The Astrophysical Journal Letters, 997:L7 (7pp), 2026 January 20	Lander et al.

https://doi.org/10.3847/2041-8213/ae31f5

The Astrophysical Journal Letters, 997:L7 (7pp), 2026 January 20
© 2026. The Author(s). Published by the American Astronomical Society.
Signatures of Magnetic Flux Expulsion from Neutron Star Cores
S. K. Lander^1, K. N. Gourgouliatos^2, Z. Wadiasingh^3,4,5 aa, and D. Antonopoulou^6
1		School of Engineering, Mathematics and Physics, University of East Anglia, Norwich, NR4 7TJ, UK; samuel.lander@uea.ac.uk
2		Laboratory of Universe Sciences, Department of Physics, University of Patras, Patras, Rio 26504, Greece
3		Department of Astronomy, University of Maryland, College Park, MD 20742, USA
4		Astrophysics Science Division, NASA Goddard Space Flight Center, Greenbelt, MD 20771, USA
5		Center for Research and Exploration in Space Science and Technology, NASA/GSFC, Greenbelt, MD 20771, USA
6		Jodrell Bank Centre for Astrophysics, Department of Physics and Astronomy, The University of Manchester, UK
Received 2025 October 30; accepted 2025 November 23; published 2026 January 13
Abstract
Shortly after a neutron star is born, the protons in its core begin to form a superconductor. In terrestrial materials, the hallmark of superconductivity is an associated expulsion of magnetic flux, but whether this expulsion process can be effective in neutron stars remains an open question—one with major implications for the phenomenology of pulsars and magnetars. Earlier theoretical arguments suggested flux must be trapped within the core, yet models of magnetars rely on it being expelled from the core and confined to the crust, where it can evolve on kiloyear timescales. We show that if expulsion is not complete, a qualitatively new evolutionary branch for neutron stars arises, which can account for the properties of newly discovered long-period radio transients and fast radio bursts in older environments. One recently proposed model that could create such field topologies has additional implications for gravitational-wave emission and predicts a characteristic energy release that, if observed, will corroborate the role of reconnection at the onset of superconductivity and can constrain the superconducting proton gap.
Unified Astronomy Thesaurus concepts: Neutron star cores (1107); Stellar magnetic fields (1610); Radio transient sources (2008)
Materials only available in the online version of record: data behind figure

1.		Introduction
Neutron star (NS) cores host the only known superconductor in the Universe outside terrestrial laboratories (B. Haskell & A. Sedrakian 2018). Superconductivity does not set in simultaneously across the whole NS core because of the strong radial dependence of the critical onset temperature T_c. The transition begins in a thin shell in the star’s outer core, which slowly expands both outwards (stopping at the boundary with the crust) and inwards, as more regions cool below T_c. If the local magnetic field B is weaker than some critical field H_c, the minimum-energy state in the superconducting shell will be dictated by the Meissner effect: the expulsion of magnetic flux, leaving B = 0 except in a surface layer (M. Tinkham 2004). While on Earth this process is well understood and near- instantaneous, NSs present a different environment. Their intense B (typically 10^12–10^15 G) is intrinsic to the star, the result of a persistent relative flow between the protons and electrons (in a background of free neutrons), inside which the superconductor forms and expands; see Figure 1. In the laboratory case, only electrons can become superconducting; in NSs, it is the proton fluid. The strong gradients in density ρ and in T_c, together with the temperature dependence of H_c, lead to a broad normal–superconducting transition; there is no static equilibrium for the system to reach for at least decades, and potentially far longer (W. C. G. Ho et al. 2017).
A long-standing uncertainty about whether effective flux expulsion is possible in NSs has led to a major tension in the
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literature. Microscopic arguments (G. Baym et al. 1969a) suggest that the process is so slow that the flux will be trapped inside the core and quantized into thin flux tubes, even if B < H_c. Models of pulsar rotational evolution and spin-up glitches often explicitly or implicitly assume that the magnetic field threads the entire star (B. Haskell & A. Melatos 2015; D. Antonopoulou et al. 2022). On the other hand, the extensive body of literature that explains magnetar activity with Halldrift-driven crustal field evolution (e.g., R. Hollerbach & G. Rüdiger 2002; J. A. Pons & U. Geppert 2007) requires a B = 0 inner (crust–core) boundary condition, i.e., that flux is completely expelled from an outer layer of the core, without giving any detailed justification for this assumption. The uncertainty stems from the fact that the Meissner “effect” is simply a statement about the minimum-energy equilibrium state for the system, and does not dictate the process by which this is reached. Recent research examining the gradual transition to superconductivity in the NS core proposed a new mechanism that could realize flux expulsion on astrophysically relevant timescales (S. K. Lander 2024). In addition to complete flux expulsion from the outer core, which can potentially resolve the tension between previous theoretical work and magnetar modeling, another possible outcome is a B = 0 shell that is broken by one or two large holes penetrated by B (see below). In this Letter, we examine some important observational consequences of such a model: a latetime energy injection following a supernova, changes to the mature star’s rotational properties, and a revised prognosis for the detection of continuous gravitational waves from pulsars like the Crab. We moreover compare simulations of crustal magnetic field evolution for full Meissner expulsion with two different partially expelled geometries. While the latter are

