Pith. sign in

REVIEW 3 major objections 5 minor 1 cited by

Generalizing the relativistic precession model of quasi-periodic oscillations through anharmonic corrections

T0 review · 3 major / 5 minor · reviewed 2026-08-16 · deepseek-v4-flash

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

Pith's one-line read The standard harmonic treatment of quasi-periodic oscillations omits a quadratic radial correction that becomes important near the innermost stable orbit.

desk verdict A real theoretical extension of the RPM with honest statistics, but the fitted amplitudes violate the perturbative regime and the decoupling step is inconsistent, so the 'necessary' claim isn't yet established. read the letter →

arxiv 2504.18403 v1 pith:HFZKKDJS submitted 2025-04-25 gr-qc astro-ph.GA

Roberto Giambò , Orlando Luongo , Marco Muccino , Alessandro Rossi This is my paper · ORCID
classification gr-qcastro-ph.GA PACS 04.70.-s04.70.Bw04.40.Dg02.70.Uu
keywords quasi-periodicoscillationsrelativisticprecessionmodelanharmoniccorrectionsradialepicyclicfrequencyKerrspacetimeneutronstarsHelmholtzoscillatorMCMCparameterestimation
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 claims that the standard relativistic precession model of quasi-periodic oscillations, which treats orbit perturbations as harmonic oscillators, is missing a term that cannot be discarded in advance: expanding the effective potential to third order produces a radial restoring force quadratic in the displacement, $\propto \delta r^2$, in any stationary, axisymmetric, asymptotically flat spacetime. Near the innermost stable circular orbit this quadratic term rivals the harmonic one and shifts the radial epicyclic frequency, while the polar mode stays approximately decoupled. The authors test this by fitting eight neutron star sources first in the harmonic approximation in Schwarzschild, Schwarzschild--de Sitter, and Kerr spacetimes, then with the anharmonic correction in Kerr. They conclude that the anharmonic correction is physically necessary and statistically relevant but still insufficient to fully account for the observed quasi-periodic oscillations, so the relativistic precession model itself needs a more fundamental revision.

What carries the argument

The load-bearing object is the third-order Taylor expansion of the effective potential for a test particle orbiting in a generic stationary, axisymmetric spacetime. Stability kills the linear terms at the equilibrium orbit, and the surviving third-order terms reduce the radial perturbation to an undamped Helmholtz oscillator, $\delta r'' + \alpha_0 \delta r + \alpha_1 \delta r^2 = 0$, with $\alpha_0$ the harmonic radial epicyclic coefficient and $\alpha_1$ the quadratic coefficient. The exact solution is $\delta r(t) = A + B\,\mathrm{sn}^2(\omega t, k)$, where $\mathrm{sn}$ is the Jacobi elliptic sine, and the angular frequency is built from the complete elliptic integral of the first kind; its small-$\alpha_1$ expansion gives the amplitude-dependent frequency shift above. This object carries the argument because it converts the geometric statement that the quadratic correction is universal into a concrete shift of the predicted periastron-precession frequency that can be compared with the observed $f_L$--$f_U$ relation.

What would settle it

Integrate the full geodesic equations numerically for the best-fit anharmonic Kerr parameters of one source, compute the resulting $f_L$--$f_U$ curve without the small-amplitude expansion, and compare it with the paper's Eq. (24) against the same data; a disagreement larger than the quoted uncertainties would show the anharmonic frequency formula, not the overall model, is responsible for the residual discrepancy.

Watch

Extended reading notes

Core claim

Regardless of the spacetime geometry, the leading anharmonic correction to the radial epicyclic oscillation is quadratic in the radial displacement, $\propto \delta r^2$, and it is not negligible near the innermost stable circular orbit. Expanding the effective potential about a circular equatorial orbit to third order, the paper shows that the coupling terms between the radial and polar perturbations are subdominant, leaving the radial equation as an undamped Helmholtz oscillator, $\delta r'' + \alpha_0 \delta r + \alpha_1 \delta r^2 = 0$. The exact Jacobi-elliptic solution makes the radial epicyclic frequency amplitude dependent, with leading correction $\Omega_r = \sqrt{\alpha_0}\,\left(1 - \frac{5A^2\alpha_1^2}{12\alpha_0^2}\right) + \mathcal{O}(\alpha_1^3)$ for small $\alpha_1$. Fitting the lower and upper quasi-periodic frequencies of eight neutron stars shows that anharmonic Kerr models become viable for some sources, but not for all; in the harmonic limit six of the eight sources prefer Schwarzschild--de Sitter over Kerr, which the authors interpret as a failure of the harmonic approximation and of the Kerr hypothesis within the standard relativistic precession model.

