# Continuous Gravitational Wave Emissions from Neutron Stars with Pinned Superfluids in the Core

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## Abstract

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## 1. Introduction

## 2. Setting the Stage: Hydrodynamic Perturbations

## 3. Magnetic Field and Pinning

## 4. Perturbation Equations for the Pinned Configuration

## 5. Continuous Wave Emission

## 6. Results

## 7. Summary and Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Note

1 | Apart from the uncertainties regarding the extent of the S-wave gap for superfluidity, further reduction in the pinning strength may be due to the fact that the mutual orientation between a vortex and the fluxtubes is expected to fluctuate locally, thereby, giving rise to a decrease of the effective pinning force, similarly to what happens for vortices that are randomly oriented with respect to the principal axis of the Coulomb lattice in the inner crust [21]. |

## References

- Abbott, B.P. et al. [The LIGO Scientific Collaboration and the Virgo Collaboration] GW170817: Observation of Gravitational Waves from a Binary Neutron Star Inspiral. Phys. Rev. Lett.
**2017**, 119, 161101. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Abbott, B.P. et al. [The LIGO Scientific Collaboration and the Virgo Collaboration] Multi-messenger Observations of a Binary Neutron Star Merger. Astrophys. J. Lett.
**2017**, 848, L12. [Google Scholar] [CrossRef] - Abbott, B.P. et al. [The LIGO Scientific Collaboration and the Virgo Collaboration] GW170817: Measurements of Neutron Star Radii and Equation of State. Phys. Rev. Lett.
**2018**, 121, 161101. [Google Scholar] [CrossRef] [Green Version] - Güven, H.; Bozkurt, K.; Khan, E.; Margueron, J. Multimessenger and multiphysics Bayesian inference for the GW170817 binary neutron star merger. Phys. Rev. C
**2020**, 102, 015805. [Google Scholar] [CrossRef] - Mondal, C.; Gulminelli, F. Can we decipher the composition of the core of a neutron star? Phys. Rev. D
**2022**, 105, 083016. [Google Scholar] [CrossRef] - Lasky, P.D. Gravitational wave astronomy. In Multimessenger Astronomy in Practice: Celestial Sources in Action; Filipović, M.D., Tothill, N.F.H., Eds.; IOP Publishing: Bristol, UK, 2021; pp. 1–9. [Google Scholar] [CrossRef]
- Piccinni, O.J. Status and Perspectives of Continuous Gravitational Wave Searches. Galaxies
**2022**, 10, 72. [Google Scholar] [CrossRef] - Riles, K. Searches for Continuous-Wave Gravitational Radiation. arXiv
**2022**, arXiv:2206.06447. [Google Scholar] - Lasky, P.D. Gravitational Waves from Neutron Stars: A Review. Publ. Astron. Soc. Austr.
**2015**, 32, e034. [Google Scholar] [CrossRef] - Glampedakis, K.; Gualtieri, L. Gravitational Waves from Single Neutron Stars: An Advanced Detector Era Survey. In Physics and Astrophysics of Neutron Stars; Rezzolla, L., Pizzochero, P., Jones, D.I., Rea, N., Vidaña, I., Eds.; Astrophysics and Space Science Library: Springer Cham, Switzerland, 2018; Volume 457, p. 673. [Google Scholar] [CrossRef] [Green Version]
- Haskell, B.; Schwenzer, K. Isolated Neutron Stars. In Handbook of Gravitational Wave Astronomy; Bambi, C., Ed.; Springer Nature: Cham, Switzerland, 2022; p. 12. [Google Scholar] [CrossRef]
- Bildsten, L. Gravitational Radiation and Rotation of Accreting Neutron Stars. Astrophys. J. Lett.
**1998**, 501, L89–L93. [Google Scholar] [CrossRef] [Green Version] - Bonazzola, S.; Gourgoulhon, E. Gravitational waves from pulsars: Emission by the magnetic-field-induced distortion. Astron. Astrophys.
**1996**, 312, 675–690. [Google Scholar] - Singh, N.; Haskell, B.; Mukherjee, D.; Bulik, T. Asymmetric accretion and thermal `mountains’ in magnetized neutron star crusts. Mon. Not. R. Astron. Soc.
**2020**, 493, 3866–3878. [Google Scholar] [CrossRef] - Ushomirsky, G.; Cutler, C.; Bildsten, L. Deformations of accreting neutron star crusts and gravitational wave emission. Mon. Not. R. Astron. Soc.
**2000**, 319, 902–932. [Google Scholar] [CrossRef] [Green Version] - Haskell, B.; Jones, D.I.; Andersson, N. Mountains on neutron stars: Accreted versus non-accreted crusts. Mon. Not. R. Astron. Soc.
