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Continuous gravitational waves from thermal mountains on accreting neutron stars: effect of the nuclear pasta phase

T0 review · 4 major / 6 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read Adding the nuclear pasta phase to the thermal-mountain calculation raises the gravitational-wave quadrupole moment of accreting neutron stars by up to two orders of magnitude.

desk verdict First estimate of thermal-mountain Q22 from the pasta layer, with an honest but fragile two-order enhancement that depends on uncertain transport and near-breaking strains. read the letter →

arxiv 2411.11075 v1 pith:67RARBPA submitted 2024-11-17 gr-qc astro-ph.HE

Hong-Bo Li , Lijing Shao , Cheng-Jun Xia , Ren-Xin Xu This is my paper · ORCID
classification gr-qcastro-ph.HE
keywords continuousgravitationalwavesneutronstarcrustnuclearpastathermalmountainsquadrupolemomentaccretingmillisecondX-raypulsarsequationofstatewavedetectability
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

The paper argues that the deepest solid layer of an accreting neutron star, the nuclear pasta phase, has been left out of earlier estimates of thermal mountains and that including it changes the answer by two orders of magnitude. With the pasta layer's low thermal conductivity and neutrino cooling included, lateral temperature differences survive deeper into the star, producing a quadrupole moment $Q_{22} \approx 1.7 \times 10^{39}\,\mathrm{g\,cm^2}$ at torque balance. If this is right, known accreting millisecond X-ray pulsars and nuclear-powered X-ray pulsars become strong candidate sources for the next generation of ground-based gravitational-wave detectors, well above the sensitivity of Cosmic Explorer and Einstein Telescope. The result also links pasta elasticity to a maximum shear strain of 0.1–0.4, meaning the pasta can support such mountains.

What carries the argument

The load-bearing object is the radial thermal perturbation equation (Eq. 24), solved with the pasta layer's assumed conductivity $\kappa_{\mathrm{pasta}} \approx 3 \times 10^{19}\,(T/T_8)\,\mathrm{erg\,K^{-1}\,cm^{-1}\,s^{-1}}$ and neutrino emissivity $Q_{\mathrm{neu}}^{\mathrm{pasta}} \approx 3 \times 10^{17}\,\rho_{12}\,T_9^6\,\mathrm{erg\,s^{-1}\,cm^{-3}}$ as new input at the deep crust. The resulting $\delta T/T$ feeds the crustal perturbation equations (Eqs. 40–43) for elastic displacement, and the quadrupole moment $Q_{22}$ is obtained from Eq. (47) by integrating over both the normal crust and the pasta layer. The pasta acts as an extra deep source region where the temperature perturbation is larger, and the shear-strain calculation (Eqs. 55–57) shows strains of 0.1–0.4 there, near the molecular-dynamics breaking strain.

What would settle it

Recompute the thermal perturbation with the pasta neutrino emissivity replaced by Lin et al.'s $T^8$ law ($Q \approx 8 \times 10^{21}\,T_9^8$) or with $\kappa_{\mathrm{pasta}} = 10^{21}\,\mathrm{erg\,K^{-1}\,cm^{-1}\,s^{-1}}$ and see whether $\delta T/T$ in the pasta layer stays above the crust level; if $Q_{22}$ from the integral drops below $\approx 10^{38}\,\mathrm{g\,cm^2}$, the predicted strains in Fig. 12 no longer reach CE/ET sensitivities. A direct test would be a two-year coherent search of a known AMXP at the amplitude predicted from $Q_{22} \approx 1.7 \times 10^{39}$: a null result would rule out the combination of torque balance and pasta transport used here.

Watch

Extended reading notes

Core claim

On an accreting neutron star, non-uniform nuclear burning leaves lateral temperature differences $\delta T/T$ in the crust; those differences deform the elastic crust into a thermal mountain that radiates continuous gravitational waves. This paper extends the standard calculation into the nuclear pasta phase, the deepest solid layer before the core, where the crust's thermal conductivity and neutrino cooling are different. The authors find that including this layer raises the mountain's quadrupole moment to $Q_{22} \approx 1.7 \times 10^{39}\,\mathrm{g\,cm^2}$, about two orders of magnitude above the crust-only value of $\approx 2.2 \times 10^{37}\,\mathrm{g\,cm^2}$, with the pasta layer alone contributing a fiducial quadrupole moment of $\approx 6.9 \times 10^{39}\,\mathrm{g\,cm^2}$. Under the standard torque-balance assumption, the known accreting millisecond X-ray pulsars and nuclear-powered X-ray pulsars then lie well above the sensitivity curves of Cosmic Explorer and Einstein Telescope.

