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The Meissner effect in neutron stars

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S. K. Lander

High Energy Astrophysical Phenomena · Submitted 12 Nov 2024 · 55 references

Pith's reading: The Meissner effect in a young neutron star proceeds by field-line dragging and reconnection, yielding four possible field geometries and a 10^12 to 5×10^14 G window for partial or complete flux expulsion.

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arXiv ID2411.08021 (version 1)
Pith IDVCEBZ527, Pith's permanent record of this paper and its review
CitationS. K. Lander. "The Meissner effect in neutron stars." arXiv:2411.08021 (2024). https://pith.science/paper/VCEBZ527
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Claim paper
The paper

The paper claims

The Meissner effect in a young neutron star proceeds by field-line dragging and reconnection, yielding four possible field geometries and a 10^12 to 5×10^14 G window for partial or complete flux expulsion.

The reading

When a neutron star is born, its core is hot enough that protons behave like normal charged particles. Within minutes the core cools below the temperature at which protons form Cooper pairs and become superconducting. In a laboratory, a superconductor expels magnetic field. Inside a neutron star the field is generated by currents inside the fluid, so expulsion is not automatic.

In this paper, Lander treats the early superconducting shell as a normal-MHD fluid and proposes that two things happen just as the shell forms. First, fluid motions drag magnetic field lines around, bunching them in a small region. Second, magnetic reconnection cuts and rejoins the field lines, allowing the superconducting shell to become field-free. Whether this works depends on field strength: below about 10^12 G the dragging is too slow, above about 5×10^14 G superconductivity is destroyed. Even in the best case, the geometry of a dipole field means reconnection cannot remove field from the very centre; some flux always stays trapped, with the shell pierced by polar holes or an equatorial belt in some outcomes.

The model produces a taxonomy of four final field geometries and estimates that the reconnection phase releases a few times 10^43 to roughly 5×10^46 erg a few minutes after birth. This energy release is a possible observational signature of the still-unconstrained proton pairing gap. The paper is honest that the crucial flow and reconnection phases are not simulated; only semi-analytic estimates are provided.

Extended reading notes

Core claim

Section 3.6 states '1012 G ≲ B0 ≲5 × 1014 G' as the range in which partial or complete Meissner expulsion could occur, and the abstract claims 'complete Meissner expulsion is but one of four possible generic scenarios for the magnetic-field geometry, and can never expel magnetic flux from the centre of the star.' If correct, flux always remains in the core, Meissner states are not the default outcome for weak fields, and up to ~5e46 erg can be released by reconnection at the superconducting transition.

Load-bearing premise

The load-bearing premise is that at the onset of superconductivity there exists a fluid flow, unmodeled in the paper, that advects the initially dipolar field into a geometry amenable to reconnection before the superconducting shell thickens. Section 3.4: 'The detailed dynamics will inevitably be complex and are beyond the scope of this first analysis, and we will just assume that they act to drive the magnetic field in the superconducting region towards its minimum-energy state of full expulsion.' If this flow is absent or too weak (below ~10^12 G), scenario 1i and its energy release do not occur.

Original abstract

We present the first model aimed at understanding how the Meissner effect in a young neutron star affects its macroscopic magnetic field. In this model, field expulsion occurs on a dynamical timescale, and is realised through two processes that occur at the onset of superconductivity: fluid motions causing the dragging of field lines, followed by magnetic reconnection. Focussing on magnetic fields weaker than the superconducting critical field, we show that complete Meissner expulsion is but one of four possible generic scenarios for the magnetic-field geometry, and can never expel magnetic flux from the centre of the star. Reconnection causes the release of up to $\sim 5\times 10^{46}\,\mathrm{erg}$ of energy at the onset of superconductivity, and is only possible for certain favourable early-phase dynamics and for pre-condensation fields $10^{12}\,\mathrm{G}\lesssim B\lesssim 5\times 10^{14}\,\mathrm{G}$. Fields weaker or stronger than this are predicted to thread the whole star.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 4,520 input tokens.

What the abstract itself states, as probabilities: claims first 0.95 · states limitation 0.84 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.98 · compares with prior results 0.09 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; .