**2006**, 373, 1423–1439. [Google Scholar] [CrossRef] [Green Version] - Woan, G.; Pitkin, M.D.; Haskell, B.; Jones, D.I.; Lasky, P.D. Evidence for a Minimum Ellipticity in Millisecond Pulsars. Astrophys. J. Lett.
**2018**, 863, L40. [Google Scholar] [CrossRef] - Mendell, G. Superfluid Hydrodynamics in Rotating Neutron Stars. II. Dissipative Effects. Astrophys. J.
**1991**, 380, 530. [Google Scholar] [CrossRef] - Andersson, N.; Sidery, T.; Comer, G.L. Mutual friction in superfluid neutron stars. Mon. Not. R. Astron. Soc.
**2006**, 368, 162–170. [Google Scholar] [CrossRef] - Antonelli, M.; Haskell, B. Superfluid vortex-mediated mutual friction in non-homogeneous neutron star interiors. Mon. Not. R. Astron. Soc.
**2020**, 499, 3690–3705. [Google Scholar] [CrossRef] - Seveso, S.; Pizzochero, P.M.; Grill, F.; Haskell, B. Mesoscopic pinning forces in neutron star crusts. Mon. Not. R. Astron. Soc.
**2016**, 455, 3952–3967. [Google Scholar] [CrossRef] [Green Version] - Alpar, M.A. Flux-Vortex Pinning and Neutron Star Evolution. J. Astrophys. Astron.
**2017**, 38, 44. [Google Scholar] [CrossRef] - Anderson, P.W.; Itoh, N. Pulsar glitches and restlessness as a hard superfluidity phenomenon. Nature
**1975**, 256, 25–27. [Google Scholar] [CrossRef] - Haskell, B.; Melatos, A. Models of pulsar glitches. Int. J. Mod. Phys. D
**2015**, 24, 1530008. [Google Scholar] [CrossRef] - Ducci, L.; Pizzochero, P.M.; Doroshenko, V.; Santangelo, A.; Mereghetti, S.; Ferrigno, C. Properties and observability of glitches and anti-glitches in accreting pulsars. Astron. Astrophys.
**2015**, 578, A52. [Google Scholar] [CrossRef] [Green Version] - Ray, P.S.; Guillot, S.; Ho, W.C.G.; Kerr, M.; Enoto, T.; Gendreau, K.C.; Arzoumanian, Z.; Altamirano, D.; Bogdanov, S.; Campion, R.; et al. Anti-glitches in the Ultraluminous Accreting Pulsar NGC 300 ULX-1 Observed with NICER. Astrophys. J.
**2019**, 879, 130. [Google Scholar] [CrossRef] [Green Version] - Ruderman, M. Crust-breaking by neutron superfluids and the Vela pulsar glitches. Astrophys. J.
**1976**, 203, 213–222. [Google Scholar] [CrossRef] - Jones, D.I. Gravitational waves from rotating strained neutron stars. Class. Quantum Gravity
**2002**, 19, 1255–1265. [Google Scholar] [CrossRef] [Green Version] - Keitel, D.; Woan, G.; Pitkin, M.; Schumacher, C.; Pearlstone, B.; Riles, K.; Lyne, A.G.; Palfreyman, J.; Stappers, B.; Weltevrede, P. First search for long-duration transient gravitational waves after glitches in the Vela and Crab pulsars. Phys. Rev. D
**2019**, 100, 064058. [Google Scholar] [CrossRef] [Green Version] - Abadie, J. et al. [The LIGO Scientific Collaboration] Search for gravitational waves associated with the August 2006 timing glitch of the Vela pulsar. Phys. Rev. D
**2011**, 83, 042001. [Google Scholar] [CrossRef] [Green Version] - Abbott, R. et al. [The LIGO Scientific Collaboration] Narrowband Searches for Continuous and Long-duration Transient Gravitational Waves from Known Pulsars in the LIGO-Virgo Third Observing Run. Astrophys. J.
**2022**, 932, 133. [Google Scholar] [CrossRef] - Bennett, M.F.; van Eysden, C.A.; Melatos, A. Continuous-wave gravitational radiation from pulsar glitch recovery. Mon. Not. R. Astron. Soc.
**2010**, 409, 1705–1718. [Google Scholar] [CrossRef] - Warszawski, L.; Melatos, A. Gravitational-wave bursts and stochastic background from superfluid vortex avalanches during pulsar glitches. Mon. Not. R. Astron. Soc.
**2012**, 423, 2058–2074. [Google Scholar] [CrossRef] [Green Version] - Melatos, A.; Douglass, J.A.; Simula, T.P. Persistent Gravitational Radiation from Glitching Pulsars. Astrophys. J.
**2015**, 807, 132. [Google Scholar] [CrossRef] [Green Version] - Chamel, N. Neutron conduction in the inner crust of a neutron star in the framework of the band theory of solids. Phys. Rev. C
**2012**, 85, 035801. [Google Scholar] [CrossRef] - Chamel, N. Entrainment in Superfluid Neutron-Star Crusts: Hydrodynamic Description and Microscopic Origin. J. Low Temp. Phys.
**2017**, 189, 328–360. [Google Scholar] [CrossRef] - Andersson, N.; Glampedakis, K.; Ho, W.C.G.; Espinoza, C.M. Pulsar Glitches: The Crust is not Enough. Phys. Rev. Lett.
