Microphysical determination of neutron-vortex pinning to proton fluxoids in neutron-star cores: what sets the fluxoid number $N_{\rm p}$ and the drag-to-lift ratio $\xi(N_{\rm p})$ governing pulsar glitch rises

In the superfluid, superconducting core of a neutron star, quantized neutron vortices can pin to magnetic fluxoids (flux tubes) of the proton superconductor. Sourie & Chamel (2020) showed that the mean number of fluxoids pinned per vortex, $N_{\rm p}$, is a key parameter controlling the mutual-friction coupling of the core superfluid to the rest of the star, and hence the shape of the spin-up at the onset of a pulsar glitch: small $N_{\rm p}$ yields a fast rise with overshoot (Vela-like), while large $N_{\rm p}$ yields a smooth, slower rise (Crab-like). However, $N_{\rm p}$ and its spatial distribution are essentially unknown, and the drag-to-lift ratio $\xi$ entering the mutual-friction coefficients was treated with an unknown scaling exponent $\alpha$ ($\xi\propto N_{\rm p}^{\alpha}$, $\alpha=1$ or $2$ from two conflicting 1990s estimates). The problem is to establish the microphysical basis of core vortex-fluxoid pinning: determine, from stellar magneto-thermal and superconducting phase-transition history, what sets $N_{\rm p}$ and its distribution (and why Vela-like and Crab-like pulsars differ); compute the microscopic $\xi(N_{\rm p})$ from electron scattering off vortex-fluxoid clusters; and quantify how relaxing the simplifying assumptions of the 2020 treatment (straight rigid vortices, aligned fluxoids, neglected buoyancy and entrainment, Newtonian dynamics, time-independent $N_{\rm p}$) modifies the mutual-friction coefficients and the predicted glitch-rise dynamics. No microphysical determination of $N_{\rm p}$ or of the $\xi(N_{\rm p})$ scaling is known to date; the two conflicting 1990s estimates must be adjudicated from the cited sources.

Background

A pulsar is a rapidly rotating, highly magnetized neutron star whose spin is observed through radio (and sometimes X-ray or gamma-ray) pulses. Pulsar timing occasionally reveals sudden spin-ups called glitches, generally attributed to the neutron superfluid interior: quantized vortex lines of the neutron superfluid are pinned in the solid crust, and a sudden collective unpinning transfers angular momentum to the observable crust. In the dense core, the protons are widely expected to form a type-II superconductor, meaning that magnetic flux penetrates it in quantized flux tubes called fluxoids, each carrying magnetic flux in units of $hc/2e$. The neutron vortices (each carrying circulation quantum $\kappa=h/2m_{\rm n}$) inevitably coexist and interact with this fluxoid array, and a neutron vortex can pin to one or more fluxoids. The strength of this coupling is central to a long-standing question: whether the core superfluid, long thought to be coupled to the crust too quickly to participate in glitches, can in fact be involved. Phenomenologically, the coupling between the neutron superfluid and the charged component (crust plus core protons and electrons) is parameterized by mutual friction, characterized by coefficients $\mathcal{B}$ and $\mathcal{B}'$ that are set by the microphysical forces on a moving vortex: the Magnus (lift) force from relative superfluid flow and a dissipative drag force, whose ratio is the drag-to-lift coefficient $\xi>0$. When a vortex pins $N_{\rm p}$ fluxoids, drag arises mainly from the scattering of the degenerate electrons off the magnetic fields of the pinned fluxoids. Two decades-old estimates disagree on the scaling: Ding, Cheng & Chau (1993, ApJ 408, 167) obtained $\xi\propto N_{\rm p}^{2}$ for fluxoids superimposed on the vortex, while the vortex-cluster model of Sedrakian & Sedrakian (1995, ApJ 447, 305) gives $\xi\propto N_{\rm p}$. Timing observations sharpen the issue. The 2016 Vela glitch rose in less than roughly 12 seconds with a post-glitch overshoot of the spin rate (Ashton et al. 2019), whereas several Crab glitches show an instantaneous partial spin-up followed by a delayed spin-up over hours to days (e.g. Shaw et al. 2021). Sourie & Chamel (2020, MNRAS Letters 493, L98), building on their force calculation for a vortex pinned to $N_{\rm p}$ fluxoids (2020, MNRAS 493, 382), incorporated fluxoid pinning into a three-component (pinned core superfluid, non-pinned core superfluid, charged component) Newtonian model of the glitch rise. They found that $N_{\rm p}$ is the key parameter: for $1\leq N_{\rm p}\leq N_{\rm p}^{\rm crit}$ (a threshold fixed by the mutual-friction coefficients, with the scaling exponent $\alpha$ of $\xi\propto N_{\rm p}^{\alpha}$ left free at 1 or 2) the rise is fast with an overshoot, while larger $N_{\rm p}$ produces a smooth, slower spin-up — plausibly distinguishing Vela from Crab. But the number of pinned fluxoids, its spatial distribution inside a given star, and the physical reason for its difference between pulsars were left undetermined: they depend on the prior magneto-thermal evolution of the internal magnetic field and on the nature of the superconducting phase transition, neither of which is settled. The treatment also assumed straight, infinitely rigid vortices with fluxoids aligned along them, neglected buoyancy forces and dynamical entrainment (the coupling of neutron and proton flows whereby each fluid drags a fraction of the other), used Newtonian gravity, and held $N_{\rm p}$ fixed in time. The authors noted that allowing $N_{\rm p}$ to evolve could explain spin-down precursors and delayed spin-ups, but left the time-dependent pinning dynamics and the local dynamics of vortex-fluxoid clusters unmodelled.