Load-bearing premise

The pasta layer's thermal conductivity and neutrino emissivity are set by hand, with no sensitivity study; if the real pasta conducts heat more efficiently or cools faster, the temperature asymmetry that creates the mountain shrinks and the quadrupole enhancement mostly disappears.

Editorial extensions

If this is right

  • With pasta included, the required quadrupole moment for torque balance is $Q_{22} \approx 1.7 \times 10^{39}\,\mathrm{g\,cm^2}$, so known AMXPs and NXPs sit above the Cosmic Explorer and Einstein Telescope sensitivity curves.
  • The crust-only prediction of $\approx 2.2 \times 10^{37}\,\mathrm{g\,cm^2}$ keeps most sources undetectable by current and near-future detectors; the pasta phase is what changes the target list.
  • The pasta layer can support shear strains of 0.1–0.4, similar to molecular-dynamics breaking strains, so the mountains it produces are not immediately limited by crust failure.
  • Increasing accretion rate or impurity parameter raises $Q_{22}$ in both models, so the brightest prospects among known LMXBs are the higher-accretion-rate systems.

Reading between the lines

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

  • The two-order enhancement is not a robust lower bound: the pasta conductivity and neutrino cooling are chosen without a sensitivity study, and published alternatives (higher $\kappa$, or a $T^8$ fast-cooling law) would shrink $\delta T/T$ and $Q_{22}$; an independent simulation or measurement of these coefficients would decide whether the enhancement survives.
  • If the pasta layer does raise $Q_{22}$ this much, known AMXPs become prime targets for a directed search at Cosmic Explorer and Einstein Telescope; a non-detection would pressure either the torque-balance explanation of spin equilibrium or the pasta transport assumptions.
  • The same perturbation machinery could be applied to the pasta region for magnetic or elastic mountains, not just thermal mountains, since the deep layer's high shear modulus changes how any deformation is supported.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 6 minor

Summary. This paper presents a forward model of thermal mountains on accreting neutron stars that explicitly includes the nuclear pasta layer. The authors construct a background stellar model with a DH core and an HZ accreted crust, solve the spherically symmetric heat equation with crustal heating, neutrino cooling, and pasta microphysics, then solve the linearized thermal perturbation equation for lateral temperature variations. These variations are fed into the elastic perturbation equations to compute the l=m=2 quadrupole moment Q22 and the associated gravitational-wave strain for known AMXPs and NXPs. The central result is that including the pasta layer raises Q22 from about 2.2x10^37 g cm^2 to about 1.7x10^39 g cm^2, making the sources detectable by CE and ET. The paper also computes shear strains in the crust and pasta and relates the maximum sustainable Q22 to the breaking strain.

Significance. If the two-order enhancement is robust, the paper identifies a qualitatively new implication of nuclear pasta for continuous-wave astronomy: known accreting neutron stars become high-priority targets for third-generation detectors, and detection or non-detection of these sources would provide indirect constraints on pasta transport properties. The paper's strengths include its use of the established Ushomirsky/Osborne-Jones/Hutchins framework, explicit perturbation equations, a clear numerical pipeline, and comparison with analytic scalings. It also makes a falsifiable prediction of h0 for known sources. The central claim, however, rests on two adopted pasta transport coefficients that span an order of magnitude or more in the cited literature, and the manuscript does not quantify how Q22 depends on them. The strain-consistency check in Section 6 also appears to require strains that exceed the quoted breaking-strain range at its lower end. These issues are fixable with targeted calculations, so the paper is a promising contribution rather than a completed proof of the headline claim.