Load-bearing premise

The load-bearing premise is that the radial oscillation amplitude is small and is captured by a single parameter per source; posterior amplitudes that reach several kilometers are comparable to the orbital radius of a neutron star, and at such amplitudes the small-displacement expansion used to derive the frequency shift is outside its validity range.

Editorial extensions

If this is right

  • Any analysis that stops at the harmonic order will misestimate the radial epicyclic frequency for orbits near the innermost stable circular orbit, so masses and spins inferred from such fits carry a systematic error.
  • In the anharmonic model the radial frequency depends on the oscillation amplitude, so fitting quasi-periodic oscillation data requires choosing an amplitude function rather than a purely geometric frequency relation.
  • Polar oscillations remain approximately decoupled from radial ones, so the two-frequency structure of the relativistic precession model survives the extension.
  • The anharmonic correction is not enough: Kerr remains physically excluded for most of the eight sources, and the harmonic preference for Schwarzschild--de Sitter does not disappear, so the model must be modified beyond this term.

Reading between the lines

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

  • Editorial inference: if the quadratic radial correction is genuinely spacetime-independent, the same frequency shift should show up in other epicyclic models of quasi-periodic oscillations, such as those applied to black holes or white dwarfs, giving an independent test of the effective-potential expansion.
  • Testable extension: the amplitude dependence of the radial frequency turns the oscillation amplitude into a measurable parameter, and the posterior values of several kilometers suggest a fully non-perturbative treatment of the geodesics may be needed before the residual tension is judged real.
  • Possible connection: comparing the fitted amplitude parameter for each source with observed variability or flux modulation could separate anharmonic orbital effects from non-geodesic disk physics such as pressure gradients or magnetic fields.
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 an extension of the relativistic precession model (RPM) for quasi-periodic oscillations (QPOs) by including third-order terms in the Taylor expansion of the effective potential. The authors derive a cubic (Helmholtz-like) radial oscillator, compute the resulting amplitude-dependent radial epicyclic frequency using Jacobi elliptic functions, and apply the model to eight neutron-star QPO sources with MCMC fits, comparing against harmonic Schwarzschild, Schwarzschild–de Sitter, and Kerr models. They conclude that anharmonic corrections are physically necessary but insufficient to fully reconcile the RPM with observations, and that the standard RPM needs a more fundamental revision.

Significance. If the central claim were established, the paper would make a useful contribution: it identifies a simple, geometry-independent feature of the perturbed geodesic equations—the leading anharmonic radial correction is quadratic in the radial displacement—and shows that this term grows relative to the harmonic term near the ISCO. The explicit elliptic-function solution and the full Table I of MCMC results are useful technical resources, and the paper is honest in reporting that anharmonic corrections do not resolve all tensions. The main significance is therefore conditional: the phenomenological evidence for the necessity of anharmonic terms is weakened by the fact that the amplitude A0 is a free parameter fitted to each source, and the fitted amplitudes often violate the small-oscillation assumption on which the expansion is based.