Problem Statement

Establish the microphysical basis of neutron-vortex pinning to proton fluxoids in the superfluid core of a neutron star, as required to predict — rather than fit — the diversity of pulsar glitch rises. Specifically, with one overarching objective (determining the pinned mutual-friction microphysics and its consequences for glitch-rise dynamics), answer: (1) What sets the mean number $N_{\rm p}$ of fluxoids pinned per neutron vortex and its spatial distribution inside a given star, given the star's prior magneto-thermal evolution and the nature of the superconducting phase transition — in particular, what explains the observed difference in rise shapes between Vela-like and Crab-like pulsars, and does a difference in fluxoid arrangement account for it? (2) What is the microscopic dependence of the drag-to-lift ratio $\xi$ on $N_{\rm p}$ — resolving the conflict between the $\xi\propto N_{\rm p}^{2}$ superimposed-fluxoid estimate and the $\xi\propto N_{\rm p}$ vortex-cluster estimate, and giving the absolute magnitude — from electron scattering off vortex-fluxoid clusters? (3) How do the mutual-friction coefficients and the predicted glitch-rise dynamics change when the simplifying assumptions of the 2020 treatment are relaxed: straight, infinitely rigid vortices; fluxoids aligned with vortices; neglected buoyancy; neglected entrainment coupling between the fluids on the short rise; Newtonian (rather than general-relativistic) dynamics; and time-independent $N_{\rm p}$ — including whether an evolving $N_{\rm p}$ explains observed spin-down precursors and delayed spin-ups? Each of the listed relaxations is to be assessed separately, and its combined effect with the others reported as well. Taken together, the objective of (1)-(3) is to integrate their results into a quantitative, predictive connection from stellar evolution history and vortex-fluxoid microphysics to the observed rise shapes (fast rise with overshoot versus smooth or delayed rise) of Vela-like and Crab-like pulsars.