major comments (4)
  1. [§3.3, Eq. (20)] The pasta thermal conductivity is fixed at kappa_pasta ≈ 3x10^19 (T/T8) erg K^-1 cm^-1 s^-1, yet the same section cites published values spanning roughly 10^17 (Deibel et al. 2017) to 10^21 (Horowitz & Berry 2008; Schneider et al. 2016) erg K^-1 cm^-1 s^-1. Because Eq. (24) governs how the lateral temperature perturbation diffuses into the pasta shell, and because Q22 is computed from δT/T through Eqs. (40)–(47), this single choice directly sets the magnitude of the claimed two-order enhancement. The manuscript contains no sensitivity study, and Section 7 lists only the density dependence of κ as future work, not the order-of-magnitude spread. I request a parameter sweep in κ with a statement of the range over which the enhancement remains at least one order of magnitude.
  2. [§3.2, Eq. (14)] The neutrino cooling law in the pasta is approximated as Q_pasta_neu ≈ 3x10^17 ρ12 T9^6, explicitly setting aside Lin et al. (2020), who report fast neutrino cooling of pasta with Q ≈ 8x10^21 T9^8. No justification is given for preferring the weaker T^6 law in the pasta region. Neutrino cooling appears as the damping term in the thermal perturbation equation, Eq. (24), so adopting the Lin et al. law will reduce δT/T in the pasta at a given temperature. The paper should either demonstrate that fast cooling is inoperative in accreted pasta or recompute δT/T and Q22 with the Lin et al. cooling law; the latter is a decisive test of the headline result.
  3. [§6, Fig. 14 and following text] With pasta, the computed strain components reach 0.1–0.4, while the paper quotes breaking strains of 0.04 (Baiko & Chugunov 2018) to 0.1 (Horowitz & Kadau 2009). At the lower end, the maximum sustainable quadrupole implied by Q_pasta_22 = 1.7x10^39 (σ/0.1) g cm^2 is only about 7x10^38 g cm^2, below the value in Eq. (51). The statement that the large breaking strain should support mountains therefore does not hold over the full quoted breaking-strain range. Please report the minimum σmax required by Eq. (51) and show whether the sources in Figs. 11–12 remain detectable if Q22 is capped at the strain limit.
  4. [§5.2, Fig. 12] The detectability discussion compares predicted strains only with projected CE/ET sensitivity curves. For known AMXPs and NXPs, targeted continuous-wave searches have published upper limits in the relevant frequency band; comparing h0 from Eq. (51) with those limits provides a direct, falsifiable check of the pasta model. The manuscript should include such a comparison, or explain why it is not applicable, because a predicted strain above an existing upper limit would require a revision of the model rather than a detection prediction.
minor comments (6)
  1. [§2, Eqs. (1)–(5)] The density is written with units of erg cm^-3; these should be g cm^-3.
  2. [Abstract] The phrase quadruple moment should be quadrupole moment; the same typo appears in several places in the text.
  3. [§5, Eq. (26)] The sentence defining the gravitational-wave frequency reads f_GW^2 = 2ν; it should read f_GW = 2ν.
  4. [§3.3, Eq. (20)] The symbol T8 is used without definition; T9 is defined after Eq. (13), so T8 should be defined as T/10^8 K.
  5. [Fig. 11] The legend and caption use AMPs; this should be AMXPs for consistency with the text.
  6. [Figs. 13–14] The text and captions describe maximum values but plot the absolute value of σr⊥, which is negative; the captions should state explicitly that |σr⊥| is shown.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: Q22 is forward-modeled from disclosed transport inputs; the pasta enhancement is conditional on parameter choices, not fitted, and the sole self-citation (Xia et al. 2023) is corroborated and non-load-bearing.