Figure 1. A neutron star is mainly composed of neutrons, protons, and electrons (gray, indigo, and red balls, respectively). The magnetic field (black lines with arrows) is sourced by a relative flow of protons and electrons. Minutes after the star’s birth, protons in a layer (green) in the outer core become superconducting. Unless B > H_c, the star’s minimum-energy state requires the field to be expelled from this layer: the Meissner effect. This expulsion can, however, be incomplete—e.g., here we sketch a polar hole, where B ≠ 0. Such field configurations have several interesting observational implications, as we will discuss.

inferred from the model in S. K. Lander (2024), such states are conceivable endpoints of any operative Meissner effect in NSs, and present a qualitatively new evolutionary path: late-onset and protracted magnetic activity in the crust. This delayed activation of old NSs could solve the mystery of some recently discovered long-period radio transients (M. Caleb et al. 2022; N. Hurley-Walker et al. 2022; F. A. Dong et al. 2025; Y. W. J. Lee et al. 2025; Z. Wang et al. 2025) and Fast Radio Bursts (FRBs) associated with aged host environments (F. Kirsten et al. 2022; C. J. Law et al. 2024; T. Eftekhari et al. 2025; V. Shah et al. 2025).
1.1.		The Onset of Superconductivity
The production of a macroscopic B = 0 region in the core has typically been associated with very slow mechanisms for flux transport (e.g., G. Baym et al. 1969a; P. B. Jones 1991; V. V. Kocharovsky et al. 1996; V. Graber et al. 2015), i.e., implicitly assuming that the process to achieve Meissner expulsion must be secular and dissipative. Expulsion via ohmic decay, for example, will take ∼10^9 yr for a field whose characteristic length scale is the stellar radius R* ≈ 10 km (G. Baym et al. 1969a, 1969b). Such slow expulsion suggests that magnetic flux tubes will thread the interior of every observed NS—even in the likely case where B < H_c. A comparison of cooling and (ohmic) flux dissipation timescales concluded that for NSs younger than 10^6 yr, any field-free regions will likely be ≲10^3 cm in extent (W. C. G. Ho et al. 2017). However, the Meissner effect is not intrinsically dissipative, leaving open other possibilities for flux expulsion.
A universal obstacle for any flux-expulsion mechanism is that the nascent shell of superconductivity will be crossed by field lines, and so creating a B =0 shell without substantial flux rearrangement would produce radial discontinuities in the field, which violate ∇ · B = 0. The radial field could, however, be rearranged into an angular component through advection via a suitable fluid flow v:
^B = × (v × B), (1) t
where |v| is only limited by the (high) local Alfvén speed. If this results in a suitably pinched field geometry, reconnection can then act to produce a B = 0 shell, provided that these processes are faster than the spreading of the superconducting region. Advection of this form was recently explored for a simplified axisymmetric model (S. K. Lander 2024), which argued—by comparing cooling and reconnection timescales, and using the restriction that for B > H_c flux expulsion is not energetically favorable—that dynamical Meissner expulsion is possible only when
10^12 G B 5 × 10^14 G. (2)
Even within this range, the field geometry and fluid flow need not be conducive to complete reconnection. A complex, smallscale field geometry is the most likely to undergo effective reconnection, perhaps requiring just vestigial fluid motions remaining from the supernova (S. K. Lander 2024). By contrast, the most challenging geometry to expel is probably an axisymmetric dipolar field, because this would require a kilometer-scale fluid circulation whose radial component would be resisted by the star’s stable stratification. Nonetheless, it is a geometrically simple first approximation that allows us to set limits on when an advection-driven Meissner effect may occur. As well as the possible outcomes of full or no flux expulsion, shearing and reconnection of B at the incipient shell of superconductivity can lead to a broken B = 0 superconducting shell pierced with field lines through either two roughly cylindrical B ≠ 0 holes (one at each pole), or one equatorial band (S. K. Lander 2024). The size of these B ≠ 0 regions in the shell may be calculated from conservation of magnetic flux and the requirement that B ^ H_c; the result is holes whose cross sectional area is proportional to B. Whether or not this B = 0 shell is complete, however, there must always be an inner trapped core of magnetic field that will be compressed until it reaches B = H_c. Magnetic flux conservation then yields an estimate of the final radius R_in of this inner core:
Rin R_* B/Hc . (3)
Within the advective model of flux expulsion, a full range of more complex flux distributions is also possible, depending on the pre-superconducting field geometry and the fluid flows. As there may also be different mechanisms for producing anisotropic flux distributions, in what follows we examine a few generic outcomes and discuss ways to test and constrain these observationally.
2.		Results
We first examine one specific prediction of the advection- driven expulsion mechanism, which might render the model observationally testable. We then look at observational manifestations of NSs with and without flux expulsion. Independent of the expulsion mechanism, there is an intriguing possibility of macroscopic B = 0 regions and possibly of more complex geometries than a fully flux-free NS core. Such field topologies will have major ramifications for both electromagnetic and gravitational radiation from NSs, but also for their magnetorotational evolution. We numerically investigate the long-term magnetic field evolution under both the typical “magnetar” B = 0 crust-core boundary condition, implying full Meissner expulsion in the outermost core region, as well as two partially expelled geometries, as discussed above. The results are especially important in view of recent discoveries that challenge NSs as the source of some high-energy transients. Last, we touch upon the rotational implications of such field topologies.