**2012**, 109, 241103. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Chamel, N. Crustal Entrainment and Pulsar Glitches. Phys. Rev. Lett.
**2013**, 110, 011101. [Google Scholar] [CrossRef] [Green Version] - Montoli, A.; Antonelli, M.; Haskell, B.; Pizzochero, P. Statistical Estimates of the Pulsar Glitch Activity. Universe
**2021**, 7, 8. [Google Scholar] [CrossRef] - Montoli, A.; Antonelli, M.; Pizzochero, P.M. The role of mass, equation of state, and superfluid reservoir in large pulsar glitches. Mon. Not. R. Astron. Soc.
**2020**, 492, 4837–4846. [Google Scholar] [CrossRef] [Green Version] - Haskell, B.; Khomenko, V.; Antonelli, M.; Antonopoulou, D. Crust or core? Insights from the slow rise of large glitches in the Crab pulsar. Mon. Not. R. Astron. Soc.
**2018**, 481, L146–L150. [Google Scholar] [CrossRef] - Montoli, A.; Antonelli, M.; Magistrelli, F.; Pizzochero, P.M. Bayesian estimate of the superfluid moments of inertia from the 2016 glitch in the Vela pulsar. Astron. Astrophys.
**2020**, 642, A223. [Google Scholar] [CrossRef] - Sourie, A.; Chamel, N. Force on a neutron quantized vortex pinned to proton fluxoids in the superfluid core of cold neutron stars. Mon. Not. R. Astron. Soc.
**2020**, 493, 382–389. [Google Scholar] [CrossRef] - Gügercinoğlu, E.; Alpar, M.A. Vortex Creep Against Toroidal Flux Lines, Crustal Entrainment, and Pulsar Glitches. Astrophys. J. Lett.
**2014**, 788, L11. [Google Scholar] [CrossRef] [Green Version] - Sourie, A.; Chamel, N. Vortex pinning in the superfluid core of neutron stars and the rise of pulsar glitches. Mon. Not. R. Astron. Soc.
**2020**, 493, L98–L102. [Google Scholar] [CrossRef] - Pizzochero, P.M.; Montoli, A.; Antonelli, M. Core and crust contributions in overshooting glitches: The Vela pulsar 2016 glitch. Astron. Astrophys.
**2020**, 636, A101. [Google Scholar] [CrossRef] - Ciolfi, R. Modelling the magnetic field configuration of neutron stars. Astron. Nachrichten
**2014**, 335, 624. [Google Scholar] [CrossRef] [Green Version] - Lander, S.K. Magnetic Fields in Superconducting Neutron Stars. Phys. Rev. Lett.
**2013**, 110, 071101. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Sur, A.; Haskell, B. The impact of superconductivity and the Hall effect in models of magnetised neutron stars. Publ. Astron. Soc. Austr.
**2021**, 38, e043. [Google Scholar] [CrossRef] - Sur, A.; Cook, W.; Radice, D.; Haskell, B.; Bernuzzi, S. Long-term general relativistic magnetohydrodynamics simulations of magnetic field in isolated neutron stars. Mon. Not. R. Astron. Soc.
**2022**, 511, 3983–3993. [Google Scholar] [CrossRef] - Ruderman, M.; Zhu, T.; Chen, K. Neutron Star Magnetic Field Evolution, Crust Movement, and Glitches. Astrophys. J.
**1998**, 492, 267–280. [Google Scholar] [CrossRef] [Green Version] - Ciolfi, R.; Rezzolla, L. Twisted-torus configurations with large toroidal magnetic fields in relativistic stars. Mon. Not. R. Astron. Soc.
**2013**, 435, L43–L47. [Google Scholar] [CrossRef] - Jones, D.I. Gravitational wave emission from rotating superfluid neutron stars. Mon. Not. R. Astron. Soc.
**2010**, 402, 2503–2519. [Google Scholar] [CrossRef] [Green Version] - Andersson, N.; Comer, G.L. Relativistic fluid dynamics: Physics for many different scales. Living Rev. Relativ.
**2021**, 24, 3. [Google Scholar] [CrossRef] - Haskell, B.; Sedrakian, A. Superfluidity and Superconductivity in Neutron Stars. In The Physics and Astrophysics of Neutron Stars; Rezzolla, L., Pizzochero, P., Jones, D.I., Rea, N., Vidaña, I., Eds.; Astrophysics and Space Science Library; Springer Nature: Cham, Switzerland, 2018; Volume 457, p. 401. [Google Scholar] [CrossRef] [Green Version]
- Gavassino, L.; Antonelli, M. Thermodynamics of uncharged relativistic multifluids. Class. Quantum Gravity