full rationale

The claimed result is a forward model, not a fitted one: Q_pasta_22 = 1.7e39 g cm^2 (Eq. 51) is obtained by integrating the crustal perturbation equations (Eqs. 40-47) with the deltaT/T profile from the linearized heat equation (Eq. 24), where the pasta enters only through the in-text transport choices kappa_pasta = 3e19 (T/T8) (Eq. 20) and Q_pasta_neu = 3e17 rho12 T9^6 (Eq. 14). These inputs are disclosed as assumptions, not calibrated to the target claim. Figure 10 shows a computed Mdot-dependence, and the crust-only result Q_crust_22 = 2.2e37 g cm^2 (Eq. 50) agrees in order of magnitude with the independent results of Ushomirsky et al. (2000) and Jones & Hutchins (2024). The only substantive self-citation, Xia et al. (2023) for mu_pasta = 1e30 erg cm^-3 (Sec. 5.1), is corroborated by Caplan et al. (2018) and Pethick et al. (2020) and does not drive the headline Q22 (the fiducial estimate of Eq. 49 is mu-independent). The paper itself flags its limitations in Sec. 7: the density-independent kappa_pasta (Eq. 20), the Cowling approximation, and earlier (Sec. 3.2) the deliberate setting-aside of Lin et al. (2020)'s T^8 fast-cooling law. These, together with the unaudited spread in published kappa_pasta (1e17-1e21) and Q_pasta_neu, mean the two-order enhancement is conditional on parameter choices. That is a robustness/correctness concern, not a circular one, because no equation reduces by definition to its own input and no fitted parameter is relabeled as a prediction.

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

No new particles, forces, or conserved quantities are introduced; all inputs are existing nuclear-matter properties. The free parameters are choices of asymmetry, impurity, and pasta transport coefficients, each with substantial published uncertainty.

free parameters (5)
  • fnuc = 0.1
    Assumed lateral variation of nuclear heating deposited in capture layers; Q22 scales linearly with this input and it is not derived from first principles or data.
  • Qimp = 1 (also 5, 10)
    Impurity parameter in the crust; observational constraints allow up to 100 (Ootes et al. 2019), and larger Qimp increases Q22.
  • κ_pasta prefactor = 3e19 erg K^-1 cm^-1 s^-1
    Thermal conductivity of nuclear pasta chosen by hand, density-independent, within the 1e17 to 1e21 range of published estimates.
  • Q_pasta_neu prefactor = 3e17
    Neutrino emissivity in pasta assumed to follow bremsstrahlung-like T^6 scaling; no derivation is given, and Lin et al. (2020) provide a faster T^8 law.
  • central density ρc = 1.496e15 g cm^-3
    Chosen to produce a 1.84 solar mass star; a model parameter for the background structure.
assumptions (6)
  • domain assumption The crust and nuclear pasta phase are solid, with Coulomb coupling Γ > 175 (Sec. 2).
    If the pasta layer is not elastic, it cannot support a static mountain and the quadrupole calculation changes.
  • domain assumption The deep electron-capture layers of the Haensel & Zdunik (1990) accreted-crust EOS deposit heat and drive the thermal perturbation (Sec. 3.1).
    Jones & Hutchins (2024), cited in Sec. 7, argue such deep capture layers may not exist in modern EOS, which would remove the assumed heat source.
  • standard math The Cowling approximation is valid, neglecting the perturbation of the gravitational potential (Sec. 5.1).
    The paper notes Ushomirsky et al. (2000) found a full calculation enhances Q22 by 25 to 200 percent; this is a known approximation in perturbation theory.
  • domain assumption The core is isothermal and perfectly conducting with zero temperature perturbation at the bottom boundary (Sec. 4.1).
    Justified by the high core conductivity; reasonable for the model.
  • standard math Newtonian gravity suffices for the crust and pasta structure, with TOV used only for the core (Sec. 2).
    Standard in this line of work; the crust is a small fraction of the star.
  • domain assumption The pasta shear modulus is μ ≈ 1e30 erg cm^-3, taken from Caplan et al. (2018), Pethick et al. (2020), and Xia et al. (2023) (Sec. 5.1).
    External molecular-dynamics and analytic estimates; the value has its own uncertainty.