We present the first model aimed at understanding how the Meissner effect in a young neutron star affects its macroscopic magnetic field. In this model, field expulsion occurs on a dynamical timescale, and is realised through two processes that occur at the onset of superconductivity: fluid motions causing the dragging of field lines, followed by magnetic reconnection. Focussing on magnetic fields weaker than the superconducting critical field, we show that complete Meissner expulsion is but one of four possible generic scenarios for the magnetic-field geometry, and can never expel magnetic flux from the centre of the star. Reconnection causes the release of up to $\sim 5\times 10^{46}\,\mathrm{erg}$ of energy at the onset of superconductivity, and is only possible for certain favourable early-phase dynamics and for pre-condensation fields $10^{12}\,\mathrm{G}\lesssim B\lesssim 5\times 10^{14}\,\mathrm{G}$. Fields weaker or stronger than this are predicted to thread the whole star.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 We present the first model aimed at understanding how the Meissner effect in a young neutron star affects its macroscopic magnetic field. contribution 1.00 0.93 0.07 asserted 0.95
s1 In this model, field expulsion occurs on a dynamical timescale, and is realised through two processes that occur at the onset of superconductivity: fluid motions causing the dragging of field lines, followed by magnetic reconnection. method 0.86 0.79 0.12 asserted 1.00
s2 Focussing on magnetic fields weaker than the superconducting critical field, we show that complete Meissner expulsion is but one of four possible generic scenarios for the magnetic-field geometry, and can never expel magnetic flux from the centre of the star. result 1.00 0.90 0.49 asserted 1.00
s3 Reconnection causes the release of up to $\sim 5\times 10^{46}\,\mathrm{erg}$ of energy at the onset of superconductivity, and is only possible for certain favourable early-phase dynamics and for pre-condensation fields $10^{12}\,\mathrm{G}\lesssim B\lesssim 5\times 10^{14}\,\mathrm{G}$. result 1.00 0.88 0.12 asserted 0.89
s4 Fields weaker or stronger than this are predicted to thread the whole star. result 0.99 0.82 0.12 asserted 0.96

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 0.41.