major comments (3)
  1. [§III.B, Eqs. (15)–(19)] The derivation of the decoupled radial equation is internally inconsistent. The bullet list following Fig. 1 states that β1 δr δθ is about 20% of β0 δθ at every r and therefore “cannot be neglected,” yet the next paragraph concludes that “the above considerations enable us to remove the couplings between the two ODEs” and drops the polar equation. Dropping α2 δθ² from Eq. (16a) makes the radial equation independent of θ, but the polar equation (16b) still contains β1 δr δθ, so the two coordinates are not decoupled. The abstract’s claim that polar oscillations “remain approximately decoupled from radial ones” is contradicted by the paper’s own estimate. The model predictions use only the radial equation, so this might be harmless if the polar motion is treated as a spectator, but that should be stated explicitly and the justification for discarding β1 must be provided.
  2. [§III.C and Table I] The anharmonic frequency shift, Eq. (24) and its small-amplitude expansion (26), is valid only when δr is small enough that the cubic truncation of the effective potential is accurate and the elliptic-solution reality condition is satisfied. The MCMC fits infer A0 values of 5.05, 9.18, and 13.45 km for GX 17+2 (A1–A3), 6.77 km for 4U0614+091 (A1), and several km for other sources. For M ≈ 1.4–2 M⊙, these amplitudes are comparable to the orbital radii inferred from the observed f_U ≈ 0.8–1.2 kHz (roughly 10–25 km), and near the ISCO α0 → 0 while α1 does not, so the controlling ratio α1 A0/α0 can become of order unity or larger. In that regime Eq. (24) is not the radial epicyclic frequency of the underlying geodesic, the particle would leave the perturbative region or cross the ISCO, and the fitted A0 cannot be interpreted as the amplitude of a small oscillation. The paper should impose a validity constraint such as |α1 A0| ≪ α0 at every sampled point, or demonstrate explicitly that the best-fit regions satisfy this condition.
  3. [§IV and Table I] The statistical evidence does not support the qualitative claim that anharmonic corrections are “physically necessary” as stated in the abstract and Sect. V. The corrections introduce an extra free parameter A0 per source with no independent constraint, so part of the improvement in ln L0 is necessarily a fit. In Table I, the anharmonic models improve the fit decisively only for 4U0614+091 (A1) and mildly for GX 340+0 (A1/A2), while for GX 17+2 they are far worse than the harmonic SdS fit, and for GX 5-1, Sco X1, and 4U1608-52 they barely change ln L0 despite nonzero A0. Given the validity problem identified above, the improved likelihood for 4U0614+091 and GX 340+0 does not establish physical necessity; it shows that a model with an unconstrained amplitude parameter can absorb part of the tension. An independent estimate of A0, or a prior derived from the small-oscillation condition, is needed before this conclusion can be drawn.
minor comments (5)
  1. [§III.B, Eq. (15)] The step from Eq. (15) to Eqs. (16) is sufficient but not necessary: setting each bracketed factor to zero guarantees that the sum vanishes, but other solutions of the sum equation exist. The text should clarify that this is an ansatz restricting to solutions for which both brackets vanish separately.
  2. [§III.C, Eq. (24)] There is a typo in the line after Eq. (24): “Ωr = Ωr =√α0” should read “Ωr = √α0”.
  3. [§III.C heading] The heading “Anharmonicity vs radial epyciclic frequency” contains a typo: “epyciclic” should be “epicyclic”.
  4. [Fig. 2 caption] The caption of Fig. 2 contains “ad A3”; this should be “and A3”.
  5. [§IV, Table I] The anharmonic corrections are applied only to the Kerr spacetime, while the harmonic SdS results are taken from previous work. The statement that SdS remains a viable replacement is therefore not a model comparison on equal footing; an anharmonic SdS analysis would be needed to make that comparison symmetric.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the anharmonic correction is derived from the geodesic potential, and the fitted amplitude A0 is an explicit parameter, not a disguised input.

full rationale

The central anharmonic result is derived rather than assumed: Eq. (13) is a Taylor expansion of the effective potential, Eqs. (16) and (19) follow from the geodesic normalization condition, and Eq. (20) is an independent exact Helmholtz solution whose parameters are matched to the potential coefficients. The claim that the leading anharmonic correction is quadratic in δr is a mathematical consequence of the vanishing of the linear terms at equilibrium, not an input. The MCMC comparison introduces A0 explicitly as a free parameter with priors in Sec. IV, and the text says the corrected frequencies "can be used to fit the data"; it does not present the best-fit f_L as an out-of-sample prediction. Model selection uses AIC and BIC, which penalize the extra parameter. The harmonic benchmarks are taken from published papers [35,75]; although these overlap with the present authors, they are external data analyses of the standard epicyclic formulas, not a uniqueness theorem or an ansatz, and the anharmonic derivation does not depend on them. The main legitimate concerns are correctness issues, not circularity: footnote 2 asserts without proof that fourth-order terms are negligible, the step from Eq. (15) to Eq. (16) sets each bracketed factor to zero as a sufficient rather than necessary condition, and the fitted A0 values (e.g., 5.05 km for GX 17+2 A1, 6.77 km for 4U0614+091 A1) can violate the small-δr assumption underlying the truncated oscillator. Because these do not reduce the model's output to its input by construction, the circularity score is 0.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The central quantitative result rests on three fitted parameters per source (M, j, A0) plus the SdS R0 in the comparison models. The anharmonic frequency shift is not a parameter-free prediction: it depends on the ad hoc amplitude function A(r). The derivation also assumes decoupling of the radial and polar modes and neglect of fourth-order terms without proof.