2.1.		Energy Release at Superconductivity Onset
In the particular case of dynamical flux expulsion, an electromagnetic signature is expected at the onset of superconductivity. For example, the kind of shearing motion needed to bring an axisymmetric dipolar B into a geometry amenable to reconnection causes an approximate doubling of a typical field line’s length and, therefore, of the magnetic energy; see S. K. Lander (2024) for detailed calculations. Such an increase in magnetic energy, in order to minimize the total free energy of the system, is a known property of superconducting systems (J. F. Annett 2004). But any subsequent reconnection event would then return B to a less-sheared, lower-energy configuration. Two very different recent examples of such potential flux-expelled end-state equilibria are the microscopic calculations of T. S. Wood & V. Graber (2022) and the global calculations of M. Das et al. (2025). In both cases, however, a dynamical mechanism is needed to realize the equilibrium state; if this happens at the onset of superconductivity, we expect reconnection to cause a sudden magnetic energy release. This could be as large as 10^46–10^47 erg for a dipolar geometry, but likely weaker for more complex and small-scale field geometries (S. K. Lander 2024).
This reconnection energy release in the core will cause local heating and shake field lines that extend through the still- molten crust and out to the magnetosphere. If this complex transfer of magnetic energy is radiatively efficient in producing a detectable signal, the reconnection event could ultimately be seen as a peculiar transient, possibly involving delayed energy injection into the supernova remnant. Given the vast zoo of optical transients (M. R. Drout et al. 2014), such signals may already have been detected. Alternatively, the reconnection event might be more readily identifiable if a binary NS merger produces a massive magnetized NS remnant: a cleaner environment for signals to transit. Many short gamma-ray bursts exhibit X-ray extended emission (lasting ∼ 10^2–10^3 s), whose physical origin is uncertain and requires delayed energy injection (Y. Kaneko et al. 2015). Secure association of such signals would allow us to constrain the maximum value of T_c and, therefore, the proton pairing gap model.
2.2.		Continuous Gravitational-wave Emission
Recent studies on the nondetection of continuous gravitational waves from various pulsars have put some stringent and physically interesting bounds on their ellipticity ε (B. Abbott et al. 2008). For example, the Crab pulsar has ε ≲ 3 × 10^-5 (R. Abbott et al. 2022), and this limit will continue to drop with advances in gravitational-wave interferometry. The Crab’s inferred dipole surface magnetic field B_dip = 4 × 10^12 G. If its ε is primarily due to an average interior field B^¯int threading the entire core, ellipticity calculations for superconducting NSs (S. K. Lander 2014) allow us to infer that
B¯int 10^15 G 300Bdip. (4)
We can also invert this logic to assess the prospects for a future gravitational-wave detection from the Crab, and how the Meissner effect alters this prognosis. A typical NS could plausibly (A. P. Igoshev et al. 2021) harbor a field Bint Bdip ; let us take, for example, B_int = 20B_dip for the Crab, close to typical magnetar values. If the Meissner effect has expelled the field from a complete shell of the star, it will cease once an inner core of B = Hc, Rin 0.4R_* (see Equation (3)) has been formed. However, the ellipticity for a superconducting core scales as BR_i^3_n, so although B has been increased substantially through flux compression, the overall effect on ε is dominated by the decrease in R_in. As a result, the ellipticity for the Meissner-expelled model ε_M is substantially lower than that of the non-Meissner model ε_nM (i.e., where field lines thread the entire core):
M 3 × 10 ^7, nM 2 × 10 ^6. (5)
In this scenario, if the Crab hosts a Meissner-expelled region, we would expect weak gravitational-wave emission, but notable magnetar-like activity (given that a crustal field of strength 20B_dip would have evolved substantially over the lifetime of the Crab). The latter has, however, not been seen from the Crab despite over 50 yr of monitoring, leading us to conclude that if the Crab has a strong B_int it must thread the entire core. Its associated gravitational-wave signal would then be relatively strong; below the sensitivity of current instruments by a factor of ∼10, but potentially detectable by nextgeneration instruments.
2.3.		Crustal Field Evolution
In the absence of flux expulsion, B is expected to penetrate from the crust into the core (G. Baym et al. 1969a), which should be incorporated as a B ≠ 0 inner boundary condition in numerical evolutions of the crustal magnetic field. Doing so results in a slowly evolving crustal field, without any particularly dramatic features (D. Viganò et al. 2013). By contrast, the commonly employed B = 0 inner boundary condition results, for young NSs (10^3–10^4 yr) with crustal fields ≳10^13 G, in an active Hall drift that produces local patches of intense field; conversely, for older sources the Hall effect saturates (K. N. Gourgouliatos & A. Cumming 2014) and eventually ohmic decay dominates, smoothing out smallscale features and weakening the field (see, e.g., J. A. Pons & U. Geppert 2007). This fits with the theoretical picture of canonical magnetars as young and highly magnetized NSs whose activity is powered by crustal magnetic-field evolution;

