvorov , A. G., & Kokkotas , K. D. 2024, , 110, 023035, 10.1103/PhysRevD.110.023035

    2024 doi

  34. [45]

    W., & Watts , A

    Steiner , A. W., & Watts , A. L. 2009, , 103, 181101, 10.1103/PhysRevLett.103.181101

    2009 doi

  35. [46]

    E., & Watts , A

    Strohmayer , T. E., & Watts , A. L. 2005, , 632, L111, 10.1086/497911

    2005 doi

  36. [47]

    2006, , 653, 593, 10.1086/508703

    ---. 2006, , 653, 593, 10.1086/508703

    2006 doi

  37. [48]

    G., Kuan , H

    Suvorov , A. G., Kuan , H. J., & Kokkotas , K. D. 2022, , 664, A177, 10.1051/0004-6361/202244082

    2022 doi

  38. [49]

    1989, N onradial O scillations of S tars, 2nd edn

    Unno , W., Osaki , Y., Ando , H., Saio , H., & Shibahashi , H. 1989, N onradial O scillations of S tars, 2nd edn. (Tokyo: University of Tokyo Press). http://adsabs.harvard.edu/abs/1989nos..book.....U

    1989

  39. [50]

    2012, , 420, 3035, 10.1111/j.1365-2966.2011.20177.x

    van Hoven , M., & Levin , Y. 2012, , 420, 3035, 10.1111/j.1365-2966.2011.20177.x

    2012

  40. [51]

    L., & Strohmayer , T

    Watts , A. L., & Strohmayer , T. E. 2006, , 637, L117, 10.1086/500735

    2006 doi

  41. [52]

    N., Arras , P., & Burkart , J

    Weinberg , N. N., Arras , P., & Burkart , J. 2013, , 769, 121, 10.1088/0004-637X/769/2/121

    2013 doi

  42. [53]

    N., Arras , P., Quataert , E., & Burkart , J

    Weinberg , N. N., Arras , P., Quataert , E., & Burkart , J. 2012, , 751, 136, 10.1088/0004-637X/751/2/136

    2012 doi

  43. [54]

    Wright , G. A. E. 1973, , 162, 339, 10.1093/mnras/162.4.339

    1973 doi

  44. [55]

    2001, , 546, 469, 10.1086/318234

    Wu , Y., & Goldreich , P. 2001, , 546, 469, 10.1086/318234

    2001 doi

  45. [56]

    2024, , 970, 6, 10.3847/1538-4357/ad4ee1

    Xiao , S., Zhang , Y.-Q., Zhu , Z.-P., et al. 2024, , 970, 6, 10.3847/1538-4357/ad4ee1

    2024 doi

  46. [57]

    G., Kouveliotou , C., et al

    Younes , G., Baring , M. G., Kouveliotou , C., et al. 2017, , 851, 17, 10.3847/1538-4357/aa96fd

    2017 doi

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.