free parameters (4)
  • A0 = per source; e.g., 5.05 km for GX 17+2 (A1)
    Radial oscillation amplitude parameter in the anharmonic models; fitted by MCMC with no independent observational constraint. The frequency shift in Eq. (26) scales as A0², so A0 absorbs the unknown physical amplitude.
  • M = per source; e.g., 2.827 M⊙ for GX 17+2 (A1)
    Central mass parameter of the Kerr metric, fitted to the QPO frequencies. It is a physical parameter of the source but is free in the fit.
  • j = per source; e.g., 0.270 for GX 17+2 (A1)
    Dimensionless spin parameter of the Kerr metric, fitted to the QPO frequencies.
  • R0 = per source; e.g., 21.53e-5 km^-2 for GX 17+2 (H-SdS)
    Curvature parameter in the Schwarzschild-de Sitter metric used for the harmonic comparison; values taken from Ref. [35], not recomputed here.
assumptions (5)
  • domain assumption The effective potential expansion is truncated at third order, and fourth-order terms are asserted to be negligible.
    Footnote 2 in Sect. III.B states fourth-order terms are negligible without a quantitative demonstration; this truncation is needed to obtain the Helmholtz oscillator.
  • domain assumption The lower QPO frequency is identified with the periastron precession f_phi - f_r and the upper frequency with the Keplerian frequency f_phi, following the standard RPM.
    Eqs. (12) take this identification from previous RPM literature; if it fails, the entire fit is void.
  • ad hoc to paper Radial and polar oscillations decouple despite the beta1 coupling being reported as 20% of the polar harmonic term.
    Sect. III.B removes the couplings after stating beta1 delta r delta theta cannot be neglected; the radial equation is then solved independently.
  • domain assumption The test particle moves on a circular equatorial orbit in Kerr, and the perturbation is small enough for the expansion to hold.
    Eqs. (6) and (8) set the equilibrium and perturbative regime; large posterior A0 values challenge this assumption.
  • ad hoc to paper The priors M in [0,10] M_sun, j in [-1,1], A0 in [0,15] km are wide enough to find the global likelihood maximum.
    Sect. IV states these priors are 'large enough' but no convergence or robustness tests are shown.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Generalizing the relativistic precession model of quasi-periodic oscillations through anharmonic corrections." pith.science (2026). https://pith.science/paper/HFZKKDJS

@misc{pith2026250418403,
  author       = {Pith},
  title        = {Pith review of: Generalizing the relativistic precession model of quasi-periodic oscillations through anharmonic corrections},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HFZKKDJS}},
  note         = {Machine review of arXiv:2504.18403}
}
abstract

We critically reanalyze the relativistic precession model of quasi-periodic oscillations, exploring its natural extension beyond the standard harmonic approximation. To do so, we show that the perturbed geodesic equations must include anharmonic contributions arising from the higher-order expansion of the effective potential that cannot be neglected \emph{a priori}, as commonly done in all the approaches pursued so far. More specifically, independently of the underlying spacetime geometry, we find that in the radial sector the non-negligible anharmonic correction is quadratic in the radial displacement, i.e. $\propto \delta r^2$, and significantly affects the radial epicyclic frequency close to the innermost stable circular orbit. Conversely, polar oscillations $\delta \theta$ remain approximately decoupled from radial ones, preserving their independent dynamical behavior. To show the need of anharmonic corrections, we thus carry out Monte Carlo-Markov chain analyses on eight neutron star sources of quasi-periodic oscillations. Afterwards, we first work out the outcomes of the harmonic approximation in Schwarzschild, Schwarzschild--de Sitter, and Kerr spacetimes. Subsequently, we apply the anharmonic corrections to them and use it to fit the aforementioned neutron star sources. Our findings indicate that the standard paradigm requires a systematic generalization to include the leading anharmonic corrections that appear physically necessary, although still insufficient to fully account for the observed phenomenology of quasi-periodic oscillations. Accordingly, we speculate on possible refinements of the relativistic precession model, showing the need to revise it at a fundamental level.

Figures

Figures reproduced from arXiv: 2504.18403 by the authors.

Figure 1
Figure 1. FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: Normalized 1–D LLH functions (solid lines) and the 1– [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: Continued Fig [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: ), the parameters R0 and j display concordant signs. This feature seems to oddly relate quantum [PITH_FULL_IMAGE:figures/full_fig_p010_4.png]

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. The macroscopic precession model of quasi-periodic oscillations for rotating compact objects

    gr-qc 2026-07 conditional novelty 6.0 of 10

    Treating QPO-emitting disk clumps as spinning test bodies reproduces the observed twin kHz QPOs without the effective de Sitter term, with fits preferring n≈2 thin-disk structures for Schwarzschild and n≈1 for Kerr.

Reference graph

Works this paper leans on

89 extracted references · 46 canonical work pages · cited by 1 Pith paper

  1. [1]

    B. P. Abbott, R. Abbott, T. D. Abbott, M. R. Aber- nathy, F. Acernese, K. Ackley, C. Adams, T. Adams, P. Addesso, R. X. Adhikari, et al., Phys. Rev. Lett. 116, 061102 (2016), 1602.03837

    arXiv 2016

  2. [2]

    Akiyama, A

    Event Horizon Telescope Collaboration, K. Akiyama, A. Alberdi, W. Alef, K. Asada, R. Azulay, A.-K. Baczko, D. Ball, M. Balokovi´ c, J. Barrett, et al., ApJ Lett. 875, L1 (2019), 1906.11238

    arXiv 2019

  3. [3]