References

1. Sourie, A.; Chamel, N. (2020). 'Vortex pinning in the superfluid core of neutron stars and the rise of pulsar glitches.' Monthly Notices of the Royal Astronomical Society: Letters 493, L98-L102. doi:10.1093/mnrasl/slaa015, https://doi.org/10.1093/mnrasl/slaa015 (arXiv:2001.09668, https://arxiv.org/abs/2001.09668)
2. Sourie, A.; Chamel, N. (2020). 'Force on a neutron quantized vortex pinned to proton fluxoids in the superfluid core of cold neutron stars.' Monthly Notices of the Royal Astronomical Society 493, 382-389. doi:10.1093/mnras/staa253, https://doi.org/10.1093/mnras/staa253 (arXiv:2001.08964, https://arxiv.org/abs/2001.08964)
3. Sourie, A.; Chamel, N. (2021). 'Vortex pinning in the superfluid core of relativistic neutron stars.' Monthly Notices of the Royal Astronomical Society 503, 1407-1417. doi:10.1093/mnras/stab606, https://doi.org/10.1093/mnras/stab606 (arXiv:2102.11054, https://arxiv.org/abs/2102.11054)
4. Ding, K. Y.; Cheng, K. S.; Chau, H. F. (1993). 'Magnetic field decay from the core of neutron stars - Effects of interpinning of 3P2 neutron superfluid and 1S0 proton superconducting fluid.' The Astrophysical Journal 408, 167. doi:10.1086/172577, https://doi.org/10.1086/172577
5. Sedrakian, A. D.; Sedrakian, D. M. (1995). 'Superfluid Core Rotation in Pulsars. I. Vortex Cluster Dynamics.' The Astrophysical Journal 447, 305. doi:10.1086/175876, https://doi.org/10.1086/175876
6. Ashton, G.; Lasky, P. D.; Graber, V.; Palfreyman, J. (2019). 'Rotational evolution of the Vela pulsar during the 2016 glitch.' Nature Astronomy 3, 1143-1148. doi:10.1038/s41550-019-0844-6, https://doi.org/10.1038/s41550-019-0844-6
7. Shaw, B.; Keith, M. J.; Lyne, A. G.; Mickaliger, M. B.; Stappers, B. W.; Turner, J. D.; Weltevrede, P. (2021). 'The slow rise and recovery of the 2019 Crab pulsar glitch.' Monthly Notices of the Royal Astronomical Society: Letters 505, L6-L10. doi:10.1093/mnrasl/slab038, https://doi.org/10.1093/mnrasl/slab038
8. Thong, K. H.; Melatos, A.; Drummond, L. V. (2023). 'Stability of interlinked neutron vortex and proton flux-tube arrays in a neutron star - III. Proton feedback.' Monthly Notices of the Royal Astronomical Society 521, 5724-5737. doi:10.1093/mnras/stad927, https://doi.org/10.1093/mnras/stad927 (arXiv:2305.04482, https://arxiv.org/abs/2305.04482)
9. Celora, T.; Khomenko, V.; Antonelli, M.; Haskell, B. (2020). 'The effect of non-linear mutual friction on pulsar glitch sizes and rise times.' Monthly Notices of the Royal Astronomical Society 496, 5564-5574. doi:10.1093/mnras/staa1930, https://doi.org/10.1093/mnras/staa1930 (arXiv:2002.04310, https://arxiv.org/abs/2002.04310)
10. Haskell, B.; Antonelli, M.; Pizzochero, P. (2022). 'Continuous Gravitational Wave Emissions from Neutron Stars with Pinned Superfluids in the Core.' Universe 8(12), 619. doi:10.3390/universe8120619, https://doi.org/10.3390/universe8120619
11. Zhou, S.; Gügercinoğlu, E.; Yuan, J.; Ge, M.; Yu, C. (2022). 'Pulsar Glitches: A Review.' Universe 8(12), 641. doi:10.3390/universe8120641, https://doi.org/10.3390/universe8120641 (arXiv:2211.13885, https://arxiv.org/abs/2211.13885)
12. Lander, S. K.; Gourgouliatos, K. N.; Wadiasingh, Z.; Antonopoulou, D. (2026). 'Signatures of Magnetic Flux Expulsion from Neutron Star Cores.' The Astrophysical Journal Letters 997, L7. doi:10.3847/2041-8213/ae31f5, https://doi.org/10.3847/2041-8213/ae31f5 (arXiv:2411.08020, https://arxiv.org/abs/2411.08020)
13. Sedrakian, A.; Rau, P. B. (2026). 'Spin effects in superfluidity, neutron matter and neutron stars.' arXiv:2604.02782, https://arxiv.org/abs/2604.02782
14. Gügercinoğlu, E. (2026). 'Superfluid Vortex Lines-Magnetic Flux Tubes Interaction and Braking Indices of Pulsars.' arXiv:2603.01582, https://arxiv.org/abs/2603.01582
15. Goglichidze, O. A.; Gusakov, M. E. (2025). 'Non-purely transverse Magnus force in superconducting neutron stars.' arXiv:2505.05628, https://arxiv.org/abs/2505.05628
16. Goglichidze, O. A.; Gusakov, M. E. (2026). 'Neutron contribution to the force on a proton vortex in superconducting neutron-star matter.' Physical Review C 114, 015807. doi:10.1103/hgmj-2s35, https://doi.org/10.1103/hgmj-2s35
17. Gil Granados, J. A.; Muñoz Mateo, A.; Viñas, X. (2025). 'Half-vortex states in the rotating outer core of neutron stars.' Physical Review C 111, 065802. doi:10.1103/PhysRevC.111.065802, https://doi.org/10.1103/PhysRevC.111.065802 (arXiv:2505.06713, https://arxiv.org/abs/2505.06713)
18. Gangopadhyay, B. K. (2026). 'Coupled Vortex Flux Tube Modes and Pinning Induced Relaxation in Neutron Star Interior.' EPJ Web of Conferences 356, 02015. doi:10.1051/epjconf/202635602015, https://doi.org/10.1051/epjconf/202635602015