**2020**, 37, 025014. [Google Scholar] [CrossRef] [Green Version] - Graber, V.; Cumming, A.; Andersson, N. Glitch rises as a test for rapid superfluid coupling in neutron stars. ArXiv
**2018**, arXiv:astro-ph.HE/1804.02706. [Google Scholar] [CrossRef] - Celora, T.; Khomenko, V.; Antonelli, M.; Haskell, B. The effect of non-linear mutual friction on pulsar glitch sizes and rise times. Mon. Not. R. Astron. Soc.
**2020**, 496, 5564–5574. [Google Scholar] [CrossRef] - Charbonneau, J.; Zhitnitsky, A. Novel mechanism for type I superconductivity in neutron stars. Phys. Rev. C
**2007**, 76, 015801. [Google Scholar] [CrossRef] [Green Version] - Link, B. Instability of superfluid flow in the neutron star core. Mon. Not. R. Astron. Soc.
**2012**, 421, 2682–2691. [Google Scholar] [CrossRef] [Green Version] - Sedrakian, A.; Clark, J.W. Superfluidity in nuclear systems and neutron stars. Eur. Phys. J. A
**2019**, 55, 167. [Google Scholar] [CrossRef] [Green Version] - Leinson, L.B. Vortex lattice in rotating neutron spin-triplet superfluid. Mon. Not. R. Astron. Soc.
**2020**, 498, 304–309. [Google Scholar] [CrossRef] - Pili, A.G.; Bucciantini, N.; Del Zanna, L. General relativistic models for rotating magnetized neutron stars in conformally flat space-time. Mon. Not. R. Astron. Soc.
**2017**, 470, 2469–2493. [Google Scholar] [CrossRef] - Castillo, F.; Reisenegger, A.; Valdivia, J.A. Magnetic field evolution and equilibrium configurations in neutron star cores: The effect of ambipolar diffusion. Mon. Not. R. Astron. Soc.
**2017**, 471, 507–522. [Google Scholar] [CrossRef] [Green Version] - Sur, A.; Haskell, B.; Kuhn, E. Magnetic field configurations in neutron stars from MHD simulations. Mon. Not. R. Astron. Soc.
**2020**, 495, 1360–1371. [Google Scholar] [CrossRef] - Pili, A.G.; Bucciantini, N.; Del Zanna, L. Axisymmetric equilibrium models for magnetized neutron stars in General Relativity under the Conformally Flat Condition. Mon. Not. R. Astron. Soc.
**2014**, 439, 3541–3563. [Google Scholar] [CrossRef] [Green Version] - Khomenko, V.; Antonelli, M.; Haskell, B. Hydrodynamical instabilities in the superfluid interior of neutron stars with background flows between the components. Phys. Rev. D
**2019**, 100, 123002. [Google Scholar] [CrossRef] [Green Version] - Andersson, N.; Glampedakis, K.; Haskell, B. Oscillations of dissipative superfluid neutron stars. Phys. Rev. D
**2009**, 79, 103009. [Google Scholar] [CrossRef] [Green Version] - van Eysden, C.A.; Melatos, A. Gravitational radiation from pulsar glitches. Class. Quantum Gravity
**2008**, 25, 225020. [Google Scholar] [CrossRef] [Green Version] - Sidery, T.; Passamonti, A.; Andersson, N. The dynamics of pulsar glitches: Contrasting phenomenology with numerical evolutions. Mon. Not. R. Astron. Soc.
**2010**, 405, 1061–1074. [Google Scholar] [CrossRef] [Green Version] - Thorne, K.S. Multipole expansions of gravitational radiation. Rev. Mod. Phys.
**1980**, 52, 299–339. [Google Scholar] [CrossRef] - Riley, T.E.; Watts, A.L.; Bogdanov, S.; Ray, P.S.; Ludlam, R.M.; Guillot, S.; Arzoumanian, Z.; Baker, C.L.; Bilous, A.V.; Chakrabarty, D.; et al. A NICER View of PSR J0030+0451: Millisecond Pulsar Parameter Estimation. Astrophys. J. Lett.
**2019**, 887, L21. [Google Scholar] [CrossRef] - Raaijmakers, G.; Greif, S.K.; Hebeler, K.; Hinderer, T.; Nissanke, S.; Schwenk, A.; Riley, T.E.; Watts, A.L.; Lattimer, J.M.; Ho, W.C.G. Constraints on the Dense Matter Equation of State and Neutron Star Properties from NICER’s Mass-Radius Estimate of PSR J0740+6620 and Multimessenger Observations. Astrophys. J. Lett.
**2021**, 918, L29. [Google Scholar] [CrossRef] - Jones, P.B. The Alignment of the Crab Pulsar Magnetic Axis. Ap&SS
**1975**, 33, 215–230. [Google Scholar] [CrossRef] - Easson, I.; Pethick, C.J. Stress tensor of cosmic and laboratory type-II superconductors. Phys. Rev. D
**1977**, 16, 275–280. [Google Scholar] [CrossRef] - Haskell, B.; Priymak, M.; Patruno, A.; Oppenoorth, M.; Melatos, A.; Lasky, P.D. Detecting gravitational waves from mountains on neutron stars in the advanced detector era. Mon. Not. R. Astron. Soc.
**2015**, 450, 2393–2403. [Google Scholar] [CrossRef] [Green Version] - Giliberti, E.; Cambiotti, G.; Antonelli, M.; Pizzochero, P.M. Modelling strains and stresses in continuously stratified rotating neutron stars. Mon. Not. R. Astron. Soc.
**2020**, 491, 1064–1078. [Google Scholar] [CrossRef]