    Leizerovich, S

    M. Leizerovich, S. J. Landau, and C. G. Sc´ occola, Phys. Lett. B 855, 138844 (2024), 2312.08542

    arXiv 2024

  4. [4]

    Renevey (2019), 1905.13720

    C. Renevey (2019), 1905.13720

    arXiv 2019

  5. [5]

    Ishak, Living Reviews in Relativity 22 (2018), ISSN 1433-8351

    M. Ishak, Living Reviews in Relativity 22 (2018), ISSN 1433-8351

    2018

  6. [6]

    P. S. Joshi, D. Malafarina, and R. Narayan, Classical and Quantum Gravity 28, 235018 (2011), ISSN 1361-6382

    2011

  7. [7]

    P. S. Joshi, D. Malafarina, and R. Narayan, Classical and Quantum Gravity 31, 015002 (2013), ISSN 1361-6382

    2013

  8. [8]

    Bueno, P

    P. Bueno, P. A. Cano, R. A. Hennigar, and ´Angel J. Mur- cia, Regular black holes from thin-shell collapse (2024), 2412.02740

    arXiv 2024

Show all 89 references
  1. [9]

    Luongo (2025), 2504.09987

    O. Luongo (2025), 2504.09987

    2025

  2. [10]

    Bardeen, in Proceedings of the 5th International Con- ference on Gravitation and the Theory of Relativity (1968), p

    J. Bardeen, in Proceedings of the 5th International Con- ference on Gravitation and the Theory of Relativity (1968), p. 87

    1968

  3. [11]

    Bambi, Regular Black Holes: Towards a New Paradigm of Gravitational Collapse (Springer Nature Singapore, 2023), ISBN 9789819915965

    C. Bambi, Regular Black Holes: Towards a New Paradigm of Gravitational Collapse (Springer Nature Singapore, 2023), ISBN 9789819915965

    2023

  4. [12]

    Volonteri, M

    M. Volonteri, M. Habouzit, and M. Colpi, Nature Re- views Physics 3, 732 (2021), 2110.10175

    2021 arXiv

  5. [13]

    Pen and A

    U.-L. Pen and A. E. Broderick, Monthly Notices of the Royal Astronomical Society 445, 3370 (2014)

    2014

  6. [14]

    Rebusco, Publ

    P. Rebusco, Publ. Astr. Soc. Japan 56, 553 (2004), astro- ph/0403341

    2004

  7. [15]

    M. A. Abramowicz, W. Klu´ zniak, J. E. McClintock, and R. A. Remillard, The Astrophysical Journal 609, L63–L65 (2004), ISSN 1538-4357

    2004

  8. [16]

    T¨ or¨ ok, M

    G. T¨ or¨ ok, M. A. Abramowicz, W. Klu´ zniak, and Z. Stuchl´ ık, A&A436, 1 (2005)

    2005

  9. [17]

    T¨ or¨ ok, P

    G. T¨ or¨ ok, P. Bakala, E.ˇSr´ amkov´ a, Z. Stuchl´ ık, M. Ur- banec, and K. Goluchov´ a, ApJ 760, 138 (2012), 1408.4220

    2012 arXiv

  10. [18]

    Boshkayev, D

    K. Boshkayev, D. Bini, J. Rueda, A. Geralico, M. Muc- cino, and I. Siutsou, Gravitation and Cosmology 20, 233 (2014), 1412.8214

    2014 arXiv

  11. [19]

    Van der Klis, Compact Stellar X-ray Sources , vol

    M. Van der Klis, Compact Stellar X-ray Sources , vol. 39 of Cambridge Astrophysics (Cambridge University Press, 2006)

    2006

  12. [20]

    F. K. Lamb and S. Boutloukos, Accreting Neutron Stars in Low-Mass X-Ray Binary Systems (Springer Nether- lands, 2008), p. 87–109, ISBN 9781402065446

    2008

  13. [21]

    Stella and M

    L. Stella and M. Vietri, Nuclear Physics B Proceedings Supplements 69, 135 (1999)

    1999

  14. [22]

    Boshkayev, D

    K. Boshkayev, D. Bini, J. Rueda, A. Geralico, M. Muc- cino, and I. Siutsou, Gravitation and Cosmology 20, 233–239 (2014), ISSN 1995-0721

    2014

  15. [23]

    Boshkayev, D

    K. Boshkayev, D. Bini, J. Rueda, and M. Muccino, As- tron. Rep. 59, 441–446 (2015)

    2015

  16. [24]

    T¨ or¨ ok, K

    G. T¨ or¨ ok, K. Goluchov´ a, M. Urbanec, E. ˇSr´ amkov´ a, K. Ad´ amek, G. Urbancov´ a, T. Pech´ aˇ cek, P. Bakala, Z. Stuchl´ ık, J. Hor´ ak, et al., The Astrophysical Journal 833, 273 (2016)