What was examined, and what was not

sources read
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not examined
  • full text
  • figures and tables
  • references and citing sentences
  • relations to other papers
levels run
  • P abstract
not run
  • S sections
  • T body sentences
  • C citations
  • R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures
Figure 1. Left: critical temperature for proton superconductivity, using an approximate model similar to that of Ho et al. (2012). Assuming an isothermal core cooling according to the approximation of Page et al. (2006) and overlaying these (horizontal) lines for different ages, we find that the first thin shell of superconducting matter forms after 170 seconds at the dimensionless radius marked R, and the superconducting shell reaches the crust-core boundary after 22 minutes. A few other ages are shown, and in the absence of magnetic field the entire core will become superconducting after 42…
Figure 2. Geometry of a field line distorted from its original length 1 due to shearing in the incipient superconducting shell (the thick purple circle). to need to worry about dissipative effects, and we can assume infinite conductivity of the matter, with the field lines perfectly frozen into the fluid. Consider an infinitessimally thin fluxtube of cross-sectional area running along the entire interior length 1 of a magnetic field line, before the onset of superconductivity. It is likely safe to assume, to leading order, that any internal distortion of the field line will have no effect on…
Figure 3. Four different scenarios for the rearrangement of flux due to the expanding region of superconductivity, all for < . An initially dipolar poloidal magnetic field (top row) may be advected by fluid motions (middle row) and then experience reconnection (bottom row). The (in)efficacy of these processes leads to four post-condensation field configurations. but that even within this range, expulsion depends on the fluid flow and reconnection properties of the star at the point of formation of the incipient superconducting shell. It is therefore now instructive to consider four limiting…
Figure 4. A purely poloidal field in hydromagnetic equilibrium in an = 1 polytropic star. The surface of the star is at a radius = , in the usual dimensionless units used in the Lane-Emden equation. Colourscale shows the magnitude of the field; white lines show contours of constant value of the streamfunction, equivalently, the magnetic field lines. Summarising the arguments of this subsection, the kind of reconnection event required to produce a Meissner-expelled magnetic field will release somewhere in the range Emag ≈ × (2.9 × 1041 − 7.2 × 1046) erg (44) of energy, where the parameters ,…
Figure 5. Length of unstretched field lines 1 (solid line) and the additional stretched component 2 (dashed line), in dimensionless units divided by , as a function of the constant value of the streamfunction labelling each field line, from the straight field line running from pole to pole (0 = 0), past the start of the closed field line region 0 ≈ 2.6 to the maximum value at the centre of the closed-field-line region 0 ≈ 3.0. easiest to visualise in the equatorial region of closed field lines), a single spoke passes through the same field line in two places, so writing = () would map one…
Figure 6. Energy per field line as a function of value = 0. Unstretched field shown with the solid line, dashed (dotted) lines show the additional magnetic energy for each field line due to the stretched piece of the field when R = 0.8 (0.7). Left-hand panel shows the increase in energy by stretching field lines towards = /2; right-hand panel shows the corresponding energy increases resulting from stretching field lines towards = 0. position of R 0.9 0.85 0.8 0.79 0.75 0.7 Emag increase from field lines stretched towards equator 0.5380 0.6267 0.7212 0.7411 0.8233 0.9271 Emag increase from…
Figure 7. Magnetic flux through a spherical surface of constant radius, as a function of dimensionless radius in the star. The onset of superconductivity in our fiducial model is marked. We can also calculate magnetic energy from equation (56) in physical units, which entails multiplying by 2 p 3 ∗ , giving us: E 0 mag = 6.5 × 1041
Figure 8. Opening angles for ≠ 0 regions within the superconducting shell, in scenarios 1ii (eq, the dashed line) and 2 (open, the solid line), as a function of field strength. open increases with field strength until it reaches the equator. eq begins at the equator, = /2, so for increasing field strength its value decreases, as it expands out towards the symmetry axis running from pole to pole. 5 LATER EVOLUTION 5.1 Field evolution in NS cores The expansion of the superconducting shell, in every case except scenario 3, drives an amplification of the magnetic field in the ≠ 0 regions, as descr
Figure 9. Inner and outer radii of the final superconducting shell, as a function of field strength. -1.0 -0.5 0.0 0.5 1.0 -1.0 -0.5 0.0 0.5 1.0 -1.0 -0.5 0.0 0.5 1.0 -1.0 -0.5 0.0 0.5 1.0 -1.0 -0.5 0.0 0.5 1.0 -1.0 -0.5 0.0 0.5 1.0
Figure 10. The final structure of the superconducting region, after flux compression has finished, for scenario 2. Scenario 1ii has an identical structure in this 2D plane but rotated through /2, so that the Northern and Southern polar ‘holes’ in the Meissner shell then become a single equatorial ring where the Meissner effect is not present; scenario 1i only has the central region with magnetic field and no breaks in the shell. For this end state, the average field strength of the inner region will be = 5 × 1014 G. Note that whilst the crust (outside the dashed circle but within the solid…
Figure 11. A purely toroidal field has field lines everywhere perpendicular to poloidal lines, and so orthogonal to the page in this meridional plot, and is non-zero within some region like the blue one plotted here. At the onset of superconductivity (left), the toroidal field on either side of the incipient superconducting shell can be matched to zero by having a surface current at both inner and outer boundaries. As the shell expands (right), a complete Meissner-expelled shell can be formed without any additional dynamics other than compression of flux in the crust, and in the inner core…
Figure 12. Sketch of location of toroidal component in each scenario. Left to right: 1i, 1ii, 2, 3. of superconductivity, but we believe the mostly likely result is a small region of toroidal field confined to the crust alone. All these possibilities are sketched in figure 12. Note the similarity of scenario 3, where poloidal fields are all open in the core and the toroidal field is confined to the crust, to earlier equilibrium models for a magnetic field threading the entire stellar interior and < (Lander 2014). This more realistic geometry deserves more attention, however, as the toroidal…

Figures reproduced from arXiv:2411.08021 by the authors.

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