Figure 2. Magnetic field in the crust (here scaled up in thickness by a factor of 8 for clarity) after 0.1 Myr of evolution, where the inner boundary at the base of the crust is: B = 0 everywhere (left), field lines penetrating into the core in a band around the equator (middle), and field lines penetrating into the core through two polar holes (right). Poloidal field lines are in black; toroidal field magnitude is shown with the color scale.
(The data used to create this figure are available in the online article.)

a B = 0 inner boundary is, however, critical for producing such activity. The dynamical Meissner effect (S. K. Lander 2024) is the first scenario to provide a physically plausible justification for this B = 0 condition. However, as discussed above, it also allows for other crust–core boundary conditions, where the B = 0 shell is broken by regions threaded by field lines: in the simplest cases, either an equatorial band or two polar holes. Figure 2 compares the result of 0.1 Myr of evolution of an initially poloidal field with surface strength 5 × 10^13 G, for these three classes of boundary condition. With the usual B = 0 inner boundary (left-hand panel), the crustal field at this late stage is smooth, quasi-stationary, and relatively weak: the maximum value of the toroidal field B^max = 4 × 10^13 G. By contrast, the simulation with B = 0 around the poles but B ≠ 0 in an equatorial band (middle panel) remains very dynamic, displaying sharp features in the poloidal field lines and B^max = 1.6 × 10^14 G close to the transition region between the B = 0 and B ≠ 0 surfaces. Above this region, small closed magnetic-field loops are progressively pushed out of the star. All these features, which are shared with the corresponding simulation featuring polar B ≠ 0 holes (right-hand panel), are qualitatively new and suggest the existence of a class of NS that becomes more active at ∼100 times the age of conventional magnetars. In the virtually untwisted magnetospheres of such old NSs, localized disruptions—like those arising from our simulations—might manifest themselves through radio emission (A. J. Cooper & Z. Wadiasingh 2024); see Figure 3.
In fact, such qualitatively different “old magnetars” may already have been observed. The recent unexpected discoveries of very long-period intermittent radio transients (N. Hurley-Walker et al. 2022; Y. W. J. Lee et al. 2025; Z. Wang et al. 2025) present a theoretical challenge: while a promising explanation is that of late-time activity of highly magnetized NSs, especially given magnetar-like inferred field strengths for two of them (M. Caleb et al. 2022; F. A. Dong et al. 2025), there is no clear mechanism to drive such behavior within the standard models of NS magnetism (P. Beniamini et al. 2023). Similarly, this new possible population of NSs could be behind the emission of some individual repeating FRBs that are associated with old sources/environments (S. P. Tendulkar et al. 2021), given their apparent nontrivial delay time with respect to star formation in their host galaxies (F. Kirsten et al. 2022; C. J. Law et al. 2024; T. Eftekhari et al. 2025; V. Shah et al. 2025). There are several indications of a connection between the long-period radio transients and FRB- like emitting sources (P. Beniamini et al. 2020; Y. Men et al. 2025); the “old magnetar” evolutionary path, which opens if partial Meissner expulsion is possible, naturally links these phenomena to our understanding of NS populations. Finally, such NSs could also manifest in some high-mass X-ray binaries, as standard accretion models for the formation of long-period NSs also appeal to the existence of sustained magnetar-like fields for ≫10^4 yr (X. D. Li & E. P. J. van den Heuvel 1999).