    2016

  17. [25]

    Boshkayev, J

    K. Boshkayev, J. A. Rueda, and M. Muccino, Main pa- rameters of neutron stars from quasi-periodic oscillations in low mass X-ray binaries (Fourtheent Marcell Gross- man Meeting, 2017), pp. 3433–3440

    2017

  18. [26]

    M. C. Miller, F. K. Lamb, and D. Psaltis, ApJ 508, 791 (1998), astro-ph/9609157

    1998 arXiv

  19. [27]

    Titarchuk, I

    L. Titarchuk, I. Lapidus, and A. Muslimov, ApJ 499, 315 (1998), astro-ph/9712348

    1998 arXiv

  20. [28]

    F. K. Lamb and M. C. Miller, ApJ 554, 1210 (2001), astro-ph/0007460

    2001 arXiv

  21. [29]

    P. C. Fragile, G. J. Mathews, and J. R. Wilson, ApJ 553, 955 (2001), astro-ph/0007478

    2001 arXiv

  22. [30]

    M. A. Abramowicz and W. Klu´ zniak, A&A 374, L19 (2001), astro-ph/0105077

    2001 arXiv

  23. [31]

    Titarchuk and K

    L. Titarchuk and K. Wood, ApJ Lett. 577, L23 (2002), astro-ph/0208212

    2002 arXiv

  24. [32]

    Kato, Publ

    S. Kato, Publ. Astr. Soc. Japan 57, L17 (2005), astro- ph/0505215

    2005

  25. [33]

    Kato, Publ

    S. Kato, Publ. Astr. Soc. Japan 64, 139 (2012), 1207.5882

    2012 arXiv

  26. [34]

    Banerjee, JCAP 2022, 020 (2022), 2201.00679

    I. Banerjee, JCAP 2022, 020 (2022), 2201.00679

    2022 arXiv

  27. [35]

    Boshkayev, O

    K. Boshkayev, O. Luongo, and M. Muccino, Phys. Rev. D 108, 124034 (2023), 2212.10186

    2023 arXiv

  28. [36]

    Boshkayev, A

    K. Boshkayev, A. Idrissov, O. Luongo, and M. Muccino (2023), 2303.03248

    2023 arXiv

  29. [37]

    van der Klis, F

    M. van der Klis, F. Jansen, J. van Paradijs, W. H. G. Lewin, E. P. J. van den Heuvel, J. E. Trumper, and M. Szatjno, Nature 316, 225 (1985)

    1985

  30. [38]

    Middleditch and W

    J. Middleditch and W. C. Priedhorsky, ApJ 306, 230 (1986)

    1986

  31. [39]

    Klu´ zniak, J

    W. Klu´ zniak, J. P. Lasota, M. A. Abramowicz, and B. Warner, A&A 440, L25 (2005), astro-ph/0503151

    2005 arXiv

  32. [40]

    C. M. Zhang, H. X. Yin, and Y. H. Zhao, Publ. Astr. Soc. Pacific 119, 393 (2007), astro-ph/0502102

    2007 arXiv

  33. [41]

    T. M. Belloni, A. Sanna, and M. M´ endez, MNRAS 426, 1701 (2012), 1207.2311

    2012 arXiv

  34. [42]

    Bambi, D

    C. Bambi, D. Malafarina, and N. Tsukamoto, Phys. Rev. D 89, 127302 (2014), 1406.2181

    2014 arXiv

  35. [43]

    Middleditch, ApJ Lett

    J. Middleditch, ApJ Lett. 257, L71 (1982)

    1982

  36. [44]

    Larsson, A&A 181, L15 (1987)

    S. Larsson, A&A 181, L15 (1987)

    1987

  37. [45]

    Larsson, A&A 217, 146 (1989)

    S. Larsson, A&A 217, 146 (1989)

    1989

  38. [46]

    J. M. Bonnet-Bidaud, M. Mouchet, C. Busschaert, E. Falize, and C. Michaut, A&A 579, A24 (2015), 1506.05891

    2015 arXiv

  39. [47]

    Bambi, A

    C. Bambi, A. D. Dolgov, and A. A. Petrov, JCAP 2009, 013 (2009), 0806.3440

    2009 arXiv

  40. [48]

    Boshkayev, A

    K. Boshkayev, A. Idrissov, O. Luongo, and D. Mala- farina, Mon. Not. Roy. Astron. Soc. 496, 1115 (2020), 2006.01269

    2020 arXiv

  41. [49]

    Capozziello, S

    S. Capozziello, S. Gambino, and O. Luongo (2025), 2503.21987

    2025 arXiv

  42. [50]