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Pith. "Pith review of Continuous gravitational waves from thermal mountains on accreting neutron stars: effect of the nuclear pasta phase." pith.science (2026). https://pith.science/paper/67RARBPA

@misc{pith2026241111075,
  author       = {Pith},
  title        = {Pith review of: Continuous gravitational waves from thermal mountains on accreting neutron stars: effect of the nuclear pasta phase},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/67RARBPA}},
  note         = {Machine review of arXiv:2411.11075}
}
read the original abstract

As density increases, the shape of nuclei transitions to non-spherical ``nuclear pasta" structures. The physical properties of the nuclear pasta, such as thermal conductivity and elasticity, have implications for detecting continuous gravitational waves from a rapidly rotating neutron star. In this work, we investigate the effect of the nuclear pasta on the quadruple moment, and find out that, compared with previous work, the quadrupole moment contributing to continuous gravitational-wave radiation can be up to two orders of magnitude larger. We also discuss the relationship between the quadruple moment and the maximum shear strain. Considering the properties of nuclear pasta, we study the detectability of known accreting neutron stars and compare predicted results to the detectable amplitude limits. These sources are well above the sensitivity curves for Cosmic Explorer and Einstein Telescope detectors. Our work advances the understanding of the properties of nuclear pasta and a possible mechanism for continuous gravitational waves.

Figures

Figures reproduced from arXiv: 2411.11075 by the authors.

Figure 1
Figure 1. Mass number A (top) and nuclear charge Z (bot￾tom) with increasing density. The data are taken from [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Density as a function of radius, with a central density of ρc = 1.496 × 1015g cm−3 . This model represents a NS with mass of 1.84 M⊙ and includes a nuclear pasta thickness of 0.43 km and a crust thickness of 0.77 km. 3. THERMAL STRUCTURE OF AN ACCRETING NEUTRON STAR In this section, we calculate a steady-state ther￾mal structure based on the Newtonian formulation (Ushomirsky et al. 2000; Osborne & Jones 2020; Hutchi… view at source ↗
Figure 3
Figure 3. The upper panel shows the temperature profiles against the radius without the pasta phase. Different accre￾tion rates are chosen. The lower panel shows the heat flux versus the radius. We set the impurity parameter Qimp = 1. 3.5. Results: Background thermal structure In the upper panel of [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (9 more)
Figure 5
Figure 5. Figure 5: Temperature profiles versus the radius with pasta phase for different accretion rates. The impurity parameter is Qimp = 1. The vertical dash-dotted line is the nuclear pasta-crust interface. 10.8 11.0 11.2 11.4 11.6 11.8 r [km] 2.5 3.0 3.5 4.0 4.5 T [10 8 K] Qimp = 1 Q…
Figure 6
Figure 6. Figure 6: Temperature profiles for different impurity pa￾rameters with nuclear pasta. The accretion rate is M˙ = 0.5M˙ Edd. The vertical dash-dotted line has the same mean￾ing as in [PITH_FULL_IMAGE:figures/full_fig_p006_6.png]
Figure 7
Figure 7. Figure 7: Temperature perturbations in the crust for different accretion rates with a fixed impurity parameter Qimp = 1 (left) and for impurity parameter with a fixed accretion rate M˙ = 0.5M˙ Edd (right). Both panels have fnuc = 0.1, fcomp = 0, and ℓ = 2. 10.75 11.00 11.25 11.5…
Figure 8
Figure 8. Figure 8: Same as [PITH_FULL_IMAGE:figures/full_fig_p008_8.png]
Figure 9
Figure 9. Figure 9: The absolute value of quadrupole moment as a function of the accretion rates without nuclear pasta for different impurity parameters. We set fnuc = 0.1 and fcomp = 0. In the inner crust, the fiducial quadrupole moment is defined as Q inner fid ≡ Z ∂ ln ρ ∂ ln T [PITH_…
Figure 11
Figure 11. Figure 11: GW strain versus frequency for mountains in known pulsars. Closed squares denote AMXPs, and open diamonds indicate NXPs. The solid curves show the nom￾inal strain noise sensitivity for various GW detectors. The predicted results are from Eqs. (26) and (50). 10 100 100…
Figure 12
Figure 12. Figure 12: Similar to [PITH_FULL_IMAGE:figures/full_fig_p010_12.png]
Figure 13
Figure 13. Figure 13: The maximum values of the different compo￾nents of the shear strain without nuclear pasta as a function of the radius. The dashed line is 0.39 σrr(r), the dashdot line denotes 0.45 σr⊥(r), and the solid line shows 0.18 σΛ(r). The background model has M˙ = 0.5M˙ Edd, f…
Figure 14
Figure 14. Figure 14: Same as [PITH_FULL_IMAGE:figures/full_fig_p012_14.png]

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