2.4.		Rotational Dynamics
Most models of NS rotation (in contrast with magnetar modeling) assume a continuous B threading the core and connecting it to the crust. This provides fast electromagnetic coupling between all charged core components and the crust, irrespective of whether the protons are superconducting, which justifies treating them as being in corotation. If the neutrons are normal, they will also be strongly coupled: either near- instantaneously via the strong interaction (for normal protons), or due to particle collisions (for superconducting protons)— still a relatively rapid process compared with most other timescales of interest, e.g., spin-down (see M. A. Alpar et al. 1984; B. Haskell & A. Sedrakian 2018, and references therein).
The presence of a macroscopic B = 0 core region could drastically change this picture. Spin-down is then likely achieved by the formation of a viscous crust–core boundary layer. The associated Ekman flow should reduce the coupling timescale compared to that of viscous diffusion, although realistic modeling remains challenging. It is conceivable that a magnetically decoupled region (the B = 0 shell and, for full Meissner expulsion, the inner B ≠ 0 core too) would deviate from “rigid” corotation with the crust (see A. Melatos 2012 and K. Glampedakis & P. D. Lasky 2015 for core superrotation in other setups); this problem deserves a careful treatment (including, e.g., the various interfaces) to assess whether it could lead to a long-lived rotational lag, or affect the response to fast spin changes, like pulsar glitches.
Once the neutron fluid becomes superfluid, it predominantly couples to the charged component via its rotational quantized- circulation vortices. These vortices become strongly magnetized due to entrainment if protons are superconducting, leading to electron-vortex relaxation timescales of ∼1–10 s (M. A. Alpar et al. 1984). Thus, the core superfluid maintains differential rotation but is close to corotation and spins down with the local charged component. If the flux is not expelled but instead is confined in flux tubes, then their interaction with vortices must be taken into account. This has been extensively studied in the literature (e.g., A. Sourie & N. Chamel 2020a), mostly in relation to mode damping, arguments against long- period NS precession (B. Link 2003), and the angular momentum reservoir available to generate large glitches as in the Vela pulsar (N. Andersson et al. 2012). These results could change drastically in a B = 0 region, since flux tubes will no longer coexist with neutron vortices there, and so should be reevaluated for the different Meissner scenarios discussed here. For example, in the case of a B ≠ 0 equatorial band, vortices could naturally pin to a different number of flux tubes depending on the pulsar, giving rise to different glitch

Figure 3. Two possible ways to “observe” the Meissner effect in a neutron star (black circle) via disruptions to its external magnetic-field lines. Left: at an age of a few minutes, the reconnection of magnetic field lines (blue)—e.g., as may be required to create a magnetar—releases a huge amount of energy and shakes field lines crossing the nebula (red-yellow disk), potentially producing an observable transient (light blue blob). Right: at an age of around 100 kyr, crustal magnetic-field evolution above an incomplete Meissner-expelled layer causes localized disruptions (wiggly field line) to an otherwise charge-starved dipolar magnetosphere, resulting in radio emission (orange cone) modulated by the star’s rotation (green arrow).