    Kurmanov, K

    Y. Kurmanov, K. Boshkayev, T. Konysbayev, M. Muc- cino, O. Luongo, A. Urazalina, A. Dalelkhankyzy, F. Belissarova, and M. Alimkulova, Phys. Dark Univ. 48, 101917 (2025), 2503.18028

    2025 arXiv

  43. [51]

    Ashoorioon, M

    A. Ashoorioon, M. B. Jahani Poshteh, and O. Luongo (2024), 2408.07762

    2024 arXiv

  44. [52]

    Kurmanov, K

    Y. Kurmanov, K. Boshkayev, T. Konysbayev, O. Luongo, N. Saiyp, A. Urazalina, G. Ikhsan, and G. Suliyeva, Phys. Dark Univ. 46, 101566 (2024), 2404.15437

    2024 arXiv

  45. [53]

    Boshkayev, T

    K. Boshkayev, T. Konysbayev, Y. Kurmanov, O. Luongo, 13 M. Muccino, A. Taukenova, and A. Urazalina, Eur. Phys. J. C 84, 230 (2024), 2307.15003

    2024 arXiv

  46. [54]

    Boshkayev, T

    K. Boshkayev, T. Konysbayev, Y. Kurmanov, O. Luongo, M. Muccino, H. Quevedo, and A. Urazalina, Eur. Phys. J. Plus 139, 273 (2024), 2306.15050

    2024 arXiv

  47. [55]

    Boshkayev, T

    K. Boshkayev, T. Konysbayev, Y. Kurmanov, O. Lu- ongo, and D. Malafarina, Astrophys. J. 936, 96 (2022), 2205.04208

    2022 arXiv

  48. [56]

    D’Agostino, R

    R. D’Agostino, R. Giamb` o, and O. Luongo, Phys. Rev. D 107, 043032 (2023), 2204.02098

    2023 arXiv

  49. [57]

    Kurmanov, K

    E. Kurmanov, K. Boshkayev, R. Giamb` o, T. Konysbayev, O. Luongo, D. Malafarina, and H. Quevedo, Astrophys. J. 925, 210 (2022), 2110.15402

    2022 arXiv

  50. [58]

    Boshkayev, T

    K. Boshkayev, T. Konysbayev, E. Kurmanov, O. Lu- ongo, D. Malafarina, and H. Quevedo, Phys. Rev. D 104, 084009 (2021), 2106.04932

    2021 arXiv

  51. [59]

    J. E. McClintock, R. Narayan, S. W. Davis, L. Gou, A. Kulkarni, J. A. Orosz, R. F. Penna, R. A. Remil- lard, and J. F. Steiner, Classical and Quantum Gravity 28, 114009 (2011), 1101.0811

    2011 arXiv

  52. [60]

    Koloˇ s, M

    M. Koloˇ s, M. Shahzadi, and Z. Stuchl´ ık, European Phys- ical Journal C 80, 133 (2020)

    2020

  53. [61]

    C. B. Singh, S. Mondal, and D. Garofalo, Monthly No- tices of the Royal Astronomical Society 510, 807–814 (2021), ISSN 1365-2966

    2021

  54. [62]

    Bellavita, F

    C. Bellavita, F. Garc´ ıa, M. M´ endez, and K. Karpouzas, Monthly Notices of the Royal Astronomical Society 515, 2099–2109 (2022), ISSN 1365-2966

    2022

  55. [63]

    Stuchl´ ık, M

    Z. Stuchl´ ık, M. Koloˇ s, and A. Tursunov, Publications of the Astronomical Society of Japan 74, 1220 (2022)

    2022

  56. [64]

    Zhang, M

    Y. Zhang, M. M´ endez, F. Garc´ ıa, K. Karpouzas, L. Zhang, H. Liu, T. M. Belloni, and D. Altamirano, Monthly Notices of the Royal Astronomical Society 514, 2891–2901 (2022), ISSN 1365-2966

    2022

  57. [65]

    Stella and M

    L. Stella and M. Vietri, ApJ Lett. 492, L59 (1998), astro- ph/9709085

    1998

  58. [66]

    Stella and M

    L. Stella and M. Vietri, Physical Review Letters 82, 17–20 (1999), ISSN 1079-7114

    1999

  59. [67]

    Stella, M

    L. Stella, M. Vietri, and S. M. Morsink, ApJ Lett. 524, L63 (1999), astro-ph/9907346

    1999 arXiv

  60. [68]

    S. E. Motta, T. Belloni, L. Stella, G. Pappas, J. Casares, A. T. Mu˜ noz-Darias, M. A. P. Torres, and I. V. Yanes- Rizo, Monthly Notices of the Royal Astronomical Society 517, 1469 (2022)

    2022

  61. [69]