**Figure 1.**

**Left:**Lateral view of our simplified configuration, in which the neutron star has been sliced along the vertical plane containing the toroidal magnetic field region (its width goes from about $0.4{R}_{*}$ to $0.8{R}_{*}$). The two axes are the rotation axis of the neutron star aligned with the angular velocity $\mathsf{\Omega}$ and the $(\theta =\pi /2,\varphi =0$), while the dark red region corresponds to the strong pinning region as indicated in the text.

**Right**: Cartoon of the triaxial deformation induced by the presence of pinned vortices in the two (diametrically opposed) strong pinning regions. For perfect pining, the force ${\mathbf{f}}_{p}$ in (7) is directed outwards when the superfluid spins faster than the normal component. Both sketches are not to scale.

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**MDPI and ACS Style**

Haskell, B.; Antonelli, M.; Pizzochero, P.
Continuous Gravitational Wave Emissions from Neutron Stars with Pinned Superfluids in the Core. *Universe* **2022**, *8*, 619.
https://doi.org/10.3390/universe8120619

**AMA Style**

Haskell B, Antonelli M, Pizzochero P.
Continuous Gravitational Wave Emissions from Neutron Stars with Pinned Superfluids in the Core. *Universe*. 2022; 8(12):619.
https://doi.org/10.3390/universe8120619

**Chicago/Turabian Style**

Haskell, Brynmor, Marco Antonelli, and Pierre Pizzochero.
2022. "Continuous Gravitational Wave Emissions from Neutron Stars with Pinned Superfluids in the Core" *Universe* 8, no. 12: 619.
https://doi.org/10.3390/universe8120619