signatures (A. Sourie & N. Chamel 2020b). By contrast, in the fully Meissner-expelled or polar-hole case, vortices might not be pinned at all, resolving contradictions with long-period precession and some glitch models (B. Link 2006; B. Haskell et al. 2013). Finally, because vortices move outwards as an NS spins down, it has long been thought (M. Ruderman et al. 1998) that their pinning to flux tubes could transport magnetic flux outwards too. The precise nature of this secular “expulsion” in old NSs will be directly affected by the different Meissner scenarios in the star’s early life.
3.		Discussion
The existence of an unmagnetized region in the core of NSs is crucial to drive significant crustal field evolution within 10^4 yr; without any mechanism for flux expulsion, the currently accepted paradigm for magnetar activity could be seriously undermined. Dissipative processes are too slow to achieve the required Meissner effect (G. Baym et al. 1969a; W. C. G. Ho et al. 2017), but an advection-reconnection process could produce an energetically favorable, flux-free, outer NS core on a dynamical timescale (S. K. Lander 2024) and thus provide the first theoretical support for this important assumption. Furthermore, this mechanism could potentially be observed: through an energy injection in the late supernova phase (see Figure 3), differences in NS rotational behavior, and gravitational-wave emission from mature NSs.
Our simulations of NS crustal field evolution for the cases of partial or complete flux expulsion from the outer core imply that there are three different observational incarnations of a highly magnetized NS. Only one of these is a classical magnetar: the case of full Meissner expulsion. Within the dynamical Meissner model, this suggests their internal fields are in the range 0.01 B_int,14 H_c,14 5. Interestingly, this is not very different from the range of B_dip values inferred for known magnetars, 0.06 ≲ B_dip,14 < 8 (N. Rea et al. 2013), with just one widely quoted outlier value: 2 × 10^15 G for SGR 1806–20. This was, however, calculated during an epoch of enhanced spin-down; the star has since stabilized to a lower spin-down rate from which B_dip = 7.7 × 10^14 G is estimated (G. Younes et al. 2017).
We predict that neither NSs with Bint > Hc, nor those with a substantially large-scale field after the birth phase, will display magnetar activity—although their substantial magnetic distortions could make them promising sources of gravitational waves. Finally, NSs possessing a partially Meissner-expelled shell with “holes” become active once substantially older than typical magnetars, potentially powering some newly discovered radiative events that are otherwise theoretically challenging to explain. The growing number of very-long-period radio transients, and the expected discovery of many well- localized FRBs by upcoming instruments, will provide critical constraints on the existence of the “old magnetar” class we predict, and reveal details of their properties that can be used to better understand NS interiors.
Acknowledgments
This work was supported by computational time granted from the National Infrastructures for Research and Technology S.A. (GRNET S.A.) in the National HPC facility—ARIS— under project ID pr017008/simnstar2. We thank Bryn Haskell, Mike J. Moss, and Brad Cenko for helpful discussions on some of these topics. Z.W. acknowledges support by NASA under award Nos. 80GSFC21M0002 and 80GSFC24M0006. D.A. acknowledges support from a UKRI (STFC/EPSRC) fellowship (EP/T017325/1). This work made use of the NASA Astrophysics Data System.
Data and Code availability
The code used to evolve Equation (A1) is available upon reasonable request to the first author. The datasets generated from these evolutions are available as the data behind Figure 2.
Appendix
Stellar Model and Numerical Method
A.1. The Onset of Superconductivity
The critical temperature T_c for proton superconductivity varies throughout the core of a NS, peaking at a value ∼(3–7) × 10^9 K (W. C. G. Ho et al. 2015), which is attained only minutes after the star’s formation. The range of peak values for T_c reflects differing approaches to calculating the
energy gap. Here we adopt an approximation (W. C. G. Ho et al. 2012) to one particular energy gap model (J. M. C. Chen et al. 1993), yielding T_c as a function of mass density ρ. We find a relativistic hydrostatic equilibrium by solving the TOV equations with the SLy4 equation of state (F. Douchin & P. Haensel 2001), giving us ρ and hence Tc as a function of radius. Finally, combining this with a standard cooling prescription (D. Page et al. 2006) allows us to track the expansion of the superconducting shell as a function of time. This gives a first thin shell of superconductivity forming at a radius of 0.79R*, a temperature T = 6.8 × 10^9 K, and a stellar age of 170 s. The shell spreads rapidly, enveloping half of the core radius (in the absence of magnetic-field effects) after ∼1 day. Another important quantity is H_c, which varies by a factor of ∼5 within the core (K. Glampedakis et al. 2011), but not monotonically, and also depends upon the equation of state, the nature of core superconductivity, and the geometry of B. We cannot account for all these features and retain any generality in the model, so for simplicity we approximate H_c as a constant for our calculations, adopting a representative value Hc = 5 × 10^14 G.
A.2. Magnetic Field Evolution in NSs
Magnetic field evolution in the NS core is complex, and the timescale of the dominant mechanisms remains contentious (see, e.g., M. E. Gusakov et al. 2020, and references therein). This, in turn, makes it difficult to attribute any NS activity unambiguously to evolution in the core. By contrast, the crust is conceptually simple. Completely generally, the electric current is due to a relative flow of positive and negative charges. In the crust, the positive charges are ions locked in a crystalline lattice, so that only the electrons are mobile. The electric current, and therefore B, depend only on the electron velocity, and from this restriction one may derive the equation
^2
^B =	^1	× ^c ( × B) × B ^c × B e