    K. Rink, I. Caiazzo, and J. Heyl, Monthly Notices of the Royal Astronomical Society 517, 1389–1397 (2022), ISSN 1365-2966

    2022

  62. [70]

    Bambi and S

    C. Bambi and S. Nampalliwar, EPL (Europhysics Let- ters) 116, 30006 (2016), 1604.02643

    2016 arXiv

  63. [71]

    S. E. Motta, T. Munoz-Darias, A. Sanna, R. Fender, T. Belloni, and L. Stella, MNRAS 439, L65 (2014), 1312.3114

    2014 arXiv

  64. [72]

    S. E. Motta, T. M. Belloni, L. Stella, T. Mu˜ noz-Darias, and R. Fender, MNRAS 437, 2554 (2014), 1309.3652

    2014 arXiv

  65. [73]

    J. M. Bardeen, W. H. Press, and S. A. Teukolsky, ApJ 178, 347 (1972)

    1972

  66. [74]

    Hu, Journal of Sound and Vibration 293, 462 (2006), ISSN 0022-460X, URL https://www.sciencedirect

    H. Hu, Journal of Sound and Vibration 293, 462 (2006), ISSN 0022-460X, URL https://www.sciencedirect. com/science/article/pii/S0022460X05006693

    2006

  67. [75]

    Boshkayev, T

    K. Boshkayev, T. Konysbayev, Y. Kurmanov, M. Muc- cino, and H. Quevedo, MNRAS 531, 3876 (2024), 2312.03630

    2024 arXiv

  68. [76]

    A. R. Liddle, MNRAS 377, L74 (2007), astro- ph/0701113

    2007

  69. [77]

    Srinivasan, Bulletin of the Astronomical Society of India 30, 523 (2002)

    G. Srinivasan, Bulletin of the Astronomical Society of India 30, 523 (2002)

    2002

  70. [78]

    Lo and L.-M

    K.-W. Lo and L.-M. Lin, ApJ 728, 12 (2011), 1011.3563

    2011 arXiv

  71. [79]

    Qi, N.-B

    B. Qi, N.-B. Zhang, B.-Y. Sun, S.-Y. Wang, and J.-H. Gao, Research in Astronomy and Astrophysics 16, 60 (2016), 1408.1654

    2016 arXiv

  72. [80]

    Boutloukos, M

    S. Boutloukos, M. van der Klis, D. Altamirano, M. Klein- Wolt, R. Wijnands, P. G. Jonker, and R. P. Fender, ApJ 653, 1435 (2006), astro-ph/0608089

    2006 arXiv

  73. [81]

    Wijnands, M

    R. Wijnands, M. M´ endez, M. van der Klis, D. Psaltis, E. Kuulkers, and F. K. Lamb, ApJ Lett.504, L35 (1998), astro-ph/9806050

    1998 arXiv

  74. [82]

    P. G. Jonker, M. van der Klis, J. Homan, M. M´ endez, W. H. G. Lewin, R. Wijnands, and W. Zhang, MNRAS 333, 665 (2002), astro-ph/0202420

    2002 arXiv

  75. [83]

    Homan, M

    J. Homan, M. van der Klis, P. G. Jonker, R. Wijnands, E. Kuulkers, M. M´ endez, and W. H. G. Lewin, ApJ568, 878 (2002), astro-ph/0104323

    2002 arXiv

  76. [84]

    P. G. Jonker, M. van der Klis, R. Wijnands, J. Homan, J. van Paradijs, M. M´ endez, E. C. Ford, E. Kuulkers, and F. K. Lamb, ApJ 537, 374 (2000), astro-ph/0002022

    2000 arXiv

  77. [85]

    M´ endez and M

    M. M´ endez and M. van der Klis, MNRAS 318, 938 (2000), astro-ph/0006243

    2000 arXiv

  78. [86]

    A. M. Cherepashchuk, T. S. Khruzina, and A. I. Bogo- mazov, MNRAS 508, 1389 (2021), 2109.00967

    2021 arXiv

  79. [87]

    M´ endez, M

    M. M´ endez, M. van der Klis, R. Wijnands, E. C. Ford, J. van Paradijs, and B. A. Vaughan, ApJ Lett. 505, L23 (1998), astro-ph/9807281

    1998 arXiv

  80. [88]

    M´ endez and M

    M. M´ endez and M. van der Klis, ApJ Lett. 517, L51 (1999), astro-ph/9903303

    1999 arXiv

  81. [89]

    E. C. Ford, P. Kaaret, K. Chen, M. Tavani, D. Barret, P. Bloser, J. Grindlay, B. A. Harmon, W. S. Paciesas, and S. N. Zhang, ApJ Lett. 486, L47 (1997), astro- ph/9706100

    1997

Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.