(A1)

where the first term on the right-hand side is Hall drift, the second ohmic decay, and ρe is the charge density (P. Goldreich & A. Reisenegger 1992). Taking a simple poloidal field as the initial condition, we evolve Equation (A1) numerically, specializing to axisymmetry for simplicity. We use a finite difference method, discretizing the numerical domain in r and cos . The resolution is typically 100 × 100, but we have experimented in several cases with doubling the resolution to confirm the numerical convergence of our results. We use a central-difference scheme for spatial derivatives, and three- and five-point stencils for second and third derivatives, respectively. The second-order Adams–Bashforth method is used for numerical time integration. The timestep is dynamically adapted using the Courant condition:
r

(A2)

vmax where Δr is the grid spacing and vmax the maximum electron velocity. The profiles for ρ_e and σ, and the numerical code used here, are the same as in K. N. Gourgouliatos & S. K. Lander (2021), except that for the present work we do not allow for plastic flow.
The outcome of the dynamical Meissner process depends on the geometry of the star’s magnetic field and any fluid flows present at the onset of superconductivity. For simplicity, however, we assume a large-scale flow advecting an initially dipolar field. For this scenario, we use three families of boundary condition at the crust–core interface: a fully expelled magnetic field, where the radial field B_r at the inner boundary is zero; a magnetic field containing two holes of half-opening angle arccos 0.8	37° where the radial field is nonzero; and a boundary condition where there is a nonzero radial field in a band at the equator, starting at arccos 0.3 = 73° and ending at arccos 0.3 = 108°. In all three cases, the radial field at the inner boundary is static, but the tangential field components (B_θ and Bf) are free to evolve. At the outer surface of the crust and into the exterior, we assume a vacuum magnetic field, i.e., no electric current, using a multipole fit, which serves as the external boundary condition. Note that although our boundary conditions are based on simplified axisymmetric endpoints predicted by the dynamical Meissner effect (S. K. Lander 2024), any other mechanism resulting in partial flux expulsion might induce similar long-term field evolution.
The three evolutions, whose endpoints at 0.1 Myr are plotted in Figure 2, proceed as follows. All three show significant evolution and the development of a toroidal-field component in the outer crustal region within 1 kyr; this component begins to extend deeper into the crust up until an age of ∼10 kyr. The two partially expelled cases also show a corresponding development of toroidal field in the innermost part of the crust, at the boundaries of the holes where field lines pass into the core. In these two cases, after ∼10 kyr, the inner and outer regions with toroidal field meet in the middle and interact, and the toroidal component becomes more intense, reaching a peak value just above the holes and at an age of around 50 kyr. There is also considerable disruption to the poloidal-field component above this region, with small loops of magnetic field rising up out of the crust. This activity is sustained right up until the end of our simulations, at 0.1 Myr, so it would clearly continue beyond this point, although numerical restrictions prevent us from following it further. By contrast, the standard boundary condition, of field expelled from the core, shows limited evolution beyond ∼20 kyr, with the field having relaxed to a quasi-equilibrium state by the end of the simulation, at 0.1 Myr.
ORCID iDs
Z. Wadiasinghaa https://orcid.org/0000-0002-9249-0515
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